TERRESTRIAL HYDROMETEOROLOGY - Universidad ZamoranoTERRESTRIAL HYDROMETEOROLOGY W. JAMES...

TERRESTRIAL HYDROMETEOROLOGY

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COMPANION WEBSITE

This book has a companion website:

www.wiley.com/go/shuttleworth/hydrometeorology

with Figures and Tables from the book for downloading

Shuttleworth_ffirs.indd iiShuttleworth_ffirs.indd ii 11/3/2011 10:06:59 AM11/3/2011 10:06:59 AM

TERRESTRIAL HYDROMETEOROLOGY

W. JAMES SHUTTLEWORTH

A John Wiley & Sons, Ltd., Publication

Shuttleworth_ffirs.indd iiiShuttleworth_ffirs.indd iii 11/3/2011 10:06:59 AM11/3/2011 10:06:59 AM

This edition first published 2012

© 2012 by John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Shuttleworth, W. James.

Terrestrial hydrometeorology / W. James Shuttleworth.

p. cm.

ISBN 978-0-470-65938-0 (hardback) – ISBN 978-0-470-65937-3 (paper) 1. Hydrometeorology–

Textbooks. I. Title.

GB2803.2.S58 2012

551.57–dc23

2011041765

A catalogue record for this book is available from the British Library.

Set in 10/12.5pt Minion Pro by SPi Publisher Services, Pondicherry, India

1 2012

Shuttleworth_ffirs.indd ivShuttleworth_ffirs.indd iv 11/3/2011 10:07:00 AM11/3/2011 10:07:00 AM

This book is dedicated with love and gratitude to Hazel, Craig, Matthew, Nicholas, Jonathan and Amy for all the

good and worthwhile things they have brought into my life.

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Contents

Foreword xvi

Preface xviii

Acknowledgements xix

1 Terrestrial Hydrometeorology and the Global Water Cycle 1

Introduction 1

Water in the Earth system 2

Components of the global hydroclimate system 4

Atmosphere 5

Hydrosphere 8

Cryosphere 9

Lithosphere 9

Biosphere 10

Anthroposphere 10

Important points in this chapter 12

2 Water Vapor in the Atmosphere 14

Introduction 14

Latent heat 14

Atmospheric water vapor content 15

Ideal Gas Law 16

Virtual temperature 17

Saturated vapor pressure 18

Measures of saturation 20

Measuring the vapor pressure of air 21


3 Vertical Gradients in the Atmosphere 25

Introduction 25

Hydrostatic pressure law 26

Adiabatic lapse rates 27

Dry adiabatic lapse rate 27

Moist adiabatic lapse rate 28

Environmental lapse rate 28

Vertical pressure and temperature gradients 29

Potential temperature 30

Virtual potential temperature 31

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viii Contents

Atmospheric stability 32

Static stability parameter 32


4 Surface Energy Fluxes 36

Introduction 36

Latent and sensible heat fluxes 37

Energy balance of an ideal surface 38

Net radiation, Rn 38

Latent heat flux, lE 39

Sensible heat flux, H 39

Soil heat flux, G 39

Physical energy storage, St 40

Biochemical energy storage, P 40

Advected energy, Ad 41

Flux sign convention 41

Evaporative fraction and Bowen ratio 45

Energy budget of open water 46


5 Terrestrial Radiation 48

Introduction 48

Blackbody radiation laws 49

Radiation exchange for ‘gray’ surfaces 51

Integrated radiation parameters for natural surfaces 52

Maximum solar radiation at the top of atmosphere 54

Maximum solar radiation at the ground 56

Atmospheric attenuation of solar radiation 58

Actual solar radiation at the ground 59

Longwave radiation 59

Net radiation at the surface 62

Height dependence of net radiation 63


6 Soil Temperature and Heat Flux 66

Introduction 66

Soil surface temperature 66

Subsurface soil temperatures 67

Thermal properties of soil 68

Density of soil, rs 69

Specific heat of soil, cs 70

Heat capacity per unit volume, Cs 70

Thermal conductivity, ks 70

Thermal diffusivity, as 71

Formal description of soil heat flow 71

Thermal waves in homogeneous soil 72


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Contents ix

7 Measuring Surface Heat Fluxes 77

Introduction 77

Measuring solar radiation 77

Daily estimates of cloud cover 77

Thermoelectric pyranometers 78

Photoelectric pyranometers 79

Measuring net radiation 80

Measuring soil heat flux 81

Measuring latent and sensible heat 82

Micrometeorological measurement of surface energy fluxes 83

Bowen ratio/energy budget method 83

Eddy correlation method 85

Evaporation measurement from integrated water loss 87

Evaporation pans 88

Watersheds and lakes 89

Lysimeters 90

Soil moisture depletion 91

Comparison of evaporation measuring methods 91


8 General Circulation Models 96

Introduction 96

What are General Circulation Models? 96

How are General Circulation Models used? 98

How do General Circulation Models work? 100

Sequence of operations 100

Solving the dynamics 102

Calculating the physics 103

Intergovernmental Panel on Climate Change (IPCC) 104


9 Global Scale Influences on Hydrometeorology 107

Introduction 107

Global scale influences on atmospheric circulation 107

Planetary interrelationship 109

Latitudinal differences in solar energy input 109

Seasonal perturbations 109

Daily perturbations 109

Persistent perturbations 109

Contrast in ocean to continent surface exchanges 109

Continental topography 109

Temporary perturbations 110

Perturbations in oceanic circulation 110

Perturbations in atmospheric content 110

Perturbations in continental land cover 110

Latitudinal imbalance in radiant energy 110

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x Contents

Lower atmosphere circulation 111

Latitudinal bands of pressure and wind 111

Hadley circulation 112

Mean low-level circulation 113

Mean upper level circulation 115

Ocean circulation 116

Oceanic influences on continental hydroclimate 118

Monsoon flow 118

Tropical cyclones 119

El Niño Southern Oscillation 120

Pacific Decadal Oscillation 122

North Atlantic Oscillation 123

Water vapor in the atmosphere 123


10 Formation of Clouds 128

Introduction 128

Cloud generating mechanisms 129

Cloud condensation nuclei 131

Saturated vapor pressure of curved surfaces 132

Cloud droplet size, concentration and terminal velocity 133

Ice in clouds 134

Cloud formation processes 135

Thermal convection 135

Foehn effect 136

Extratropical fronts and cyclones 138

Cloud genera 140


11 Formation of Precipitation 143

Introduction 143

Precipitation formation in warm clouds 144

Precipitation formation in other clouds 146

Which clouds produce rain? 148

Precipitation form 149

Raindrop size distribution 150

Rainfall rates and kinetic energy 151

Forms of frozen precipitation 151

Other forms of precipitation 152


12 Precipitation Measurement and Observation 155

Introduction 155

Precipitation measurement using gauges 156

Instrumental errors 157

Site and location errors 157

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Contents xi

Gauge designs 160

Areal representativeness of gauge measurements 162

Snowfall measurement 165

Precipitation measurement using ground-based radar 168

Precipitation measurement using satellite systems 171

Cloud mapping and characterization 171

Passive measurement of cloud properties 172

Spaceborne radar 173


13 Precipitation Analysis in Time 176

Introduction 176

Precipitation climatology 177

Annual variations 177

Intra-annual variations 177

Daily variations 180

Trends in precipitation 181

Running means 182

Cumulative deviations 183

Mass curve 184

Oscillations in precipitation 186

System signatures 187

Intensity-duration relationships 189

Statistics of extremes 190

Conditional probabilities 195


14 Precipitation Analysis in Space 198

Introduction 198

Mapping precipitation 199

Areal mean precipitation 200

Isohyetal method 200

Triangle method 202

Theissen method 202

Spatial organization of precipitation 203

Design storms and areal reduction factors 205

Probable maximum precipitation 207

Spatial correlation of precipitation 209


15 Mathematical and Conceptual Tools of Turbulence 213

Introduction 213

Signature and spectrum of atmospheric turbulence 213

Mean and fluctuating components 216

Rules of averaging for decomposed variables 217

Variance and standard deviation 219

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xii Contents

Measures of the strength of turbulence 220

Mean and turbulent kinetic energy 220

Linear correlation coefficient 221

Kinematic flux 223

Advective and turbulent fluxes 225


16 Equations of Atmospheric Flow in the ABL 231

Introduction 231

Time rate of change in a fluid 232

Conservation of momentum in the atmosphere 234

Pressure forces 235

Viscous flow in fluids 236

Axis-specific forces 239

Combined momentum forces 242

Conservation of mass of air 243

Conservation of atmospheric moisture 244

Conservation of energy 245

Conservation of a scalar quantity 246

Summary of equations of atmospheric flow 247


17 Equations of Turbulent Flow in the ABL 248

Introduction 248

Fluctuations in the ideal gas law 248

The Boussinesq approximation 249

Neglecting subsidence 250

Geostrophic wind 251

Divergence equation for turbulent fluctuations 252

Conservation of momentum in the turbulent ABL 252

Conservation of moisture, heat, and scalars

in the turbulent ABL 254

Neglecting molecular diffusion 255


18 Observed ABL Profiles: Higher Order Moments 259

Introduction 259

Nature and evolution of the ABL 259

Daytime ABL profiles 261

Nighttime ABL profiles 263

Higher order moments 265

Prognostic equations for turbulent departures 265

Prognostic equations for turbulent kinetic energy 269

Prognostic equations for variance of moisture and heat 271


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Contents xiii

19 Turbulent Closure, K Theory, and Mixing Length 277

Introduction 277

Richardson number 277

Turbulent closure 279

Low order closure schemes 280

Local, first order closure 281

Mixing length theory 283


20 Surface Layer Scaling and Aerodynamic Resistance 289

Introduction 289

Dimensionless gradients 290

Obukhov length 292

Flux-gradient relationships 293

Returning fluxes to natural units 294

Resistance analogues and aerodynamic resistance 296


21 Canopy Processes and Canopy Resistances 300

Introduction 300

Boundary layer exchange processes 301

Shelter factors 306

Stomatal resistance 308

Energy budget of a dry leaf 310

Energy budget of a dry canopy 311


22 Whole Canopy Interactions 316

Introduction 316

Whole-canopy aerodynamics and canopy structure 317

Excess resistance 319

Roughness sublayer 321

Wet canopies 323

Equilibrium evaporation 325

Evaporation into an unsaturated atmosphere 327


23 Daily Estimates of Evaporation 334

Introduction 334

Daily average values of weather variables 335

Temperature, humidity, and wind speed 335

Net radiation 337

Open water evaporation 339

Reference crop evapotranspiration 341

Penman-Monteith equation estimation of ERC

342

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xiv Contents

Radiation-based estimation of ERC

344

Temperature-based estimation of ERC

345

Evaporation pan-based estimation of ERC

346

Evaporation from unstressed vegetation: the Matt-Shuttleworth

approach 348

Evaporation from water stressed vegetation 353


24 Soil Vegetation Atmosphere Transfer Schemes 359

Introduction 359

Basis and origin of land-surface sub-models 359

Developing realism in SVATS 362

Plot-scale, one-dimensional ‘micrometeorological’ models 364

Improving representation of hydrological processes 367

Improving representation of carbon dioxide exchange 368

Ongoing developments in land surface sub-models 370


25 Sensitivity to Land Surface Exchanges 380

Introduction 380

Influence of land surfaces on weather and climate 381

A. The influence of existing land-atmosphere interactions 383

1. Effect of topography on convection and precipitation 383

2. Contribution by land surfaces to atmospheric

water availability 385

B. The influence of transient changes in land surfaces 385

1. Effect of transient changes in soil moisture 385

2. Effect of transient changes in vegetation cover 388

3. Effect of transient changes in frozen precipitation cover 389

4. Combined effect of transient changes 391

C. The influence of imposed persistent changes in land cover 392

1. Effect of imposed land cover change on near

surface observations 392

2. Effect of imposed land-cover change on

regional-scale climate 393

3. Effect of imposed heterogeneity in land cover 395


26 Example Questions and Answers 404

Introduction 404

Example questions 404

Question 1 404

Question 2 405

Question 3 407

Question 4 408

Question 5 410

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Contents xv

Question 6 411

Question 7 412

Question 8 414

Question 9 416

Question 10 418

Example Answers 418

Answer 1 418

Answer 2 420

Answer 3 420

Answer 4 425

Answer 5 426

Answer 6 427

Answer 7 429

Answer 8 432

Answer 9 434

Answer 10 437

Index 441

COMPANION WEBSITE

This book has a companion website:

www.wiley.com/go/shuttleworth/hydrometeorology

with Figures and Tables from the book for downloading

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Foreword

As a doctoral student of hydrology in the 1970s my only exposure to the

meteorological aspects of the hydrologic cycle was a few introductory chapters in

hydrology textbooks. These were limited in scope because class emphasis was on

the surface and subsurface water flow. Coverage of precipitation and evaporation

were also limited to single brief chapters, and there was no exposure to the

interface between meteorology and hydrology. Since then there has been a

complete transformation. The discipline of hydrometeorology has evolved

rapidly due to advances in observational technologies and large scale modeling,

both stimulated by the scientific need to address emerging issues such as climate

change and the requirement to provide water resources to a growing global

population. International programs such as the Global Energy and Water Cycle

EXperiment (GEWEX) initiated by the World Climate Research Programme and

the Biospheric Aspects of the Hydrologic Cycle (BAHC) initiated by the

International Geosphere Biosphere Program heightened interest in the coupling

between atmospheric and the terrestrial systems. But the many scientists and

students involved in this new interdisciplinary research had to gain their

knowledge of the two fields in piecemeal fashion. A textbook was obviously

needed to bridge the two disciplines.

Being a visionary in the field, Professor Jim Shuttleworth recognized this void

and accepted the challenge. For over a decade he devoted himself to developing

courses and writing a book, Terrestrial Hydrometeorology, that specifically

addresses the topic of hydrometeorology as a unified component within the Earth

system, with appropriate emphasis on hydrometeorology in the terrestrial

environment where people live.

The resulting book contains twenty-six chapters which provide excellent

coverage of key elements of the hydrologic cycle associated with the coupling of

the atmosphere with land surfaces. Coverage of the energy cycle and its role,

including the feedback via the water cycle, are extensively and clearly addressed in

the first ten chapters. A thorough discussion of precipitation formation, measure-

ment and analysis follows in chapters ten through fourteen. The latter sections of

the book provide in-depth coverage of the atmospheric boundary layer dynamics

and turbulent transfer that play a primary role in feedbacks in the exchange of

water and energy between land and near surface atmosphere. This section of the

book demonstrates Shuttleworth’s creative contribution to thinking in the theory

of soil moisture and evapotranspiration processes. The final chapter rounds off

this ideal textbook by providing example questions and answers that students can

use to test their understanding.

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Foreword xvii

It is exciting to finally have a single textbook on terrestrial hydrometeorology

that is balanced, timely and elegant, and that will be appropriate for use in graduate

courses for many years to come. Terrestrial Hydrometeorology will provide

opportunity for atmospheric science and hydrology programs to develop courses

that satisfy their cross-disciplinary educational requirements and, in this way, play

an important role in the education of the new generation of interdisciplinary

scientists who investigate the complex role of the hydrologic cycle in the climate

system. For many academics such a book would be the capstone publication of

their career but, knowing Jim Shuttleworth as I do, I am certain that we can expect

more such creative contributions in the future!

Soroosh Sorooshian

Director of the Center for Hydrometeorology and Remote Sensing

Distinguished Professor of Civil and Environmental Engineering

and Earth System Science at the University of California – Irvine

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Preface

Water is the medium through which the atmosphere has most influence on human

wellbeing and terrestrial surfaces have significant influence on the atmosphere.

Hitherto atmospheric and hydrologic science and practice have largely developed

separately. To hydrologists, meteorological variables were monitored and used as

independent forcing in models of hydrological responses. But hydrologists now

understand that near-surface meteorology is itself in part determined by how

surface water moves and how much water the land surface returns to the

atmosphere as evaporation. Consequently, graduate-level hydrological training

must now include relevant aspects of meteorological science. To meteorologists,

atmospheric exchanges with the land surface were regarded as boundary conditions

that could be calculated simply, with corrections to weather forecast models then

made by assimilating meteorological observations. But because about half the

energy that drives the atmosphere enters from below, meteorological forecasts

beyond a few days, and climate predictions in particular, require models that

include adequately realistic sub-models of surface hydrology and associated

energy exchanges. Consequently, graduate-level meteorological training must

now include relevant aspects of hydrological science. In fact the relationship

between hydrologists and meteorologists and their need to speak a common

scientific language is such that it is now recognized that a new science discipline

that overlies the land-atmosphere interface is needed, and courses that teach this

new discipline of Hydrometeorology are now being created at universities.

Hitherto there has been no single graduate-level text with sufficient breadth

across the hydrological and meteorological sciences that provides understanding

with adequate depth in both disciplines for use in hydrology departments to teach

relevant aspects of meteorology, in meteorological departments to teach relevant

aspects of surface hydrology, and to serve as an introductory text to teach the

emerging discipline of hydrometeorology. The primary intended readership of

this book is, therefore, graduate students studying surface water hydrology,

meteorology, and hydrometeorology. However this book could be used in relevant

advanced undergraduate courses and it will likely also find broader readership

among scientists seeking to broaden their education.

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Acknowledgments

This book was written in response to a need for an appropriate text for use in

teaching a core course in the University of Arizona Hydrometeorology Program.

That course was based on an existing course taught by the author in the

Hydrology and Water Resources Program that had evolved over the years in

response to students’ needs and students’ input. The resulting syllabus ultimately

determined the content of this book. I would, therefore, like to thank all the

many students who contributed to that evolution and who have in this way

participated in the definition of Terrestrial Hydrometeorology, both the subject

and the text book. The manuscript was largely written while the author was on

sabbatical leave as a Fellow of the Joint Centre for Hydro-Meteorological

Research (JCHMR) which is located in the Centre for Ecology and Hydrology

(CEH), Wallingford, UK. I am grateful for the support and the friendship I

received from everyone at the CEH, and in particular I would like to thank

Richard Harding, Eleanor Blyth, Bob Moore, Martin Best, and Alan Jenkins for

facilitating my pleasant and rewarding time at JCHMR. The NSF Science and

Technology Center for Sustainability of semi-Arid and Riparian Areas (SAHRA)

provided partial financial support during that period and also subsequently

supported the refinement of this text through the patience and precision of

Annisa Tangreen who provided copy editing support for the manuscript. I am

pleased to acknowledge the support of SAHRA, and Annisa in particular.

I would also like to thank my good friend and ex-colleague, John Gash, who

carefully checked for typographic errors in equations and made a final review of

the manuscript, and I am also happy to acknowledge Xu Liang of University of

Pittsburg for advice and her input to the review of SVATS given in Chapter 24.

Finally, I would like to give my wholehearted thanks to my wife, Hazel, for her

immense patience through the many exacting hours it took to prepare this text,

for her cheerfulness when the writing was difficult, and the gin-and-tonics we

shared when it was just too difficult!

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Landprecipitation

113

Land

Riverslakes178

Soil moisture122Ocean

1,335,040

Surface flow

Ground water flow

Oceanevaporation 413

Oceanprecipitation

373

Ocean to landwater vapor transport

40

Atmosphere12.7

Groundwater15,300

Permafrost22

Vegetation

Percolation

40

Evaporation, transpiration 73

Ocean

Ice26,350

Plate 1 The global annual average hydrological cycle including estimates of the main water reservoirs (in plain font in

units of 103 km3) and of the flow of moisture between stores (in italics in units of 103 km3 yr−1). (From Trenberth et al., 2007,

published with permission.)

Shuttleworth_bins.indd 1Shuttleworth_bins.indd 1 11/3/2011 5:39:26 PM11/3/2011 5:39:26 PM

Plate 2 A schematic diagram of the physical and physiological processes represented in the second generation Simple

Biosphere (SiB2) soil vegetation atmosphere transfer scheme. (From Colello et al., 1998, published with permission.)


lE

lE

lEH

H

H Sr

Lu

Lu

LuSr

Sr lEH

Lu

SrlE

H

Lu

Sr

P S Ld

Runoff

Runoff

RunoffRunoff

Bare soil

Snow pack

Deep drainageDeep drainage

Deep drainage

Deep drainage

Plate 3 Schematic diagram of second generation one-dimensional SVATs in which a plot-scale micrometeorological model

with an explicit vegetation canopy was applied at grid scale.

Mixedvegetation

grid squares

S

lE

lElE

lE

H

HH

H

Sr

SrSr

Sr

Snow pack

Water table

Bare soil

P

m

1-m

Fractional precipitation on each grid

Topography

Lu

Lu

Lu

Lu

Ld

Plate 4 Schematic diagram of SVATS with improved representation of hydrologic processes.


Vegetationdynamics Vegetation

growth cycle

CO2

N2

lE

lE

lE

H

H

H Sr

Lu

Lu

LuSr

Sr lEH

Lu

SrlE

H

Lu

Sr

PS Ld

Runoff

Snow pack Runoff

RunoffRunoff

Bare soil


Deep drainage

Deep drainage

Plate 5 Schematic diagram of SVATS with improved representation of vegetation related processes, including CO2

exchange and ecosystem evolution.

Mixed vegetationwith vegetation

dynamics

SLd

(CO2, N2,...)

Dataassimilation

m

1-m


(CO2, N2,...)

LuLu

SrSr

HH

lElE

Snowpack

Routing

Routing

Plate 6 Schematic diagram of potential future developments in SVATS.


Cultivated Systems:Areas in which at least30% of the landscapeis cultivated

(a)

MEDITERRANEAN FORESTS,WOODLANDS, AND SCRUS

TEMPERATE FORESTSTEPPE AND WOODLAND

TEMPERATE BROADLEAFAND MIXED FORESTS

TROPICAL ANDSUB-TROPICAL DRY

BROADLEAF FORESTS

FLOODED GRASSLANDSAND SAVANNAS

TROPICAL AND SUB-TROPICALGRASSLANDS, SAVANNAS,

AND SHRUBLANDS

TROPICAL AND SUB-TROPICALCONIFEROUS FORESTS

DESERTS

MONTANE GRASSLANDSAND SHRUBLANDS

TROPICAL AND SUB-TROPICALMOIST BROADLEAF FORESTS

TEMPERATECONIFEROUS FORESTS

BOREALFORESTS

TUNDRA

Conversion of original biomesLoss by1950

Fraction of potential area converted−10 0 10 20 30 40 50 60 70 80 90 100%

Loss between1950 and 1990

Projected lossby 2050*

(b)

Plate 7 (a) Land areas which were more than 30% cultivated in 2000. (b) Projected change in original land cover by 2050

given by biome according to the four Millennium Ecosystem scenarios. (Redrawn from MEA, 2005, published with

permission.)


60N

30N

EQ

30S

60S180 120W 60W 0 60E 120E 180

−0.03

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Plate 8 Geographical distribution of land-atmosphere ‘coupling strength’ (i.e., the degree to which anomalies in soil

moisture can affect rainfall generation and other atmospheric processes) averaged for eight GCMs in the GLACE study

(Redrawn from Koster et al., 2006, in which paper ‘coupling strength’ is defined, published with permission.)

60

(W m−2) (mm d−1)

4

3

2

1

0.5

−0.5

−1

−2

−3

−4

June June

Latent Heat Flux Precipitation

July July

August August

40

30

20

10

−10

−20

−30

−40

−60

Plate 9 Map of differences in monthly average latent heat flux (W m-2) and precipitation (mm d-1) given when a

description of interactive vegetation cover was introduced into the Weather Research and Forecasting (WRF) model

coupled with the Noah land surface model to substitute for prescribed changes in vegetation cover. The modeled domain

covers the contiguous US between 21°N–50°N and 125°W–68°W. (Redrawn from Jiang et al., 2009, published with

permission.)


Temperature (January)

(c)

−1 −0.8−0.6−0.4−0.2 0.2 0.4 0.6 0.8 1

115E39S

36S

33S

30S

27S

24S

21S

18S

15S

120E 125E 130E 135E 140E 145E 150E

(a)

42S

39S

36S

33S

30S

27S

24S

21S

18S

15S

12S

115E 120E 125E 130E 135E 140E 145E 150E −3 −1 1 3−0.5 0.5

(b)

39S

36S

33S

30S

27S

24S

21S

18S

15S

115E 120E 125E 130E 135E 140E 145E 150E

Total Rainfall (January)

Plate 10 Simulated changes in climate made with the MM5 mesoscale model using natural (1788) and current (1988)

vegetation cover in Australia: (a) areas where vegetation cover was changed; (b) simulated change in total rainfall in January

(in mm); and (c) simulated change in average temperature in January (in °C). (Redrawn from Narisma and Pitman, 2003,



Terrestrial Hydrometeorology, First Edition. W. James Shuttleworth.

© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

Introduction

Water is not the most common molecule on planet Earth, but it is the most

important. Life started in water and cannot survive long without it; it makes up

approximately 60% of animal tissue and 90% of plant tissue. The most important

greenhouse gas in the atmosphere is water vapor. If it were not present the Earth’s

surface temperature would be several tens of degrees cooler, and predicting the

effect of changing atmospheric water content is arguably the greatest challenge

facing those who seek to predict future changes in climate. It is the continuous

cycling of water between oceans and continents that sustains the water flows over

land which in large measure determine the evolution of landscapes. The ability of

water to store energy in the form of latent heat or because of its high thermal

capacity means that moving water as vapor or fluid transports large quantities of

energy around the globe. The presence of frozen water on land as snow also has a

major impact on whether energy from the Sun is captured at the Earth’s surface or

is reflected back to space. In fact, it is hard to think of a process or phenomenon

important to the way our Earth behaves in which the presence of water is not

significant.

Hydrologists originally considered hydroclimatology to be ‘the study of the

influence of climate upon the waters of the land’ (Langbein 1967). This definition

is now outdated because it implies too passive a role for land surface influences on

the overlying atmosphere. The atmosphere is driven by energy from the Sun, but

about half of this energy enters from below, via the Earth’s surface. Whether that

surface is ocean or land matters and, if land, the nature of the land surface also

matters because this affects the total energy input to the atmosphere and the form

in which it enters. In practice, the science of hydroclimatology is often concerned

with understanding the movements of energy and water between stores within the

Earth system. Because climate is the time-average of weather, strictly speaking

1 Terrestrial Hydrometeorology and the Global Water Cycle

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2 TH and the Global Water Cycle

hydroclimatology emphasizes the time-average movement of energy and water.

Such movement occurs in two directions, both out of and into the atmosphere.

Consequently, the present text is motivated not only by the need to understand the

global and regional scale atmospheric features that affect the weather in a specific

catchment, but also to understand how the surface-atmosphere exchanges that

operate inside a catchment contribute along with those from nearby catchments to

determine the subsequent state of the atmosphere downwind.

Broadly speaking, hydrometeorology differs from hydroclimatology in much

the same way that meteorology differs from climatology. Hydrometeorologists

therefore tend to be more interested in activity at shorter time scales than

hydroclimatologists. They are particularly concerned with the physics, mathematics,

and statistics of the processes and phenomena involved in exchanges between

the atmosphere and ground that typically occur over hours or days. Sometimes

these short-term features are described statistically. Hydrometeorologists may, for

example, analyze precipitation data to compute the historical statistics of intense

storms and flood hazards. However, hydrometeorologists are also interested

in seeking basic physical understanding of surface exchanges of water and energy.

This commonly involves the study of processes that act in the vegetation covering

the ground, or the soil and rock beneath the ground, or in lower levels of the

atmosphere where most atmospheric water vapor is found. The present text

includes some description of the statistical approaches used in hydrometeorology

but gives greater prominence to providing an understanding of fundamental

hydrometeorological processes.

Water in the Earth system

Although there have been several studies which have attempted to quantify where

water is to be found across the globe, the magnitude of the Earth’s water reservoirs

and how much water flows between these reservoirs still remains poorly defined.

Table 1.1 gives estimates of the size of the eight main reservoirs together with the

approximate proportion of the entire world’s water stored in each reservoir and

an estimate of the turnover time for the water. The magnitude of the groundwater

reservoir and the associated residence time is complicated by the fact that a large

proportion of the water in this reservoir is ‘fossil water’ stored in deep aquifers

which were created over thousands of years by slow geo-climatic processes. The

amount of such fossil water stored is very difficult to estimate globally. Defining

a residence time for oceans is also complicated. This is because oceans usually

have a fairly shallow layer of surface water on the order of 100 m deep that

interacts comparatively readily with the atmospheric and terrestrial reservoirs,

but this layer overlies a much deeper, slower moving, and more isolated reservoir

of saline water.

Clearly oceans are by far the largest reservoir of water on Earth, which means

that a vast proportion of water on the Earth is salt water. The majority of Earth’s


TH and the Global Water Cycle 3

freshwater supply is currently stored in the polar ice caps, as glaciers or permafrost,

or as groundwater. Freshwater lakes, rivers, and marshes contain only about 0.01%

of Earth’s total water. The water present in the atmosphere is very small indeed,

only about 0.001%. However, the water exchanged between this atmospheric

reservoir and the oceanic and land reservoirs is comparatively large, on the order

of 100 km3 per year for land and 400 km3 per year for oceans. Consequently, there

is a rapid turnover in atmospheric water and the atmospheric residence time is

low.

Figure 1.1 illustrates the annual average hydrological cycle for the Earth as a

whole, together with an alternative set of estimates of water stores made by

combining observations with model-calculated data. It is clear that the simple

concept of a hydrological cycle that merely involves water evaporating from the

ocean, falling as precipitation over land then running back to the ocean is a poor

representation of the truth. There are also substantial hydrological cycles over the

oceans which cover about 70% of the globe, and over the continents which cover

the remainder, as well as water exchanged in atmospheric and river flows between

these two.

On average there is a net transfer from oceanic to continental surfaces because

the oceans evaporate about 413 × 103 km3 yr −1 of water, which is equivalent to about

1200 mm of evaporation, but they receive back only about 90% of this as

precipitation. Some of the water evaporated from the ocean is therefore transported

over land and falls as precipitation, but on average about 65% of this terrestrial

precipitation is then re-evaporated and this provides some of the water subsequently

falling as precipitation elsewhere over land. On average about 35% of terrestrial

precipitation returns to the ocean as surface runoff, but the proportion of terrestrial

precipitation that is re-evaporated and the proportion leaving as surface runoff

Table 1.1 Estimated sizes of the main water reservoirs in the Earth system, the

approximate percentage of water stored in them and turnover time of each reservoir

(Data from Shiklomanov, 1993).

Volume (106 km3)

Prcentage of total

Approximate residence time

Oceans (including saline inland seas)

∼1340 ∼96.5 1000 – 10 000 years

Atmosphere ∼0.013 ∼0.001 ∼10 daysLand: Polar Ice, glaciers, permafrost

∼24 ∼1.8 10 – 1000 years

Groundwater ∼23 ∼1.7 15 days – 10 000 yearsLakes, swamps, marshes ∼0.19 0.014 ∼10 yearsSoil moisture ∼0.017 0.001 ∼50 daysRivers ∼0.002 ∼0.0002 ∼15 daysBiological water ∼0.0011 ∼0.0001 ∼10 days



varies significantly both regionally and with season. Area-average runoff in the

semi-arid south western USA is, for example, commonly just a few percent. When

averaged over large continents and over a full year, variations in the fraction of

precipitation leaving as runoff are less. Table 1.2 gives an example of the estimated

annual water balance for the continents (Korzun 1978). Runoff ratios in the range

of 35 to 45% are the norm, but the extensive arid and semi-arid regions of Africa

reduce average runoff for that continent. Fractional runoff in the form of icebergs

from Antarctica is hard to quantify but may be 80% because sublimation from the

snow and ice covered surface is low.

Components of the global hydroclimate system

Understanding the hydroclimate of the Earth does not merely require knowledge

of hydrometeorological process in the atmosphere. Several different components

of the Earth system interact to control the way near-surface weather variables vary

Landprecipition

113

Land

RiversLakes178

Soil moisture122

Ocean

Ocean1 335 040

Surface flow

Ground water flow

Oceanevaporation 413

Oceanprecipitation

373

Ocean to landWater vapor transport

40

Atmosphere12.7

Groundwater15 300

Permafrost22

Vegetation

Percolation

40

Evaporation, transpiration 73

Ice26,350

Figure 1.1 The global annual average hydrological cycle including estimates of the main water reservoirs (in plain font in

units of 103 km3) and of the flow of moisture between stores (in italics in units of 103 km3 yr−1). (From Trenberth et al., 2007,

published with permission.) See Plate 1 for a colour version of this image.



in time and space. It is helpful to recognize the nature of these components from

the outset and to appreciate in general terms how they influence global hydrocli-

matology. For this reason we next consider salient features of the atmosphere,

hydrosphere, cryosphere, and lithosphere, biosphere, and anthroposphere.

Atmosphere

The air surrounding the Earth is a mixture of gases, mainly (~80%) nitrogen and

(~20%) oxygen, but also other minority gases such as carbon dioxide, ozone, and

water vapor which have an importance to hydroclimatology not adequately

reflected by their low concentration. Compared to the diameter of the Earth

(~20,000 km), the depth of the atmosphere is small. The density of air changes

with height but about 90% of the mass of the atmosphere is within 30 km and

99.9% within 80 km of the ground.

The atmosphere is (almost) in a state of hydrostatic equilibrium in the vertical,

with dense air at the surface and less dense air above; there is an associated change

in pressure. The temperature of the air changes with height in a very distinctive

way and this can be used to classify different layers or ‘spheres’. Figure 1.2 shows

the vertical profile of air temperature in the US Standard Atmosphere (US Standard

Atmosphere, 1976) as a function of height and atmospheric pressure. Starting

from the surface, the main layers are the troposphere, stratosphere, mesosphere,

and thermosphere, separated by points of inflection in the vertical temperature

profile that are called ‘pauses’. Near the ground, air temperature falls quickly with

height for reasons which are discussed in more detail later. Higher in the

atmosphere the air is warmed by the release of latent heat when water is condensed

in clouds and, in the upper stratosphere, it is also warmed by the absorption of a

portion of incoming solar radiation. There is then further cooling through the

mesosphere, but some further warming at the very top of the atmosphere where

most of the Sun’s gamma rays are absorbed.

Table 1.2 Estimated continental water balance (Data from Korzun, 1978).

Precipitation (mm year−1)

Evaporation (mm year−1)

Runoff (mm year−1)

Runoff as percentage of precipitation

Africa 740 587 153 21Asia 740 416 324 44Australia 791 511 280 35Europe 790 507 283 36North America 756 418 339 45South America 1595 910 685 43



The relative concentration of atmospheric nitrogen, oxygen and other inert

gases is uniform with height, but most ozone is found in the middle atmosphere

where it absorbs ultraviolet radiation to warm the air. The concentration of carbon

dioxide falls away in the mesosphere and the vast majority of atmospheric water

vapor is found within 10 km of the ground, mainly in the lower levels of the

troposphere. The fact that water vapor content falls quickly with height is strongly

related to the fall in temperature with height. The amount of water vapor that air

can hold before becoming saturated is less at lower temperatures and water is

precipitated out as water droplets or ice particles in clouds. The concentrations of

liquid and solid water in clouds and that of other atmospheric constituents,

including solid particles such as dust particles, sulfate aerosols, and volcanic ash,

all vary substantially both in space and with time.

As previously mentioned, the residence time for water in the atmosphere is

short, about 10 days. In fact, a comparatively short response time is a general

feature of the atmosphere that distinguishes it from the other components of the

climatic system. Air has a relatively large compressibility and low specific heat and

density compared to the fluids and solids that make up the hydrosphere,

cryosphere, lithosphere and biosphere. Because air is more fluid and unstable, any

20

40

60

80

100

0160 180 200 220

Temperature (K)

240 260 280 300100

50

10

5

1

0.5

0.1

0.05

0.01

0.005

0.001

0.0005

0.0001Thermosphere

Mesosphere

Mesopause

Stratosphere

Stratopause

Hei

ght (

km)

Pre

ssur

e (k

Pa)

Troposphere

TropopauseFigure 1.2 Idealized vertical

temperature profile for the

US Standard Atmosphere

showing the most important

layers, ‘spheres’, and the

‘pauses’ that separate them.



perturbations generated by changes in the inputs that drive the atmosphere

typically decay with time scales on the order of days to weeks.

Differential heating by the Sun causes movement in the atmosphere that is

complicated by the rotation of the Earth, the Earth’s orbit around the Sun, and

inhomogeneous surface conditions. Consequently, the air in the troposphere

undergoes large-scale circulation which, on average, is organized at the global

scale. There are substantial perturbations within this circulation associated with

weather systems at mid-latitudes, and also pseudo-random turbulent motion in

the atmospheric boundary layer and near ‘jet streams’ higher in the atmosphere.

Figure 1.3 shows how contributions to the variance of atmospheric movements in

the atmospheric boundary layer change as a function of frequency. Most movement

occurs at low frequencies. The first peak in this figure is associated with movement

linked to the annual cycle and is in response to seasonal changes in solar heating,

while the third peak is linked to the daily cycle of heating. The large contribution

to variance at the time scales of days to weeks is the result of the large-scale

disturbances associated with transient weather systems. At lower frequencies

atmospheric variance is therefore mainly associated with horizontal features

within the atmospheric circulation. The fourth maximum in variance, which

occurs at timescales of an hour or less, is different because it is due to small-scale

turbulent motions. Such turbulent variations occur in all directions, but their

influence on the vertical movement of atmospheric properties and constituents is

particularly important. Understanding this influence on the vertical movement is

a critical aspect of hydrometeorology.

10

0.01 0.1 1 10

1 day1 month

Frequency (day−1)

Kin

etic

ene

rgy

(m−2

s−2)

1 year

1 hour 1 min. 1 sec.

100 1000 10 000

20

30

40

50

Figure 1.3 Approximate

spectrum of the contributions

to the variance in the

atmosphere for frequencies

between 1 second and 1000

days.



Hydrosphere

The liquid water in oceans, interior seas, lakes, rivers, and subterranean waters

constitute the Earth’s hydrosphere. Oceans cover about 70% of the Earth’s surface

and therefore intercept more total solar energy than land surfaces. Most of the

energy leaves oceanic surfaces in the form of latent heat in water vapor, but this is

not necessarily the case for land surfaces. Consequently, maritime air masses are

very different to continental air masses. The atmosphere and oceans are strongly

coupled by the exchange of energy, matter (water vapor), and momentum at their

interface, and precipitation strongly influences ocean salinity. The mass and

specific heat of the water in oceans is much greater than for air and understanding

this difference is very important in the context of seasonal changes in the atmos-

phere. The oceans represent an enormous reservoir for stored energy. As a result,

changes in the sea surface temperature happen fairly slowly and this moderates the

rate of change of associated features in the atmosphere, thereby greatly aiding

seasonal climate prediction.

The oceans are also denser than the atmosphere and have a larger mechanical

inertia, so ocean currents are much slower than atmospheric flows, and oceanic

movement at depth is particularly slow. The atmosphere is heated from below by

the Sun’s energy intercepted by the underlying surface, but oceanic heating is from

above. Consequently, there is a profound difference in the way buoyancy acts in

these two fluid media. The higher temperature at the surface of the sea means

oceanic mixing by surface winds tends to be suppressed, and such mixing is lim-

ited to the active surface layer that has a thickness on the order of 100 m. A strong

gradient of temperature below this surface layer separates it from the deep ocean.

The response time for oceanic movement in the upper mixed layer is weeks to

months to seasons. In the deep ocean, however, movement due to density variations

associated with changes in temperature and salinity occur over time scales from

centuries to millennia. There are eddies in the upper ocean but turbulence is in

general much less pronounced than in the atmosphere. Ocean currents are

important because they move heat from the tropical regions, where incidental

solar radiation is greatest, toward colder mid-latitude and polar regions where

radiation is least. Currents in the upper layer of the ocean are driven by the

prevailing wind patterns in the atmosphere. Ocean flow is from east to west in the

tropics (in response to the trade winds), poleward on the eastern side of continents,

then back toward the equator on the western side of continents.

Lakes, rivers, and subterranean waters make up the remainder of the hydro-

sphere. They can have significant hydrometeorological and hydroclimatological

significance in continental regions, particularly at regional and local scale. The

contrast between the influence on the atmosphere of open water on the one hand

and land surfaces on the other is significant. This is responsible for ‘lake effect’

snowfall in the US Great Lakes and ‘river breeze’ effects near the Amazon River,

for example. River flow into oceans also has an important influence on ocean

salinity near coasts.



Cryosphere

The areas of snow and ice, including the extended ice fields of Greenland and

Antarctica, other continental glaciers and snowfields, sea ice, and areas of permafrost,

are the Earth’s cryosphere. The cryosphere has an important influence on climate

because of its high reflectivity to solar radiation. Continental snow cover and sea ice

have a market seasonally, and this can give rise to significant intra-annual and perhaps

interannual variations in the surface energy budgets of frozen polar oceans and conti-

nents with seasonal snow cover. Gradual warming in polar regions has the potential to

give rise to similar changes in surface energy balance over longer time periods. The

low thermal diffusivity of ice can also influence the surface energy balance at high lati-

tudes, because ice acts as an insulator inhibiting loss of heat to the atmosphere from

the underlying water and land. Near-surface cooling also gives rise to stable atmos-

pheres, which inhibit convection and contribute to cooler climates locally.

The large continental ice sheets do not change quickly enough to influence seasonal or interannual climate much, but historical changes in ice sheet extent

and potential changes in the future extent of ice sheets are important because they

are associated with changes in sea level. If substantial melting of the continental ice

sheets occurs, altered sea level could change the boundaries of islands and

continents. Since many inhabited areas are close to such boundaries, sea level

change will likely have serious consequences for human welfare that are dispro-

portionate to the fractional area of land affected. The effect of global warming on

ice sheets is considered a major threat for this reason.

Lithosphere

The lithosphere, which includes the continents and the ocean floor, has an almost

permanent influence on the climatic system. There is large-scale transfer of

angular momentum through the action of torques between the oceans and the

continents. Continental topography affects air motion and global circulation

through the transfer of mass, angular momentum, and sensible heat, and the

dissipation of kinetic energy by friction in the atmospheric boundary layer.

Because the atmosphere is comparatively thin, organized topography in the form

of extended mountain ranges that lie roughly perpendicular to the preferred

atmospheric circulation, such as the Rocky Mountains in North America and the

Andes Mountains in South America, can inhibit how far maritime air penetrates

into continents and thus affect where clouds and precipitation occur.

The transfer of mass between the atmosphere and lithosphere is mainly as water

vapor, rain, and snow. However, this may sometimes also occur as dust when vol-

canoes throw matter into the atmosphere and increase the turbidity of the air. The

ejected sulfur-bearing gases and particulate matter may modify the aerosol load

and radiation balance of the atmosphere and influence climate over extended

areas. The soil moisture in the active layer of the continental lithosphere that is



accessible to the atmosphere via plants can have a marked influence on the local

energy balance at the land surface. Soil moisture content affects the rate of

evaporation, the reflection by soil of solar radiation, and the thermal conductivity

of the soil. Because soil moisture tends to change fairly slowly it can provide a

land-based ‘memory’ with an effect on the atmosphere broadly equivalent to that

of slowly changing sea surface temperature.

Biosphere

Terrestrial vegetation, continental fauna, and the flora and fauna of the oceans

make up the biosphere. It is now recognized that the nature of vegetation covering

the ground is not only influenced by the regional hydroclimate, but also itself

influences the hydroclimate of a region. This is because the type of vegetation

present affects the aerodynamic roughness and solar reflectivity of the surface, and

whether water falling as precipitation leaves as evaporation or runoff. The rooting

depth of vegetation matters because it determines the size of the moisture store

available to the atmosphere. Changes in the type of vegetation present may occur

in response to changes in climate, and modern climate prediction models attempt

to represent such evolution. Imposed changes caused by human intervention

through, for example, large-scale deforestation or irrigation also occur and these

can be extensive and alter surface inputs to the atmosphere of continental regions.

The behavior of the biosphere influences the carbon dioxide present in the

atmosphere and oceans through photosynthesis and respiration. It is essential to

include description of these influences when models are used to simulate global

warming. For this reason, advanced sub-models describing the biosphere in mete-

orological models seek to represent the energy, water, and carbon exchanges of the

biosphere simultaneously. Water and carbon exchanges are linked by the fact that

the water transpired and the carbon assimilated by vegetation occurs by molecular

diffusion through the same plant stomata. Models of the biosphere are often

referred to as Land Surface Parameterization Schemes (LSPs) or Soil Vegetation

Atmosphere Transfer Schemes (SVATs); see Chapter 24 for greater detail. Figure 1.4

shows an example of the Simple Biosphere Model (SiB; Sellers et al., 1986), an

example of a SVAT that is currently widely used.

Because the biosphere is sensitive to changes in climate, detailed study of past

changes in its nature and behavior as revealed in fossils and tree rings and in pollen

and coral records is important as a means of documenting the prevailing climate

in previous eras.

Anthroposphere

The word anthroposphere is used to describe the effect of human beings on the

Earth system. For much of our existence human impact on the environment was

small, but as our numbers grew our impact on the atmosphere and landscape



expanded. With the start of the industrial revolution in the late eighteenth century,

humans developed the ability to harness power from fossil fuels and transitioned

from mostly observers to participants in global change. We have significantly

altered the biogeochemical cycles of carbon, nitrogen, sulfur, and phosphorus.

Alteration of the carbon cycle has changed the acidity of the oceans and is chang-

ing the climate of Earth. Chemical inventions such as chlorinated fluorocarbons

(CFCs) have altered the ozone in the stratosphere and the amount of ultraviolet

light reaching the Earth’s surface. The footprint of our chemical activities is now

found in the air, water, land, and biota of Earth in the form of naturally occurring

and human-created molecules.

The anthroposphere has expanded to occupy land for dwellings and agriculture.

Human dwellings now occupy about 8% of ice-free land and about three-quarters

of the land surface has been altered by humans in some way. As mentioned above

and discussed in more detail in later chapters, changing the nature of the land

Figure 1.4 A schematic

diagram of the physical and

physiological processes

represented in the second

generation Simple Biosphere

(SiB2) soil vegetation

atmosphere transfer scheme.

(From Colello et al., 1998,


See Plate 2 for a colour

version of this image.



surface is important for hydrometeorology because it alters the way energy from

the Sun enters the atmosphere from below and, if sufficiently extensive, land-use

change has impact on regional and perhaps global climate and weather. Examples

of such change include urban heat islands and changes to regional evaporation due

to the building of large dams, extensive irrigation, and land-cover change such as

deforestation.

When compared with most natural changes in other spheres of the globe,

change in the anthroposphere is happening very rapidly. This is partly because

human population has increased quickly over the past few centuries and still is

today, but also because strides in technology have empowered humans to

directly and indirectly effect change to the environment in new and different

ways, and because as society develops the per capita demand for energy

increases hugely.

Important points in this chapter

● Hydrometeorology: hydrometeorology (and this text) concerns the physics,

mathematics, and statistics of processes and phenomena involved in

exchanges between the atmosphere and ground that typically occur over

hours or days, and how the time average of these exchanges combine to

define hydroclimatology.

● Water reservoirs: the size of the Earth’s water reservoirs are poorly defined

but include the oceans (~ 95.6%), groundwater (~2.4%), frozen water (1.9%),

and water bodies, soil moisture, atmospheric water, rivers and biological

water (in total ~ 0.01%).

● Water cycle: as a global average about 90% of oceanic evaporation falls back

to the oceans as precipitation, the remainder being transported over land;

and about 55–65% of the precipitation falling over land re-evaporates

(depending on the continent) leaving 35–45% to runoff back to the ocean in

rivers and icebergs.

● Atmosphere:

— Constituents and structure: About 80% N2 and 20% O

2 and other minority

gases (CO2, O

3, H

2O, etc.), 99.9% of which are within 80 km of the ground

in the troposphere, stratosphere, mesosphere, thermosphere, with most

water vapor in the lower troposphere within 10 km of the ground.

— Circulation: Differential heating by the Sun causes global circulation in

the troposphere which moves energy toward the poles and which is

complicated by the Coriolis force but is, on average, organized.

— Variance: Contributions to the variance of the atmosphere arise at

frequencies linked to the seasonal cycle, transient weather systems,

and the daily cycle, with these contributions separated by a distinct

spectral gap from those at higher frequencies that are associated

with turbulence.



● Hydrosphere:

— Extent and importance: Oceans cover ~70% of the Earth and the solar

energy they intercept is mainly used to evaporate water vapor into the

atmosphere; they have a large thermal capacity and act as ‘memory’ in the

Earth system that influences season climate.

— Structure: Oceans have a surface layer 10s–100s m deep warmed by the

Sun’s energy in which there are wind-driven ocean currents, this layer

being separated by a strong thermal gradient from the deep ocean which

moves very slowly in response to changes in temperature and salinity.

— Currents: Upper ocean currents move heat from the tropics to polar

regions: ocean flow is east to west in the tropics, poleward on the eastern

side of continents, then back toward the equator on the western side of

continents.

● Cryosphere: comprises the polar ice fields and glaciers that change slowly

and transient continental snow/ice fields with a strong seasonal influence on

climate.

● Lithosphere: organized topography perpendicular to atmospheric flow can

inhibit penetration of maritime air into continents, and aerosols from volca-

noes can alter the radiation balance over extensive areas.

● Biosphere: vegetation cover affects aerodynamic roughness and reflection of

solar energy and by intercepting rainfall and accessing water in the soil

through roots, also whether precipitation leaves as evaporation or runoff.

● Anthroposphere: human population is now large enough to influence cli-

mate, mainly by changing the concentrations of gases in the atmosphere and

by modifying land cover over large areas.

References

Colello, G.D., Grivet, C., Sellers, P.J., & Berry, J.A. (1998) Modeling of energy, water, and

CO2 flux in a temperate grassland ecosystem with SiB2: May–October 1987. Journal of

Atmospheric Sciences, 55 (7), 1141–69.

Langbein, W.G. (1967) Hydroclimate. In: The Encyclopaedia of Atmospheric Sciences and

Astrogeology (ed. R.W. Fairbridge), pp. 447–51. Reinhold, New York.

Korzun, V.I. (1978) World Water Balance of the Earth. Studies and Reports in Hydrology, 25.

UNESCO, Paris.

Sellers, P.J., Mintz, Y., Sud, Y.C. & Dalcher, A. (1986) A simple biosphere model (SiB) for use

within general circulation models. Journal of Atmospheric Sciences, 43, 505–531.

Shiklomanov, J.A. (1993) World fresh water resources. In: Water in Crisis: A Guide to the

World’s Fresh Water Resources (ed. P.H. Gleick), pp. 13–24. Oxford University Press, New

York.

Trenberth, K.E., Smith, L., Qian, T., Dai, A. & Fasullo, J. (2007) Estimates of the global water

budget and its annual cycle using observational and model data. Journal of

Hydrometeorology, 8, 758–69.

US Standard Atmosphere (1976) US Government Printing Office, Washington, DC.


Water Vapor in the Atmosphere2

Introduction

Hydrometeorologists are commonly concerned with quantifying the amount of

water in the atmosphere in vapor, liquid, and solid form and with seeking to

describe the way energy and water move vertically in the atmosphere toward and

away from the ground. In this chapter we consider the basic definitions and

important concepts needed for this.

Latent heat

The molecules that make up ice are held rigidly together in close proximity by

intermolecular forces. In liquid water the molecules are also close together but,

because they are at a higher temperature, they move around and their average

separation is therefore somewhat greater. In water vapor, molecular separation is

very much larger: molecules in water vapor are typically separated by about ten

molecular diameters. As water molecules move farther apart, the forces that bind

them reduce quickly with distance and at ten molecular diameters these forces are

much smaller than when the molecules are in near contact. Viewed in this way, the

transition from ice to liquid water and then to water vapor can be viewed as a

temperature related increase in the separation of molecules in the face of the

attractive intermolecular forces acting between them.

To move against a force requires work, and therefore energy has to be given to

separate water molecules to give changes in phase. The amount of energy needed

is directly related to the number of molecules present and therefore to the mass of

water that changes phase. The amount of energy needed for ice-to-liquid water

transition is called the latent heat of fusion and is 0.333 MJ kg−1. Because the change




Water Vapor in the Atmosphere 15

in separation for transition from liquid water to water vapor is much larger, the

latent heat of vaporization for liquid water is also greater. The amount of energy

needed also depends on the temperature at which the phase changes occur.

Molecules in warm water are already a little farther apart than in cold water, so

rather less energy is needed and the latent heat of vaporization is slightly lower at

higher temperatures. For this reason, l, the latent heat of vaporization of water,

is temperature dependent and when the temperature, TC, is given in °C, l is

calculated by:

−= − 12.501 0.002361 MJ kgCTl

(2.1)

Atmospheric water vapor content

As discussed in Chapter 1, the atmosphere is a complex mixture of gases with

nitrogen and oxygen being the dominant constituents. However, in the tropo-

sphere, water vapor is a particularly important constituent, with a variable con-

centration that is typically a few percent. When describing the concentration of

water vapor in the air, it is convenient to speak in terms of air comprising just

two constituents, i.e., water vapor and ‘dry air.’ Dry air is the general description

of all the other gases present. The sum of these two constituents is then referred

to as ‘moist air.’

If the densities of water vapor, dry air, and moist air in an air sample are rv, r

d

and ra, respectively, the proportion of water vapor in the moist air can be charac-

terized in several different ways. One is as the mixing ratio, r, which is defined as

the ratio of the mass of water vapor to the mass of dry air in the moist air sample,

expressed in terms of densities as:

v

d

r =rr

(2.2)

Another common way to define the water vapor content of an air sample is as the

ratio of the mass of water vapor to the mass of moist air. This ratio, q, is called the

specific humidity of the air, expressed in terms of densities as:

( )v v

a v d

q = =+

r rr r r

(2.3)

Because rv is always considerably less than r

d and r

a, the numerical values of mix-

ing ratio and specific humidity are usually very similar. Strictly speaking, both

r and q are dimensionless quantities, but often q is referred to as having units of

kg kg−1 or, more often, gm kg−1. The specific humidity of air in the troposphere is

generally in the range 0–30 gm kg−1.


16 Water Vapor in the Atmosphere

Ideal Gas Law

In the seventeenth and eighteenth centuries it was discovered that there are simple

relationships between the pressure, volume, and temperature of gases. Boyle’s Law,

for example, states that at constant temperature, the absolute pressure, P, and the

volume, V, of a gas are inversely proportional. Similarly, Charles’s Law states that at

constant pressure, the volume of a given mass of an ideal gas increases or decreases

by the same factor as its temperature increases or decreases, providing the

temperature, T, is measured in K. These two laws can be combined to give the

Ideal Gas Law, which has the form:

PV nRT= (2.4)

where R is the universal gas constant equal to 8.314 J mol−1 K−1, and n is the number

of moles of gas considered. One mole of gas is defined as comprising 6.02 × 1023

molecules: this number is called Avogadro’s number. Strictly speaking, Equation

(2.4) only applies for an ‘ideal’ gas in which the molecules are treated as non-

interacting point particles engaged in a random motion that obeys the conserva-

tion of energy. In practice, however, the real gases that make up the atmosphere

approximate the behavior of an ideal gas closely for the range of temperatures and

pressure found in the troposphere.

The mass of a mole of one specific gas, Mg, is called the gram molecular weight

of the gas. If a volume, V, contains n moles of gas, it therefore has a mass (nMg),

and the sample of gas has a density rg = (nM

g)/V. This means Equation (2.4) can be

re-written as:

g gP R T= r

(2.5)

where Rg = (R/M

g) is the gas constant for the specific gas. It is convenient to use this

second form of the ideal gas law when describing moist air. The gram molecular

weight of water is 18 grams per mole and if dry air is assumed to comprise

78% nitrogen and 22% oxygen, the gram molecular weight of dry air is 29 grams

per mole. Consequently, the specific gas constants of water vapor and dry air are

461.5 J kg−1 K−1 and 286.9 J kg−1 K−1, respectively.

Since the molecules making up a gas are typically separated by about ten

molecular diameters at normal temperatures and pressures, only about one

thousandth of the volume of the gas is actually occupied by gas molecules. Gases

are therefore largely empty space and this means that for a gas that is a mixture of

molecules, the contribution to the pressure made by one constituent gas in the

mixture is independent of the pressure contribution made by the other gases

present. This result is Dalton’s law of partial pressures, and it means that the ideal

gas law can be applied not only to the mixture of gases but also separately to each

constituent in a gas mixture.



Thus, if a sample of moist air has a total pressure, P, the contribution to that

pressure given by the water vapor molecules it contains, e, called the vapor pressure

of the moist air, is given by:

v ve R T= r

(2.6)

where rv is the (variable) density of the water vapor in the moist air mixture and R

v

is the gas constant for water vapor. Similarly, if we treat the remainder of the moist

air as being one gas, the contribution to the total pressure of this dry air compo-

nent of the moist air gas mixture is (P-e), given by:

( ) ( )a v dP e R T− = −r r

(2.7)

where ra is the density of the moist air mixture and R

d is the gas constant of the dry

air. Like the mixing ratio and specific humidity, the vapor pressure of the air is a

measure of the amount of water vapor present in a sample of moist air and it is

frequently used as such in this text.

The different measures of atmospheric water vapor content are of course inter-

related. Because (Rd /R

v) = 0.62, the relationship between the specific humidity and

vapor pressure of the air can be shown to have the form:

( )( )

0.62

0.381

d vv

a d v

e R R eq

P eP e R R= = =

−⎡ ⎤− −⎣ ⎦

rr

(2.8)

with q in kg kg−1. However, because e is usually much less than P it is often accept-

able to assume that, to an accuracy of a few percent:

0.622e

q rP

= =

(2.9)

Virtual temperature

Combining Equations (2.6) and (2.7) gives:

( ) 1 1v v

a v d v v a d

d a

RP R T R T R T

R

⎡ ⎤⎛ ⎞= − + = + −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

rr r r r

r

(2.10)

This equation can be re-written as:

a d vP R T= r

(2.11)



where Tv is called the virtual temperature. It is the temperature that dry air

would have if it had the same density and pressure as the moist air, and is given by:

1 1v v

v

d a

RT T

R

⎡ ⎤⎛ ⎞= + −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

rr

(2.12)

Substituting Equation (2.2) and recalling that (Rd/R

v) = 0.62, this last equation

becomes:

1 0.61 v

T T r= +⎡ ⎤⎣ ⎦ (2.13)

and because q and r are numerically similar, it is usually acceptable also to write:

1 0.61 v

T T q= +⎡ ⎤⎣ ⎦ (2.14)

In the next chapter we use this definition of virtual temperature to adjust the

temperature profile of the atmosphere so as to compensate for the effect on atmos-

pheric buoyancy of density variations associated with height-dependent changes

in water vapor concentration.

Saturated vapor pressure

The net evaporation rate from a water surface is the difference between two

exchange rates: the rate at which molecules are being ‘boiled off ’ from the water

surface minus the rate at which molecules of water already present in the air above

the surface are recaptured back into the water, see Fig. 2.1.

Condensation Vaporization

Water vapor

Liquid waterFigure 2.1 Vaporization and

capture components of the

evaporation process.



According to Kevin-Boltzmann statistics, the number of molecules that acquire

enough energy to break the intermolecular bonds in the water and enter the over-

lying air is related to temperature, i.e.:

1“Boil off ” rate exp

nk

kT

⎛ ⎞≈ −⎜ ⎟⎝ ⎠

l

(2.15)

where k and k1 are constants and n is the number of molecules per unit volume.

On the other hand, the rate of capture of molecules is directly related to concentra-

tion of vapor molecules in the overlying air, i.e.:

( )2“Capture” rate 1 'k r e≈ −

(2.16)

where k2 is a constant and r′ is the fraction of water vapor molecules colliding with

the surface that are reflected without capture. If the (temperature dependent) boil

off rate exceeds the (concentration dependent) capture rate, there is net evapora-

tion and liquid water leaves the surface and enters the air as water vapor.

But what would happen if the air above the evaporating water surface was enclosed

so that evaporated molecules were not able to diffuse away higher into the atmos-

phere? Gradually the concentration of molecules in the air would rise until such time

as the rate of capture equaled the rate of boil off. There would then be no net loss of

water molecules, evaporation would cease; the air is then be said to be saturated.

Because net evaporation is the difference between two rates, there is a well-defined

concentration of water vapor at which the net exchange is zero. This concentration

depends on the temperature-dependent boil off rate. Were the temperature higher,

for example, the boil off rate would increase and exchange equilibrium would be

established with the saturated air having a higher concentration of water vapor.

The relationship between the saturated vapor pressure, esat

, and temperature has

been defined by experiment and several empirically determined relationships

have been proposed. Here we select the following:

⎛ ⎞= ⎜ ⎟+⎝ ⎠

17.27 0.6108exp kPa

237.3

C

sat C

Te

T

(2.17)

when the temperature is given in °C. Note that elsewhere in this text the temperature,

T, is usually expressed in K. Here, T C is used to represent temperature to emphasize

that in this empirical formula the value of temperature must be expressed in °C.

The gradient of the relationship between saturated vapor pressure and temperature

is often used in equations describing evaporation rate and, when used in this way,

this gradient is usually represented by Δ. Differentiating Equation (2.17) gives:

( )1

2

4098 kPa °C

237.3

sat sat

CC

de e

dT T

−Δ = =+

(2.18)



Figure 2.2 illustrates how esat

and Δ change as a functions of TC. Both variables

change substantially over the normal range of temperatures. The fact that esat

and

Δ have specified relationships with temperature is very important in hydrometeor-

ology: it means that additional equations are available to link the surface exchanges

of water vapor and heat.

At a particular temperature, the density of water vapor in saturated air, rv,sat

, can

be specified from the saturated vapor pressure using Equation (2.6). Consequently,

corresponding values of saturated mixing ratio, rsat

, and saturated specific humid-

ity, qsat

, can be defined by substituting this value of rv,sat

into Equations (2.2) and

(2.3), respectively.

Measures of saturation

Figure 2.3 is helpful, not only because it defines the extent to which the atmosphere

is saturated, but also because later it is used in important methods to measure

the vapor pressure of moist air. This diagram is for the example case of an atmos-

phere with a temperature of 35°C that is 60% saturated.

Probably the most common way to specify the extent to which air is saturated is

to specify its relative humidity, RH. Relative humidity is defined as the ratio of the

actual vapor pressure of the air to the saturation vapor pressure at air temperature

and is normally expressed as a percentage. To good accuracy the relative humidity

can also be calculated as the ratio of the mixing ratio of the air to saturation mixing

ratio at air temperature, or as the ratio of specific humidity of the air to the satura-

tion specific humidity at air temperature, thus:

100 % 100 % 100 %sat sat sat

qe rRH

e r q

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (2.19)

A second important measure of atmospheric humidity content is the vapor pressure

deficit, D. The vapor pressure deficit is defined as the difference between saturation

vapor pressure at air temperature and the vapor pressure of the air, i.e.:

8

6

4

2

0

0.5

0.4

0.3

0.2

0.1

0.00 10 20

T c �C

30 400 10 20

T c �C

30 40e sat

(kP

a)

Δ (k

Pa�

C– 1

)

Figure 2.2 Variation in

saturated vapor pressure of water

and the gradient of that variation

as a function of temperature

in °C.



( )satD e e= −

(2.20)

Knowing the value of D for air is particularly important when calculating

evaporation rate because it provides a direct measure of how much additional

water vapor the atmosphere can hold.

Measuring the vapor pressure of air

The dew point of air, Tdew

(see Fig. 2.3) is a measure of the vapor pressure of the air.

Dew point is defined as the temperature to which air must be cooled at constant

pressure for it to saturate. It can be shown by inverting Equation (2.17) that for air

with vapor pressure e (in kPa), the dew point of the air in °C is given by:

( )( )

ln 0.49299

0.0707 0.00421 lndew

eT

e

+=

−

(2.21)

A dew point hygrometer measures the humidity of air by cooling an initially clear

mirror until its surface becomes clouded by water condensing onto the mirror.

The measured temperature of the mirror when this first occurs is the dew point

temperature of the air and the vapor pressure of the air (which is, by definition,

equal to the saturated vapor pressure at dew point) can be then calculated using

Equation (2.17).

8

6

4

2

00 10 20

T C �C

30 40

Vapor pressureof the air (e)

Vapor pressure deficitD = (esat– e)

Saturated vaporpressure at air

temperature (esat)

Relative humidityRH = (e/esat) x 100%

Airtemperature

e sat

(kP

a)

“Dew point”temperature

Figure 2.3 Measures of the

extent of atmospheric saturation

and temperatures used when

measuring the vapor pressure of

moist air.



Determining Twet

, the wet bulb temperature of a sample of air (see Fig. 2.3) while

also measuring air temperature is arguably the most common way that atmospheric

humidity is determined. In this context, the measured air temperature is generally

called the dry bulb temperature, Tdry

. The names wet bulb and dry bulb temperature

are because air temperature can be measured using a mercury thermometer with

a dry ‘bulb’ (i.e., mercury reservoir), while a second mercury thermometer whose

bulb is moist is used to measure Twet

. In practice, the wet bulb is usually covered

with a moist cloth sheath that is shaded from the Sun’s rays. Preferably both

thermometers should also be aspirated, i.e., have air drawn over their mercury

reservoirs, using a fan.

Wet bulb temperature is defined to be the temperature to which air is cooled

by evaporating water into it at constant pressure until it is saturated. It is help-

ful to consider the relationship between vapor pressure of air, and wet bulb and

dry bulb temperature by imagining a volume, V, of air overlying a thin layer of

water inside a container that thermally isolates the air from its surroundings.

Initially this air, which has a pressure P, has vapor pressure e and temperature

Tdry

. Some of the water in the thermally isolating container evaporates using

energy taken from the air itself to provide the required latent heat. Consequently,

the temperature of the air is progressively reduced. Eventually the air in the

container saturates. The now saturated air has the temperature Twet

and its

vapor pressure is equal to esat

(Twet

), the saturated vapor pressure at this tem-

perature, see Fig. 2.4.

From Equation (2.9), it follows that the initial and final specific humidity of

the air in the container are q = 0.622(e/P) and qsat

(Twet

) = 0.622[esat

(Twet

) /P)],

Air cools toprovide latent

heat

Vapor pressureof the air (e)

Air temperatureor “Dry bulb”temperature

Air moistensdue to

evaporation

“Wet bulb”temperature

403020100

TC �C

8

6

4

2

0

e sat

(kP

a)

Figure 2.4 Illustrating how

the air in a thermally

insolating container overlying

a thin layer of water saturates

and also cools to provide the

energy needed to evaporate

the water.



respectively. Because the air in the container is thermally isolated, the heat removed

from the air as it cools must equal the latent heat required to evaporate water to

raise the specific humidity of the air to saturation, i.e.:

( ) ( )– – a dry wet p a sat wetV T T c q T q V⎡ ⎤= ⎣ ⎦r r l

(2.22)

where cp (= 1.013 kJ kg−1 K−1) is the specific heat at constant pressure for air.

Rewriting Equation (2.22) in terms of vapor pressure:

( ) ( )0.622 – –

a sat wet

a p dry wet

e T ec T T

P

⎡ ⎤⎣ ⎦=r l

r

(2.23)

This last equation can be re-written as the so-called wet bulb equation, which takes

the form:

( ) ( )– sat wet dry wet

e e T T T= − γ

(2.24)

In this last equation the term (Tdry

– Twet

) is often called the wet bulb depression and

γ is the psychrometric constant calculated from:

0.622

pc P⎛ ⎞

γ = ⎜ ⎟⎝ ⎠l

(2.25)

Actually the psychrometric ‘constant’ is not constant because it varies with

atmospheric pressure and also to some extent with temperature, because the latent

heat of vaporization of water has temperature dependency, see Equation (2.1). For

a temperature of 20°C and pressure of 101.2 kPa, the value of γ is 0.0677.

Wet and dry bulb temperatures are often routinely measured at climate stations

and if measured using aspirated thermometers the value of γ calculated by

Equation (2.25) should be used to calculate the vapor pressure. However, if the

thermometers are not aspirated (which is often the case), a different, empirically

determined value of γ * is required to substitute for γ when calculating the atmos-

pheric vapor pressure from the wet bulb equation.


● Latent heat: separating water molecules in liquid water to give water vapor

requires energy (∼2.5 MJ kg−1) which is called the latent heat of vaporization

and which reduces with temperature by about 0.1% per °C.

● Atmospheric water content: is quantified in terms of the ratio of the density

of water vapor to that of the (dry) air, called the mixing ratio (r), or the



ratio of the density of water vapor to that of moist air, called the specific

humidity (q). The vapor pressure of air (e) is also a measure of atmospheric

water content.

● Ideal gas law: the temperature (T), volume (V) and pressure (P) of a gas

are related by the ideal gas law which can be written as PV = rgR

gT where

rg and R

g = R/M

g are the density and (gas-specific) gas constant, respectively,

and Mg is its gram molecular weight. When applied to both the dry air and

water vapor portions of moist air, to good accuracy this gives the result

q = r = (0.622 e/P).

● Virtual temperature: Tv = T(1+0.61q) is the temperature that dry air would

have if it had the same density and temperature as the moist air.

● Saturated vapor pressure: the maximum vapor pressure of air when

saturated, esat

, and the rate of change of esat

with temperature, Δ, are both

well-defined functions of the temperature.

● Measures of saturation: two measures in common use are the relative

humidity, RH = (100e)/ esat

, and the vapor pressure deficit D = (esat

– e).

● Measuring the vapor pressure: two ways to measure vapor pressure are:

— Dew point hygrometer: Dew point is obtained as the temperature of an

initially clear mirror cooled until its surface becomes clouded by dew.

Equation (2.17) is then applied to calculate e from Tdew

.

— Wet bulb psychrometer: Two (preferably aspirated) thermometers, one

dry and one wet, measure the dry bulb and wet bulb temperatures,

Tdry

and Twet

, respectively; vapor pressure is calculated from e = esat

( Twet

) –

γ (Tdry

– Twet

) where esat

(Tw) is the saturated vapor pressure at wet bulb tem-

perature and γ is the psychrometric constant.


Vertical Gradients in the Atmosphere3

Introduction

Air nearer to the ground must support the column of air above it and so has to

exert an upward force per unit surface area (i.e., an upward pressure) that is equal

and opposite to the downward gravitational force exerted by all the overlying air.

Because the mass of overlying air reduces with distance from the ground, air pres-

sure reduces with height. Air temperature is related to pressure and density

through the ideal gas law, Equation (2.5), so air temperature necessarily also

changes with height. In the absence of any disturbing influences such as heat

inputs from the Sun or surface, the atmosphere would therefore settle into a stable

condition with hydrostatic vertical gradients of pressure, density, and temperature,

all of which can be calculated.

Heat flow occurs when there is non-uniformity in the spatial distribution of

heat in a medium, with movement away from regions with higher temperature

toward regions with lower temperature. But the temperature gradient that is

established in a hydrostatic atmosphere is not associated with vertical heat flow.

Rather it is deviations from this temperature gradient that are associated with the

vertical movement of heat. Consequently, it is convenient to define a variable

that is directly related to vertical heat flow, potential temperature, which is a

combination of both local temperature and pressure, and to calculate the vertical

profile of this potential temperature to diagnose the thermal stability of the

atmosphere.

Because the gram molecular weight of water vapor is less than the average

gram molecular weight of the other gases (mainly nitrogen and oxygen) that

make up air, the density of air with more water vapor is less than that of air with

less water vapor. The concentration of atmospheric water vapor can change with

height, and height dependent variations in density associated with changes in

fractional vapor content complicate the relationship between the density,




26 Vertical Gradients in the Atmosphere

temperature, and pressure in a hydrostatic atmosphere. In practice, it is simpler

mathematically to allow for the effect of vertical changes in vapor content by

calculating and using the temperature profile of an equivalent dry atmosphere

with the same density as the actual moist atmosphere, i.e., the profile of virtual

temperature, see Equation (2.14) and associated text. In this way, the effect of

vertical changes in water vapor content can be allowed for by making a (usually

small) adjustment to the profile of potential temperature to give the virtual

potential temperature profile. The several hydrostatic profiles are discussed in the

following sections.

Hydrostatic pressure law

Figure 3.1 illustrates how pressure differs between two levels, z and z + δz, in the

atmosphere. Consider a thin rectangular volume of air of cross-sectional area

A and depth δz. This volume of air has a mass [A δz ra], where r

a is the density of

the moist air. At the bottom of this volume the pressure is P and at the top it is

P + δP. (Note that δP will be negative because the air below the volume must exert

a higher upward force across the area A to balance the additional gravitational

force acting on the mass of the air between the two levels). The forces exerted at

the top and bottom of the volume of air are A.(P + δP) and A. P, respectively, and

additional downward gravitational force is g.A.ra.(δz), where g is the acceleration

due to gravity, i.e., 9.81 m s−2. Therefore:

[ ] aAP A P P gA z− + =δ ρ δ

(3.1)

Hence:

aP g zδ = − ρ δ

(3.2)

Mass = [A. δz. ra] P + δP

P

A

δz

Figure 3.1 Basis of the

hydrostatic pressure law.


Vertical Gradients in the Atmosphere 27

In the limit of small δz this means:

a

Pg

z

∂= −

∂ρ

(3.3)

Adiabatic lapse rates

The word adiabatic is used to describe changes in which there is no net change in

energy. Here it is used to indicate that we are defining how the temperature of a

parcel of air of volume V would change if it were moved vertically in the atmos-

phere in such a way that there was no net change in the internal energy of the

parcel. The vertical movement of a buoyant air parcel might, for example, be

approximately adiabatic if its ascent is rapid and there is no time for the parcel to

exchange energy with the surrounding air.

The first law of thermodynamics states that the heat added to a system is the

sum of the change in internal energy plus the work done by the system on its sur-

roundings. When applied to the case of an air parcel of volume V moving a small

distance δz in the vertical and undergoing associated small changes δP in pressure

and δT in temperature, this law implies:

p

m

aV H* V c T V Pδ = −ρ d d

(3.4)

where δH* is the heat added per unit volume of air and cp

m is the specific heat at

constant pressure of moist air. The specific heat of moist air varies slightly, but

because the amount of water vapor in moist air is typically just a few percent, it is

generally considered acceptable to use cp = 1.013 kJ kg−1K−1, i.e., the specific heat at

constant pressure for dry air in Equation (3.4) instead of cp

m.

Dry adiabatic lapse rate

When vertical movement of the moist air is adiabatic and the air remains

unsaturated, (δH*) = 0. Equation (3.4) can then be rewritten as:

a pP c T=δ r d

(3.5)

Combining this last equation with Equation (3.2) gives:

δ δ= −p

gT z

c

(3.6)

To a good approximation, both g and cp are constants in the lower atmosphere.

Consequently, for vertical movements of an unsaturated air parcel that occur adiabati-

cally in the atmosphere, the resulting rate of temperature change is constant, i.e.:



T

z

∂= −Γ

∂ (3.7)

where Γ = g/cp is the dry adiabatic lapse rate which is conventionally defined to be

positive, and which has the value 0.00968 K m−1 or 9.68 K km−1.

Moist adiabatic lapse rate

When vertical movement of moist air is adiabatic but the air is saturated,

condensation can occur in the air parcel as it ascends. If its specific humidity

decreases by an amount δqsat

, latent heat is released and the internal energy of the

parcel changes by an amount δH* = (l .ra.δq

sat). Equation (3.4) therefore becomes:

aa sat pV q V c T V P= −λρ δ ρ δ δ

(3.8)

Combining this last equation with Equation (3.2), rearranging in the limit gives an

equation similar to Equation (3.7), i.e.:

m

T

z

∂= −Γ

∂ (3.9)

where Γm

is the moist adiabatic lapse rate given by:

sat

m

p

q

c z

⎛ ⎞∂Γ = Γ −⎜ ⎟∂⎝ ⎠

λ

(3.10)

Because the temperature of the atmosphere normally decreases with height the

saturated specific humidity also decreases with height, hence the rate at which the

temperature falls in a parcel of air ascending in a saturated atmosphere is less than

in an unsaturated atmosphere, i.e., Γm

< Γ. Because qsat

depends on temperature,

the moist adiabatic lapse rate also depends on temperature. For a specific value of

saturated vapor pressure, because the value of qsat

depends on pressure, see

Equation (2.9), Γm

also depends on atmospheric pressure.

Environmental lapse rate

The actual measured rate at which atmospheric temperature changes away from the

ground on a particular day at a particular place depends on the history of heat inputs

and outputs to the air overhead. The locally observed atmospheric lapse rate, which

is called the environmental lapse rate, Γe, therefore varies with location and time.

Near the surface the lapse rate may well be approximately constant, often with

a value intermediate to the dry and moist adiabatic lapse rates. The temperature



profile in the lower troposphere for the US Standard Atmosphere shown in Fig. 1.2

illustrates that this is the case on average, the average near-surface environmental

lapse rate in this case being 6.5 K km−1. However, it is important to emphasize that

the behavior shown in Fig. 1.2 is a temporal and spatial average for all of the USA

and the actual environmental lapse rate on any particular day and at any particu-

lar place will differ from this average profile. The actual environmental lapse rate

and, especially, its relationship to the dry adiabatic lapse rate and the moist

adiabatic lapse rate can affect cloud and precipitation formation, as discussed

in Chapter 10.

Vertical pressure and temperature gradients

Recall that the vertical gradients of atmospheric pressure and temperature are

necessarily linked via the ideal gas law. If Ra is the specific gas constant for moist

air, then by combining Equation (2.5) with Equation (3.2) and rearranging, it can

be shown that the small change in pressure over a vertical distance δz is given by:

a

gPz

P R T= −

δ δ

(3.11)

If Γlocal

is the local environmental lapse rate, then δT = Γlocal

δz. Using this relation-

ship to substitute for δz in Equation (3.11), it follows that the small changes in

pressure and temperature with height are related by:

a local

gP T

P R T

⎛ ⎞= ⎜ ⎟Γ⎝ ⎠

δ δ

(3.12)

If Γlocal

is constant through a portion of the atmosphere then by taking the limit

of Equation (3.12) and integrating between two levels where the air temperatures

(in degrees K) are T1 and T

2, and air pressures P

1 and P

2, it can be easily shown that

across this region:

2

2 1

1

a local

g

RTP P

T

Γ⎛ ⎞= ⎜ ⎟⎝ ⎠

(3.13)

2

2 1

1

a localR

gPT T

P

Γ

⎛ ⎞= ⎜ ⎟⎝ ⎠

(3.14)

Figure 3.2 illustrates the lapse rate and the way atmospheric pressure and air

density consequently change through the lower troposphere for the US Standard

Atmosphere for which Γlocal

= 6.5 K km−1.



Potential temperature

For an adiabatic atmosphere, Γlocal

= g/cp and Equation (3.14) can therefore be

re-written as:

2

2 1

1

a

p

R

cPT T

P

⎛ ⎞= ⎜ ⎟⎝ ⎠

(3.15)

This equation can be used to correct temperature variations in the atmosphere for

the effect of the hydrostatic pressure gradient. If such corrections are made, any

remnant variations in this corrected temperature profile are those which may result

in vertical heat flow as discussed earlier. It is convenient to use Equation (3.15)

to renormalize the observed temperature profile to correspond to a specific value

of pressure, i.e., 100 kPa. When corrected in this way the resulting temperature is

called the potential temperature, q. Potential temperature is therefore defined to be

the temperature that a parcel of air anywhere in the atmosphere would have if it

were to be brought adiabatically to a pressure of 100 kPa. It is calculated from the

actual temperature and pressure using the equation:

100a

p

R

c

TP

⎛ ⎞= ⎜ ⎟⎝ ⎠θ

(3.16)

For a parcel of air moving up or down in an adiabatic atmosphere the value of the

potential temperature is conserved and remains constant with height. To a good

approximation, the vertical gradient of the potential temperature can be calculated

from that for temperature using:

d

T

z z

∂ ∂= + Γ

∂ ∂θ

(3.17)

150 250 0 50 100 0.00

2

4

Hei

ght (

km)

6

8

10

0

2

4

Hei

ght (

km)

6

8

10

0

2

4

Hei

ght (

km)

6

8

10

0.5

Air density (kg m−3)Pressure (kPa)Temperature ( �K)

1.0 1.5

Figure 3.2 Temperature

lapse rate, and pressure and

density variations through

the troposphere of the US

Standard Atmosphere.



Virtual potential temperature

As well as correcting for the influence of the hydrostatic pressure gradient on

temperature, it is also possible to make a simple correction for the additional

effect of changes in water vapor content on local atmospheric density

(buoyancy) by calculating qv, the virtual potential temperature at any level.

This is done using a definition analogous to Equation (3.16) but expressed in

terms of the virtual temperature as defined by Equation (2.14). Virtual

potential temperature is therefore defined relative to the virtual temperature,

Tv = T(1 + 0.61q), by:

100a

p

R

c

v vT

P

⎛ ⎞= ⎜ ⎟⎝ ⎠θ

(3.18)

To a good approximation, the vertical gradient of virtual potential temperature

can be calculated from that for virtual temperature using:

v v

d

T

z z

∂ ∂= + Γ

∂ ∂θ

(3.19)

Figure 3.3 shows the calculated profiles of potential temperature and virtual

potential temperature for simple example gradients of temperature and humid-

ity. Note that in well-mixed portions of the atmospheric boundary layer (ABL),

the vertical gradients of humidity and potential temperature (but not actual

temperature) are often small. During the day a reversal in potential temperature

often then separates this well-mixed layer from the atmosphere above where

the air is typically drier than in the ABL. Although the actual temperature of

this overlying air may decrease with height, it typically falls at a rate less than

100

0.0 2.5 5.0

Mixing ratio (g kg−1)

7.5

Pre

ssur

e (k

Pa)

90

80

70

60

100

250 260 270

Temperature (K)

290280

Potentialtemperature

Temperature

Virtual potentialtemperature

300

Pre

ssur

e (k

Pa)

90

80

70

60

Figure 3.3 Simplified

daytime profiles of humidity

and temperature through the

atmospheric boundary layer

(both shown as thick black

lines) and the associated

potential temperature and

virtual potential temperature

calculated from these two.

Potential temperature is

shown as a thin black line and

the virtual potential

temperature as a gray line.



the dry adiabatic lapse rate, so the potential temperature of the overlying air is

greater than the air in the ABL. In general (and in Fig. 3.3), virtual potential

temperature is typically just a few degrees greater than potential temperature,

the difference being directly related to the specific humidity of the atmosphere,

see Equation (2.8).

Atmospheric stability

If a parcel of air moves up or down in the atmosphere (perhaps in response to

atmospheric turbulence) it is likely that it will find itself in a new environment

whose temperature and density differs from that which would have resulted from

adiabatic warming or cooling of the parcel. Any resulting density difference

between the parcel and the air that surrounds it will give rise to a buoyancy force

which will influence further vertical movement of the parcel.

If after moving, the parcel has a higher temperature and is less dense and lighter

than the air it displaces, it is said to be unstable in its new location and further

upward movement will be encouraged and downward movement inhibited. If

after moving the parcel has a lower temperature and is more dense and heavier

than the air it displaces, it is said to be stable in its new location and further upward

movement will be inhibited and downward movement encouraged. If after

moving, the parcel has the same temperature and density as the air it displaces, it

is said to be neutral with respect to the environment in its new location and further

movement is neither encouraged nor inhibited.

Static stability parameter

From Archimedes principle, if a parcel of air with density ra′ moves vertically and

displaces the same volume of surrounding air with density ra, it will be subject to

a force such as to cause a buoyant acceleration, ap, given by:

( )a a

p

a

a g′ −

= −ρ ρ

ρ

(3.20)

Assuming the pressure of both parcels is the same, substituting for densities using

Equation (2.11) this last equation becomes:

( )v v

p

T Ta g

T

− ′= −

(3.21)

where Tv’ and T

v are the virtual temperatures of the parcel and the surrounding air,

respectively. If vertical air movement is viewed in terms of small displacements,



δz, along the vertical gradients of virtual temperature, Tv, or virtual potential

temperature, qv, buoyant acceleration can be written as:

v

p d

v

Tga z

T z

⎛ ⎞ ∂⎛ ⎞= − + Γ⎜ ⎟⎜ ⎟ ⎝ ⎠∂⎝ ⎠

δ

(3.22)

or:

v

p

v

ga z

z

⎛ ⎞ ∂= −⎜ ⎟ ∂⎝ ⎠

θδ

θ

(3.23)

The static stability parameter, s, is defined from buoyant acceleration using this last

equation and has the form:

v

v

gs

z

⎛ ⎞ ∂= −⎜ ⎟ ∂⎝ ⎠

θθ

(3.24)

The parameter s provides a quantitative measure of the static stability of atmos-

phere at any level in terms of the potential temperature gradient at that level. Thus,

the atmosphere is said to be:

(a) Unstable: when s < 0, i.e., when (∂qv�∂z) < 0 or (∂T

v�∂z) < −Γ

d

That is, when virtual temperature falls more quickly than the dry

adiabatic lapse rate – in which condition the atmosphere is said to be

superadiabatic.

(b) Neutral: when s = 0, i.e., when (∂qv�∂z) = 0 or (∂T

v�∂z) = −Γ

d

That is, when virtual temperature falls at the dry adiabatic lapse – in which

condition the atmosphere is said to be adiabatic.

(c) Stable: when s > 0, i.e., when (∂qv�∂z) > 0 or (∂T

v�∂z) > −Γ

d

That is, when virtual temperature falls less quickly than the dry

adiabatic lapse rate – in which condition the atmosphere is said to be

subadiabatic.

In addition, certain subadiabatic (stable) atmospheric conditions are further

distinguished by their gradients of virtual temperature, as follows:

(1) if Tv is constant with height, the atmosphere is said to be isothermal;

(2) if Tv increases with height, there is said to be an inversion,

Figure 3.4 illustrates how the vertical gradient of virtual temperature is used

to characterize atmospheric conditions. In Chapter 20 an alternative measure

of atmospheric stability is defined based on the turbulent fluxes through

the atmosphere.




● Hydrostatic atmosphere: in the absence of external influences the atmosphere

would have hydrostatic vertical gradients of pressure, density, and tempera-

ture. Deviations from the temperature gradient in hydrostatic conditions

control the thermal stability of the atmosphere and it is convenient to

calculate potential temperature or, if water vapor content varies, virtual

potential temperature profiles that are directly related to thermal stability.

● Hydrostatic pressure law: the rate of change of pressure with height is given

by the product of local air density and the acceleration due to gravity.

● Dry adiabatic lapse rate: if moist air moving vertically cools adiabatically

but remains unsaturated, it cools at a rate Γ = g/cp, i.e., 0.00968 K m−1 or

9.68 K km−1.

● Moist adiabatic lapse rate: if moist air moving vertically cools adiabatically

in a saturated atmosphere, it cools at a rate Γm

which is less than Γ.

● Environmental lapse rate: the actual rate at which air temperature falls away

from the ground is determined by the history of heat inputs/outputs to it, but

it may be approximately constant and in the ‘US Standard Atmosphere’ is

6.5 K km−1.

● Vertical pressure and temperature gradients: are linked by the ideal gas

law hence, if Γlocal

is the local environmental lapse rate, the temperatures

and pressures at two levels (T1 and T

2 and P

1 and P

2, respectively) are

related by:

0290 295 300 305 310

Virtual temperature (K)

Subadiabatic

Superadiabatic

Inversion

Isothermal

Adiabatic

200

400Hei

ght (

m) 600

800

1000

Figure 3.4 Classification

of atmospheric stability

conditions based on the

vertical gradient of virtual

temperature.



( ) ( )( )2 1 2 1

a localR g

T T P PΓ

=

(3.25)

● Potential temperature: the temperature that a parcel of air in the atmosphere

would have if it were to be brought adiabatically to a pressure of 100 kPa, q, is:

( )( )100

a pR c

T P=θ

(3.26)

The gradient of potential temperature is that for actual temperature minus Γ.

● Virtual potential temperature: analogous to potential temperature, qv is:

( )( )(1 0.61 ) 100

a pR c

vT q P= +θ

(3.27)

The gradient of virtual potential temperature is that for virtual temperature

minus Γ.

● Atmospheric stability: the gradient of qv determines atmospheric stability,

thus:

Unstable: (∂qv/∂z) < 0 Neutral: (∂q

v/∂z) = 0 Stable: (∂q

v/∂z) > 0 (3.28)

Some stable conditions are further distinguished:

Isothermal: Tv is constant with height: Inversion: T

v increases

with height (3.29)


Introduction

Terrestrial surfaces influence and are influenced by the overlying atmosphere

through the exchange of energy, water, and other atmospheric constituents. In this

chapter we consider the exchange of energy that occurs mainly in the form of sur-

face fluxes of radiant energy, latent heat (when water vapor evaporates from or

condenses onto the land), sensible heat (that warms or cools the air in contact with

the surface), and heat that diffuses into or out of the ground.

The surface flux of any entity is the amount of that entity flowing through and

normal to the surface in unit time, per unit surface area. In the case of energy flux

exchange with terrestrial surfaces, this is the rate of flow of energy per unit area

of land surface. In Système International (SI) units, surface energy fluxes are

expressed in units of Joules per second per square meter, but one Joule per second

is one Watt, consequently the units of surface energy fluxes are Watts per square

meter (W m−2). For terrestrial surfaces, the maximum rates of surface energy

transfer are constrained by the incoming energy from the Sun. At the top of the

atmosphere, the time average energy flux arriving from the Sun when directly

overhead is ∼1366 W m−2: this value is called the Solar Constant. Typically 25–75%

of the Sun’s energy is absorbed as it travels through the atmosphere depending

mainly on cloud cover, so the incoming energy arriving at the surface as solar

radiation could be ∼1000 W m−2 in clear sky conditions at midday near the

equator. However, it is typically much less than this at other times of day and at

higher latitudes, and there is no incoming solar radiation during the night.

The energy incoming as solar radiation is shared between several surface fluxes,

consequently the order of magnitude of such energy fluxes is typically a few 10s

to a few 100s of W m−2.

4 Surface Energy Fluxes




Surface Energy Fluxes 37

Terrestrial surfaces often have heterogeneous land cover and the nature of this

cover can substantially alter the amount of solar radiation captured at the surface

and the way in which this is then shared as surface energy fluxes. In hydrometeorology

and hydroclimatology, heterogeneous terrestrial surfaces are often imagined as

being made up of a patchwork of ideal surfaces, with each ideal patch assumed to be

opaque to radiation and acceptably homogeneous in terms of those surface

characteristics that influence surface energy fluxes. It is further assumed that above

a certain level in the atmospheric boundary layer (say 50–100 m) the atmosphere is

well-mixed so atmospheric variables can be considered independent of the

underlying patch. This height is called the blending height. The land surface

characteristics that influence surface energy exchange and are assumed to be

homogeneous across an ideal patch include the vegetation-dependent reflection

coefficient for solar radiation, thermal emissivity, aerodynamic roughness, and

the ability to capture and store precipitation on plant canopies or in the soil accessible

by plant roots. Often in atmospheric models ideal patches are also assumed to be

horizontal, and even if not horizontal, it is at least assumed that the flow of energy

into or out of the atmosphere is vertical.

Latent and sensible heat fluxes

In SI units the evaporation flux, E, has the units of kilograms of evaporated water

per second per square meter of land surface. Conveniently, because the density of

water is close to 1 kg l−1, 1 kg s−1 m−2 of evaporated water is equivalent to 1 mm s−1

depth of water evaporated. However, surface evaporation rates of the order 1 mm s−1

do not occur in the natural world: evaporation rates are more typically on the

order of a few mm day−1 because the latent heat needed to support evaporation is

constrained by the Earth’s surface energy balance.

It is very common to quantify evaporation rates from terrestrial surfaces not in

terms of mass flow but in terms of the flow of energy leaving the evaporating surface

as latent heat of vaporization in the water vapor. To express the rate of evaporation

as the latent heat flux, λE, in W m−2, the evaporation flux E, in kg s−1 m−2, is multiplied

by λ, the latent heat of vaporization of water in J kg−1. Hydrologists more used to

working in terms of the mass balance of water rather than the surface energy

balance find it useful to remember that an evaporation rate of 3.5 mm d−1 is

equivalent to a daily average latent heat flux of 100 W m−2, and an evaporation rate

of 1 mm d−1 is equivalent to a daily average latent heat flux of 28.6 W m−2 (i.e., about

30 W m−2).

The flow of energy as latent heat away from or toward the land surface is very

important, but it is not the only way the energy can be exchanged with the

atmosphere. A second important way is by the surface directly warming or cool-

ing the air in contact with the surface, with heat then either diffusing outward to

the air above or inward from the air above, respectively. The associated flow of

energy is called the sensible heat flux because it is associated with changes in air



temperature and the temperature of air is something that humans can more

readily ‘sense’ than its latent heat content.

Energy balance of an ideal surface

Figure 4.1 illustrates the energy budget of a sample volume with unit horizontal

area that intersects a horizontal, uniform ‘ideal’ terrestrial surface that comprises

soil with overlying vegetation permeated by air. Outgoing vertical energy fluxes

are defined at some level above the vegetation called the reference level, and at

some depth below the soil surface. Horizontal fluxes are defined parallel to the

wind at the edges of the sample volume. The several energy components involved

in defining the energy balance of this sample volume are as follows.

Net radiation, Rn

The driving input to the surface energy balance is the net flux of radiant energy,

over all wavelengths, at the upper surface of the sample volume. This flux is called

the net radiation, Rn. The net radiation is itself a balance between four compo-

nents: specifically incoming and outgoing radiation in the shortwave band called

solar radiation, and the incoming and outgoing radiation in the wavelength band

determined by temperatures typical of the Earth surface and the lower atmosphere

called longwave radiation, see Fig. 4.2. Because the position of the Sun changes, the

strength of incoming solar radiation varies greatly with time of day leading to a

marked diurnal variation in the net radiation flux. Daytime net radiation is domi-

nated by the solar radiation balance (except at high latitude in winter), while

nighttime net radiation is determined by the longwave radiation balance.

The nature of the surface radiation balance and how net radiation can be quanti-

fied is discussed in greater detail in Chapter 5.

Figure 4.1 Energy balance of

a sample volume selected to

lie through a horizontally

uniform land surface. Rn is

the net radiation, λE, H and

G are the latent heat, sensible

heat, and soil heat fluxes,

respectively; St and P are the

physical and biochemical

energy stored within the

sample volume; and Ain

and

Aout

are horizontally advected

energy entering and leaving

the volume, respectively.G

lE

Rn

H

AoutAin

St

P



Latent heat flux, lE

As described above, the latent heat flux is the flow of energy as latent heat away

from the surface if there is evaporation, or toward the surface if there is condensation.

During the day, evaporation is often the dominant energy flux into the atmosphere

from water surfaces or from moist soil or crops, but sometimes there is a downward

latent heat flux at night with condensation at the surface as dew or frost.

Sensible heat flux, H

Warming of the overlying air by an outgoing sensible heat flux occurs if the

temperature of the surface is higher than that of the overlying air. Conversely,

there is cooling of overlying air and an incoming sensible heat flux when the

surface temperature is less than the air temperature. Because incoming solar

radiation during the day raises the temperature of the surface, the daytime sensible

heat flux is often (but not always) outward. Commonly at night when the surface

cools there is a net outward flux of longwave radiation, the sensible heat flux is

inward to help support this.

Soil heat flux, G

When the soil surface is warmed by solar radiation or indirectly by the warming

air during the day, heat is transferred downward by thermal conduction into the

soil. At night, heat is then conducted back to the surface when the temperature of

the top of the soil cools. This flow of heat is called the soil heat flux. There is a

Figure 4.2 Surface radiation

balance established between

incoming solar radiation to

the surface, S, which arrives

in both the direct solar beam

and after scattering in the

atmosphere (Sd), the reflected

outgoing solar radiation, Sr ,

and the outgoing upward and

incoming downward

longwave radiation, Lu and L

d ,

respectively.Solar radiation Longwave radiation

SrS

Sd

So

Lu Ld



tendency for the diurnal cycle in soil heat flux to average out over the day. Because

of the change in average air temperature between summer and winter, there is also

a seasonal cycle of soil heat flux upon which the diurnal cycle is superimposed.

This longer cycle also tends to average out over the year. It is more general to speak

of a substratum heat flux into the underlying medium rather than soil heat flux

because in some cases, e.g., paddy fields, the underlying medium may be water.

Soil heat flux is discussed in greater detail in Chapter 6.

Physical energy storage, St

Some energy is stored within the sample volume because of the thermal capacity

of its contents. The amount of energy stored will change with time as the tempera-

ture of the air or vegetation changes or if the humidity of the air changes. In prac-

tice, this storage term is often neglected for short crops. However, the change in

physical energy storage can become significant in comparison with the latent and

sensible heat fluxes in the case of tall (forest) vegetation, because there is more

biomass and more air in the deeper sample volume.

The amount of energy stored per unit time per unit area in the interfacial layer

between the level z1 in the soil and the reference level z

2 in the atmosphere is cal-

culated by:

( )ρ ρ λ⎡ ⎤∂ ⎛ ⎞ ∂⎡ ⎤= +⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠∂ ∂⎣ ⎦⎣ ⎦

∑∫ ∫2 2

1 1

[change in temperature] [change in humidity]

z z

t i i i aiz z

c T dz q dzt t

S (4.1)

In Equation (4.1), the first term is the energy associated with temperature changes

and the index i corresponds to contributions that arise from (i) the layer of soil and

roots that is above z1 but below the soil surface; (ii) the vegetation (including

trunks, branches, and leaves) between the soil surface and z2; and (iii) the air

which permeates the vegetation and lies above it up to the level z2. The second

term is the change in latent heat energy associated with changes in the humidity of

the air that permeates the vegetation or lies just above it up to the level z2.

When attempts are made to estimate physical energy storage for forest stands it

is usually considered sufficient to measure changes in the temperature and humid-

ity at a few sample levels in the air in and above the canopy, and to measure changes

in temperature in a sample of trunks and large branches at depths considered

characteristic.

Biochemical energy storage, P

The photosynthesis and respiration of any vegetation present in the volume sam-

ple involves the capture or release of energy. In practice, the amount of energy



involved in plant photochemistry is usually comparatively small (on the order of

2% of the incoming solar energy). For this reason this energy term is often

neglected in hydrometeorology.

Advected energy, Ad

The net advected energy, Ad = A

in – A

out (see Fig. 4.2), is also usually neglected for

homogeneous ideal surfaces, but can become significant in ‘oasis’ situations.

Flux sign convention

The sign convention most often used for the several terms in the energy budget

summarized above is biased toward the value of fluxes being positive in daytime

conditions. Consequently:

(a) all radiation fluxes are defined as being positive when directed toward the

surface (this applies to Rn and its component fluxes S, S

d, S

r, L

u and L

d, see

Fig. 4.2).

(b) all the other vertical energy fluxes are defined as being positive when

directed away from the surface (this applies to λE, H, and G)

(c) the storage terms are defined as being positive when they take in energy

(this applies to St and P)O

(d) if considered, the net advected energy, Ad, is defined as positive when it

brings energy into the sample volume.

Consistent with this sign convention, the overall energy budget for the sample

volume through an ideal surface shown in Fig. 4.1 is written as:

( ) ( )n t tR G A S P E H− + − − = +l (4.2)

The terms on the left of Equation 4.2 are usually grouped together in this way

because in daytime conditions they together define the energy available in the

energy budget that is shared between the latent and sensible heat on the right. For

this reason this set of terms is sometimes referred to as the available energy, A,

which is defined by:

( )n t tA R G A S P= − + − − (4.3)

The magnitudes and, in the case of vertical energy fluxes, the direction of terms

in the surface energy budget are characteristically different in daytime and

nighttime conditions, and they also depend strongly on whether the surface is

bare soil or covered with vegetation and also on how much moisture is available

in the soil.



Figure 4.3 shows representative values of (vertical) energy fluxes for the energy

budgets of dry soil and wet soil in daytime and nighttime conditions. In these

example cases the downward solar radiation, S, was assumed to be 350 W m−2 for

the daytime example, and the net longwave flux (equal to the net radiation flux at

night) was assumed to be –75 W m−2. Evaporation is usually small at night and the

nighttime latent heat flux is arbitrarily set to zero in this figure. It is further

assumed that the net advected energy, Ad, is zero and the biochemical storage, P,

and the physical storage, St, are assumed negligible because there is no vegetation

present.

The greater thermal conductivity of moist soil means that the soil heat flux is

greater for the wet soil case than the dry soil case, i.e., for the wet soil G is greater

when energy is conducted into the soil during the day and out again at night. Dry

soil also reflects more incoming solar radiation than wet soil so the daytime net

radiation flux is somewhat less for the dry soil example. The most obvious differ-

ence between these dry and wet soil examples is in the way the available energy is

partitioned between latent and sensible heat. In the dry soil example when there is

no water available, there is no daytime latent heat flux. However, outgoing latent

Wet soilreflects less

solar so Rn isgreater

Rn A lE H G St

Moist soil - day Moist soil - night

No latent heat(H = A)

Latent heatdominant More soil heat

storage for wetsoil

−100

100

200

300

400

500

0

Rn A lE H G St

−100

100

200

300

400

500

0

Dry soil - day

Ene

rgy

(W m

−2)

Ene

rgy

(W m

−2)

Rn A lE H G St

−100

100

200

300

400

500

0

Dry soil - night

Ene

rgy

(W m

−2)

Rn A lE H G St

−100

100

200

300

400

500

0

Ene

rgy

(W m

−2)

Figure 4.3 Representative daytime and nighttime surface energy budget for dry and wet soil.



heat flux dominates outgoing sensible heat during the day for wet soil. The outgo-

ing net radiation flux at night, which is entirely longwave radiation, is supported

partly by energy returning to the surface as soil heat flux and partly by an inward

flux of sensible heat flux, the latter being the more important contribution in the

dry soil example.

Figure 4.4 shows representative daytime and nighttime surface energy budgets for

a short crop and for forest. In each case there is plenty of water available to the veg-

etation in the soil. Again, downward solar radiation, S, is assumed to be 350 W m−2

for the daytime examples, nighttime net radiation flux is assumed to be −75 W m−2,

and nighttime evaporation is set to zero. For short annual crops (including

grass), both of the storage terms, St and P are negligible, but in the case of forest

physical energy storage can be significant. In Fig. 4.4, both the short and forest

vegetation are assumed to fully cover the ground and as a result the soil heat flux

is small in both cases. But G is particularly low for forest vegetation because the

leaf cover is greater. Forests usually reflect less solar radiation than short annual

crops because the top of the canopy is rougher, consequently the net radiation

input tends to be higher during the day. For well-watered crops, most of the avail-

able energy is used for evaporation during the day and the outgoing latent heat

Forest reflectsless solar so Rn

is greater

Rn A lE H G St

Moist forest - day Moist forest - night

Latent heatdominant

Latent heatLess dominant Little

soil heatstorage

−100

100

200

300

400

500

0

Rn A lE H G St

−100

100

200

300

400

500

0

Moist crop - day Moist crop - night

Ene

rgy

(W m

−2)

Ene

rgy

(W m

−2)

Rn A lE H G St

−100

100

200

300

400

500

0

Ene

rgy

(W m

−2)

Rn A lE H G St

−100

100

200

300

400

500

0E

nerg

y (W

m−2

)

Figure 4.4 Representative daytime and nighttime surface energy budget for short crop and forest when there is plenty of

water available in the soil.



flux dominates over sensible heat. However, forests tend to be more conservative

with respect to transpiring water than annual crops. Perhaps this is because indi-

vidual plants have to survive for many years, which is not necessarily the case for

crops. Consequently, for forests, the proportion of available energy leaving as sen-

sible heat during the day is often greater and often the daytime latent heat and

sensible heat fluxes are roughly equal.

Figure 4.5 shows representative field measurements of the diurnal variation in

the surface energy budget for a well-watered agricultural crop and well-watered

−200

00 04

Rn

lE

G

H

08 12

Time of day (hr)

16 20 24

Ene

rgy

flux

(W m

−2)

200

(a)

400

600

0

−20000 04

Rn

lE

G + S

H

08 12Time of day (hr)

16 20 24

200

Ene

rgy

flux

(W m

−2)

(b)

400

600

0

Figure 4.5 Field

measurements of the daily

variation in the surface

energy budget for (a) barley

and (b) Douglas fir forest.

(Redrawn from Arya, 1988,

after Long et al., 1964 and

McNaughton and Black,

1973, published with

permission.)



forest. For the well-watered crop (a) daytime latent heat flux typically consumes

70–90% of net radiation, as in this example, and the sensible and soil heat fluxes

can be of similar magnitude. Notice that this example data shows evaporation con-

tinuing into the evening with the energy in excess of net radiation primarily being

provided by a downward sensible heat flux. The behavior shown is also broadly

typical for forest vegetation, with the outgoing daytime fluxes of latent and sensi-

ble heat more similar than for the crop. In the figure showing the forest (b), the soil

heat flux and physical energy storage terms are plotted as a sum. In practice the

physical storage term is likely to be the greater of these two.

Evaporative fraction and Bowen ratio

The ratio of the latent heat flux to the sum of the sensible heat flux and latent heat

flux is called the evaporative fraction, EF. If the instantaneous values of sensible and

latent heat are H and λE, respectively, the instantaneous value of the evaporative

fraction is therefore given by:

( )FEE

H E=

+l

l (4.4)

Similarly, if the time average values of sensible and latent heat over a specific time

period are H and Eλ respectively, FE , the evaporative fraction of the average

fluxes over this same period is given by:

( )FEE

H E=

+l

l (4.5)

It is important to recognize that, because the ratio EF may change rapidly with

time, the time-average of instantaneous values of EF does not reliably give the daily

evaporative fraction of the average fluxes, the evaporative fraction for a day must

be calculated from the all-day average fluxes.

The ratio of the sensible heat flux to the latent heat flux is called the Bowen

Ratio, β. If the instantaneous values of sensible and latent heat are H and λE,

respectively, the instantaneous value of the Bowen Ratio is therefore given by:

HE

=bl

(4.6)

Similarly, if the time-average values of sensible and latent heat over a specified

time period are H and Eλ respectively, the time-average Bowen Ratio over this

same period, b , is given by:

HE

=bl

(4.7)



Again, because β is a rapidly changing ratio, the value of b is not reliably given by

taking the time average of the instantaneous values of β. The average value of β

must be calculated from the two time-average fluxes.

Energy budget of open water

Measuring the energy balance components for expanses of water is difficult partly

because of the practical problems involved in mounting and maintaining relevant

equipment, but also more fundamentally because the fluid to fluid interface

problem is poorly specified. But often it is only the surface heat fluxes (especially

the latent heat flux) exchanges that are needed and attempts have been made to

measure these using micrometeorological methods or, in the case of latent heat,

using a water balance approach to determine the net evaporation.

The evaporation flux from open water is very strongly related to wind speed and

to the difference between the (saturated) vapor pressure at the water surface and

the vapor pressure at some level (usually 2 m) above the surface. Given the

comparative simplicity of the physics describing the exchange in the atmosphere

between the water surface and air, and the difficulty involved in making

measurements, evaporation rates are sometimes estimated from semi-empirical

equations that were derived by calibration against prior careful measurements.

Because near surface air is progressively modified as it moves across a water

surface to an extent that depends on the distance traveled, these semi-empirical

equations are also expressed in terms of the surface area, Aw, of the evaporating

water.

If the measured wind speed at 2 m is U2, and the surface temperature of the

evaporating water is Ts, for small water areas such that 0.5 m < A

w0.5 < 5 m (includ-

ing evaporating pans), an estimate of the evaporation in mm d−1 is:

( )0.066

2 3.623 w sat sE A e T e U− ⎡ ⎤= −⎣ ⎦ (4.8)

where e is the vapor pressure at 2 m and esat

(Ts) is the saturated vapor pressure at

temperature Ts. For larger areas where 50 m < A

w0.5< 100 km, such as lakes, an

estimate of the evaporation in mm d−1 is:

( )0.05

2 2.909 w sat sE A e T e U− ⎡ ⎤= −⎣ ⎦ (4.9)


● Ideal surfaces: in many applications, including hydrological and

meteorological models, the land surface is assumed to be made up of a

patchwork, each patch being homogeneous in terms of surface characteristics

that influence surface energy fluxes.



● Latent and sensible heat flux: natural evaporation rate (typically a few mm

per day) is converted to latent heat flux by multiplying by the latent heat of

vaporization which has units of W m−2 (1 mm d−1 is equivalent to 28.6 W m−2).

The flow of heat leaving the surface and directly warming the air (in the same

units) is called the sensible heat flux.

● Energy balance: the energy budget of a volume with unit horizontal area that

intersects a horizontal, uniform ideal terrestrial surface that comprises soil

with overlying vegetation permeated by air is (lE + H) = (Rn − G + A

t − S

t − P)

where:

lE is the latent heat flux (see above);

H is the sensible heat flux (see above);

Rn is the net radiation (the net flux of radiant energy at all wavelengths);

G is the soil heat flux (the flow of heat into or out of the soil by conduction);

At is advected energy (the energy advected horizontally in the air by wind);

St is the storage (the change in energy stored in vegetation, air and soil);

P is the biochemical storage (energy stored by photosynthesis/respiration).

● Sign convention: radiation fluxes are positive toward the surface; other verti-

cal energy fluxes are positive away from the surface; storage terms are positive

when energy is absorbed, and advection positive when energy is brought in.

● Difference values of fluxes: examples given are:

— Dry Soil versus Wet Soil: During the day there is no latent heat flux for dry

soil and net radiation is greater for wet soil because less solar energy is

reflected; wet soil has higher thermal conductivity so soil heat fluxes are

greater.

— Moist Crop versus Moist Forest: Forests reflect less solar radiation so day-

time net radiation is higher, but also transpire less so latent heat is less

dominant than for crops; soil heat fluxes are small under dense

vegetation.

● Evaporative fraction and Bowen ratio: evaporative fraction is the ratio of the

latent heat flux to the sum of latent heat plus sensible heat (called the avail-

able energy); Bowen ratio is the ratio of the sensible heat to the latent heat

flux.

● Open water evaporation: is related to wind speed and the difference between

the vapor pressure of the air and the (saturated) vapor pressure at the water

surface by empirical formulae that change with the evaporating area.

References

Arya, S.P. (1988) Introduction to Micrometeorology. Academic Press, San Diego.

Long, I.F., Monteith, J.L., Penman, H.L. and Szeicz, G. (1964) The plant and its environ-

ment. Meteorologische Rundschau, 17, 97–102.

McNaughton K.G. & Black, T.A. (1973) A study of evapotranspiration from a Douglas fir

forest using the energy balance approach. Water Resources Research. 9 (6), 1579–90,

doi:10.1029/WR009i006p01579.


Introduction

Surface energy transfer as electromagnetic radiation is a very substantial

component of the Earth−atmosphere system; it is important because it is the driving

force for hydroclimatic movement. In comparison with other energy exchanges that

are slower because they involve transfers via the physical movement of molecules

or portions of air, radiation transfer occurs at the speed of light, c = 3 × 108 m s−1

and is effectively instantaneous. As Fig. 5.1 shows, electromagnetic radiation

covers a wide spectrum of wavelengths. Note that in physics it is conventional to

use the symbol l to describe wavelength and this convention is adopted in this

chapter, although elsewhere in this text the symbol l is used to describe the latent

heat of vaporization of water.

When considering the terrestrial radiation balance at the surface, it is helpful to

remember that anything with a temperature above absolute zero emits radiation

with a spectrum and at a rate that reflects the temperature of the emitting entity.

In the case of terrestrial radiation two radiators are important, the Sun, and the

Earth’s surface and atmosphere. The Sun has a temperature of around 6000 K and

emits shortwave or solar radiation, while the Earth’s surface and lower atmosphere

typically has a temperature of ∼290 K and emits thermal or longwave radiation.

Consequently, at the Earth’s surface the majority of the radiation exchange is via

radiation which lies in the wavelength range 0.1–100 μm (1 μm = 10−6 m). Most

is in the visible (0.39–0.77 μm), near infrared (0.77–25 μm) and far infrared

(25–1000 μm) wavebands, but there is also some in the ultraviolet (0.001–0.39 μm)

waveband. The visible portion of the spectrum is the radiation we see, but we are

aware of infrared radiation because it warms us, and ultraviolet radiation because

it tans our skin.

Figure 5.2 shows that in practice most solar energy is in the wavelength range

0.15–4 μm while most of the energy in terrestrial radiation is in the wavelength

5 Terrestrial Radiation




Terrestrial Radiation 49

range 3–100 μm. The wavelength at which there is most energy in these two

spectra differs by a factor of about twenty and there is little overlap between them.

This fact is extremely important because it allows us to consider the two streams

of radiation separately.

Blackbody radiation laws

The radiant flux density of a surface is defined to be the amount of radiant energy

integrated over all wavelengths emitted or received by unit area of surface per unit

time. In common with other energy fluxes, the flux of radiant energy is expressed

in units of W m−2. When describing radiation from natural surfaces it is simplest

first to consider the laws which describe radiation emitted from an idealized

emitting and absorbing surface called a blackbody, then to make corrections to

allow for the relative imperfections of real natural surfaces. A blackbody is an

ideal (standard) surface that emits maximum radiation at all wavelengths in all

00.0

0.2

0.4

0.6

1.0

0.8

1

Wavelength (mm)

Nor

mal

ized

flux

den

sity

(dim

ensi

onle

ss)

10 100Figure 5.2 Normalized

spectra of shortwave and

longwave radiation.

Figure 5.1 Spectrum of radiation with bands defined in terms of wavelength in μm.

10−3 10−2 10 100 100010−1 1

X rays and γ rays Ultraviolet Near infrared Far infrared Radar, TV, Radio

Thermal radiation

Violet 0.39 μm Red 0.77 μm

Visible



directions and absorbs all incident radiation. The area across a small hole in

a container whose internal surface temperature is uniform approximates the

behavior of a black body.

The spectral irradiance, i.e., radiant energy emitted per unit wavelength by a

blackbody, Rl, expressed as function of the surface temperature Ts (in K) is shown

in Fig. 5.3 and is given by Planck’s Law, which has the form:

1

5 2exp 1s

CR

CT

=⎡ ⎤⎛ ⎞ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

l l

λ

(5.1)

where C1 = 3.74 × 10−16 W m−2 and C

2 = 1.44 × 10−2 K. The wavelength, λ

max, at

which most blackbody radiation is emitted is given by Wein’s Law, which has the

form:

max

2897

sT=l

[with l in μm and TS in K] (5.2)

The total flux of radiation, R, emitted by a blackbody per unit area of surface per

unit time is given by the Stefan-Boltzmann Law, which has the form:

4

0

= sR R d T∞

= σ∫ l l (5.3)

in which σ = 5.67 × 10−8 W m−2 K−4 is the Stefan−Boltzmann constant, R is in W m−2

and Ts is in K. Together Equations (5.2) and (5.3) require that as the surface tem-

perature rises, the maximum wavelength at which most radiation is emitted

Figure 5.3 Radiant energy

per unit wavelength emitted

by a blackbody with a surface

temperature of 5777 K.

0.00

500

2000

1500

0.5 1.0

Wavelength (μm)

Spe

ctra

l irr

adia

nce

(Wm

−2 μ

m−1

)

2.0 2.5 3.01.5

1000



becomes less and the amount of radiation emitted increases. Consequently, there

is more radiation emitted per unit area from the Sun than from the Earth’s surface

and it is emitted at shorter wavelengths.

Radiation exchange for ‘gray’ surfaces

The amount of radiation emitted from and absorbed by real surfaces is less than

that from a perfect blackbody and the extent of this reduction is quantified in

terms of parameters that characterize the natural surface. Emitting surfaces that

are imperfect blackbodies are sometimes called gray surfaces. Figure 5.4 illustrates

the emission and absorption of energy by a gray surface.

Definitions of the radiation properties that characterize gray surfaces are as

follows:

● The emissivity, e l: the ratio of radiant energy flux emitted at given wavelength by

a gray surface relative to that emitted by a blackbody at the same temperature.

● The absorptivity, a l: the proportion of radiant energy incident on a surface

at given wavelength that is absorbed.

● The reflectivity, r l: the proportion of radiant energy incident on a surface at

given wavelength that is reflected.

● The transmissivity, t l: the proportion of radiant energy incident on a sur-

face at given wavelength that is transmitted to a subsurface medium.

Because all of the energy incident on a surface must go somewhere, it follows that

(al + rl + tl) = 1, which in turn means al, rl and tl necessarily must all lie in the

range zero to one. Also, there must be equality between the energy absorbed and

the energy emitted otherwise the temperature of a body hanging isolated inside an

evacuated, isothermal container would rise or fall continuously. The Kirchoff ’s

Principle follows as a consequence of this and states that:

=l la e (5.4)

Greysurface

Greysurface

Emitted radiation Incident radiation

Same temperature

Blacksurface

All radiationabsorbed

R x ελ

R x αλ R x τλ

R x ρλR

R R

Blacksurface

Figure 5.4 The difference between blackbody and gray surfaces in terms of the parameters that characterize their relative

behavior.



Integrated radiation parameters for natural surfaces

As mentioned above, most of the solar (shortwave) radiation in the Earth system

is in the wavelength range 0.15–4.0 μm while most of the terrestrial (longwave)

radiation is in the wavelength range 3–100 μm. Because there is so little overlap

between the wavelengths that define solar and terrestrial radiation, it is possible to

define integrated values of reflection coefficient and emissivity for natural (gray)

terrestrial surface, as follows:

● The albedo, a, of a natural surface is the integrated reflectivity of the surface

for radiation incident over the frequency range 0.15–4.0 μm

● The surface emissivity, e, of a natural surface is the integrated emissivity of

the surface over the frequency range 3–100 μm

The surface emissivity of many natural surfaces is in the range 0.90–0.99, or

90–99%. The value of daily average albedo for natural surfaces varies and depends

on the nature of the surface. Table 5.1 gives typical values for some terrestrial

surfaces important in hydrometeorology. Note that the albedo for forests is about

half that typical of bare soil and agricultural crops. In fact, this is more generally

the case for ‘tufty’ vegetation that has a rough canopy with significant depressions

that can trap solar radiation more easily.

More solar radiation is reflected when the angle of incidence of the solar beam

is low in the morning and evening, and the albedo is therefore substantially greater

at these times. Figure 5.5 shows some examples of how albedo varies with solar

altitude. However, because there is usually much more incoming solar radiation in

the middle of the day than in the early morning and late evening, the daily average

value of albedo is biased toward lower midday values.

The values for albedo given in Table 5.1 are typical daily average values but it is

important to recognize that the reflection coefficient for solar radiation can change

significantly from place to place even if the vegetation cover is the same. Table 5.2

shows the observed range of values for albedo and emissivity for different surfaces.

Table 5.1 Typical all day average values of the albedo for selected

land covers.

Surface Type Typical Value of albedo

Open water surfaces ∼0.08 (∼8%)Fresh snow ∼0.8 (∼80%)Dirty, old snow ∼0.4 (∼40%)Bare soil and agricultural crops ∼0.23 (∼23%)Forest ∼0.12 (12%)



Figure 5.5 Typical

variation in albedo as a

function of solar altitude

for selected surfaces.

00

0.2

Clean snow

Dirty snow

Pasture/crop

ForestWater

0.4

0.6

0.8

1.0

30 60

Solar altitude (�)

Alb

edo

(dim

ensi

onle

ss)

90

Table 5.2 Range of reported values for albedo and emissivity for different surfaces.

Surface Type Specification Albedo (a) Emissivity (e)

Water Small zenith angle 0.03–0.1 0.92–0.97Large zenith angle 0.1–0.5 0.92–0.97

Snow Old 0.4–0.7 0.82–0.89Fresh 0.45–0.95 0.9–0.99

Ice Sea 0.3–0.4 0.92–0.97Glacier 0.2–0.4

Bare Sand Dry 0.35–0.45 0.84–0.9Wet 0.2–0.3 0.91–0.95

Bare soil Dry clay 0.2–0.35 0.95Moist clay 0.1–0.20 0.97Wet fallow field 0.05–0.1

Paved Concrete 0.17–0.27 0.71–0.88Black gravel road 0.05–0.10 0.88–0.95

Grass 0.16–0.26 0.9–0.95

Agricultural Crops Wheat, rice, etc. 0.1–0.25 0.9–0.99Orchards 0.15–0.2 0.9–0.95

Forests Deciduous 0.1–0.2 0.97–0.98 Coniferous 0.05–0.15 0.97–0.99



The value of the albedo also strongly depends on the altitude of the Sun and so

varies through the day.

The reflection coefficient for electromagnetic radiation incident on land sur-

faces in the solar waveband changes with the wavelength of the radiation and this

behavior is important both in the context of remote sensing and when building

numerical models of land surface exchanges. Figure 5.6 shows examples of the

variation in reflection coefficient with wavelength for fresh green vegetation, dry

(dead) vegetation, and soil. The reflection coefficient for soil changes with wave-

length, but in a much less dramatic way than it does for green leaves. The distinct

change in the reflection coefficient for plant leaves above and below about

0.72 μm is associated with the absorption of Photosynthetically Active Radiation

(PAR), i.e., that portion of incoming solar radiation that plants use to provide the

energy they need to carry out photosynthesis. In Fig. 5.6, for example, the ratio of

the reflection coefficient at 0.65 and 0.85 μm is about 1.2 for soil and about 1.5 for

dry (dead leaves) but is much greater for actively transpiring green leaves. Some

remote sensing systems measure the relative reflection coefficient at selected wave-

lengths above and below 0.72 μm and use this distinct difference in the ratio of the

measured reflection coefficients to diagnose the extent to which vegetation covers

the soil beneath. The difference in reflection coefficient for leaves above and below

0.72 μm is so distinct that some advanced models of land surface exchanges also

choose to recognize it in their computations and they separately model the absorp-

tion and reflection of visible light in wavebands below and above this wavelength.

Maximum solar radiation at the top of atmosphere

As mentioned earlier, the flux of solar energy at all wavelengths incident on unit

area normal to solar beam at the outer edge of atmosphere when the Earth is at its

mean distance (one astronomical unit) is called the ‘solar constant’, So. In fact the

Figure 5.6 Typical

variation of spectral

reflectance with wavelength

for green vegetation, dry

(dead) vegetation, and soil.

0.5

Green vegetation

Dry vegetation

Soil

0.0

0.2

0.4

0.6

1.0

Wavelength (μm)

Ref

lect

ance

(di

men

sion

less

)

1.5 2.0 2.5



amount of radiation emitted by the Sun fluctuates by a few tenths of a percent during

the 11 year sunspot cycle and is also believed to have changed more gradually at

much longer timescales with implications for the Earth’s climate. However, for

many purposes it is possible to assume the solar constant is constant with the value

1367 W m−2. The spectrum of solar radiation reaching the Earth is close to being

a pure blackbody spectrum, but it is not perfectly so. In part this is because there

are variations in temperature across the surface of the Sun. However, by integrat-

ing the energy in the observed solar spectrum at the top of the atmosphere and

equating this to that from a blackbody, the effective blackbody solar temperature

has been calculated to be 5777 K.

Because Earth is in an elliptical orbit with the Sun as one focus, see Fig. 5.7, the

distance between the Sun and the Earth changes with time of year around its mean

value of 1.496 × 108 km (i.e., one ‘astronomical unit’, AU). Because of this, and

because radiation density follows an inverse square law with distance from the

radiation source, the maximum radiant energy reaching the top of the atmosphere

also changes seasonally. However, the seasonal change in Sun-Earth distance is

only about 3% so it is possible to adequately parameterize the yearly cycle in solar

radiation reaching the Earth using a simple, sinusoidal multiplicative factor which

is called the eccentricity factor, dr, and which is given for each day of the year, D

y, by:

21 0.033cos

365r yd D⎛ ⎞= + ⎜ ⎟⎝ ⎠

p (5.5)

Figure 5.7 The elliptic orbit of the Earth with the Sun as one focus illustrating the change in distance between the Sun and

the Earth and declination of the Earth with season.

Autumnal equinox22/23 October

δ = 0�

Vernal equinox20−21 March

δ = 0�

Winter solstice20−21 December

δ = –23.5�

Summer solstice20−21 Juneδ = +23.5�

1 astronomical unit

1 astronomical unit

1.017 astronomical unit 0.983 astronomical unit



Consequently, the energy reaching the top of the atmosphere as solar radiation

normal to the solar beam is calculated by:

top r oS d S= (5.6)

D is the day of the year (sometimes inaccurately called the Julian day), with D = 1

on 1 January and D = 365 on 31 December.

Maximum solar radiation at the ground

The starting point for calculating how much solar energy reaches the ground is to

calculate the energy that would reach the surface if there were no intervening

atmosphere. The amount of solar radiation which would be received per unit area

over a specified time interval t1 to t

2 on a horizontal surface on Earth were there no

intervening atmosphere is called the insolation, I, which is calculated from:

= ∫1

2

cos( ).

t

o rt

I S d dtq (5.7)

where q is the solar zenith angle, i.e., the apparent angle of the Sun relative to the

normal angle to the surface at the specific location.

The need is, therefore, to calculate the solar zenith angle. Doing this is complex,

because the solar zenith angle depends not only on the latitude of the site for

which the calculation is made (because the Sun is on average closer to overhead

nearer the equator), and the time of day (because the Earth rotates each day), but

also because it depends on the solar declination, d. The solar declination is the

angle between the rays of the Sun and the plane of the Earth’s equator. The axis of

rotation of the Earth is at an angle of ∼23.5° with respect to the plane in which the

Earth moves around the Sun, see Fig. 5.7. Consequently, the solar declination

changes with time of year. It is zero at the vernal and autumnal equinox, and

around 23.5° and -23.5° at the summer and winter solstice, respectively. The value

of δ can be calculated (in radians) for each day of the year, Dy, from:

20.4093sin 1.405

365yD

⎛ ⎞= −⎜ ⎟⎝ ⎠pd (5.8)

For a site at latitude f (positive in the northern hemisphere; negative in the southern

hemisphere), the cosine of the solar zenith angle required in Equation (5.7) to

calculate the insolation is given by:

cos sin sin cos cos cos= +q f d f d ω (5.9)



where w is the hour angle, i.e., the angle representing time of day, which is given by:

( )12 (in radians)

12

h−= πw (5.10)

where h is time of day in hours in local time. Thus, if there were no atmosphere, on a

particular day the daily total incoming solar energy received at the ground integrated

over daylight hours between t1 and t

2, i.e., the daily insolation I

o, is given by:

= +∫2

1

(sin sin cos cos cos )o

t

o rt

I S d dtf d f d w (5.11)

The values of t1 and t

2 are best expressed in terms of the equivalent hour angle that

defines both the beginning and end of the day. This angle is called the sunset hour

angle, ws, which can be calculated from;

arccos ( tan tan ) [radians]s = −w f d (5.12)

with the day length, N, in hours then following immediately from:

= 24 [hours]sN wp (5.13)

The total solar energy which would be received per unit area between sunrise and

sunset on a horizontal surface at latitude f if there were no intervening atmos-

phere is then obtained by integrating Equation (5.11) between -ws and +w

s as:

2 1

037.7 ( sin sin cos cos sin ) [ MJ m d ]d

r s sS d − −= +w wf d f d (5.14)

where dr, d, and w

s are given by Equations (5.5), (5.8), and (5.12), respectively. When

estimating evaporation rates, it is sometimes convenient to write Equation (5.14)

in terms of an equivalent depth of evaporated water, thus:

1

015.39 ( sin sin cos cos sin ) [ mm d ]d

r s sS d −= +φ δ φ δ ωw (5.15)

It is important when estimating daily average evaporation rates (see Chapter 23)

that this absolute upper limit on evaporation rate (from which estimates of actual

evaporation rates can be made) can always be calculated solely on the basis of

knowledge of the latitude of the site and the day of the year.



Atmospheric attenuation of solar radiation

In the previous section we considered the solar radiation that would be incident

on the ground if there were no atmosphere. However, the amount, spectrum and

directionality of solar radiation at the ground are all significantly altered by

interactions through the atmosphere, which occur in two main ways, by scattering

and by absorption. Some of the radiation is scattered by the gas molecules that

make up the air, and this is called Rayleigh scattering. Solar radiation is scattered

preferentially at lower wavelengths by gas molecules and it is the resulting scattered

blue radiation that we see and associate with clear sky above us. Additional

scattering of solar radiation occurs due to the atmospheric aerosols such as dust

and smoke in the air.

All scattering processes alter the direction of solar radiation and some radiant

energy is scattered back and does not reach the surface. In addition, some radiant

energy is absorbed from the solar beam as it passes through the atmosphere, giving

rise to atmospheric warming. Absorption occurs preferentially in wavelength

bands that correspond to excited states in important minority gases in the air, nota-

bly ozone in the stratosphere, which mainly absorbs ultraviolet radiation, but also

water vapor, carbon dioxide and other so-called radiatively active gases. Figure 5.8

shows how the incoming spectrum of solar radiation is progressively eroded

through the atmosphere by these several scattering and absorbing processes in

clear sky conditions.

If the sky is not clear, substantial energy in the solar beam is lost through the

atmosphere because of the presence of clouds. The ice particles and water droplets

in clouds interfere strongly. They scatter solar radiation mainly backward into

Figure 5.8 The progressive

loss of energy in the solar

beam by scattering and

absorption as it passes

through the atmosphere in

typical clear sky conditions

showing the representative

spectra (a) of extraterrestrial

radiation, (b) after absorption

by ozone, (c) after Rayleigh

scattering, (d) after aerosol

interactions, and (e) after

absorption by H2O, CO

2, etc.

0.5

(e)

(d)

(c)

(a)(b)

Rel

ativ

e irr

adie

nce

(W m

−2 p

er μ

m)

1.0

Wavelength (μm)

1.5



space and they also absorb solar energy. Thick clouds will reflect as much as 70%

of solar radiation and absorb a further 20%, while transmitting just 10%. As a

result, the proportion of solar radiation absorbed, which is typically just 25% in

cloudless conditions, is about 75% for overcast skies.

Actual solar radiation at the ground

The complex scattering and absorbing properties of the atmosphere can and often

are represented explicitly in meteorological models. Commonly in hydrology,

however, atmospheric loss of solar radiation is parameterized more simply either

in terms of an estimate of the fractional cloud cover, c, on a particular day, or the

number of hours with bright sunshine, n, in a day lasting N hours. In terms of

fractional cloud cover, the actual daily total solar radiation, Sd, is given by:

0 [ (1 ) ]d d

s sS a c b S= + − (5.16)

And in terms of bright sunshine hours by:

0 d d

s snS a b SN

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦ (5.17)

Ideally, empirical values of as and b

s would be derived locally by comparing

estimates from Equation (5.16) with measurements of Sd on overcast days to give

as and on days with continuous bright sunshine to give (a

s+ b

s). Typical values

derived in this way are as = 0.25 and b

s = 0.5, corresponding to a 25% and 75% loss

of energy in clear sky and overcast conditions, respectively. These values of as and

bs are often assumed in the absence of any locally calibrated values.

As already discussed, once the solar radiation reaches the Earth’s surface, a

proportion is reflected, depending on the albedo of the surface. Consequently, the

net daily solar radiation, Sn

d, is less than Sd and is given by:

(1 )d dnS a S= − (5.18)

where a is the daily average value of albedo described earlier.

Longwave radiation

The terrestrial surface emits thermal radiation following the Stefan−Boltzman

Law, with the surface temperature Ts in Equation (5.3) being T

surface, the effective

temperature of the land surface, and an appropriate value for the surface emissiv-

ity, esurface

, see Table 5.2. Thus, there is an upward flux of radiant energy in the

longwave waveband, Lu, which is given by:

4 u surface surfaceL T= - e s (5.19)



However, much of this radiation is re−absorbed in selected wavebands by gases in

the atmosphere, including water vapor, carbon dioxide, the oxides of nitrogen,

methane and ozone, see Fig. 5.9. This is the basis of the so−called greenhouse effect.

However, according to the Kirchoff ’s principle, gases which absorb energy at a

particular wavelength also re−emit energy at the same wavelength and some of

this radiation is emitted downward toward the surface, so there is an associated

downward flux of radiant energy in the longwave wave band, Ld , which is given by:

4 d atmos atmosL T= ε s (5.20)

At any instant, the net exchange of longwave radiation at the surface, Ln, is the

difference between Lu and L

d, i.e.:

n u dL L L= − (5.21)

Figure 5.10 illustrates the spectrum of the upward and downward longwave

streams for a hypothetical case with a surface temperature of 288 K and a cloudless

Figure 5.9 Absorption spectra of radiatively active gases in the lower atmosphere as a function of wavelength.

0.10%

0%

0%

100%

100%

0%100%

0%

100%

N2O

H2O

Totalatmosphere

O2 and O3

CO2

100%

0.2 0.3 0.4 0.6 0.8 1.51 2

Wavelength (μm)

3 4 5 6 8 10 20 30



atmosphere at 263 K. The net longwave flux is upward (outward) partly because

the surface temperature is higher than the effective average temperature of the

atmosphere above, and partly because the effective emissivity of the overlying

atmosphere is less than that of the surface.

In practice, there is some linkage between the temperature of the surface and air

temperature, and a great deal of the downward longwave radiation originates in

the lower atmosphere comparatively close to the surface. For this reason, both the

upward and downward longwave radiation fluxes are linked to near-surface (2 m)

air temperature and in clear sky conditions there is therefore often a reasonably

strong approximately linear relationship between air temperature and both upward

and downward longwave radiation.

When clouds are present, more of the outgoing longwave radiation is absorbed

and returned to the surface and this is the basis of the simple empirical formula

much used in hydrological applications for estimating the daily average net

longwave radiation, Ln

d, which has the form:

4 dn airL f T= − e¢s (5.22)

where Tair

is the daily average air temperature. Because water vapor makes an

important contribution to the absorption of outgoing longwave radiation and

emission of downward radiation, the effective emissivity, e′, in Equation (5.22)

depends on the humidity content of the air and is estimated by:

0.34 0.14 de= −e¢ (5.23)

where ed is the daily average vapor pressure in kPa. The factor f is an empirical

cloud factor which is calculated from (Sd/S

dclear), i.e., the ratio of the estimated surface

solar radiation given by Equation (5.16) or (5.17) in ambient conditions to the

Figure 5.10 The net

exchange of longwave

radiation between a surface

at 288 K and a cloudless

atmosphere at 263 K.

(Redrawn from Monteith and

Unsworth, 1990, published

with permission.)0

5 10 15 20

288 K

263 K

25Wavelength (μm)

Irra

dien

ce (

W m

−2 p

er μ

m)

10

20

CO2 CO2O3H2O H2O H2O



value estimated by the same equation for clear sky conditions. Two empirical

formulae are used to calculate f, one applicable in humid conditions and one in

arid conditions. In humid conditions:

( )(1 ) i.e., ; or

ds ss s

ds s s sclear

a n N ba c bSfa b a bS

+⎛ ⎞ ⎡ ⎤+ −= ⎢ ⎥⎜ ⎟ + +⎝ ⎠ ⎣ ⎦

(5.24)

but in arid conditions:

(5.25)

Net radiation at the surface

As described in Chapter 2, the all−day average net radiation at the surface is the

sum of the downward and reflected solar radiation and the net longwave radia-

tion as illustrated in Figure 4.2, and the daily average net radiation flux, Rn

d, is

given by:

d d dn n nR S L= + (5.26)

Equation (5.26) is written in terms of daily total values but it is of course possible

to write equations describing the instantaneous radiation balance, as follows:

( ) ( )n r d uR S S L L= + + + (5.27)

or:

(1 )n nR S a L= − + (5.28)

The value of Rn varies greatly through the day. Net longwave radiation is negative

and usually fairly constant through the day, with the net solar radiation provid-

ing a positive input during the daylight hours. The typical diurnal cycle of net

radiation therefore has a temperature dependent negative offset, upon which is

superimposed a positive diurnal input of solar radiation whose magnitude

depends on the latitude of the site, the day of the year, fractional cloud cover on

that day, and, not least, time of day. The resulting diurnal pattern of net radiation

may therefore have a strong seasonal dependence at high latitude as illustrated

in the case of the radiation balance for Bergen, Norway at 60°N, which is shown

in Fig. 5.11.

+⎛ ⎞ ⎡ ⎤+ −= − − −⎢ ⎥⎜ ⎟ + +⎝ ⎠ ⎣ ⎦

( )(1 )1.35 0.35 i.e., 1.35 0.35; or,1.35 0.35

ds ss s

ds s s sclear

a n N ba c bSfa b a bS



Height dependence of net radiation

The main focus of interest in this chapter is on radiation exchange at terrestrial

surfaces. However, when considering the energy balance of the atmospheric

boundary layer, it is important to recognize that net radiation can change with

height above the ground. This is inevitable because there is absorption of solar

radiation as it passes through the air, and there is absorption and emission of long-

wave radiation at all levels in the atmosphere. During the day, solar radiation

Figure 5.11 Diurnal variation in net solar radiation, upward and downward radiation, and net radiation at Bergen Norway

on (a) 13 April 1968 and (b) 11 January 1968. (Redrawn from Monteith and Unsworth, 1990; after Gates, 1980, published

with permission.)

00

–100

Rad

iatio

n flu

x (W

m−2

)

200

400

600

0

03 06 09 12 15 18 21 24

Netradiation

Netradiation

Upward longwaveradiation

Upward longwaveradiation

(a)

Solarradiation

Solarradiation

Downward longwaveradiation

Downward longwaveradiation

Local time (hr)

00

–100

Rad

iatio

n flu

x (W

m−2

)

200

400

0

03 06 09 12 15 18 21 24

(b)

Local time (hr)



normally dominates the net radiation flux and in the absence of cloud, fog, or

heavy aerosol contamination, the loss of solar radiation through the clear air

boundary layer is comparatively small. The profile of longwave radiation through

the boundary layer can be influenced by variations in the concentration of radia-

tively active atmospheric constituents and the heat content of the air. During the

night, longwave radiation dominates and the nighttime potential temperature

profile may induce vertical changes in the net radiation flux. Regardless of its

cause, energy conservation means vertical divergence of the net radiation flux will

be associated with local heating or cooling of the air, the amount being directly

related to the rate of change of net radiation with height and given by:

v np

Rc

t z∂ ∂

=∂ ∂q

r (5.29)


● Separation of wavebands: most solar radiation is in the wavelength range

0.15–4 μm and most longwave radiation in the Earth system is in the wave-

length range 3–100 μm, this allows separate consideration of the two streams.

● Blackbody radiation laws: describe radiation for an idealized emitting and

absorbing blackbody surface, and include Planck’s Law describing the

spectrum; Wein’s Law, giving the wavelength of peak emission; and Stefan-

Boltzmann Law, giving the total flux of radiation.

● Gray surfaces: are imperfect blackbodies for which radiation exchange at

wavelength l is calculated from blackbody radiation using multiplicative

factors, i.e., emissivity (el) for emitted radiant energy; absorptivity (al) for

absorbed radiant energy, reflectivity (rl), for emitted radiant energy; and

transmissivity (tl) for transmitted radiant energy. Kirchov’s principle requires

that el = al.

● Integrated parameters for natural surfaces: because there is little overlap

between the waveband for solar and longwave radiation, radiation exchange

for natural surfaces is often described by wavelength integrated surface

properties, i.e., albedo (a), the integrated reflectivity for solar radiation, and

emissivity (e), the integrated emissivity for longwave radiation.

● Typical values of parameters: daily average values albedo are typically ∼8%

for water, ∼80% for fresh and ∼40% for dirty snow, ∼23% for bare soil and

agricultural crops, and ∼12% for forests; emissivity is typically ∼95 ± 5%.

● Top of atmosphere solar radiation: maximum value changes by about ±1.6%

with day of year around the ‘solar constant’ So = 1367 W m−2.

● Maximum solar radiation at the ground: if there were no atmosphere the

daily total solar radiation reaching the ground could be calculated using

an (albeit complex) formula from the latitude of the site (f), solar declination



(d – a function of day of year), and the sunset hour angle, (ws – a function of

latitude and day of year).

● Atmospheric attenuation: typically 25% of solar radiation is absorbed for

clear skies and 75% for overcast skies (attenuation is by scattering by gas

molecules and dust, and absorption by radiatively active gases and cloud):

solar radiation at the ground can be estimated either from fractional cloud

cover or from the fraction of daytime hours with bright sunshine.

● Net longwave radiation: longwave exchange with the surface can be

estimated using a version of the Stefan-Boltzmann Law that includes an

effective emissivity (dependent on humidity) and a cloud cover correction

factor (estimated from the attenuation of solar radiation) that is different for

humid and arid conditions.

● Net radiation: is obtained by adding the net solar radiation (allowing for

albedo) to the net longwave radiation; at the daily time scale it has a tempera-

ture dependent negative (longwave) offset upon which is superimposed a

positive solar radiation input whose magnitude depends on latitude, day of

the year, fractional cloud cover, and time of day.

References

Gates, D.M. (1980) Biophysical Ecology. Springer-Verlag, New York.

Monteith, J.L. & Unsworth, M.H. (1990) Principles of Environmental Physics. Edward

Arnold, London, UK.


Introduction

The flow of energy into temporary storage in the soil that underlies the interface

between terrestrial surfaces and the atmosphere can be a substantial component

(∼25%) of the midday surface energy budget for bare soil surfaces. It can also be a

significant component of the daytime surface energy budget for vegetated surfaces,

depending on the extent to which the plants shade the soil. The surface temperature

of the soil is determined by the need to continuously maintain a balance between

the energy fluxes of radiation, latent heat, and sensible heat into the overlying air

and the flow of energy by thermal conduction into the soil. Consequently, the

surface temperature of soil is a dependent variable. However, energy flow within

the soil and associated changes in below ground soil temperature are determined

by changes in the surface temperature of the soil. In this sense, soil surface

temperature is the forcing variable that determines thermal behavior in the soil.

The present chapter considers soil temperatures and heat flow in soil from the

perspective that soil surface temperature is a forcing variable that varies with time.

Soil surface temperature

Measuring area-average soil surface temperature is difficult because it varies

greatly from place to place and can change quickly with time. The surface

temperature of typically rough bare soil surfaces can vary over short distances

during daylight hours depending on the relative orientation between the local soil

surface and the solar beam. Spatial variability in plant shading complicates this

relationship for soil covered by vegetation. Attempts have been made to measure

soil surface temperature using conventional mercury bulb thermometers

6 Soil Temperature and Heat Flux




Soil Temperature and Heat Flux 67

positioned to be in close contact with the soil. But the presence of a thermometer

alters the surface energy balance and therefore the surface temperature that is

being measured. Fine thermocouple thermometers positioned on the soil surface

disturb the surface energy balance less, but the issue of how to obtain an adequate

sample to determine the area-average temperature remains. In principle, the

average temperature can be measured over areas of a few square meters of soil by

positioning a radiometer above it and deducing the surface temperature from the

measured thermal radiation emitted. However, soil surfaces are not blackbody

emitters so soil emissivity has to be assumed and poor knowledge of this can

systematically bias such a radiometric measurement by several degrees.

The soil surface temperature is determined by the surface energy balance. As a

result, for dry exposed soils in fairly calm conditions and with clear skies, the ampli-

tude of the daily cycle in soil surface temperature can be very large, perhaps greater

than 30°C. This is noticeably larger, by perhaps a factor of two, than the daily cycle

in air temperature measured at 2 m. Incoming radiant energy plays a major role in

the surface energy balance that determines the surface temperature of bare soil and

the timing of the daily cycle in soil temperature, therefore, tends to follow that in

net radiation, albeit with some lag (typically less than an hour). For a wet soil surface,

much of the energy outgoing to the atmosphere is as latent heat and the sensible heat

flux is less. There is, therefore, less need for a large difference in temperature between

the soil surface and the overlying air to support the sensible heat flux. Consequently,

the amplitude of the daily cycle in soil surface temperature is much smaller. But solar

radiation is still the dominant term in available energy for wet bare soil so the timing

of the daily cycle in soil temperature still tends to follow that in net radiation.

When there is vegetation overlying the soil, the magnitude and timing of the

daily cycle in soil surface temperature differs from that for bare soil. Much of

the incoming radiant energy is captured by the vegetation canopy and returned to

the atmosphere before it reaches the soil. The presence of vegetation also enhances

turbulent mixing in the air near the ground thus reducing the difference in

temperature between air adjacent to the soil surface and that in the atmosphere

above. Since there is much less solar radiation reaching the soil surface, the solar

cycle has less influence on surface energy balance and soil surface temperature, the

latter being thus more similar to near-surface air temperature. Consequently, the

daily cycle in vegetation-covered soil surface temperature has reduced amplitude

relative to that for bare soil in the same meteorological conditions, and the timing

of the cycle tends to follow that in air temperature and so typically lags the solar

radiation cycle by several hours.

Subsurface soil temperatures

The temperature of soil below the surface is somewhat easier to measure than

soil surface temperature providing the thermometers are carefully inserted with

minimum disturbance to the soil structure. Commonly, small thermometers are



used which are inserted horizontally into the vertical edge of a cautiously dug soil

pit, so that they sample soil at least 0.1 m away from the edge of the pit. If the soil

is uniform, heat flow in the soil tends to average out some of the spatial heteroge-

neity in temperature present at the surface, so horizontal variations in measured

subsurface temperatures are less extreme.

Subsurface soil temperatures are determined by heat flow into and out of the

soil in response to changing surface temperature. Consequently, the magnitude

and timing of the daily cycle in subsurface temperatures are necessarily different

for dry soil and wet soils, and for vegetation-covered soils. Not surprisingly, the

magnitude of the cycle in subsurface temperatures is greater for bare soil than for

soil covered by vegetation, and is later relative to the cycle in the solar radiation,

depending on depth. Figure 6.1 shows profiles of soil temperature measured at

selected hours of the day in bare soil and in the soil beneath a nearby crop of

potatoes. On this day, the range of variation in the surface temperature measured

beneath the potato crop is around 15°C and less than the 22°C range for the bare

soil. The range of subsurface temperatures measured at the same depth in the two

profiles reflects this difference in the surface temperature cycle. In both cases, the

magnitude of diurnal variation progressively decreases with depth. This last

feature is evident in Fig. 6.2, which also more clearly shows how the phase-lag of

the cycle in subsurface soil temperature increases with greater depth.

Thermal properties of soil

The thermal properties of soil determine how the magnitude and phase of the soil

heat flux and subsurface soil temperatures respond to changes in the soil surface

temperature. All the relevant soil properties are strongly dependent on the

moisture content of the soil. This is because air-filled pores present in dry soil are

progressively filled with water as the moisture content of soil increases, and the

40

50

30

20

10

0

Dep

th (

cm)

10 15 20 25 30 35

04 00 08 20 12 16

Local time (hr)

Bare soil

Soil temperature (�C)

(a)

40

50

10 15 20 25 30 35

30

20

10

004 00 08 20 12 16

Local time (hr)

Potatoes

Dep

th (

cm)

Soil temperature (�C)

(b)

Figure 6.1 Profiles of soil

temperature measured at

selected hours during the day

beneath (a) a bare soil surface

and (b) a crop of potatoes.

(Redrawn from Monteith and

Unsworth, 1990, published

with permission.)



thermal properties of air are profoundly different to those of the water which

replaces it. Values of the soil properties discussed below are given in Table 6.1,

together with those for air and water for purposes of comparison.

Density of soil, rs

The local density of soil in kg m−3 is the mass of unit volume of soil. As just

mentioned, the density of soil changes greatly with the moisture content of the soil

24 06 12 18 24 06 12

Local time (hr)

18 24 06 12 18 24

20

30

40

50

Tem

pera

ture

(�C

)

Figure 6.2 The time

variation in subsurface soil

temperature beneath a sandy

loam soil with bare surface

measured for three days at

depths of 2.5 cm (full line), 15

cm (broken line), and 30 cm

(dotted line). (Redrawn from

Monteith and Unsworth,

1990, after Deacon, 1969,


Table 6.1 Mass density and thermal properties of soils, air, and water.

Material ConditionMass density rs (kg m−3 × 103)

Specific heat cs (J kg−1 K−1 × 103)

Heat capacity Cs (J m−3 K−1 × 106)

Thermal conductivity ks (W m−1 K−1)

Thermal diffusivity as (m2 s−1 × 10−6)

Sandy Soil (40% porosity)

DrySaturated

1.602.00

0.801.48

1.282.98

0.302.20

0.240.74

Clay Soil (40% porosity)

DrySaturated

1.602.00

0.891.55

1.423.10

0.251.58

0.180.51

Clay Soil (80% porosity)

DrySaturated

0.301.10

1.923.65

0.584.02

0.060.50

0.100.12

Air 20°C, still 0.0012 1.00 0.0012 0.026 21.5Water 20°C, still 1.00 4.19 4.19 0.58 0.14



as the air in the pores of the soil is replaced by water which is much denser,

see Table 6.1.

Specific heat of soil, cs

The specific heat of soil in J kg−1 K−1 (1 J kg−1 K−1 is the same as 1 J kg−1 °C−1) is the

amount of heat absorbed or released in raising or lowering the unit mass of soil by

1 K. Because dry soil is porous, its specific heat is only about 25–40% of the specific

heat of water, but the specific heat of soil typically almost doubles when the soil

becomes saturated, see Table 6.1.

Heat capacity per unit volume, Cs

The heat capacity per unit volume in J m−3 K−1 (1 J m−3 K−1 is the same as 1 J m−3 °C−1)

is the amount of heat absorbed or released in raising or lowering unit volume

of soil by 1 K. It is the product of the density of the soil with its specific heat,

thus:

s s sC c= r

(6.1)

Because both rs and c

s separately increase with soil moisture content, there is an

even greater proportional increase in the value of Cs as the moisture content of the

soil increases, see Table 6.1.

Thermal conductivity, ks

Thermal conduction of heat in soil is described by a simple diffusion equation

with the form:

soil

z s

TG k

z= −

∂∂

(6.2)

where Gz (in W m−2) is the local vertical soil heat flux at depth z below the surface

of the soil (z is measured downward), ∂Tsoil

/∂z is the vertical temperature gradient

at depth z in K m−1, and ks is the local thermal conductivity of the soil in W m−1 K−1

(1 W m−1 K−1 is the same as 1 W m−1 °C−1). The negative sign is required on the

right hand side of this equation because soil heat flux is defined to be positive

when directed away from the surface (see Chapter 4), and this occurs when soil

temperature decreases with depth below ground. As is the case for other soil

properties, there is a marked change in thermal conductivity when soil moisture

content increases, see Table 6.1.



Thermal diffusivity, as

When describing soil heat flow, it is useful to define a renormalized form of the

thermal conductivity called the thermal diffusivity, as, which is defined by the

equation:

ss

s

kC

=a

(6.3)

The soil moisture dependency of as, is less dramatic than for k

s, see Table 6.1.

When the diffusion equation describing soil heat flow (Equation (6.2)) is

rewritten in terms of thermal diffusivity rather than thermal conductivity it

becomes:

soilz s s

TG Cz

∂= −

∂a

(6.4)

Formal description of soil heat flow

Figure 6.3 illustrates the energy budget for a thin horizontal element of soil of

thickness dz and cross sectional area A located at a depth z beneath the soil surface.

The soil heat flux into the element from above is Gz and that out from below G

z+dz.

Consequently, over a period of time dt, the element receives a net input of soil heat

flux [A.(Gz − G

z+dz).dt]. Over this same period of time, the temperature of the soil

element rises by dTsoil

. This takes an amount of heat equal to [Cs.A.dz.dT

soil], and

energy conservation requires that:

( )s soil z z zC A z T A G G t+= − dd d d

(6.5)

Hence:

( )soils z z zTC z G Gt += − d

dd

d (6.6)

Expanding the right hand side of Equation (6.6) using Taylor’s theorem in the limit

of small dz gives:

.soil zs z zT GC z G G zt z

⎛ ⎞= − +⎜ ⎟⎝ ⎠d

d dd

∂∂

(6.7)



Hence:

1soil z

s

T Gt C z

= −∂ ∂

∂ ∂

(6.8)

Substituting G from Equation (6.4) into this last equation gives:

1soil soils s

s

T TCt C z z

⎡ ⎤= ⎢ ⎥⎣ ⎦a

∂ ∂∂∂ ∂ ∂

(6.9)

In the particular case of heat flow in homogeneous soil for which both Cs and a

s

are constant with depth below ground, Equation (6.9) simplifies to:

2

2

soil soils

T Tt z

= a∂ ∂

∂ ∂ (6.10)

Thermal waves in homogeneous soil

Equation (6.9) can be solved numerically to provide a general description of the

evolution of soil temperature and therefore (from Equation (6.4)) soil heat flux in

response to a prescribed time series of soil surface temperature when the depth

dependency of Cs and a

s is defined. However, much can be learned about the

mechanics of soil heat flow by investigating the analytic solution of Equation (6.10)

which applies for homogeneous soil. It is also instructive to consider the case of

a simple sinusoidal variation in soil surface temperature, ,0tsoilT , described by the

expression:

,0 0( )

sin 2tsoil m a

t tT T TP−⎡ ⎤= + ⎢ ⎥⎣ ⎦

p

(6.11)

where Tm

is the mean temperature of the soil surface, Ta is the amplitude of the

sinusoidal variation in soil surface temperature, t is time in seconds, and P and t0

Soil surface

Area = A

Temperature = Tsoil

Air

Soil

z+δz z Gz

Gz+dz

Figure 6.3 Energy budget

for a thin horizontal element

of soil of thickness бz, cross

sectional area A, and

temperature Tsoil

located at a

depth z beneath the soil

surface, with soil heat flux Gz

entering from above and Gz+бz

leaving from below.



both in seconds are respectively the period of the sinusoidal variation and a time

slip introduced to adjust its phase such that ,0tsoil mT T= when t = t

0. It can be shown

by substitution into (6.10) that the expression:

, 0( )

exp sin 2t zsoil m a

t tz zT T TD P D

−− ⎡ ⎤⎡ ⎤= + −⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦p

(6.12)

describes the behavior of soil temperature as a function of time t and depth z when

Equation (6.11) is the upper boundary condition at the soil surface, with:

0.5

sPD ⎛ ⎞= ⎜ ⎟⎝ ⎠ap

(6.13)

D has units of distance and is called the damping depth. Figure 6.4 shows the daily

variation in soil temperature as a function of soil depth calculated from Equation

(6.12) in response to a sinusoidal cycle of temperature of amplitude 15°C around a

mean temperature of 40°C with a phase delay of 6 hours when the damping depth

is 0.1 m. The variation with depth of the amplitude and phase of calculated soil

temperature can be compared with that measured for a bare sandy loam surface

shown in Fig. 6.2.

The amplitude of the soil temperature wave and its phase relative to the surface

temperature wave changes with depth and are controlled by the thermal diffusivity

of the soil and period of the surface temperature cycle via the value of D. Because

damping depth is related to the square root of the period of the surface wave, the

depth of penetration is much greater for the seasonal cycle in surface temperature

than for the daily cycle in temperature. In the case of dry sand (as = 0.24 × 10−6 m2 s−1),

D is about 0.08 m for the daily cycle but about 1.6 m for the yearly cycle. In the case

of wet clay (as = 0.51 × 10−6 m2 s−1), the value of D is about 0.12 m for the daily cycle

but about 2.24 m for the yearly cycle.

18 24Time (hr)

30 36 42 48126020

30

40

50

60

Soi

l tem

pera

ture

(�C

)

Figure 6.4 Calculated soil

temperature at 2 cm (solid

line), 15 cm (dotted line), and

30 cm (broken line) for

damping depth of 0.1 m.



The soil heat flux can be calculated at any depth in the soil using Equation (6.2)

or (6.4). However, it is the soil heat flux into the soil surface that is most often

required for use in calculating the energy budget of a sample volume lying between

ground level and a reference level in the atmosphere above. By differentiating

Equation (6.12) and substituting z = 0, it can be shown that for a homogeneous soil

and assuming the simple sinusoidal variation in surface temperature in Equation

(6.11), the surface soil heat flux is given by:

00

2 ( ) 2sin 2

8a s

zT k t tGD P=

−⎡ ⎤= +⎢ ⎥⎣ ⎦pp

(6.14)

Comparing Equation (6.14) with Equation (6.11) illustrates a general feature of

the relationship between soil heat flux and soil surface temperature. Specifically,

the sinusoidal wave in soil heat flux is advanced by one eighth of a cycle with

respect to the wave in surface temperature. This means that the peak soil heat

flow is approximately 3 hours earlier than the peak soil surface temperature

for the daily cycle, and the peak in soil heat flow is 1.5 months earlier than the

peak soil surface temperature for the yearly cycle. Physically this is because

there is most conduction of heat into the soil when the soil surface temperature

is rising rapidly in the morning for the daily cycle and in spring for the yearly

cycle, and most conduction of heat out of the soil when the soil surface

temperature is falling quickly in the evening for the daily cycle and in autumn

for the yearly cycle.

Equation (6.14) also shows that the magnitude of the wave in surface soil heat

flux is inversely proportional to the damping depth and therefore inversely

proportional to the square root of the period of the surface temperature wave,

see Equation (6.13). This means the amplitude of the soil heat flux wave associated

with the yearly cycle is (365)0.5 times less (i.e., about 19 times less) than the

amplitude of the daily cycle. The yearly cycle in soil heat flux is therefore about 20

times less than the daily wave but penetrates about 20 times deeper.

In fact the value of damping depth determines many interesting features of soil

heat flow as follows.

● The amplitude of temperature wave falls to e−1 (~0.37) of its surface value at

depth D

● The velocity with which temperature maximum and minimum appear to

propagate downward through the soil is given by (2πD/P)

● The temperature wave is π radians (180°) out of phase with the surface wave

at depth (πD)

● From Equation (6.14), the maximum surface heat flux is (√2.Ta.k

s)/D

which is the heat flow that would be maintained through a thickness

(√2.D) of soil if one side were maintained at (Tm

+Ta) and the other at

(Tm

–Ta). For this reason (√2.D) has been called the effective depth of soil

heat flow



● For a layer of soil with thickness equal to the effective depth, the net flow of

heat into the soil during one half cycle (i.e., the integral of Gz=0

from −π/4 το

3π/4) would raise the temperature by 1 K.

The values of D for three types of soil are shown in Fig. 6.5 as a function of

volumetric water content for the daily and yearly cycles. D increases quickly as the

volumetric water content changes from 0 to 0.1 for sandy and clay soils, and

rapidly reaches values of 120-150 mm, these being typical of values often found

in the field. However, for peat soils D remains in the range 3-5 cm regardless of

volumetric water content, again consistent with observations that indicate organic

soils heat and cool slowly.


● Soil surface temperature: spatial variability makes measurement difficult;

the amplitude of daily cycle is large (∼30°C) for bare, dry soil in calm, clear

sky conditions (more than air temperature) with timing linked to radiation,

but less for wet soil and also less and linked to air temperature for vegetation-

covered soil.

● Subsurface soil temperature: is easier to measure with carefully inserted

thermometers, and is driven by soil surface temperature and so differs for

dry, wet, and vegetation-covered soil, with magnitude of daily cycle reducing

in size and phase progressively delayed with depth.

● Thermal properties of soil: The density (rs), specific heat (c

s), and thermal

diffusivity (as), and especially the heat capacity per unit volume (C

s), and ther-

mal conductivity (ks) of soil (defined in the text) are all strongly dependent on

moisture content.

● Thermal conduction: the soil heat flux (Gs) into soil away from the surface is

described by a simple diffusion equation as ks (or a

sC

s) times the negative

gradient of soil temperature with depth.

0 0.2 0.4

Volumetric water content

0.6 0.80

1

2

3

Annual cycle

damping depth (m

)

Dai

ly c

ycle

dam

ping

dep

th (

cm)

15

10

5

0

Clay soil

Sandy soil

Peat soilFigure 6.5 Change of

damping depth volumetric

water content for typical

sandy, clay, and peat soils for

the daily and yearly surface

temperature cycles.



● Soil heat flow equation: the rate of change of soil temperature at depth is

given by the divergence of the equation describing thermal conduction, and

for homogeneous soil this becomes the product of the thermal diffusivity

with the second partial derivative of the soil temperature with depth.

● Thermal wave in soil temperature: assuming a sinusoidal daily cycle in soil

surface temperature and homogeneous soil, the thermal wave in the soil is

sinusoidal and relative to the surface wave has an amplitude less by a factor

(z/D) and phase (in radians) delayed by (z/D), where D (= √(Pas/π) is the

damping depth and P is the period of the wave in seconds.

● Damping depth: D is 0.08 m and 0.12 m for the daily soil surface temperature

cycle, and 1.6 m and 2.24 m for the yearly cycle for dry sandy soil and wet

clay, respectively.

● Surface soil heat flux: is advanced by one eighth of a cycle with respect to

sinusoidal variations in soil surface temperature, i.e., by 3 hours for the daily

cycle and 1.5 months for the yearly cycle.

References

Deacon, E.L. (1969) Physical processes near the surface of the Earth. In: World Survey of

Climatology, Vol. 1, General Climatology (ed. H.E. Landsberg). pp. 39–104. Elsevier,

Amsterdam.

Monteith, J.L. & Unsworth, M.H. (1990) Principles of Environmental Physics. Edward

Arnold, London.


Introduction

This chapter provides an overview of some of the most commonly used methods

by means of which important terms in the surface energy budget are measured.

Most measurement methods provide an estimate of the average rate of flow of

energy into or out of the Earth’s surface over a specified period. However, latent

heat flux is also frequently measured in the form of the net evaporative water loss

from sample areas of the terrestrial surface.

Measuring solar radiation

The most common early approach used to derive surface solar radiation was to

estimate how much of the calculable solar energy entering the top of the atmosphere

was absorbed before it reached the ground. This required estimates or

measurements of cloud cover. Subsequently, instruments were devised which

measured the incident solar energy from the warming it induced when incident

on a near blackbody surface or from the number of electrons it mobilized in a

semiconductor.

Daily estimates of cloud cover

Estimates of cloud cover are made with the human eye by trained observers at

meteorological stations looking upward at the sky overhead. When estimates are

made in this way, they are usually expressed either in oktas (eighths of the sky) or

in tenths and given to the closest whole number value. A value of 0 refers to clear

7 Measuring Surface Heat Fluxes





sky while 8 oktas, or 10 on the decimal scale, indicates overcast sky. Estimates

made in this way are, of course, only representative of conditions within the range

of visibility of the observer. Problems associated with this method include inability

to make observations when the visibility is very low, when it is foggy for example,

and difficulty in estimating the correct fractional cover for clouds that are near the

visual horizon. Quite often just one visual estimate of cloud cover is made each

day, often at the same time as other daily measurements (such as rainfall), although

several observations through the day are required to give an estimate of the daily

average because there is often a marked diurnal variation in cloud cover.

John Francis Campbell (1822–1885) invented the first simple instrument to

estimate daily average cloud cover indirectly in the form of the number of hours of

the bright sunshine expressed as a fraction of the maximum number of hours for

which bright sunshine is feasible on the day observations are made. The device he

invented, which is often called the Campbell-Stokes recorder, comprises a sphere of

glass that serves to focus the Sun’s rays onto a card, see Fig. 7.1a. When the Sun is

exposed it has sufficient energy to burn the card and as it moves in the sky, the

length of the burnt trace on the card can be later measured and interpreted in

terms of the time without cloud during the day, see Fig. 7.1b. At the time of writing,

this is by far the most common radiation instrument used at agro-climate stations

worldwide.

Thermoelectric pyranometers

Figure 7.2 shows a thermoelectric sensor of solar radiation often called a Kipp

pyranometer. When solar radiation is incident on a surface which has properties

close to those of a blackbody, the surface temperature rises. The warmed surface

loses energy to its surroundings and an equilibrium is established between the

incoming solar energy and this outgoing heat loss. The equilibrium temperature of

the absorbing surface is related to the strength of the solar beam. In the Kipp

Length of burnsare measured

(a) (b)

Photograph of a burnt card

Figure 7.1 (a) Campbell-

Stokes sunshine hour

recorder; (b) a card after it has

been removed from the

recorder showing burns made

on the card by the focused

solar beam when there was

bright sunshine during the

day. The length of the burns

are measured to determine

the number of bright sunshine

hours. (From Fairmount

Weather Systems, 2010.)


Measuring Surface Heat Fluxes 79

pyranometer, solar radiation is measured using the thermoelectric effect from the

difference in temperature between the surface heated by solar radiation and the

body of the instrument. A thermopile comprising many bi-metallic junctions

connected in series is used to generate the voltage produced. Glass domes are used

to provide protection for the surface receiving solar radiation and to inhibit

convection from the blackened surface. These domes also act as a filter, restricting

incoming energy to that arriving in the solar waveband. Many of the pyranometers

in common use today are thermoelectric sensors that use this approach.

Photoelectric pyranometers

Photoelectric pyranometers use silicon photovoltaic detectors that provide an

electrical output proportional to the radiant energy falling on the semiconductor.

The sensitivity of photovoltaic detectors is not uniform with wavelength and they

are not sensitive to the full spectrum of the energy incident in the solar beam, see

Fig. 7.3a. Photoelectric pyranometers therefore require careful calibration against

a standard sensor before use. It is assumed that any changes in the incoming solar

spectrum, in changing meteorological conditions, does not greatly influence the

calibration of the instrument. Commercial photoelectric pyranometers (see 7.3b)

are, however, less expensive and easier to use than other pyranometers and they are

popular because of this. After careful calibration, the error in the measurement

they provide can be less than 5% under most conditions of natural daylight.

By carefully introducing filters above the active surface, photovoltaic sensors

can be designed to measure the incoming energy in selected wavebands. The most

common requirement is for sensors which measure incoming energy in the 0.4 μm

to 0.7 μm waveband, this being the range of wavelengths used by plants for photo-

synthesis. Such sensors are usually referred to as quantum sensors.

Treated surfaceabsorbs solar energy

Glass domes protect andfilter solar radiation

(a)

(b)

Hot surface warmed byradiation

Cool surface in metal atair temperature

“Thermopile” measurestemperature difference

between surfaces

+ −

Figure 7.2 (a) Operating principle of a thermoelectric sensor for measuring solar radiation; (b) Kipp and Zonen

pyranometer which uses the thermoelectric method. (From Kipp and Zonen, 2010.)



Measuring net radiation

Net radiometers differ from pyranometers in that they measure the difference

between the incoming and outgoing radiant energy in both the solar and the

longwave wavebands. Many net radiometers currently in use measure the net

difference in energy input to two blackened surfaces, one facing up and one down.

Using a thermopile, analogous to the approach used in thermoelectric pyranometers,

the difference in temperature between the two surfaces generates a voltage, see

Fig. 7.4a. The surfaces are commonly protected from the environment (especially

precipitation) by polythene domes which allow radiation of all wavelengths to reach

0.05

0.10

0.15

0.20

0.25

00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Solar irradiation curve outside atmosphereSolar irradiation curve at sea levelCurve for blackbody ct 5900�k

O3

O3

O2H2O

H2O

H2OH2O

H2O

H2OH2O, CO2

H2O, CO2H2O, CO2

50

100

Wavelength (μ)

1.8 2.0

Pyranometer sensor

(a)

(b)

2.2 2.4 2.6 2.8 3.0 3.2

Per

cent

rel

ativ

e re

spon

se to

irra

dian

ceS

pect

ral i

rrad

ianc

e (S

λ) -

W m

−2 A

−1

Figure 7.3 (a) Wavelength

dependent response of a

LI-CORR LI-200 photovoltaic

pyranometer compared with

the spectrum of solar

radiation above and below

the atmosphere; (b) LI-COR

pyranometer which uses the

photovoltaic method.

(LI-COR Environmental,

2010; after Federer and

Tanner, 1966.)



the two active surfaces, see Fig. 7.4b. Sometimes these domes are inflated using dry

nitrogen at slightly greater than atmospheric pressure.

Net radiation can also be obtained by measuring all four components of net

radiation (i.e., upward and downward shortwave radiation, and upward and

downward longwave radiation) separately. Instrument packages can be obtained

comprising four separate radiometers, two pyranometers to measure the solar

fluxes and two pyrgeometers to measure the longwave fluxes, see Fig. 7.4c. These

radiometers are thermoelectric sensors operating similarly to thermoelectric

pyranometers but with appropriate wavelength filtering to select the required

wavebands.

Measuring soil heat flux

Soil heat flux is measured using soil heat flux plates. These are circular disks a few

centimeters in diameter and a few millimeters thick made of material with a

thermal conductivity that is broadly similar to that of soil. The assumption is that

because the thermal conductivity is similar, when the disk of material is inserted

horizontally into the soil the flow pattern of heat in the soil is not greatly disturbed.

There are shortcomings in this assumption because the thermal conductivity of

soil changes substantially with soil moisture content. A thermopile with the bimetal

(b)

(c)

Polythene domes

Treated surfacesabsorbs radiant

energy at allwavelengths

Dry Nitrogento inflatedomes

“Thermopile” measurestemperature difference

between surfaces

Upward radiation(all wavelengths)

Downward radiation(all wavelengths)

(a)

+ −

Figure 7.4 (a) Schematic diagram of a thermoelectric net radiometer which measures the difference in radiant energy at all

wavelengths arriving from above and below; (b) A simple thermoelectric net radiometer; (c) Kipp and Zonen CNR 4

thermoelectric net radiometer which measures all four components of net radiation using four separate sensors. (From Kipp

and Zonen, 2010.)



junctions alternately near the top and bottom of the soil heat plate measures the

temperature difference across the soil heat flux plate. Because the thermal

conductivity and thickness of the soil heat flux plate are known, the heat flow per

unit area through the plate can be calculated and this is assumed to be the same as

that which would be flowing through the volume of soil that the plate replaces.

Soil heat flux plates cannot realistically be used on the surface of the soil so they

are normally installed at some depth (typically 5 cm) below the soil surface, see

Fig. 7.5. Because the damping depth of heat flow in soils is typically on the order

of 5–15 cm, this means the soil heat flux measured by the plate is not a good

estimate of the surface soil heat flux. Therefore, an attempt is made to estimate the

loss or gain of heat in the layer of soil between the plate and the surface. To do this

at least one and usually more thermometers are inserted into the soil to measure

the rate of change of soil temperature between the soil flux plate and the surface.

Often a pit with a vertical edge is dug and these thermometers and the soil heat

flux plate are inserted sideways into the soil through the edge of the pit to minimize

soil disturbance.

If the heat capacity per unit volume of the soil, Cs, is known, the corrected soil heat

flux at the surface is estimated from the soil heat flux measure by the plate using:

= =

δ= +

δ0

soilz z d s

TG G C

t (7.1)

where Gz=0

and Gz=d

are respectively the soil heat fluxes at the surface and at depth

d at which the soil heat flux plate is inserted, and dTsoil

is the average change in the

temperature of the soil layer above the plate over the period dt for which estima-

tion is required.

Measuring latent and sensible heat

There are two general ways in which the fluxes of latent and sensible heat are

measured. One is to determine one or both of these energy fluxes by making

meteorological measurements in the turbulent air just above the surface through

Soil heatflux plate

Soil surface

Soil

Air

Soilthermometers

Gz =0

Gz =0

~5 cm

Figure 7.5 Arrangement of a

soil heat flux plate and soil

thermometers when used to

estimate the heat flux at the

soil surface.



which they pass. The second way is, by measuring the water lost by evaporation

from the surface, to obtain the latent heat flux, and then to deduce the sensible

heat flux from the surface energy budget. These two different approaches are

discussed separately.

Micrometeorological measurement of surface energy fluxes

When energy fluxes are measured using micrometeorological techniques, the

measurement is of vertical flow made at a point with a sensor or sensors mounted

typically at a height of a few meters or tens of meters on a pole or tower in the

turbulent airstream comparatively close to the ground. Energy exchange is assumed

to be vertical and the energy fluxes are assumed independent of height and to be a

weighted average of the surface fluxes originating upwind of the instrumentation.

The upwind area sampled is called the fetch, see Fig. 7.6. The proportional

contributions from areas that lie within the fetch depend on the buoyancy of the

atmosphere, the height of the sensor(s) used, and on the aerodynamic roughness

of the upwind surface. Several different micrometeorological methods have been

used in the past but here attention is focused on the two still in common use.

Bowen ratio/energy budget method

One method that has been much used to measure latent and sensible heat fluxes is

based on the surface energy budget. The approach is theoretically simple. It relies

on the fact that it is always possible to provide an estimate of the sum of the latent

and sensible heat fluxes at any point from Equation (4.2) providing all the remain-

ing terms in the surface energy balance equation can be measured, thus:

+ =H E Al (7.2)

where A is the available energy given by Equation (4.3). To deduce the latent and

sensible heat fluxes separately, a second equation describing their interrelationship

is required and the ratio of the sensible to the latent heat fluxes, b, the Bowen ratio,

is used to provide this second relationship. Consequently, the approach is called

the Bowen ratio/energy budget method.

To estimate the ratio of the flux of sensible heat to the flux of latent heat it is

assumed that for height of the order of meters to tens of meters above the ground,

the transfer processes responsible for moving sensible heat vertically are the same

Wind direction

“Fetch” of instrument A

A

Figure 7.6 Upwind fetch of

micrometeorological sensors.



as those that move latent heat vertically, and that these are equally effective in each

case. Recognizing the need to express temperatures in terms of virtual potential

temperature when describing flow in a hydrostatic atmosphere (see Chapter 3), the

rate of vertical sensible heat flow between two levels is assumed to be proportional

to the difference in atmospheric heat content at these two levels, i.e., proportional

to the difference in (rac

pq

v). Similarly, the rate of water vapor flow between the

same two levels is assumed to be proportional to the difference in humidity content

between the two levels, i.e., proportional to the difference (raq), and latent heat

flow is then proportional to this difference multiplied by λ. Consider two levels

where the potential temperatures are qv1 and q

v2, respectively, and the specific

humidities are q1 and q2, respectively. If the processes for the transfer of heat and

water vapor are the same between these levels, then β is given by:

⎡ ⎤−⎣ ⎦=⎡ ⎤−⎣ ⎦

2 1

2 1

( ) ( )

( ) ( )

a p v a p v

a a

c c

q q

r q r qb

r l r l (7.3)

Substituting for q using Equation (2.9), this equation can be rewritten as:

Δ= =

ΔvH

E eq

b gl

(7.4)

where Δθv and Δe are the differences in virtual potential temperature and vapor

pressure between levels 1 and 2, respectively, and γ is the psychrometric constant

defined by Equation (2.25). Because the sum and the ratio of the sensible heat and

latent heat fluxes are known, individual values of these two fluxes can be calcu-

lated by combining Equations (7.2) and (7.4) to give:

( )=+1

AElb

(7.5)

and:

= −H A El (7.6)

Measuring the difference in air temperature and humidity between two heights

required to calculate the Bowen ratio from Equation (7.4) can be difficult if these

differences are comparable in size with systematic errors in the instruments used.

But field systems have been designed to minimize the effect of sensor errors; one

approach (Fig. 7.7) is to regularly interchange pairs of temperature and humidity

sensors between the two levels. In practice, measuring the difference in humidity

is usually more difficult than measuring the difference in temperature and an

effective way to minimize the effect of humidity measurement errors is to duct air

alternately from the two levels to a common humidity sensor (Fig. 7.8). The effect

of any instrumental offset error then cancels out in the measured difference.



Eddy correlation method

In the atmosphere a few meters or tens of meters above the ground, vertical

movement of atmospheric entities is almost entirely by turbulent transport. Near

a horizontal surface the mean wind, u, is parallel to the ground so at any point, P,

the average wind vector perpendicular to the ground, w, is near zero (Fig. 7.9).

However, the turbulent eddies in the air cause pseudo-random fluctuations in the

vertical wind, w′, and in other atmospheric variables. In particular, there are

fluctuations q′ around the mean value of specific humidity, q, and fluctuations qv′

around the mean value of virtual potential temperature, qv.

Upward movement of water vapor in the turbulent field means that on average

there is a correlation between fluctuations of higher than average humidity and

movement of air away from the surface, i.e., a correlation with positive fluctuations

in vertical wind speed. Similarly lower than average fluctuations in average

humidity are on average correlated with negative fluctuations in vertical wind

speed. Integrating the product of the instantaneous value of w′ with the

instantaneous fluctuation in the volumetric latent heat content of the air, (l ra q′),

gives the time average outward flux of latent heat, λE, i.e.:

= ′ ′ aE q wl l r (7.7)

(b)Reversing motor

housing

Vane

StopWater

reservoir

Radiationshield

Thermopilecables

Plug(a)

Wet-bulb Inner shieldOuter shieldDry-bulb

Plug

PVC Tee

PVC Tee

Aspiration motor

(Viewed from the frontof the system)

PVC Pipe

Figure 7.7 (a) Schematic diagram of a Bowen ratio measuring system with interchanging temperature and humidity

sensors; (b) A Bowen ratio measuring system with interchanging sensors used over short vegetation. (From McCaughey,

1981, published with permission.)



Similarly, integrating the product of the instantaneous value of w′ with the

instantaneous fluctuation in the volumetric heat content of the air, (ra c

p q

v′) gives

the time average sensible heat flux, H, i.e.:

a p vH c w= ′ ′r q (7.8)

Thus, to measure the latent and sensible heat fluxes it is necessary to take

simultaneous, co-located measurements of the wind speed perpendicular to the

Water vapor measurement

(a)

(b)

Cooledmirror

DewPT.

Pump

Adjustableflow meters

2 liter containersextends time constantof vapor measurement

Air intakesat two heights

Sounoid valvecontrols whichinput goes tocooled mirror.switched every

2 minutes

Figure 7.8 (a) Schematic

diagram of a Campbell

Scientific Bowen ratio/energy

budget system for measuring

latent and sensible heat with

humidity measured by

ducting air alternately to a

common sensor; (b)

Schematic diagram of the

system used to duct air to the

common humidity sensor.

(From Campbell Scientific,

1987.)



surface, atmospheric humidity, and temperature above the ground. Rapid response

sensors are required to capture high frequency fluctuations and the sensors must

have stable calibration over the period over which the time-average product is

calculated. To make the measurement, sometimes electronic hardware is used to

rapidly interrogate the sensors and to compute the fluctuating component of each

measured value and their instantaneous cross products, and then to average and

store the resulting fluxes, preferably over long periods. However, now that data

storage is inexpensive, simply storing the data for later analysis is an alternative.

In practice, the turbulent eddies responsible for the flux transfer occur over a

wide range of frequencies, with the spectrum of contributing frequencies being

determined by:

● ambient horizontal wind speed – there is more transfer at higher frequencies

when wind speeds are greater;

● sensor mounting height – there is more transfer at lower frequencies when

sensors are mounted farther from the surface;

● aerodynamic roughness of the surface – there is more transfer at lower

frequencies over rougher surfaces such as forest than smoother surfaces such

as bare soil; and

● atmospheric buoyancy – there is more transfer at lower frequencies in

unstable atmospheres.

In eddy correlation measuring systems, measurements are usually attempted using

sensors that can resolve frequencies from about ten cycles per second to about ten

cycles per hour.

Evaporation measurement from integrated water loss

Evaporation can be measured as the net water loss to the atmosphere from a

terrestrial surface over a given time period. If required, the time average latent heat

Figure 7.9 Mean

and vertical wind

speed components

parallel to and

perpendicular to

the surface. The

mean perpendicular

wind is zero, with

fluctuations in

vertical wind speed

caused by turbulent

eddies.



can then be calculated from this. The approach is to define a sample of the

evaporating surface with known area for which the water balance can be closed,

i.e., to define a sample for which all the water entering or leaving can be measured

or adequately estimated. The water balance equation for such a sample volume is

illustrated in Fig. 7.10, and the evaporation from the sample is calculated from:

( )− + Δ += − ri ro s Lv V V V

E PA

(7.9)

where E is the (required) evaporation loss from and P is measured precipitation

input to the sample volume, both in mm depth of water; A is the surface area of the

sample, in m2; Vri and V

ro are the ‘runin’ to and ‘runoff ’ from the sample volume,

respectively, measured in liters; ΔVs is the measured or estimated change in water

stored in the sample volume, in liters; and VL is the unmeasured ‘loss’ from the

system, in liters. VL is therefore the error in the evaporation arising from poor

water balance closure.

Evaporation pans

The first measurements of evaporation were of the evaporation from the surface of

samples of water held in a container exposed to the atmosphere. Measurement

from well-specified containers, usually called evaporation pans, is still much used

to provide an approximate index of the atmospheric conditions that influence the

evaporation rate from well-watered crops and soils. Many designs for evaporation

pans have been documented, one that has been widely adopted as a standard is the

US Weather Bureau ‘Class A’ pan (Fig. 7.11). This is a cylindrical ‘raised pan’

mounted above ground with an area of 1.21 m2 and a depth of 0.255 m which is

made of 22-gauge galvanized iron or monel metal, and which is mounted horizon-

tally 0.15 m above ground on a wooden platform with soil build up to be within

Precipitationinput, P

Evaporationoutput, E

“Runin”, Vri

“Runoff”, Vro

Surface area, A

Storage, VsSamplevolume

Leakage, VL

Figure 7.10 Water balance

of a sample volume used to

measure the net water loss to

the atmosphere as

evaporation by measuring the

other inputs and outputs to

the volume.



0.05 m beneath the pan. Water is added to the pan as needed so that the water level

is always 0.05–0.075 m below the rim of the pan.

The measurements provided by evaporation pans are usually significantly

greater than the evaporation from nearby surfaces such as crops and lakes. This is

mainly because the energy balance of the pan is not representative. Some of the

energy for evaporation comes from radiant energy incident on the sides of the

pan, conduction through the sides or bottom of the pan, and/or energy transferred

from the air as it blows over the (usually cooler) water surface. For this reason

empirical correction factors, called pan coefficients, are used to correct the

measured evaporation rates downward to give a better estimate of the evaporation

rates for nearby vegetation, soil, or water surfaces. Pan coefficients can be in the

range 0.3 to 1.0 but are more typically in the range 0.65 to 0.85. Their value depends

on wind speed, relative humidity and where the pan is located, relative to the area

for which estimates are required, see Chapter 23. If, for example, evaporation

estimates are sought for an irrigated grass crop located in a dry landscape, the crop

factor appropriate for a pan located within the cropped area is closer to unity than

the pan factor for a pan located at the edge of the crop that is exposed to less humid

air from the surrounding dry area.

Watersheds and lakes

Evaporation has been inferred from the water balance of watersheds and lakes but

this is difficult to do with high accuracy because it is often deduced as a compara-

tively small residual in a water balance in which other terms dominate. Although

errors in measured runoff can be just a few percent, the errors in estimating area-

average precipitation and water storage can be significantly higher, say 5–15%

because of the sampling errors which can be significant over large areas, see

Chapter 12. In addition, unmeasured groundwater leakage is a systematic error

that is hard to estimate. As a result, errors in estimates of evaporation made from

the water balance of catchments and lakes may be as high as 20–30%.

However, comparative studies of the water balance of carefully selected and

well-maintained paired catchments have given definitive evidence of evaporation

Figure 7.11 US Weather

Bureau ‘Class A’ pan (From

Illinois State Water Survey

http://www.isws.illinois.edu/

atmos/statecli/Instruments/

panevap.jpg.)



differences for different vegetation covers. Figure 7.12 shows the results of a good

example, the Plynlimon catchments, which demonstrate a clear difference in the

area-average evaporation loss from two adjacent catchments, one entirely grassland

and one partially forested, in the high rainfall maritime climate of Wales.

Lysimeters

Lysimeters are sometimes regarded as providing the standard evaporation

measurement against which alternative measurement methods can be validated.

High quality lysimeter systems are expensive, however, and their use requires

great care. Accurate lysimeters have been much used in research to calibrate

some of the empirical formulae used to estimate evaporation from irrigated

crops, see Chapter 23. The method requires isolating a sample volume of soil and

vegetation, typically 0.5–2.0 m in diameter, in a container from which there is no

leakage and from which runoff is measured. The incoming water as precipitation

is measured, and the change in the water inside the container determined, often

by weighing the whole lysimeter. Lysimeters must be installed carefully to leave

the natural soil and vegetation undisturbed, and if a crop is grown in the

GrasslandForest

Severn

Wye

1975 1980 1985

Year

Evaporation

Cum

ulat

ive

tota

ls(m

m)

Precipitation

Severn (65% forest; 35% grassland)

Wye (100% grassland)

20000

10000

0

Figure 7.12 Results of the Plynlimon paired catchment study demonstrating the additional water loss as evaporation for

the partly forested catchment.



surrounding area, an identical crop must be grown in the lysimeter to maintain

a similar soil moisture status.

Soil moisture depletion

Evaporation is sometimes measured by monitoring incoming precipitation and the

change in soil moisture beneath sample areas of vegetation and soil. The change in

soil moisture can be determined in different ways, including using neutron probes,

capacitance probes, and time-domain reflectometer sensors. Adequate spatial

sampling is required for an accurate estimate of evaporation, and drainage must

either be easily quantified or negligible. This method is particularly useful for

comparing evaporation from undisturbed plots of different crops. The method

becomes more accurate if there are co-located measurements of soil tension to

determine the average zero flux plane which separates regions in the soil profile in

which movement is primarily upward from those where it is downward, see Fig. 7.13.

Comparison of evaporation measuring methods

Attributes of the several evaporation measuring methods described above are

given in Table 7.1 along with their relative strengths and weaknesses, the scale at

which measurement is made, and likely errors in each method.

Water lost to evaporation

Watermovementdownwards

Watermovementupwards

Finalprofile

Initial soilmoisturecontentprofile

Soil moisture content Soil water potential

Final soilmoisturecontentprofile

Average “zero flux” plane

Dep

th b

elow

soi

l sur

face

Depth below

soil surface

Initialprofile

Water lost to drainageB

B

A

A

Figure 7.13 Measuring evaporative water loss using soil water depletion with regions in the profile with upward and

downward water loss defined by the zero flux plane obtained from measurements of soil water potential. (From

Shuttleworth, 1993, published with permission.)


Tabl

e 7.

1 Su

mm

ary

of

attr

ibu

tes

of

dif

fere

nt

met

ho

d f

or

mea

suri

ng

evap

ora

tio

n a

lon

g w

ith

op

inio

ns

on

th

eir

rela

tive

str

engt

hs

and

wea

kn

esse

s, t

he

scal

e at

wh

ich

mea

sure

men

t is

mad

e, a

nd

lik

ely

erro

rs i

n e

ach

met

ho

d.

Mic

rom

eteo

rolo

gica

l m

etho

dsBr

ief

Des

crip

tion

Ass

umpt

ions

Stre

ngth

s an

d w

eakn

esse

sSc

ale

of

Mea

sure

men

tEr

ror

Bow

en R

atio

– E

nerg

y Bu

dget

Calc

ulat

e ev

apor

atio

n as

the

late

nt h

eat f

rom

the

surfa

ce

ener

gy b

udge

t usin

g th

e ra

tio o

f sen

sible

to la

tent

he

at (B

owen

ratio

) der

ived

fro

m th

e ra

tio b

etw

een

atm

osph

eric

tem

pera

ture

an

d hu

mid

ity g

radi

ents

m

easu

red

over

a fe

w m

eter

s ab

ove

the

vege

tatio

n.

Assu

mes

the

turb

ulen

t diff

usio

n co

effic

ient

for s

ensib

le h

eat a

nd

late

nt h

eat a

re th

e sa

me

in th

e lo

wer

atm

osph

ere

in a

ll co

nditi

ons

of a

tmos

pher

ic

stab

ility

, and

sam

ple

plot

sca

le

mea

sure

men

ts o

f ene

rgy

budg

et

com

pone

nts

(net

radi

atio

n, s

oil

heat

) are

repr

esen

tive

of

upw

ind

cond

ition

s.

Fairl

y w

ell e

stab

lishe

d m

etho

d w

hich

is

ava

ilabl

e as

rela

tivel

y in

expe

nsiv

e pr

oprie

tary

sys

tem

s th

at c

an b

e us

ed

for b

oth

shor

t cro

ps a

nd n

atur

al

vege

tatio

n bu

t is

prob

lem

atic

ove

r ta

ll ve

geta

tion

whe

n at

mos

pher

ic

grad

ient

s ar

e lo

w a

nd c

omm

only

ca

nnot

be

used

dur

ing

hour

s ar

ound

da

wn

and

dusk

hou

rs w

hen

the

Bow

en ra

tio is

min

us o

ne.

Fiel

d sc

ale

Erro

rs a

ssoc

iate

d w

ith a

ssum

ptio

ns

and

repr

esen

tativ

enes

san

d th

e er

rors

in

requ

ired

supp

lem

enta

ryse

nsor

s im

ply

over

all e

rror

s ca

n be

~5–

15%

.Ed

dy c

orre

latio

nCa

lcul

ate

the

evap

orat

ion

as 2

0 to

60

min

ute

time

aver

ages

from

the

corr

elat

ion

coef

ficie

nt

betw

een

fluct

uatio

ns in

ve

rtic

al w

inds

peed

and

at

mos

pher

ic h

umid

ity

mea

sure

d at

hig

h fre

quen

cy

(~10

Hz)

and

at t

he s

ame

plac

e, a

few

met

ers

abov

e th

e ve

geta

tion.

Assu

mes

tran

sfer

of w

ater

va

por i

s al

l via

turb

ulen

t tr

ansf

er a

t the

sam

ple

poin

t, bu

t doe

s no

t occ

ur in

tu

rbul

ence

with

ass

ocia

ted

time

scal

es le

ss th

an ~

0.1

seco

nds

or g

reat

er th

an th

e se

lect

ed

aver

agin

g tim

e.

The

pref

erre

d m

etho

d fo

r fie

ld s

cale

m

easu

rem

ents

in re

sear

ch

appl

icat

ions

, giv

es ro

utin

e un

supe

rvis

ed d

ata

colle

ctio

n us

ing

reas

onab

ly e

xpen

sive

pro

prie

tary

lo

gger

and

co-

loca

ted

sens

ors,

but

pron

e to

sys

tem

atic

und

eres

timat

ion

of fl

uxes

so

best

use

d to

mea

sure

Bo

wen

ratio

, with

eva

pora

tion

then

de

duce

d fro

m s

urfa

ce e

nerg

y bu

dget

Fiel

d sc

ale

Syst

emat

icun

dere

stim

atio

n up

to

25%

can

occ

ur

in th

e ba

sic

mea

sure

men

t, re

duce

d to

rand

om

erro

rs ~

5–15

% if

se

nsib

le h

eat a

lso

mea

sure

d an

d en

ergy

bal

ance

us

ed

Wat

er lo

ss

mea

sure

men

tsBr

ief

Des

crip

tion

Ass

umpt

ions

Stre

ngth

s an

d w

eakn

esse

sSc

ale

of

Mea

sure

men

tEr

ror

Evap

orat

ion

pan

Dire

ct m

easu

rem

ent o

f the

ch

ange

in w

ater

leve

l ove

r tim

e fo

r a s

ampl

e of

ope

n w

ater

in a

“pa

n” w

ith

wel

l-spe

cifie

d di

men

sion

s an

d si

ting.

Assu

mes

that

the

rela

tions

hip

betw

een

the

mea

sure

d ev

apor

atio

n fro

m p

ans

(with

pr

escr

ibed

cha

ract

eris

tics)

and

th

e ac

tual

eva

pora

tion

from

the

adja

cent

are

a ca

n be

cal

ibra

ted,

an

d th

at th

is c

alib

ratio

n is

tr

ansf

erra

ble

betw

een

diffe

rent

lo

catio

ns a

nd c

limat

es

Met

hod

is lo

ng e

stab

lishe

d an

d w

ell-r

ecog

nize

d, s

impl

e to

un

ders

tand

and

impl

emen

t, an

d re

ason

ably

inex

pens

ive;

but

bec

ause

it

fund

amen

tally

relie

s on

the

valid

ity

of a

n ex

trap

olat

ed c

alib

ratio

n fa

ctor

pr

evio

usly

def

ined

els

ewhe

re, i

t is

prim

arily

use

d fo

r cro

p ev

apor

atio

n es

timat

es ra

ther

than

het

erog

eneo

us

natu

ral v

eget

atio

n co

vers

Plot

sca

le

(ass

umed

repr

esen

tativ

eat

fiel

d sc

ale)

Varie

s with

relia

bilit

y an

d re

leva

nce

of

calib

ratio

n fa

ctor

, bu

t ~10

–20%

erro

rs

are

poss

ible

for

crop

s, w

ith g

reat

er

erro

rs li

kely

for

natu

ral v

eget

atio

n be

caus

e ca

libra

tion

may

be

unkn

own


Wat

er b

alan

ce

of c

atch

men

tTh

e un

mea

sure

d di

ffere

nce

betw

een

othe

r mea

sure

d co

mpo

nent

s of

the

catc

hmen

t wat

er b

alan

ce,

incl

udin

g in

com

ing

prec

ipita

tion,

sur

face

(and

pr

efer

ably

als

o gr

ound

wat

er) o

utflo

w, a

nd

the

chan

ge in

soi

l wat

er

stor

age.

Assu

mes

all

the

othe

r co

mpo

nent

s of

the

catc

hmen

t w

ater

bal

ance

can

be

mea

sure

d as

spa

tial a

vera

ges

with

su

ffici

ent a

ccur

acy

for

evap

orat

ion

to b

e re

liabl

y ca

lcul

ated

as

perh

aps

a sm

all

diffe

renc

e be

twee

n th

em.

Giv

es a

n ar

ea-a

vera

ge m

easu

rem

ent

for n

atur

al v

eget

atio

n co

vers

for a

hy

drol

ogic

ally

sig

nific

ant r

egio

n w

hich

rela

tes

to w

ater

reso

urce

is

sues

, but

are

a-av

erag

e m

easu

rem

ent o

f the

oth

er w

ater

ba

lanc

e te

rms

can

be e

xpen

sive

and

di

fficu

lt, e

spec

ially

gro

undw

ater

flow

an

d so

il w

ater

sto

rage

, con

sequ

ently

on

ly lo

nger

tim

e-av

erag

e es

timat

es

are

poss

ible

Catc

hmen

t sca

leVa

ries

with

qua

lity

of im

plem

enta

tion

and

size

and

nat

ure

of c

atch

men

t, bu

t er

rors

as

low

as

~10

–20%

may

be

achi

evab

le in

re

sear

chca

tchm

ents

with

pe

rsis

tent

car

e

Lysi

met

ryM

easu

res

the

chan

ge in

w

eigh

t ove

r tim

e of

an

isol

ated

pre

fera

bly

undi

stur

bed

sam

ple

of s

oil

and

over

lyin

g ve

geta

tion

whi

le s

imul

atan

eous

ly

mea

surin

g in

com

ing

prec

ipita

tion

to a

nd

outg

oing

dra

inag

e fro

m th

e sa

mpl

e

Assu

mes

that

the

sam

ple

of s

oil

and

over

lyin

g ve

geta

tion

on

whi

ch m

easu

rem

ents

are

mad

e ar

e re

pres

enta

tive

of th

e pl

ot o

r fie

ld fo

r whi

ch e

vapo

ratio

n m

easu

rem

ent i

s re

quire

d in

te

rms

of s

oil w

ater

con

tent

and

ve

geta

tion

grow

th a

nd v

igor

.

If th

e so

il an

d ve

geta

tion

sam

ple

is

trul

y re

pres

enta

tive

(diff

icul

t to

achi

eve)

, the

lysi

met

er is

wid

ely

acce

pted

as

bein

g an

unp

aral

lele

d st

anda

rd a

gain

st w

hich

to c

ompa

re

and

valid

ate

othe

r eva

pora

tion

mea

sure

men

ts/m

odel

s of

cro

p ev

apor

atio

n, b

ut m

oder

n hi

gh

prec

ison

lysi

met

ers

are

very

ex

pens

ive

(~$5

0k) a

nd re

quire

ex

pert

sup

ervi

sion

.

Sam

ple

scal

e (a

ssum

edre

pres

enta

tive

at p

lot o

r fie

ld

scal

e)

Stat

e of

the

art

lysi

met

ers

can

prov

ide

daily

m

easu

rem

ents

with

hi

gh a

ccur

acy

(few

%),

but e

rror

s ca

n ea

sily

bec

ome

subs

tant

ial (

few

×

10%

) with

un

repr

esen

tativ

esa

mpl

ing.

Soil

moi

stur

e de

plet

ion

Mea

sure

the

chan

ge in

w

ater

con

tent

of a

re

pres

enta

tive

sam

ple

of

undi

stur

bed

soil

and

vege

tatio

n w

hile

si

mul

atan

eous

ly m

easu

ring

inco

min

g pr

ecip

itatio

n an

d ru

n-on

and

runo

ff an

d es

timat

ing

deep

dra

inag

e fo

r the

sam

ple

plot

Assu

mes

soi

l wat

er m

easu

ring

devi

ce (r

esis

tanc

e bl

ocks

, te

nsio

met

ers,

neut

ron

prob

es,

time-

dom

ain

refle

ctom

eter

s, ca

paci

tanc

e se

nsor

s)

adeq

uate

ly d

eter

min

e ch

ange

in

soi

l wat

er, t

he e

ffect

s of

dee

p ro

ots

and

sens

or p

lace

men

t are

sm

all,

and

deep

dra

inag

e ca

n be

est

imat

ed a

dequ

atel

y.

Mea

sure

men

t is

reas

onab

ly

inex

pens

ive

and,

in p

rinci

ple,

re

pres

enta

tive

of th

e of

ten

crop

co

vere

d pl

ot in

whi

ch it

is

impl

emen

ted,

but

dis

turb

ance

dur

ing

inst

alla

tion

of s

oil w

ater

sen

sors

and

de

ep ro

ots

exte

ndin

g be

low

the

mea

sure

men

t dep

th c

an n

egat

ivel

y in

fluen

ce th

e m

easu

rem

ent,

and

deep

dra

inag

e is

har

d to

est

imat

e.

Plot

sca

le

(ass

umed

repr

esen

tativ

eat

fiel

d sc

ale)

Varie

s w

ith q

ualit

y of

impl

emen

tatio

n bu

t err

ors

of

~5–

15%

like

ly

achi

evab

le w

ith

TDR

or n

eutr

on

prob

es; (

soil

capa

cita

nce

and

cond

uctiv

ityse

nsor

s no

t yet

ac

cura

te e

noug

h.




● Measuring solar radiation: either estimated from cloud cover (reported by

an observer, or from a Campbell-Stokes recorder that focuses the Sun’s rays

to burn a card when skies are clear); or measured using a Thermoelectric

pyranometer (from the heating it induces on a blackened surface), or a

Photoelectric pyranometer (from the electrical output, from a silicone voltaic

detector, that it generates).

● Measuring net radiation: either measured as the difference in temperature

between blackened surfaces using a thermopile in a net radiometer, or by

measuring all four components of net radiation separately using two

pyranometers to measure (upward and downward) solar radiation and

two pyrgeometers to measure (upward and downward) longwave

radiation.

● Measuring soil heat flux: measured using the temperature difference across

a soil heat flux plate (a disk a few centimeters in diameter, a few millimeters

thick, with thermal conductivity similar to soil) inserted typically ~5 cm

below ground, with thermometers above to estimate flux loss in the soil

between the surface and plate.

● Micrometeorological measurement of latent and sensible heat: two

techniques remain in common use – both involve deploying sensors meters

or tens of meters above the ground.

— Bowen ratio/energy budget method. H and lE are deduced by simul-

taneously measuring (a) all the other components of the surface energy

budget, to determine the sum of H and lE; and (b) the gradients of

temperature (strictly virtual potential temperature) and humidity (often

measured as vapor pressure), to determine the ratio of H to lE. Individual

values are calculated from the sum and ratio of the two. Sometimes, when

gradients are small, accuracy is improved by interchanging sensors

between levels or ducting air from different levels alternately to a common

sensor.

— Eddy correlation method. H and lE are deduced by multiplying the

instantaneous fluctuation in vertical wind speed with the instantaneous

fluctuation in the volumetric heat content of the air to give the

instantaneous value of H, and with the instantaneous fluctuation in

the volumetric latent heat content of the air to give the instantaneous

value lE. Time-average values are found by integrating these

instantaneous flux values. Rapid response sensors with stable

calibration are required, these typically being interrogated several

times per second.

● Integrated water loss measurement of evaporation: involves defining a sam-

ple of the evaporating surface for which all the water entering or leaving can

be measured over the sampling period. Common measurements include the

following:



— Evaporation pans. The measured evaporation from a sample of water

held in a prescribed container (e.g., the ‘Class A’ pan) may be used as an

indication of that from nearby surfaces such as crops and lakes, but

requires correction using an empirical pan coefficient in the range 0.3–1.0

(but usually 0.65–0.85).

— Watersheds and lakes. Evaporation may be inferred from the measured

water balance of watersheds and lakes, but this is difficult because it is

deduced as a small residual in a water balance in which other terms

dominate, so errors may be 20–30%.

— Lysimeters. Evaporation is determined from detailed measurements of

the water balance for a sample of soil and vegetation that is representative

of its surroundings, but ensuring this can be a challenge. High quality

weighing lysimeters are considered the (albeit expensive) standard

against which alternatives can be validated.

— Soil moisture depletion. The measurements of precipitation and the

change in the soil moisture stored in the soil profile (measured using

neutron probes, capacitance probes, time-domain reflectometers, etc.)

can be used to estimate evaporation.

References

Campbell Scientific (1987) Bowen Ratio Instrumentation, Campbell Scientific, Logan, Utah,

online at: http://www.campbellsci.com/documents/manuals/bowen.pdf.

Fairmount Weather Systems (2010) Meldreth, Hertfordshire, UK, online at: http://www.

fairmountweather.com/.

Federer, C.A. & Tanner, C.B. (1966) Sensors for measuring light available for photosynthe-

sis. Ecology, 47, 654–7.

Kipp and Zonen (2010) Delft, The Netherlands, online at: http://www.kippzonen.

com/?category/111/Home.aspx.

LI-COR Environmental (2010) Lincoln, Nebraska, online at: http://www.licor.com/env/

Products/Sensors/200/li200_description.jsp.

McCaughey, J.H. (1981) A reversing temperature-difference measurement system for

Bowen ratio determination. Boundary-Layer Meteorology, 21 (1), 47–55.

Shuttleworth, W.J. (1993) Evaporation. In: Handbook of Hydrology (ed. D. Maidment),

pp. 4.1–4.53. McGraw-Hill, New York.


Introduction

Among the many acronyms used by scientists arguably ‘GCM’ is currently the best

recognized and most widely used, not only by meteorologists and hydrometeor-

ologists but also by other scientific disciplines, politicians and policymakers, and

by members of the public. Many non-experts think the acronym is shorthand for

‘Global Climate Model’ because it is widespread interest in global climate change

that has fostered its popularity. But the acronym predates this widespread interest

and in reality it is shorthand for General Circulation Model.

This chapter is written to provide an introduction to this aspect of present-day

atmospheric science for hydrometeorologists who will not necessarily become

experts in the specialized field of atmospheric modeling, but who need to be

familiar with the basic nature of GCMs and their strengths and shortcomings.

Because changes in the hydroclimate of the Earth are predicted, hydroclimatolo-

gists and hydrometeorologists need a basic understanding of the models used to

make such predictions.

What are General Circulation Models?

General Circulation Models are complex computer programs written to

describe how the air in the atmosphere moves, or circulates, around the globe.

To do this they include in their code the equations that describe the conserva-

tion and movement of momentum, energy, and the mass of atmospheric con-

stituents (including water vapor) which are discussed in later chapters. They

8 General Circulation Models




General Circulation Models 97

also include equations describing the processes that transfer momentum,

energy, and mass into and out of the atmosphere from oceans and continents,

and the energy incoming to the atmosphere as shortwave radiation from the

Sun and outgoing as longwave radiation to deep space. Aspects of these

processes were discussed in earlier chapters. GCMs also include in their

code equations that describe the evolution with time of atmospheric constitu-

ents. This may require description of chemical reactions but much more

commonly it requires that the phase changes for atmospheric constituents are

described, the description of the phase changes of water being the most

important need.

When describing the computational mechanics that GCMs use to describe the

atmosphere, it is helpful first to draw an analogy with the computer programs

that are set up to manage bank accounts. GCMs manage atmospheric bank

accounts for volumes of atmosphere each of which is defined by the area of the

Earth’s surface it overlies (specified as lying between latitude and longitude limits)

and by a height range above the ground. Although not strictly cubes, one might

visualize these volumes as being equivalent to bricks which when cemented

together form the whole atmosphere. The bank accounts maintained by the GCM

for each volume of atmosphere contain wealth in several different ‘currencies’, the

most important being the energy, momentum, and mass of atmospheric

constituents.

Extending the bank account analogy one step further, at regular intervals a

banking program would update the bank accounts it maintains to reflect changes

in the amount held in each currency in each account as a result of financial

transactions between them, and the interest earned on deposits and value lost by

inflation. In a broadly similar way, having built this basic descriptive framework

for the atmosphere, at regular intervals GCMs represent how much energy,

momentum, and mass is moved between the volumes of atmosphere it has defined

and the extent to which there are internally generated increases or decreases

within the volume.

Figure 8.1 illustrates how a GCM might partition the atmosphere into

segments and some of the exchanges and internal processes that are represented

in the model. The size of the areas of atmosphere represented in each column

of air, which is usually called the grid scale of the GCM, and the height range of

each volume within each column are mainly determined by computational

constraints. The computer memory available to store values of atmospheric

variables for each volume element in part determines the size of the volume

elements selected. However, just as important, if the computer program is to

remain stable, the time step at which updates are made to the variables in each

volume element falls rapidly as the grid scale and height ranges decrease.

Hence, the run time of atmospheric simulations also rapidly increases as the

grid scale and height ranges decrease. In practice, for stable operation a time

step of 20–30 minutes is required when GCMs have a grid scale of a few hundred

kilometers.



IN THE ATMOSPHERICCOLUMN

Wind vectorsHumidityCloudsTemperatureHeight

Vertical exchangebetween levels

AT THE SURFACEGround temperature,

water and energyfluxes

Horizontal exchangebetween columns

Figure 8.1 Partition of the

atmosphere in GCMs in

Cartesian coordinates and the

exchanges and internal

processes represented in the

model. (From Henderson-

Sellers and McGuffie, 1987,


How are General Circulation Models used?

General Circulation Models (GCMs) are currently used in three main ways, their

application having evolved historically as atmospheric scientists have come to

realize additional ways in which they can be used. GCMs were originally developed

in support of Numerical Weather Prediction (NWP), their purpose in this

application being to provide a physically realistic description of the circulation of

the atmosphere that is responsible for moving weather systems across the Earth’s

surface. National and international weather forecasting centers continue develop-

ing and applying GCMs extensively for this important purpose. The process used

is essentially one of extrapolation. Weather forecast centers first use as much

observational data as they can routinely obtain to help define a measurement-

influenced description of the atmosphere, much of this data being obtained via the

World Weather Watch (WWW) system managed by the World Meteorological

Organization (WMO). This becomes the initial state specified in the GCM when

used for NWP, with updating of initial states and subsequent weather predictions

based on each initiation typically being made at six hourly intervals.

There are always observational errors in measured variables and these must be

recognized when the initial model state is defined, otherwise the GCM will

become unstable when run forward in time. To cope with this issue, the process

known as four-dimensional data assimilation (4DDA) is used. This involves



making a compromise between the model-calculated fields defined by the GCM at

the time of the initiation (which are, of course, consistent with the equations used

in the GCM), and any observations available at that time. The values of the weather

variables specified in the initial state of the GCM thus become a weighted average

of that predicted by the GCM based on an earlier initiation and those currently

observed. Plausible values are assumed for errors in the observational data and

model calculated values, and these are used to define the weighted average initial

fields. If the observed values are substantially different to model calculated values

they are rejected as implausible and not used in the initiation.

The GCM is then run forward in time for some days ahead and in this way

numerical predictions of actual weather made for the future. In practice, there are

always shortcomings in the description of the atmosphere represented in the GCM

not least because the representation is made at grid scale using approximate

equations that parameterize processes that occur at much smaller scale.

Consequently, the accuracy of weather predictions degrades with time ahead.

Currently GCMs used for weather prediction can make reasonable predictions for

a few days ahead, and some weather centers attempt forward look predictions for

as long as 8–10 days ahead. Thus, when used for NWP, GCMs extrapolate observa-

tions forward in time using a physically realistic representation of atmospheric

circulation to predict the actual future weather for periods of several days ahead.

Some time ago it was realized that the process of NWP could provide an

important byproduct. The six hourly initiation process using data assimilation

calculates values of the meteorological variables used in the GCM everywhere in

the atmosphere. These values are in significant part based on model predictions,

but they are at least in part influenced by observations. Globally available fields of

weather variables cannot be routinely obtained at regular six hourly intervals in

any other way. But they are needed, and for this reason model-calculated data

based on the assimilation of observations into GCMs have now become widely

used as a source of data in their own right. Thus, the initiation fields of GCMs used

for NWPs are often stored and can be made available as a data source which has

the advantage of being broadly consistent with the laws of atmospheric physics (to

the extent these are represented in the GCM), while also reflecting relevant

observations available at the time of initiation to the extent allowed by the data

assimilation process.

One feature of such model-calculated data when provided as time series is that

the amount and nature of observational data that influence the data product

through data assimilation can change with time. Consequently, the relative

influence of observations versus model in the data also changes with time. Over

the years weather forecast centers have sought to use more and more observations

to better define the initial states used for their forecasts, and the amount of remote

sensing data so used has also greatly increased. As a result more recent

model-calculated data are arguably a better reflection of reality. An important

shortcoming of model-calculated data arises because weather forecast centers are

always striving to improve the realism with which atmospheric processes are

represented in their model. Consequently, improved versions of the GCMs used



for NWP are therefore adopted and applied. Because the data delivered by data

assimilation is influenced by the nature of the GCM used, this means there are

discontinuities in the derived global data sets associated with model changes. For

this reason, some major modeling centers have deliberately created model-

calculated data sets by applying consistent data assimilation procedures to

historical observational data using the same recent model to provide more

consistent data series. These data are called reanalysis data. Thus, when GCMs are

used to create model-calculated reanalysis data, a consistent data assimilation

procedure is used with the same version of the GCM to calculated globally available

data sets of weather variables from historical data records.

As experience in the use of GCMs in NWP grew, confidence also grew that

although their ability to predict actual weather decayed after several days, their

ability to make reasonable predictions of the statistics of weather, i.e., climate, at a

particular place was retained. This confidence sparked a whole new branch of

atmospheric science, namely climate prediction. When GCMs are used in this way,

they are used to predict a sequence of weather events which, it is assumed, is

representative of those that will occur, or that would occur given prescribed

changes in the factors that influence weather. Among the factors that may change

future climate are the concentration of gases that are involved in the absorption

and emission of radiation in the atmosphere and the nature of the vegetation

covering land areas of the globe.

Thus, climate prediction involves running GCMs for longer periods than weather

predictions, at least for seasons and commonly for several years, decades, centuries,

and even millennia. An initial global field of weather variables must still be provided

when a GCM is used for climate prediction and different patterns of simulated

weather sequences will result for different initiation fields. For this reason, climate

predictions are now usually made with several different initiation fields selected

(perhaps randomly) to be typical from those made available by operational NWP.

The resulting suite of predictions given by a GCM with different initiation fields is

called an ensemble. The mean of such an ensemble might be adopted as the average

climate prediction for the GCM, and the variability of the ensemble considered a

measure of the GCM dependent accuracy of the prediction. Thus, when GCMs are

used for climate prediction the purpose is to simulate the average statistics of weather

across the globe over long periods and to quantify the changes induced in these statistics

in response to prescribed changes in the influences that determine weather.

How do General Circulation Models work?

Sequence of operations

The operational sequence used in running GCMs is illustrated in Fig. 8.2. As

described in the previous section, the first step is to define an initial state for all of

the variables whose evolution is being simulated using 4DDA.



The equations coded in the GCM are then applied sequentially in two groups at

each time step. During the first set of calculations the processes that control changes

in modeled variables within each simulated volume (such as radiation divergence,

phase changes, and the input or loss of energy, momentum, and mass at the top and

bottom of each atmospheric column) are held constant. The conservation laws and

ideal gas law are applied to compute how energy, mass, and momentum are

re-distributed between the many volume elements over the time step and how

the equivalent meteorological variables describing the state of the atmosphere, the

state variables, are altered as a result. Applying this set of equations with fixed flux

divergences at each grid point is sometimes called solving the dynamics.

In the next step in GCM operation the state variables are held constant and the

processes that give rise to internal changes in state variables (the divergence terms)

are calculated in anticipation of their application during the next time step. Making

these calculations is sometimes called calculating the physics although some of the

processes described may actually be chemical or biological in nature. If the effect

of changing influences on weather are being investigated, imposed changes in, for

example, the concentration of radiatively active gases or the representation of

surface vegetation are imposed while calculating the physics. In the real world,

equations involved in solving the dynamics and solving the physics apply simulta-

neously rather than in sequence, and the need to apply them in sequence in a

GCM run is a compromise which can give rise to model instability. For this reason

it is usually necessary to apply some form of smoothing procedure to the

divergences calculated during the physics calculations before the next time step.

This two stage sequence of calculations, i.e., first solving the dynamics and then

calculating physics, is then repeated successively as the GCM runs forward in time

Initiationi.e. using 4DDA to define initial values of state variables globally

Solving the dynamicsi.e. calculating updates values of the state variables by solving

conservation laws assuming fixed values of divergences

Calculating the physicsi.e. calculating updates values of divergences for each volume

element assuming fixed state variables

Smoothing divergences

Stop and output

i.e. applying smoothing to maintain model stability

Run timeoutputs

Figure 8.2 Sequence of

operations during a GCM run.



until the model run reaches some predefined stopping point. Selected calculated

fields are output as the model proceeds that provide the required description of

the evolving atmosphere.

Solving the dynamics

GCMs use two different ways to store state variables. The first is that illustrated in

Fig. 8.1 in which the state variables are stored as individual values of atmospheric

variables for each of the defined atmospheric volume elements as specified by lines

of latitude and longitude using (currently 0.5° to 5°) grid scale and the (currently

5–25 m) vertical height ranges. When this is done, state variables are said to be

stored on a Cartesian Grid.

The alternative way to store the values of state variables is on a Spectral Grid, see

Fig. 8.3. When this is done, instead of storing individual values of atmospheric

Transformation to grid space samples fieldaround zones of Iatitude and longitude

Each atmospheric layer heldand moved in spectral space

Spectral truncationrestricts information

Each surface is transformedinto sampled grid space

representation

sp nplatitude

longitude

Vertical exchangesin grid space

0� 360�

Surface fluxes arecomputed in grid space

(2)

(1)

(3)

(4)

Figure 8.3 Representation of

the atmosphere in GCMs in

spectral coordinates and the

interchange Cartesian

coordinates required to

calculate the physics. (From

Henderson-Sellers and

McGuffie, 1987, published

with permission.)



variables, the GCM stores information on the global pattern of state variables at

each level in the atmosphere in the form of a Fourier series. In this series, each

term has a wavelength an integral multiple of which corresponds to the distance

around the Earth. By expressing the longitudinal distribution of state variables in

this way between time steps, it is possible to capture most (but not all) of the spatial

variability using less computer storage by truncating the Fourier series after a

specified number of wavelengths. The number of wavelengths before truncation

varies with the application of the GCMs, longer climate prediction runs having

truncation earlier than shorter NWP runs. When using a spectral grid, movement

within each vertical layer of the atmosphere can be calculated in spectral space and

this computation can be efficient. However, vertical movement and calculating the

physics for each atmospheric column requires transposition of the state variables

into coordinate space. Using a spectral grid is most effective when describing state

variables that vary smoothly across the globe. Truncating the Fourier series in a

spectral description can give rise to artificial wavelike features and unrealistic

divergences and, as computer memory becomes more available, the technical

advantages of using a spectral description of state variables become less

significant.

Calculating the physics

Calculation of the physics is made for each atmospheric column extending from

the surface upward to a defined level, which is regarded as being the top of the

atmosphere. The GCM code contains subroutines that compute the divergence of

energy, momentum and the mass of atmospheric constituents, the surface

exchanges of these variables, and buoyant exchanges between cells at different

levels in this column. Typically GCMs will include at least the following

subroutines.

● Radiation scheme: The radiation transfer scheme calculates the propagation

and reflection of shortwave and longwave radiation through each layer in the

atmospheric column from the modeled profiles of temperature, humidity,

cloud amount and the concentration of aerosols, ozone, and radiatively

active gases such as water vapor and carbon dioxide.

● Boundary-layer scheme: The boundary-layer scheme uses first order, height-

integrated representations of the surface energy and momentum exchanges

between the lowest model level represented in the GCM and the ground,

based on the modeled vertical gradients of the variables which control these

exchanges.

● Surface-parameterization scheme: The surface-parameterization scheme pro-

vides a description of shortwave and longwave exchange, momentum cap-

ture, and how the available energy at the surface is shared as heat fluxes

for sea, sea-ice, land-ice, snow-covered land, and several snow-free

land-cover types.



● Convection scheme: The convection scheme computes the exchange of buoy-

ant thermals and compensation flows between modeled layers, and calcu-

lates the amount of convective precipitation generated as these thermals rise

and cool.

● Large-scale precipitation scheme: The large-scale precipitation scheme

calculates if each of the modeled layers has become saturated and removes

any excess water as rain, if the air temperature is greater than freezing point,

or as frozen precipitation, if the air temperature is less than freezing point.

Intergovernmental Panel on Climate Change (IPCC)

The Intergovernmental Panel on Climate Change (IPCC) is a scientific

intergovernmental body tasked to evaluate the risk of climate change caused by

human activity. The panel was established in 1988 by the World Meteorological

Organization (WMO) and the United Nations Environment Programme (UNEP),

two organizations of the United Nations. The IPCC shared the 2007 Nobel Peace

Prize.

The IPCC does not carry out research, nor does it monitor climate or related

phenomena. A main activity of the IPCC is publishing special reports on topics

relevant to the implementation of the UN Framework Convention on Climate

Change, an international treaty that acknowledges the possibility of harmful

climate change. The IPCC bases its assessment on the most recent scientific,

technical and socio-economic information produced worldwide relevant to the

understanding of climate change. IPCC reports are widely cited in almost any

debate related to climate change.

The IPCC First Assessment Report appeared in 1990 with a supplementary

report in 1992, and there have been subsequent reports in 1995, 2001 and 2007.

The key conclusions of the most recent IPCC report (IPCC, 2007) are as

follows:

● Warming of the climate system is unequivocal.

● Most of the observed increase in globally averaged temperature since the

mid-twentieth century is very likely due to the observed increase in anthro-

pogenic (human-caused) greenhouse gas concentrations.

● Anthropogenic warming and sea-level rise would continue for centuries due

to the timescales associated with climate processes and feedbacks, even if

greenhouse gas concentrations were to be stabilized, although the likely

amount of temperature and sea-level rise varies greatly depending on the

intensity of fossil fuel burning during the next century.

● The probability that this is caused by natural climatic processes alone is less

than 5%.

● World temperatures could rise by between 1.1 and 6.4°C (2.0 and 11.5°F)

during the twenty-first century, and:



— sea levels will probably rise by 18 to 59 cm (7.08 to 23.22 in).

— there is a confidence level >90% that there will be more frequent warm

spells, heat waves and heavy rainfall.

— there is a confidence level >66% that there will be an increase in droughts,

tropical cyclones and extreme high tides.

● Both past and future anthropogenic carbon dioxide emissions will continue

to contribute to warming and sea-level rise for more than a millennium.

● Global atmospheric concentrations of carbon dioxide, methane, and nitrous

oxide have increased markedly as a result of human activities since 1750 and

now far exceed pre-industrial values over the past 650,000 years.

In IPCC statements ‘most’ means greater than 50%, ‘likely’ means at least a 66%

likelihood, and ‘very likely’ means at least a 90% likelihood.


● What are General Circulation Models (GCMs)? Computer programs which

describe atmosphere circulation using equations that portray conservation

and movement of atmospheric constituents, incoming solar radiation and

outgoing longwave radiation, and transfers between the atmosphere and

oceans and continents. Currently GCMs describe changes in the atmosphere

every 20–30 minutes at points separated by 10s–100s of kilometers in the

horizontal and by 10s-1000s of meters in the vertical.

● How are GCMs used? To give:

— numerical weather prediction (NWP) in which an observation-based

description of the current atmosphere is extrapolated forward for several

(typically 3–8) days to calculate the actual weather likely to occur over

this period.

— globally-available atmospheric ‘data’ provided by GCMs as a byproduct of

the process necessary to define the initial state of the atmosphere prior to

an NWP run, with four-dimensional data assimilation used to define a

weighted average description between that previously predicted by the

GCM and all relevant observations available at the time of initiation.

— climate prediction in which it is assumed that, although they are not able

to predict actual weather for more than a few days, GCMs are able to

predict the statistics of weather, i.e., climate, for periods up to many cen-

turies ahead.

● How do GCMs Work? Once an initial state of the atmosphere has been

defined, equations coded in the GCM are applied at each time step in two

groups sequentially;

— the dynamics, equations that control the model’s state variables solved

assuming fixed values for variables (such as flux divergences) that deter-

mine changes in state variables; and



— the physics, equations that calculate the controlling values to be applied in

the subsequent dynamics, these being calculated using the most recently

updated state variables.

The dynamics are then again solved and the physics re-calculated, and so on

until the run is complete.

● Solving the dynamics: GCMs use two ways to store and solve state variables,

either as individual values for each volume elements stored in a Cartesian

Grid, or on a Spectral Grid as a (truncated) series of Fourier coefficients that

describe the longitudinal variation in state variables.

● Calculating the physics: the divergence of state variables is calculated

throughout the atmospheric column, along with the buoyant exchanges

between cells at different levels and the surface exchanges of state variables

using (at least) the following subroutines: (a) Radiation scheme; (b) Boundary-

layer scheme; (c) Surface-parameterization scheme, (d) Convection scheme;

and (e) Large-scale precipitation scheme.

● Intergovernmental Panel on Climate Change (IPCC): an intergovernmental

body tasked with evaluating the risk of climate change caused by human

activity based on the most recent scientific, technical and socio-economic

information produced worldwide relevant to the understanding of climate

change. The 2007 IPCC Assessment Report stated that warming of the

climate system is unequivocal, and most of the observed increase in globally

averaged temperatures since the mid-twentieth century is very likely due to

the observed increase in anthropogenic (human-caused) greenhouse gas

concentrations.

References

Henderson-Sellers, A. & McGuffie, K. (1987) Climate Modelling Primer. Wiley & Sons,

Chichester, UK.

IPCC (2007) The Intergovernmental Panel on Climate Change, World Meteorological

Organization, Geneva 2, Switzerland, online at http://www.ipcc.ch/


Introduction

The purpose of this chapter is to give general insight into why hydrological

catchments located in different regions of the world have characteristically

different hydrometeorological context. Although there is always very substantial

variability in the atmosphere, the air in the troposphere undergoes large-scale

circulation which is on average organized at the global scale. The global patterns

created lead to distinguishable regional differences in hydroclimate. Such

organization happens because there are large-scale influences on atmospheric

behavior that have a discernable consequence.

As mentioned in the previous chapter, the process of four-dimensional data

assimilation, by means of which available weather observations are merged into

GCMs, has allowed synthesis of time series of model-calculated fields of atmos-

pheric variables that are available globally. The availability of these fields facilitates

a fusion of existing and more recent understanding of large-scale atmospheric cir-

culation. Much of the description of atmospheric behavior given in this chapter

draws on knowledge provided by this useful byproduct of the regular and repeated

initiation of the GCM at numerical weather prediction centers.

Global scale influences on atmospheric circulation

Differential heating by the Sun is the primary cause of the general circulation of the

atmosphere. The spatial pattern of atmospheric heating is greatly influenced by the

relative geometry of the Sun and the Earth and how this changes with time, but it is

also influenced by regional differences in the terrestrial controls involved in

transferring solar energy into the atmosphere. Cartoons in Fig. 9.1 illustrate the

9 Global Scale Influences on Hydrometeorology




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Global Scale Influences on Hydrometeorology 109

most important controls on atmospheric circulation described below. In approximate

descending order of importance, these controls can be classified as follows.

Planetary interrelationship

Latitudinal differences in solar energy input

The Earth rotates around an axis that is (when averaged over the whole year)

perpendicular to the plane in which the Earth moves around the Sun. Consequently,

the solar energy received per unit area is, on average, greatest at the equator and

least at the poles. This difference causes atmospheric circulations that transfer

energy within the moving atmosphere from low to high latitude.

Seasonal perturbations

Because the axis of rotation of the Earth on any particular day is not perpendicular

to the plane in which the Earth moves around the Sun (see Fig. 5.7 and Fig. 9.1),

the latitude where there is most and least solar radiation changes with season. This

results in persistent regular seasonal changes in circulation patterns.

Daily perturbations

The rotation of the Earth means there is a regular diurnal cycle in the longitude at

which there is maximum input of solar radiation. At any latitude, the magnitude

and timing of this daily cycle of energy changes with season.

Persistent perturbations

Contrast in ocean to continent surface exchanges

On average about half of the solar energy reaching the Earth enters the atmosphere

from the Earth’s surface, mainly in the form of surface latent and sensible heat

fluxes. Because water is freely available at the ocean surface but not necessarily at

continental surfaces, there is a characteristic difference in the relative magnitude

of these two different fluxes for these two surfaces. The average aerodynamic

roughness of oceans is also less than that of continents. This contrast in surface

exchanges of energy and momentum influences regional hydroclimate.

Continental topography

Atmospheric circulation mainly occurs in the troposphere. In some locations the

height of continental topography is of the same order as the depth of the troposphere

and this can influence regional flow patterns to some extent, particularly when

topography is organized in mountain chains that lie roughly perpendicular to

atmospheric flow (e.g., the Rockies and Andes).



Temporary perturbations

Perturbations in oceanic circulation

Oceans cover about two-thirds of the globe and are strongly coupled to the

overlying atmosphere through sea-surface temperature. Regular changes in sea-

surface temperature that occur annually in response to solar heating influence the

frequency of tropical storms. Changes at longer timescales such as occur in El

Niño and La Niña events or during the Pacific Decadal Oscillation (PDO) are

associated with consequential shifts in atmospheric circulation that influence

terrestrial hydrometeorology.

Perturbations in atmospheric content

Some important impacts of activity on continental surfaces arise indirectly through

associated changes in atmospheric constituents that participate in the process that

control radiation transfer through the atmosphere. Erupting volcano and natural or

human-induced atmospheric pollution can alter aerosol concentrations and

influence regional and global hydroclimate by altering the absorption of solar

radiation. Changes in hydroclimate also occur in response to natural or human-

induced changes in the concentration of the radiatively active gases such as carbon

dioxide that control the transfer of longwave radiation through the atmosphere.

Perturbations in continental land cover

The general contrast between the surface exchanges of oceanic and continental

surface can be modified by changes in continental land cover. Such changes alter

the albedo and hence solar radiation capture, the aerodynamic roughness of the

surface and hence momentum capture, and the partition of available energy

between latent and sensible heat fluxes. Changes in land cover may occur naturally

in response to changing climate, or they may result from large-scale intervention

that alters the vegetation present over land areas, such as deforestation.

Latitudinal imbalance in radiant energy

As discussed in Chapter 5, all the radiant energy entering the Earth system from

the Sun is within a spectrum that is determined by the temperature of the Sun and

is mainly in the wavelength range of 0.15 to 4 μm. The intensity with which solar

radiation enters the top of the atmosphere is strongly determined by latitude, with

more arriving at the equator and less at the poles. On the other hand, most of the

energy leaving the Earth system is in the longwave, mainly in the wavelength

range of 3 to 100 μm. The spectrum and amount of outgoing radiation is

determined by the temperature of the surface of the Earth and overlying

atmosphere at the latitude at which the longwave radiation leaves. The temperature



of the Earth falls from the equator toward the poles but the latitudinal variation in

the amount of radiant energy leaving as longwave radiation is much less than

the latitudinal variation in the amount of energy received as shortwave radiation,

see Fig. 9.2.

On average across the surface of the globe, there is a near perfect balance

between incoming and outgoing radiant energy. At low latitudes, however, there is

more radiant energy incoming as shortwave radiation than is leaving as longwave

radiation; consequently surplus energy is available. At high latitudes the reverse is

true, when averaged over the year there is more outgoing radiant energy as long-

wave radiation than incoming radiant energy as shortwave radiation. This discrep-

ancy causes the atmospheric general circulation, because to support the persistent

latitudinal imbalance in radiant energy transfer, energy must be moved from low

latitudes to high latitudes in the atmosphere and oceans.

Lower atmosphere circulation

Latitudinal bands of pressure and wind

When mariners first began to travel the globe using sailing ships, they soon realized

that, on average, there are characteristic patterns of wind flow at different latitudes,

and they learned how to exploit these in their travels. Close to the equator they found

regions with little wind where progress was difficult under sail. In these regions

the ensuing state of inactivity could make the travelers dull, listless and depressed.

90900

50

100

150

Wat

ts m

−2200

250

300

350

Wat

ts m

−2

0

50

100

150

200

250

300

350

705070 50 40 30

North South

Net shortwave

Surplus heat energy transferredby atmosphere and oceans

to higher latitudes

Surplus

Def

icit D

eficit

Net longwave

Latitude

403020 2010 100

Figure 9.2 Balance

between average net

shortwave and

longwave radiation

from 90° North to

90° South.



They called these regions the Doldrums. North or south of the Doldrums the winds

became steadier and were more consistently from the east: southeasterly winds

north of the equator and northeasterly winds south of the equator. These winds were

important because their consistent presence meant international trade using sailing

ships could flourish and they became known as the Trade Winds. At about 30° on

either side of the equator, again there was sometimes little wind. These areas were

called the horse latitudes, because here ships could become becalmed and, on occa-

sion, any horses on board were thrown overboard in order to conserve precious

water. Farther from the equator still, at latitudes near 60°N and 60°S, average winds

became strongly westerly. Pressure shows similar time-average consistent tendencies

with latitude, with low pressure at the equator, high pressure at 30°N and 30°S, low

pressure again at 60°N and 60°S, and higher pressure at the poles, see Fig. 9.3.

Hadley circulation

In the early 1700s, George Hadley, an English lawyer and amateur meteorologist,

argued that solar heating creates upward motion of equatorial air and air from

Mid-latitude cell

Mid-latitude cell

Polar cell

Polar cell

Westerlies

Inter-tropicalconvergence

30�N

30�S

60�S

60�N

0�

Westerlies

High pressure

High pressure

Low pressure

Northeasterly trades

Southeasterly trades

Hadley cell

Hadley cell

Figure 9.3 Variation in time-average wind speed and pressure with latitude.



neighboring latitudes must flow in to replace it and that the easterly component of

the trade winds was associated with the rotation of the Earth. The idea of

symmetrical cells on either side of the equator became accepted but it was later

realized that in fact there are several cells on either side of the equator, see Fig. 9.3.

However, because the latitude of maximum solar heating is strongly seasonal,

there is a strong seasonality in the Hadley cell and circulation is only briefly

symmetrical at the spring and fall equinoxes. Otherwise, the dominant latitudinal

circulation comprises three main cells, with the strong equatorial Hadley cell being

of central importance, see Fig. 9.4. There is a strong, single summer Hadley cell in

each hemisphere with the location of rising air shifting with the thermal equator.

This is balanced by falling air in the winter hemisphere. The regular seasonal shift

in the location of rising and falling air between the two hemispheres means the

circulation pattern appears symmetrical as an annual average.

Mean low-level circulation

Outside the tropics, the rotation of the Earth is a fundamental influence on atmos-

pheric general circulation, see Fig. 9.5a. Near-surface wind flow, which would oth-

erwise be purely pressure driven, includes circular flows induced by the Coriolis

force. In the northern hemisphere, the circulation induced is clockwise around

regions of high pressure and counterclockwise around regions of low pressure. In

the southern hemisphere the circulation sense is reversed, i.e., circulation is coun-

terclockwise around high pressure and clockwise around low pressure. In the

60S 60N10

8

6

4

2

30S 30N

FERREL FERRELHADLEY HADLEY

(a)

Pre

ssur

e (m

b 10

3 )

00

00

0

84

12 16

−2

[V](DJF)

−1 −2

(b)

60S 60N10

8

6

4

2

30S 30N

Equator

FERREL FERRELHADLEY HADLEY

Pre

ssur

e (m

b 10

3 )

0

0

0 02

−4−8

−12

−16

0

[V](JJA)

−1

2

Figure 9.4 Mean latitude

average circulation of the

atmosphere (a) December to

February, (b) June to August.

Values on the streamlines are

total mass circulation

between that streamline and

the zero streamline. (From

Rassmusson et al.,1993,




annual average flow pattern (Fig. 9.5a) this influence is most obvious over the

oceans which are aerodynamically smoother than the continents, with annual

average circulation around the high pressure zones apparent at 30°N and 30°S. But

it is possible to distinguish circulation around low pressure at 60°N.

There is a pronounced seasonal variation in the location and strength of the

circulation associated with the Coriolis force within the general pattern of

atmospheric circulation as the location of the overhead Sun moves north and

south. The greater presence of continental surfaces in the northern hemisphere

also has an impact on the seasonal variations in circulation and pressure,

18090S

60S

6060

60

40

40

40

40

−40

−100

−40

−40

−40

−40

−40−80

−40

−20

−20

0

0 0

0

00

20

20

30S

30N

60N

90N

EQ

120W 120E 18060W 60E

Latitude

Insufficient data

Annual average streamflow and pressure

0

(a)

Insufficient data

90S

60S

30S

30N

60N

90N

EQ

40

40

−40

−80

−80

60

−100100

−20

−20

−20 −40

40

0

0

0

00

0

0

0

040

40

40

100 100

Northern hemisphere winter

HighLow

Monsoonflow

(c)

0

0

0

0

000

60

80

−80

−60

−40

−20

−20

−120

60

20

20

60 60

Insufficient data

90S

60S

30S

30N

60N

90N

EQ

40

40

40

Northern hemisphere summer

High Low

Monsoonflow

(b)

Figure 9.5 Atmospheric streamflow (each barb = 2 m s−1) and mean sea level pressure difference (in mb) across the globe

(a) as an annual average, (b) in the northern hemisphere summer, and (c) in the northern hemisphere winter.

(Redrawn from Peixoto and Oort, 1992, published with permission.)



see Fig. 9.5b and c. In the northern hemisphere summer the oceanic subtropical

highs are farther north and more intense and there are regions of low pressure

over the warm continents which contribute to the creation of monsoon flows.

In the northern hemisphere winter there is a reversal in the pressure differences

between the oceans and the continents with pronounced low pressure regions

in the northern seas reflecting the more persistent presence of storms in this

region.

Mean upper level circulation

Higher in the troposphere, at about 20 kPa or 10 km, the atmospheric circulation

intensifies and simplifies (Fig. 9.6). The most prominent features are bands of

strong westerly winds in both hemispheres in the subtropics and middle latitudes

where the tropospheric jet streams are found. There is some seasonality in this

pattern (not shown), with intensification of the flow in the winter hemisphere. In

the northern hemisphere there is a tendency for the upper level winds to have a

90N

60N–400

200

600

400

400

–2000

–600

–600

600

–800

–400

a

00

30N

EQ

30S

60S

90S180 120W 60W 0

Latitude

60E 120E 180

Figure 9.6 Annual average atmospheric streamflow (each barb = 5 m s−1) at 200 mb (10 km). (From Peixoto and Oort,

1992, published with permission.)



wave number of two and to move toward the equator as they flow over continents,

and away from the equator as they flow over oceans. This tendency is more

pronounced in the northern hemisphere winter. Within this average flow pattern,

the jet stream is continually developing, meandering, and decaying giving relatively

less consistent lower-level winds. The jet streams have a fundamental influence on

hydrometeorological variability.

Ocean circulation

As mentioned in Chapter 1, the oceans are the principal source of ‘memory’ in the

hydroclimatic system because of their capability to store and subsequently release

large amounts of heat energy. The thermal structure of the oceans is fundamentally

different to that of the atmosphere. Energy from the Sun is captured near the ocean

surface and some warms the atmosphere from below. As a result, air nearer the

ocean surface tends to be warmer and lighter in daytime conditions, and the

atmosphere is unstable and well-mixed throughout the boundary layer as a result.

On the other hand, solar energy warms the oceans from above, so the temperature

of the water is higher nearer the surface and the surface layer of the ocean is stable.

Some mixing of energy downward does occur as a result of the turbulence caused

by surface winds but the depth to which this occurs is limited. Consequently, the

oceans are divided into a mixed layer (typically between 100 m and 1000 m deep)

which is separated from the deep ocean below by the thermocline, i.e., the steep

negative temperature gradient that gives a stable interface between these two layers

and suppresses mixing, see Fig. 9.7.

At low latitudes, solar heating is strong and fairly constant through the year, and

the stability of the thermocline is able to keep the mixed layer fairly shallow with

limited seasonality in its depth. In middle latitudes the strength of the solar heating

changes seasonally. In the hemispherical summer, the mixed layer is again fairly

shallow as at low latitudes. However, in the hemispherical winter solar heating is

less, so the distinction between the mixed layer and deep ocean is reduced; because

the temperature gradient and stability of the thermocline is therefore less, surface

winds can mix warmer surface water to greater depth.

Sea-surface temperature (SST) provides an important lower boundary

condition on the atmosphere over the ocean surfaces that cover 70% of the globe.

As discussed in Chapter 1, there is a strong coupling between oceans and

atmosphere because of the effective exchange of energy fluxes and momentum.

Since air near open water is close to saturation, the surface temperature of the

ocean defines not only the lower boundary condition for sensible heat transfer

but also for latent heat transfer. The isotherms of annual average SST run roughly

east-west across the large oceans, from greater than 29°C at the equator to almost

–2°C near the poles where the presence of salt in the ocean depresses the freezing

point, see Fig. 9.8. However, isotherms are modified near continents in response

to wind-driven currents in the upper few hundred meters of the ocean. In the

most southerly latitudes of the southern hemisphere (not shown in Fig. 9.8), the



strong and persistent westerly winds and absence of continental barriers give rise

to the Antarctic circumpolar current which is continuous around the globe.

Elsewhere in the Atlantic Ocean and Pacific Ocean, away from the equator,

anticyclonic gyres occur which involve currents that carry warm water toward

the poles on the eastern side of continents and currents that take cooler water

away from the poles on the western side of continents. At the equator the narrow

currents are east-west but these can change in strength rapidly in response to

influences such as the El Niño Southern Oscillation phenomena described later.

2000

0 10 20 30�C 0 10 20�C

1000

0Surface Surface

Temperature �C

Mixedlayer

Mainthermocline

zone

Seasonalthermocline(Summer)

Low latitudes Mid latitudes

Winter

Upperzone

Deepzone

Dep

th (

m)

Figure 9.7 Typical ocean

temperature profiles for

tropical and temperate

regions. (Redrawn from

Picard and Emery, 1982,

published with

permission.)

60E40S

20S

20Agu

hius

Som

ali

(Sea

sona

lR

ever

sal)

Kuros

hio

E. A

ustr

alia

n

Per

u

Guiana

Can

ary

Bra

zil

S. Eq. S. Eq.

California

20

24

28

28

8

28

28

2726

16

12

24

24

20

2420

26

26

27

16 16

2420

1612

86

29

16

20N

40N

60N

EQ

60W 0120E 120W180

Ben

guel

a

Gul

f Stre

am

Figure 9.8 Annual average

sea-surface temperatures and

extratropical ocean surface

currents. (From Rassmusson

et al.,1993, published with

permission.)



In addition to the wind-driven surface currents, the thermohaline circulation is

caused by changes in density associated with the temperature and salinity of the

sea water. This originates at the poles as vertical flow that sinks to the middle

ocean or lower where it becomes horizontal. It originates as a result of density

increases likely due in part to direct cooling, but also as a result of increased salin-

ity occurring when sea water freezes and ejects additional salt into nearby saline

water. Although the downward branches of the thermohaline circulation are

largely restricted to high latitudes, the compensating upward branches occur more

widely across the globe. The thermohaline cycle is very slow, typically 1600 years,

but there is evidence that more rapid changes occur at time periods of decades

with an associated influence on climate at this time scale.

Oceanic influences on continental hydroclimate

Monsoon flow

The seasonal change in near-surface air temperature is markedly different between

oceanic and continental surfaces. Oceans have a large heat capacity and efficient cou-

pling with the atmosphere and this has the effect of moderating the seasonal cycle in

the temperature of the overlying air. The heat capacity of land surfaces is less so the

seasonal cycle can be much larger. Figure 9.9 shows isotherms of the difference

between the near-surface air temperature in January relative to that in July. Although

there are reductions of around 30°C in air temperature over the Arctic Ocean, else-

where changes in near-surface air temperature over oceans are on the order of 5–10°C.

However, the mid-continental reduction in near-surface air temperature from July to

January is more typically on the order of 40–50°C. Notice that the penetration of

maritime air into the European continent is greater than it is into the North American

continent presumably because there is less topographic obstruction to onshore flow

and the seasonal difference in near-surface air temperature is less as a result.

One important consequence of the substantial difference in temperature

between summer and winter over land surfaces relative to that over adjacent ocean

is the occurrence of monsoon air flow in the tropics. The largest monsoon system

is the Asian-Australian monsoon system (Fig. 9.10) which, recognizing the popu-

lation distribution of the world, is arguably therefore the hydroclimate phenome-

non that has most impact on humankind. In the northern hemisphere winter

season, there is a low level flow of dry, cool air from cold continent to the warmer

ocean and precipitation over the land is low. However, during the northern hemi-

sphere summer season the direction of flow is reversed, and warm, moist air flows

from the ocean over the warm land where the resulting upward motion of the

heated air produces heavy precipitation during the monsoon season. Monsoonal

flows of a broadly similar nature occur elsewhere in West Africa and in the North

American Monsoon System (NAMS), the latter being responsible for much of the

summer rain falling in northern Mexico and the southwestern USA.



Tropical cyclones

Tropical cyclones are the most energetic transient weather systems in the tropics.

They are areas of intense convergence which, when mature, have a central core

which is considerably warmer than the surrounding air. They are designated tropi-

Figure 9.9 Global

distribution of change in

near-surface air temperature

between July to January in °C.

(From Peixoto and Oort,


permission.)

180 180120 12060W

20

−20−20−10

10

00

15

−10−20

−30

−40

−20−30

−30

−40

−50

−5

5

−5

10

1010

5

5

55

5

90S

60

30

30

60

90NDifference (�C) January minus July

EQ

60E

Latitude

0

60E

28

28 28

27

27

27

28

28

28

28

27

February

August

120E 120W 60W

40N

20N

20S

40S

EQ

40N

20N

20S

40S

EQ

0180

60E 120E 120W 60W 0180

Figure 9.10 Monsoon component of atmospheric circulation showing the departure of the monthly mean surface

circulation during February and August from its annual mean and 27°C and 28°C sea-surface isothermals for the

midsummer month in each hemisphere. (From Rassmusson et al.,1993, published with permission.)



cal storms when they have sustained winds of ∼18 m s−1 and hurricanes or typhoons

(the name changes regionally) when sustained winds reach 33 m s−1 or greater.

For tropical cyclones to generate, three conditions must occur simultaneously.

Together these restrict the oceanic regions over which tropical cyclones can

originate. The first requirement is that the ocean must be sufficiently warm: SSTs

of at least 26–27°C are required. Second, because significant rotations can only be

generated where there is a significant Coriolis force, tropical storms cannot form

within 5–8° of the equator even if the SST is high enough. Finally, a small change

of wind with height is required if the storm is to persist. Together these three

criteria restrict the potential source areas for tropical cyclones to those shown in

Fig. 9.11. Once established, tropical cyclones generally move westward and toward

the poles. When they reach land they can have a major and usually detrimental

impact before eventually decaying because they no longer have access to the latent

heat energy they extract from warm oceanic waters.

El Niño Southern Oscillation

The annual average sea-surface temperature distribution shown in Fig. 9.8 reveals

that equatorial SSTs are significantly modified by wind-driven oceanic currents. In

particular, the Peru current brings cold polar waters from the Antarctic to the

equator and SST in the eastern Pacific is lower than it would otherwise be. The

trade winds blowing across the Pacific support an easterly equatorial surface cur-

rent and the waters warmed by solar radiation during this transit gather to form

the western Pacific warm pool (Fig. 9.12a). Because the sea surface temperature in

the warm pool is higher than elsewhere in the equatorial Pacific, there is more

convection in the atmosphere above, and higher precipitation and latent heat

release. On average, the prevalent atmospheric ascent in this region draws in air,

including air from across the Pacific. The trade winds are thus enhanced, and an

unstable equilibrium is established in equatorial Pacific air-sea interactions char-

acterized by warmer water and more convection in the west Pacific and cooler

water and less convection in the east Pacific.

60E 120E

27.5�C(Feb)

27.5�C(Feb)

27.5�C(Feb)

27.5�C(Aug) 27.5�C(Aug) 27.5�C(Aug)

120W 60W

40N

40S

20N

20S

EQ

0180

Figure 9.11 Primary regions where tropical storms are initiated and their subsequent primary paths together with the

27.5°C sea-surface temperature isotherm for August. (From Rassmusson et al.,1993, published with permission.)



From time to time there is less upwelling of cold water in the eastern Pacific and

therefore a relative warming of the SST in this region. This phenomenon is called

El Niño, from the Spanish for ‘the little boy’, referring to the Christ child, because

the phenomenon is usually noticed around Christmas off the west coast of South

America. Different theories have been advanced for why the upwelling of cold water

in the eastern Pacific is suppressed but as yet no definitive explanation as been

established. However, when this surface warming in the western Pacific occurs,

there is greater atmospheric convection locally and the resulting advection of air to

support this ascent moderates the strength of the trade winds. This causes equatorial

Pacific surface temperature pattern to be less extreme, with less warm water

concentrated in the warm pool and more warm water farther east. The presence of

El Niño tends to be self-supporting for a period of 6 to 18 months because the

anomalously warm water and reduced trade winds leads to moderation of the

equatorial current that would otherwise drive warming water westward (Fig. 9.12b).

Normal conditions

El Niño conditions

Increasedconvection

Convective loop

120°E

Equator

80°W

120°E

Equator

(b)

(a)

80°W

Figure 9.12 Water

temperatures in the Pacific

Ocean during (a) non-El

Niño and (b) El Niño years.

(From http://www.cotf.edu/

ete/modules/elnino/crwhatis.

html.)



The opposite phenomenon when the SST in the eastern Pacific is colder than

average is called La Niña, which means ‘the little girl’. The east-west movement

in Pacific SSTs that occurs in El Niño and La Niña conditions necessarily results in

shifts in the position of maximum convection in the equatorial Pacific, and this in

turn gives rise to large-scale changes in the general circulation of the atmosphere.

By correlating the changes in regional climate at locations around the world that

are generated by such changes in atmospheric flow with observed fluctuations in

the SST in the tropical Pacific, statistical relationships have been developed that

can be used to make seasonal predictions in those regions when El Niño and

La Niña have influence.

Pacific Decadal Oscillation

The Pacific Decadal Oscillation (PDO) is a long-lived El Niño-like pattern of

Pacific climate variability, see Fig. 9.13. While the two climate oscillations are simi-

lar in that they have spatial climate fingerprints, they have very different behavior

in time. Two main characteristics that distinguish PDO from El Niño/Southern

Oscillation (ENSO) are:

● persistence: during the twentieth century PDO events persisted for 20 to 30

years while typical ENSO events persisted for 6 to 18 months; and

● location of impact: the climate fingerprints of the PDO are most visible in the

North Pacific and North America with secondary signatures in the tropics,

but the opposite is true for ENSO.

Several studies have found evidence for two full PDO cycles in the past century.

Cool PDO regimes prevailed from 1890 to 1924 and again from 1947 to 1976,

while warm PDO regimes dominated from 1925 to 1946 and from 1977 through

the mid-1990s and beyond.

−0.6

−0.2

0.0

0.2

0.4

0.8PDO

Warm phase Cool phase

Figure 9.13 Typical

wintertime Sea Surface

Temperature (colors), Sea

Level Pressure (contours) and

surface wind stress (arrows)

anomaly patterns during

warm and cool phases of


(From http://jisao.

washington.edu/pdo/.)



North Atlantic Oscillation

Westerly winds blowing across the Atlantic bring moist air into Europe. In years

when westerlies are strong, summers are cool, winters are mild and rain is fre-

quent. If the westerlies are suppressed, the temperature is more extreme in sum-

mer and winter leading to heat waves, exceptional frosts and reduced rainfall.

A permanent low-pressure system over Iceland, called the Icelandic Low, and a

permanent high-pressure system over the Azores, called the Azores High, control

the direction and strength of westerly winds into Europe. The relative strengths

and positions of these pressure systems vary from year to year and this variation is

known as the North Atlantic Oscillation (NAO), see Fig. 9.14. A large difference in

the pressure at the two stations, i.e., a year when the NAO index is high, leads to

increased westerly winds and, consequently, cool summers and mild, wet winters

in central Europe. When the NAO index is low, the westerly winds are suppressed

and central Europe has cold winters with the storm tracks farther south, toward

the Mediterranean Sea, resulting in more storm activity and rainfall in southern

Europe and North Africa.

Water vapor in the atmosphere

The global distribution of water vapor in the atmosphere and how that

concentration is changing with time is the most important hydroclimatological

consequence of atmospheric general circulation because it reflects the regional

surface water balance. Over the ocean, and over many land areas where water is

readily available, the specific humidity of air near the surface is strongly linked to

near-surface air temperature and changes from about 18 g kg−1 at the equator to

Figure 9.14 The North Atlantic Oscillation is the anomalous difference between the polar low and the subtropical high

during the winter season (December through March). (a) A positive NAO index phase corresponds to a stronger than usual

subtropical high pressure center and a deeper than normal Icelandic low, resulting in more and stronger winter storms

crossing the Atlantic Ocean on a more northerly track. (b) A negative NAO index phase corresponds to a weak subtropical

high and a weak Icelandic low, resulting in fewer and weaker winter storms crossing on a more west-east pathway. (From

http://www.ldeo.columbia.edu/res/pi/NAO/.)



less than 1 g kg−1 at the poles. In desert regions where there is little water available

in the soil, the specific humidity is less (Fig. 9.15a). The total amount of water in

the whole atmospheric column (in kg m−2), which is sometimes (wrongly) called

the precipitable water, is, in numerical terms, approximately proportional to the

vertically integrated specific humidity. Figure 9.15b shows that precipitable water

also decreases from the equator to the poles and is higher over oceans than land.

This figure also shows that the penetration of moister maritime air into continents

90N

60

a

(a)

30

EQ

30

60

90S180 120 60W 0 60E 120 180

90N

60

30

EQ

30

60

90S

90N

60

30

EQ

30

60

90S180 120 60W 0 60E 120 180

gm kg–1

gm kg–1

Near-surfacespecific humidity

18

10

5 54.9

4.5

(b)

4.5

Vertical averagespecific humidity

11.5

22.5

3

44.54.5

4

3

4

1.52.5

21.5

4.5

1

0.50.25

1

12 11

43

2

43

5

2

46

1012

1416

1614

12

8

6

6

6

4

2

1818

18

8

18

1410

8

Figure 9.15 Global

distributions of annual

average (a) near-surface

specific humidity in units

of g kg−1; and (b) vertical

average specific humidity

in units of g kg−1.

(Redrawn from Peixoto

and Oort, 1992, published

with permission.)



in westerly airstreams at middle latitudes is influenced by topographic features

such as the Rocky Mountains and Andes Mountains.

Figure 9.16a shows atmospheric uplift of moisture (and associated cooling) that

is large in the Hadley cell at the equator but also in mid-latitude low pressure sys-

tems. Downward transport is greatest in the subtropical anticyclones. Figure 9.16b

shows areas with large-scale moisture convergence in the atmosphere (where pre-

cipitation is in excess of evaporation) are associated with the river basins of the

world’s major rivers.

Figure 9.16 Global

distributions of annual

average (a) vertical transport

of water vapor at the 85 KPa

level in units of 10−6 kg m−2 s−1,

negative values indicate

transport upwards; and

(b) horizontal water vapor

transport in units of 0.1 m

yr−1. (Redrawn from Peixoto

and Oort, 1992, published

with permission.)




● Influences on atmospheric circulation: are in three general classes:

— Planetary interrelationship, primarily:

Solar energy input (solar radiation is greater at the equator than the poles)

Seasonal perturbations (latitude of maximum solar radiation changes

with season)

Daily perturbations (the daily cycle in solar radiation changes with season)

— Persistent perturbations, primarily:

Ocean versus continent (energy return to the atmosphere differs with

surface)

Continental topography (high relief can interfere with tropospheric flow)

— Temporary perturbations, primarily:

Oceanic circulation (temporary fluctuations in SST alter atmospheric flows)

Atmospheric content (changing air content alters atmospheric radiation

transfer)

Land cover (large-scale land-cover change alters surface energy balance)

● Latitudinal radiation imbalance: atmospheric and oceanic circulation must

occur to redress the imbalance between the strong latitudinal variation in

incoming solar radiation and the weaker latitudinal variation in outgoing

longwave radiation loss.

● Lower atmosphere circulation— Latitudinal bands of pressure and wind: on average, pressure is low at the

equator with southeasterly (northeasterly) trade winds north (south) of

the equator; there is high pressure and less wind at 30°N and 30°S, but

stronger westerly winds at 60°N and 60°S.

— Hadley Circulation: there is a strong, single summer Hadley cell with

rising air located at the shifting thermal equator with falling air in the

winter hemisphere.

— Mean low-level circulation: Coriolis force gives clockwise (counterclock-

wise) circulation around high (low) pressure in the northern hemisphere

most noticeable over oceans around high pressure at 30°N and 30°S. The

circulation is reversed in the southern hemisphere. There is a seasonal

shift in the circulation pattern as the thermal equator moves north-south

that is different between hemispheres because the northern hemisphere

has a larger land area.

— Mean upper-level circulation: at ∼10 km atmospheric circulation intensi-

fies with strong westerly winds in both hemispheres in the subtropics and

middle latitudes where the tropospheric jet streams continually develop,

meander and decay giving less consistent lower-level winds and hydro-

meteorological variability.

● Ocean circulation— Thermal structure: oceans have a (∼100–1000 m deep) mixed layer

separated from the deep ocean by the thermocline, a steep negative

temperature gradient.



— Sea-surface temperature (SST): isotherms run roughly east-west from

∼29°C at the equator to -2°C near the poles, but are modified near conti-

nents by wind-driven surface currents carrying warm water toward

(cooler water from) the poles on the eastern (western) side of

continents.

— Thermohaline circulation: originates near the poles (from increased

salinity due to sea water freezing) as vertical flow that becomes horizontal

in the middle/lower ocean, with compensating upward branches widely

spread across the globe: it is very slow (∼1000s years) but has decadal

changes that influence climate.

● Oceanic influences on continental hydroclimate— Monsoon flow: difference in near-surface air temperature in winter gives

flow of dry, cool air from a colder continent to a warmer nearby ocean,

but in summer warm, moist air flow from ocean to warmer land where

ascent gives heavy precipitation.

— Tropical cyclones: originate over ocean that has SST >26–27°C and is at

least 5–8° north and south of the equator and generally move westward

and toward the poles giving a major detrimental impact if they reach

land.

— El Niño–Southern Oscillation (ENSO): Pacific trade winds support easterly

equatorial flow and warmed waters form an eastern Pacific warm pool

above which there is convection, with ascent in part supported by

enhancing the trade winds. From time to time there is relative warming

of the normally cooler SST in the eastern Pacific (El Niño) with greater

atmospheric convection locally. The resulting advection moderates the

strength of the trade winds, hence there is more warm water farther east

and El Niño tends to be self-supporting for a period of 6 to 18 months.

The shift in the center of convection has consequences on climate globally.

The opposite phenomenon when the SST in the eastern Pacific is colder

than average is La Niña.

— Pacific Decadal Oscillation (PDO): a long-lived El Niño-like pattern of

Pacific climate variability that persists for 20–30 years and which has

most impact in the North Pacific and North America.

— North Atlantic Oscillation (NAO): a variation in the strength and position

of the Icelandic Low and the Azores High which control the direction and

strength of westerly winds in the Atlantic which in turn modify the

seasonal climate of Western Europe.

References

Peixoto J.P. & Oort, A.H. (1992) Physics of Climate. Springer-Verlag, New York.

Picard, G.L. & Emery, W.J. (1982) Descriptive Physical Oceanography, 4th edn. Pergamon

Press, New York.

Rassmusson, E.M., Dickinson, R.E., Kutzbach, J.E. & Cleaveland, M.K. (1993) Climatology.

In: Handbook of Hydrology (ed. D. Maidment) pp. 2.1–3.1. McGraw-Hill, New York.


Introduction

On average, about 60% of the Earth is cloud-covered. Clouds are extremely

important in hydrometeorology because when they move in the atmosphere they

transport substantial amounts of water from one location to another. If present,

they also have a major impact on the absorption of solar radiation and modify the

surface energy balance and thereby the input of water and energy into the

atmosphere from below. Tall cumulus and stratocumulus clouds tend to shade and

inhibit solar radiation reaching the ground, but high altitude cirrus cloud tends to

blanket and inhibit the loss of longwave radiation. In seeking to predict climate

change, one of the biggest challenges is to predict how any additional water

evaporated into the atmosphere will be expressed in terms of modified cloud

cover – will that cloud reduce solar radiation input or increase the retention of

outgoing longwave radiation?

Three things are required for clouds to form. One is the presence of moisture

in the air in sufficient quantity to result in cloud if the air is cooled. This is often

the limiting constraint on cloud formation in arid and semi-arid regions such as

the southwestern US and northern Mexico. It is difficult to condense water from

air unless there is something for the water to condense on to, so a second

possible constraint on cloud formation is a lack of cloud condensation nuclei

(CCN), i.e., entities in the atmosphere around which condensation can begin.

The third important need for cloud development is a mechanism by means of

which air can be cooled sufficiently to allow condensation of water vapor. In

practice, the cooling required is usually associated with atmospheric movement,

as discussed next.

10 Formation of Clouds




Formation of Clouds 129

Cloud generating mechanisms

Differential buoyancy in the air, and the convective thermals that result, provide

an important mechanism by means of which air can be cooled and clouds

formed. The process involved is comparatively simple (Fig. 10.1a). A parcel of

unsaturated air whose temperature is higher than the air that surrounds it is

buoyant and rises. As it does so, it cools. As long as it remains buoyant relative

to the surrounding air, it continues to rise; if it cools enough to become saturated,

Figure 10.1 Mechanisms for

atmospheric cooling to give

cloud: (a) convective ascent;

(b) horizontal movement of air

masses; and (c) near-surface

mixing of saturated air with

different temperatures.

Temperature = T2Mixing ratio = r

T2 ≥ Tz; but r = rsat (at T2)

(b)

Cooler air massor

topographyTemperature = T1

Mixing ratio = rT1 ≥ T0; but r < rsat (at T0)

Temperature ofsurrounding air = T0

Temperature ofsurrounding air = Tz



T1 ≥ T0; but r < rsat (at T0)

T2 ≥ Tz; but r = rsat (at T2)

i.e., the air is warmer thanthe surrounding airand is unsaturated

i.e., the air is still warmer thanthe surrounding air

but is now saturated

(a)

Buoyant ascent

(c)

Cool moist air

Mixed moist air

Warm moist air Saturated

TC �C0

0

2esat (

KP

a)

4

6

8

20 40



cloud formation begins. Once cloud formation is underway and water vapor is

being converted to liquid water or ice, the latent heat so released further

increases the temperature of the parcel helping it to remain buoyant and thus

aiding further ascent and cooling. The cloud created in this way is called

convective cloud.

A second important way that vertical movement and associated cooling can

give rise to condensation and cloud is as a result of large-scale atmospheric

movement (Fig. 10.1b). Moving masses of air may have different temperatures and

different buoyancy. Thus, a more buoyant warm air mass moving horizontally

may, for example, impinge on and ‘float’ over a cooler air mass, or a less buoyant

cold air mass moving horizontally may impinge on a warmer air mass and ‘sink’

beneath it, pushing it upward. Thirdly, air moving horizontally may sometimes

come up against topography and be forced upward. In these examples it is the

large-scale horizontal movement of the atmosphere that is resulting in ascent and

cooling of the air, the net result being that unsaturated air moves upward, cools

and becomes saturated to give cloud.

Another way that wisps of cloud can be formed is very different because it

doesn’t involve ascent. If two portions of air that are both close to saturation but

with different temperatures are mixed, the resulting air mixture will have the

average temperature and average water content. However, because the saturated

vapor pressure curve shown in Fig. 2.2 is not linear, the resulting water content of

the mixed air may be greater than that of saturated air at the new average

temperature. Consequently, water condenses out (Fig. 10.1c). This mechanism is

not a major source of cloud but is the reason why we see sea frets and wisps of

cloud over wet forest after rain.

These different mechanisms for atmospheric condensation result in very

different types of cloud, which are subsequently associated with different patterns

of rainfall, as follows.

● Convective cloud: Because the mechanism involved in cloud formation is

associated with strong vertical ascent over fairly small horizontal areas, the

resulting cloud can be quite tall with horizontal dimensions typically on the

order of a few hundred to a few thousand meters. Moreover, being associated

with local rapid ascent, any precipitation associated with clouds of this type

tends to be heavy and local.

● Frontal cloud: The nature of the processes involved in the formation of frontal

clouds means the resulting cloud cover tends to have wide spatial coverage,

typically over areas on the order of a few kilometers to several tens of

kilometers or greater. Being associated with gradual uplift, the associated

precipitation is often widespread and steady but is usually lighter than that

for convective clouds.

● Surface mixing: The wisps of clouds produced by mixing saturated air with

different temperatures have very limited extent and are not usually associated

with precipitation.



Cloud condensation nuclei

There are always numerous small particles, called aerosols, suspended in the atmos-

phere. Some are important because they act as the condensation nuclei required for

cloud droplet formation. Aerosols are classified according to their size as:

● Aitken Nuclei, if they have diameter in the range 0 to 0.2 μm;

● Large Nuclei, if they have diameter in the range 0.2 to 2 μm;

● Giant Aerosols, if they have diameter greater than 2 μm.

Typically the concentration of aerosols in the troposphere is on the order of 1012

per cubic meter but it can be as high as 1014 per cubic meter downwind of pollution

sources. Higher in the atmosphere, aerosol concentration falls off to about 1010 per

cubic meter at 15 km. The concentration of aerosols falls off rapidly with the

size of the aerosols so the concentration of Aitken nuclei greatly exceeds that for

giant nuclei.

Aerosols originate in many different ways but their origin can be broadly classi-

fied according to whether they result from natural phenomena or human activity.

The main source of aerosols and the amount of aerosols these sources produce

annually are listed in Table 10.1. On average natural sources are dominant,

Table 10.1 Sources of atmospheric aerosols and estimates of

the amount of aerosols produced in tonnes per year.

Natural phenomena (tonnes per year)

Airborne Sea salt 1000Gas to particle conversion 570Windblown dust particles 500Natural forest fires 35Meteoric debris 20Volcanoes (a highly variable source) 25Total from natural sources >2150

Human activities (tonnes per year)Gas to particle conversion 275Industrial processes 56Fuel combustion (stationary sources) 44Solid waste disposal 3Transportation 3Miscellaneous 28Total from human activities ∼410



although locally aerosol cloud condensation nuclei concentration can be greatly

enhanced by industrial activity and agricultural practice, including the burning of

agricultural crops and forests.

Saturated vapor pressure of curved surfaces

Chapter 2 discussed how evaporation involves two processes, the escape of

water molecules from a liquid surface when they have sufficient energy to

overcome the forces that bind them, and the capture of a proportion of the

molecules above the surface when they collide with it. These two competing

exchanges achieve equilibrium when the air above a water surface is saturated and

the vapor pressure of the air at saturation depends on the binding energy of the

molecules in the water. The lower the binding energy, the greater is the number of

molecules in the water with sufficient energy to escape and the greater is the boil

off rate. If the boil off rate is higher, the equilibrium between rates occurs when the

concentration of water vapor in the air is higher and consequently the saturated

vapor pressure is higher.

If the evaporating water surface is curved, molecules leaving the surface are on

average farther away from the molecules that remain in the water, and the effective

binding force is therefore reduced, see Fig. 10.2. Consequently, water molecules

escape more easily from a curved surface and the saturated vapor pressure of air

above a curved surface is higher as a result. The increase in saturated vapor pres-

sure above a curved surface with radius r relative to the saturated vapor pressure

above a flat water surface is desat

, where:

=−

2vsat

w v

Sder

rr r

(10.1)

where rw and r

v are the densities of water and water vapor, respectively, and S is

the surface tension of water. Therefore, in principle, the difference in saturated

vapor pressure goes to infinity as the radius of the curved surface goes to zero.

Figure 10.2 The difference

in binding force on escaping

water molecules above a

curved and flat water surface

and its consequences on

saturated vapor pressure.

Escaping molecule

More escape and capture

Higher saturated vaporpressure

Lower saturated vaporpressure

Less molecular bondsfor a curved surface

Less escape and capture



The form of Equation (10.1) means that the saturated vapor pressure at which

condensation on to a surface can occur is less for surfaces with larger radius than

it is for surfaces with smaller radius.

Cloud droplet size, concentration and terminal velocity

Because saturated vapor pressure is higher over curved surfaces, condensation in

clouds is preferentially onto the larger aerosols, i.e., onto the fraction of aerosols that

are most rare, particularly if they are hydrophilic. Because of this, the cloud droplet

density in clouds is typically a thousand times less than the density of atmospheric

aerosols, i.e., about 109 per cubic meter rather than 1012 per cubic meter.

The ratio of the area of a cloud particle to its volume is greater for smaller

particles than for larger particles. As a result, if growth is only by condensation

(as opposed to by collisions), smaller drops increase their diameter more quickly

than the larger drops. Hence, not only does the mean value of the particle diameter

increase with time in a cloud, but also the spectrum of diameters present in the

cloud narrows because the smaller diameter particles grow quicker.

Table 10.2 shows typical values for the radius of aerosol particles, cloud

condensation nuclei, cloud droplets and raindrops in a cloud, their likely

concentration, and terminal velocity. For comparison, the order of magnitude of

uplift in developing clouds is 1 m s−1.

How do clouds work? Clouds occur when atmospheric uplift causes air to cool,

saturate, and condense water vapor onto cloud condensation nuclei. The smallest

cloud droplets or ice particles are swept upward by the uplift while still growing.

When they have grown larger they can hang, apparently stationary, in the rising air

because their terminal velocity is similar to the uplift. A fraction of these water

droplets or ice particles may become large enough to fall out of the cloud but these

quickly evaporate. Eventually, the cloud particles become larger still, large enough

Table 10.2 Typical values for the radius, concentration and terminal velocity of the

airborne entities present in clouds.

Airborne entity

Radius (mm) Concentration

(per m3) Terminal

velocity (m s-1)

Atmospheric aerosols 0-2 1012 Various but smallTypical cloud condensation nuclei

0.1 109 10−7

Typical cloud droplet 10 109 10−2

Large cloud droplet 50 106 ∼0.27Cloud/rain borderline droplet

100 – ∼0.7

Raindrop 1000 103 ∼7



to have a terminal velocity such that they fall through the cloud despite uplift,

colliding with and capturing water from the smaller particles as they do so. The

resulting large cloud particles so formed which fall out of the cloud may then

be big enough to reach the ground as rain or snow before they can evaporate.

In strongly convective conditions, uplift in the cloud can become so rapid that it is

hard for particles to fall against it. As a result, they continue to grow until they

eventually become much larger and can fall out of the cloud to reach the ground,

usually as heavy rain.

Ice in clouds

Broadly speaking there are three types of clouds which are distinguished by the air

temperature in which they exist, i.e., cold clouds when the air temperature is –40°C

or less, made up almost entirely of ice particles; warm clouds when the air

temperature is greater than 0°C, made up almost entirely of water droplets; and

mixed clouds between these two temperatures which are a mixture of ice particles

and water droplets, see Fig. 10.3.

The ice particles present in mixed and cold clouds can be formed or grow in

different ways, the most important being:

● Spontaneous nucleation: at low temperatures (< –40°C) a chance aggregation

of enough water molecules can spontaneously freeze into an ice particle;

● Heterogeneous nucleation: ice grows around a pre-existing ice nucleus

inside a water droplet;

● Contact nucleation: a pre-existing supercooled water droplet makes contact

with a pre-existing ice particle and is captured to increase the size of the ice

particle; and

● Bergeron-Findeison process: water evaporated from water droplets freezes

directly on to pre-existing ice particle.

Figure 10.3 Temperature ranges and constituents for different cloud types.

Mixed clouds

Water droplets

Ice particle

−50�C −40�C −30�C −20�C −10�C 0�C 10�C

Cold clouds

Dominant in thehigher atmosphere

and at high latitudes

Dominant in thelower atmosphere

and at low latitudesSupercooled droplets

Warm clouds



The relative importance of these processes can influence the overall structure and

shape of the cloud and the extent and severity of any precipitation that may be

generated by them.

Cloud formation processes

Thermal convection

Whether cloud develops, and the extent and form in which it develops, depends

on the lapse rate of the advected atmosphere, i.e., on the Environmental Lapse

Rate (Ge). The Environmental Lapse Rate is the lapse rate of the incoming air mass

and so is location-specific and changes with time because it is determined by the

history of inputs and outputs to the atmosphere upstream. It might be measured

using a radiosonde attached to a balloon released into the atmosphere to provide

data for weather forecasts, including forecasts of cloud cover. The Environmental

Lapse Rate will usually include an inversion in potential temperature at the

tropopause which can stop the ascent of moist air and thus limits the vertical

extent of cloud. Near the ground the Environmental Lapse Rate is rarely equal to

(and is usually less than) the dry adiabatic lapse rate, G, this being the approximate

rate at which buoyant air cools when it ascends quickly, see Chapter 3.

The potential for cloud formation depends on the height dependent interrela-

tionship between three lapse rates, specifically:

● The environmental lapse rate, Ge – the (measured or model-calculated)

lapse rate of the air overlying the location of interest into which cloud might

be seeded.

● The dry adiabatic lapse rate, G – the rate at which air cools if is moved

upward in the atmosphere adiabatically (see Chapter 3).

● The moist adiabatic lapse rate, Gm

– the rate at which saturated air cools if it

is moved upward in the atmosphere adiabatically.

Recall that the dry adiabatic lapse rate (G = g/cp) is constant with height but that

the moist adiabatic lapse rate, Gm

, in addition to being smaller than G, increases

with temperature (see Equation 3.10). In practice, temperature decreases with

height inside a cloud, consequently the moist adiabatic lapse rate and therefore the

rate of cooling of the atmosphere increases with height.

Figure 10.4 illustrates the rate of cooling of a buoyant parcel of air (created by

surface warming) that is seeded into the atmosphere with three different

Environmental Lapse Rates which we refer to as Case 1, 2, and 3. In all three cases

the air parcel initially rises at the dry adiabatic lapse rate. In Case 1, the air is so dry

that the dry adiabatic rate intersects the environmental lapse rate before the air has

cooled enough to saturate, consequently no cloud develops. In Cases 2 and 3 the

air is initially moister. Consequently, the parcel rises and cools at the dry adiabatic



lapse rate until it reaches the lifting condensation level, i.e., the height at which the

temperature of the parcel has cooled enough for it to saturate. At this level cloud

formation begins.

In Cases 2 and 3 the parcel then continues to rise, but now being saturated

inside the cloud, it cools at the moist adiabatic rate. The rate of cooling is therefore

less, but the cooling rate gradually increases as the air temperature decreases with

height inside the cloud. These two cases are different because the environmental

lapse rates into which the buoyant moist air is seeded are different. In Case 2 the

moist adiabatic lapse rate intersects the environmental lapse rate before the tropo-

pause so ascent and cloud formation stops at this level. However, in Case 3 the

environmental lapse rate is such that the moist adiabatic lapse rate does not inter-

sect the environmental lapse rate before the tropopause. In Case 2 cloud develop-

ment is inhibited by loss of buoyancy, but in Case 3 tall cumulus tower clouds can

develop with a consequently greater potential to generate stronger uplift inside the

cloud and heavier precipitation.

Foehn effect

When a moist, unsaturated air mass moving horizontally impinges on a topo-

graphic barrier, the air is forced to ascend (Fig. 10.5a). If ascent is sufficiently

rapid, the moist air first cools at approximately the dry adiabatic lapse rate

Figure 10.4 The mechanism of ascent and cloud formation for the three different cases described in the text.

Virtual temperature Virtual temperature

Hei

ght a

bove

the

grou

nd

Hei

ght a

bove

the

grou

nd

Hei

ght a

bove

the

grou

nd

Case 2

Virtual temperature

Case 3Case 1

Dryadiabaticlapse rate

Moistadiabaticlapse rate

Moistadiabaticlapse rateEnvironmental

lapse rateEnvironmental

lapse rateEnvironmental

lapse rate



Liftingcondensation

level



(Fig. 10.5b) until it reaches the cloud condensation level when cloud starts to

form. It then continues rising and cooling at the moist adiabatic lapse rate,

giving more cloud and possibly precipitation preferentially on the windward

side of the barrier.

Once the air reaches the top of the topographic barrier, it begins to fall and

warm again and, being still in cloud, does so at the moist adiabatic lapse rate.

Assuming some moisture was lost as precipitation during the ascent, the water

vapor content (mixing ratio) of the air is now less than before the ascent.

Consequently, the air becomes unsaturated at a lower temperature and at a

higher level, so the cloud begins to dissipate with a higher cloud base down-

wind of the barrier than upwind of the barrier. Once it becomes unsaturated

the descending air continues to warm, now at the dry adiabatic lapse rate. The

net effect of this forced transit over a barrier is that some of the water vapor

originally in the air mass is precipitated mainly on the windward side, and the

air downwind of the barrier is ultimately drier and warmer than it was on the

upwind side.

Figure 10.5 (a)

Diagrammatic illustration of

the Foehn effect; (b) Cooling

and warming rates of moist,

unsaturated air forced to pass

over a topographic barrier

assuming some of the

moisture in the air is lost by

precipitation between the

upwind and downwind sides.

Moist airflow

Cloud development

Precipitation

Forced upliftgiving cooling at dryadiabatic lapse rate

Environmentallapse rate

Leewardcloud baseWindwardcloud base

Moistadiabaticlapse rate

Dry

adiabatic

lapse rate

Warmer,drier air

Warming at dryadiabatic lapse rate

Topographic barrier

Cooler,moist air

Height

Virtual temperature

Saturation level

(a)

(b)



In certain circumstances significant turbulent eddies can develop downwind of

a topographic barrier, and clouds may form at the crests of the waves or at the top

of the large-scale eddies so caused (Fig. 10.6).

Extratropical fronts and cyclones

These weather systems normally result from the interaction of two air masses with

characteristically different temperatures. The interface between two air masses is

called a cold front if colder air displaces and moves beneath warm air (Fig. 10.7a)

and a warm front if warm air displaces and moves over cold air (Fig. 10.7b). In cold

fronts the interface between the two air masses tends to be steeper than for warm

fronts so the resulting patterns of cloud and precipitation are less extensive in nature.

An approaching cold front is usually associated with increasing wind speed and

reducing pressure, accompanied by increasing cloud that becomes lower and

produces more precipitation as the front passes. If the warm air is unstable,

scattered heavy storms may occur. After passage of the front, the air temperature

falls sharply, pressure rises rapidly, the wind direction changes, and the weather

generally becomes cooler and drier.

Figure 10.6 Development

of wave clouds over and

downwind of a topographic

barrier showing as (a) the

mountain wave or helm

cloud, as (b) the subsequent

wave clouds, and as (c) the

rotor cloud.

Figure 10.7 Cross-section

of a typical (a) cold front and

(b) warm front. In both

cases the vertical scale is

exaggerated.

(a)

(b)

Cold air

Cold air

Warm air

Warm air



Warm fronts tend to move more slowly than cold fronts and they are usually less

well-defined because the interface is shallower. As the warm air moves over the

cold air, a broad band of cloud develops which may extend several hundred

kilometers ahead of the front and likely gives rise to precipitation. If the warm air

is moist and stable the precipitation increases gradually as the front approaches,

but if it is moist and unstable taller cloud will likely occur and often heavy storms

will result as well.

Because the interface between air masses tends to be unstable, it frequently

evolves into a spiraling stream as a result of the rotation of the Earth. The resulting

weather system is called a cyclone. Cyclones vary greatly but Figs 10.8 and 10.9

show a typical life cycle.

In the initial stages the winds on either side of the stationary front are blowing

in opposite directions (Fig. 10.8a). As a result of small disturbances in the shear or

other irregularities in surface roughness or surface heating, the front may gradu-

ally assume a wave-like shape (Fig. 10.8b), which may persist and increases in

amplitude. In due course, a frontal wave evolves with well-defined cold and warm

fronts that in the northern hemisphere has a counterclockwise flow pattern

(Fig. 10.8c). Because the cold front portion usually moves quicker than the warm

front portion, the cold front eventually overtakes the warm front to become an

occluded front (Fig. 10.8d), at which time the cyclone achieves maximum inten-

sity. Later the occlusion gradually disappears and a new cyclone may be formed.

Figure 10.8 Evolution of a typical northern hemisphere cyclone as illustrated by surface weather maps, with the wind aloft

shown parallel to isobars of constant pressure; (a) stationary front with wind shear across the interface; (b) and (c) a frontal

wave with growing amplitude; (d) warm front being overtaken by the cold front to create an occluded front.

(a)High

pressure

Warm air

Cool air

Highpressure

Lowpressure

Warm air

Cool air

(b)

(c)

Warm air

Cool air

A A´

B B´

Highpressure

Lowpressure

Warm air

Cool air

Cool air

C´

Highpressure

Lowpressure

(d)

C



Figure 10.9 illustrates typical cloud and precipitation behavior in the vertical

cross-sections along the lines A-A′ and B-B′ in Fig. 10.8c, and along C-C′ in

Fig. 10.8d. The cross-section along line B-B′ shows the expected behavior for the

two separated fronts moving to the right, while the line A-A′ shows the behavior

in a region where the warm air has risen and is still rising above the cold air. The

behavior shown in the cross-section C-C′ illustrates the behavior when the two

fronts have become occluded and cyclonic activity is at its peak.

Cloud genera

Classification of clouds is not straightforward. Cloud morphology and cloud

height are the usual basis for defining cloud genera. The form of cloud is indicated

by the names cumulus meaning ‘heaped’, stratus meaning ‘layered’, and cirrus

meaning ‘fibrous’. However, the designation nimbus, which indicates rain clouds,

is additionally used as a qualifier. Cumulonimbus, for example, describes a cloud

with a heaped form that produces precipitation, a designation that is almost syn-

onymous with a thunderstorm. Cumulus clouds may extend through the tropo-

sphere, and stratus, cirrus, and cumulus may have a height related prefix, such as

alto, which implies medium height. The main cloud genera currently used are

given in Table 10.3, along with some important cloud characteristics associated

Figure 10.9 Vertical

cross-section along (a) the

lines A-A′ in Figure 10.8c; (b)

the lines B-B′ in Figure 10.8c;

and (c) the line C-C′ in

Figure 10.8d.

Warm air

Cool airCool air

Warm air

Warm air

Cool air

Cool airCool air

Cool air

A-A´

B-B´

(a)

(b)

(c)

C-C´



with these genera. Sometimes an additional word of Latin origin is added as a

qualification of these genera, such as lenticularis meaning ‘lens-shaped’.


● Requirements for cloud: three things are required: (a) sufficient atmospheric

moisture to produce cloud if the air is cooled; (b) cloud condensation nuclei

(CCN) on which to condense water vapor; and (c) a mechanism to cool air

to generate condensation, this usually being some form of atmospheric

uplift.

● Cloud generation: the two main mechanisms that give the required atmos-

pheric uplift to cool air and produce cloud are:

— buoyant ascent of parcels of warmed moist air giving convective cloud

that is tall and has horizontal dimensions of order 100s to 1000s of

meters; and

— large-scale horizontal movement in the atmosphere that results in air

moving upward (a warm air mass rising over colder air; a cold air mass

pushing beneath a warmer air mass forcing it up; or air that is forced to

rise over topography) giving shallower frontal cloud with horizontal

dimensions of order 1 to 10s of km.

Near-surface mixing of saturated air with different temperatures can also

give small wisps of cloud as in sea frets and above rain-soaked forests.

● Cloud condensation nuclei: in air, there are typically ∼1012 m−3 aerosols that

are potential condensation nuclei for cloud droplets, these being mainly of

natural origin, but with local enhancements by human activities. Aerosol

concentration falls rapidly with diameter from Aitken Nuclei (<0.2 μm) to

Giant Aerosols (>2 μm)

Table 10.3 Cloud genera based on morphology and height of occurrence.

Category Type (abbreviation) Base height (km) Depth (km)

Typical base temperature (°C)

Water content (g m−3)

High Cirrus (Ci) 5–13 0.6 –20 to −60 0.05Cirrocumulus (Cc) 5–13 –20 to −60Cirrostratus (Cs) 5–13 –20 to −60

Medium Altocumulus (Ac) 2–7 0.6 –30 to +10 0.1Altostratus (As) 2–7 0.6 –30 to +10 0.1

Low Stratus (St) 0–0.5 0.5 –5 to +20 0.25Stratocumulus (Sc) 0.5–2 –5 to +15Nimbostratus 1–3 2.0 –15 to +10 0.5

Vertical Cumulus 0.5–2 1.0 –5 to +15 1.0 Cumulonimbus 0.5–2 6.0 –5 to +15 1.5



● Vapor pressure near curved surfaces: saturated vapor pressure above curved

surfaces is higher because water molecules can escape more readily from

them so condensation is easier onto droplets with larger radius.

● Droplet concentration: cloud droplet density is about 1000 times less than

aerosol density (∼109 m−3 versus ∼1012 m−3) because condensation is easier

onto larger radii aerosols that are much less common (see last two points).

● How do clouds work?: smaller cloud particles with lower terminal velocity

are swept upward or hang almost stationary in the atmospheric uplift in

clouds until some particles grow large enough to fall quickly through the

cloud, gathering water from smaller particles as they do so. These leave the

cloud and some may be big enough to reach the ground before evaporating.

● Cloud types: three types of cloud are distinguished by their temperature,

cold clouds (< –40°C), mixed clouds (–40°C to 0°C) and warm clouds (>0°C).

● Cloud formation: three important cloud formation mechanisms described

in the text are Thermal convection, the Froehn effect, and Extratropical fronts

and cyclones.

● Cloud genera: cumulus means ‘heaped’, stratus means ‘layered’, and cirrus

means ‘fibrous’, with nimbus indicating rain clouds used as a qualifier, and

alto and cirro respectively used as a prefix to indicate ‘medium’ and ‘high’.


Introduction

As discussed in the last chapter, clouds are made up of cloud droplets, ice particles,

or a mixture of the two, depending on the temperature of the cloud and therefore

on its height and location. It is very difficult for these basic constituents to reach

the ground because they are small in size and so have very low terminal velocity;

there is also often uplift in the cloud that counteracts the tendency they have to

fall. Even droplets or particles that are just large enough to fall out of the cloud are

likely to evaporate before they reach the ground, once they leave the saturated

atmosphere inside the cloud.

Moreover, cloud particles do not grow rapidly enough by simple condensation

to fall as precipitation. Some other growth mechanism is required. As discussed in

more detail below, there are two general ways cloud particles can grow more

quickly. The first is via some form of collision. The name given to growth processes

that involve collision is different depending on the phase of the cloud particles

involved. When water droplets collide with other water droplets and grow as a

result, the process is called coalescence. This can occur in both warm clouds and

mixed clouds. When ice particles grow by colliding with other ice particles, the

growth process is called aggregation. This can occur in both cold clouds and mixed

clouds. When an ice particle grows by colliding with and freezing the water from

(or riming) water droplets, the process is called accretion. This process can only

occur in mixed clouds. All of these collision processes require that there is air

motion within the cloud, but this is a common phenomenon in clouds.

A second way that ice particles grow in mixed clouds is by the Bergeron-

Findeisen process. In this process (described in greater detail later) ice particles

grow at the expense of water droplets in the cloud because there is a difference in

the saturated vapor pressure for ice and water at the same temperature. This growth

process does not necessarily require internal movement in the cloud, but it can

Formation of Precipitation11




144 Formation of Precipitation

obviously only give growth in mixed cloud where both phases of water coexist. It is

worth noting that there are more growth processes available for mixed clouds, i.e.,

all the collision processes as well as the Bergeron-Findeisen process, and their ability

to produce precipitation is therefore greater than for warm clouds or cold clouds.

Precipitation formation in warm clouds

It is clear from the discussion above that, in clouds that ultimately produce pre-

cipitation, the growth of cloud particles is complex, and that it is particularly so for

mixed clouds. A full description of the microphysics of cloud evolution is beyond

the scope of this text. However, it is helpful to consider at least the simplest case of

droplet growth in warm clouds to provide basic insight into the complex process

of collision growth.

By definition, the air temperature in a warm cloud is above 0°C. Consequently,

warm clouds are found below the 0°C isotherm, which means they have a limited

depth at middle and high latitudes where they make only a small contribution to

precipitation. But their contribution to tropical precipitation and to warm season

precipitation can be appreciable. In warm clouds, coalescence is the only collision

mechanism available for cloud particle growth. It occurs between droplets or drops

of different size that are, therefore, falling at different rates. The larger drop falls more

quickly and collides with and potentially captures the slower moving smaller drop.

The likelihood of two water droplets colliding can be expressed in terms of an

impact parameter, y, which is equal to the distance between the geometric centers of

the two droplets (Fig. 11.1a). Collision is likely if y =0 and unlikely if y >> (R + r),

where R is the radius of the larger ‘collector’ drop and r is the radius of the smaller

drop with which the collector drop collides. However, the distinction between

whether there is or is not a collision is blurred because complications arise, some of

which are illustrated in Fig. 11.1b. The collision efficiency, E1, is defined to be the

effective area for which collision of the two cloud droplets is certain relative to the area

[π(R + r)2] for which collision would be certain in the absence of these complicating

processes. The value of E1 varies from near zero when both cloud droplets are small to

near unity when both are large (Fig. 11.2). However, not all collisions will result in the

two droplets coalescing because, having collided, they can then subsequently break

apart. So it is necessary to multiply E1 by the coalescence efficiency, E

2, which decreases

markedly for droplets of similar size, to give the collection efficiency, Ec, with which a

falling droplet will collide with and then absorb another droplet, hence Ec = (E

1E

2).

Attempts have been made to estimate E1, E

2, and E

c theoretically, but it is very difficult

to validate these theoretical estimates experimentally.

Given a (perhaps approximate) description of the collection of droplets by a

single droplet, the next step is to imagine a large droplet of radius R falling though

a field of smaller droplets that are of uniform radius r and equally distributed in

space. Assuming R >> r, it can be easily shown that the resulting rate of growth in

the radius R is given by:


Formation of Precipitation 145

( )−=

4

R r c

w

v v wEdRdt r

(11.1)

where vR and v

r are the terminal velocities of the droplets with radii R and r, respec-

tively, w is the liquid water content of the cloud per unit volume, rw is the density

of liquid water, and Ec is the collection efficiency for droplets with radii R and r.

Figure 11.1 (a) Collision of

a large cloud droplet with a

small cloud droplet (b)

complicating process that

means the impact factor, y,

is not a perfect diagnostic

of collision likelihood.

(a)

R

y

r

Complicating process:

Thin air layer keeps dropletsseparate

Wake of large drop allowssmall droplet to coalescence

Small droplet caught in airflow around large droplet

(b)

Figure 11.2 Typical profiles

of collision efficiency for two

falling droplets with different

radii. Note the low magnitude

of the collision efficiency for

a small collector drop.

0.001

0 0.25 0.5 0.75 1.00

Medium collectordrop (e.g. 20 μm)

Large collectordrop (e.g. 30 μm)

Small collectordrop (e.g. 10 μm)

Ratio of radius of smaller drop to larger drop

Col

lisio

n ef

ficie

ncy

0.01

0.1

1.0



If R is much greater than r, then vR is much greater than v

r and if (for the purposes

of this illustration) it is also assumed that the large droplet is a perfect sphere, then

vR is proportional to R0.5 and it follows that:

0.5dR wRdt

≈

(11.2)

Equation (11.2) implies that drop growth occurs at an accelerating rate.

Although the model just described illustrates some relevant features of droplet

growth in a warm cloud, it is clearly overly simplistic. In fact the collision between

droplets has stochastic aspects that involve spatial and temporal dependency. But

more realistic statistical collision models have been created which simulate droplet

growth reasonably well. These show that the predisposition of bigger droplets to

get bigger means that after about 15 minutes the droplet distribution becomes bi-

modal, and that after some time drops with radius greater than 500 μm are formed

which are able to fall from the warm cloud.

Precipitation formation in other clouds

In cold clouds, aggregation of ice particles is the only mechanism available for par-

ticle growth. The physics involved in the aggregation is broadly similar to that of

coalescence although there are additional complicating mechanisms involved. The

terminal velocities of ice crystals in clouds tends to be very slow and to vary with the

shape of the particles, with the more complex shapes such as plate-shaped crystals

having little variation in velocity with increasing size. Generally the terminal veloci-

ties of pure ice crystals are small, usually less than 0.1 m s−1 and commonly around

0.05 m s−1 and the range of terminal velocities is narrow. This lowers the opportunity

for pure ice particles to grow by aggregation. The solid nature of ice particles also

means they have a tendency to bounce off each other – hence collection efficiency

is further reduced. For all these reasons, the opportunity for ice particle growth by

aggregation to precipitation-sized entities in cold clouds is somewhat limited.

In mixed clouds that occur above the 0°C isotherm, but which have a tempera-

ture greater than –40°C, all the growth processes mentioned in the introduction

can occur, i.e., coalescence of water droplets, aggregation of ice particles, and

accretion of water droplets onto ice particles, along with the Bergeron-Findeisen

process. Aggregation can be more effective in mixed clouds if the ice particles have

a thin film of supercooled water on their surface. This makes them ‘sticky’ because

the thin layer of water between the colliding ice particles freezes instantaneously.

The efficiency with which this aggregation occurs seems to be greater in the tem-

perature range –4°C to 0°C. Snowflakes are formed by this process with the largest

snowflakes produced in the warmest regions of clouds.

Although the terminal velocities of ice particles are small, differential motion of ice

particles and water droplets within mixed clouds results in collisions, so accretion is a



comparatively efficient mechanism by which particle growth can yield precipitation-

sized drops. In the cold portions of a cloud the freezing of water onto cold ice particles

can be so rapid that the process is sometimes called ‘dry growth’. If the water content

of a mixed cloud is high, the water gathered by accretion onto ice particles can be

considerable, and there may be opportunity for several water drops to combine

together before freezing, giving layers of ice. The resulting particle can become quite

large and quite dense, resulting in hail. If the water content of the cloud is low, accre-

tion is slower and air can be trapped in the particle giving graupel.

In mixed clouds, the Bergeron-Findeisen process can occur and is the most

important mechanism responsible for producing precipitation-sized particles, to

the extent that in some meteorological models it is the only process represented.

In part this importance is because the large majority of precipitating clouds extend

upward above the 0°C isotherm and there is, therefore, an appreciable thickness in

which ice particles and (supercooled) water droplets coexist. In addition, tall con-

vective clouds may occur that seed ice particles; these fall and grow rapidly at the

expense of the water droplets in lower, warmer clouds with higher liquid water

content. But the main reason why the Bergeron-Findeisen process is so important

is because it does not require air movement in the cloud or mechanical collision.

It is therefore very efficient and is considered capable of producing more intense

rainfall such as that which occurs from cumulonimbus cloud.

The process itself is simple, see Fig. 11.3. It relies on the fact that there is a dif-

ference between the saturated vapor pressure for water vapor over ice and water

vapor over water. Consider evaporation from supercooled water on the one hand

and sublimation from ice on the other. For the reasons discussed in detail in

Chapter 2, because the water in ice is more rigidly bound, it takes more energy to

release a molecule to the vapor state and, as a result, the saturated vapor pressure

is less for ice than water, see Table 11.1.

Figure 11.3 illustrates how the Bergeron-Findeisen Process operates for an

example case in which the air in the mixed cloud has a temperature of -10°C. At

this temperature, the saturated vapor pressure over water is 0.286 kPa, but that

over ice is 0.260 kPa. The air inside the cloud will establish a vapor pressure that is

intermediate to these two, at (say) 0.278 kPa. The air is therefore unsaturated with

respect to water, and water vapor is evaporated into the air from the water drop-

lets. However, the air is supersaturated with respect to ice. Consequently, water

Figure 11.3 Ice particle

growth by vapor transfer in

the Bergeron-Findeisen

Process. Ice particle

esat (−10) = 0.260 kPa esat (−10) = 0.286 kPa

eair = 0.278 kPa

Water droplet

Vapor flow



vapor is frozen out of the air and directly deposited on to the ice particles. Because

there is no physical movement of cloud particles involved, the transfer of vapor

and resulting ice particle growth can be rapid.

Which clouds produce rain?

Most clouds do not produce rain. This can be seen easily by comparing satellite

derived maps of cloud cover with simultaneous maps of precipitation intensity

derived from rainfall radar, both of which are now readily available on the Internet.

Clouds do not produce rain if they are:

● too short lived, or have not yet been in existence long enough for drops of

sufficient size to escape from the cloud and fall to the ground;

● too shallow to allow vertical motion that encourages particle growth;

● too high, which implies that they have low moisture content because they are

cold, they have limited internal vertical motion, and any precipitation gener-

ated has abundant opportunity to evaporate before reaching the ground.

Clouds that are more likely to produce precipitation have the opposite characteristics

to those given above, specifically they:

● have existed for some time, so they more likely have greater dynamical activ-

ity which includes differential vertical motion, and there has been more time

for multiple collisions to have occurred between component cloud particles;

Table 11.1 Saturated vapor pressure over water and ice as a function of temperature.

Temperature (°C) Saturated vapor pressure

over water (kPa) Saturated vapor

pressure over ice (kPa)

0 0.611 0.611–2 0.527 0.517–4 0.454 0.437–6 0.390 0.369–8 0.334 0.310–10 0.286 0.260–12 0.244 0.218–14 0.207 0.182–16 0.175 0.151–18 0.148 0.125–20 0.124 0.104



● have significant depth, implying that there is greater opportunity for cloud

particle growth, especially if the cloud has appreciable thickness between the

0°C to –40°C isotherms where all the particle growth processes described

above can operate;

● have high liquid-water content, this being a critical requirement in the case

that the cloud is below the 0°C isotherm.

Precipitation form

Meteorologists have defined many different classes of precipitation, see Table 11.2.

The most basic distinction between precipitation forms is between (a) liquid pre-

cipitation falling as rain or drizzle, which are distinguished mainly by drop size,

and (b) frozen precipitation, which is distinguished by morphology and by whether

it is melting or otherwise when it reaches the ground. In fact the distinction between

rain and drizzle is somewhat confused because although mainly based on drop

size, with rain having drops greater than 0.5 mm, this distinction may be colored

by a subjective measure of how widespread the precipitation is. Meteorologists

sometimes speak of ‘widespread drizzle’ but isolated ‘light rain’, for example. Rain,

which is the most common form of liquid precipitation, can result from several

different processes, but mainly results either from coalescence in shallow, low

clouds at warm latitudes, or elsewhere as the melted remnant of ice particles fall-

ing from clouds. Drizzle is most common in temperate latitudes, near coasts or on

high mountains as precipitation from stratified clouds, but it can also result from

coalescence in comparatively warm shallow clouds.

Table 11.2 Recognized forms of precipitation.

Precipitation form Description

Liquid precipitation

Rain Drops of water with diameter >0.5 mm but smaller if widely scatteredDrizzle Fine drops of water with diameter <0.5 mm but close togetherFreezing rain or drizzle Rain or drizzle the drops of which freeze on impact with a solid surface

Frozen precipitation Snowflakes Loose aggregate of ice crystals, often adopting a hexagonal form, most of which are branched

Sleet Partly melted snow flakes, or snow and rain falling togetherSnow pellets, soft hail, graupel White, opaque grains of ice or conical with diameters of 2–5 mmSnow grains, granular snow, graupel Very small, white, opaque grains of ice which are flat or elongated with a

diameter generally <1mmIce pellets Transparent or translucent pellets of ice, spherical or irregular, with

diameter <5 mmHail Small pieces of ice with diameters >5 mm commonly comprising

alternate layers of clear and opaque ice Ice prisms Unbranched ice crystals in the form of needles, column, or plates



Raindrop size distribution

There is always a range of drop sizes present in any individual storm and it is

quite common to have drops that fit the criterion of rain (> 0.5 mm) and drizzle

(<0.5 mm) in the same storm. Observational studies of drop size distributions

reveal that for drops with diameter greater than 1 mm, the number of drops is

often found to fall off exponentially at a rate which is approximately related to

the rainfall rate and follows the Marshall-Palmer equation, which has the form:

−=( ) e DoN D N l

(11.3)

where N(D) is the number of drops of diameter D per unit volume, D is the drop

diameter in millimeters, No is a constant in drops per cubic meter per millimeter

of drop diameter, and lD, is a function of rainfall intensity. In this expression, typi-

cal empirical values might be No ∼ 8000 drops m−3 mm−1 and l

D ≈ 4.1R−0.21, where

R is the rainfall rate in mm hr−1.

Figure 11.4 shows observed forms of drop size distribution at different rainfall

rates that are typical of those found for a young cloud. Older clouds tend to

provide less small drops because the bigger drops in the cloud will have grown

preferentially at the expense of smaller drops. However, the largest drops, with

diameters greater than 3 mm for example, can become unstable, and older clouds

may therefore give raindrops which are the smaller fragments created when

large drops break up.

Figure 11.4 An example of

observed raindrop diameter

(D) distributions during a

rainfall event in which

rainfall rate changed with

time, together with curves

computed from the Marshall-

Palmer equation. (From

Sumner, 1988, after Shiotsuki,


permission.)

102

103

104

101

0

: 6.4 mm/hr

: 46.5

: 24.0

: 84.0 mm/hr

: 90.5

: 97.0

: 1.4 mm/hr

: 6.2

: 1.4

N (

m−3

mm

−1)

1 2 3 0 1 2 3 4 0 1 2 3 4 5

M-P40 mm/hr

M-P100 mm/hr

D (mm)

M-P10 mm/hr

M-P5 mm/hrM-P

1 mm/hr



Rainfall rates and kinetic energy

The terminal velocity of a perfectly spherical water droplet is proportional to the

square root of its diameter. However, as raindrops fall they vibrate and, being fluid,

they deform and flatten (Fig. 11.5) and, as a result, for drops with average dimen-

sions of the order 0.8 to 4 mm, the observed terminal velocities of raindrops are

described by:

0.67( ) 3.86v D D= (11.4)

where v(D) is the terminal velocity in m s−1 for a droplet with average dimension

of D mm.

The rainfall rate, R in mm s−1, is given by the integral of the product of raindrop

volume with the terminal velocity of the drop, weighted by the number of drops of

given diameter per unit volume, i.e., by:

( ) ( )3

0

6

DR N D v D dD∞ ⎡ ⎤⎛ ⎞π= ⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦∫

(11.5)

in which N(D) and v(D) are given by equations (11.3) and (11.4), respectively.

In a similar way, the kinetic energy flux of the rain falling to the ground, EKE

, is

proportional to half the integral of the product of raindrop mass with terminal

velocity squared, again weighted by the number of drops of given diameter per

unit volume, i.e., by:

( ) ( )∞ ⎡ ⎤⎛ ⎞π

= ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦∫

32

0

12

KE wDE N D v D dDr

(11.6)

The equations used in hydrology to estimate soil erosion due to the impact of

raindrops are an empirical form of the relationship between equations (11.5) and

(11.6) with an assumed drop size distribution.

Forms of frozen precipitation

Numerous forms of frozen precipitation are recognized (Table 11.2), but as is the

case for liquid precipitation, two broad classes of frozen precipitation can be rec-

ognized. These are associated with the mechanisms by means of which the frozen

Figure 11.5 Typical shape of

large rain droplets when they

have achieved terminal

velocity. (From Smith, 1993,

after Doviak and Zrnic, 1984,


8 mm 7.35 mm 5.80 mm 5.30 mm 3.45 mm 2.7 mm



precipitation is created. Snow and sleet (sleet being just melting snow) generally

originate in low or multi-layer stratiform clouds in cold weather. As a result, their

occurrence is usually more widespread than other forms of frozen precipitation

and they tend to occur over longer duration. The other types of frozen precipita-

tion are different forms of ice particles. These can be produced from a wide range

of clouds, but because ice pellets and hail are mainly produced by clouds in which

convection is important, they tend to be more intense and more localized than

frozen precipitation which falls as snow. Figure 11.6 illustrates some of the basic

forms of solid precipitation together with the symbols and codes meteorological

observers use to represent them.

Other forms of precipitation

In its most general sense, the word precipitation can include the release of water

from the atmosphere to the ground or to vegetation on the ground in liquid or

solid form by any mechanism. Clouds are not necessarily an intermediary.

Although the precipitation provided by alternative processes does not necessarily

make a major contribution to the water cycle at the global scale, it can make a

significant seasonal contribution at local and perhaps regional scale in particular

locations.

Dew and frost are notable alternative forms of precipitation, which in moist,

cool climates can have significant influence on the water balance in the winter

season. In many cases the amount of water deposited as dew or frost during a

Figure 11.6 Basic forms of

solid precipitation, from top

to bottom, plate, stellar

crystal, column, needle,

spatial dendrite, capped

column, irregular crystal,

graupel, ice pellet, and

hailstone.



winter night roughly equals the amount of water evaporated during the subse-

quent daylight hours. The form of the water deposited, whether dew or frost,

depends on temperature, but the process of deposition is much the same in each

case. Nighttime loss of longwave radiation lowers the temperature of the ground

and/or vegetation and the saturated vapor pressure of the air in contact with

these; at the dew point, water or ice is deposited on the surface. The movement of

water vapor toward the surface is by turbulent diffusion. Consequently, ambient

wind speed plays a role in controlling the amount of water or ice deposited. In

light winds (0.5 to 1 m s−1), the near-surface atmosphere becomes strongly stable

and turbulent transport of vapor toward the surface and hence deposition dew or

frost is restricted. At higher wind speeds (above 3 m s−1, for example) turbulent

transport is more effective and, providing the ground is cold enough, this gives

greater deposition.

Mist and fog can also give an input of water to the surface which is sometimes

called fog drip because it may be observed as water dripping from solid surfaces

(particularly forest vegetation) that is projecting into warm, moist orographic

cloud. This phenomenon can result in a significant hydrological input, and it is not

uncommon in the mid-latitude coastal regions of western North America, Europe,

and New Zealand. Substantial precipitation also arrives as fog drip in mountain-

ous regions on eastern coasts in the tropics, indeed in some regions of Queensland

in Australia it has been reported that as much as 40% of precipitation arrives in

this way. Fog drip is particularly important from an ecological perspective in

mountainous coastal regions with little rain in Namibia and Chile, for example,

and on isolated mountains in Brazil where it provides the primary source of water

to sustain isolated ecosystems that would otherwise perish.


● Cloud particle growth: water droplets and ice particles grow large enough

to fall from clouds as precipitation either by collision processes (called coa-

lescence between droplets, aggregation between ice particles, and accretion

between particles and droplets), or via an intermediate vapor phase when

ice particles grow at the expense of droplets in the Bergeron-Findeisen

process.

● Warm clouds: growth in warm clouds (> 0°C) is simplest to understand

because it can only occur by coalescence, but it illustrates features common

to collision growth in mixed and cold clouds, including:

— particle collisions do not always result in particle merger (see Fig. 11.1b)

— collision with merger is most efficient for larger particle collisions; and

— when large particles fall, collision growth occurs at an accelerating rate.

● Cold and mixed clouds: in cold clouds growth is only by aggregation and

particles are small and fall slowly, hence the opportunity to produce precipi-

tation is low, but in mixed clouds not only can all collision processes occur,



but also growth by the very efficient (often dominant) Bergeron-Findeisen

process. Hence the potential for producing precipitation size particles is

much higher.

● Which clouds produce rain?: clouds are more likely to produce rain if they

have existed for some time, have significant depth, and have high water

content.

● Precipitation form: there is distinction between (a) liquid precipitation

falling as rain or drizzle (distinguished mainly by drop size); and (b) frozen

precipitation (distinguished by morphology and whether melting or other-

wise at ground level).

● Raindrop size: ground observations suggest the number of drops falls off

exponentially with drop size at a rate related to a rainfall rate, approximately

following the Marshall-Palmer equation, i.e., −=( ) e DoN D N l

● Rain rate and kinetic energy: rain rate and the kinetic energy of rain can be

written as integrals of (empirical expressions for) terminal velocity and drop-

let size distribution: this forms the basis for soil erosion equations used in

hydrology.

● Frozen precipitation: numerous forms of frozen precipitation are recognized

but there is marked distinction between snow/sleet (from multi-layer,

stratiform clouds in cold weather) and ice pellets/hail (most intense from

convective cloud).

● Other forms of precipitation: deposition of water from the atmosphere as

dew or frost can be significant in moist, cool climates in winter, and the inter-

ception of mist or cloud by vegetation can be important locally, especially in

coastal regions.

References

Doviak, R. & Zrnic, D. (1984) Doppler Radar Weather Observations. Academic Press, New York.

Shiotsuki, Y. (1974) On the flat size distribution of drops from convective raincloud. Journal

of the Meteorological Society of Japan, 52 (1), 42–59.

Smith, J.A. (1993) Precipitation. In: Handbook of Hydrology. (ed. D. Maidment), pp. 3.1–4.1.

McGraw-Hill, New York.

Sumner, G.N. (1988) Precipitation Process and Analysis. John Wiley and Sons, Bath, UK.


Introduction

The measurement of precipitation appears simple, and indeed measurements of

point precipitation that are prone to systematic errors of the order of 10% have

been made for a long time, in some cases for centuries. But measurement of point

precipitation with greater accuracy than this is more difficult, and measurement of

area-average precipitation is much more difficult, particularly if the average value

required is for a large area. There are three main ways in which precipitation is

currently measured or estimated from observations. By far the most common way

is using rain gauges. Gauge measurements have been made for a long time so

gauge data has the distinct advantage that long time series are available in some

places. But gauge data also have disadvantages. They are point samples and are

consequently prone to random sampling errors. They are also prone to systematic

errors induced by selection of the gauge site and studies have identified wind-

related errors inherent to the instruments themselves.

In recent decades, ground-based radar systems have been developed that are

capable of estimating precipitation. Precipitation estimates provided by radar have

the advantage that they are real time estimates made over a large area around the

radar sensor. These attributes are extremely useful in the primary role of rainfall

radar, that is in providing information for use in short-term weather forecasting.

From the hydrological standpoint radar observations have the advantage that they

are often used to calculate area-average estimates of precipitation over pixel areas

of about 10–20 km2 rather than being point measurements. However, they have

the distinct disadvantage that they are inaccurate, even when calibrated by an

underlying gauge network and, being a recent development, there are as yet no

long-term records of radar-estimated precipitation.

Indirect estimates of precipitation are also made from remote sensing data using

several different approaches. Again there are problems with measurement accuracy

12 Precipitation Measurement and Observation





and longevity of record, but also other issues, including those associated with the

intermittency of satellite observations. However, recognizing the vast oceans and

inaccessible land areas where ground-based measurements are difficult, remote

sensing is probably the only means by which precipitation observations might be

feasible with global coverage. In future, one alternative approach to providing

globally available precipitation estimates might be to merge all available data from

ground-based radar and satellite systems into a meteorological model by using

four-dimensional data assimilation.

Precipitation measurement using gauges

Measuring precipitation using a rain gauge is simple in principle. It requires a fun-

nel of known area to gather the rainfall, a collector to store the water gathered,

some means for measuring the amount of water stored in the collector (such as a

measuring cylinder), and an observer to write down the amount of water meas-

ured. This is the way most measurements of rainfall and some measurements of

snowfall are still made worldwide. Such manual measurements are made at regular

daily, weekly, or monthly intervals, often at 9:00 a.m. local time. The values are

given in equivalent rainfall depth over the sampling interval, in inches in the USA,

but in millimeters elsewhere, often quoted to an accuracy of 0.01 inch or 0.1 mm,

respectively, or designated a ‘trace’ if the rainfall depth is less than this amount.

Although the operators making such manual measurements are trained to make

measurements with care, clearly unquantifiable operator errors can occur from

time to time.

Gauges were used, and standards of gauge design were independently defined

in different countries, before errors associated with specific design and mounting

were properly appreciated. The result is that country-specific recommendations

on gauge design and site selection are not necessarily ideal. Examples of such spec-

ifications are given in Table 12.1. These recommendations remain in place despite

greater understanding of the weaknesses involved. The need for continuity of

record is an important inhibition on change, because long-term records are very

valuable in the context of hydrological design and when documenting precipita-

tion climate.

Table 12.1 Example of national gauge and gauge mounting recommendations.

Country Funnel diameter

Mounting height of funnel top

USA 203.2 mm (8 in) 787 mm (31 in)UK (since 1866) 127 mm (5 in) 300 ± 20 mm (12 ± 0.75 in)Australia and Canada 203.2 mm (8 in) 305 mm (12 in)


Precipitation Measurement and Observation 157

As discussed in detail below, this means that most of the rainfall and snowfall

data currently available are subject to systematic error. Rainfall data measured

using gauges have been and still are systematically low, by about 5–10% on average

and the systematic errors in snowfall data are greater than this. In general terms,

more precipitation is measured when gauges have the top of their funnel mounted

nearer to the ground. This is mainly because being nearer to the ground tends to

reduce the wind speed and hence wind-related gauge errors. However, splash-in of

rainwater to gauges nearer the ground from the surrounding area may also contribute

to a higher catch. In hydrological applications the systematic under-measurement of

measured precipitation is implicitly accommodated (usually without recognition) in

the value of rainfall-runoff coefficients, or in the parameters used in the models that

describe the relationship between precipitation and runoff.

Instrumental errors

Basic instrumental errors can occur even with a device as simple as a rain gauge.

The most obvious of these is the possibility that some of the collected water evapo-

rates before it is measured, this being most likely if measurement is as an average

value over long periods. In practice the now-usual design of gauges, which involves

a funnel feeding the collecting vessel via a thin tube, minimizes this error by main-

taining a more humid environment where the water is stored. Nonetheless, if the

period between measurements is very long, observers may choose to introduce a

known volume of buoyant oil which spreads across the surface of the water in the

container to inhibit evaporation. In periods of light rainfall, evaporation losses

from the layer of water that wets the funnel itself (sometimes known as wetting

errors) may be significant in percentage terms. In humid atmospheres, the oppo-

site problem can occur, with water condensed onto cold metal gauges to increase

the apparent rainfall. This latter problem is most likely to occur at high latitude

and coastal sites, but can be minimized by choice of funnel material. In the case of

gauges used for measuring rainfall intensity, timing errors can arise. Delays of up

to 10 minutes may occur with the tipping bucket design (discussed later) as the

bucket is filled, particularly for storms that occur after a long dry period.

Site and location errors

The representativeness of rainfall measurements can easily be affected by both

local characteristics within 10 m of the site where the gauge is located and by major

obstructions to the wind flow at some distance upwind of the gauge. The funda-

mental reason for errors is because the proportion of rain measured by a gauge,

the ‘catch’, depends on the wind speed and direction immediately above the top of

the gauge. This in turn depends on obstructions nearby and at some distance.

Typically gauges that are mounted above the ground and exposed to wind miss

about 5 to 10% of the incident rain, depending on wind speed and direction of fall



of raindrops, but persistent flow features in the local turbulent field (Fig. 12.1)

alter the wind flow across the top of the gauge and the average path of the rain-

drops and snowflakes entering, or failing to enter, the funnel.

Larger surrounding features such as trees, walls, and buildings upwind of the

gauge can induce persistent flow distortions in the overall wind field to which the

gauge is exposed, or they may shade the gauge from rain. Consequently, selecting

a gauge site is a compromise between potential over-exposure to high wind speed

which worsens the local catch errors, and the possibility that large upwind obstruc-

tions shade the gauge and reduce the catch. To reduce the effect of large-scale

upwind obstructions it is often recommended that the angle such obstacles sub-

tend with the ground when viewed from the gauge site is less than 30°.

Minimizing the percentage loss of catch due to local flow distortion is a less

tractable problem, but efforts have been made to do this at exposed sites. In gen-

eral, the approach is to establish wind flow across the gauge that is as near as pos-

sible horizontal while seeking to reduce the wind speed. Figure 12.2 illustrates

some of the methods that have been recommended for this purpose. Figure 12.2a

illustrates the method recommended by the UK Meteorological Office in which a

turf wall is built 1.5 m from and surrounding the gauge with height equal to that of

the gauge. The assumption is that the wind flow is moved upward and has time to

become horizontal before passing across the top of the gauge. An alternative and

Gauge

Gauge

Deflection of air by rain gauge (horizontal flow)

(a)

Upward deflection over gauge (turbulent flow)

Gauge

(b)

(c)Figure 12.1 Air flow around

a standard rain gauge

standing on the ground: (a)

with no nearby obstruction

and (b) and (c) with nearby

obstruction at different

distances up wind. (From

Sumner, 1988, published with

permission.)



arguably superior method for achieving horizontal flow across the top of the gauge

is to place it in a pit which is as deep as the gauge is tall, with a (plastic) grid over

the remainder of the pit that simulates the aerodynamic properties of the sur-

rounding ground (Fig. 12.2b).

At some locations gauges must be mounted relatively high above the ground to

ensure they are not buried by accumulating snow in winter months. To do this,

gauges are mounted on poles above the ground with, for example, the top of the

gauge at 31 inches in the USA and 2 m in the USSR. Clearly, mounting at height

exacerbates the problem of loss of catch due to wind flow and, to minimize this,

shields can be mounted around the gauge with the purpose of slowing the wind

D

1.2m

150mm

HorizontalCB

1.5m

300mm

1.5m

BC

150mm

1.2m

D

Turf wall for use at exposed rain gauge sites

(a)

(b)

(c)

Nipher precipitation gauge

Figure 12.2 Some methods used to minimize the effect of turbulence around a gauge: (a) a turf wall surrounding the

gauge; (b) a covered pit into which the gauge is put; (c) a Nipher gauge with shield to break up the local wind field. (From

Sumner, 1988, published with permission.)



and encouraging local horizontal flow. Figure 12.2c shows the Nipher precipita-

tion gauge which is one example of such shielding. A second gauge, the Alter or

Tretyakov shield gauge, has slats mounted in a circle around the gauge which are

hinged at the top so they blow toward the gauge on the windward side.

Gauge designs

There are many designs for simple gauges that measure precipitation from the

depth of the water stored in a container between manual measurements. The

range is from simple, inexpensive, pole-mounted, graduated plastic cylinders to

more expensive brass or copper gauges with precision-engineered funnel in a

metal surround with adjusting feet to allow accurate leveling of the top of the fun-

nel. The former are preferred by the amateur on the basis of cost, the later by

professional organizations tasked with providing accurate and consistent gauge

measurements.

Over the last half century, there have been efforts to move away from manual

measurement toward automatic recording, and this transition is accelerating with

the advent of digital recording and remote data capture technology. The difficul-

ties involved in frequent manual sampling meant that automatic recording gauges

were used first in applications where measurement of precipitation timing and

intensity were required.

Siphon and chart recorders, which date back to the nineteenth century, were the

first systems designed for automatic recording of precipitation (Fig. 12.3). Such

gauges are outwardly similar to professional standard, manually read gauges but,

instead of collecting the rain delivered by the funnel in a container, the water is

temporarily collected in a chamber whose volume is equivalent to a rainfall depth

typically of 10 mm. When the chamber is full it siphons and empties before again

being filled by the rain from the collector funnel. If the natural siphoning is used

to empty the chamber, drainage can take 10–20 seconds, which means that some

of the rain may not be measured over this period, this being especially important

in heavy rainfall.

In a natural siphon rainfall recorder, a float chamber on top of the water in the

collection chamber is mechanically connected to a pen that touches a chart which

rotates around a drum, see Fig. 12.3a. The chart typically records for either 1 or 7

days. It is removed after this time for interpretation and a replacement chart

installed. When there is no rain the chart has a level trace (Fig. 12.3b). When it is

raining the rate of upward movement of the pen on the chart is proportional to the

intensity of rainfall. When the storage chamber siphons the chart trace falls rapidly

to indicate an empty chamber and then starts to rise again if rainfall persists. Some

chart recording gauges incorporate a tilting mechanism which removes the pen

from the chart during the siphon, and some include a container which collects the

rainfall that would otherwise be lost during the siphon. The majority of historical

high time resolution precipitation data currently available were gathered using

chart recording rain gauges.



The tipping bucket rain gauge shown in Fig. 12.4 provides a measure of rainfall

rate at high time resolution without the need for laborious interpretation of charts.

Such gauges are again outwardly similar to professional standard, manually read

gauges, but the water gathered by the funnel is in this case fed into a simple see-

saw (or titter-totter) mechanism comprising two buckets of known volume. The

bucket volume is selected to define the resolution in precipitation amount required.

Chart

(a)

(b)

Pen and trace

Float

Chamber

Siphon tube

103 9 15 21

Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday3 9 15 21 3 9 15 21 3 9 15 21 3 9 15 21 3 9 15 21 3 9 15 21 3 9 15 21

9876543210

Figure 12.3 (a) Internal mechanism of a natural siphon rainfall recorder and (b) a typical chart produced by the recorder.

(From Sumner, 1988, published with permission.)



The water fills one bucket and, when this is full, the bucket assembly tips, emptying

the first bucket and simultaneously positioning the second bucket under the funnel

outlet. The second bucket then fills and when it is full the system tips in the

opposite direction. Each time a tip occurs, an electrical signal is sent to a digital

recorder that either records the time of the tip if high frequency intensity

measurements are required, or the number of tips over a predefined interval if

integrated rainfall amount is required. The precision of the measurement is limited

by the volume of the bucket. Smaller buckets give higher precision, but they can

pose problems in intense rainfall when water loss may occur because the tip rate is

high. The simplicity of the mechanism involved in the tipping bucket rain gauge

means it is a robust, reliable and much used instrument. However, in recent years

the advent of devices (strain gauges/load cells) with electronic output that are

capable of accurately measuring the weight of water stored in the chamber below

a gauge funnel has resulted in an alternative to the tipping bucket rain gauge that

has few moving parts and potentially requires less servicing because of this.

Areal representativeness of gauge measurements

Gauges provide point samples of the precipitation at a particular place. Comparison

with other data sampling challenges puts the rainfall sampling issue into

Splash guard

Splash guard

Cable torecorder

Mercoid switch

(a)Tipping bucket

(b)

Figure 12.4 A tipping bucket rain gauge: (a) simplified diagram of internal mechanism; (b) external view of the gauge

assembly. (From Sumner, 1988, published with permission.)



perspective. The polls used to gauge public opinion, for example, typically sample

a few thousand out of several tens of millions of people, and so are a sample of

order 0.1%. In comparison, the rain gauge network used in the United Kingdom

has about two gauges with funnel area 0.0127 m2 to sample an area of 100 km2

(a 5 × 10−12 % sample), and the network used in the United States of America has a

sampling density approximately one tenth of this.

The global challenge of providing estimates of area-average precipitation is

further exacerbated by the fact that gauge densities are very substantially less

than those just mentioned for the UK and USA. The location of available gauges

is also heavily biased, with most gauges located in wealthier, developed coun-

tries. Even in countries where there are many gauges, the gauge sample is heavily

biased toward centers of population and, because of this, to lowland sites. This is

an issue because much of the surface water used to provide water resources for

human use falls as precipitation in sparsely populated regions with significant

topography.

When seeking to evaluate the representativeness of available gauge data, typical

questions asked are, ‘How representative are a set of gauges in giving average rain-

fall when there is systematic spatial variability which may be related to topography

and/or mean rainfall gradients?’, or, ‘What is the minimum gauge density needed

to sample spatial rainfall pattern and total water volume of a rain event?’ In gen-

eral, there is a need for a denser network when seeking to determine short-term

rainfall totals. For example, in a temperate climate a study in which a 9-gauge

network was used to measure rainfall across a small, 20 m grid gave gauge-to-

gauge variations of ±5% in the measured monthly total, but ±8% in single storm

totals. Studies in the semi-arid climate of southern Arizona, where much of the

rain falls in thunderstorms, suggest that one gauge every 2.4 km is needed to pro-

vide an adequate estimate of the annual water balance for a catchment of area

25 km2 (Sumner 1988).

When installing a gauge network to document precipitation pattern and

area-average precipitation, the optimum sampling pattern should recognize

known systematic variations. In flat regions and in urban environments, it is

generally considered best to install gauges on a rectangular grid, with a gauge

at each intersection of the grid. In regions with steep topography, which is often

the case for experimental catchments in which hydrological studies are made,

the recommended procedure is to define ‘hill slope elements’ on the basis

of these having similar slope and aspect, and to locate a rain gauge in the mid-

dle of each. If there is evidence of a rainfall gradient, or such a gradient might

realistically be expected (because the ground is sloping, or sampling is away

from a coast where weather systems move onshore), the preferred sampling

will be along the gradient.

One strategy that can be used is first to make an educated guess at the most

appropriate sampling strategy for the situation under study, then collect data for a

trial period and analyze the data, and then optimize the gauge arrangement, if

necessary. The correlation coefficient between all the gauges in the network for the

duration over which rainfall estimates are required may be used as a numerical



basis for deciding gauge separation. For example, Fig. 12.5 shows the distance

decay curve for correlation coefficients of daily rainfall measured using a gauge

network deployed on a rectangular grid near the coast of Tanzania. Average curves

are drawn for correlation coefficient as a function of distance for gauges lying

perpendicular and parallel to the coast and for all the gauges in the network. Note

the different rates of decay in correlation coefficient. If some value of correlation

coefficient is selected as being acceptable for specifying the separation of gauges,

the required average separation of gauges corresponding to this value can be

deduced. In the case shown in Fig. 12.5, the separation is different for gauges

arranged parallel and perpendicular to the coast.

Inter-gauge correlations have been used to make recommendations on

ideal gauge densities. The (albeit rather crude) guidelines from the World

Meteorological Organization are given in Table 12.2. A second approach used

to define the recommended minimum gauge densities is to consider standard

errors for different network areas and densities. Table 12.3 shows an example

of this approach used to define the number of randomly positioned gauges

needed to give an ‘adequate’ measure of monthly average precipitation for

different areas.

0.9

0.8

0.7

0.6

0.5r

Cor

rela

tion

coef

ficie

nt b

etw

een

gaug

es (

dim

ensi

onle

ss)

0.4

0.3

0.2

0.1

0.00 10 20 30 40

Separation of gauges (km)

50 60 70 80 90 100

Coastal All

Parallel to coast

Normal to coast

Regional

Network

Figure 12.5 Distance Decay

curve for the correlation

coefficient, r, between gauges

in a network installed in a

rectangular array near the

Tanzanian coast with distance

D from the coast. (From

Sumner, 1983, published with

permission.)



Snowfall measurement

Three major problems associated with measuring snowfall with gauges are as

follows:

(1) Snow is easily blown by wind, much more so than is rain. Consequently, the

under-measurement due to near-gauge turbulence is exacerbated and this

problem is further compounded by snow that has been deposited in the

Table 12.2 Recommended minimum rain gauge densities for different types of

topographic and climate regions as recommended by the World Meteorological

Organization.

Nature of the area

Area per gauge (km2)

Normal tolerance

Extreme tolerance in adverse conditions

Small mountainous islands with irregular precipitation

25 –

Mountainous regions in temperate, Mediterranean, and tropical areas

100–250 250–2000

Flat regions in temperate, Mediterranean, and tropical areas

600–900 900–3000

Arid and polar areas 1500–10 000 –

Table 12.3 Minimum number of randomly

located gauges needed for adequate estimation of

monthly rainfall (Data from Stephenson, 1968).

Area (km2) Number of gauges needed

100 5200 6500 81000 102000 135000 1710 000 2025 000 2550 000 29100 000 33500 000 36



gauge subsequently being blown out. Sometimes ‘blow fences’ are set up

around the gauge to protect the gauge from wind. These shielding fences

may perhaps be made from horizontal plastic strips mounted between

poles. Sheltering of this type can reduce the under-measurement of snow,

but does not eliminate it, see Fig. 12. 6.

(2) Snow (and other types of frozen precipitation) takes time to melt, so precipi-

tation intensity measurements are difficult and sublimation of the precipita-

tion stored in the gauge prior to melting can add to under-measurement

problems. The only solution to this problem is to use some form of heating

in the gauge.

(3) Snow can completely cover the gauge and the surrounding area in heavy

storms. The only solution is to mount a large gauge well above the ground

and accept greater wind-related errors because wind speed is greater farther

from the ground.

Because measuring snow with gauges is so difficult, observers are obliged to seek

alternative methods to measure snowfall. If the observer is present at the time of

measurement, one technique used is to push the inverted funnel from a conven-

tional gauge into the accumulated snow cover on the ground, then to remove the

funnel and snow it contains, melt the snow, and measure the water so formed.

Sometimes a snow board might be used if the snowpack is deep, this being a thin,

white, slightly rough board that is left on the snow pack after a storm to act as a

(new) reference level. The inverted funnel method described above might then be

used to measure the subsequent snow deposited.

One very common approach to measuring snowfall is simply to measure the

height of the snow and to assume a density for the snow pack. Often the snow is

assumed to have a density of 10% of that of water. A snow board might again be

used with this approach to allow measurement of increments in the snow pack.

More than 80% of Canadian meteorological stations use this approach. Another

simple approach is to deploy a container of known weight containing antifreeze

and subsequently to weigh the container after snow accumulation has occurred.

Figure 12.6 Mean fractional

catch as a function of wind

speed for rainfall, and

snowfall when measured with

shielded and unshielded

gauges. (Adapted from from

Larson and Peck, 1974.)



Radioactive techniques have been used to measure snow fall. Experimental

measurements have been made which involve installing a radiation source

such as cobalt-60 or caesium-137 at ground level and a radiation detector

above the ground. The reduction in detector counts as snow is deposited above

the source and below the detector is a measure of snow water equivalent.

However, this method is dangerous and expensive, and it has not gained

acceptance. Much more successful is the measurement of regional snow cover

over large areas made by flying an aircraft along predefined tracks and measuring

the gamma ray emission from the ground. A baseline gamma ray emission

must be established along the flight tracks used before snow falls, and subse-

quent flights must accurately follow the same flight path later in the season.

Ground truth measurements are also required at sample locations along the

flight path to improve the accuracy of the estimated area-average snowpack.

In the USA, the National Operational Hydrologic Remote Sensing Center

routinely undertakes successful measurements of snow cover across the north-

ern states in this way.

In some situations, estimates of snowpack are required to guide the manage-

ment of the water resources that will become available later in the season when the

snow melts and enters rivers. For this purpose, snow courses have been established

at preselected sample locations as indices of snowfall, and empirical relationships

have been established between measured melt season river flow and the snow

cover measured at these locations. This management technique is used in the

western mountain ranges of the USA which are source areas for the rivers that

provide water to heavily populated areas downriver. But the technique is labor

intensive. It involves observers accessing selected snow courses in the mountains,

inserting long tubes which take a core of the snowpack present, and weighing the

tube before and after insertion (Fig. 12.7).

Over recent decades an alternative to snow courses has been developed in the

form of snow pads or snow pillows. These are increasingly common in the western

USA. The technique involves ‘weighing’ the amount of snow resting on a thin

Figure 12.7 Snow course

measurement: (a) inserting

the sampling probe; and

(b) weighting the probe and

snow contained in it after

insertion. (From US National

Atlas, 2011.)



inflated pliable pad put in place before snow was present (Fig. 12.8). The weight of

the snow is determined from the measured change in pressure in the snow pad,

and this and auxiliary information are then transferred by remote data capture

techniques to the centers responsible for monitoring snowpack. Such sites form

an already extensive and growing SNOTEL network in the USA. Although there

are issues with SNOTEL sensors related to the representativeness of the measure-

ments and associated with ‘bridging’ of snow influencing the measured pressure,

this technique has the advantage that sensors can be installed in remote locations

during the summer months when they are more accessible and then left to operate

with minimum supervision through the winter months.

Precipitation measurement using ground-based radar

In principle measuring precipitation using radar is simple. The approach involves

sending out a pulse of energy from a (usually revolving) dish and detecting the

‘echo’ or ‘return’ that is received from airborne hydrometeors such as rain, hail,

snow, etc. The time it takes for the echo to return allows the radar to determine the

Figure 12.8 Hardware

components installed at a

typical SNOTEL site. (From

US National Atlas, 2011.)



distance to the hydrometeors, while the strength of the return signal provides

information on the amount of hydrometeors present (Fig. 12.9a). In some radar

systems minor difference in the wavelength caused by the Doppler effect are used

to allow the system also to determine the speed of the hydrometeors moving in the

wind relative to the radar station.

Because the energy pulse used for detection is at microwave frequency (with

wavelength of order 10 cm), it is the average density of hydrometeors in the sample

Figure 12.9 (a) Basic operation of a radar system used to measure the presence of hydrometeors in the atmosphere from

which estimates of precipitation at the ground are made; (b) The NEXRAD system which is the basis of the weather radar

system in the USA; (c) The array of NEXRAD systems deployed in the USA and the maximum range they could sample

over. (From NOAA, 2010, and Firstweather, 2010.)



volume that is detected. The average returned power, P, received at the radar

receiver from a hydrometeor-filled atmosphere at range, r, between the radar and

the measured sample is given by:

=2

r rC A ZPr

(12.1)

where Cr is a constant determined by the design of the system and dependent on

the beam width, antenna gain, wavelength, and pulse duration, etc; Ar is a factor

representing the signal attenuation during its transit through the atmosphere; and

Z is the radar reflectivity factor for the volume of atmosphere sampled by the radar

beam which is usually expressed in units of mm6 m−3.

The estimated rainfall rate in mm hr−1, R, is related to the radar reflectivity fac-

tor, Z, by a semi-empirical power law often called the Z-R relationship with the

form:

bR aZ= (12.2)

At microwave wavelengths, the return signal is generated by Rayleigh scattering,

which means the strength of the return signal expressed in the form of the radar

reflectivity factor is proportional to the sixth power of the diameter of the hydro-

meteors. Consequently, the sensitivity of the system is strongly influenced by the

unknown range of hydrometeor diameters present in the sample. The values of the

empirical parameters a and b in Equation (12.2) are determined by the very vari-

able size spectrum of the hydrometeors and are poorly defined. For the WSR-88D

network that covers the USA (see, for example, http://www.erh.noaa.gov/ohrfc/

ZRlisting.shtml) typical values for a are in the range 130–300 and for b in the

range 1.4–2.0, which implies the system calibration might vary by substantial fac-

tors between rainstorms. A further source of error in the system is that the hydro-

meteors are detected well above the surface so there is potential for them to

evaporate to a variable extent as they fall to the ground. Because the calibration of

the system is inherently poor, when used to estimate rainfall for hydrological

applications radar observations must be continuously recalibrated by merging the

radar estimates with observations from an underlying network of gauges. The

resulting data fields so created are called merged products.

The primary application of radar data is to support weather forecasting and

NEXRAD systems of the type shown in Fig. 12.9b have been deployed for this

purpose across the USA in the network shown in Fig. 12.9c. Notwithstanding the

serious calibration issues associated with radar-based precipitation observations,

they have important properties that are potentially of great value for hydrological

science. The data provided as merged radar-gauge products are area-average esti-

mates of precipitation typically for 4 km by 4 km pixels that are available with high

temporal resolution for intervals of the order of minutes. Such data could greatly

enhance skill in flood forecasting and once they have been available for long

enough, also water resource estimation. However, the very different nature of



these data mean rainfall-runoff relationships and the parameters used in hydro-

logical models will need recalibration.

Precipitation measurement using satellite systems

At the present time the quest to provide estimates of precipitation from satellite

observations is an extremely active area of hydrometeorological research. The

motivation for this research is that estimates derived in this way are likely the only

means by which measurement-based (as opposed to model-based) estimates of

precipitation can be made for the entire globe, including the two-thirds of the

globe covered with ocean and for the vast areas of land that are hard to access.

Exploratory methods have reached the stage where they can provide estimates

that are at least of qualitative value and increasingly also of quantitative value

when averaged over large areas and long periods and they are used to guide the

development of GCMs. Because the rate of development in this field of hydrome-

teorology is currently so rapid, it is likely that detailed documentation of current

methods will be quickly outdated. Consequently, the discussion below is given in

general terms. Broadly speaking, three main methods are used, which are based

around:

(1) cloud mapping and characterization;

(2) passive measurement of cloud properties; and

(3) spaceborne radar.

Modern methods for deriving precipitation from remotely sensed information

often use a combination of more than one of these three basic approaches.

Cloud mapping and characterization

This was the method used in the earliest efforts to derive estimates of precipitation

from satellite data. The basis of the approach is to calibrate and exploit empirical

relationships between the precipitation at a location, and the extent and nature of

the overlying cloud as identified from space. Cloud indices are derived based on

remotely sensed characteristics which are then empirically related to rainfall

intensity. In general terms, the estimated rainfall falling over a defined period, Rsat,

is calculated from an equation with the form:

[ ], ( )satR f c i a= ¢

(12.3)

where f´ denotes an empirical function, c is the area of cloud measured by the

satellite, i denotes a cloud type, and a its altitude. This index of cloud development

can be related to available rain gauge data over the same period and, once



calibration has been made, future satellite imagery used to make estimates of

probable precipitation.

Estimates of rainfall rate are still made using knowledge of the extent and

nature of cloud but it has become usual to merge such knowledge with additional

meteorological information derived from satellite observations. The TOVS

precipitation estimate and the AIRS precipitation product are examples. Data from

the Television Infrared Operational Satellite (TIROS) Operational Vertical Sounder

(TOVS) instruments on board polar-orbiting platforms and AIRS instrument

aboard the Earth Observing System Aqua polar-orbiting satellite are both processed

to provide meteorological statistics. These two precipitation products then infer

precipitation from deep extensive clouds using a multiple regression relationship

between collocated rain gauge measurements and the several satellite data streams

(related to cloud volume, cloud-top pressure, fractional cloud cover, and relative

humidity profile). The relationship used is allowed to vary with season and latitude

and separate relationships are used for ocean and land.

Passive measurement of cloud properties

The essential basis of this approach is to calibrate and exploit empirical relation-

ships between precipitation at a particular location and physical characteristics of

the overlying cloud measured by remote sensing systems. The most commonly

used physical characteristic is the brightness temperature of the top of cloud as

diagnosed by the measured outward longwave radiation (OLR). Colder cloud top

temperature implies deeper clouds with more convective ascent and a greater like-

lihood of rain. Estimates based on cloud top temperature have greatest probability

of success in regions and at times where convective rainfall is dominant, i.e., in the

tropics, and in the summer season elsewhere.

One significant satellite precipitation estimate based on this approach is the

Global Precipitation Index (GPI), which uses three-hourly infrared images provided

by three geostationary satellites. An empirical relationship based on data from the

Global Atmospheric Research Programme (GARP) Atlantic Tropical Experiment

(GATE) is used for this product. For a brightness temperature ≤ 235 K, a rain rate of

3 mm hr−1 is assigned, while for a brightness temperature >235 K, a rain rate of

0 mm/hour is assigned. The GPI estimate works best over space and time averages

of at least 250 km and 6 hours, respectively, and in oceanic regions with deep

convection. Several other precipitation products based on OLR are also available.

Some, such as the PERSIANN precipitation product, have developed

interrelationships using neural networks rather than using fixed formulae or linear

regression.

At microwave frequencies, the atmosphere is almost transparent where no

precipitating clouds occur. Where there is high humidity and particularly where

there are precipitating clouds, microwave emissivity and absorption is greater.

Consequently, data from satellite sensors operating in the microwave waveband

can contribute information relevant to remote sensing-based precipitation



products. One microwave-based data product is the Goddard Profiling Algorithm

(GPROF) fractional occurrence of precipitation which gives the fraction of area

with precipitation on a 0.5° by 0.5° grid over ocean. The algorithm applies a

Bayesian inversion method to the observed microwave brightness temperatures

using an extensive library of cloud-model-based relations between hydrometeor

profiles and microwave brightness temperatures. Each hydrometeor profile is

associated with a surface precipitation rate.

There is value in seeking to use the many alternative sources of remotely sensed

data relevant to estimating precipitation now available to provide a single best esti-

mate. The Global Precipitation Climatology Project (GPCP) seeks to derive such a

preferred satellite precipitation data set by selecting and merging data from many

sources.

Spaceborne radar

Good progress has been made toward remote sensing precipitation using

space-based radar broadly similar to that used in ground-based systems, and

there are plans to develop this approach further. The precipitation radar used

in the Tropical Rainfall Measuring mission (TRMM) was the first spaceborne

instrument designed to provide three-dimensional maps of storm structure.

These measurements yield information on the intensity and distribution of the

rain, on rain type and storm depth, and on the height at which the snow melts

into rain. TRMM had a horizontal resolution at the ground of about five kilom-

eters and a swath width of 247 kilometers. One important feature was its ability

to provide vertical profiles of the rain and snow from the surface to a height of

~20 kilometers. The TRMM radar was able to detect fairly light rain rates down

to about 0.7 mm hr−1, but it was less successful when detecting intense rain

rates.

Providing enough power to detect the weak return echo from the raindrops

when seen from orbital height is a fundamental challenge with spaceborne

radar. From the standpoint of provide routing information globally, the fact

TRMM had only intermittent coverage of the same location is also problematic.

However, multiple deployment of spaceborne radar systems is not a realistic

option for economic reasons. The proposed solution is the Global Precipitation

Measurement (GPM) mission. This will involve a core precipitation-measuring

observatory with both dual-frequency precipitation radar and a high-resolution,

multi-channel passive microwave rain radiometer. This observatory will serve

as the calibration reference system for a constellation of up to eight support

satellites conceived as being relatively small spacecraft that carry a single

high-resolution, multi-channel passive microwave rain radiometer which is

identical to that on the core satellite. In this way it is anticipated that the

GPM mission will frequently sample the diurnal variation in rainfall by

capitalizing on some satellite orbits that are synchronized with the sun and

others that are not.




● Precipitation gauges: provide point observations by gathering precipitation

falling into a funnel of known area and storing it for later measurement

either manually or automatically, but preferred gauge designs and site rec-

ommendations vary between countries and most gauge data are systemati-

cally low by about 5–10% for rainfall, and by much greater than this (perhaps

~50%) for snowfall.

● Gauge errors: may include basic instrumental errors (e.g., evaporation loss)

but are mainly caused by wind blowing across the gauge top reducing the

catch, although these can be reduced by improved (but rarely implemented)

gauge mounting (see text).

● Gauge design: most gauges are still operator read, but most historical high

time resolution precipitation data have been gathered using siphon and chart

recorders, but recording systems which provide electronic output such as

when using a tipping bucket or a strain gauge/load cell are now preferred.

● Areal representativeness of gauges: gauge networks provide a poor (at best

5 × 10−12%) sample and are mainly biased toward wealthy, developed coun-

tries and population centers, but in new networks strategies that sample

likely systematic spatial influences (see text) can give improvements.

● Snowfall measurement: measurement of snowfall using gauges is problem-

atic because catch reduction due to wind is much greater and frozen precipi-

tation takes time to melt and might cover the gauge completely, so it is likely

preferable to measure snow depth and assume snow density or to measure it

as in snow courses, or to weigh the snow cover using snow pads/pillows.

● Ground-based radar estimates: provide real time estimates of area-average

precipitation over 10–20 km2 pixels but they are inaccurate unless calibrated

by underlying gauges and as yet do not exist as long-term records.

● Satellite precipitation estimates: is an active area of hydrometeorological

research with activity in three general areas, i.e., (a) cloud mapping and char-

acterization; (b) passive measurement of cloud properties; and (c) space-

borne radar (see text for details), all of which involve developing empirical

relationships between remotely sensed variables and surface calibration data,

with some data products now using merged information from several

remotely sensed variables.

References

Firstweather (2010) Online at www.1stweather.com/global/radar/index.shtml.

Larson, L.W. & Peck, E.L. (1974) Accuracy of precipitation measurements for hydrological

modeling. Water Resources Research, 10, 857–63.

NOAA (2010) Online at www.magazine.noaa.gov/stories/mag103.htm.



Stephenson, P.M. (1968) Objective assessment of adequate numbers of rain gauges for esti-

mating areal rainfall depths. Proceedings of the Berne International Association of

Hydrological Sciences General Assembly, IAHS Publ. No. 78, 252–64.


US National Atlas (2011) online at: http://www.nationalatlas.gov/articles/climate/a_snow.

html.


Introduction

Analysis of variations in precipitation with time at a specified location is undertaken

in many ways, the nature of which is determined by their purpose. This chapter

provides an overview of some of the more important of these purpose-driven

analyses and the associated approaches used. One major reason for analysing

historical precipitation records is to characterize the precipitation climatology of a

specific geographical location in present day climate. This is expressed in terms of

the normal expectation for the amount, intensity, and timing of precipitation.

A second motivation for precipitation analysis is to discern the existence (or

otherwise) of any long term trends or periodic oscillations through time that may

be occurring within the local precipitation climatology. Sometimes analysis has

been made of the relationship between rainfall amount and duration, or of the

within-storm timing of rainfall with a view to identifying the system signature of a

storm to identify the atmospheric mechanism that caused it.

The most practical motivation for sophisticated statistical analyses of

precipitation records is to provide the basis for the design of infrastructure or water

management systems. Such analyses might, for example, seek to aid agricultural

management by estimating the likelihood of severe drought, or the ‘reliability’ of

receiving precipitation above the minimum value required to grow a crop. Or on

the basis of past observations, they might seek to define the frequency and

magnitude of extreme events and associated flood discharge to aid the design of

drainage systems.

The descriptions of analyses of precipitation in time that follow are written in

general terms without defining the source of the precipitation measurements.

However, because the analysis methods described generally assume the existence

of long precipitation data records, in practice such analyses are usually made using

gauge data.

13 Precipitation Analysis in Time




Precipitation analysis in time 177

Precipitation climatology

Annual variations

The amount and variability in annual total precipitation is arguably the most

important aspect of the precipitation at a location because it determines the nature

of the region and its viability for human habitation. The basic information required

includes mean and standard deviation of annual totals and any discernable trends

in these.

Long-term trends in annual total precipitation are of considerable interest to

hydroclimatologists concerned with global science because trends may specula-

tively result from human intervention in the Earth system globally, regionally, or

locally. At the global scale, possible intervention mechanisms include climate

change due to increasing concentrations of radiatively active atmospheric gases of

human origin; at the regional scale, large-scale change in vegetation cover caused

by deforestation or overgrazing; and at the local scale, changes in precipitation due

to modified aerosol loading associated with upwind burning of agricultural land.

Similarly, fluctuations in annual total precipitation may speculatively be linked to

observable and perhaps predictable variations in aspects of the Earth system, such

as the regional climate impacts that result from oceanic phenomena such as El

Niño-Southern Oscillation (ENSO), see Chapter 9.

Intra-annual variations

The dominant cause of intra-annual variations within the precipitation climatology

at a location is the seasonal change in the regional pattern of atmospheric

circulation discussed in Chapter 9. These changes determine the atmospheric

processes operating to generate precipitation and to a significant extent also the

short-term character of contributing precipitation events. The general nature of

seasonal changes in major precipitation patterns is illustrated in Fig. 13.1. Key

features include the season to season changes in the strength and location of the

westerly wind belts north and south of the equator, the north-south movement

and changing pattern of the precipitation band associated with atmospheric ascent

in the intertropical convergence zone, and the strong seasonal change in

precipitation associated with the northerly to southerly reversal in wind direction

in monsoon systems, especially the southeast Asian monsoon system.

At mid-latitudes the general features of the intra-annual variations in precipitation

that result from these large-scale movements in precipitation climate are as follows:

● The western margins of continents are dominated by precipitation associated

with oceanic depressions, which give copious rain in all seasons but a marked

maximum in the relevant autumn and winter months in each hemisphere

when the westerly winds strengthen.


178 Precipitation analysis in time

● The eastern margins of continents have variable exposure to oceanic and

continental influences and so a more variable seasonal pattern, but with a

marked tendency to winter snow when the upwind continent is cold.

● Inner continental areas have penetration of oceanic weather in winter

months with associated rain and snow, but a greater tendency to convective

rain showers in summer.

July

Seasonal change instrength and

location of westerlyair streams

Precipitationcaused by

seasonal reversalin monsoon air flow

Seasonal changein location ofintertropical

convergence

January

High

High

High

HighHighHigh

High

Low

Low

HighHigh

Low

Low

Subtropical and continental (summer) high pressure areas

Major areas of precipitation generation in westerly belts

Belt of precipitation generation along intertropical convergence

Figure 13.1 Seasonal

changes in the location and

strength of the global

precipitation pattern.

(Redrawn from Sumner,

1988, after Bartholomew,


permission.)



In tropical regions, there is solar radiation driven seasonality in precipitation with

at least one wet season caused by:

● the north-south movement of the central ascent in the intertropical conver-

gence zone, see Fig. 9.4;

● the large-scale reversal in wind direction in regions affected by monsoons

and the southeast Asian monsoon in particular, see Fig. 9.10;

● the seasonal cycle in the strength of oceanic influences, notably the rate of

occurrence of tropical storms, see Fig. 9.11.

The need to provide measures of the intra-seasonal variation in precipitation is

most critical in tropical and subtropical regions because in these regions the often

marked seasonality has significance for human welfare; the potential for

catastrophe is high because agricultural systems are more marginal and water

resource infrastructure tends to be less resilient. In such regions there may, for

example, be a need to understand the extent and timing of seasonality in

precipitation for planning and advisory purposes, but the available data to do this

are often sparse. Consequently, there is a need to interpolate from limited data

using contour graphing techniques or multiple regression techniques to describe

relationships with features that can influence the precipitation, such as distance

from the sea, altitude, and latitude.

Seasonality can be expressed in visual form in several ways including as:

● maps of the percentage of precipitation falling in each month of the year,

called isomers;

● maps of the ratio of precipitation falling in each month relative to one twelfth

of the annual average precipitation, called pluviometric coefficients;

● polar diagrams of the monthly rainfall, with the angular direction indicating

the month of the year, and distance from the origin proportional to the

monthly average rainfall; or

● ‘pie’ diagrams of the monthly rainfall, with angle subtended as a fraction of

360° for each month proportional to the monthly percentage contribution of

annual rainfall.

One numerate way to express seasonality in precipitation is by calculating a

Seasonality Index, an example being:

12

1

1

1.83 12a

nna

XSI X

X =

= −∑ (13.1)

where Xa is the total annual precipitation and X

n are the individual total monthly

precipitation values. Values of SI <0.2 indicate a ‘very equitable’ precipitation cli-

mate, values in the range 0.6–0.8 a ‘seasonal climate’, while those with SI >1.2 arise

if almost all the precipitation falls in one month.



A further characteristic of seasonal climate, of value for agricultural planning, is

the time of onset of seasonal rainfall, but there is no universally accepted way of

defining this. Some popular choices include:

● specifying the date on which the precipitation exceeds an arbitrary amount

(e.g., 1 inch or 25 mm of rain);

● specifying the date on which the precipitation exceeds a selected fraction of

the estimated potential evapotranspiration for the remainder of the growing

season;

● fitting a Markov chain model (see later in this chapter) to the precipitation

pattern of early rains to define a date when there is a change in the probabil-

ity of rainfall after antecedent rain.

Daily variations

Although it is comparatively simple to demonstrate the overall intra-annual

variation in precipitation for most places, daily precipitation is far more

variable. The probability distribution for daily precipitation is always positively

skewed and often quite strongly so. Zero precipitation is quite common – there

is no guarantee of rain every day even during a wet season in the humid

tropics. Anomalously high daily rainfall can also occur, occasionally reaching

values as high as 1 m of rain per day at some locations. The derivation of a

robust mean daily rainfall is therefore statistically futile. The median daily

precipitation, i.e., the value for which occurrence of greater of less precipitation is

equally probable, is arguably a more stable measure of daily precipitation

climate. It may also be advantageous to specify daily rainfall in terms of days

with greater than a set amount of precipitation, e.g., defining days with precipi-

tation greater than 0.25 mm per day as ‘rain days,’ and days with more than 1 mm

as ‘wet days’.

Because the convection process can be important over continents, a distinct

diurnal cycle in precipitation is commonly observed throughout the year in

tropical regions and during the summer months in temperate regions. Figure 13.2

shows an example of this for Manaus, Brazil. In some situations, such as that

shown in Fig. 13.2, it is the time of first occurrence of rainfall that is most obviously

linked to peak convective activity in the middle of the day. This is because severe

storms, once started, tend to be self-supporting and can last into the evening.

The daily pattern of variation in precipitation can also be complicated by the

presence of mountains. Moreover, near coasts or the edges of large lakes, differential

surface heating gives rise to diurnal changes in local air flows which translate into

ascent, and the timing of these flows can impact the diurnal pattern of precipitation.

Interestingly, over oceans, peak convective activity and associated precipitation is

often at night, a feature which has been ascribed to the cooling of cloud tops by

outward radiation at night.



Trends in precipitation

Strictly speaking, the mean values of annual, seasonal, or monthly precipitation is

statistically useful only when the probability distribution they follow is normally

distributed. However, as mentioned above, over short periods such as a day, rainfall

data are always positively skewed. Trends and oscillations in average precipitation

over shorter periods are therefore hard to discern although it may be possible to

identify them in median values if there are enough samples in the sample period to

define these. More usually when seeking to identify trends and oscillations, the

approach is to define methods that involve some form of averaging or summing of

precipitation data over longer periods because this gives values whose probability

distribution is closer to normal.

To illustrate this by example, consider the data given in Table 13.1 which are the

values of the monthly total precipitation, Pi, for July at Musoma, Tanzania between

1931 and 1960. These values follow the strongly skewed distribution shown in Fig. 13.3.

The ten year mean of July precipitation of these data, PM

, for the periods 1931–1940,

1941–1950 and 1951–1960 are given in Table 13.2 along with the standard deviation,

sM

, and coefficient of variation, CM

, for each ten year period computed from:

2( )i MM

P PN

−=s (13.2)

and:

100M M

M

CP

= s (13.3)

0.01 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Local time (hours)

Hourly average precipitation at manaus

Pre

cipi

tatio

n (m

m h

r−1 )

18 19 20 21 22 23 24

0.1

0.2

0.3

0.4

0.5

0.6

Figure 13.2 Average diurnal

variation in rainfall for four

years (2000, 2002, 2003, 2004,

2005) at Manaus, Brazil.



respectively, where N is the number of years, in this case 10. These values suggest

that there may be some form of trend in these data.

Running means

One commonly used way to smooth out variations in a time series and make

trends in the data more obvious is to use a running mean. The mean is taken over

an odd number of years centered on the year of interest, i.e., over (2n+1) years

where n = 0, 1, 2 etc. Thus, a derived series of values, Pi′, is calculated from the

initial series, Pi, using the equation:

( 2 )

( 2 )

1

(2 1)

j i n

i jj i n

P Pn

= +

= −

=+ ∑¢ (13.4)

Table 13.1 Monthly total precipitation for July at Musoma, Tanzania from

1931 to 1960 (Data from East African Meteorological Department, 1966).

Year Rainfall (mm) Year

Rainfall (mm) Year

Rainfall (mm)

1931 45 1941 0 1951 31932 23 1942 0 1952 31933 4 1943 0 1953 291934 2 1944 13 1954 1451935 0 1945 18 1955 371936 13 1946 9 1956 1031937 2 1947 38 1957 01938 2 1948 1 1958 201939 0 1949 11 1959 11940 31 1950 181 1960 3

Table 13.2 Mean, standard deviation, and coefficient of variation over 10-year

periods of the monthly total precipitation for July at Musoma, Tanzania from 1931 to

1960 derived from the data given in Table 13.1.

Period Mean (mm) Standard deviation (mm)

Coefficient of variation (%)

1931–1940 12.2 15.7 1291941–1950 27.1 55.3 2041951–1960 34.4 49.9 145



This series is valid for values of i in the range (n+1) to (N–n-1), where N is the

number of data elements in the original time series. Figure 13.4a shows the time

series of values of total July precipitation at Musuma, Tanzania and Fig. 13.4b the

time series of the 5-year running average value of these same data. The year-to-

year fluctuations in Fig. 13.4b are much smoother than in Fig. 13.4a, and a trend

toward a wetter period during the 1950s clearly emerges.

Cumulative deviations

Significant and sustained shifts in precipitation can be demonstrated by accumu-

lating the deviations in yearly precipitation about the period mean. Yearly devia-

tions above (positive) or below (negative) the period mean are accumulated

through the data period, either as absolute amounts or as percentages of the mean

precipitation. A plot is then made as a function of time of the accumulated devia-

tions, Pj,a

, or the accumulated percentage residuals, Pj,p

, given by:

,1

( )j

j a ii

P P P=

= −∑ (13.5)

and:

,1

100( )

j

j p ii

P P PP =

= −∑ (13.6)

where Pi are the time series of precipitation undergoing analysis and P is the

mean precipitation for the period of interest. Figure 13.5 shows the results of a

cumulative percentage analysis for annual precipitation at several tropical sites

00

5

Fre

quen

cy o

f rai

nfal

l

10

15

20

10 20 30 40 50 60 70 80

Monthly rainfall (mm)

90 100 110 120 130 140 150 160 170 180 190

Figure 13.3 Frequency

distribution for the July

precipitation at Musoma,

Tanzania from 1931 to 1960

given in Table 13.1.



and indicates there was a marked change in annual precipitation in the decade

1901–1910, with some indication of a reversal in this trend beginning in the

1930s or 1940s.

Mass curve

The mass curve approach is an alternative way to reveal long-term sustained

trends. Here the analysis involves accumulating and plotting total precipitation

throughout the data period. The accumulated precipitation total plotted for year j

is given for the precipitation time series, Pi, by:

,1

j

j m ii

P P=

= ∑ (13.7)

01931 1936 1941 1946

Year

(a)

1951 1956

50Ju

ly ra

infa

ll (m

m)

100

150

200

01931 1936 1941 1946

Year

(b)

1951 1956

50

July

rain

fall

(mm

)

100

150

200

Figure 13.4 (a) Time series

of total July precipitation at

Musoma, Tanzania between

1931 and 1960 given in Table

13.1; (b) Five-year running

mean of these same data.



Freetown

Vizagpatam

HabanaT - 47.20 In.

TrinidadT - 61.95 In. T - 38.94 In.

T (1881 1936)-152.0 In.

−100

100

200

300

0

−200

−100

100

200

0

−200

−100

100

200

0

−100

100

200

0

BogotaT - 39.72 In.

Georgetown Q.T - 30.35 In.

HonoluluT - 29.29 In.

Townsville Q.T - 44.6 In.

1861–70 1871–80 1881–90 1891–00 1901–10 1911–20 1921–30 1931–40 1861–70 1871–80 1881–90 1891–00 1901–10 1911–20 1921–30 1931–40

1871–80 1881–90 1891–00 1911–20 1921–30 1931–40 1881–90 1891–00 1901–10 1911–20 1921–30 1931–40 1941–50

Figure 13.5 Cumulative percentage deviations from the annual mean precipitation for several tropical sites. (Redrawn

from Kraus, 1955, published with permission.)



Any distinct and sustained change is revealed by a change in slope of the resulting

graph, which might be quantified by fitting a linear regression to selected portions

of the mass curve. Figure 13.6 shows an example of this type of analysis made at

two Australian sites indicating that there was a noticeable increase in precipitation

at these two sites in the mid-1940s relative to the period 1890 to 1970.

Oscillations in precipitation

Precipitation time series may include periodicity at several different frequencies.

Initial identification of possible contributing periodicity might be sought using

serial autocorrelation. For a precipitation time series Pi, comprising N data points

which has a mean value P , the series correlation, CL, for a time lag L is given by:

1

2

1

( )( )

( )

i N

i i li

L i N

ii

P P P PC

N P P

=

+=

=

=

− −=

−

∑

∑ (13.8)

When L = 1, CL = 1. If the time interval between data samples is short (e.g., daily)

the correlation for small time lags may well be significant because successive daily

precipitation amounts are not truly independent. For longer time periods and

Figure 13.6 Use of the mass

curve to reveal changes in the

annual precipitation at

Cootamundra (open circles)

and Mount Victoria (filled

circles) in New South Wales,

Australia. (Redrawn from

Sumner, 1988, after Cornish,


permission.)



longer sample periods, there will likely be a progressive and marked decrease in CL

with increasing time lags. However, if there is periodicity present in the data series,

the magnitude of the correlation coefficient will increase again for a time lag that

matches the period of one of the fluctuations present. A graph of serial correlation

coefficient versus time lag is called the correlogram.

Identification of periodicity in a precipitation time series can also be made using

harmonic (Fourier) analysis. This entails fitting a mathematical function P f of time

t (in units appropriate to the problem) to a time series with the general form:

1

( ) cos( )k n

fk k

kP t P P kt

=

=

= + −∑ f (13.9)

where Pk and f

k are the amplitude and phase assigned to the k th harmonic in the

harmonic series so that it adequately represents P f. The maximum value of n is

determined by the requirement that there is at least one full sinusoidal cycle of the

corresponding term in the time period for which data are available. An earlier

autocorrelation analysis might be used to guide the selection of terms in Equation

(13.8) for which fitting is made. The contribution of each harmonic term to the

total variance is subsequently found by expressing the variance for each harmonic

as a proportion of the total variance. Not every harmonic term in the series can

necessarily be associated with an identifiable physical mechanism.

As an alternative to harmonic analysis, numerical filters can be applied to the

precipitation time series to identify and enhance the contributions from influences

with different periodicity. Figure 13.7 shows an example of this approach in which

digital filters are used to identify fluctuating contributions superimposed on the

moving average value, in this figure corresponding to periodicity between 6 and 8

years, between 8 and 13 years and between 13 and 30 years. There is a possible

association with the sunspot cycle when the 8 to 13 year filter is applied to these

data, with the northern hemisphere (as represented by the west coast of the USA)

out of phase with the southern hemisphere (as represented by the east coast of

Australia). There is also some suggestion of association with the 18 year lunar

cycle revealed by applying the 13–30 year filter. Once trends and fluctuations in

observed precipitation records have been clearly established, they might be

extrapolated into the future for forecast purposes.

System signatures

The variation with time of precipitation intensity within a precipitation event is

called the system signature of the storm that gave rise to the precipitation. As might

be expected from the discussion given in Chapter 11, there is an important

distinction between convective and synoptic-scale storms. Observations of

precipitation rate during storms indicates that not only do most large-scale

synoptic systems produce longer duration and lower intensity precipitation than

convective storms, but also the distribution of precipitation intensity through the



storm is usually different. Longer duration events tend to have a relatively even

distribution of precipitation with time, or to have higher intensity precipitation

toward the end of the storm. On the other hand, shorter duration events which are

mostly convective in origin tend to have more intense rainfall near the start of the

storm associated with a downdraft, although if there are several intense cells in the

storm, the pattern is more complex. Figure 13.8 gives the precipitation rate through

a single convective storm measured at four sites and shows a characteristic higher

rainfall rate near the beginning of the storm with a second less intense cell passing

the gauges about 30 minutes later.

Because precipitation events with similar origin have different rainfall totals

and different duration, when comparing several storms it is convenient to

re-normalize the mass curve such that the Y axis displays the percentage of total

rainfall and the X axis the percentage of time through the storm. Figure 13.9a

shows an example mass curve for a frontal storm without re-normalization, while

Fig. 13.9b compares the percentage mass curves for several storms most of which

are convective in nature, but one of which is frontal in nature. Approximately

speaking, for convective storms about 50% of the total storm rainfall falls in the

first quarter of the storm and 90% within the first half of the storm. In contrast,

because the rainfall rate is more uniform, only about half of the storm rainfall falls

in the first half of frontal storms.

Figure 13.7 Fluctuations in

annual rainfall from 1887 to

1960 for the west coast of the

USA when subject to filters

(described in the text) shown

as full lines. Also shown (as

broken lines) are results using

data from eastern Australia.

(Adapted from Vines, 1982.)



Intensity-duration relationships

Observations of rainfall intensity suggest that there exists a basic, inverse, non-

linear relationship between different intensities of rainfall and the duration over

which different rainfall intensities persist. Most analysis of rainfall intensity has

been directed at the occurrence of maximum intensity over different durations

because this gives a measure of extreme precipitation at the location of interest. As

discussed later, probabilities may be assigned to such extreme values so that likely

volumes of rainfall over an area may be estimated and interpreted in terms of

00

12

24

36

48

60

72

84

96

108

120

20Time after start (minutes)

Rai

nfal

l int

ensi

ty (

mm

hr−

1 )

40 60

Figure 13.8 Rainfall

intensity through a convective

storm measured at four sites

in Kampala, Uganda.



permission.)

018:00 24:00

5 Aug. 1973 6 Aug. 197306:00

Tot

al r

ainf

all (

mm

)

10

Date and time

20

30

40

50(a) (b)

00 20 40 60 80

Percentage of total time

Frontalstorm

Convective storms

Per

cent

of t

otal

rai

nfal

l

20

40

60

80

100

Figure 13.9 (a) Mass curve

for a frontal storm in

Lampeter, Wales; (b)

Percentage mass curve for

eight convective storms and

one frontal storm in Dar es

Salaam, Tanzania. (Redrawn

from Sumner, 1988, published

with permission.)



storm drainage, for example. The most important result to emerge from such

studies is that the maximum intensity attained for different durations at a location

are interrelated, and that this interrelationship can be represented relatively simply

both mathematically and graphically.

The form of the relationship between maximum intensity and duration changes

with region and the assumed form will generally need to be calibrated over a

portion of the range before application. One commonly adopted assumed

relationship between maximum intensity and its duration is the McCullum model

which has the form:

nI kt −= (13.10)

where I is the intensity (mm hr−1) that is sustained for a duration t (in hours), and

k and n are location dependent constants obtained by calibration. Interestingly,

this relationship (which yields a straight line when values are plotted on log-log

graph paper) seems to apply at the global scale and also within individual storms

both of convective and frontal nature. Figure 13.10 shows the greatest magnitude

rainfalls and their duration for the world as a whole, while Fig. 13.11 shows

intensity-duration relationships (a) at Dar es Salaam, Tanzania for several storms

of convective origin and (b) at Lampeter in Wales for several storms of frontal

origin. Note the very different axes used in parts (a) and (b) of Fig. 13.11.

Statistics of extremes

Calibrating intensity-duration relationships over the usually longer duration and

lower intensity portion of their range and then using the resulting curve to estimate

the duration of more intense rainfalls (mentioned above) is one example of a more

general approach. Hydrologists and hydrometeorologists are often concerned with

1

0.01

0.1

1

10

100

10

Rai

nfal

l dep

th (

m)

100 1 000

Duration (minutes)

Minutes Hours Days Months Years

10 000 100 000 1 000 000

Figure 13.10 Relationship

between the greatest

magnitude rainfalls and

duration for the world as a

whole. (Redrawn selecting

extreme values from

Brutsaert, 2005, after WMO,


permission.)



estimating extremes, i.e., the frequency of events for which the probability of

occurrence occurs at the ends of the frequency distribution, say the 10% highest

or lowest probability events. The frequent challenge faced is to how use a

comparatively short run of data to estimate the longer term probabilities of very

low probability events.

0.010.1

0.2

0.5

1.0

2.0

5.0

10.0

20.0

50.0

100.0

0.02 0.05 0.1 0.2Duration (hours)

Inte

nsity

(m

m h

r−1 )

0.5 1.0 2.0 3.0

31.12.68 20.4.71

14.11.69

24.11.69

6.11.69

(a)

0.1

0.2

0.5

1.0

2.0

5.0

10.0

20.0

50.0

Inte

nsity

(m

m h

r−1 )

Duration (hours)

18.9.73

28.11.73

12.11.7310.7.74

13.11.74

0.01 0.02 0.05 0.10 0.20 0.50 1.0 2.0 5.0 10.0 20.0

(b)

Figure 13.11 Maximum

intensity-duration

relationships (a) at Dar es

Salaam, Tanzania for several

convective storms and (b) at

Lampeter, Wales for several

frontal storms. (Redrawn

from Sumner, 1978, published

with permission.)



A typical precipitation frequency distribution is not usually normal but rather is

positively skewed, with a large number of lower magnitude events and fewer high

magnitude events, see, for example, Fig. 13.12a. This applies both to within-storm

intensities and to long period total rainfall. Often the main concern is to estimate the

probability of events that occur at the limbs of the distribution, or to estimate the prob-

ability of events with magnitude greater than a prescribed amount. Figure 13.12b

is derived from the hypothetical probability distribution shown in Fig. 13.12a and

shows the accumulated probabilities of exceeding a certain magnitude, and conse-

quently approaches one on the extreme left. The curve may be reversed to obtain the

probability of less than a given magnitude.

Although the cumulative probability in Fig. 13.12b reflects the probability

curve, it is difficult to extrapolate to the critical extremes that often occur at return

periods greater than the period for which the data are available. To aid in this, it is

helpful to ‘straighten’ the curve graphically by adopting the statistical frequency

00.999

0.8

0.4

0.2

0.1

0.01

1.0

0.8

0.6

0.4

0.2

0

0

2

4

6

8

20 40 60 80 100

Magnitude (X)

(c)

(b)

(a)

Pro

babi

lity

ofle

ss th

an X

Pro

babi

lity

ofle

ss th

an X

Fre

quen

cyof

X

Figure 13.12 Different ways

of depicting a probability

distribution when

investigating the statistics of

extremes: (a) a hypothetical

observed frequency

distribution; (b) the

equivalent cumulative

probability curve; (c) the

same cumulative probability

curve plotted on graph paper

that ‘straightens’ the

distribution.



distribution that best fits the observed frequency distribution for the data during

the period for which observations are available. Once this has been done, estimates

of the probabilities of extreme occurrences or the probability of exceeding limits

can be made graphically or algebraically.

Over the years, the selection of an appropriate probability distribution to use in

such design problems has been the subject of much debate among engineering

hydrologists and civil engineers. Five broad groupings of possible distributions are

used:

(1) Conventional frequency distributions (e.g., normal, Poisson, gamma);

(2) Grumbel distributions (there are numerous);

(3) Pearson distributions (particularly Types I and III);

(4) Extremal distributions (Types I, II, III); and

(5) Transformal distributions (e.g., logarithmic and polynomial transforms).

However, the selection of a particular probability distribution is as much an art

as a science, and is a decision that is aided by experience but influenced by

personal preference. Hydrometeorological understanding has little to offer

to aid such a detailed choice. For this reason, extended discussion of the

appropriateness of particular assumed probability distributions for particular

tasks is beyond the scope of a text such as this, which is concerned with

understanding hydrometeorological and hydroclimatological phenomena and

processes.

Nonetheless, an example of the general approach used is appropriate, and a

thirty year time series of annual rainfall for Musoma, Tanzania is used for this

purpose, see Table 13.3. The first step is to rank the data in either ascending or

descending order on the basis of the magnitude of the annual rainfall. This is done

in ascending order from 1 to 30 in Table 13.3. On the basis of this set of observa-

tions, the return period, T, for an event with the magnitude corresponding to rank

m is given by:

( 1)nTm+= (13.11)

where n is the number of years in the data series, in this case 30. Alternatively, the

probability, P, of an event of rank m being equaled or exceeded is:

( 1)

mPn

=+

(13.12)

The use of (n+1) in these equations implies that the period for which data are

available is merely a sample and neither the highest ranked sample value, nor the

lowest, represents the true highest or lowest values in the population as a whole.

It is possible to assign a notional probability for each value of annual precipitation

from Equation (13.10), and these can be plotted to describe the sampled probability

distribution. In Fig. 13.13 such a plot is made on log-normal graph paper for the

data in Table 13.3. It is approximately a straight line, implying the probabilities are



approximately normal. By extrapolating the line in Fig. 13.13 (either by eye or by

fitting a linear regression), the probability can be estimated to an annual rainfall

amounts outside the range for which observations are available.

Alternative options for assumed distributions (and associated graph papers)

could be used as alternatives to assuming a normal distribution to give plots

equivalent to Fig. 13.13 for the Grumbel or Pearson distributions. In practice the

selection between these alternative assumptions would likely be made based on

which gave the best appearance of a straight line. But the fact that several different

assumptions can be made and that several give at least reasonable linearity is

significant. It demonstrates the fundamental limitation on the accuracy of the

estimates made using the approach of the statistics of extremes, because the

probability distribution always has to be assumed. Perhaps several estimates using

Table 13.3 Annual total rainfall at Musoma, Tanzania from 1931 to 1960 ranked in

order of increasing precipitation amount (Data from East African Meteorological

Department, 1966).

RankAmount

(mm) Year RankAmount

(mm) Year RankAmount

(mm) Year

1 442 1934 11 713 1941 21 893 1955 2 467 1949 12 714 1932 22 932 1947 3 550 1933 13 760 1945 23 949 1951 4 613 1953 14 772 1956 24 954 1950 5 624 1938 15 774 1940 25 998 1944 6 637 1939 16 782 1952 26 1015 1954 7 646 1931 17 823 1958 27 1026 1937 8 650 1946 18 850 1948 28 1039 1957 9 680 1935 19 852 1942 29 1128 196010 711 1943 20 883 1959 30 1184 1936

0% 10%

100

200

800

1000

50%

Estimated percentage probability

Tota

l ann

ual r

ainf

all (

mm

)

90% 99%

Figure 13.13 Annual total

rainfall at Musoma, Tanzania

between 1931 and 1960 given

in Table 13.3 plotted on

log-normal probability paper

against the notional

probability of each value as

calculated by Equation (3.11).



different assumed probability distributions should be made and the mean value

of the resulting ensemble of estimates used, with the range of estimates given

using different assumptions then providing an approximate lowest estimate of the

error implicit in the calculation. An estimated probability given by the statistics

of extremes approach is compromised if the sample of precipitation in the

observational record used is not representative of the precipitation climate at the

location of interest. It also involves the implicit assumption that the precipitation

climate is not changing.

Conditional probabilities

Up to this point the analysis perspective we have adopted has been that precipita-

tion events are independent of each other. In reality, however, precipitation cli-

mates often can be viewed as having ‘wet spells’ and ‘dry spells’. This is evident in

the fact that precipitation data commonly reveal seasonal dependency and some-

times evidence of periodicity. A major cause of shorter term persistence is the fact

that the weather conditions and weather systems involved in the production of

precipitation themselves have some persistence. Indeed, much of the skill in

weather forecasting depends on this fact.

The presence of persistence in the precipitation measured at a particular loca-

tion can be sought, and the extent to which it occurs quantified, using simple

stochastic techniques. A long time series of observed precipitation is analyzed to

count the number of times a ‘wet day’ or ‘rain day’ follows a preceding period

comprising a specified number of days with rain or a specified number of days

without rain. In this way a set of conditional probabilities are derived that charac-

terize the level of persistence in the data record. Figure 13.14 shows a simple exam-

ple. In this case the probability that a wet period will be further extended by a day

with rain, or a dry period further extended by a day without rain is plotted for

Selangor, Malaysia.

The probabilities shown in Fig. 13.13 are examples of conditional probabilities,

in this case for daily rainfall. The four relevant basic conditional probabilities are

the probability that day n is wet if day (n−1) was wet; the probability that day n is

wet if day (n−1) was dry; the probability that day n is dry if day (n−1) was dry; and

the probability that day n is dry if day (n−1) was wet. Such conditional probabili-

ties (or those of greater complexity) are the basis of a widely used mathematical

model of persistence, the Markov chain. This model, whose detailed description is

beyond the scope of this text, begins by calculating a number of equations derived

to calculate the probability of a wet day or days after a wet or a dry day, of wet or

dry spells of different lengths, etc. One important application of models of persis-

tence such as the Markov chain model is to generate synthetic time series of pre-

cipitation which have the same level of persistence as the data series originally

used to calibrate the model. Such long time series might be used in design studies

for hydraulic structures.




● Annual precipitation: determines the nature of a region and its viability for

human habitation: long-term trends in its value may result from human

intervention in the global system, while fluctuations may be predictable (e.g.,

ENSO, NAO, etc.).

● Intra-annual precipitation: variations caused by seasonal changes in atmos-

pheric circulation, especially in the ITCZ, monsoons systems, and the

strength and location of westerly wind bands, give:

— at mid-latitudes, characteristic differences between winter and summer

climates depending on continental location (see text for details); and

— in the tropics, strong seasonality in precipitation, wind direction, and

tropical storms (see text for details).

These are documented visually using isomers, pluviometric coefficients, and

polar diagrams or using numerate seasonality indices.

● Daily precipitation: generally has a strong diurnal variation linked to atmos-

pheric convection, and daily total values are very variable and have a highly

skewed probability distribution and so are problematic to analyze statisti-

cally, except perhaps as a median.

● Precipitation trends: are identified by methods that involve averaging/

summing precipitation, including: (a) running means; (b) cumulative

deviations; and (c) mass curves (see text for details).

● Oscillations in precipitation: are identified by methods that include:

(a) autocorrelation to define a correlogram; (b) harmonic analysis; and

(c) numeric filters (see text for details).

● System signatures: because the timing of in-storm precipitation is usually

different for frontal and convective storms, mass curves are used to investigate

the primary atmospheric mechanism giving rise to a storm.

0

50

60

70

80

90

Pro

babi

lity

(%)

1 2 3

Dryspell

Wetspell

4 5 6

Length of spell (days)

7 8 9 10 11 12 13 14 15

Figure 13.14 The

probability that a spell of

weather will be extended by

a further day at Selangor,

Malaysia deduced by

counting the times this has

occurred in an observed

precipitation data series.


1988, after Yap, 1973,




● Intensity-duration relationships: observations of rainfall intensity suggest

an inverse non-linear relationship between the maximum intensity, I, of

rainfall and duration, t, over which that intensity persists that applies to

storm totals for regions (and the globe) and within-storm intensity (I = k t−n

is often assumed).

● Statistics of extremes: in hydrological design, the frequency of rare

precipitation events is estimated by fitting an assumed probability distribution

to data records of limited duration, then extrapolating to extreme probabilities

(not sampled) to estimate the likelihood of the event (see text for an example).

● Conditional probabilities: because weather systems have finite lifetimes,

stochastic techniques can be used to calibrate numerical models of day-to-

day persistence in precipitation data records which can then be used to

calculate long synthetic precipitation records that have the same persistence

characteristics.

References

Bartholomew, J.C. (1970) Advanced Atlas of Modern Geography. J.C. Bartholomew,

Edinburgh.

Brutsaert, W. (2005) Hydrology: An Introduction. Cambridge University Press, Cambridge.

Cornish, P.M. (1977) Changes in seasonal and annual rainfall in New South Wales. Search,

8, 38–40.

East African Meteorological Department (1966) Monthly and Annual Rainfall in Tanganyika

and Zanzibar during years 1931 to 1960, East African Meteorological Department,

Nairobi.

Kraus, E.B. (1955) Secular changes in tropical rainfall regimes. Quarterly Journal of the

Royal Meteorological Society, 81, 198–210.

Sumner, G.N. (1978) The prediction of short-duration storm intensity maxima. Journal

of Hydrology, 37, 91–100.


Vines, R.G. (1982) Rainfall patterns in the western United States. Journal of Geophysical

Research, 87 (C9), 7303–11.

WMO (1986) Manual for Estimation of Probable Maximum Precipitation. Operational

Hydrology Report No. 1, WMO-No 332, World Meteorological Organization, Geneva.

Yap, O. (1973) The persistence of wet and dry spells in Sungei Buloh, Selangor. Meteorological

Magazine, 102, 240–45.


Introduction

There is substantial spatial variation in precipitation fields due to differences in

the type and scale of atmospheric processes that cause precipitation, and to local

or regional influences such as topography and the wind direction at the time the

precipitation was produced. For this reason the temporal analyses of precipitation

described in Chapter 13 are site or at least regionally specific because the intensity

and probability analyses described are for point precipitation data. However, as

mentioned in Chapter 12, gauge data are a poor representation of area-average

precipitation, and assuming they are representative can be dangerous, particularly

in the case of short duration samples and in climates prone to convective storms.

Adjacent gauges are not always consistent, even monthly and annual average

precipitation data may vary significantly, and this is true even in areas with

comparatively low topography.

The movement of storms relative to the ground and the fact that they develop

and decay influences the precipitation pattern on the ground. Some gauges may

experience very heavy rainfall while others see no rain for a particular storm.

Ideally, estimates of the precipitation falling in a particular area would track and

accurately model the passage of a rain storm over the area and calculate the

distribution of the precipitation in space and time. By convoluting this with

the shape and topography in a drainage basin, the area-average precipitation could

be computed. Doing this is at least difficult, and perhaps impossible given the

inherently chaotic nature of the precipitation-producing processes in the

atmosphere. An approximate alternative is to derive empirical models based on

observations that relate intensity to storm area and to use these to estimate mean

areal precipitation.

Notwithstanding the problems involved with analyzing the spatial organization

and distribution of precipitation, the subject has demanded and continues to demand

14 Precipitation Analysis in Space




Precipitation Analysis in Space 199

great attention. Not least this is because area-average precipitation is a major influence

on human settlement and development. Understanding the spatial organization of

precipitation is also crucial to hydrologists and civil engineers concerned with the

design of infrastructure. On the one hand, long-term area-average precipitation rates

determine available surface water resources while, on the other hand, short-term

area-average rates are needed when designing flood protection systems.

Mapping precipitation

Mapping precipitation is a straightforward and visually convenient way to illustrate

the organization of precipitation in space. However, precipitation is a meteorological

variable that is difficult to map, especially from gauge data which is the most

common source of the information used, because the point samples available may

be unrepresentative. Because precipitation is so spatially variable, there may be

extreme values which are missed, especially if the gauge network used is sparse

and uneven. There is also likelihood that a network of gauges will mis-sample local

systematic variations in precipitation associated with topography. In principle,

measurements made with radar would not be subject to the same shortcomings

but, as discussed in Chapter 12, the errors associated with radar measurement are

so large that such data only become reliable when re-normalized to an underlying

gauge network. Hence, the problems associated with gauge sampling remain.

Consequently, gauge data remain the basis of precipitation mapping, and for many

parts of the world are likely to remain so for some considerable time.

To use gauges for mapping it is necessary to assume precipitation is a spatially

continuous variable with no dramatic discontinuities, and to assume that it is

adequately sampled by the gauge network used. Correlation tests between gauge

measurements might provide some level of reassurance that a comparatively

smooth field with point to point correlation is being adequately sampled. In

principle the mapping process is then simple. It involves drawing isohyets, i.e.,

lines of constant precipitation, between the point precipitation data available.

Figure 14.1 provides a simple example of the process used.

The accuracy with which isohyets can be drawn will depend on the density of

the gauge network because spurious high or low values can easily dominate the

resulting map if the gauge network is sparse. The assumptions used when drawing

smoothed isohyets also influences accuracy and, if isohyets are drawn manually,

may be subjective. The presence of topography in the area mapped can also affect

accuracy because its influence on precipitation rate is rarely properly sampled and

will therefore be poorly acknowledged in isohyetal maps. However, with knowl-

edge of the underlying topography, a skilled operator or computer program can

make some recognition of the tendency toward higher precipitation at higher

locations when specifying smooth contours of precipitation. In areas with very

strong relief the influence on isohyets of making such allowances for height might

even match the influence of the available gauge data.



A commonly encountered issue when preparing isohyet maps is the fact that, as

a result of instrumental or human error, precipitation data is rarely 100% complete.

In the absence of a particular gauge measurement, an estimate of the missing value

is normally made from that measured at nearby gauges. In areas with low relief

the arithmetic mean of the values at three to four nearby stations might be adopted.

However, in areas with high relief a preferable alternative is to take the average of

these station values weighted by the long-term average ratio between stations.

Areal mean precipitation

Estimates of area-average precipitation from gauge network data are a focus of interest

within hydrology because they provide the basis for water resource estimation or

extreme event (flood or drought) forecasting. The estimate required is the total

volume of water falling on a specific drainage basin, i.e., the area of the basin multiplied

by the area-average precipitation depth. In areas with low relief where there is an even

distribution of gauges, a simple arithmetic average will provide an adequate estimate.

However, such conditions are rarely met and one of several alternative ways of taking

a weighted mean of the gauge measurements available is required. Weights can be

assigned in two general ways, either using mapping methods that involve or are

equivalent to drawing isohyets (including the computational equivalents such as

kriging techniques or reciprocal-distance-squared methods), or geometric methods

such as the Triangle method or Theissen method in which areas are defined over

which precipitation is assumed uniform. In practice, studies suggest that these

methods tend to produce comparable results especially if applied over a long period.

Isohyetal method

The basis of this method is straightforward, it involves:

(1) constructing isohyets across the catchment based on measured values,

including making allowance for relief if appropriate;

Figure 14.1 The process of

drawing isohyets from gauge

data which involves

estimating the location of

points with the required value

between pairs of gauge

measurements then drawing

smoothed isohyets from

these.

10 mm 12 mm 11mm

8 mm

Trueinformation

9 mm

7 mm

Smoothedisohyet

(subjective, or subjectto assumption)



(2) measuring or computing the area between isohyets to provide the weights

to be used when calculating the average value; and

(3) creating the weighted average by adding the products of the area between

isohyets in the catchment with the mean precipitation between isohyets.

Figure 14.2 illustrates the approach used in the Isohyetal method for an example

catchment. Using this method with these data and this example catchment,

Sumner (1988) calculated a catchment average precipitation of 17.3 mm.

Computer calculations of mean rainfall over selected catchments based on

mapping can also be made using kriging methods or the simpler (but arguably as

effective) Reciprocal-Distance-Squared method, both of which are similar in

concept to the Isohyetal method. In the Reciprocal-Distance-Squared method, a

value of precipitation, Pi (in mm), is assigned to each area element in the catchment

into which the catchment is subdivided by the computer program. This value is

the average of the nearest three gauges weighted by the square of the inverse

distances, d1, d

2, and d

3 (in m) between the element and the three gauges (Fig. 14.3).

Thus Pi is calculated from:

2 2 2

1 1 2 2 3 3

2 2 2

1 2 3

( ) ( ) ( )

( ) ( ) ( )i

P d P d P dP

d d d

− − −

− − −

+ +=

+ +

(14.1)

Figure 14.2 Calculation of area-average precipitation using the Isohyetal Method. (From Sumner, 1988, published with

permission.)

10

Basin margin

Contours

Isohyets (mm)

Point values (mm)

15 20

14.9

12.7

27.2

25

26.1

19.0

27.326.5

25.8

24.2

10.89.2

5.610

7.1

5

5

10 15

15.0

2018.5

25

7.14.7



Triangle method

This method involves:

(1) constructing a network of triangles as near equilateral as possible, with one

of the available gauges at the apex of each triangle;

(2) measuring or computing the area of each triangle or portion of each trian-

gle that is within the basin to provide the weights to be used when calculat-

ing the average value;

(3) if needed, making a ‘best guess’ of the precipitation for any area of the

catchment not included in a triangle; and

(4) calculating the weighted average by adding the products of the area of each

triangle with the mean precipitation for the three gauges at the apex of

each triangle.

Figure 14.4 illustrates the approach used in the Triangle Method for an example



Theissen method

The Theissen method is often the preferred method for estimating area-average

precipitation. The method involves:

(1) constructing a network of polygons by drawing the perpendicular bisector

of the line joining each pair of gauges;

Figure 14.3 Calculating

area-average precipitation

using the reciprocal-

distance-squared method.

(Precipitation P1, P

2, and P

3

in mm, distance d1, d

2, and

d3 in m.)

P1P2

d1 d2

d3

P3



(2) measuring or computing the area of each polygon or portion of each poly-

gon that is within the basin to provide the weights to be used when calcu-

lating the average value; and

(3) calculating the weighted average by adding the products of the within-

basin area of each polygon with the precipitation measured by each gauge.

Figure 14.5 illustrates the approach used in the Theissen method for an example



Spatial organization of precipitation

Analysis of individual storm events as measured using dense networks of gauges

suggests that storms generally comprise one or more cells of high intensity

precipitation embedded in a surrounding field of lower intensity precipitation. The

precipitation pattern generated by individual storms measured by gauges differ

significantly from one another but, based on analyses made by many observers and

at many sites, opinion is that on average storms tend to be (a) elliptical in shape,

(b) more elongated when larger (i.e., the ratio of the major axes to the minor axis

of the ellipse is greater for bigger storms), and (c) organized into groups which

themselves are either larger pseudo-elliptical areas or linear bands (a linear band

Y T

M

15.0

O

U

L

E

K

N

F

CG

PQ

W

V

X

R

J

I

H

D

B

27.326.5

24.2

25.8

18.5

A

7.1

9.2 10.8

19.05.6

4.7

S

26.1

12.7

27.214.9

7.1

Figure 14.4 Calculation of


using the Triangle Method.

(From Sumner, 1988,




being the asymptotic limit of an ellipse). There are usually bands of precipitation

(clusters of intense cells) in frontal systems, spiral bands of storms in tropical

cyclones, and individual intense cells in convective conditions (Fig. 14.6a).

Thus, on average, the characteristic form of a basic single cell storm deduced

from surface gauge observations has an outer boundary which is approximately

elliptical in shape, with a series of successively smaller ellipses inside corresponding

to increasing intensity. The rate of change in intensity rises toward the middle of the

storm, i.e., the slope of the upper surface of the intensity pattern is concave upward,

steeper toward the center of the storm and dying out gradually and intermittently

toward the edges of the storm (Fig. 14.6b).

J15.0

CD

I

9.2

E

K

G

F

B

27.326.5

24.2 25.8

18.5

A

7.1

10.819.0

5.6

4.7

H

26.1

12.7

27.214.9

7.1

Figure 14.5 Calculation of


using the Theissen Method.

(From Sumner, 1988,


Frontal

Cyclonic

Intensity

Convective

Direction ofstorm movement

(a) (b)

Figure 14.6 (a) Typical

arrangement of elliptical

storms in different

meteorological conditions;

(b) characteristic

precipitation field in a basic

single cell storm.



Design storms and areal reduction factors

The general relationship found in many parts of the world between the area-

average precipitation in a storm, the storm area, and the peak intensity at its focus

is used as the basis for defining and calibrating design storms for climatological

regions. To define design storms, observations of individual precipitation events

gathered from a dense gauge network in the region are analyzed to specify the

average shape of the intensity distribution when normalized to the peak intensity

in each storm. Because storms with more persistent precipitation tend to have

greater storm area and also tend to be rarer, the relationships which describe the

precipitation intensity distribution relative to the peak intensity are different for

classes of observed storms when subdivided by storm duration and/or storm

return period. Hence, the results of such analyses are expressed in terms of the

relationship between the peak storm precipitation and the area-average

precipitation for storms subdivided either on the basis of return period (Fig. 14.7)

or on the basis of duration (Fig. 14.8).

Figure 14.7 Point area precipitation relationships for Lund, Sweden by return period. (From Niemczynowicz, 1982,




The so-called point area relationships just described may be used to aid design

of water management infrastructure such as flood control systems. In Chapter 13

the use of the statistics of extremes was discussed and, as an example, the

probability of rainfall rates being greater than a defined value was estimated on the

basis of a previously measured time series of precipitation. This method provides

an estimate for the precipitation rate at a point. For small drainage basins, it might

be acceptable to assume the precipitation rate with this probability falls uniformly

across the basin and storm drainage from the basin calculated as the basis for

flood management. However, for larger drainage basins assuming uniform

precipitation is unrealistic. Rainfall greater than a specified amount might occur

with the estimated probability somewhere in the basin, but the precipitation

elsewhere in the basin during the rain storm will be less than this. The point area

relationship corresponding to the area of the basin and storm return period and

duration can be used to make a first order estimate of the (reduced) basin average

precipitation required for the storm drainage calculation.

Point area relationship analysis can be extended more generally by deriving

average areal reduction factors (ARFs) for a region. Two approaches have been

used. Storm centered analyses focus on defining the ratio of the peak observed

rainfall to the area-average rainfall for different storms, the area of the storm

therefore being variable. A more commonly used approach is to make a fixed area

Figure 14.8 Point area precipitation relationships for Lund, Sweden by duration. (From Niemczynowicz, 1982, published

with permission.)



analysis. In this case ARFs are calculated for entire drainage basins by relating the

area-average precipitation (calculated using the Theissen method, for example)

for a chosen storm duration and for an annual time series of extreme events in a

number of different selected catchments. In each case, the ratio between the aver-

age precipitation for the catchment and the areal maximum precipitation is calcu-

lated and an overall mean then calculated from these values for all the catchments

analyzed. Figure 14.9 shows an example of the ARFs obtained in this way in terms

of duration and rain area. Apparently in this example the ARF is considered inde-

pendent of return period. Area-average precipitation can be simply obtained by

taking the product of observed point depth precipitation with the ARF for an

appropriate duration and area.

Probable maximum precipitation

A further measure of extreme precipitation for a region that might be helpful

in infrastructure design is the concept of probable maximum precipitation

(PMP). Although the name implies PMP is a statistical measure, it is largely

0.25 0.5 1 2 3 4 6

Duration, D (hours)

12 24 48 96 192

99

98

97

96

9594

9392

9190

87.5

85

82.5

80

75

70

6560

5550

4540

3530

10 000

5000

2500

1000

500

250

100

50

25

10

Are

a, A

(km

2 )

Figure 14.9 Areal reduction

factor related to rain area and

duration. (Redrawn from

Sumner, 1988; after Rodda

et al., 1976, published with

permission.)



a physical estimate of what might be the greatest possible precipitation given a

certain set of extreme atmospheric conditions. PMP is a hypothetical concept

which is defined as ‘the analytically estimated greatest possible depth of precipi-

tation that is physically possible and reasonably characteristic over a geographi-

cal region at a certain time of year’. PMP is usually defined with respect to a

given area, often a drainage basin, and includes estimates of the inflow of mois-

ture over the basin and the maximum likely amount of that moisture which

could be precipitated.

One atmospheric variable likely to exert control on the PMP is W, the total

precipitable water in the atmosphere overlying the region. W can be calculated (in

mm) from data measured during a radiosonde ascent through the atmosphere

(see http://amsglossary.allenpress.com/glossary/search?id=precipitable-water1)

by taking the integral:

1.

top

ground

P

P

eW dpg P e

=−∫ (14.2)

where g is the acceleration due to gravity, e and P are respectively the vapor pres-

sure and atmospheric pressure (in kPa) measured as a function of height by the

radiosonde, and Pground

and Ptop

are the atmospheric pressure at ground level and

the top of ascent (when contact is lost or the balloon bursts). Total precipitable

water can also be estimated from surface dew point assuming the moist adiabatic

lapse rate prevails throughout the atmosphere.

The name total precipitable water is inaccurate because not all of the water in the

atmosphere can be precipitated by any known mechanism. Consequently, in addi-

tion to depending on W, the calculation of PMP needs to recognize and make

allowance for realistic restrictions on the rate of convergence of water vapor

toward a storm and the maximum effect of vertical motion within a storm. One

approach used to estimate the PMP is to adopt (and if necessary transpose from

elsewhere) models of real extreme storms to estimate these additional restrictions,

but then to index these to local extreme values of W. However, the assumptions

and generalizations made when adopting the storm model approach are such that

a sometimes preferred technique involves the use of actual storm occurrences,

which are then ‘maximized’ to become an extreme storm for the area using the

highest observed surface dew points and most extreme morphological conditions.

Available regional depth-area-duration and maximum intensity information as

well as local isohyetal maps may used, or these may adopted from areas that are

similar. Time factors such as season, time of day and storm duration may also be

taken into account. In regions with topography the estimation of PMP is much

more difficult. Several different methods have been attempted but as yet none has

universal acceptance. Figure 14.10 shows an example map of all season average

probable maximum precipitation for the eastern USA.

The performance of PMP estimates can be evaluated against maximum

observed storm totals. Studies of this type in the USA (e.g., Reidel and Schreiner,

1980) suggest that observed maximum precipitation is approximately 60% of PMP.



Spatial correlation of precipitation

In hydrometeorology and hydroclimatology, spatial analyses of precipitation are of

interest as a means for studying the dominant precipitation-producing mecha-

nisms and prevailing moisture flows in a region. For example, if an analysis was

made of the correlation between the precipitation measured at a point with that

measured at progressively increasing distances, it might be possible to deduce

information about average storm size and typical storm separation. For example,

consider a correlation analysis across a featureless plain for a period when there

was no consistent direction of air flow and precipitation was produced by ran-

domly located convective storms. In this case the spatial pattern of the correlation

might reveal concentric rings of positive correlation whose width reflected the

spatial dimensions of storms, and areas of negative correlation whose diameter

reflected the separation between storms (Fig. 14.11).

In more usual conditions such a plot of correlation coefficient can reveal evidence

of the prevailing direction of rain-bearing winds and of the location of features in the

landscape that are associated with atmospheric activity that generates precipitation.

119°

25°

29°

33°

37°

41°

45°

25°

29°

33°

37°

41°

45°

127° 123° 119° 115° 111° 107° 103° 99° 95° 91° 87° 83° 79° 75° 71° 67°

115° 111° 107° 103° 99° 95° 91° 87° 83° 79° 75°

14

14

15

16

15 16

16

16

1717

15

16

14

13

12

11

10.31011

1213

14

15

16

18

19

20

21

22

23

2424.6

1718

1920

21

22

232424.6

All-seasonPMP

estimate(for 6 hours,200 miles2)

17

Figure 14.10 All season average probable maximum precipitation calculated in inches (1 inch = 25.4 mm) for the eastern

USA for a 6 hour period and an area of 518 km2. (From Smith, 1993; after Hanson et al., 1982, published with permission.)



Figure 14.12a illustrates a good example of this for Queensland, Australia. The

contours of the correlation coefficient between one selected inland gauge and other

gauges in the region show the effect of coherence between precipitation measured

along the line of the Great Divide and coastal precipitation associated with sea breeze

activity, and also strong correlation along the dominant direction of moisture flow.

Ascent Ascent

Decent

~ Storm separation

Distance

~ Storm size

Cor

rela

tion

coef

ficie

nt

Figure 14.11 Schematic

diagram of the hypothetical

variation in correlation

coefficient versus distance for

long term average

precipitation relative to one

location if produced by

randomly occurring

convective storms over a

moist, flat, featureless plain.

(a)

25On shore

winddirection

Line of thegreatdivide0.25

0.25

0.25

0.250.25

0.25

0.25

0

0

−0.2

5

0.5

0.5

0.25

0

0

0 0.25

(b)

Bandsparallelto coast

0

Figure 14.12 (a) Contours of correlation coefficient for observed precipitation in Queensland, Australia relative to a single

point, and (b) contours of composite correlation obtained by superimposing the correlation coefficients for individual

gauges in the same region as (a). (Redrawn from Sumner and Bonell, 1988, published with permission.)



In some situations where the gauge density is high enough, it may also be possible to

create contour maps of composite correlation coefficient in which the correlation

fields for each gauge are superimposed. Figure 14.12b shows an example of a contour

map of composite correlation coefficient for this same area of Queensland showing

correlated bands of precipitation inland away from the coast separated by 40–45 km.


● Mapping precipitation: drawing isohyets of precipitation involves esti-

mating the location of points with the required precipitation value between

pairs of gauge measurements then drawing smoothed isohyets through these

points.

● Area mean precipitation: is the value calculated as a weighted mean of

available precipitation observations, with weighting assigned in one of

several possible ways including using (a) the Isohyetal, (b) the Triangle, and

(c) the Theissen methods, although the Theissen method is often preferred

and all methods tend to produce similar results, especially if applied over a

long period.

● Spatial organization: storms tend to be organized in pseudo-elliptical or

linear groups, with individual storms elliptical in shape, more elongated

when larger, and with the rate of change in intensity increasing toward the

middle of the storm.

● Design storms: statistical analysis of observed storms results in the definition

of regional relationships between peak storm intensity and area-average

precipitation called areal reduction factors which are used in hydrological

design and which vary with storm duration and/or storm return period.

● Probable maximum precipitation (PMP): the hypothetical estimate for a

region of the greatest possible precipitation that might occur in extreme

atmospheric conditions over a specified area and period of time: it is variously

calculated but may be an overestimate relative to observed maximum

precipitation rates.

● Spatial correlation: spatial characteristics of the atmospheric mechanisms

involved in precipitation release in a region can be revealed by analysis of the

correlation between observations from a rain gauge network (see text).

References

Hanson, E.M., Schreiner, L.C. & Miller, J.F. (1982) Application of probable maximum pre-

cipitation estimates – United States east of the 105 meridian. Hydrometeorological Report

52, National Weather Service, NOAA, Washington, DC.

Niemczynowicz, J. (1982) Areal intensity-duration-frequency curves for short-term rainfall

events in Lund. Nordic Hydrology, 13, 193–204.



Reidel, J. & Schreiner, L. (1980) Comparison of generalized estimates of probable maximum

precipitation with greatest observed rainfalls. NOAA Technical Report NWS 25, NOAA,

Washington, DC.

Rodda, J.C., Downey, R.A. & Law, F.M. (1976) Systematic Hydrology. Newnes-Butterworth,

London.

Smith, J.A. (1993) Precipitation. In: Maidment, D. (ed.) Handbook of Hydrology, pp. 3.1–4.1.



Sumner, G.N. & Bonell, M. (1988) Variations in the spatial organization of daily rainfall

during the Queensland wet season. Theoretical and Applied Climatology, 39 (2), 59–74.


Introduction

The movement of momentum downward from the atmosphere into the Earth’s

surface and the transport of water vapor, heat, and minority gases between the

surface and the overlying atmosphere are primarily by the mechanism of turbulent

transport. Consequently, a basic understanding of the turbulence that occurs in

the atmospheric boundary layer is a necessary component of hydrometeorological

understanding. This chapter provides an introduction to the concept of atmos-

pheric turbulence and to the mathematical tools used when deriving equations

that describe it.

Signature and spectrum of atmospheric turbulence

If a fast response sensor of any weather variable is placed a few meters above the

ground in daytime conditions, the measurement it provides will reveal evidence of

apparently haphazard variability in the atmosphere. Figure 15.1 shows an example.

In this case the measurement recorded on the chart recorder trace shown is of

horizontal wind speed made with an anemometer capable of responding to

changes at a time scale of greater than 10 seconds. The measured wind speed is

recorded for about two and a half hours from 12:00 to 14:30 local time.

What does this chart recorder trace disclose? Careful inspection of Fig. 15.1

reveals the following.

a. There are clearly visible quasi-random fluctuations in the measured wind

speed, the pattern of which is not regular and not wave like.

b. The mean value of the measured wind speed is changing with time from around

6 m s−1 between 12:00 and12:30 to around 5 m s−1 between 14:00 and 14:30.

15 Mathematical and Conceptual Tools of Turbulence





c. The magnitude of the variability as characterized by variance of the trace

over 30 minutes decreases between 12:00 and 14:30.

d. There is arguably some evidence of structure in the variations, with small

peaks separated by about 1 minute superimposed on larger peaks at about

3–5 minutes, and some evidence of variations at about 8 to 10 minutes.

Because the measurement is of horizontal wind speed, the fact that there are

periods with faster and slower horizontal wind speed suggests the presence of

structures in the air flow that take about a minute to a few tens of minutes to

pass the fixed sensor. These, therefore, have a horizontal size on the order of

tens to thousands of meters (Fig. 15.2) and are sometimes referred to as

turbulent eddies.

Thus, observations made with a weather sensor mounted above the ground (in

this case an anemometer) show that there are haphazard variations in the meas-

13:000

5

10

Local time

Structure at 3–13 minsPeaks at about 1 min.

Mean 5 m s−1

Mean 6 m s−1

Win

d sp

eed

(m s

−1)

Change in variance

12:00 14:00

Figure 15.1 Trace of

horizontal wind speed

measured with an

anemometer. (Redrawn

from Stull, 1988,

published with

permission.)

1000’s m

Turbulent structures with differenthorizontal dimensions within a body

of air moving horizontally

10’s m

5 − 6 m s−1Figure 15.2 Schematic

diagram of turbulent eddies

circulating within an air

stream which is itself moving

at the mean wind speed.


Mathematical and Conceptual Tools of Turbulence 215

ured value around a gradually changing background. These are symptomatic of a

field of turbulence above the ground that involves parcels of air of variable size and

longevity moving horizontally and vertically in an apparently haphazard way in

the atmospheric boundary layer. Such turbulence is generated partly by friction as

the moving air stream moves across the rough surface and, in daytime conditions,

partly also by buoyancy.

Time series of measurements such as those shown in Fig. 15.1 can be analysed

using Fourier analysis to define the frequency spectrum of component

contributions to the variations in weather variables. The results of such an analysis

were described in Chapter 1 for a much wider range of frequencies, see Fig. 1.3.

Here we consider the more restricted range of frequencies from fractions of a

second to about 100 days shown in Fig. 15.3. In this figure three distinct peaks are

visible, as follows:

a. at around 100 hours there is variability that is associated with weather

systems which typically influence local conditions for a few days;

b. at around 24 hours there is variability that is associated with difference in

conditions between day and night; and finally,

c. there is variability across the range of frequencies less than about 10 to

15 minutes.

A feature that is apparent in Fig. 15.3, which is crucially important from the

standpoint of describing turbulence in the atmosphere, is the distinct lack of

variability for frequencies with periodicity between about 30 and 90 minutes.

0.001

0.001

0.01

0.01

0.1

0.1

Eddy frequency (cycles per hour)

Time period (hours)

Rel

ativ

e sp

ectr

al in

tens

ity

10

10

100

100

1000

1000

1 day

Synoptic scale Turbulent scalesSpectral gap

Weathersystems

1 minute

Microscale eddies

1

1

Figure 15.3 A typical

frequency spectrum for the

variability in atmospheric

variables measured above the

ground in the turbulent field

in the atmospheric boundary

layer.



This portion of the spectrum is called the spectra gap in variability and its

presence provides an opportunity to divide the representation of atmospheric

variability into two halves.

To the left of the spectral gap, variations are primarily associated with synoptic

scale features and with the more gradual change that occurs through the daily

cycle, such as the change in the mean value of wind speed between 12:00 and 14:00

already noticed in Fig. 15.1. To the right of the spectral gap, variations correspond

to the haphazard variability at higher frequency that are apparent in Fig. 15.1 and

are associated with turbulence. They reflect the movement of the parcels of air that

are continuously being created and destroyed in the turbulent field which, as they

move, provide the primary mechanism by means of which water vapor, heat,

momentum and chemical constituents are transported between the surface and

the atmosphere. Some important characteristics of turbulence to bear in mind in

what follows are:

● it is irregular and appears random, or at least pseudo-random, and because

of this a deterministic description of turbulence at all spatial scales is not

feasible;

● it is diffusive, and in most situations where turbulence occurs, the turbulent

transfer of energy, water vapor, momentum and atmospheric entities is much

more effective than transfer by molecular diffusion;

● it is three-dimensional, and entities such as plumes and vortexes can play a

significant role; and finally,

● it is continually being created and destroyed.

Mean and fluctuating components

Separating variations in atmospheric entities into low-frequency and high-

frequency variations is used as the basis for representing the turbulent atmos-

phere mathematically. Variations to the left of the spectral gap are described by

equations which are firmly based on physical principles such momentum, mass

and energy conservation, and which explicitly represent, or ‘resolve’, changes in

the ‘mean flow’ of atmospheric entities. Describing the haphazard turbulent vari-

ations that occur at higher frequencies that are to the right of the spectral gap in

Fig. 15.3 is less tractable. The most common approach is to represent their net

effect in the form of less well-grounded empirical equations, which are often

called ‘ parameterizations’. In later chapters it is shown that such parameterizations

are sometimes written in the form of empirical functions of the mean values of

atmospheric entities, these functions having been derived and calibrated by field

studies.

The first step toward writing a mathematical description of atmospheric changes

and movement is to re-write the value of each of the several atmospheric variables

in a form that explicitly recognizes that the variable has poorly described turbulent



variations that are superimposed on better described variation in their mean

values. Figure 15.4 illustrates this separation for the case of the wind speed

component, u, along the X axis selected to be parallel to the ground. All atmos-

pheric entities show similar variability in a turbulent field and can be similarly

re-written with separate mean and fluctuating components, see Table 15.1.

Rules of averaging for decomposed variables

It is useful that over a time period T of around 20–60 minutes all atmospheric

variables can be considered as being made up of the mean value over that period

and a fluctuating component which by definition has an average value of zero

when averaged over the period T. This allows simplifications when deriving

equations. Table 15.2 documents some of the more important mathematical results

Turbulentfluctuation

Mean flow

Time

u�(t )

u(t )

Win

d sp

eed

u

Figure 15.4 Separation of

the time dependent

horizontal wind speed along

the X axis, u(t), into a time

dependent turbulent

fluctuation component,

u′(t), and a mean flow

component, u–.

Table 15.1 Atmospheric variables and their decomposition into components

representing the mean value of the variable and the fluctuating component

associated with turbulence.

Variable Symbol and decomposition into components

Wind speed parallel to the ground, along direction of the mean wind

u(t ) = u– + u ′(t )

Wind speed parallel to the ground, perpendicular to the direction of the mean wind

v (t ) = v– + v ′(t )

Wind speed perpendicular to the ground w (t ) = w– + w ′(t )Virtual potential temperature qv(t )= q– + qv′(t )Specific humidity q(t ) = q– + q ′(t )Atmospheric constituent, e.g. CO2 c(t ) = c– + c ′(t )



Table 15.2 Averaging rules for time-dependent variables A and B

re-written in terms of their mean and fluctuating components and a time

independent constant C.

Number Averaging rule Number Averaging rule

A1 C_ = C A6 ⎛ ⎞ =⎜ ⎟⎝ ⎠

dA d Adt dt

A2 ( )A B A B+ = + A7 A At t

∂ ∂⎛ ⎞ =⎜ ⎟⎝ ⎠∂ ∂

A3 =( )CA C A A8 ∂ ′ ∂ ′⎛ ⎞ =⎜ ⎟⎝ ⎠∂ ∂a at t

A4 ( )A A= and ( )B B= A9 ∂ ′ ∂ ′⎛ ⎞ =⎜ ⎟⎝ ⎠∂ ∂a a

A At t

A5 =( ) AB A B A10 ⎛ ⎞∂ ′ ∂ ′=⎜ ⎟∂ ∂⎝ ⎠

2 2( ) ( )a at t

that follow when two time dependent variables, A and B, are decomposed into

mean and fluctuating components and the mean value of their fluctuating

components is zero. In Table 15.2, A and B are as follows:

= + ′A A a (15.1)

and

= + ′B B b (15.2)

while C is a fixed constant that does not vary over the averaging period used to

define the mean and fluctuating components of A and B. In this table the presence

of an overbar over a variable implies that an average of that variable is taken over

the time period T.

One important (though obvious) result that immediately follows from these

rules is that because:

= + ′ = + ′ = + ′( ) ( ) ( )A A a A a A a (15.3)

it follows that:

′ = 0a (15.4)



From Equation (15.4), it also follows that:

′ = ′ =( . ) 0B a B a (15.5)

and, by analogy, that:

′ = ′ =( . ) 0A b Ab (15.6)

Equations (15.5) and (15.6) can be used to demonstrate an important result. If one

takes the time-average of the cross product of two atmospheric variables, thus:

= + ′ + ′ = + ′ + ′ + ′ ′( ) ( )( ) ( )AB A a B b A B a B A b a b (15.7)

and then substitutes Equations (15.5) and (15.6) it follows that:

= + ′ ′( ) AB A B a b (15.8)

Thus, the time-average of the cross product of two atmospheric variables is equal

to the sum of two terms, the product of their mean values plus the time-average of

the instantaneous product of their fluctuating components over the period T. This

result is known as Reynolds averaging and it provides the basis for defining and

calculating measures of the strength of atmospheric turbulence and turbulent

fluxes, as described below. It is important to recognize that although Equations

(15.5) and (15.6) show the time-average of the product of a mean value with a

turbulent fluctuation is zero, in general, the time-average of the product of two or

more turbulent fluctuations cannot be assumed to be zero, thus ′ ′ ≠ 0a a , ′ ′ ≠ 0a b ,

′( ′) ≠2 0a b , ′ ′ ≠2 2( ) ( ) 0a b , etc.

Variance and standard deviation

The variance of an atmospheric variable, A, which has been re-written in terms of

mean and fluctuating parts, is formally defined by:

σ = + ′ − 22( ) (( ) )A A a A (15.9)

Multiplying out this equation and (for the purpose of illustrating their application)

applying the rules of averaging as used when deriving Equation (15.8), it follows

that:

σ = + ′ + ′ − + ′ +

= + ′ + ′ − − ′ +

2

2

( ) ( )( ) 2 ( )

2 ( ) 2 2

A A a A a A A a A A

A A A a a A A Aa A A (15.10)



hence:

σ = ′2 2( ) ( )A a (15.11)

but note that in this case Equation (15.11) also follows directly from Equation

(15.9). From Equation (15.11), the standard deviation of A is given by:

σ = ′ 2( )A a (15.12)

Measures of the strength of turbulence

One measure of the strength of atmospheric turbulence is sm

, the square root of

the sum of the variances of the three orthogonal components of wind speed, u

parallel to the mean horizontal wind, v perpendicular to the mean horizontal

wind, and w in the vertical direction. sm

is calculated from:

σ = ′ + ′ + ′2 2 2( ) ( ) ( )m u v w (15.13)

Recall how the strength of the turbulence was judged to vary with time in Fig. 15.1.

Other measures of strength of turbulence can also be defined, including the turbu-

lent intensity, I, which is defined by normalizing sm

by the magnitude of the mean

wind vector, Um

, at the point where sm

was measured. Because Um

is given by:

= + +2 2 2( ) ( ) ( )mU u v w (15.14)

the turbulent intensity is given by:

′ + ′ + ′=+ +

2 2 2

2 2 2

( ) ( ) ( )

( ) ( ) ( )

u v wIu v w

(15.15)

Mean and turbulent kinetic energy

The kinetic energy, EK, of a body of mass M moving with a speed V is given by:

21

2kE MV= (15.16)

When defining the kinetic energy of air in the atmosphere it is usual to normalize

by the density of air to give the turbulent energy per unit mass and to separate the

kinetic energy associated with the mean air flow and the turbulent fluctuations



individually. The Mean Kinetic Energy, MKE, i.e., the energy per unit mass

associated with mean flow in the atmosphere, is given by:

+ +=2 2 2( ) ( ) ( )

2

u v wMKE (15.17)

While the Turbulent Kinetic Energy, TKE, i.e., the energy per unit mass associated

with turbulent fluctuations in the atmosphere, is given by:

′ + ′ + ′=2 2 2( ) ( ) ( )

2

u v wTKE (15.18)

As explained in greater detail in later chapters, the TKE is generated in the

atmospheric boundary layer by the mechanical forces acting between the

atmosphere and an aerodynamically rough surface as it moves, and by forces

associated with atmospheric buoyancy, the latter being enhanced in unstable

conditions but suppressed in stable conditions. TKE is always being destroyed by

friction in the atmosphere and it is the balance between the rate of production of

turbulence and its destruction which determines the amount of turbulent kinetic

energy present at any point and time. Figure 15.5 shows how the TKE typically

changes through the day, while Fig. 15.6 shows typical profiles for TKE as a

function of height in different conditions of atmospheric stability.

Linear correlation coefficient

The covariance, CA,B

, between two variables A and B is defined by the expression:

= − −,

( )( )A BC A A B B (15.19)

08:000

0.2

0.4

0.6

0.8

1.0

1.2

10:00 12:00 14:00

Local time of day (hr.)

Turb

ulen

t kin

etic

ene

rgy

per

met

er(m

2 s−2

)16:00 18:00

Figure 15.5 A typical

diurnal variation in the range

of values for the linear density

of TKE in the lower

atmosphere.



Consequently, if A and B are expressed in mean and fluctuating parts, the

covariance is:

= + ′ − + ′ − = ′ ′,

(( ) )(( ) ) ( )A BC A a A B b B a b (15.20)

In practice, covariance is most commonly used in the form of the Linear Correlation

Coefficient, rA,B

, which is the covariance normalized by the standard deviation of

the two variables, thus:

′ ′=σ σ,

( )A B

a b

a br (15.21)

The value of rA,B

always lies in the range from −1 to +1 and indicates the degree of

commonality between variations in the two variables. Thus, if the two variables are

perfectly correlated (i.e., they vary together in the same direction) then rA,B

= 1,

and if they varied together but in opposite directions rA,B

= −1. In fact perfect

correlation between variations in the values of atmospheric variables is rare, but

significant and important correlations can and do occur in the atmospheric

boundary layer. For example, if one of the variables is the vertical wind speed, w,

and the second is the virtual temperature of the air, qv, then if air that is warmer

than average tends to move upward and air that is colder than average tends to

move downward, their covariance will likely be greater than zero. In this situation,

hotter air is moved upward in the upward fluctuations, to be replaced by cold air

that is moved downward in downward fluctuations, so there is a net flow of energy

upward. Consequently, although on average there is no net vertical motion of the

air and w_

= 0, there is a flow of heat away from the surface that is associated solely

0

0

0.5

1.0

1.5

2.0

1 2 3

TKE per m (m2 s−2)

Bothbuoyant and

frictionalproduction

of TKE

UnstableH

eigh

t (km

)

4 0

0

0.5

1.0

1.5

2.0

1 2 3


Stable

Hei

ght (

km)

4

Frictionalproduction

and buoyantdestruction

of TKE

0

0

0.5

1.0

1.5

2.0

1 2 3


Frictionalproduction

of TKE

Neutral

Hei

ght (

km)

4

Figure 15.6 Typical examples of the variation of the linear density of TKE per unit mass with height in different conditions

of atmospheric stability.



with the fact that the variations in w and qv are correlated. The resulting flow of

heat is called the turbulent flux of sensible heat. We return to this point below.

In the atmospheric boundary layer (ABL), there are substantial correlations

between the fluctuations in atmospheric variables such as virtual potential

temperature, qv, and specific humidity, q, and also between these two variables and

the vertical wind speed, w. The strength of these correlations changes with height

and time of day and are associated with height dependent differences in the ABL

itself and with the transport of turbulent fluxes through it. Figure 15.7 shows

typical variations with height (normalized such that the height of the ABL is unity)

in the linear correlations coefficients rqv,w, r

q,w and rqv,q

in daytime conditions.

During the day and near the ground, there is usually a significant positive correlation

between fluctuations in vertical wind speed, and those in virtual potential temperature

and specific humidity. These are associated with the upward flow of sensible and latent

heat that is transported by turbulent fluctuations away from the surface. There is also

correlation between fluctuations in virtual potential temperature and specific humidity

near the ground reflecting the fact that the air in parcels moving upward tend to be

both warmer and moister than average, while those moving downward tend to be both

cooler and drier than average. At the top of the ABL the situation is entirely different.

The sign of rqv,w, and rqv,q

, are negative implying a downward flow of heat energy, in the

opposite direction to that of moisture which is still upward.

Kinematic flux

The frame of reference we adopt when describing atmospheric flows in the ABL

has the Z axis perpendicular to and positive away from the ground, the X axis

parallel to the ground along the direction of the mean wind, and the Y axis parallel

to the ground and perpendicular to the direction of the mean wind.

−0.5

rqv q

rqv w

rq w

0.5

1.0

0

Correlation coefficient

Hei

ght r

elat

ive

to to

p of

AB

L

00.5

Figure 15.7 Typical daytime

vertical profiles of correlation

coefficients through the ABL.



The flux of any entity is the average value of the product of the volume density

relevant to that entity with the velocity of the air in the required direction. For

example, the flux of water vapor in the X direction over a specified period is the

time-average of:

(mass of water vapor per unit volume) × (velocity of air in X direction)

This water vapor flux, ρ( )aq u , therefore has units of (kg m−3) × (kgwater

kg−1) ×

(m s−1), or kgwater

m−2 s−1. Similarly, the sensible heat flux in the Y direction is the

time-average of:

(heat content per unit volume of air) × (velocity in Y direction)

Consequently, this sensible heat flux, ( )a p vc vr q , has units of (kg m−3) × (J kg−1 K−1) ×

(K) × (m s−1), or J m−2 s−1. And the flux of the Y component of momentum (recall

momentum is a vector) transferred in the Z direction is the time-average of:

(momentum flux in the Y direction per unit volume of air) × (velocity in Z direction)

Hence, this momentum flux, ( )au wr , has units of (kg m−3) × (m s−1) × (m s−1), or

kg m−1 s−2.

However, such entities as ‘heat content per unit volume of air’ and ‘momentum in

the Y direction per unit volume of air’ are either rarely measured or they are

unmeasureable. Rather, it is the equivalent entities, namely ‘virtual potential tem-

perature’ or ‘velocity in the Y direction’ that are measured instead. A product of a

measurable atmospheric entity with a velocity component can thus be related to a

true flux with appropriate physical dimensions, but it has the advantage that it

appears naturally when cross products between atmospheric variables are defined.

Partly because of this, but also because (as shown later) doing so simplifies the

suite of equations that describe atmospheric flow and makes them more comparable

with each other, it is convenient to redefine the fluxes of mass, sensible heat,

momentum and moisture and minority constituents. This is done by dividing by the

density of moist air or, in the particular case of sensible heat, by the product of the

density of moist air with the specific heat of air. It is viable to do this because changes

in the density of air are typically 10% or less through the depth of the atmospheric

boundary layer. The equivalent flux so defined is called the kinematic flux.

Thus, each true flux expressed in appropriate physical units can be associated

with a kinematic flux in different units that is the cross product of a measurable

atmospheric variable with a measureable velocity component. Table 15.3 lists the

true fluxes and their units, the measurable atmospheric entities equivalent to each

flux, the relationship between each pair of actual and kinematic fluxes, and the

dimensions of the equivalent kinematic flux. The description of the theory of tur-

bulence that follows in later chapters is given using kinematic fluxes. However, it

is important to remember that each kinematic flux must ultimately be recast back

into a true flux in appropriate units, as for example when used in equations

describing surface energy exchange.



Table 15.3 The interrelationship between true and kinematic fluxes and their respective units.

Actual flux Units of actual flux

Measurable variable

Relationship of kinematic flux to actual flux

Units of kinematic flux

Mass kgair m−2 s−1 Velocity =k

a

MM

ρm s−1

Sensible heat J m−2 s−1 Potential temperature

kp a

HH

c ρ= K m s−1

Momentum kg m−1 s−2 Velocity (in prescribed direction)

ka

ττρ

= m2 s−2

Moisture kgwater m−2 s−1 Specific humidity k

a

EE

ρ= kgwater kgair

−1 m s−1

Constituent kgconstit m−2 s−1 Relative density of

constituentk

a

CC

ρ= kgconstit kgair

−1 m s−1

Advective and turbulent fluxes

As just described, the kinematic version of the fluxes of mass, sensible heat,

momentum and moisture and minority constituent are given by taking the time-

average of the product of a relevant atmospheric variable with the velocity

component in the direction of interest. Taking as an example the kinematic

sensible heat moving in the direction of the Z axis, the kinematic vertical flux of

sensible heat flux is the time-average product of qv with w. Separating q

v and w

into the mean and fluctuating components defined over the averaging period, the

total sensible heat flux is therefore calculated from:

= = + ′ + ′( ) ( )( )k v v vH w w wq q q (15.22)

This last equation differs from Equation (7.8) in that it no longer includes (rcp)

because it describes the kinematic flux of sensible heat, and it also allows the

possibility of a sensible heat flux associated with the mean vertical flow of air at

mean air temperature. By analogy with Equation (15.8), Equation (15.22) becomes:

= + ′ ′( ) ( ) ( )v v vw w wq q q (15.23)

The first term in Equation (15.23) has dimensions of kinematic sensible heat flux and

describes the mean flow, in this case of thermal energy. It is called the Advective Flux

or Mean Flux and it calculates the possible vertical heat flow that might occur via

transfers that happen on the low frequency side of the spectral gap (Fig. 15.3) if there

were a finite vertical wind speed over the time period for which averaging is made.



The second term in Equation (15.23) also has dimensions of kinematic sensible

heat flux (compare Equation (7.8) ) and is called the Turbulent Flux or Eddy Flux,

in this case of thermal energy. It describes the vertical transfer that results if there

is a positive Linear Correlation Coefficient, rqv,w, between virtual potential

temperature and vertical wind speed. As Fig 15.7 shows, such correlation is quite

common. In fact the turbulent flux usually dominates the advective flux of sensible

heat in the vertical direction because mass conservation requires that the time-

average of mean vertical wind speed is small over flat surfaces. In the Y direction

the mean wind speed is zero by definition, so again transfer along this axis can

only be as turbulent flux. However, in the case of air blowing horizontally over

heterogeneous surfaces (e.g., patches of irrigated crop growing in an arid

landscape) where the near-surface temperature can differ from one location to the

next, transfers along the X axis, via the advective sensible heat flux term, may be

considerable.

Turbulent fluxes occur because turbulence is not truly random. If it were

random, positive excursions in the product of two atmospheric variables would be

cancelled out by negative excursions at some other time during the averaging

period. To further illustrate how such fluxes can arise, consider the example shown

in Fig. 15.8 which is for vertical sensible heat flux in (a) daytime conditions when

there is a superadiabatic temperature profile above the ground; and (b) nighttime

conditions when there is a stable inversion in temperature.

In case (a), a turbulent eddy giving a positive excursion of vertical wind speed

moves a parcel of air upward which is subsequently warmer than its surroundings.

In this case a sensor mounted between the levels labeled 1 and 2 would

simultaneously measure a positive fluctuation in vertical wind speed and a positive

fluctuation in temperature, so the product (qvw) would be positive. Conversely, a

0

Case (a)

w� positiveq� positive

w�positiveq�negative

w� negativeq� negative

w�negativeq� positive

Z Z

Level 2

Level 1

Level 2

Level 1

Netupwardheat flux

(a) (b)

Netdownwardheat flux

q 0Case (b)

q0 0

q q

Figure 15.8 The correlation

between positive and negative

fluctuations in vertical wind

speed, and higher and lower

temperature fluctuations in

(a) daytime conditions with a

superadiabatic temperature

profile, and (b) nighttime

conditions with a stable

inversion in temperature.



turbulent eddy that gives a negative excursion of vertical wind speed moves

a parcel of air downward that is subsequently cooler than its surroundings. In this

case the same sensor would simultaneously measure a negative fluctuation in

vertical wind speed and a negative fluctuation in temperature, and the product

(qvw) would again be positive. Thus, both eddies contribute positively to the time-

average value of the product (qvw) and the kinematic eddy flux of sensible heat is

positive. Hence, there is heat flow away from the surface, consistent with the fact

that it is warmer than the overlying air in the ABL.

In case (b) the situation is reversed. Positive excursions of vertical wind speed

move air upward that is subsequently cooler than its surroundings and negative

excursions of vertical wind speed move air downward that is subsequently

warmer than its surroundings. In both cases the product (qvw) is negative, so

the time-average value of the product (qvw) and the kinematic eddy flux of

sensible heat is negative. There is heat flow from the warmer air toward the

cooler surface.

The discussion just given might wrongly be interpreted as implying that vertical

excursions and associated transport are of similar magnitude and duration. But

this is not the case. Very commonly, measurements taken at a point above the

ground during the day in unstable superadiabatic conditions show strong short-

lived positive excursions in (qvw) superimposed on longer intervening periods

with (qvw) less than zero, see Fig. 15.9. Figure 15.10 shows that the frequency of

occurrence for qv′, w′, and (q

vw)′ are typically skewed in such conditions for both

qv′ and w′, but are particularly strongly skewed for the product (q

vw)′. Such

observations suggest that in this case much of the turbulent transport is occurring

via rapidly ascending ‘convective plumes’ which have limited extent in the

horizontal plane and therefore limited duration at a particular point. This is

confirmed by Fig. 15.11, which shows the normalized frequency distribution of

0

0

1

2

20 40 60

Time (seconds)

Meanvalue of(w� q�)

(w�

q�)

(m s

−1 K

)

80 100

Figure 15.9 Time variation

of (qvw) in unstable,

superadiabatic daytime

conditions. (Redrawn from

Stull, 1988, published with

permission.)



−2 −1 00

50Fre

quen

cy o

f occ

urre

nce

100

150

200

250

w �(m s−1)

2 3 −0.5 00

50

Fre

quen

cy o

f occ

urre

nce

100

150

200

250

300

q�(K)

0.5 1 −0.3 00

100

Fre

quen

cy o

f occ

urre

nce

200

300

400

500

600

700

w�q� (m s−1K)

0.3 0.6

Figure 15.10 Frequency of occurrence of the fluctuations qv′, w′, and (q

vw)′ in unstable superadiabatic daytime

conditions. (Redrawn from Stull, 1988, published with permission.).

−80

0

1

2

1

3

−60 −40 −20 80604020

Angle of attack on vertical wind sensor(degrees relative to horizontal)

Nor

mal

ized

flux

-ang

le p

roba

bilit

yfr

eque

ncy

dist

ribut

ion

0

Figure 15.11 The

normalized flux-angle

distribution of evaporation

for three day’s data collected

over a pine forest (positive

flux values only). Updraughts

are positive. (Redrawn from

Gash and Dolman, 2003,

published with permission.).

evaporation with the angle of the wind vector from the horizontal. There is a

bi-modal distribution with the maximum flux being carried by downward moving

eddies between −5° and −10°, and upward moving eddies between 20° and 25°.

Idealized profiles of turbulent fluxes sensible heat, momentum, and moisture in

daytime and nighttime conditions are shown in Fig. 15.12. During the day the

profiles are large and usually stay roughly constant or change linearly with height

through the ABL. At night the fluxes are much less and fall off rapidly with distance

from the ground.




● Turbulent eddies: fast response sensors of atmospheric variables reveal

quasi-random fluctuations around a mean value whose variance changes with

time, which are not regular and not wave-like, and which suggest there are

structures in the air flow that are sometimes referred to as turbulent eddies.

● The spectral gap: Fourier analysis of observed atmospheric fluctuations

shows the variability associated with time scales of hours to days to months

is separated from variability at higher frequencies by a spectral gap

corresponding to time scales of 30 to 90 minutes in which there is limited

variability.

● Decomposed variables: because there is a spectral gap, atmospheric variables

can be written as being a slowly varying 20–60 minute average (described by

equations based on physical principles) with superimposed haphazard

turbulent fluctuation at higher frequencies that have zero mean value.

00 0 0

Free atmosphere

Surface layer

Capping inversion

Residual layer

Stable boundary layer

Nighttime profiles of turbulent fluxes

1

Hei

ght (

km)

2

(b)

Free atmosphere

Surface layer

Mixed layer

Entrainment layer

Daytime profiles of turbulent fluxes

Heat flux (w�q�)

00 0 0

1

Hei

ght (

km)

2

Momentum flux (w �u�) Moisture flux (w �q�)

Heat flux (w�q�) Momentum flux (w �u�) Moisture flux (w �q�)

(a)

Figure 15.12 Idealized profiles of the turbulent fluxes of sensible heat, momentum and moisture as a function of height

(a) in daytime conditions through the convective mixed layer, and (b) in nighttime conditions through the stable boundary

layer. The typical range of variability in momentum and moisture fluxes is shown in gray.



● Averaging rules: simplifying rules apply when the time-average is taken of

decomposed atmospheric variables, decomposed variables multiplied by a

constant, and derivatives and combinations of decomposed variables

(Table 15.2).

● Reynolds averaging: the cross product of two atmospheric variables is equal

to the product of their mean values plus the time-average of the instantane-

ous product of their fluctuating components: the time-average of the product

of two or more turbulent fluctuations cannot be assumed to be zero.

● Mean and turbulent kinetic energy: Mean Kinetic Energy (MKE) and

Turbulent Kinetic Energy (TKE) per unit mass of air are given by the sum of

the squares of the mean wind speed components and sum of the squares of

the turbulent wind speed components, respectively: TKE is not conserved.

● Correlation of variables: in the ABL there are substantial height- and time-

dependent correlations between the fluctuations in atmospheric variables

(e.g., virtual potential temperature, specific humidity, and vertical wind speed)

that are associated with the transport of turbulent fluxes within the ABL.

● Kinematic fluxes: to simplify the development of equations describing

atmospheric flow described in later chapters it is convenient to work in terms

of kinematic fluxes (e.g., Hk, E

k,, t

k,, etc), these being the fluxes in natural

units (e.g., H, E, t, etc) divided by r or, in the case of sensible heat, by (rcp).

● Advective and turbulent flux: a kinematic flux is the time-average of the

product of an atmospheric variable with the velocity component in the

direction of interest. Reynolds averaging identifies kinematic flux as the sum

of an advective flux (the product of their mean values), and a turbulent flux

(the time-average of the instantaneous product of their fluctuating

components).

References

Gash, J.H.C. and Dolman, A.J. 2003. Sonic anemometer (co)sine response and flux meas-

urement: I. The potential for cosine error to affect flux measurements. Agricultural and

Forest Meteorology, 119, 195–207.

Stull, R.B. (1988) An Introduction to Boundary Layer Meteorology (Atmospheric Sciences

Library). Kluger Academic Publishers, Dordrecht, Netherlands.


Introduction

This chapter introduces the set of equations that are used to describe the

movement and evolution of the atmosphere and its constituents at any point in

time. Later these equations are developed to provide a description of mean

atmospheric flow and atmospheric turbulence by separating the variables used

into mean and fluctuating components and then applying the Reynolds averaging

described in the chapter 15.

One of the equations meteorologists use to describe the atmosphere is the

ideal gas law introduced in Chapter 1. Otherwise, the set of equations used are

simply the conservation laws for mass, momentum, energy and atmospheric

constituents, including moisture. However, to those unfamiliar with them, these

conservation equations may appear complex because they typically involve

many terms. The need to include several terms arises because the conservation

laws are being applied in a complex situation, i.e., in a moving fluid which has

viscosity, which is subject to a gravitational force and constrained to the surface of

a rotating Earth, and which has constituents some of which can undergo phase

changes.

Nonetheless, it is important to understand that the equations are fundamen-

tally just conservation laws, and the approach used to define them in this chapter

reflects this. In each case, the rate of change with time of the local concentration

of each conserved entity is first defined, recognizing that it is the rate of change

appropriate in a moving fluid that is required. Then the several physical processes

that can give rise to changes in local concentration of the conserved entity are

each separately identified and expressed algebraically. The required conservation

equation follows immediately by setting the rate of change in local concentration

equal to the sum of the terms that describe how it might be altered.

16 Equations of Atmospheric Flow in the ABL





Because the equations describing the movement and evolution of atmospheric

properties are complex, sometimes they can be written more efficiently using

shorthand representations, including using the ‘summation convention’ and

‘vector algebra’. Use of such representations has merit for the specialist because it

allows conciseness. However, their use requires first creating familiarity with

the efficient representation adopted, and to those who are not fluent in the

language of the selected representation this can inhibit ready understanding.

In this chapter, the primary goal is to convey understanding of how the basic

equations that describe the atmosphere arise. For this reason, use of such efficient

representations is largely avoided and clumsiness in representation is accepted

if this is more likely to improve comprehension. However, vector algebra equa-

tions are occasionally introduced when this is unlikely to confuse.

For reasons of conciseness, not all the equations sought are derived indepen-

dently and completely in this chapter. Often it is possible to derive an equation in

one dimension and its form in the other two dimensions follows by analogy. Also,

the need for some terms in one conservation equation can be easily recognized by

analogy with equivalent terms in another conservation equation.

Time rate of change in a fluid

Most readers will understand the distinction between the total derivative and

the partial derivative of a variable. However, for those less confident in this

branch of mathematics it is useful to review why the rate of change of an

atmospheric variable with time is written as the sum of more than one term. To

do this the rate of change of momentum in the direction of the mean wind

parallel to the ground is used as an example.

In this chapter the selected frame of reference is defined such that the Z axis

is perpendicular to and positive away from the ground, the X axis is parallel to

the ground pointing east along the line of latitude, and the Y axis is parallel to the

ground along the line of longitude. Consider the rate of change with time of u

(the velocity along the X axis) at a point where the velocity of the moving air

has three components, u, v, and w along the X, Y, and Z axes respectively. For

simplicity, first consider the case when the air is only moving along the X axis.

Figure 16.1 illustrates that, in this case, there are two reasons why there may be a

change in the value of u at a particular point.

First, there may be a change in the value u at the point due to some (as yet

undefined) local force acting at that point (Fig. 16.1). This will give rise to a

change in the velocity component along the X axis which is represented by the

partial derivative of u with respect to time. However, even if there were no local

force acting, the air is moving past the point and the velocity along the X axis

within the moving body of air may not be constant. Consequently, the velocity in

the moving air as it passes at one time may be different to that within the air as it

passes at a later time (Fig. 16.1). The rate of change in the value of u resulting solely


Equations of Atmospheric Flow in the ABL 233

from changes in the velocity field in the moving air as it passes the point is given

by the product of the local velocity in the X direction, u, with the gradient of the

velocity u with respect to X in the moving air. Thus, in this simple case the total

rate of change in u with time is given by the sum of two partial derivative terms,

i.e., by:

∂ ∂= +

∂ ∂du u u

udt t x

(16.1)

However, more generally, the value of the instantaneous velocity component u

within the moving air may be changing not only in the direction of the X axis but

also in the direction of the Y and Z axes, and the air may not just be moving only

in the direction of the X axis. The expression for the total rate of change in u with

time therefore recognizes these two additional possible causes of change and

the full expression for the total derivative is:

du u u u uu v w

dt t x y z

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

(16.2)

Note that if the vector algebra representation were used, the last equation would be

written more concisely as:

⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂= + ∇ ∇⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦( . ) where is the vector operator , ,

du uv u

dt t x y z

(16.3)

X axis

Final velocityfield at time t+dt

Initialvelocityfield attime t

Acceleration due to an imposedforce changing the otherwise

constant field

Acceleration due to moving in avelocity field that changes with

position

Initialposition in

velocityfield

Finalposition in

velocityfield

u(velocityalong theX axis)

u(velocityalong theX axis)

u

x X axisx

Forceat x

u → u + dut u + dux

δux

δx

dut


diagram illustrating the

contributions to velocity

changes in a moving

fluid field.



Entirely analogous arguments can be used to define the total derivative of the v

and w components of velocity with time. Moreover, when applied to a unit volume

of air, Newton’s second law of motion requires that the acceleration along each of

the three axes must be equal to the total force along each axis divided by the

density of air, thus:

∂ ∂ ∂ ∂= + + + =

∂ ∂ ∂ ∂x

a

Fdu u u u uu v w

dt t x y z r

(16.4)

∂ ∂ ∂ ∂= + + + =

∂ ∂ ∂ ∂y

a

Fdv v v v vu v w

dt t x y z r

(16.5)

∂ ∂ ∂ ∂= + + + =

∂ ∂ ∂ ∂Z

a

Fdw w w w wu v w

dt t x y z r

(16.6)

where Fx, F

y, and F

Z are (at this point in time unspecified) axis-specific forces

acting on the parcel of air of unit volume. These three equations describe the

conservation of momentum along the three axes. To include them among the

suite of equations describing the movement and evolution of the atmosphere,

the next step is to identify all the possible ‘force’ terms whose sum causes

change in each velocity component. This procedure is described in the next

section.

In the discussion above, the change in velocity with time along three axes was

used to illustrate how the conservation equations that describe the movement

and evolution of the atmosphere are put together. In this case, the starting point

was the conservation of momentum, but the conservation equations for scalar

quantities (i.e., mass, energy, moisture and other atmospheric constituents) can

also be used as starting points. However, specifying these conservations is easier

than formulating the momentum conservation equations, and can be done by

analogy. For this reason, it is the derivation of the equations for momentum

conservation in the atmosphere which is described in more detail in the next

section.

Conservation of momentum in the atmosphere

To write the equations describing momentum conservation it is necessary to

identify all the possible ‘forces’ that can give rise to changes in the momentum in

the X, Y, and Z directions, to formulate these in mathematical form, and then to

include them as terms on the right hand side of Equations (16.4), (16.5) and (16.6).

In the next three sections the three terms that can change momentum are

considered separately.



Pressure forces

The most obvious reason why there may be change in kinetic momentum (i.e.,

velocity) in the atmosphere is as a result of pressure differences. If there is a gradient

of pressure in a certain direction there will be a difference in the force acting in that

direction on the opposite sides of parcels of air, and acceleration will result. Here

the example of a pressure gradient in the X direction is used as an example.

Consider the elementary volume of air shown in Fig. 16.2 which has a thickness

dx in the X direction and a cross-sectional area A perpendicular to the X direction.

If pressure is not uniform in the X direction, the pressure on the two sides of

the element at locations x and x+dx will be different and equal to P and P′, respectively. The mass of the element is (r

aAdx), so Newton’s law of motion

requires that:

∂ = − ′∂

( ) ( )a

uA x P P A

tr d

(16.7)

Because the volume element is thin, P′ can be estimated from P and the thick-

ness dx by taking the first two terms in a Taylor expansion, thus:

∂′ = +∂P

P P xx

d

(16.8)

Combining Equations (16.7) and (16.8) gives:

∂ ∂= −

∂ ∂1

a

u P

t xr

(16.9)

Clearly analogous derivations can be made for the effect of pressure gradients on

kinematic momentum along the Y and Z directions. Consequently, the three pres-

sure gradient related terms that must be included on the right hand side of

Equations (16.4), (16.5), and (16.6) are respectively:

⎛ ⎞ ∂ ∂⎛ ⎞= = −⎜ ⎟⎜ ⎟ ⎝ ⎠∂ ∂⎝ ⎠1x

a pressure apressure

F u P

t xr r

(16.10)

x

Surface area = A

pressure

�uP P �

x + dx

�t

Figure 16.2 The acceleration

due to a gradient of pressure

in the X direction.



⎛ ⎞ ∂ ∂⎛ ⎞= = −⎜ ⎟⎜ ⎟ ⎝ ⎠∂ ∂⎝ ⎠1y


F v P

t yr r

(16.11)

⎛ ⎞ ∂ ∂⎛ ⎞= = −⎜ ⎟⎜ ⎟ ⎝ ⎠∂ ∂⎝ ⎠1z


F w P

t zr r

(16.12)

Were we to use vector algebra formulation for Equations (16.10), (16.11), and

(16.12) these three equations would be written more concisely as a single vector

equation, thus:

⎛ ⎞ ∂⎛ ⎞= = − ∇⎜ ⎟⎜ ⎟ ⎝ ⎠∂⎝ ⎠ pressurepressure

1

a a

F vP

tr r

(16.13)

Viscous flow in fluids

A second way that the velocity component of a small parcel of air might change at

a particular point is if, as a result of the molecular movements in the air, there is a

net transfer to that point of momentum in the direction of interest. In other words

if, for example, there is more momentum in the X direction diffusing into the par-

cel by molecular diffusion than is diffusing out of the parcel, the local velocity of

the parcel in the X direction will increase. We therefore expect that one of the

‘force’ terms needed on the right hand side of Equations (16.4), (16.5), and (16.6)

will quantify the effect of any imbalance in the amount of momentum in each

direction entering and leaving at the point where the equations are applied. The

next step is, therefore, to consider the equations which describe the molecular dif-

fusion of momentum in air and, from these, to define the term that calculates the

local imbalance for each axis.

The equation describing molecular transfer of momentum in fluids was written

by Newton many centuries ago. It is framed in terms of the viscosity of the fluid,

which is a basic molecular property of the fluid (in the present case, air) that is a

measure of the fluid’s internal resistance to deformation. In other words, it defines

the ease with which (hypothetically) parallel layers of fluid can slip past each other.

Figure 16.3 illustrates a two-dimensional example in which a fluid is undergoing

smooth, streamlined, laminar (i.e., not turbulent) flow in the X direction between

two very large parallel plates separated by a distance h.

Consider the variation in velocity in the Z direction which is perpendicular to

the two plates. In this case, the velocity u in the X direction varies uniformly from

zero at the lower plate to Uh at the upper plate and:

hUu

z h

∂=

∂ (16.14)



The resistance to continued motion between the plates per unit area, which is

called the shearing stress,t, is proportional to the (in this case uniform) gradient

of the velocity in the fluid. More generally, Newton proposed that even if the

gradient were not uniform, the local shearing stress between layers of fluid at z

is proportional to the local gradient in velocity along the X direction at that

point, i.e., that:

∂=

∂a

u

zt mr

(16.15)

where m is a property of the fluid called the dynamic viscosity. In the application

for which Equation (16.15) is required here, it is preferable to re-write the equation

to provide a description of the diffusion in air of the kinematic flux of momentum,

tk (= t /r

a), by defining the kinematic viscosity, u (= m /r

a). The resulting equation

has the form:

∂=

∂k

u

zt u

(16.16)

Note that the dimensions of tk in Equation (16.16) are (m s−1)(m s−1) as they must

be because the kinematic flux of momentum τ = ′ ′k

u w .

The above description is of a steady state in which the rate at which momentum

in the X direction is diffusing vertically does not change with distance along the Z

axis. In this example the desired term to be included in the sum of ‘forces’ on the

right hand side of Equation (16.4) that correspond to unbalanced diffusion of

horizontal momentum would be zero. However, it is when there is imbalance in

diffusion of horizontal momentum that is of general interest. This will occur when

the gradient of u in the Z direction is not uniform. Then the flow of momentum in

the X direction entering a small parcel of air at the point of interest from below will

not necessarily equal that leaving from above, and the parcel will change its veloc-

ity. Figure 16.4 illustrates this in the X–Z plane for the one-dimensional case of

non-uniform flow in the X direction.

Moving plate

Fixed plateX

Z

u

Uh

tFigure 16.3 Velocity and

shearing stress generated by

laminar flow in a fluid

between when two plates, one

stationary and one moving in

the X direction at a velocity Uh.



Consider the small element of air shown in Fig. 16.4 which has cross-

sectional area A and thickness dz and therefore has a volume V = (Adz). There

is a flow of kinematic momentum (velocity) in the X direction of tk per unit

area from below the volume element that is different to the flow tk′ per unit

area leaving from above. This difference will generate an acceleration of the

volume V in the X direction that is equal to the difference between the two

kinematic fluxes, thus:

∂ = ′ −⎡ ⎤⎣ ⎦∂ k k

uV A

tt t

(16.17)

Because the volume element is thin, tk′ can be estimated from t

k and the thickness

dz by taking the first two terms in a Taylor expansion, thus:

∂= +

∂' k

k kz

z

tt t d

(16.18)

Substituting Equation (16.18) and V = (Adz) into Equation (16.17) gives:

( ) k

k k

uA z A z

t z

⎡ ⎤∂τ∂ ⎛ ⎞δ = τ + δ − τ⎢ ⎥⎜ ⎟⎝ ⎠∂ ∂⎣ ⎦

(16.19)

and substituting Equation (16.16) into Equation (16.19) and simplifying gives:

2

2

u u

t z

∂ ∂= υ

∂ ∂ (16.20)

The above analysis only considers the diffusion along the Z axis of velocity (i.e.

kinematic momentum) in the X direction, but analogous analyses can be made for

diffusion along the X and Y axes of kinematic momentum in the X direction.

Consequently, the total rate of change in kinematic momentum in the X direction

associated with molecular diffusion processes that must be included as a ‘forcing

Thickness = dz

Surface area = A

Acceleration

Different gradientsand hence different

fluxes

u

tk

tk�

Figure 16.4 The flow of

momentum in the X direction

transferred by molecular

diffusion into and out of a

small volume when the

magnitude of the horizontal

velocity is changing along the

Z axis.



term’ on the right hand side of Equation (16.4) has contributions from diffusion

along all three axes, as follows:

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= = + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

2 2 2

2 2 2

viscosityviscosity

x

a

F u u u u

t x y zu

r

(16.21)

Similar equations can also be readily derived for kinematic momentum (velocity)

in the Y and Z direction with analogous form, i.e.

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= = + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

2 2 2

2 2 2

viscosityviscosity

y

a

F v v v v

t x y zu

r

(16.22)

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= = + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

2 2 2

2 2 2

viscosityviscosity

z

a

F w w w w

t x y zu

r

(16.23)

Were we to use vector algebra representation, the last three equations could be

written more concisely as a single vector equation thus:

⎛ ⎞ ∂⎛ ⎞= = ∇⎜ ⎟⎜ ⎟ ⎝ ⎠∂⎝ ⎠2

viscosityviscosity

a

F vv

tu

r

(16.24)

Note that the physical process that underlies the linear diffusion equation

describing transfer of momentum by molecular diffusion is very similar to those

that underlie the transfer by molecular diffusion of scalar quantities such as energy,

moisture, and other minor constituents of air. Later in this chapter conservation

equations similar to Equations (16.4), (16.5), and (16.6) are written for these scalar

quantities, and the contribution of molecular diffusion toward changes in the local

concentration of such scalar quantities must be included among the terms on the

right hand side of these conservation equations. It is possible to draw analogy with

the above analysis to define the required terms directly, but the diffusion coefficient

for each scalar quantity is different. Consequently, the molecular diffusion terms

to be included in the conservation equations for heat (which is framed in terms of

the virtual potential temperature, qv), for moisture (which is framed in terms of

the specific humidity, q), and for an unspecified scalar quantity with concentration

c are (uq∇2q

v), (u

q∇2q), and (u

c∇2c), respectively.

Axis-specific forces

In addition to the terms on the right hand side of Equations (16.4), (16.5), and

(16.6) associated with pressure gradients and viscosity, there are additional ‘force’

terms that are specific to each axis. In the frame of reference we have adopted, the

axis-specific force in the Z direction can be immediately identified as the force of



gravity acting toward the surface. Consequently, the axis-specific acceleration in

the Z direction required in Equation (16.6) is:

axis specific

wg

t

∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠

(16.25)

However, because the selected frame of reference is stationary on the surface of the

Earth, there are also axis-specific ‘forces’ causing acceleration, namely the X and Y

components of the Coriolis force which arises because angular momentum must be

conserved.

All bodies rotating around an axis have angular momentum that must be con-

served. The angular momentum of the body is defined as the triple product of the

mass of the body, multiplied by the distance from the axis around which the body

is rotating, multiplied by the speed at which the body is moving. Conservation of

angular momentum applies to the parcel of air at latitude q shown in Fig. 16.5 that

is constrained to move in a plane parallel to the surface of the Earth and that is

moving with an apparent velocity u in the X direction as viewed by an observer

who is stationary on the Earth’s surface.

Because the Earth is rotating with an angular velocity, ω, the true velocity of the

parcel as observed by an independent observer in space is:

= + ωtrue

u u r

(16.26)

where r is distance between the parcel of air at latitude q and the axis of rotation of

the Earth. Angular momentum must be conserved in the frame of reference of this

independent observer and in this frame of reference the angular momentum, G, of

the parcel of air with volume V and density ra is:

Γ = ( )a true

V rur

(16.27)

wr

Angular velocity, w

Radius, r

Latitude, q

uutrue = u + (w r )

Figure 16.5 True speed of a

body along a line of latitude

relative to the apparent speed

when viewed from a frame of

reference fixed on the surface

of the Earth.



Consider next a particular case shown in Fig. 16.6 in which a parcel of air with the

same volume and density initially has no velocity when viewed from a frame of

reference stationary on the surface of the Earth (i.e., u = 0). In this case, Equations

(16.26) and (16.27) give the angular momentum, Γ′, of the parcel as:

Γ′ = ω ( )a

V r rr

(16.28)

Suppose this parcel now begins to move with a fixed velocity v in the Y direction.

At a short time, dt, later the parcel has moved a distance (vdt) along the Y axis and,

because the parcel of air is constrained to move in a plane parallel to the surface of

the Earth, the distance between the parcel and the axis of rotation of the Earth has

changed to (r+dr), where dr = −v dt sin(q), see Fig. 16.6. (Note the radius r

decreases in this case, so dr is negative.) Because angular momentum must be

conserved, the (true) velocity of the parcel must also have changed, by an amount

[dt(∂u/∂t)], such that the new angular momentum is equal to Γ′, i.e.

{ } { }⎡ ⎤− ∂ ∂ + − =⎣ ⎦( ) sin( ) ( ) sin( ) ( )a a

V r v t u t t r v t V r rr d q d w d q r w

(16.29)

It can be shown by multiplying out the left hand side of Equation (16.29) and can-

celling terms that:

∂⎛ ⎞ =⎜ ⎟⎝ ⎠∂axis specific

2 sin( )u

vt

w q

(16.30)

In similar way, if a parcel of air, previously stationary in a frame of reference sta-

tionary on the Earth, suddenly begins to move with a velocity u along the X axis,

angular momentum conservation requires that it then accelerates along the Y axis

in order to move farther from the axis of rotation. In this case geometric

Z

Y

X

vdtvdt

dr

dt

r +dr

r +dr

r

q

�u�t

Note: In this case dr has a negative value:, i.e. dr = − [v.dt. sin (q)]

Figure 16.6 The rate of

change in velocity in the X

direction required to

conserve angular momentum

when a parcel of air moves in

the Y direction at a velocity v.



consideration of the required relationship between the resulting acceleration and

the change in velocity gives:

∂⎛ ⎞ = −⎜ ⎟⎝ ⎠∂axis specific

2 sin( )v

ut

w q

(16.31)

For simplicity when writing Equations (16.30) and (16.31) we have considered

only simple specific cases, but these two equations apply in general and define the

axis-specific forces for the X and Y axes. For consistency with other terms on the

right hand side of Equations (16.4), (16.5) and (16.6), these three equations are

re-written as:

⎛ ⎞=⎜ ⎟⎝ ⎠

axis specific

x

a

Ffv

r

(16.32)

⎛ ⎞= −⎜ ⎟⎝ ⎠

axis specific

y

a

Ffu

r

(16.33)

⎛ ⎞= −⎜ ⎟⎝ ⎠

axis specific

z

a

Fg

r

(16.34)

where:

ω θ= 2 sin( )f

(16.35)

Combined momentum forces

As previously mentioned, to define the equations which together describe the

movement and evolution of momentum in the atmosphere the total rate of change

of momentum along each axis, i.e., Equations (16.4) to (16.6), are combined with

those that describe possible forces that may give rise to change, i.e., Equations

(16.10) to (16.12), Equations (16.21) to (16.23), and Equations (16.32) to (16.34),

as follows:

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = − + υ + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

1

a

du u u u u P u u uu v w fv

dt t x y z x x y zr

(16.36)

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = − − + υ + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

1

a

dv v v v v P v v vu v w fu

dt t x y z y x y zr

(16.37)

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = − − + υ + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

1

a

dw w w w w P w w wu v w g

dt t x y z z x y zr

(16.38)



Conservation of mass of air

The rate of change with time in ra, the density of the parcel of air with volume V

shown in Fig. 16.7 is given by the difference between the incoming and outgoing

fluxes of air mass along all three coordinates. The contribution from along the

x axis is given by:

+

∂⎛ ⎞ = −⎜ ⎟⎝ ⎠∂( )a

a x x x

x

V A u ut

d

rr

(16.39)

where A is the cross-sectional area of the parcel of air in the plane perpendicular

to the X axis. By taking the first two terms in a Taylor expansion, this can be re-

written as:

∂ ⎛ ⎞⎛ ⎞ ∂ ∂⎡ ⎤= − + = −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠∂ ∂ ∂⎝ ⎠⎣ ⎦a

a x x a

x

u uV A u u x V

t x x

rr d r

(16.40)

In three dimensions, the total change in density is therefore:

⎛ ⎞∂ ∂ ∂ ∂= − + +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

a

a

u v w

t x y z

rr

(16.41)

This is the Continuity Equation for the mass of air and applies everywhere in the

atmosphere.

It can be shown that in atmospheric domains where f is the maximum frequency

of pressure waves and cs is the speed of sound, and where typical air velocity is less

than 100 m s−1 and length scale is less than 12 km, (cs2/g), and (c

s2/f ), pressure forces

are able to equilibrate density fluctuations in the atmosphere sufficiently quickly

Changing Internal Density, rwin the Volume V = A dz

Cross SectionalArea, A,

Perpendicularto the x axis

Wz + dz

ux + dx

Vy + dyWz

Vy

ux

Figure 16.7 Axial

contributions to the time rate

of change of mass in a parcel

of air.



for the left hand side of Equation (16.41) to be negligible in comparison with the

right hand side. Such conditions apply in the atmospheric boundary layer (ABL).

Consequently in the ABL the continuity equation for mass of air can be simplified

to become:

0u v w

x y z

∂ ∂ ∂+ + =∂ ∂ ∂

(16.42)

This equation can be re-written in the vector format as:

. 0v∇ = (16.43)

Conservation of atmospheric moisture

The time rate of change of the total moisture concentration at a particular point in

the atmosphere is equal to the sum of two terms. The first is the transfer of mois-

ture to that point by molecular transport. This is equivalent to the transfer of a

component of momentum by molecular transfer described earlier and is repre-

sented by including an analogous term in the continuity equation for moisture.

The second contribution is from a source/sink term, Sq

total, that corresponds to the

possible creation or destruction of water molecules by chemical means.

Consequently, the continuity equation for moisture takes the form:

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂⎛ ⎞∂ ∂ ∂

= υ + +⎜ ⎟∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2 +

total total total total

totaltotal total totalq

q

a

q q q qu v w

t x y z

Sq q q

x y z r

(16.44)

which equation can be written more concisely in vector format as:

∂+ ∇ = υ ∇

∂2. +

totaltotalqtotal total

q

a

Sqv q q

t r

(16.45)

This equation for total moisture might be split into two separate equations which

describe the conservation of water vapor and the conservation of liquid/solid

water (such as cloud droplets) in the atmosphere separately, thus:

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = υ + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2 + +

q v

q

a a

S Eq q q q q q qu v w

t x y z x y z r r

(16.46)

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = υ + + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2 +

ll l l l l l lq v

q

a a

S Eq q q q q q qu v w

t x y z x y z r r

(16.47)



where q is the specific humidity of the air, q′ is the liquid/solid water content per

unit mass, Ev is the rate of creation of water vapor by evaporation/sublimation of

liquid/solid water (in kg m−3 sec−1), and Sq and S

ql are possible separate source/

sinks terms (in kg m−3 sec−1) corresponding to chemical formation for vapor and

liquid/solid water, respectively. Equations (16.46) and (16.47) can also be written

more concisely in vector format similar to Equation (16.45), see Table 16.1.

Conservation of energy

The time rate of change of potential temperature of a parcel of air in the atmos-

phere is equal to the sum of three terms, namely, the (by now familiar) rate of

inflow or outflow of thermal energy by molecular transfer processes, energy

entering or leaving the parcel of air as radiation, and energy that is released or

absorbed as a result of phase changes between water vapor and liquid or solid

water within the parcel. Hence, the required equation can be written (for concise-

ness here in vector format) as follows:

θ

λ∂θ+ ∇θ = υ ∇ θ − ∇ −

∂2 1

. v

n

a p a p

Ev R

t c cr r

(16.48)

where Ev is the moisture evaporated within the parcel (in kg m−3 sec−1), l is the

latent heat associated with the phase change from liquid/solid to water vapor, ra

and cp are respectively the density and specific heat at constant pressure of moist

air. In this equation net radiation has three axial components and nR is the net

radiation vector and the second term on the right hand side of Equation (16.48),

the divergence of the net radiation flux, is made up of three terms, one for each axis,

thus;

∂⎡ ⎤∂ ∂∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦

( )( ) ( )1 1=

n yn x n z

n

a pm a pm

RR RR

c c x y zr r

(16.49)

Among these three terms it is the third, that associated with the vertical flux of

net radiation, which is usually dominant, and in the ABL and over an ‘ideal’

surface, it is assumed that there is no change in horizontal net radiation transfer,

see Equation (5.28).

Conservation of a scalar quantity

The conservation equation for any scalar quantity c (e.g., the concentration of

carbon dioxide) can be written by equating the total derivative of the quantity

to two terms one of which represents the divergence of the flux transferred by



molecular transfer while the second represents all possible sources or sinks of

the quantity (which might include chemical reactions that occur at some level

in the atmosphere, as in the case of ozone). Consequently, the general conser-

vation for such scalar quantities (for conciseness here in vector format) takes

the form:

2. q c

cv c c S

t

∂ + ∇ = υ ∇ +∂

(16.50)

Table 16.1 The suite of equations that describe the evolution of atmospheric variables in

the atmosphere.

Ideal Gas Law:

P = raRdTvConservation of mass:

In general In the ABL

ρ ρ ⎛ ⎞∂ ∂ ∂ ∂= − + +⎜ ⎟⎝ ⎠∂ ∂ ∂ ∂a

au v w

t x y z∂ ∂ ∂+ + =∂ ∂ ∂

0u v wx y z

Conservation of momentum:

υρ

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

1

a

du u u u u P u u uu v w fv

dt t x y z x x y z

υρ

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = − − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

1

a

dv v v v v P v v vu v w fu

dt t x y z y x y z

υρ

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = − − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

1

a

dw w w w w P w w wu v w g

dt t x y z z x y z

Conservation of moisture:

υρ ρ

∂ + ∇ = ∇∂

2. + + q vq

a a

Sq Ev q q

t

(vapor)

υρ ρ

∂ + ∇ = ∇ −∂

2. + ll

ql l vq

a a

Sq Ev q q

t (liquid/solid)

Conservation of energy:

θθ λθ υ θ

ρ ρ∂ + ∇ = ∇ − ∇ −∂

2 1. v

na p a p

Ev R

t c c

Conservation of a scalar quantity:

υ∂ + ∇ = ∇ +∂

2. c cc

v c c St



Summary of equations of atmospheric flow

In addition to the several conservation equations introduced above, the suite of

basic equations describing the atmosphere also includes the ideal gas law for moist

air here given in terms of virtual temperature (see Equation (2.11) and associated

text), i.e., in the form:

=a d v

P R Tr

(16.51)

where Rd is the gas constant for air (287 J kg K) and T

v = T(1 + 0.61q) is the virtual

temperature. Table 16.1 summarizes the resulting set of equations used to describe

the movement and evolution of atmospheric variables.


● Prognostic equations: the set of equations that describe atmospheric flow at

a point in time, which are often called prognostic equations, are just the local

conservation equations for each atmospheric variable, plus the ideal gas law.

● Time rate of change in fluids: there are two ways in which a property of a mov-

ing fluid measured at a point can change, either in response to mechanisms

acting within the fluid itself (e.g., forces, internal diffusion, or source/sink pro-

cesses), or because the property is not constant with distance inside the fluid

as it moves past the point. Their sum is the total rate of change with time.

● Momentum conservation: the prognostic equations for kinematic velocity

are given by applying momentum conservation equations along three axes,

with the total rate of change of momentum equated to the sum of ‘forces’

associated with pressure gradients, molecular diffusion of momentum, and

axis-specific (Coriolis and gravity) forces.

● Mass conservation: the continuity equation for air mass is given by equating

the local rate of change in air density to the difference between incoming and

outgoing air flow along all three axes. In the ABL, changes in air density

equilibrate quickly (at about the speed of sound) and are often neglected.

● Conservation of other variables: the prognostic equations for other varia-

bles (e.g., moisture, temperature, CO2) are given by equating their total rate

of change to the relevant mechanisms by means of which they might be

changed (e.g., atmospheric sources/sinks, phase changes, radiation

divergence).


Introduction

The set of equations which describe the movement and evolution of the atmosphere

in the ABL, at any point in time, were introduced in Chapter 16. In this chapter

these equations are re-written to describe the mean flow for time-average values of

atmospheric variables, including the influence on these mean flow equations of

turbulent fluctuations on the high frequency side of the spectral gap, see Fig. 15.3

and associated text. Doing this involves expressing the value of each variable

described by an equation introduced in Chapter 16 in terms of a mean and a

fluctuating part, then applying the Reynolds averaging rules (see Table 15.2) to

derive the equivalent equation for mean flow, and finally introducing any

simplifications and approximations that are appropriate in the ABL.

Fluctuations in the ideal gas law

Consider first the effect of re-expressing the ideal gas law in terms of atmospheric

variables which recognize mean and fluctuating components. It is convenient to

rearrange the equation P = ra R

d T

v into the form P/R

d = r

aT

v before substituting

P = P—

+ P ', = + ′a a a

r r r and = + ′v v v

T T T to give:

′+ = + ′ + ′ = + ′ + ′ + ′ ′( )( )

a a v v a v v a a v a v

d d

P PT T T T T T

R Rr r r r r r

(17.1)

Averaging this equation it becomes:

′+ = + ′ + ′ + ′ ′

a a a a vv v vd d

P PTT T T

R Rr r r r

(17.2)

17 Equations of Turbulent Flow in the ABL




Equations of Turbulent Flow in the ABL 249

and after applying Reynolds averaging to remove the time average of fluctuating

components, the equation reduces to:

= + ′ ′d a d av vP R RT Tr r

(17.3)

However, in practice, ′ ′a vTr is very much less than a v

Tr in the ABL and it can

safely be neglected, consequently:

=v a v

P R Tr

(17.4)

This equation merely says that the ideal gas law applies to average values, which is

as it should be, since it was observation of average values that originally stimulated

its discovery.

It is useful later to subtract Equation (17.3) from Equation (17.2) and then

divide by Equation (17.4) to give:

′ ′′= +a v

a v

TP

P T

rr

(17.5)

In the ABL, pressure fluctuations are rarely if ever observed to be greater than

0.01 kPa and because mean pressure is on the order of 100 kPa, (P/P—

) is on the

order of 10−4. On the other hand, fluctuations in mean temperature, which is itself

on the order of 300 K, are typically on the order of 1 K, hence ( / )v v

T T is on the

order of 33 × 10−4. Consequently it is possible to neglect (P ′/P— ) in comparison with

the other terms and write:

′ ′ ′≈ ≈a v

a v

T

T

r qr q

(17.6)

Using this equation, density fluctuations in the ABL (which are otherwise hard to

measure) can be estimated from the measurable fluctuations in temperature.

The Boussinesq approximation

Starting from the equation for the conservation of momentum in the vertical

direction, Equation (16.38), with the molecular flow term written in vector form

for conciseness, multiplying by ra, recalling m = (r

a u), and then expressing all the

variables as the sum of mean and fluctuating parts gives:

+ ′ ∂ + ′+ ′ = − + ′ − + ∇ + ′

∂2( ) ( )

( ) ( ) ( )a a a a

d w w P Pg w w

dt zr r r r m

(17.7)



Dividing this last equation by a

r and rearranging gives:

∂ ∂∂ ∂

⎛ ⎞′ ′ ⎛ ⎞+ ′ ′+ = − − − + + ∇ + ′⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

2( ) 1 11 ( )a a

a

a a a a

d w w P Pg g w w

dt z z

r rr u

r r r r

(17.8)

The mean vertical pressure gradient in the atmosphere (around which turbulent

fluctuations occur) is in hydrostatic equilibrium, and is described by Equation (3.3).

Consequently the third term on the right hand side of Equation (17.8) is zero.

Equation (17.6) shows that fractional fluctuations in density can be estimated

from fractional fluctuations in temperature, and are of the order 10−2. Fractional

fluctuations in density can therefore be neglected in comparison with unity on

the left hand side of the equation, but must be retained in the first term on the

right hand side of the equation where they can be estimated from temperature

fluctuations. Hence, Equation (17.8) becomes:

∂∂

+ ′ ′ ′= − − + υ∇ + ′2( ) 1

( )a

d w w Pg w w

dt z

qq r

(17.9)

The approximation procedure just used, in which ‘density fluctuations are neglected

in the inertia (storage) term but are retained in the buoyancy term’, is the Boussinesq

approximation. In an equation of atmospheric flow, implementing the Boussinesq

approximation involves simultaneously replacing each occurrence of ra by

ar and

each occurrence of g by g [(r—aa

+ r′a)/r— ] or g[1 − q ′/q

—].

Neglecting subsidence

Observations in the ABL show that the value of w—, the mean vertical wind speed

(which is sometimes referred to as the rate of ‘subsidence’), is usually small and

commonly less than 0.1 m s−1. On the other hand, the magnitude of fluctuations

around this mean value are much greater and on the order of several meters per

second. For this reason it is often assumed acceptable when writing equations

describing momentum conservation to ‘neglect subsidence’, i.e., to retain terms

involving w′ in the equation while removing those involving w—. With this assump-

tion Equation (17.9) would, for example, simplify to:

∂∂

′ ′ ′= − − + ∇ ′21

a

dw Pg w

dt z

qu

q r

(17.10)



Geostrophic wind

Consider Equations (16.36) and (16.37) which describe momentum conservation

in the X and Y directions when applied to atmospheric flow above the atmospheric

boundary layer. In this case the atmosphere is assumed to be in a steady state and

consequently the time differentials on the left hand side of Equations (16.36) and

(16.37) are zero. Also in this case, terms describing molecular diffusion can be

neglected in comparison with other terms in the equations, i.e., in these equations:

= = ∇ = ∇ =2 20; 0; 0; 0du dv

u vdt dt

u u

(17.11)

Equations (16.36) and (16.37) can therefore be re-written as:

∂∂

= −1

g

a

PU

f yr

(17.12)

∂∂

=1

g

a

PV

f xr

(17.13)

where f = 2ωsin(θ), with q is the latitude and ω angular velocity of the Earth.

The wind speed components Ug and V

g are components of the Geostrophic Wind

which is generated by the large-scale pressure gradients, see Fig. 17.1. Thus, mean

atmospheric flow above the ABL is parallel to the isobars, with low pressure on the

left in the northern hemisphere and low pressure on the right in the southern

hemisphere.

Low

High

YP P + ΔP

P + 2ΔP

Vg

Ug

G

X

Figure 17.1 Axial components of the geostrophic wind

in the northern hemisphere in a region with pressure

gradients.



Divergence equation for turbulent fluctuations

Expressing Equation (16.42), the equation describing the conservation of mass (or

divergence equation) that is relevant in the ABL, in terms of mean and fluctuating

parts gives:

∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ ′ + ′ + ′ ′ ′ ′

+ + = + + + + + =( ) ( ) ( )

0u vu u v v w w u v w w

x y z x x y y z z

(17.14)

Averaging, this equation gives:

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

′ ′ ′+ + + + + = 0

u u v v w w

x x y y z z

(17.15)

From Reynolds averaging rules, the second, fourth, and sixth terms in the last

equation are zero, consequently:

∂ ∂ ∂∂ ∂ ∂

+ + = 0u v w

x y z

(17.16)

While subtracting Equation (17.16) from Equation (17.15) gives the (obvious but

later useful) result:

∂ ∂ ∂∂ ∂ ∂

′ ′ ′+ + = 0

u v w

x y z

(17.17)

Thus, in the ABL the continuity equation holds separately for both the mean and

the fluctuating components of kinematic velocity, i.e., ∇ =. 0u and ∇ ′ =. 0.u

Conservation of momentum in the turbulent ABL

In the following, derivation of the required mean flow equation is illustrated

for the case of the Z axis with those for the X and Y axes then written later by

analogy. Starting from Equation (16.38) and applying the Boussinesq approxima-

tion, gives:

2 2 2

2 2 2

11

w w w w P w w wu v w g

t x y z z x y z

⎛ ⎞′⎛ ⎞+ + + = − − − + υ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

qrq a

(17.18)



( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )2 2 2

2 2 2

1 1

a

w w w w w w w wu u v v w w

t x y z

P P w w w w w wg

z x y z

+ ′ + ′ + ′ + ′+ + ′ + + ′ + + ′

∂⎛ ⎞+ ′ + ′ + ′ + ′′⎛ ⎞= − − − + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∂ ∂ ∂ ∂∂ ∂ ∂

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

q urq

(17.19)

Multiplying out and averaging the last equation, gives:

2 2 2 2 2 2

2 2 2 2 2 2

1 1

a a

w w w w wwu u u u

t t x x x x

w w w w w ww wvv v v w w ww

y y y y z z z z

wP P w w w wwg g

z z x x y y z z

′ ′ ′+ + + + ′ + ′

′ ′ ′ ′+ + + ′ + ′ + + + ′ + ′

⎛ ⎞′ ′ ′ ∂ ′ ∂ ∂ ′= − + − − + + + + + +⎜ ⎟⎝ ⎠

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

q ur rq

(17.20)

Applying Reynolds averaging rules removes terms 2, 4, 5, 8, 9, 12, 13, 16, 18, 20, 22,

and 24 from this equation. Rearranging the remaining terms then gives:

2 2 2

2 2 2

1

a

w w w w P w w w w w wu v w g u v w

t x y z z x y zx y z

⎛ ⎞⎛ ⎞⎛ ⎞ ′ ′ ′+ + + = − − + + + − ′ + ′ + ′⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂

ur

(17.21)

It is now appropriate to re-write the final term in this last equation so that its

relationship to turbulent fluxes becomes more obvious, as follows.

Multiplying Equation (17.17) by w′ and taking the time average gives:

∂ ∂ ∂∂ ∂ ∂

′ ′ ′′ + ′ + ′ = 0u v w

w w wx y z

(17.22)

Subtracting this zero identity from Equation (17.21) and re-expressing the resulting

equation in more concise form gives:

21

a

w w w u v wdw Pg w u v w w w w

dt z x y z x y z

⎛ ⎞′ ′ ′ ′ ′ ′= − − + ∇ − ′ + ′ + ′ + ′ + ′ + ′⎜ ⎟

⎝ ⎠u

r∂ ∂ ∂ ∂ ∂ ∂∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ (17.23)

Expanding each atmospheric variable in mean and fluctuating parts this becomes:



The final term in this equation can now be simplified by recognizing that:

∂ ∂∂∂ ∂ ∂

′ ′′ ′= ′ + ′

( ) w uu wu w

x x x

(17.24)

with similar equations relevant for (v′w′) and (w′w′). Consequently Equation

(17.23) can be re-written as:

2( ) ( ) ( )1

a

u w v w w wdw Pg w

dt z x y z

⎛ ⎞′ ′ ′ ′ ′ ′= − − + ∇ − + +⎜ ⎟

⎝ ⎠ur

∂ ∂ ∂∂∂ ∂ ∂ ∂

(17.25)

Starting from the equations describing momentum conservation in the ABL at a

point in time in the X and Y directions (Equations (16.36) and (16.37) ), and

following a procedure similar to that just used for the Z direction above gives:

∂ ∂ ∂∂∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′= − + ∇ − + +⎜ ⎟

⎝ ⎠2

( ) ( ) ( )1

a

u u v u w udu Pf u u

dt x x y zu

r

(17.26)

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′= − − + ∇ − + +⎜ ⎟⎝ ⎠

21 ( ) ( ) ( )

a

dv P u v v v w vf v v

dt y x y zu

r

(17.27)

There are marked similarities between Equations (17.25), (17.26), and (17.27), the

three equations that describe the evolution of mean flow, and Equations (16.36),

(16.37), and (16.38) that describe instantaneous momentum conservation in the

atmosphere. But differences occur because the effect of turbulent fluctuations must

also be considered when describing the variation in mean quantities, and the

additional terms in the mean flow equation account for contributions from turbulent

flux (as opposed to molecular flux) divergence, e.g., the divergence ∂ ∂′ ′( )u w z in

the turbulent momentum flux ′ ′( )u w . If there is coherence between fluctuating

components of velocity these give rise to turbulent fluxes that move momentum

from one place to another, and losses/gains in these fluxes, this will cause acceleration/

deceleration in the mean flow. Table 17.1 gives the physical meaning of the three

equations describing momentum conservation for mean flow in a turbulent field.

Conservation of moisture, heat, and scalars in the turbulent ABL

Starting from the equations for conservation of moisture, heat and scalar quantities

in the atmosphere, and using an approach analogous to that used in the last section

to derive the equations describing conservation of momentum, equivalent mean

flow equations in the ABL can easily be derived. The form of these equations and

the physical meaning of component terms are given in Tables 17.2, 17.3, and 17.4,

respectively.



Table 17.1 Physical meaning of the terms in the equations describing momentum

conservation for mean flow in a turbulent field.

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − + ∇ − + +⎜ ⎟⎝ ⎠21 ( ) ( ) ( )

a

u u u u P u u v u w uu v w f v u

t x y z x x y zu

r

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ρ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − − + ∇ − + +⎜ ⎟⎝ ⎠21 ( ) ( ) ( )

a

v v v v P u v v v w vu v w f u v

t x y z y x y zu

21 ( ) ( ) ( )

a

w w w w P u w v w w wu v w g w

t x y z z x y z∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ρ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − − + ∇ − + +⎜ ⎟⎝ ⎠u

I II III IV V VI

TERM I Storage of mean momentumTERM II Advection of mean momentumTERM III Influence of Earth’s rotation and gravityTERM IV Influence of mean pressure gradientsTERM V Influence of viscous stress on mean motion (or divergence of molecular

momentum flux)TERM VI Influence of turbulent stress on mean motion (or divergence of turbulent momentum flux)

Table 17.2 Physical meaning of the terms in the equation describing moisture

conservation for mean flow in a turbulent field.

2 ( ) ( ) ( ) q

q

a a

Sq q q q E u q v q w qu v w q

t x y z x y z∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = + + ∇ − + +⎜ ⎟⎝ ⎠u

r r

I II III IV V VI

TERM I Storage of mean moistureTERM II Advection of mean moistureTERM III Mean ‘body’ source of moisture per unit volumeTERM IV Mean creation of moisture as vapor per unit volume by evaporation of other water

phasesTERM V Divergence of mean molecular moisture fluxTERM VI Divergence of turbulent moisture flux

Neglecting molecular diffusion

The Reynolds number, Re, provides a measure of the ratio between the forces

giving turbulent diffusion in a fluid relative to those giving molecular diffusion.

For air, it is defined as the ratio:



Table 17.3 Physical meaning of the terms in the equation describing heat conservation

for mean flow in a turbulent field.

( ) ( ) ( )2 n

a p a p

u v wR Eu v w

t x y z x y zc cθ

∂ θ ∂ θ ∂ θ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ρ ρ

⎛ ⎞′ ′ ′ ′ ′ ′∇+ + + = − − + ∇ − + +⎜ ⎟⎝ ⎠

q q q q u q

I II III IV V VI

TERM I Mean storage of heatTERM II Advection of heat by mean windTERM III Mean heat source from net radiation divergenceTERM IV Mean heat source by latent heat releaseTERM V Divergence of mean molecular heat fluxTERM VI Divergence of turbulent heat flux

Table 17.4 Physical meaning of the terms in the equation describing conservation

of scalars in a turbulent field.

2 ( ) ( ) ( ) c c

c c c c u c v c w cu v w S c

t x y z x y z∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = + ∇ − + +⎜ ⎟⎝ ⎠u

I II III IV V

TERM I Mean storage of scalarTERM II Advection of scalar by mean windTERM III Mean body source of scalarTERM IV Divergence of mean molecular scalar fluxTERM V Divergence of turbulent scalar flux

=ν.

ReU L

(17.28)

where U is a ‘typical velocity’, L is a ‘typical length’, and u = 1.33 × 10−5 m2 s−1 is the

viscosity of air. The Reynolds number is used to characterize the transition from

molecular to turbulent flow as velocity increases. Although defined for momen-

tum transfer, it is also a measure of the relative efficiency of other flux transfers

because the physical processes involved are similar.

Turbulent transfer in the ABL is characterized by a velocity U that is typically on

the order of 1–5 m s−1 and a length L that is typically on the order of 1–100 m.

Consequently the product (VL) is on the order of 1–500 m2 s−1 which means that

the Reynolds number in the ABL is on the order of 105 to 107. This implies

turbulent transfer is about one million times more efficient than molecular transfer

in the ABL. Note that for transfer very close to the surface (i.e., within a few



millimeters), molecular flow remains important – the consequences of this are

discussed in Chapter 21.

All equations describing mean flow in a turbulent field include two flux

divergence terms, one describing transfer by molecular transfer and the other

turbulent transfer. In Equation (17.25), for example, these terms are respectively:

Table 17.5 The suite of equations that describe the evolution of mean atmospheric flow

in the ABL including the effect of turbulent flux divergence.

Ideal Gas Law:

d a vP R Tρ=

Conservation of Mass:

In general In the ABL

aa

u v wt x y z

∂ ρ ∂ ∂ ∂ρ∂ ∂ ∂ ∂

⎛ ⎞= − + +⎜ ⎟⎝ ⎠

0u v wx y z

∂ ∂ ∂∂ ∂ ∂

+ + =

Conservation of Momentum:

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − − − + +⎜ ⎟⎝ ⎠⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − − − + +⎜ ⎟⎝ ⎠

⎛ ′ ′ ′ ′ ′ ′+ + + = − − + +

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

g

g

u u u u u u v u w uu v w f V v

t x y z x y z

v v v v u v v v w vu v w f U u

t x y z x y z

w w w w u w v w w wu v w g

t x y z x y z⎞

⎜ ⎟⎝ ⎠

Conservation of Moisture:

( ) ( ) ( )q

a

S Eq q q q u q v q w qu v w

t x y z x y z∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ρ

+ ⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − + +⎜ ⎟⎝ ⎠

Conservation of Energy:

( ) ( ) ( )n

a p

R E u v wu v w

t x y z x y zc

∂ ∂θ ∂θ ∂θ ∂ θ ∂ θ ∂ θ∂ ∂ ∂ ∂ ∂ ∂ ∂ρ

⎛ ⎞∇ + ′ ′ ′ ′ ′ ′+ + + = − − + +⎜ ⎟⎝ ⎠q

Conservation of a Scalar Quantity:

( ) ( ) ( )c

c c c c u c v c w cu v w S

t x y z x y z∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞′ ′ ′ ′ ′ ′+ + + = − + +⎜ ⎟⎝ ⎠ �

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂

⎛ ⎞⎡ ⎤⎛ ⎞ ′ ′ ′ ′ ′ ′∇ + + + +⎢ ⎥ ⎜ ⎟⎜ ⎟

⎝ ⎠⎢ ⎥ ⎝ ⎠⎣ ⎦

2 2 2

2

2 2 2

( ) ( ) ( ) or and

w w w u w v w w ww

x y zx y zu u (17.29)

Because transfer by turbulent transfer is much more efficient than that by molecular

transfer in the ABL, it is acceptable at this stage to neglect the term describing

divergence of molecular transfer in each conservation equation. Table 17.5



summarizes the resulting set of equations used to describe the movement and

evolution of mean flow in the atmosphere.


● Derivation methodology: turbulent flow equations are all derived by express-

ing variables as turbulent fluctuations superimposed on mean values and

then using Reynolds averaging rules to remove terms with zero time-average

value.

● Simplifying assumptions: often used in the ABL are:

– a v a vT T′ ′ <<r r in the ideal gas law;

– density fluctuations can be estimated from ( / ) ( / )aa′ ≈ ′rr q q ;

– (1 )aa+ ′ rr = 1 in the Boussinesq approximation;

– subsidence can often be neglected (i.e., w— ≈ 0) because |w—| << |w'|

● Geostrophic wind: because the free atmosphere above the ABL is in a steady

state, time differentials in equations describing u and v are zero and the wind

components are Ug = −(2r

aw.sin(q) )−1 (∂P/∂ y) and V

g = (2r

aw.sin(q) )−1

(∂P/∂ x).

● Divergence equation: the continuity equation in the ABL holds for both the

mean and fluctuating components of kinematic velocity, i.e., ∇ =. 0u and

∇ ′ =. 0.u

● Turbulent flux divergence: in all equations describing mean atmospheric

flow in a turbulent field the divergence of fluxes transferred by turbulent

flow must be included in addition to the divergence of fluxes transferred by

molecular flow.

● Prognostic equations: prognostic equations for mean atmospheric flow are

all similar to those for instantaneous atmospheric flow, but they include extra

terms describing turbulent flux divergence (Tables 17.1, 17.2, 17.3, and 17.4).

● Neglecting molecular flow: turbulent transfer is about a million times more

efficient than molecular transfer in the ABL, so it is acceptable to neglect the

divergence of molecular transfer fluxes in prognostic equations for mean flow.


Introduction

The equations describing the evolution of the mean values of atmospheric variables

in the turbulent ABL were introduced in Chapter 17. It is appropriate next to

investigate how these equations control atmospheric behavior by considering

typical observed changes in mean variables and turbulent fluxes in the ABL during

the course of the day. For simplicity, it is helpful to do this while assuming the ABL

overlies a flat, horizontally homogeneous surface. This makes the equations

simpler to understand because it means terms that represent the rate of change of

mean values or turbulent fluxes with distance along the X and Y axes can be

assumed small in comparison with those that describe the rate of change along the

Z axis. If the terrain is flat, it is also plausible to assume that the mean wind speed

along the Z axis (i.e., subsidence) is zero.

Nature and evolution of the ABL

In general terms, the lower atmosphere can be divided into the four main layers

which are diagnosed by the rate of change with height of virtual potential

temperature, wind speed, specific humidity and other scalar variables, as shown

in Fig. 18.1 for daytime conditions. The lowest layer, the surface layer, which has a

depth on the order of 100 meters, is strongly influenced by the aerodynamic

roughness of the underlying surface and by surface heating. In this layer, mean

atmospheric variables initially change rapidly with height but the rate of change

becomes progressively less away from the surface. During the day, air in the surface

layer is usually unstable because of surface warming but at night the surface layer

usually becomes stable as the surface cools by emitting longwave radiation.

18 Observed ABL Profiles: Higher Order Moments





The second main layer of the ABL is the mixed layer which is usually so well-

mixed by atmospheric turbulence during the day that the rate of change with

height of mean atmospheric variables is comparatively small. The mixed layer

grows deeper during the day and can reach a height of several kilometers. Above

the mixed layer there is a strong inversion in potential temperature which has the

effect of inhibiting mixing between the well-mixed turbulent ABL and the free

atmosphere above. This layer of air is often called the inversion layer and is typically

a few tens to a few hundred meters deep. It is also sometimes called the entrainment

layer because it is the entrainment of air from the free atmosphere into the ABL

which causes the mixed layer to grow through the day. In the inversion or

entrainment layer, the mean values of atmospheric variables change rapidly. Above

the inversion layer the free atmosphere then extends upward through the

troposphere. The geostrophic wind speed applies in the free atmosphere and the

air is typically warmer (in terms of potential temperature) and also usually drier

than the air in the mixed layer.

Figure 18.2 shows the typical evolution of the ABL in clear sky conditions

starting at sunrise through the subsequent day and night. The depth of the several

layers in the ABL evolve in response to surface heating by solar radiation during

the day and to surface cooling by longwave radation at night. During the day, heat

and water vapor enter the mixed layer through the surface layer. Some of the air in

the free atmsophere is captured and becomes part of the mixed layer because large-

scale turbulence in the mixed layer can generate temporary breakdowns in the

thermal inversion that otherwise inhibits downward transfer. As a result, the depth

of the mixed layer grows. Its temperature also increases partly as a result of surface

Height

h

0

Actual

Geostrophic

Free atmosphere

Entrainment layer

Mixed layer

Surface layer

q—v

u� q� c�

Figure 18.1 Typical height variation in the value of average atmospheric variables through the four main layers which

make up the ABL.


Observed ABL Profiles: Higher Order Moments 261

heating and partly as a result of the downward mixing of the warmer air from the

free atmosphere above. On the other hand, the moisture input from the surface is

mainly used to moisten the drier air that is being captured from above, so the

diurnal cycle in humidity content in the ABL is usually limited. At night, the

surface cools and a stable surface layer develops. This may acheive a height of

several hundred meters and is largely uncoupled by a stable boundary layer from

the decaying remnants of the previous day’s mixed layer. At sunrise the surface

layer again becomes unstable, and the process of mixed layer growth is reinvigorated.

Daytime ABL profiles

It is instructive to consider the equations describing conservation of heat and

moisture in the ABL during the day in clear sky conditions over uniform, flat ter-

rain. In these conditions, horizontal advection of heat and moisture, and the diver-

gence of horizontal turbulent fluxes can be neglected. Consequently, terms which

involve the partial derivative with respect to x and y can be neglected in the con-

servation equations. It is also reasonable to assume there is negligible subsidence

over the flat surface so terms involving the mean velocity of w can also be neglected.

Net radiation divergence can arguably be neglected in daytime conditions, there

are no chemical sources of water and, if there is no boundary layer cloud, terms

describing phase changes in atmospheric moisture are zero. With these several

simplifying assumptions, the equations describing the mean flow for moisture and

temperature in Table 17.5 simplify dramatically and become respectively:

∂ ∂ ′ ′= −∂ ∂

( )q w qt z

(18.1)

( )wt z

∂ ∂ ′ ′= −∂ ∂

θq (18.2)

06:00Sunrise

Height

Mixedlayer

Free atmosphere

Inversion layer

Residualmixedlayer

Stableboundary

layer

Unstable surface layer Stable surface layer

18:00Sunset

12:00 06:00Sunrise

12:0000:00

Figure 18.2 Typical diurnal

evolution of the ABL over

land under clear sky

conditions.



Note the minus sign on the right hand side of these two equations. This means that

if the moisture flux and heat flux reduce with height above the ground (i.e., their

divergence is negative) there will be an increase with time in the local moisture

and heat content in the air. Alternatively, when the moisture flux or the heat flux

increases with height and the partial derivatives are positive, the local moisture

content or temperature decreases.

Field experiments can demonstrate these last two equations in action.

Figure 18.3 shows measurements of fluxes made at different heights using aircraft

flying over a uniform study area in Oklahoma on two days in May 1983. These

daytime observations were made in clear sky conditions over terrain that was

uniform and flat. Because it takes a finite time to make each aircraft measurement

and because both the surface fluxes and the height of the ABL changes with time

of day, in these figures the fluxes measured during each flight are normalized by

the then-current value of the surface flux, and the height above the surface is

normalized by the then-current height of the ABL. Notice that there is substantial

scatter in the measurements of moisture flux. Likely this is partly because humidity

measurements are less accurate than temperature measurements.

The general behavior on the two days for which data are available is broadly

similar in each case. Both fluxes are positive near the ground and there is the input

of energy and evaporated moisture from the surface resulting from the input of

solar radiation. There is then an almost uniform fall off in the value of the heat flux

with height up to the bottom of the inversion layer, from positive values to negative

values. Over this height range, Equation (18.2) implies there is an increase in

(a) (b)

−1.0

0.0

Normalized flux

Nor

mal

ized

hei

ght

0.5

1.0

1.5 Heat fluxMoisture flux

−0.5 0.5 1.00 −1.0

0.0

Normalized flux

0.5

1.0

1.5 Heat fluxMoisture flux

−0.5 0.5 1.0 1.5 2.00

Figure 18.3 Aircraft measurements on two days in May 1983 of daytime heat and moisture fluxes made at different

heights through the ABL, shown with height normalized to the height of the ABL and fluxes normalized to the surface

fluxes at the time of measurement. (Redrawn from Stull, 1988, published with permission.)



temperature with time that is almost independent of height, consistent with the

fact that there is a well-mixed atmosphere over this height range. The heat flux

changes sign through the mixed layer revealing that this ABL warming is supported

partly by upward heat from the ground at lower levels and partly by downward

heat in the warm air entrained from above the inversion layer at higher levels. The

heat flux remains negative (i.e., downward) through and just above the inversion,

but the rate of change in heat flux becomes positive. Equation (18.2) therefore

implies that at these levels the air that is being entrained from the overlying free

atmosphere is cooling as it merges with the cooler air in the growing mixed layer.

The behavior of the moisture flux is less well-illustrated by the data, but is clearly

very different. The moisture flux remains positive through the mixed layer and

then falls off progressively through and just above the inversion layer. The change

in moisture flux with height through the mixed layer is small (slightly negative on

one day and slightly positive on the second day), so Equation (18.1) implies there is

only modest change in moisture content with time. Through and just above the

inversion layer, the moisture flux remains positive, but it falls rapidly. At these levels

moisture from the surface, which has largely passed straight through the mixed

layer, is used to moisten the drier free atmosphere air as it is entrained downward

to become part of the growing mixed layer.

In summary, the behavior of observed fluxes through and above the evolving

daytime boundary layer shows boundary layer growth involves inputs from both

below and above the inversion layer. Growth is achieved by entraining air from the

free atmosphere which is warmer and drier than the air already in the mixed layer.

Moisture that entered the mixed layer from below is largely used to moisten the

dry entrained air. Hence, the humidity of the air in the ABL is often reasonably

constant through the day. On the other hand, sensible heat is brought into the ABL

from both below and above, and there is a significant diurnal cycle in ABL

temperature during the day. This comparatively simple model of boundary layer

growth over flat uniform terrain can be modeled quite well. Figure 18.4a, for

example, shows the modeled time evolution of profiles of potential temperature,

which can be compared with observed profiles measured at Wangara, Australia on

a day in 1967, shown in Fig. 18.4b. Figure 18.4c shows modeled profiles of humidity

on the same day which can be compared with the observed profiles shown in

Fig. 18.4d.

Nighttime ABL profiles

At night, turbulence in the ABL declines and other terms in the equations

describing the mean flow remain significant in comparison. Over uniform flat

terrain the rate of change of mean values or turbulent fluxes with distance along

the X and Y axes can still be assumed to be small, but subsidence (the mean wind

speed along the Z axis) cannot necessarily be assumed to be zero. Also, because

the net radiation is now all longwave radiation, temperature variation with height


Figu

re 1

8.4

(a)

Mo

del

ed a

nd

(b

) o

bse

rved

pro

file

s o

f p

ote

nti

al t

emp

erat

ure

an

d (

c) m

od

eled

an

d (

d)

ob

serv

ed p

rofi

les

of

hu

mid

ity

on

a d

ay i

n 1

96

7 a

t W

aran

ga,

Au

stra

lia.

(R

edra

wn

fro

m S

tull

, 19

88

, pu

bli

shed

wit

h p

erm

issi

on

.)

6

0

1.0

2.0

0

1.0

2.0

Height (km)

Height (km)

1014

186

1014

18

18:0

0

09:0

0 12:0

015:0

016

:00

14:0

0

12:0

0

11:0

0

10:0

009

:00

(b)

(a)

q�

v (�

C)

q�

v (�

C)

0

1.0

2.0

Height (km)

0

1.0

2.0

Height (km)

12

34

12

18:0

0

10:0

0

11:0

0

12:0

0

14:0

0

16:0

0

09:0

009

:00

12:0

0

15:0

0

(d)

(c)

34

q�

(gm

kg

−1)

q�

(gm

kg

−1)



may mean the divergence of net radiation with height is not necessarily small. In

this case the equation describing heat conservation in Table 17.5 simplifies to:

( )n

pa

R ww

t z zc∇∂ ∂ ∂ ′ ′+ = − −

∂ ∂ ∂θ θq

r (18.3)

Thus, the equation now includes terms that describe the nighttime subsidence and

net radiation divergence. Figure 18.5a shows an example of the evolution of the

profile of potential temperature observed over a flat uniform site in Oklahoma

between 21:00 and 23:30 on a June day in 1983, and Fig. 18.5b shows a model-

calculated estimate of how the three height dependent terms in Equation (18.3)

contributed to the observed change over this period.

Higher order moments

Prognostic equations for turbulent departures

In Chapter 17, the basic equations of atmospheric flow defined in Chapter 16 were

developed to provide a suite of equations describing the evolution of mean varia-

bles. These are referred to as prognostic equations for the mean atmospheric flow

variables ū, v−, w−, −θ, q−, etc. But it is also possible to derive a similar suite of prog-

nostic equations for the turbulent departures u′, v′, w′, q ′, q′, etc., as now illus-

trated for the example of vertical velocity.

Starting from Equation (16.38) and applying the Bossinesq approximation,

expanding each atmospheric variable as mean and fluctuating components, then

multiplying out the resulting equation gives:

−30950

900

850

800

750

950

900

850

800

750

−20 −10 10 20 3020 3010 0

Turbulence

Temperature change (�C/day)Temperature (�C)

(a) (b)

Pre

ssur

e (m

b)

Pre

ssur

e (m

b)

Subsidence

Radiation

2100CDT

2230CDT

Figure 18.5 (a) observed change in the profile of potential temperature over a flat uniform site in Oklahoma between 21:00

and 23:30 on a June day in 1983; (b) model-calculated contributions to this observed change associated with subsidence and

the divergence of sensible heat and net radiation. (Redrawn from Carlson and Stull, 1986, published with permission.)



to give the result:

Subtracting from this last equation the prognostic equation for mean vertical wind

speed given in Table 17.5 gives:

∂ ∂ ′ ∂ ∂ ′ ∂ ∂ ′ ∂ ∂ ′ ∂ ∂ ′+ + + + ′ + ′ + + + ′ + ′

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂′∂ ∂ ′ ∂ ∂ ′ ∂ ∂ ′

+ + + ′ + ′ = − + − −ρ ρ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ′ ∂ ∂ ′ ∂ ∂ ′+ + + + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2 2 2 2 2 2

2 2 2 2 2 2

1 1v

a av

w w w w w w w w w wu u u u v vv vt t x x x x y y y y

w w w w PPw w w w g gz z z z z z

w w w w w wx x y y z z

θθ

υ

(18.4)

Using essentially the same procedure as above, similar prognostic equations can

be derived for turbulent fluctuations in other velocity components.

Equation (18.5) is the short-lived prognostic equation for the fluctuation w′. In

principle, it might be used in a short time-step model of turbulence. However, in

such basic form prognostic equations of turbulent fluctuations have limited value

because their descriptive ability is limited to the time of existence of a turbulent

eddy. However, the equations can be used to derive prognostic equations for

turbulent variance. Again their derivation is illustrated by example for the case of

variance in vertical velocity.

The first step is to multiply Equation (18.5) by 2w′ and then to collect terms

using the relationships:

( ) ( ) ( )2 2 2

2 2 2

1 v

av

w w w w w w w w w wu w u v w u v wvt x y z x y z x y z

u w v w w wP w w wgz x y zx y z

⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ∂ ∂ ∂ ′ ∂ ′ ∂ ′+ + + + ′ + ′ + ′ + ′ + ′ + ′⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞∂ ′ ′ ∂ ′ ′ ∂ ′ ′⎛ ⎞′ ∂ ′ ∂ ′ ∂ ′ ∂ ′= − + + + + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂∂ ∂ ∂⎝ ⎠ ⎝ ⎠

qu

rq

(18.5)

∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′= ′ = ′ = ′ = ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2 2 2( ) ( ) ( ) ( )2 ; 2 ; 2 ; 2

w w w w w w w ww w w wt t x x y y z z

(18.6)

⎛ ⎞⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ∂ ∂+ + + + ′ ′ + ′ ′ + ′ ′⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ′∂ ′ ∂ ′ ∂ ′ ∂ ′+ ′ + ′ + ′ = ′ − ′⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ′ ∂ ′ ′+ ′ + + + ′ +⎜ ⎟ ∂∂ ∂ ∂⎝ ⎠

2 2 2 2

2 2 2

2 2 2

2 2 2

( ) ( ) ( ) ( )2 2 2

( ) ( ) ( ) 1 2 2

( ) ( ) 2 2

v

av

w w w w w w wu v w w u w v w wt x y z x y z

w w w Pu v w w g wx y z z

w w w u w v ww wxx y z

qrq

u⎛ ⎞∂ ′ ′

+⎜ ⎟∂ ∂⎝ ⎠( )w w

y z

(18.7)



The last term disappears on taking the time average of this last equation and

applying Reynolds averaging rules, and the equation then becomes:

⎛ ⎞ ⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ∂ ∂+ + + + ′ ′ + ′ ′ + ′ ′⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

⎛ ⎞ ′ ′∂ ′ ∂ ′ ∂ ′ ∂ ′⎛ ⎞+ ′ + ′ + ′ = − ′⎜ ⎟ ⎜ ⎟⎝ ⎠∂ ∂ ∂ ∂⎝ ⎠

+

2 2 2 2

2 2 2

( ) ( ) ( ) ( )2( ) 2( ) 2( )

( )( ) ( ) ( ) 2 2

2

v

av

w w w w w w wu v w w u w v w wt x y z x y z

ww w w Pu v w g wx y z z

qrq

⎛ ⎞′∂ ′ ∂ ∂ ′′ + ′ + ′⎜ ⎟∂ ∂ ∂⎝ ⎠

2 2 2

2 2 2

ww ww w wx y z

u

(18.8)

Recall that the divergence of turbulent fluctuations is zero in the ABL. Consequently

Equation (18.8) still holds if the time average of the product of (w′)2 with the diver-

gence of turbulent fluctuations is added into the left hand side of Equation (18.8).

When this is done, the equation becomes:

⎛ ⎞ ⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ∂ ∂+ + + + ′ ′ + ′ ′ + ′ ′⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

⎛ ⎞∂ ∂ ∂′ ′ ′∂ ′ ∂ ′ ∂ ′ + ′ + ′ + ′+ ′ + ′ + ′⎜ ⎟∂ ∂ ∂∂ ∂ ∂⎝ ⎠′ ′ ∂ ′

= − ′∂

2 2 2 2

2 2 22 2 2

( ) ( ) ( ) ( )2( ) 2( ) 2( )

( ) ( ) ( ) ( ) ( ) ( )

( ) 22

v

av

w w w w w w ww w u w v w wvut x y z x y z

u v uw w w w w wu v w x x xx y zw Pg wq

rq⎛ ⎞⎛ ⎞ ′∂ ′ ∂ ∂ ′⎜ ⎟ + ′ + ′ + ′⎜ ⎟⎝ ⎠ ∂ ∂ ∂⎝ ⎠

2 2 2

2 2 22

ww ww w wz x y zu

(18.9)

The product rule of calculus gives the four identities:

∂ ′ ′ ∂ ′ ∂ ′= ′ + ′∂ ∂ ∂

∂ ′ ′ ∂ ′ ∂ ′= ′ + ′∂ ∂ ∂

∂ ′ ′ ∂ ′ ∂ ′= ′ + ′∂ ∂ ∂

∂ ′ ′ ∂ ′ ∂ ′= ′ + ′∂ ∂ ∂

2 22

2 22

2 22

( ) ( )( )

( ) ( )( )

( ) ( )( )

( )

u w w uu wx x x

v w w vv wy y y

w w w ww wz z z

w P P ww Pz z z

(18.10)



Using the first three identities in the fourth term of Equation (18.8) and

rearranging gives:

⎛ ⎞ ′ ′∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ′ ∂ ′+ + + = − + ′⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ′ ′ ∂ ′ ′ ∂ ′ ′− ′ ′ + ′ ′ + ′ ′ − + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ′ ∂ ′ ∂ ′+ ′ + ′ + ′⎜ ⎟∂ ∂ ∂⎝ ⎠

2 2 2 2

2 2 2

2 2 2

2 2 2

( )( ) ( ) ( ) ( ) 2 ( ) 22

( ) ( ) ( )2( ) 2( ) 2( )

2

v

a av

ww w w w w P wu v w g Pt x y z z z

w w w u w v w w ww u w v w wx y z x y z

w w ww w wx y z

qr rq

u

(18.11)

The product rule of calculus gives the results:

∂ ′ ∂ ′ ∂ ′⎛ ⎞= ′ + ⎜ ⎟⎝ ⎠∂ ∂∂⎛ ⎞∂ ′ ∂ ′ ∂ ′= ′ + ⎜ ⎟∂ ∂∂ ⎝ ⎠

∂ ′ ∂ ′ ∂ ′⎛ ⎞= ′ + ⎜ ⎟⎝ ⎠∂ ∂∂

22 2 2

2

22 2 2

2

22 2 2

2

( )

( )

( )

w w wwx xx

w w wwy yy

w w wwz zz

(18.12)

but observations in the ABL show:

⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′⎛ ⎞ ⎛ ⎞<< << <<⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂∂ ∂ ∂⎝ ⎠

22 22 2 2 2 2 2

2 2 2

( ) ( ) ( ); ;

w w w w w wx y zx y z

(18.13)

Together Equations (18.12) and (18.13) imply that the last term in Equation

(18.11) can be re-written such that the equation becomes:

⎛ ⎞ ′ ′∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ′ ∂ ′+ + + = − + ′⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛⎛ ⎞∂ ∂ ∂ ∂ ′ ′ ∂ ′ ′ ∂ ′− ′ ′ + ′ ′ + ′ ′ − + +⎜⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝

′∂−

2 2 2 2

2 2

( )( ) ( ) ( ) ( ) 2 ( ) 22

( ) ( ) ( 2 ( ) ( ) ( )

2

v

a av

ww w w w w P wu v w g Pt x y z z z

w w w u w v w w ww u w v w wx y z x y z

w

qr rq

u⎛ ⎞′ ′⎛ ⎞∂ ∂⎛ ⎞ ⎛ ⎞⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎝ ⎠⎜ ⎟⎝ ⎠

22 2w wx y z

(18.14)



This is the required prognostic equation for the variance in vertical wind speed.

Clearly analogous equations can be derived for the variance in horizontal wind

speeds u and v, and these three equations can be combined to describe the evolu-

tion of turbulent energy as described in the next section.

Prognostic equations for turbulent kinetic energy

The turbulent kinetic energy, e, provides a measure of the intensity of turbulence in

the ABL and is therefore strongly related to the turbulent transport of momentum,

heat, and moisture, see Equation (15.11) and is defined by the equation:

( )′ ′ ′= = + +2 2 21

2

TKEe u v wr

(18.15)

The prognostic equation for TKE is obtained by combining the prognostic

equations for variance along all three axes and takes the form:

⎛ ⎞ ′ ′∂ ∂ ∂ ∂ ∂ ′ ′ ∂ ′ ′ ∂ ′ ′+ + + = − − −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ∂ ∂+ ′ + + − ′ ′ + ′ ′ + ′ ′⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂− ′ ′ + ′ ′ + ′ ′ − ′ ′ + ′ ′ + ′⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) (

v

v

w u P v P w Pee e eu v w gt x y z x y z

u u uu v wP u u u v u wx y z x y z

w wv v vv u v v v w w u w v wx y z x y

qq

⎛ ⎞∂′⎜ ⎟∂⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′ ∂ ′⎛ ⎞ ⎛ ⎞⎜ ⎟− + + − υ + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎜ ⎟⎝ ⎠ ⎝ ⎠

22 2

)wwz

u e v e w e u u ux y z x y z

(18.16)

Equation (17.17) requires that the fifth term on the right hand side of Equation

(18.16) is zero. Moreover, if this equation is written in a coordinate system which

is aligned with the mean wind so that terms involving v− are zero, and applied over

a flat, homogeneous area with no subsidence so that terms involving (∂/∂x), (∂/∂y)

and w− are also zero, the equation simplifies to:

( ) 1 ( )( )

I II II I IV V VI

v

av

uw w P w ee g u wt z z z

′ ′ ′ ′ ′∂ ∂ ∂ ∂′ ′= − − − −∂ ∂ ∂ ∂

q erq

(18.17)

where e is the turbulent dissipation of TKE defined by:

⎛ ⎞⎛ ⎞∂ ′ ∂ ′ ∂ ′⎛ ⎞ ⎛ ⎞⎜ ⎟= + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎝ ⎠⎜ ⎟⎝ ⎠

22 2u u ux y z

e u

(18.18)



Because e is the sum of squared terms it is always positive, and −e, i.e., Term VI in

Equation (18.17), is always negative.

The physical meanings of the terms in Equation (18.17) are as follows:

TERM I represents temporal change in the local ‘storage’ of TKE.

TERM II is either a buoyant production or destruction term, depending on

the sign of the buoyancy flux ( )′ ′vw q which is positive giving

production during the day and negative giving destruction of

turbulence at night.

TERM III describes the redistribution of TKE by pressure fluctuations.

TERM IV describes the production of turbulence by friction which is always

positive because ( )′ ′u w is always negative while (∂ū/∂z) is always

positive, and which is greatest in regions where the momentum

flux and gradient of wind speed with height is large.

TERM V represents the turbulent transport of TKE between different

levels.

TERM VI viscous dissipation of TKE.

The presence of TERM VI in Equation (18.17) means TKE is continually being

destroyed and will decay if not also continually created either by buoyant

production (TERM II) or by frictional production (TERM IV). Within the wide

observed ranges reported by Stull (1988), Fig. 18.6 illustrates the typical observed

relative strength of terms in the prognostic equation for TKE over flat, homogeneous

terrain in daytime conditions as a function of height relative to the top of the ABL.

−1.5

0

0.2

0.4

0.6

0.8

1.0

1.2Bouyancy

Turbulence is generated bybuoyancy near the groundand throughout the ABL

but is destroyed near andthrough the inversion layer

DissipationTurbulence is destroyed atall levels but most strongly

near the ground

Shear generationTurbulence is generated by

friction, mainly near theground

TransportTurbulence near thesurface is transferred

upwards through the ABL

1.0 −0.5

Relative size of terms in the TKE budget

Hei

ght r

elat

ive

to to

p of

the

AB

L

0.5 1.0 1.5

Figure 18.6 Typical values of terms in the prognostic equation for Turbulent Kinetic Energy and their variation with

height relative to the top of the ABL in daytime conditions. Note that there is very substantial variability around these

typical values.



Figure 18.7 shows how TKE of air in the ABL is ‘spun up’ during the day and then

subsequently decays at night.

The turbulent kinetic energy in the ABL is present across a range of frequencies

and the shape of this spectrum evolves with time depending on local ambient

conditions. Figure 18.8a shows a typical spectrum for TKE in unstable conditions.

The terms in Equation (18.17) also have different spectra, and Fig. 18.8b shows the

spectra for the buoyant and shear production terms and the turbulent dissipation

term. This figure reveals that turbulence is largely produced at low frequency but

is mainly dissipated at high frequency. The energy present in large eddies provides

energy to smaller and smaller eddies until the secondary eddies so created are

small and the spatial gradients of variance in the viscous dissipation term, e, there-

fore large. The kinetic energy associated with motion in the turbulent air is then

dissipated through friction as heat.

Prognostic equations for variance of moisture and heat

The prognostic equation for the variance in the moisture content and heat

content of air in the ABL can be derived following procedures broadly analo-

gous to that used to derive Equation (18.13), the prognostic equations for the

0

0.5

0.761.18

0.40

0.30

0.18

0.05

1.05

0.93

0.68

0.430.30

0.18

0.05

0.68

0.55

0.43

0.300.18

0.05

Hei

ght (

km)

1.0

1.5

12

Day 1 Day 2

Time (hours)

Day 3

18 0 6 12 18 0 6

Figure 18.7 Measured TKE with height in daytime conditions at Wangara, Australia. (Redrawn from Yamada and Mellor,

1975, published with permission).



0.001

−0.4

−0.2

0.2

0.4

0

0.5

0

1.0(a)

(b)

0.01 0.1 1 10 100 1000

0.001 0.01 0.1 1 10 100 1000

TKE dissipated smalleddies by friction

TKE in large eddies producedby both buoyancy and shear

Rel

ativ

e si

ze o

fT

KE

term

sR

elat

ive

cont

ribut

ion

to T

KE

Energy transfer betweeneddy frequencies Dissipation

Shearproduction

Buoyantproduction

Frequency of turbulent eddy

Figure 18.8 (a) Typical example of the relative strength of the contributions to TKE at different frequencies; (b) typical

change in the relative magnitude of the production and destruction terms in the prognostic equation for TKE and the

energy transfer (between frequencies) as a function of eddy frequency.

variance in vertical wind speed. For moisture fluctuations the resulting equation

has the form:

( )2 2 2 2( ) ( ) ( )2

I II III

q q qq q q q + u. + v. + w. = q u . + q v . + q w .t x y z x y z

⎡ ⎤ ⎡ ⎤′ ′ ′ ′− ′ ′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

22 22 2 2( ) ( ) ( )2

IV V

u q v q w q q q q + x y z x y z

⎡ ⎤⎡ ⎤ ⎛ ⎞′ ′ ′ ′ ′ ′ ′ ′ ′⎛ ⎞ ⎛ ⎞⎢ ⎥− + − + +⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦ ⎣ ⎦

∂ ∂ ∂ ∂ ∂ ∂ν∂ ∂ ∂ ∂ ∂ ∂

(18.19)




TERM I represents temporal change in the local ‘storage’ of moisture

variance.

TERM II describes the advection of moisture variance.

TERM III describes the production/consumption of moisture variance.

TERM IV represents the turbulent transport of moisture variance.

TERM V viscous dissipation of moisture variance by molecular processes.

Within the wide observed ranges reported by Stull (1988), Fig. 18.9a shows the

typical relative strength over flat, homogeneous terrain in daytime conditions of

moisture variance, and Fig. 18.9b the typical relative strength of the terms in

Equation (18.19), both as a function of height relative to the top of the ABL. The

production term is positive and the dissipation term negative throughout the

ABL, and both are greatest near the inversion layer. The turbulent transport term

describes the movement of moisture variance vertically within the ABL.

The prognostic equation for variance in potential temperature has the form:

⎡ ⎤ ⎡ ⎤′ ′ ′ ′+ + + = − ′ ′ + ′ ′ + ′ ′⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦

′( ′)− +

2 2 2 2

2

( ) ( ) ( ) ( ). . . 2 . . .

I I I III

u v w u v wt x y z x y z

u v x

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂∂

q q q q q q qq q q

q ⎡ ⎤⎡ ⎤ ⎛ ⎞′( ′) ′( ′) ′ ′ ′⎛ ⎞ ⎛ ⎞⎢ ⎥+ − + +⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦ ⎣ ⎦

⎛ ⎞− ′ + ′⎜ ⎟

⎝ ⎠

22 22 2

2

I V V

2

xn

a p

wy z x y z

Rc x

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

∂ ∂∂

qq q q q q

u

q qr

⎡ ⎤+ ′⎢ ⎥

⎣ ⎦ VI

y zn nR R

y z∂

∂ ∂q

(18.20)


TERM I represents temporal change in the local ‘storage’ of temperature

variance.

TERM II describes the advection of temperature variance.

TERM III describes the production/consumption of temperature variance.

TERM IV represents the turbulent transport of temperature variance.

TERM V viscous dissipation of temperature variance by molecular

processes.

TERM VI describes the destruction of temperature variance by gradients in

radiation.

Within the wide observed ranges reported by Stull (1988), Fig. 18.10a shows the

typical relative strength over flat, homogeneous terrain in daytime conditions of



0

0

0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(a)

Relative variance in specific humidity

Hei

ght r

elat

ive

to to

p of

the

AB

L

−1000 −100 −10 0

Moleculardestruction

Turbulenttransport Production

(b)

10 100 1000

Relative strength of terms in moisture variance budget

0

0.2

0.4

0.6

0.8

1.0

1.2

Hei

ght r

elat

ive

to to

p of

the

AB

L

Figure 18.9 (a) Typical relative strength of daytime moisture variance as a function of height relative to the top of the ABL.

(b) Typical relative strength of terms in the prognostic equation for daytime moisture variance as a function of height

relative to the top of the ABL. Note that there is very substantial variability around these typical values.

the variance in potential temperature, and Fig. 18.10b the typical relative

strength of the terms in Equation (18.20), both as a function of height relative to

the top of the ABL. The production and dissipation terms are greatest near the

top and bottom of the ABL as is the (smaller) radiation term (because this is



00

0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.4

0.6

0.8

1.0

1.2

−1000 −100 −10 0

Moleculardestruction

Turbulenttransport

Radiation

Production

(b)

(a)

10 100 1000

Relative variance in potential temperature

Relative strength of terms in potential temperature variance budget

Hei

ght r

elat

ive

to to

p of

the

AB

L

0

0.2

0.4

0.6

0.8

1.0

1.2

Hei

ght r

elat

ive

to to

p of

the

AB

L

Figure 18.10 (a) Typical relative strength of daytime temperature variance as a function of height relative to the top of the

ABL. (b) Typical relative strength of terms in the prognostic equation for daytime temperature variance as a function of

height relative to the top of the ABL. Note that there is very substantial variability around these typical values.

where the gradient in potential temperature is largest). The transport term

describes the movement of temperature variance within the ABL upwards to

the top and downwards to the lower ABL where most dissipation of temperature

variance occurs.




● ABL structure: during the day the lower atmosphere has four main layers:

(a) surface layer (∼100 m), strongly influenced by the surface; (b) mixed layer

(∼ 0.5–3 km), strongly turbulent; (c) entrainment layer (∼10–100 m), a thermal

inversion which inhibits mixing; and (d) free atmosphere (several km), with

limited turbulence.

● ABL growth: warmer drier air is ‘entrained’ downward into the mixed layer

through the inversion from the free atmosphere above so the ABL grows

during the day. At night the surface cools and a stable surface layer develops

which grows upward and is a partial barrier between the ground and residual

mixed layer.

● Daytime ABL properties: the growing daytime mixed layer is heated both by

sensible heat from the surface and by entrainment of warmer air from the

free atmosphere so ABL temperature increases, but vapor from the surface is

mainly used to moisten incoming drier entrained air so ABL humidity often

sees limited diurnal change.

● Nighttime ABL properties: the time evolution of mean variables is usually

more strongly dependent on atmospheric subsidence and longwave radia-

tion flux divergence at night than during the day.

● Evolution of variances: prognostic equations for turbulent departures

(e.g., u′, v′, w′, q′, q′) can be derived similar to those describing mean values

(e.g., u−, v−, w−, q−, q−), and these are used to derive prognostic equations for

variances, e.g., (u′)2, (v′)2, (w′)2, (q ′)2 and (q′)2.

● Evolution of TKE: the prognostic equations for velocity variances combine

to give the important prognostic equation for the turbulent kinetic energy

(TKE) with terms that describe the production, destruction, and transport

of TKE.

● Conservation of TKE: because TKE is continually being destroyed by

viscosity and created by buoyancy and friction at rates which vary with

atmospheric conditions, so is the capability to transport fluxes by turbulent

diffusion.

References

Carlson, M.A. and Stull, R.B. (1986) Subsidence in the nocturnal boundary layer. Journal

of Climate and Applied Meteorology, 25, 1088–1099.

Stull, R.B. (1988) An Introduction to Boundary Layer Meteorology (Atmospheric Sciences

Library). Kluger Academic Publishers, Dordrecht.

Yamada, T. and Mellor, G. (1975) Simulation of the Wangara atmospheric boundary layer

data. Journal of the Atmospheric Sciences, 32, 2309–2329.


Introduction

The prognostic equations introduced in the last three chapters describe both the

instantaneous and mean values of atmospheric variables, but they are not by them-

selves sufficient to allow hydrometeorological modeling of turbulence in the ABL.

To complete the description it is necessary to introduce additional equations

which represent the process of turbulent transport. This chapter discusses how

this need arises and introduces the most common way in which it is met. Because

the nature of these additional equations is sensitive to whether the turbulent field

is generated by friction or by buoyancy, it is necessary also to define criteria to

quantify the origin of the turbulence present at each height in the ABL.

Richardson number

One obvious and commonly used way to quantify numerically the origin of

turbulence is by using the production terms in the prognostic equation for

turbulent kinetic energy. Equation (18.16) contains one term which describes the

production of turbulent kinetic energy by buoyancy, and three terms which

together describe production by frictional processes. The ratio of the buoyant

production term to the sum of the frictional production terms is called the

Richardson number and is used to quantify the relative importance of these two

production processes.

When expressed in a coordinate system in which the X axis is selected to lie

along the direction of the mean wind, terms that involve v, the mean velocity along

the Y axis, are zero, and the equation for the Richardson number is then:

19 Turbulent Closure, K Theory, and Mixing Length





′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

=

+ + + + +⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

( )

( ) ( ) ( ) ( ) ( ) ( )

v

vRi

wg

u u u w w wu u u v u w w u w v w wx y z x y z

qq (19.1)

If Ri is also evaluated over a flat, homogeneous area and in an atmosphere with no

subsidence, terms in Equation (19.1) that involve ( )x∂ ∂ , ( )y∂ ∂ and w are zero and

the equation simplifies to:

′ ′

′ ′

=∂∂

( )

( )

i

v

v

wgR

uu wz

qq

(19.2)

The sign of the Richardson Number depends on atmospheric stability. In the ABL,

momentum flux is toward the surface, i.e., ( ) 0u w′ ′ ≤ but the mean wind speed in

the X direction increases with height so ∂ ∂ > 0u z . Consequently, the denominator

in Equation (19.2) is negative. The value of Ri is therefore negative in unstable

(daytime) conditions when the buoyancy flux, vw ′ ′q , is positive, but it is positive

in stable (nighttime) conditions when the buoyancy flux is negative.

An alternative ‘gradient’ from of the Richardson number, Rig, is sometimes

defined from the locally measured gradients of virtual potential temperature and

wind speed by assuming that these two gradients are proportional to the buoyancy

and momentum fluxes, respectively. The differences in potential temperature, vΔq ,

and wind speed, uΔ , over the same height range, zΔ , might also be used to estimate

the gradients. The definition of the gradient Richardson number is:

(a) (b) (c)

Z Z ZStable Stable Stable

Stable

UnstableUnstable

qv qv qv

Unstable

Figure 19.1 Simplified

profiles typical of potential

temperature for (a) the

daytime ABL over short

vegetation or soil; (b) the

daytime ABL over tall

vegetation; and (c) the

nighttime ABL over short

vegetation or soil, illustrating

that there is often

inconsistency between actual

atmospheric stability in the

ABL and that expected from

the gradient form of the

Richardson number that is

derived from the gradient of

virtual potential temperature.


Turbulent Closure, K Theory, and Mixing Length 279

( )⎡ ⎤⎛ ⎞ ⎛ ⎞ Δ∂ ∂⎢ ⎥= =⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎝ ⎠ ⎥⎝ ⎠ Δ⎣ ⎦

2

2org g vv

i iv v

g guR Rz z u

qqq q

(19.3)

Although often an acceptable estimate in the surface layer, where (as described later)

the turbulent fluxes are proportional to the gradient in mean values, the gradient

form of the Richardson number can be problematic when used elsewhere in the

ABL. This is shown in Fig. 19.1, which gives possible gradients of virtual potential

temperature and identifies the nature of the stability at different heights. The latter

are often inconsistent with the local gradient of virtual potential temperature.

Turbulent closure

Starting from the basic conservation equations for atmospheric flow, prognostic

equations were introduced in Chapter 17 that describe the evolution in space and

time of mean flow variables. These equations involved terms that include not only

mean flow variables but also turbulent fluxes, i.e., the time average of double cor-

relation coefficients. Then, in Chapter 18, prognostic equations were introduced

that describe the time average of turbulent variances in equations which involve

terms that include not only mean flow variables and turbulent fluxes, but also the

time averages of triple correlation coefficients. This process can be successively

repeated to derive prognostic equations for correlation coefficients of increasingly

higher complexity which contain terms that involved mean flow variables and cor-

relations of lower complexity. However, at each new level of complexity, the num-

ber of new unknown variables introduced into the prognostic equations derived

exceeds the number of equations available to describe them. Table 19.1 illustrates

Table 19.1 Progression of the sets of prognostic equations derived to describe

correlation coefficients of velocity with increasing complexity and the number

of new equations and new unknown combinations involved in these.

‘Moment’ or ‘order’

General form of equations

Number of new equations

New variables (and number of new variables)

Zero u , v , and w , are specified in space and time

0 u , v , w , (3)

First ( )∂ ′ ′∂ =∂ ∂

.......i ji

j

u uut x

3 ′2u , 2v ′ , 2w ′ , u v¢ ¢ , u w¢ ¢ , u w¢ ¢ , (6)

Second ( ).......

i j ki j

j

u u uu ut x

∂∂=

∂ ∂¢ ¢ ¢¢ ¢ 6 3u¢ , 3v¢ , 3w¢ , 2u v¢ ¢ , 2u w¢ ¢ , 2v u¢ ¢ ,

2v w¢ ¢ , 2w u¢ ¢ , 2w v¢ ¢ , u v w¢ ¢ ¢ , (10)

etc. etc. etc. etc.



the derivation of prognostic equations of escalating complexity for the case of

velocity components.

Because the process of deriving successive ‘orders’ of prognostic equations

always results in more new unknowns than new equations, it is not possible to use

this approach to derive a suite of physically based equations that by themselves are

an independent basis for describing turbulent transport in the ABL. To allow such

a description it is always necessary to provide additional equations that relate the

unknown variables to each order. The process of providing these additional

equations to complement the prognostic equations derived at any specific order of

complexity is known as making ‘turbulent closure’.

To obtain turbulent closure at a particular order it is necessary to ‘parameterize’

higher order moments in terms of lower order moments through new equations.

In practice, these additional equations are often observation-stimulated, and are

usually approximate descriptions that are applicable in restricted regions of the

ABL and/or in particular stability conditions. The range of turbulent closure

schemes that can be proposed is limited only by human ingenuity, but a closure

scheme is only credible if its use can be shown to result in a description that is

confirmed by observations. Proposed turbulent closure schemes can be local if the

values of unknown quantities at a specific point are assumed related to each other.

Or they can be nonlocal if some unknown quantities are related to other quantities

at many points. In this text the closure scheme described is that most commonly

used in the surface layer of the ABL, i.e., local closure at first order. However the

next section first discuses closure at lower order than this.

Low order closure schemes

The lowest order closure possible is zero order closure. This is the trivial case in

which the spatial and temporal distributions of mean atmospheric variables are

specified simply as numerical or algebraic functions, i.e., as global, regional, or

local space-time maps.

Sometimes a form of closure is used in the ABL in which the variation in space

and time of mean meteorological variables is described, but without explicit

representation of turbulent transport mechanisms. This approach is sometimes

called a Slab Model and is used to describe the daytime evolution of the ABL over

homogeneous, flat surfaces. In such a model the temperature and humidity profiles

are assumed to have a fixed (but plausible) height dependency, but the mean value

of these profiles changes with time in response to the input of sensible heat and

water vapor.

A slab model is illustrated in Fig. 19.2. The surface fluxes and rate of entrainment

into the ABL are described by subsidiary equations, and the divergence of fluxes is

assumed negative and constant with height. This is consistent with assuming the

two profiles have time-independent height dependence. The rates of change of

temperature and humidity content in the ‘slab’ are then calculated from



Equations (18.1) and (18.2). In practice, slab models can provide a reasonable

approximate description of the actual behavior of the ABL in daytime conditions.

Local, first order closure

The most popular closure scheme adopted in the ABL over flat homogenous

surfaces is framed by analogy with Newton’s law for molecular viscosity and

Fourier’s and Fick’s laws for molecular diffusion of heat and mass. It involves

assuming linear relationships between turbulent fluxes and the local mean gradient

of the relevant atmospheric variable driving these fluxes. For example, in the case

of kinematic momentum flux, tk, sensible heat, H

k, and vapor flux, E

k, in the vertical

direction, the relevant local mean gradients are those of wind speed, potential

temperature, and specific humidity, respectively, and the associated general

equations describing their interrelationship are:

∂−τ = = −∂

( ) .k Muu w Kz

¢ ¢ (19.4)

∂= = −θ∂

.vk H

vH Kwz

q¢ ¢ (19.5)

∂= = −

∂.

k V

qE q w K

z¢ ¢ (19.6)

where KM

, KH, and K

V are the (at this stage undefined) eddy diffusivities for the

turbulent fluxes of momentum, sensible heat, and water vapor.


diagram of a slab model with

fixed profiles of virtual

potential temperature and

humidity, the mean values of

which change in response to

inputs from the surface and

through the entrainment

layer.

Fixed virtualtemperature

profile

Fixedhumidityprofile

Entrainment layer

Surface layer

Whole boundary layer

�qv

�t

�q

�t Whole boundary layer

He

Hs

He+ Hs Ee+ Es

Es

Ee

∝ ∝



The signs in Equations (19.4), (19.5), and (19.6) may be confusing and merit

explanation. The fluxes , vwu w q ¢ ¢¢ ¢ , and q w¢ ¢ are positive when w′ is away from

the surface, in the direction of the Z axis. But all fluxes are from higher concentra-

tions toward lower concentrations. Consequently, if the gradients of wind speed,

virtual potential temperature, and specific humidity are all positive (i.e., increase

with height), the equivalent turbulent fluxes will all be towards the surface and

their values therefore negative. The minus sign on the right hand side of Equations

(19.4), (19.5), and (19.6) is required for this reason. The (in this case kinematic)

fluxes of sensible heat and moisture (and other turbulent fluxes) are defined to be

positive in the direction of the Z axis, i.e., in the same direction as wvq ¢ ¢ , and

q w¢ ¢ , see Chapter 4. However in Equation (19.4), the momentum flux, in this case

kinematic momentum, tk, is by convention uniquely defined to be positive when

toward the surface, with opposite sign to u w¢ ¢ . This convention is helpful later

when defining friction velocity (see Equation (19.19)) and it reflects the fact that

momentum transfer is always towards the ground whereas Hk and E

k are often

away from the ground in daytime conditions.

The parameterization of turbulent fluxes in terms of mean gradients given in

Equations (19.4), (19.5), and (19.6) (and similar equations that could be written for

the turbulent fluxes of other atmospheric entities) is referred to as K Theory. It is the

simplest and most commonly used first-order closure parameterization, and it is

found from experiment to work quite well in the surface layer. However, it is known

to fail frequently if applied to describe vertical transfer within a stand of vegetation or

in the mixed layer, as illustrated in Fig. 19.3 for sensible heat. Within vegetation there

is often a positive sensible heat flux away from the ground in the lower portions of the

canopy, over a height range where the virtual potential temperature profile would sug-

gest flow toward the surface. Higher in the ABL, in the mixed layer in daytime condi-

tions, the gradient of the profile of virtual potential temperature is very small when the

sensible heat flux is perhaps large, and is initially positive near the ground and then

negative in the upper ABL. This phenomenon is known as counter gradient flow.

Hei

ght

Hei

ght

Zero gradient flow

Counter gradient flow

K Theory applicable

0 q 0 H

Figure 19.3 Typical daytime

profiles of potential

temperature through

vegetation and the overlying

ABL, and typical height

dependent profile of the

vertical sensible heat flux

showing regions where K

Theory does not apply

because flow is not linearly

related to the local gradients.



Mixing length theory

To complete a K Theory based closure scheme, it is necessary to prescribe a

functional form to the values of eddy diffusivity. Observations suggest that when

K Theory is applied in the surface layer the values of the eddy diffusivities depend

on the aerodynamic properties of the underlying surface and change with

atmospheric stability. The relationship to surface aerodynamics is explored next

and the effect of atmospheric stability considered in the next chapter.

The following discussion refers to turbulent transfer in neutral conditions in the

surface layer, i.e., when all of the turbulent kinetic energy is mechanically generated

and there is no sensible heat flux (strictly no buoyancy flux) to give buoyant

production. In these conditions the potential temperature gradient (strictly virtual

potential temperature gradient) is zero.

Consider now the movement of a parcel of air upward by amount z′ to reach the

level z which is caused by a positive fluctuation in vertical wind speed w′, as shown

in Fig. 19.4. The parcel brings with it the mean humidity q and mean wind speed

u from (z - z′). It follows that the fluctuations q’ and u’ so caused at level z are

respectively given by

qq zz

⎛ ⎞∂= −⎜ ⎟∂⎝ ⎠¢ ¢

(19.7)

and:

uu zz

⎛ ⎞∂= −⎜ ⎟∂⎝ ⎠¢ ¢

(19.8)

Assume also that fluctuations in horizontal and vertical wind speed are correlated,

i.e. that:

w cu= −¢ ¢ (19.9)

z

Height Height

Sameparcelof air

z ’ z ’

q ’ u ’

q u

z

Figure 19.4 The movement

within a turbulent field of a

parcel of air from level z′ to z,

bringing with it the mean

humidity and wind speed

relevant at level z′. Note that

in this diagram mean

humidity is assumed to

increase with height for

consistency with the profile

for wind speed, but the

converse is more usually the

case over moist surfaces in

daytime conditions.



Note that the minus sign appears in Equation (19.9) because the fluctuation w′, which is positive away from the surface, results in a fluctuation in horizontal wind

speed at the level z which is negative relative to the mean wind speed at that level.

The turbulent kinematic flux of water vapor is by definition the time average

value of the product of fluctuations in specific humidity and vertical wind speed,

and it can be obtained by substituting for q′ from Equation (19.7) and w′ from

(19.9) and averaging, i.e.

. . .q uq w z c zz z

⎧ ⎫⎧ ⎫ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪⎪ ⎪= − − −⎢ ⎥⎨ ⎬⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎪ ⎪⎩ ⎭ ⎣ ⎦⎩ ⎭¢ ¢ ¢ ¢

(19.10)

which simplifies to:

2. .( ) . q uq w c zz z

⎛ ⎞ ⎛ ⎞∂ ∂= − ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠¢ ¢ ¢

(19.11)

The basis of mixing length theory is the postulate that, providing turbulence is

solely frictional in origin, it can be characterized by a hypothetical mixing length,

l, whose value is assumed to represent the effective average vertical size of the

turbulent eddies in the turbulent field that transports atmospheric entities. In the

surface layer of the ABL the mixing length is defined such as to simplify

Equation (19.11) by:

2 2.( )c z=l ¢ (19.12)

and Equation (19.11) becomes:

2. . q uq wz z

⎛ ⎞ ⎛ ⎞∂ ∂= − ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠l¢ ¢

(19.13)

In a similar way, it can be shown that the momentum flux is given by:

2 . u uu wz z

⎛ ⎞ ⎛ ⎞∂ ∂= − ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠l¢ ¢

(19.14)

By comparing Equation (19.13) with Equation (19.6) and Equation (19.14) with

Equation (19.4) it follows that:

2 .M VuK Kz

⎛ ⎞∂= = ⎜ ⎟∂⎝ ⎠l

(19.15)

But it still remains necessary to assign a value to the mixing length, l.



Observations in the surface layer suggest that the effective average eddy size of

the turbulent eddies is influenced by proximity to the surface. In mixing length

theory, it is assumed that, in neutral conditions in the surface layer over an

aerodynamically rough surface where the roughness elements covering the surface

are quite short (e.g., grass), the mixing length is directly proportion to the height,

z, above the level of the effective sink of momentum within the roughness elements,

see Fig. 19.5, i.e. that:

2 2 2k z=l (19.16)

where k is an, at this stage, undefined constant.

Hence the eddy diffusivities KM

and KV, (and because the same physical mecha-

nism is assumed to drive the turbulent kinematic flux of sensible heat, also KH) can

be re-written as:

2 2 .M V HuK K K k zz

⎛ ⎞∂= = = ⎜ ⎟∂⎝ ⎠

(19.17)

(Note that in neutral conditions the value of KH derived from mixing length

theory is not important because (assuming K Theory does apply) there can be

no transfer of sensible heat because the gradient of virtual potential temperature

is zero.)

Focusing next on momentum transfer, substituting Equation (19.17) into

Equation (19.4) in neutral conditions gives:

( )2

2 2 uu w k zz

⎛ ⎞∂= − ⎜ ⎟∂⎝ ⎠¢ ¢

(19.18)

Mean wind speedH

eigh

t

z

0

I = kz

Figure 19.5 Effective eddy

size or mixing length for

turbulence in the surface

layer of the ABL assumed

proportional to height above

the effective sink for

momentum.



In the surface layer, it is assumed that kinetic momentum flux is constant with

height and, recalling the momentum flux tk is conventionally selected to be in the

opposite direction to the kinematic turbulent flux u w¢ ¢ , it is usual to set:

τ = = −2

*k u u w¢ ¢

(19.19)

where u* is the ‘friction velocity’. Combining Equations (19.18) and (19.19) and

taking the square root of the resulting equation gives:

∂=

∂*uu kzz

(19.20)

Integrating Equation (19.20) with the boundary condition that the mean wind

speed is zero at the aerodynamic roughness length, z0, gives:

τ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦0 0

* ln lnku z zu

k z k z

(19.21)

By simultaneously measuring the shape of this logarithmic wind profile above an

aerodynamically rough surface and the momentum transfer to that surface (to

give the value of u* from Equation (19.9)), it is possible to obtain an estimate of

k. The constant k is called the von Kármán constant and is believed to be

approximately 0.4.

Equation (19.21) provides the description of the logarithmic profile of

average wind speed observed in neutral conditions over aerodynamically

rough surfaces when covered with comparatively short roughness elements. It

is relevant over rough soil and short turf, for example. However, over taller

vegetation such as agricultural crops or forests, the apparent height at which

the average wind speed appears to go to zero when deduced by extrapolating

the observed wind speed profile measured above the canopy is greater, see

Fig. 19.6. It disappears at the level (z0 + d), where d is the zero plane displacement.

Consequently, the more general form of Equation (19.21) that is applicable

over all natural surfaces is:

⎡ ⎤−= ⎢ ⎥

⎣ ⎦0

( )* lnu z duk z

(19.22)

Equation (19.20) must also be adjusted to be consistent with this shift in the origin

of the z axis, and the resulting more general expression for the friction velocity in

neutral conditions is:

∂= −

∂( )

*uu k z dz

(19.23)



while, on the basis of mixing length theory, the general expression for eddy

diffusion coefficients applicable in neutral conditions in the surface layer of the

ABL is:

= = = −( )*M V HK K K ku z d

(19.24)

The friction velocity u* may be substituted into Equation (19.24) from Equation

(19.21) to give:

2 1

0

( )( )lnM V H

z dK K K k u z dz

− ⎡ ⎤−= = = − ⎢ ⎥⎣ ⎦

(19.25)

or substituted from Equation (19.23) to give:

2 2( )M V HuK K K k z dz

∂= = = −∂

(19.26)

Thus, in conclusion, in neutral conditions in the surface layer, substituting the

value of eddy diffusivity from Equation (19.26) can be used in Equations (19.4),

(19.5), and (19.6) to give the required first order local turbulent closure equations:

∂−τ = − = = − −

∂2 ( )

* *kuu u w ku z dz

¢ ¢

(19.27)

⎛ ⎞∂= = − − ⎜ ⎟∂⎝ ⎠

( )*

vk vH w ku z d

zq

q ¢ ¢

(19.28)

4

3

2

1

0 00

2 2 31 13

Hei

ght (

m)

4

3

2

1

0

Hei

ght (

m)

Tall crop

d

Wind speed (m s−1) Wind speed (m s−1)

Short grass

Figure 19.6 Observed

profiles of average wind speed

over short grass and tall crops

demonstrating that the

position of zero wind speed

as deduced from

measurements above the crop

is elevated by the zero plane

displacement.



⎛ ⎞∂= = − − ⎜ ⎟∂⎝ ⎠

( )*k

qE q w ku z d

z¢ ¢

(19.29)

In the next chapter this first order local closure scheme is generalized to provide

equations that apply in other conditions of atmospheric stability.


● Richardson number: is the ratio of the buoyancy to frictional production

terms in the prognostic equation for turbulent kinetic energy and is used to

quantify the origin of turbulence and atmospheric stability. When the X axis

is selected along the mean wind over a flat homogeneous area with no

subsistence, it is given by Equation (19.2).

● Gradient Richardson number: measured gradients of virtual potential

temperature and wind speed are used to estimate the Richardson Number

using Equation (19.3); this is risky because fluxes are not necessarily

proportional to gradients at all heights and in all conditions (Fig. 19.1).

● Turbulent closure: in principle it is possible to derive a succession of

prognostic equations that describe correlation coefficients in terms of mean

flow variables and correlations of lower complexity, but at each level of

complexity the number of unknown variables exceeds the number of

equations introduced to describe them. It is therefore necessary to provide

additional equations relating the unknown variables, and providing these is

known as making turbulent closure.

● K theory: is first order closure in which it is assumed kinematic fluxes (e.g.,

tk, H

k and E

k) are proportional to relevant local mean gradients (e.g., of wind

speed, potential temperature and specific humidity, respectively) with

proportionality constants (e.g., KM

, KH, and K

V, respectively) that are called

eddy diffusivities.

● Mixing length theory: assumes that in neutral conditions turbulence is

characterized by a hypothetical mixing length whose value is given by

multiplying the von Kármán constant (k ∼ 0.4) by the height above the ground

for short crops, or height above zero plane displacement for tall crops, giving

Equation (19.24).


Introduction

In the previous chapter, first order closure equations were defined by analogy with

molecular transfer processes. These equations related the kinematic turbulent

fluxes of momentum, sensible heat and latent heat to the mean profiles of wind

speed, virtual potential temperature and specific humidity through the (initially

unspecified) eddy diffusivities KM

, KH, and K

V, respectively. Observationally-

guided mixing length theory was then introduced to argue that in neutral

conditions KM

, KH, and K

V are all equal and can be expressed in terms of the von

Kármán constant, the height above the zero plane displacement, and either u* as in

Equation (19.23) or the mean wind speed and aerodynamic roughness length as

in Equation (19.24).

Here, we go further than this and seek first order closure in unstable and stable

conditions. Providing the aerodynamics of the surface exchange is expressed in

dimensionless form, the influence of stability can be accounted for using

hypothetically universal, empirical correction functions. Such functions, which

are often called stability corrections, are necessarily also parameterized in terms

of an appropriately defined dimensionless measure of atmospheric stability.

The mathematical procedure used to define this dimensionless representation of

aerodynamic transfer and dimensionless stability corrections is called surface

layer scaling.

Once first order closure based on mixing length theory has been generalized

to apply in all stability conditions, the resulting equations can be re-expressed in

an alternative form that considers the rates of surface exchange by turbulent

transfer as being controlled by a resistance to flow called aerodynamic resistance.

The representation of inhibition to surface transfers in terms of resistances is now

widely accepted and almost universally adopted in models.

20 Surface Layer Scaling and Aerodynamic Resistance





Dimensionless gradients

The theory of surface layer scaling is assumed to apply in the surface layer of the

ABL where it is also assumed that all vertical fluxes (of momentum, sensible heat

and water vapor, for example) are constant with height. The theory also applies

above uniform horizontal surfaces where there are no changes in the mean values

of atmospheric variables and fluxes of atmospheric entities in horizontal direc-

tions, and no subsidence. In these conditions, the prognostic equation for turbu-

lent kinetic energy is simplified to the form given earlier as Equation (18.17).

Remembering that because the momentum flux is assumed independent of

height in the surface layer, Equation (19.19) requires that u* is also independent of

height. Equation (18.17) can be re-written in dimensionless form by multiplying

by the factor [k(z-d)/(u*)3], as follows:

( )

′ ′ ∂ ′ ′− ∂ − −= −

∂ ∂

− ∂ − ∂ ′− ′ ′ − −

∂ ∂

3 3 3

3 3

( ) ( )( ) ( ) 1 ( )

( ) ( ) ( )* * *

I II III

( ) ( ) (

( ) ( )* *

v

v a

we w pk z d k z d k z dgt zu u u

k z d u k z d w e k zu wz zu u

qrq

−3

)

( )*

IV V VII

du

e

(20.1)

From the definition of u* in Equation (19.19), it follows that = −2 ( )

*u u w¢ ¢ and

term IV in Equation (20.1) can be simplified to:

*

( )M

uk z du z− ∂=

∂f

(20.2)

The function on the right hand side of Equation (20.2) is called the dimensionless

gradient of wind speed. When written in this dimensionless form, the wind speed

gradient in the surface layer has been normalized to allow for the local surface-

related dependency on u* and for height above the zero plane displacement. The

dimensionless function fM

on the left hand side of Equation (20.2) is as yet unspec-

ified, both in terms of functional form and purpose. However, multiplying

Equation (20.2) by (-u*

2/fM

) gives:

2 1

* *( ) ( )M

uu ku z dz

− ∂− = − −∂

f

(20.3)

Comparing this equation with Equation (19.27) suggests that the function fM

is

a dimensionless function whose reciprocal acts to change the effective average

vertical size (i.e. the mixing length) of the turbulent eddies operating in the

surface layer. In neutral stability conditions, comparison between Equations

(19.27) and (20.3) requires that fM

= 1, but in other conditions of thermal


Surface Layer Scaling and Aerodynamic Resistance 291

stability (as characterized by an as yet undefined dimensionless measure of

buoyant production) its value may change.

We return to this point later, but meanwhile next consider how it is possible

to define other dimensionless gradients of atmospheric entities, including fH

corresponding to kinematic sensible heat flux and virtual potential temperature,

and fV corresponding to kinematic moisture flux and specific humidity. Bearing

in mind the purpose these dimensionless functions serve (i.e., to modify the effec-

tive average mixing length of the turbulent eddies operating in the surface layer),

by analogy with Equation (20.3) it is possible to re-write Equation (19.28) in the

form:

−⎛ ⎞∂

= − − ⎜ ⎟∂⎝ ⎠1( ) ( )

*v

v Hw ku z dzq

q ¢ ¢ f

(20.4)

which can then be rearranged to define the dimensionless gradient of virtual

potential temperature thus:

∂−=

∂( )

*

vH

k z dzq

fq

(20.5)

where:

−=

( )*

*

v wu

q ¢ ¢q

(20.6)

Similarly Equation (19.29) can be re-written as:

− ⎛ ⎞∂= − − ⎜ ⎟∂⎝ ⎠

1( ) ( )* V

qq w ku z dz

¢ ¢ f

(20.7)

which can be rearranged to define the dimensionless gradient of specific

humidity:

∂−=

∂( )

*V

qk z dq z

f

(20.8)

where:

−=

( )*

*

q wqu¢ ¢

(20.9)

As is the case for the dimensionless wind speed gradient, when written in the

dimensionless form of Equations (20.5) and (20.8), respectively, the gradient of



virtual potential temperature and gradient of specific humidity in the surface layer

are normalized to allow for the local surface-related dependency on u* and height

above the zero plane displacement.

At this stage, the mixing length hypothesis has been rewritten such that the

effective mixing length which controls turbulent transport in the surface layer

can be modified by the value of the dimensionless functions fM

, fH and f

V.

These functions are all equal to unity in neutral conditions, but their value may

alter with the extent of buoyant production or destruction in the turbulent

field. It is next necessary to assume that the functional forms of dimensionless

functions fM

, fH and f

V are a universal feature of turbulence in the surface layer

and independent of the actual surface itself. Although these dimensionless

functions are not known, if they are indeed universally applicable they can

presumably be defined by calculation from measured gradients and fluxes at

one place using Equations (20.2), (20.5) and (20.8), and the functional forms

so defined may then be applied everywhere. But the next step is to define a

dimensionless measure of the rate of buoyant production/destruction in

terms of which the empirical functional form of fM

, fH and f

V can be defined

by experiment.

Obukhov length

Equation (20.1), i.e., the dimensionless form of the prognostic equation for

turbulent kinetic energy, can be used to define the required dimensionless

measure of buoyant production/destruction in terms of which fM

, fH and f

V can

be parameterized. In this equation, Term II describes the contribution of buoyancy

to the production or destruction of turbulence. This term can be re-written in the

form (z-d)/L where L is called the Obukhov length and defined by:

−=

3( )*

( )

v

v

uL

gk w

q

q ¢ ¢

(20.10)

Because all the other factors on the right hand side of Equation (20.10) are positive,

the Obukhov length, and therefore (z-d)/L, takes its sign as being opposite to the

sign of the kinematic sensible heat flux, v wq ¢ ¢ . Consequently, L is negative when

the heat flux is positive (often in daytime conditions), and L is positive when the

heat flux is negative (often in nighttime conditions). One physical interpretation

of the Obukhov Length is that the value (-L) corresponds to the height at which

buoyant production of turbulent kinetic energy begins to dominate over shear

(mechanical) production when the mean profiles are assumed to be logarithmic

through the whole surface layer.

Not surprisingly, the factor (z-d)/L is formally related to the flux form of the

Richardson number, at least in neutral conditions, as can be shown by rearranging

terms in (z-d)/L as follows:



32 **

*

( )

( ) (( )

( )( )

( )

v

v v

v

g wk z d g wz d

uL u uk z d

− −− = =⎛ ⎞− ⎜ ⎟⎝ ⎠−

q ¢ ¢q ¢ ¢ ) q

q

(20.11)

Substituting for tk from Equation (19.19) and then for u

* from Equation (19.20),

the denominator of the right hand side of Equation (20.11) gives the flux

Richardson number as defined by Equation (19.2). However, observations in the

ABL suggest the more general relationship shown in Fig. 20.1, with the flux

Richardson number, Rf, approximately equal to (z-d)/L within experimental error

in unstable and neutral conditions. In stable conditions they are related by:

2

( )0.74 4.7

( )

( )1 4.7

f

z dz d LRL z d

L

−⎡ ⎤+⎢ ⎥− ⎣ ⎦≈−⎡ ⎤+⎢ ⎥⎣ ⎦

(20.12)

Flux-gradient relationships

As previously stated, to generalize the application of mixing length in the surface

layer it is necessary to parameterize the dimensionless expressions fM

, fH and f

V as

functions of (z-d)/L, the selected dimensionless measure of buoyant production/

destruction. During the 1960s and 1970s, several field experiments were carried

out which sought to define the required form for these (assumed universal)

dimensionless expressions. These suggested that in neutral and stable conditions

the three functions are the same within experimental error (i.e., fM

= fH = f

V), but

that in unstable conditions, although fH and f

V are the same, f

M differs and in fact

fM

2 = fH = f

V. Eventually, a reasonable consensus emerged on the functional forms

−1.0

−0.5

25

1−1−2 2 3(z−d )

L

Unstable Stable

Rf

Figure 20.1 Approximate

relationship between the flux

Richardson number and

(z-d)/L based on

observations.



for fM

, fH and f

V within the limits of experimental error and the underlying

assumptions of surface layer scaling theory. The most widely accepted form

for these functions is given in Table 20.1 and is adopted in this text. In the absence

of any better information, it is usually considered acceptable to assume the

flux- gradient relationships for the fluxes of other atmospheric variables such as

carbon dioxide flux are the same as for sensible heat and moisture.

The general form of the surface layer wind profile is given by integrating the

functional form of fM

in Table 20.1 to give:

⎡ ⎤⎛ ⎞− −⎛ ⎞ ⎛ ⎞= + Ψ⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦*

0

( ) lnu z d z du zk z L

(20.13)

where, for values of (z-d)/L ≥ 0, Ψ is given by:

5( )z d z dL L− −⎛ ⎞Ψ =⎜ ⎟⎝ ⎠

(20.14)

and for values of (z-d)/L < 0, Ψ is given by:

π φ−

⎡ ⎤− + +⎛ ⎞ ⎡ ⎤Ψ = − − ⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦ ⎣ ⎦⎡ ⎤−⎛ ⎞+ − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2

1

1 12ln ln

2 2

2tan [ ] where x = 2

M

z d x xL

z dxL

(20.15)

Returning fluxes to natural units

In Chapter 15 the concept of kinematic units was introduced as a mechanism to

simplify the equations describing turbulent transfer, and to enhanced similarity

between equations giving opportunity to draw analogy between them. Having

Table 20.1 Form of the flux-gradient relationships adopted in this text.

Flux gradient relationship

Stable conditions

Neutral conditions Unstable conditions

φ −⎛ ⎞⎜ ⎟⎝ ⎠Mz dL

φ −⎛ ⎞= + ⎜ ⎟⎝ ⎠1 5Mz dL

1−

−⎡ ⎤⎛ ⎞= − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

14

1 16Mz dL

φ

φ −⎛ ⎞⎜ ⎟⎝ ⎠Hz dL

φ −⎛ ⎞= + ⎜ ⎟⎝ ⎠1 5Hz dL

1−

−⎡ ⎤⎛ ⎞= − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

12

1 16Hz dL

φ

φ −⎛ ⎞⎜ ⎟⎝ ⎠Vz dL

φ −⎛ ⎞= + ⎜ ⎟⎝ ⎠1 5Vz dL

1−

−⎡ ⎤⎛ ⎞= − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

12

1 16Vz dL

φ



now reached the stage of writing equations derived from turbulence theory

that can be used to calculate fluxes from mean variables and the aerodynamic

properties of natural surfaces, it is now appropriate to recast these equations

back into the natural units in which they are normally applied.

As described in Table 15.3 and associated text, the process of returning from

kinematic fluxes to actual fluxes involves (for most fluxes) substituting the true

flux divided by the density of moist air for the kinematic version of fluxes in

equations derived from turbulence theory. However, in the case of sensible heat

flux, the true sensible heat flux divided by the product of the density of moist air

with the specific heat of air must be substituted for the kinematic sensible heat

flux. Taking sensible heat flux as an example, this means that the definition of q*

given in Equation (20.6) is re-expressed in terms of the true flux of sensible heat,

H, and becomes:

−=

**a p

Hc u

qr

(20.16)

When this definition of q* is substituted into Equation (20.5), it can be rearranged

to give:

−− ∂θ

= −φ ∂

2( )

* (W m )vp

H

k z d uH c

zr

(20.17)

Similarly, rewriting the definition of q* in terms of the actual moisture flux, E, and

substituting this into Equation (20.8) and rearranging gives:

− −− ∂

= −φ ∂

2 1( )

* (kg m s )V

k z d u qEz

r (20.18)

Moisture flux is often expressed in terms of the equivalent flux of latent heat, in

which case this last equation becomes:

−− ∂

= −∂

2( )

* (W m )V

k z d u qEz

l rlf

(20.19)

and in practice, when describing near-surface energy exchange, it has become

more common to express latent heat flux in terms of the gradient of vapor pressure

rather than the gradient of specific humidity, and to use the alternative equation:

−− ∂

= −∂

2( )

* (W m )p

V

c k z d u eEz

rl

g f (20.20)

where g = (cpP)/(0.622λ) is the psychrometric constant introduced in Equation

(2.25). The equivalent equation describing the eddy diffusion of momentum flux

is obtained by multiplying both the numerator and denominator on the right hand



side of Equation (20.2) by (u*)2, rearranging, then multiplying the resulting

equation by r to give:

− ∂= =

∂2

( )*

*M

k z d u uuz

t r rf

(20.21)

Notice that the sign in this equation is different because momentum flux is defined

positive when downward toward the surface, but the fluxes of sensible and latent

heat are defined to be positive upward, away from the surface.

Resistance analogues and aerodynamic resistance

Diffusion equations are usually used to describe surface-atmosphere exchanges in

equations and numerical models. In the case of vertical flow by turbulent diffusion

of momentum and energy in the surface layer, the representation in terms of K

Theory is given using the equations:

∂=∂

.M

uKz

t r

(20.22)

ρ ∂= −∂

.H

v

pH c K

zq

(20.23)

∂= −∂

.p

V

c eE Kz

lργ

(20.24)

with the eddy diffusivities KM

, KH and K

V in Equations (20.22), (20.23) and (20.24)

specified by comparison with Equations (20.21), (20.17) and (20.20), respectively.

As discussed in more detail in the next chapter, very near the ground or very near

vegetation (and, in the case of latent heat, also inside leaves), flux transfer is largely

controlled by the molecular diffusion process. Molecular flow is described by

diffusion equations similar to those given above describing turbulent diffusion,

but the eddy diffusivities are replaced by molecular diffusivities DM

, DH and D

V,

which are properties of the air through which diffusion occurs.

When writing equations for building numerical models of surface exchange, it

is very common to write the turbulent and molecular diffusion equations that

describe flow in integrated form. Consider, for example, sensible heat flow by

turbulent diffusion in the vertical direction described by Equation (20.23) which

can be rearranged into the form:

∂=− ∂

1 V

a p H

Hc K z

qr

(20.25)



Integrating this equation over the height range z1 to z

2 assuming there is no

significant loss of sensible heat flux (i.e. no flux divergence) between these two

levels, gives:

( ) ( )ρ∂=∫ ∫ ∂−

2 2

1 1

1. .

v

Ha p

z zH dz dzK zc z z

q (20.26)

which can be re-written as:

( )= =−=

1 2

2( )1

z zv v

H ca p rH

q qr

(20.27)

where =1zvq and =2z

vq are the virtual potential temperatures at z1 and z

2 (with z

1

being closer to the surface), and

( )= ∫2

1

12( ) .1H

H

zr dzKz

(20.28)

There is analogy between Equation (20.27) and Ohm’s Law, which describes flow

of electrical current through a resistance in response to a voltage difference

between the ends of the resistance, i.e.

VoltageCurrent

Resistance=

(20.29)

The flow of sensible heat is analogous to the ‘current,’ and the difference in virtual

potential temperature analogous to the voltage difference that is driving the flow

of current. This analogy identifies 2

1( )Hr as a ‘resistance’ in Equation (20.27) that is

acting to moderate the flow of sensible heat between the two levels driven by

the difference in the virtual potential temperature between z1 and z

2. Similar

integrated diffusion equations can be written to describe the flow of momentum

and latent heat in terms of the ‘resistances’, thus:

ργ

= =−=

1 2

2

1

( )

( )

z za p

V

c e eEr

l

(20.30)

and

= =−=

2 1

2

1

( )

( )

z z

aM

u ur

t r

(20.31)

The three resistances 2

1( )Hr , 2

1( )Vr and 2

1( )Mr are in fact the aerodynamic resistances

between the levels z1 to z

2 for sensible heat, latent heat, and momentum transfer,



respectively, and are obtained by integrating the reciprocal of the appropriate

eddy diffusivity between the two levels.

Quite commonly, however, it is the total aerodynamic resistances to flow that are

of interest. These resistances act between the reference level above the canopy,

where atmospheric variables are measured, and the effective source of fluxes

within the canopy of vegetation. In neutral conditions this aerodynamic resistance

is comparatively simple to calculate for momentum transfer if mixing length

theory is assumed, and if (on the basis of extrapolating above-canopy profiles) it is

assumed that the wind speed profile goes to zero at a height z0 above the zero

plane displacement, see Fig. 19.6. In this case KM

= ku*(z-d) and f

M = 1, see

Equation (19.3). Consequently ra

M, the aerodynamic resistance between the sink

of momentum in the canopy and the height zm

, is given by:

++

⎛ ⎞ ⎡ ⎤ ⎛ ⎞−−= = =⎢ ⎥⎜ ⎟ ⎜ ⎟− ⎝ ⎠⎢ ⎥⎝ ⎠ ⎣ ⎦

∫0

1 ln( ) 1. ln

( )* * *

m

m

o

o

zzM m

a d zd z

z dz dr dzku z d ku ku z

(20.32)

In neutral conditions the wind speed profile is given by Equation (19.21), which

equation can be rearranged to give the value of u* in terms of the wind speed u

m

measured at height zm,

thus:

− ⎡ ⎤−= ⎢ ⎥

⎣ ⎦1

0

( ) ln

*m

m

z du ku

z

(20.33)

Consequently, Equation (20.32) becomes:

⎛ ⎞−== ⎜ ⎟⎝ ⎠

2

2

0

1lnM m

am

z dr

zk u

(20.34)

Similarly, if ra

H and ra

V are respectively the aerodynamic resistance to sensible and

latent heat transfer between zm

and the source/sink of these two heat fluxes in the

canopy, these resistances can be simply derived in neutral conditions. If the source/

sink heights for sensible and latent heat are assumed to be at heights z0

H and z0

V

above the zero plane displacement, respectively, the two resistances are given by:

⎛ ⎞ ⎛ ⎞⎛ ⎞− − −= =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠2

00 0

1 1ln ln ln

*

H m m ma H H

m

z d z d z dr

ku zz k u z

(20.35)

and

⎛ ⎞ ⎛ ⎞⎛ ⎞− − −= =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠2

00 0

1 1ln ln ln

*

V m m ma V V

m

z d z d z dr

ku zz k u z

(20.36)

Deriving these aerodynamic resistances in other stability conditions is more

difficult and somewhat circuitous. It involves integrating eddy diffusivities that



include the flux gradient relationships (i.e. fM

, fH and f

V) which depend on (z-d)/L,

and which therefore depend on the ambient fluxes of sensible heat flux and

momentum. But these in turn are themselves dependent on the resistances that are

being calculated.


● Dimensionless prognostic equation for TKE: in the constant flux layer above

a uniform horizontal surface the prognostic equation for turbulent kinetic

energy (TKE) in the ABL can be re-written in dimensionless form as Equation

(20.1).

● Dimensionless gradients: in the dimensionless prognostic equation for TKE

the fourth term can be used to define fM

, the dimensionless gradient of

wind speed (wind speed gradient normalized to friction velocity and height

above zero plane displacement): similar dimensionless gradients of virtual

potential temperature, fH, and specific humidity, f

V, can also be defined.

● Application of fM

, fH

and fV: in K Theory the reciprocals of the dimension-

less functions fM

, fH and f

V act as multipliers to modify the mixing length

(and thus the efficiency of turbulent transfer) in the surface layer.

● Dimensionless measure of stability: in the dimensionless prognostic equa-

tion for TKE the second term can be used to define, (z-d)/L, a dimensionless

measure of atmospheric stability, in terms of which the dimensionless

functions fM

, fH and f

V (and thus the efficiency of turbulent transfer) can be

parameterized.

● Specification of fM

, fH

and fV: the functional forms of f

M, f

H and f

V with

respect to (z-d)/L are assumed to be universally applicable and field

experiments were carried out to define them: the expressions in Table 20.1

are adopted in this text.

● From kinematic to natural units: before application equations for the

kinematic fluxes of sensible heat, latent heat and momentum derived from

K-Theory must be returned to natural units to give Equations (20.17),

(20.20), and (20.21).

● Resistances: integrating the relationships between fluxes and gradients of

atmospheric variables gives resistances (by analogy with Ohm’s law) that

relate fluxes to the differences in variables between two heights.

● Aerodynamic resistance: total aerodynamic resistance to the turbulent

transfer between a level in the atmosphere and a ‘source’ level in a canopy

can be derived in terms of wind speed, zero plane displacement, and

aerodynamic roughness, e.g., Equations (20.34), (20.35), and (20.36).


Introduction

As discussed in Chapter 17, within the ABL itself the Reynolds number is such

that movement of energy, water, and atmospheric constituents is primarily by tur-

bulent transport rather than molecular transport because of the physical scale of

the turbulent eddies involved. Approaching the surface, the efficiency of turbulent

transport reduces as the scale of the turbulence reduces and turbulent transport no

longer dominates very near the ground. If the surface is bare soil, or very near the

components that make up the canopy when the surface is vegetation-covered, the

resistances to flow are primarily determined by molecular diffusion processes; in

these cases transfers are through non-turbulent boundary layers close to the

sources or through pores in the soil or vegetation.

If the surface is vegetation-covered, as is often the case, flux exchange is complex

within the canopy and for some distance above it. Over a height range on the order

of ten times the aerodynamic roughness above the vegetation, surface-related

features influence the nature of the turbulent regime. While within the vegetation

canopy itself, flux exchange involves interplay between still turbulent vertical

diffusion of fluxes and the divergence of these fluxes through interaction with the

vegetation elements (leaves, twigs and branches) that make up the canopy. In

effect, molecular diffusion through non-turbulent boundary layers around

vegetation and (in the case of water vapor) through stomata in the leaves, act like

resistances which control transfer of those portions of the fluxes that are dissipated

or generated at each level in the canopy.

Further complications arise. The exchange of momentum between moving air

and a body is more efficient than the exchange of other entities such as heat, water

vapor and carbon dioxide. This is because momentum can be transferred not only

by molecular diffusion through the boundary layer surrounding the body, but also

21 Canopy Processes and Canopy Resistances




Canopy Processes and Canopy Resistances 301

by pressure forces. Transfer for other entities is only by molecular diffusion. The

second complication (mentioned in Chapter 19, see Fig. 19.3) is that K Theory is

a poor representation of vertical diffusion inside vegetation canopies. But it is

often still used despite this. Fortunately the negative consequences of this do not

seem to be too severe.

Boundary layer exchange processes

During the 1960s and 1970s there was considerable interest in better quantifying

the aerodynamic behavior of components of natural vegetation inside canopies,

especially leaves. These studies focused on understanding differences in the

effectiveness with which different transferred entities (e.g., momentum, heat,

water vapor and carbon dioxide) are exchanged. Many studies were made inside

wind tunnels, often using artificial replicas of leaves and twigs. There is some

question as to how directly the results obtained in wind tunnels translate into

the real-world environment. Nonetheless, such studies contributed important

background understanding – helpful when considering the whole-canopy behavior

of vegetation stands. This section describes some of the more important results

obtained.

Boundary layers develop over smooth flat surfaces immersed in a moving fluid

such as air. Figure 21.1 illustrates a simple case in which air moving horizontally at

a fixed speed encounters a thin plate (equivalent to a flat leaf, perhaps). A layer of

air, the boundary layer, develops above the plate, which becomes deeper with dis-

tance from the leading edge. There is laminar (as opposed to turbulent) wind flow

inside this boundary layer, the wind speed varying from zero at the surface of the

plate to the speed of the incoming air flow at the edge of the boundary layer.

Momentum is transferred from the moving air through the boundary layer by

molecular diffusion, and the plate experiences a ‘drag’ force as a result.

The interaction between a moving fluid and obstructions in the fluid are

parameterized in terms of the Reynolds number defined in Chapter 17

(Equation 17.28). The velocity of air in a vegetation canopy is typically ∼1.0 m s–1

and a typical dimension of a leaf is ∼0.05 m, giving a Reynolds Number of

order 104.

When the primary exchange between the air flow and the flat plate is by molec-

ular diffusion through a boundary layer, the resulting sheer stress on the plate is

called ‘skin friction’. Experimentally calibrated aerodynamic theory indicates that

the total transfer of momentum per unit area can be estimated from:

ν⎛ ⎞= ρ ⎜ ⎟⎝ ⎠

0.5

0.66 aUUl

t

(21.1)

where U is the speed of the air flow and l is the distance from the edge. If the net

exchange of momentum is expressed in terms of RM

sf, the equivalent



Wind direction

Boundarylayer

Unmodified flowvelocity

Modified flowvelocity

C

B

A

Dep

th

Velocity

Figure 21.1 Boundary layer

development above a flat plate in

horizontal wind flow. Laminar

flow occurs between A and B but

the flow is unmodified between

B and C.

boundary-layer resistance to momentum flow by skin friction per unit area of

plate, τ is given by:

−=

( 0)a sf

M

UR

t r

(21.2)

Combining Equations (21.1) and (21.2), the effective value of the boundary-layer

resistance to momentum transfer for unit area of plate becomes:

0.5

0.51.51.5 Resf

MlR

U U⎛ ⎞≈ ≈⎜ ⎟ν⎝ ⎠

(21.3)

Equation (21.3) suggests that for a flat, horizontal plant leaf with characteristic

dimension 0.05 m in a canopy air stream of 1 m s–1, RM

sf is about 100 s m–1 per unit

area of leaf. Were the leaf twice as large, the area to which momentum could be

transferred would be larger and the resistance for the leaf would be less. This is the

order of magnitude for the boundary-layer resistance for transfer of momentum

by skin friction and it is also the order of magnitude for the boundary-layer resist-

ance to exchange of other entities such as heat and water vapor that also diffuse

through a boundary layer to reach the surface.

However, if the body in the air stream presents a substantial cross-sectional area

perpendicular to air flow (e.g., a flat leaf at an angle to the flow), referred to as a

‘bluff body’ in aerospace engineering, momentum (but not other entities) can be

exchanged more efficiently via pressure forces. In this case the stress exerted via

pressure forces on the bluff body is called ‘form drag’. The momentum exchanged

per unit cross-sectional area by form drag to a bluff body in air moving at speed,

U, is given by:

= 2

2a

fc Ur

t

(21.4)

where cf is an empirically determined ‘drag coefficient’. Combining Equations

(21.1) and (21.4), if the bluff body is a flat leaf at an angle to the flow, RM

bb, the



boundary-layer resistance to momentum exchange by the form drag exchange

mechanism per unit area of leaf, is given by:

1bbM

f

Rc U

⎛ ⎞≈ ⎜ ⎟⎜ ⎟⎝ ⎠

(21.5)

Note that Equation (21.5) has been written to give the resistance per unit area of

leaf assuming the leaf has two sides. If the canopy air stream velocity is 1 m s–1

and cf = 0.2, R

Mbb is 5 s m–1, which is about an order of magnitude less than the

boundary-layer resistance would be if momentum were transferred solely by

skin friction.

In the case of momentum transfer, it is convenient to combine the effect of skin

friction and bluff body transfer processes in a single drag coefficient, cd, which,

given the form of Equations (21.3) and (21.5), should have the approximate form:

0.5

d fc c nU −≈ + (21.6)

where n is a constant. RM

, the corresponding boundary-layer resistance per unit

area of leaf for momentum transfer by both skin friction and bluff body transfer, is

then given by:

1M

d

Rc U

⎛ ⎞≈ ⎜ ⎟

⎝ ⎠

(21.7)

On the basis of the above discussion, it is clear that the drag coefficient, cd, will

be a strong function of the orientation of the leaf with respect to the wind, a

result that has been demonstrated for model leaves in wind tunnels, see Fig. 21.2,

for example.

The boundary-layer resistance for exchanges other than momentum is deter-

mined by molecular diffusion through a boundary layer. Taking sensible heat, for

example, if the ‘effective’ boundary-layer thickness through which heat has to diffuse

from a leaf to the canopy air stream is d, the boundary-layer resistance per unit area

of leaf for sensible heat, RH, is given by:

=HH

RDd

(21.8)

where DH is the molecular diffusivity for heat. However, because d is not a

measureable quantity, it is preferable to estimate boundary-layer resistance based

on a measureable characteristic dimension, d, of the vegetation instead. For this

reason, the boundary-layer resistance for sensible heat is more conveniently

parameterized in terms of the ‘Nusselt number’ (Nu), defined to be the ratio of a

characteristic dimension of vegetative element, d, to the effective boundary-layer

thickness, d, see Fig. 21.3. In terms of the Nusselt number, the boundary-layer



resistance to the transfer of sensible heat for an element of vegetation with

characteristic dimension d is then given by:

H

H

dRD Nu

=

(21.9)

Nu has been expressed in terms of empirical relationships with the Reynolds

number for selected shapes such as plates (which broadly compare with leaves)

and cylinders (which broadly compare with coniferous needles) in wind-tunnel

studies, see Table (21.1).

0.3 0.9Wind speed (m s−1)

1.5

6.6 cm

0.5 cm

4.5 cm

u

0.5

0.3

0.1D

rag

coef

ficie

nt

f

f = +90�

f = −90�

f = +23�

f = −23�

f = 0�

(c)

(a)

(b)

Figure 21.2 (a) Model leaf used to determine drag coefficient, (b) Specification of orientation of model leaf relative to

wind direction, (c) resulting values of drag coefficient versus angle and wind speed. (Redrawn from Thom, 1968,


Moleculardiffusion throughboundary layer

Effectiveboundary layer

thickness

Characteristicdimension ofvegetationelementFigure 21.3 Representation

of the Nusselt Number (Nu)

for a leaf.



The description of boundary-layer resistance to the transfer of other entities

such as water vapor and carbon dioxide is analogous to that for heat transfer, but

the boundary-layer resistances to such mass transfers are expressed in terms of

the Sherwood number. The Sherwood number is very similar in concept to the

Nusselt number and has also been expressed empirically in terms of Reynolds

number. However, the molecular diffusion coefficients differ for different

exchanged entities and all have a dependency on the temperature, Tc, of the air in °C.

The molecular diffusion coefficients are: for momentum, υ = 1.33 × 10–5

(1+0.007Tc) m2 s–1; for heat, D

H = 1.89 × 10–5 (1+0.007T

c) m2 s–1; for water vapor,

DV = 2.12 × 10–5 (1+0.007T

c) m2 s–1; and for carbon dioxide, D

C = 1.29 × 10–5

(1+0.007T c) m2 s–1.

The molecular diffusion coefficient for an entity influences not only its rate

of diffusion through a boundary layer but also the effective thickness of the

boundary layer relevant to each diffused entity. Experiments suggest that,

providing the exchange between the surface of a leaf and the canopy air stream

is dominated by forced convection, the ratio of the boundary-layer resistances

for two entities is inversely proportional to the ratio of the corresponding

molecular diffusion coefficients raised to the power 0.67 (Monteith and

Unsworth, 1990), e.g.

0.67

0.9V H

H V

R DR D

⎛ ⎞= ≈⎜ ⎟⎝ ⎠

(21.10)

Table 21.1 Empirical relationships between the Nusselt number and

Reynolds number determined for selected shapes from wind tunnel

studies (Data from Monteith and Unsworth, 1990).

Shapes Range of Reynolds number (Re) Nusselt number (Nu)

Flat plates

d

Streamline flow Re < 2 × 104 Nu = 0.60 Re0.5

d

d

Turbulent flow Re > 2 × 104 Nu = 0.032 Re0.8

Cylinders 1 to 4 Nu = 0.89 Re0.33

d

4 to 4040 to 4 × 103

4 × 103 to 4 × 104

4 × 104 to 4 × 105

Nu = 0.82 Re0.39

Nu = 0.62 Re0.47

Nu = 0.17 Re0.62

Nu = 0.024 Re0.81



and:

0.67

1.3C H

H C

R DR D

⎛ ⎞= ≈⎜ ⎟

⎝ ⎠

(21.11)

Thus, in summary, the boundary-layer resistances of vegetation components

associated with molecular diffusion through boundary layers are similar in

magnitude (but not identical) for most atmospheric transfers. However, the

boundary-layer resistance for momentum transfer is substantially less than for

other exchanges, by about an order of magnitude. As discussed later in Chapter 22,

this characteristic difference in boundary-layer resistance means the effective sink

of momentum in a vegetation canopy tends to be higher than the effective average

source/sink for other transferred entities. This, in turn, results in different values

for the aerodynamic resistance for different canopy-atmosphere transfers.

Shelter factors

The area-average boundary-layer resistance for all the leaves acting together above

a square meter of ground at a particular level in a plant canopy is given by the

parallel sum of their individual boundary-layer resistances. However, this implic-

itly supposes that each individual leaf is exposed to the same canopy air stream. In

fact the leaves inside real canopies are not isolated and, as a result, they are not all

equally exposed to the same average microclimate at any level. Rather, they are

usually clumped together to some extent, with the result that the values of atmos-

pheric variables (most significantly, wind speed) outside the boundary layer of an

individual leaf will differ from those for another leaf, and both will differ from the

values in the air stream at each level in the canopy, see Fig. 21.4.

Clearly it is not practical to describe the aerodynamics of every individual leaf in

a canopy together with their mutual interference, so some form of empirical correc-

tion is required. Studies in wind tunnels suggest that the general effect of mutual

Leaf-specificweather variables

Mean canopy airstream weather

variables

Leaf surface specificweather variables

Figure 21.4 Vegetation

clumping gives rise to differences

between the atmospheric

variables outside the boundary

layer of individual leaves.



interference is that the effective value of the area-average boundary-layer resistance

is higher than would be calculated by a parallel summation of the values for indi-

vidual leaves. One way to approximately describe the effect of mutual interference

is to assume the increase in the effective area-average boundary-layer resistance can

be allowed for by including a simple multiplicative factor called a shelter factor.

The contribution to a canopy flux at a particular level in a canopy might be

assumed proportional to the difference between the mean value of an atmospheric

variable in the canopy air stream and the mean value of that same variable at the

surface of the vegetation elements (leaves) at that level. The equations describing

the contributions to the sensible heat flux, latent heat flux, and momentum flux

generated or lost at level z are then respectively given by:

( )( )

( )

s zH z a p

H z

T Tc

R−

=d r

(21.12)

λ

−=

( )( )

( )

a p s zE z

V z

c e eR

rd

g

(21.13)

−=

(0 )( )

( )

zz a

M z

uRtd r

(21.14)

where Tz, e

z, and u

z are the temperature, vapor pressure, and wind speed of the

mean canopy air stream at level z, respectively, and Ts and e

s are the mean values of

the temperature and vapor pressure at the surface of the leaves at level z,. In

Equations (21.12) to (21.14), the mean boundary-layer resistances for heat, water

vapor, and momentum for the N leaves in a small height range dz around the level

z are respectively defined by:

−

=

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦∑

1

1

( )

H

Ni

H z H ii

AR P

R

(21.15)

−

=

⎡ ⎤= ⎢ ⎥

⎣ ⎦∑

1

1

( )N

iV z V i

i V

AR P

R

(21.16)

−

=

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦∑

1

1

( )

M

Ni

M z M ii

AR P

R

(21.17)

where Ai is the area and R

Hi, R

Vi, and R

Mi are the boundary-layer resistances per

unit area of the i th leaf, and PH, P

V, and P

M are shelter factors for heat, water vapor

and momentum, respectively. Because it is the reciprocal of the boundary-layer



resistance, the boundary-layer conductance, that scales linearly with the area of

leaf, when calculating the mean boundary-layer resistance it is necessary to first

calculate the area-weighted average of the reciprocal of the boundary resistance

for each leaf, and then to take the reciprocal of that mean.

The numerical values of shelter factors are poorly defined and are likely to vary

greatly with canopy structure but they are believed to be of the order 2 to 3. The

fact that shelter factors are so large and so poorly defined compromises the value

of early wind tunnel research into the boundary-layer resistance for individual

leaves to some extent. However, basic knowledge of the order of magnitude of

boundary-layer resistances found in those studies, and the understanding they

gave of the difference between boundary-layer resistance for momentum transfer

and for other transfers, is important when writing equations describing the whole-

canopy aerodynamic resistance, as discussed in the next chapter.

Stomatal resistance

At each level in the canopy, contributions to the overall canopy exchange of sensible

heat flux originate from the exposed surface of the vegetation elements present at

that level, and arise because of the difference between the surface temperature of

the vegetation and the temperature of the canopy air stream. As just discussed, the

magnitude of the contributions is controlled by the mean boundary-layer resistance

of the vegetation elements at each level.

Plant cells are about 90% water and would quickly desiccate and die if exposed

to an unsaturated atmosphere. For this reason, plants seek to retain water content

using surface layers that are resistant to water loss. Most leaves, for example, have

a waxy cuticle that inhibits the loss of gases such as water vapor and carbon

dioxide from their surface. But, if plants are to grow, they need to allow the cells

within leaves not only to absorb photosynthetically active radiation but also to

have access to the CO2 present in the atmosphere. They do this by gas exchange,

with internal cells gaining access to CO2 through small pores in the leaf surfaces

called ‘stomata’ which can be opened in environmental conditions favorable for

photosynthesis.

The same stomata that allow carbon dioxide to enter leaves also allow water

vapor evaporated from the moist cell walls inside the leaf to escape to the

atmosphere. Consequently, if the outside surface of vegetation canopy is dry (not

wet as after rainfall), the primary source of water leaving plants is from cells inside

leaves. This water vapor diffusing by molecular diffusion from sub-stomatal

cavities through stomata to the leaf surface and the need to diffuse through narrow

stomata inhibits the rate of water vapor flow, depending on the extent to which the

plant stomata are open. This inhibition on flow is represented in equations and

models by a resistance, the stomatal resistance, rST

, to vapor flow from inside to

just outside the leaf, see Fig. 21.5. Thus, the stomatal resistance per unit area of leaf

is used in much the same way that boundary-layer resistance is used to represent



molecular diffusion from the surface of the leaf to the air in the canopy. But it

applies only to the gaseous exchanges.

The flow of latent heat per unit area of leaf, lEl, leaving from inside of leaves to

the surface of leaves is given by:

( )l a p s i

ST

c e eEr−

=r

lg

(21.18)

where es and e

i are the vapor pressure in the stomatal cavity and at the surface of

the leaf outside the stomata, respectively, and rST

is the stomatal resistance per unit

area of leaf. In practice it is usually assumed that the air inside the leaf is saturated

at the nearby surface temperature of the leaf, Ts, see Figure 21.5. Hence e

s = e

sat(T

s)

and Equation (21.18) becomes:

( ( ) )l a p sat s s

ST

c e T eEr

−=

rl

g

(21.19)

As is the case for boundary-layer resistance (see Equation (21.16) for example), it

is the reciprocal of the stomatal resistance, i.e., the stomatal conductance, which

scales linearly with the area of leaf. Consequently the mean stomatal layer

resistance is calculated as the reciprocal of the area-weighted average of the

Boundary layer

Boundary layerresistance

Stomatalresistance

Leaf surface

Boundarylayer limit

T

Ts

RnI

HI lEI

es

e

ei =esat (Ts)

Cuticle

Epidermis

Sub-stomatalcavity

Mesophyll(Chloroplasts)

H

Figure 21.5 Cross-section of the surface of a leaf showing a stomata and the stomatal and boundary-layer resistance used

to represent the restrictions on the flows of latent heat and sensible heat.



reciprocal of stomatal resistance for each leaf. For example, ( )st zr , the mean stomatal

resistance per unit leaf area of the N leaves in a (small) height range δz around the

level z is given by:

1

1

( )N

ist z i

i ST

Arr

−

=

⎡ ⎤= ⎢ ⎥

⎣ ⎦∑

(21.20)

Energy budget of a dry leaf

On a per leaf area basis, the sensible heat and latent heat flux exchanges with the

surface of a leaf shown in Figure 21.5 are given by:

( )l s za p

H

T TH cR−

= r

(21.21)

and:

( ( ) )a pl sat s

V ST

c e T eER r

−=

+r

lg

(21.22)

Assuming the energy stored in the leaf is negligible, the surface energy budget is

given by:

l l lnE H R+ =l

(21.23)

where Rn

l is the energy falling as net radiation per unit area of leaf.

If the equivalent linear rate of change in saturated vapor pressure between Ts

and T, Δ, is defined by:

( ) ( )sat s sat

s

e T e TT T

−Δ =

−

(21.24)

then rearranging Equation (21.23), and substituting first Equations (21.21) and

then Equation (21.24) gives:

( ( ) ( ))l l sat s satn a p

H

e T e TE R cR−

= −Δ

l r

(21.25)

which equation can be written as:



( ( ) ) ( ( ) )l l sat s satn a p

H

e T e e T eE R cR

− − −= −

Δl r

(21.26)

or:

( ( ) )l lsat sa p n p

H H

e T e DE c R cR R

−+ = +

Δ Δl r r

(21.27)

where D is the vapor pressure deficit of the air outside the boundary layer of the

leaf.

Substituting Equation (21.22) into Equation (21.27) and rearranging gives:

a pln

l H

v ST

H

c DR

RE R rR

Δ +=

+Δ +

r

lg

(21.28)

If it is then assumed that the boundary-layer resistances for latent and sensible

heat are equal and represented by R (i.e., that R = RH = R

V), then Equation (21.28)

becomes:

1

a pln

l

ST

c DR

RErR

Δ +=

⎛ ⎞Δ + +⎜ ⎟⎝ ⎠

r

lg

(21.29)

This equation is the well-known and much used Penman-Monteith equation

(Monteith, 1965), which is the basis for much of the description of evaporation in

hydrometeorology. In this case the equation is applied to the energy balance for

unit area of leaf.

Energy budget of a dry canopy

Early research into how best to represent the complexity of exchanges in vegetation

canopies involved two general approaches. One approach (e.g., Waggoner and

Reifsnyder, 1968; 1969) was to build multi-layer computer models of the interaction.

Such models (see Fig. 21.6a) represent the capture of radiant energy at several levels

in the canopy, and the heat exchanges between leaves and air at these levels is

calculated from the level-average stomatal resistance to water vapor flow and the

level-average leaf boundary-layer resistance to momentum, water vapor and sensible

heat flow. Multi-layer models also often describe the aerodynamic resistance to

energy flow between each level using a form of K Theory.



A second school of thought (Monteith, 1965) preferred the much simpler big

leaf approach to describe plant canopy exchange with the overlying atmosphere

(see Fig. 21.6b). The big leaf approach had its origin in the Penman-Monteith

equation and essentially assumes that the exchange of the whole canopy can be

adequately represented by assuming that all the radiation capture and partitioning

of energy into latent and sensible heat can be described as if it occurred at a single

level, the effective source-sink height. At this level, the whole-canopy, parallel-

average values of stomatal resistance and boundary-layer resistance control the

exchange between the hypothetical big leaf and the surrounding air, these

resistances being appropriately scaled-down from the resistance for individual

leaves by dividing by the leaf area index (LAI) of the whole canopy. The

aerodynamic resistance for latent and sensible heat is then used to represent the

turbulent transfer of energy fluxes upward into the atmosphere. In a simple big-

leaf model, the canopy-average boundary-layer resistance and the aerodynamic

resistance act in series for both the latent and the sensible heat transfer, and are

often combined as a single aerodynamic resistance.

The relative merits of multi-layer computer modeling of whole-canopy

exchanges versus the simpler big leaf approach were debated throughout the late

1960s and early 1970s. However, during the 1970s, the big leaf approach gained

preference over multi-layer computer modeling, mainly because it was realized

that multi-layer canopy models require a level of detail in the specification of can-

opy properties and canopy structure that limit their use to research sites where

such detailed knowledge might be available. It was also realized that when repre-

senting and modeling whole-canopy interactions, detailed representation of

within-canopy exchanges is less important numerically than adequately represent-

ing the major controls of stomatal resistance and bulk aerodynamic transfer

between the canopy and the overlying air. The big leaf approach is now almost

H Rn lEH Rn lE

Level 1

Level 2

Level 3

Level ‘n’

(rH)1

(rH)2

(rH)3

(rH)n

(RH)n

(RH)3

(RH)2

(RH)1

(rST)n

(rST)3

(Rv)n

(rST)2

(rST)1

(Rv)3

(Rv)2

(Rv)1

(rv)3

(rv)n

(rv)2

rH rv

RH rST Rv

(rv)1

(a) (b)

Figure 21.6 Resistance scheme used to describe whole canopy surface energy flux exchanges in (a) multi-layer computer

simulation models, and (b) single source or ‘big leaf ’ models.



universally used in hydrological and meteorological modeling, sometimes, but not

always, with the inclusion of below canopy fluxes from the soil, which may also be

calculated using the Penman-Monteith Equation applied at the soil surface.

If below canopy fluxes are neglected (they can be small below a fully developed

vegetation canopy), the representation of whole-canopy surface energy balance in

the big leaf representation can be calculated by a whole-canopy version of the

Penman-Monteith Equation. In this case (see Fig. 21.7) the equations representing

the whole-canopy exchanges of sensible heat, H, and latent heat flux, λE, are

respectively:

( )s refa p H

a

T TH c

r−

= r

(21.30)

and:

( ( ) )a p sat s refH

a s

c e T eE

r r−

=+

rl

g

(21.31)

where Tref

and eref

are the temperature and vapor pressure at a reference level above

the canopy, Ts is the canopy-average leaf surface temperature, r

s is the canopy-

average leaf stomatal resistance (often called the ‘surface resistance’), and ra

H and

ra

V are the aerodynamic resistances for the transfer of water vapor and sensible

heat (which include both the canopy-average leaf boundary-layer resistances and

the aerodynamic transfer resistance for turbulent transport between the effective

Figure 21.7 The whole canopy

single source model used to

derive the whole canopy version

of the Penman-Monteith

equation.



source in the canopy and the reference level.) In a whole-canopy representation,

the total energy available to support sensible and latent heat transfer is given by:

+ =E H Al (21.32)

where A, the available energy, in addition to net radiation, includes allowance for

soil heat flux, G, and the physical and biochemical storage in the canopy, S and P,

respectively (see Chapter 4).

Equations (21.30), (21.31), and (21.32) are analogous to Equations (21.21),

(21.22), and (21.23), respectively, and the derivation of the whole-canopy Penman-

Monteith Equation therefore follows by direct analogy with that for unit area of

leaf given above. Assuming ra = r

aH = r

aV, the resulting equation takes the form:

1

a p ref

a

s

a

c DA

rErr

Δ +=

⎛ ⎞Δ + +⎜ ⎟⎝ ⎠

r

lg

(21.33)

where Dref

is the observed VPD at the reference level above the canopy.


● Turbulent and non-turbulent controls: within a vegetation canopy flux

exchange involves interplay between turbulent vertical diffusion of fluxes

and the divergence of these fluxes through non-turbulent interaction with

the vegetation elements that make up the canopy. Resistances associated with

molecular diffusion through boundary layers and stomata control the

dissipation or generation of portions of the fluxes at each level in the canopy.

● Skin friction and bluff body transfer: all fluxes from the canopy air stream to

leaves can occur by skin friction transfer (i.e. by molecular diffusion through

the non-turbulent boundary layers surrounding leaves), but momentum can

also be transferred more efficiently by pressure forces in bluff body transfer,

hence the boundary-layer resistance for momentum is about an order of

magnitude less than for other transfers, depending on the orientation of

the leaf.

● Influence of molecular diffusion coefficient: when boundary-layer resist-

ance is controlled by skin friction transfer the relevant molecular transfer

coefficient controls both rate of diffusion and thickness of the boundary

layer, hence the ratio of the boundary-layer resistances for two transfers is

inversely proportional to the ratio of their molecular diffusion coefficients

raised to the power 0.67.



● Shelter factors: mutual sheltering of leaves in a canopy raises the effective

value of boundary-layer resistances by empirical shelter factors of the order

of 2 to 3.

● Stomatal resistance: in dry canopies water mainly evaporates inside the

leaves so, in addition to boundary-layer resistance, latent heat (and CO2) flux

also has to pass through a stomatal resistance associated molecular diffusion

through the plants’ stomata.

● Single leaf Penman-Monteith equation: simultaneously solving the energy

balance equation for a single leaf and equations that control the rate of diffu-

sion of heat fluxes through boundary-layer resistances and stomatal resistance

gives the Penman-Monteith equation for a single leaf, see Equation (21.29).

● Whole-canopy Penman-Monteith equation: the single source or ‘big leaf ’

representation of energy balance for a canopy (with surface resistance taken

as the canopy average of all stomatal resistances acting in parallel, and

aerodynamic resistance as the sum of turbulent transfer resistance and canopy

average boundary-layer resistance acting in series) gives the whole-canopy

Penman-Monteith Equation using a derivation analogous to that for a single

leaf, see Equation (21.33).

References

Monteith, J.L. (1965) Evaporation and environment. Symposium of the Society for

Experimental Biology 19, 205–234; republished in Gash, J.H.C. and Shuttleworth, W.J.,

(2007) Benchmark Papers in Hydrology: Evaporation. IAHS Press, Wallingford, 521p.

Monteith, J.L. and Unsworth, M.H. (1990) Principles of Environmental Physics. Edward

Arnold, London, p. 291.

Thom, A.S. (1968) The exchange of momentum, mass, and heat between an artificial leaf

and the airflow in a wind-tunnel. Quarterly Journal of the Royal Meteorological Society,

94, 44–55.

Waggoner, P.E. and Reifsnyder, W.E. (1968) Simulation of the temperature, humidity, and

evaporation profiles in a leaf canopy. Journal of Applied Meteorology, 7, 400–409.

Waggoner, P.E. and Reifsnyder, W.E. (1969) Simulation of the microclimate in a forest.

Forest Science, 15, 37–40.


Introduction

Real vegetation canopies are extended three-dimensional entities. Even in the

case of agricultural crops in the middle of their growth season (arguably the

most uniform vegetation cover), assuming the whole-canopy interaction can

be adequately represented as if it occurred at a single effective source/sink level as

in the Penman-Monteith equation requires some faith, because observed profiles

of atmospheric variables change substantially with height, see Fig. 22.1.

However, as already discussed in Chapter 21 (see Fig. 21.6 and associated text),

the alternative approach of using numerical models to simulate multi-level

exchanges is impractical, given that such models require detailed site- and time-

specific knowledge of canopy structure that is rarely available. Multi-layer canopy

models also usually make the problematic assumption that K Theory applies

within plant canopies, as discussed in Chapter 19 (see Fig. 19.3). But, given the

necessity to resort to the single source/sink assumption, how can the knowledge

of in-canopy processes discussed in the last chapter be used to best advantage in a

big leaf model? This is one of the topics considered in this chapter.

When the big leaf representation of the plant–atmosphere interactions and

Penman-Monteith equation are adopted, surface exchanges depend on the values

of weather variables at some reference level in the ABL. However, the discussion of

ABL development given in Chapter 18 indicates that atmospheric variables in the

ABL (including those at the reference level) are themselves partly determined by

surface energy inputs. Thus, because the air in the ABL is not totally ‘free’ but rather

often partly ‘contained’ by an inversion layer, feedback processes can come in to

play, with surface exchanges not only determined by, but also in part determining,

the value of near surface weather variables. The effect of ABL feedbacks on area-

average surface exchange is also discussed in this chapter.

22 Whole-Canopy Interactions




Whole-Canopy Interactions 317

Whole-canopy aerodynamics and canopy structure

Chapter 20 described the aerodynamic behavior and aerodynamic resistance

of whole canopies when viewed from above, i.e., on the basis of measurements of

profiles of atmospheric variables made above the canopy. Chapter 21 then

discussed detailed studies of in-canopy exchanges with individual vegetation

elements (especially leaves). One result of these studies is the estimated order of

magnitude for the drag coefficient, cd, of a typical leaf. The model leaf described

in Fig. 21.2, for example, when aligned at an average angle ∼45° to the wind with

the drag coefficient reduced by a ‘shelter factor’ of order two, suggests cd ≈ 0.2. As a

result, the question arises, is a drag coefficient of this order consistent with typical

values of d and zo found for whole canopies and with their observed variation

with crop height and canopy density?

Modeling studies have been used to investigate how the effective values d and zo

for a vegetation canopy vary with canopy density and the vertical distribution of

leaves. Shaw and Pereira (1982) used a second order closure model to describe

vertical transfer within and above a modeled canopy assuming that the momen-

tum divergence at height z in the canopy is proportional to a drag coefficient

cd = 0.2 multiplied by the plant (mainly leaf) area present at each height. They

simulated the resulting wind speed profile above the model canopy and from this

calculated the values of d and zo that best described the shape of the modeled

profile assuming this had a logarithmic height dependency, compare Equation

(19.22). In the model the simple height dependent leaf area distribution L(z)

shown in Fig. 22.2 was used, with L(z) normalized such that:

0

( ).h

L z dz LAI=∫ (22.1)

Rn

1

0

1

0

u T e cZ

h

Z

h

Figure 22.1 Vertical profiles of

net radiation, (Rn), wind speed

(u), temperature (T), vapor

pressure (e) and CO2

concentration (c) that are typical

of those observed in a uniform

stand of cereal during the day

and at night.



where h is the height of the canopy and LAI is the total leaf area index of the

canopy. The variation of d and zo was explored as a function of LAI and the ratio

(zM

/h), where zM

is the height with maximum leaf area, see Fig. 22.2.

The results of this modeling study are shown in Fig. 22.3. They show that the

normalized displacement height, (d/h), increases with (zM

/h), as might be expected,

and that (d/h) also increases with the total leaf area of the modeled canopy, again

as expected. For ‘closed canopies’, i.e., when LAI is typically in the range 2 to 6,

the modeled value of (d/h) is in the range 0.5 to 0.75, depending on the position

of maximum leaf area in the canopy. This is consistent with field observations.

The normalized aerodynamic roughness, (zo/h), was modeled initially to increase

Modeled canopy

Z

h

zm

L(z)

Figure 22.2 Specification of the

simple model canopy used to

investigate the variation of zero

plane displacement and

aerodynamic roughness with leaf

area index and the height

distribution of leaf area.

0.8

0.7

0.6

0.5

0.4

0.5 1 2

LAI

4 6 8

0.8

0.6

0.4

0.2

(a)zM

h

d

h

0.5 1 2

LAI

4 6 8

0.20.40.6

0.8

(b)zM

h

0.02

0.04

0.06

0.08

0.10

0.12

0.14

z0

h

Figure 22.3 Modeled variations in (a) normalized zero plane displacement, (d/h), and (b) normalized aerodynamic

roughness, (zo/h), as a function of the leaf area index, LAI, and the normalized position of peak leaf area in the canopy,

(zm

/h). (Redrawn from Shaw and Pereira, 1982, published with permission.)



with total leaf area. At a value of LAI which varies with (zM

/h), normalized

aerodynamic roughness then decreases with LAI as meanwhile (d/h) continues

to rise.

When the height of maximum leaf area is about halfway up the canopy, i.e.,

(zM

/h) ∼ 0.5, the variation of normalized displacement height with leaf area index

is approximately described by:

0.25

1.1 ln 15

LAId h⎡ ⎤⎛ ⎞≈ +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(22.2)

Similarly, for (zm

/h) ∼ 0.5 the variation of normalized aerodynamic roughness

with leaf area index is approximately described by:

0.5

0.29 for 15

o oLAIz z h LAI⎛ ⎞≈ + ≤⎜ ⎟⎝ ⎠

¢ (22.3)

0.3 1 for 1odz h LAIh

⎛ ⎞≈ − >⎜ ⎟⎝ ⎠

(22.4)

where zo’ is the aerodynamic roughness of the underlying (soil) surface.

Excess resistance

One significant general result found in the wind tunnel studies described in

Chapter 21 is that the leaf boundary-layer resistance for the transfer of

momentum is typically about an order of magnitude less than that for other

exchanged entities, because momentum transfer can be by the efficient bluff

body process in addition to the skin friction process. This general difference

between the boundary-layer resistance for momentum and other exchanged

entities is not likely to be greatly altered by the mutual sheltering of clumped

leaves. This raises the question, how can this known difference in boundary-

layer resistances best be simply acknowledged in formulae describing the total

resistance between the surface of the leaf and the above-canopy reference level

when a big leaf representation is used?

The approach that has now almost universally been adopted for allowing for

the difference in boundary-layer resistances is to add an ‘excess resistance’ to the

aerodynamic resistance for momentum transfer when the effective aerodynamic

resistance for other exchanges is calculated. The excess resistance approach not

unrealistically assumes that the effective source/sink level for other exchanged

entities is lower in the canopy than the sink of momentum, because the rate of

divergence of downward momentum flux in the canopy is enhanced by bluff body

processes. The momentum flux therefore dissipates more quickly than it would

were only skin friction processes available. As discussed in Chapter 19, when



derived from the above canopy wind speed profile in neutral conditions, the level

of the effective sink for momentum appears to be at (d + zo), see Equation (19.22).

But the source/sink level for sensible heat and water vapor can be different, as

acknowledged in Equations (20.35) and (20.36). If the concept of excess resistance

is adopted, it is assumed that the effective source/sink level for sensible heat, for

example, is deeper in the canopy, see Fig. 22.4.

It is still assumed that the eddy diffusivities for the turbulent fluxes of momen-

tum, sensible heat, and water vapor (KM

, KH and K

V) are the same and given

by Equation (19.25) in neutral conditions. But the source/sink height for heat and

vapor in Equations (20.35) and (20.36) are at heights z0

H and z0

V above the zero

plane displacement, d, which are assumed to be less than z0, the aerodynamic

roughness length for momentum. It is also usually assumed that the excess

resistance is the same for all properties and, therefore, that other properties have

a common source sink height z0

P (= z0

H = z0

V). Consequently, in neutral conditions,

the aerodynamic resistance for sensible heat, for example, is:

2

* m

1 - 1 - ln ln

. .

H oa P P

oo o

zz d z drk u zz k u z

⎡ ⎤⎡ ⎤ ⎛ ⎞⎛ ⎞= = ⎢ ⎥⎢ ⎥ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ⎣ ⎦

(22.5)

which equation can be re-written as:

2 2

m m

1 - 1 ln ln

. .

H oa P

o o

zz drzk u k u z

⎡ ⎤⎡ ⎤= + ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦ (22.6)

Hence, ra

H = ra

M + re, where

2

m

1 ln

.

oe P

o

zr

k u z⎡ ⎤

= ⎢ ⎥⎣ ⎦

(22.7)

Resistance

Effective sinkfor momentum

Effective sinkfor heat

Wind Speed or Temperature

To’ To

re

raM

raH

Figure 22.4 The concept of

excess resistance for the case of

sensible heat in which it is

assumed the additional

resistance for heat transfer above

that for momentum can be

parameterized in a single source

model by defining it to have a

lower source/sink height, at

which height the temperature

difference with respect to the

reference level is greater.



The value of re has been measured in field studies for a wide range of natural,

permeable, fibrous surfaces. Figure 22.5 shows measured values of ln(z0/z

H) as a

function of the roughness Reynolds number (defined by Re* = (u*

zo)/ν) for a wide

range of surface types. Thus, the observed magnitude of ln(z0/z

H) for the fibrous

vegetation surfaces is of order two which, given the very substantial observational

variability in these measurements, roughly corresponds to z0

P ≈ (z0 / 10).

Substituting z0

P ≈ (z0 / 10) into Equations (20.35) while allowing the possibility

that the measurement height for temperature, zm′, might differ from that for wind

speed, zm

, gives:

⎛ ⎞⎛ ⎞− −= = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠2

0 0

1ln ln

10

H m ma

m

z d z dr

z zk u¢

(22.8)

Commonly the whole canopy aerodynamic resistance, ra, is assumed to be the

same for sensible and latent heat transfer and given by:

2

0 0

1ln ln

10

V H m ma a a

m

z d z dr r r

z zk u⎛ ⎞⎛ ⎞− −

= = = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠¢ (22.9)

Roughness sublayer

The turbulent interaction of the atmosphere can be parameterized in one dimen-

sion through the aerodynamic parameters z0 and d, and similarity theory is

assumed to be independent of the nature of the underlying surface. However, field

observations over tall vegetation suggest that very close to the surface (typically for

heights up to 50 z0) this is not true and that surface-related features (such as the

‘wakes’ generated by individual plants) can alter the local efficiency of turbulent

10010 1000 10000 100000

0

−2

2

4

Re*

(1)

(2)(3)

(4)

(5)

(6)

(7)

InZo

ZoH

Figure 22.5 Observed values of

ln(z0/z

H) as a function of the

roughness Reynolds number

Re* = (u*

zo)/ν for surface types

(1) vineyard, (2) short grass,

(3) medium grass, (4) bean crop,

(5) savanna scrub, and (6) and

(7) pine forest. (Redrawn from

Garratt, 1992, published with

permission.)



transfer. The layer over which such modification occurs is referred to as the

‘ roughness sublayer,’ see Fig. 22.6.

A realistic description of turbulent transport in the roughness sublayer requires

higher order closure representation, but in K Theory the approach adopted is to

treat the effect of the nearby surface on turbulence by re-defining the similarity

relations near the surface such that Equation (20.2), (20.5) and (20.8) respectively

become:

*

( )

*M

k z d u z d z du z L z− ∂ − −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∂

f j (22.10)

*

( )

*v

Hk z d z d z d

z L z− − −⎛ ⎞ ⎛ ⎞∂ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∂

q f jq

(22.11)

*

( )

*V

qk z d z d z dq z L z

∂− − −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∂f j

(22.12)

On the basis of observations over tall crops and trees, it has been suggested that

over a height range z < z* the empirical correction function j has the form:

* *

exp 0.7 1z d z dz z

⎡ ⎤⎛ ⎞ ⎛ ⎞− −≈ − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

j (22.13)

where z* is an empirical height range. It is assumed that j = 1 above z = z*, although

this gives an unrealistic discontinuity in j at this level. The value of (z*/z0) is very

poorly defined but has an order of magnitude of 50.

The net effect of the factor j is to enhance the eddy diffusivities for turbulent

transfer near the surface and, in this way, reduce the overall aerodynamic resistance

Height aboveground

Inertial sublayer

Roughness sublayer

Constant flux layer

Crop height, h

Zero planedisplacement,

dFigure 22.6 Location of the

roughness sublayer within the

constant flux layer above a stand

of vegetation.



for turbulent transfer. However, the magnitude of the reduction depends strongly

on the height at which the reference level is defined relative to the underlying

surface. Figure 22.7 illustrates the height-average reduction in aerodynamic

resistance in neutral conditions as a function of (normalized) reference level, with

(z*/z

0) assumed to be 50. The relative height of a reference level that is 2 m above

crop height for a 0.12 m high grass crop and a 10 m high forest stand are also

shown (expressed relative to the aerodynamic roughness length for these two

surfaces). With the assumption (z*

/z0) ≈ 50, the height-average reduction in

aerodynamic resistance for the grass crop is around 10%, but there is a reduction

of almost a factor two for a forest stand.

Wet canopies

When leaves and other components of a plant canopy are wet during and shortly

after rainfall, the source of the water evaporated from the wet portions of the can-

opy is no longer inside the leaves, rather it is from the water surfaces on the outside

of the leaves. Consequently for wet leaf surfaces the stomatal resistance is ‘shorted

out’ and is zero. Strictly speaking, the canopy average surface resistance in such

conditions should be calculated from the average surface area covered with water.

However, in practice, using the wetted area is not feasible and models of evaporation

from wet and partly wet canopies have adopted the alternative of describing the

evaporation in terms of the depth of water (in mm) stored on the canopy.

The most successful model of wet canopy evaporation is the Rutter model

(Rutter et al., 1971; 1975) and derivatives thereof (e.g., Gash, 1979). This model,

1.0

0.9

0.8

0.7

0.6

0.50 25 50 75 100 125 150

z*

z*Z0

= 10

= 50

= 100

Height of reference level [above(z-d ) in units of z0]

Hei

ght a

vera

ge r

educ

tion

fact

or

Reference level2m above forest

Reference level2m above grass

Z0

Z0

z*

Figure 22.7 Height average

reduction in aerodynamic

resistance in neutral

conditions for different

relative reference levels with

(z*/z

0) = 10, 50 and 100. The

height of 2 m reference

expressed relative to the

aerodynamic roughness

length for grass and forest are

also shown.



which is illustrated in Fig. 22.8, assumes there is a depth S of water, called

the canopy capacity, which is the minimum necessary to saturate the canopy.

The model makes a running water balance of water storage on the canopy (and

in some versions of the model, also the stems) from the difference between the

incoming precipitation intercepted by the canopy (and stems), the drip rate of

water from the canopy (or flow of water from the stems), and the intercepted water

that is evaporated.

The rate of evaporation of intercepted water, lEI, is calculated using the Penman-

Monteith equation with the surface resistance set to zero. When the calculated

amount of water stored on the canopy, C, is less than S, the evaporation is weighted

by the fractional fill of the canopy store. Thus, the calculated rate of evaporation of

intercepted water is:

( )ρΔ +⎛ ⎞λ = ⎜ ⎟⎝ ⎠ Δ + γa p ref a

I

A c D rCES

(22.14)

The water balance of the canopy is calculated from the equation:

(1 )t s IdC p f f E Ddt

= − − − λ − (22.15)

where p is the incoming precipitation rate, D is the rate of canopy drainage, fc is the

fraction of precipitation falling through holes in the canopy, and (assuming a stem

storage balance is also calculated) fs is the fraction of rain diverted to the stems of

PlE

lEIlET

S

St

Figure 22.8 The Rutter

model of rainfall interception

and evaporation.



the vegetation. One expression for the drainage rate that has been used in the

Rutter model is:

[ ]exp ( )D a b C S= −

(22.16)

with a = 0.002 mm min−1 and b = 4 mm being typical values. The total evaporation

from the canopy includes that from the intercepted water, lEI, and that from any

dry leaves, lET, and is estimated from the weighted sum:

1T IC CE E ES S

⎛ ⎞ ⎛ ⎞λ = λ + − λ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(22.17)

where

( )( )1

a p ref aT

s a

A c D rE

r r

Δ +λ =

Δ + +

r

g

(22.18)

where rs is the surface resistance of the canopy were it all dry.

The Rutter model of canopy interceptions has been adopted in the land surface

schemes of General Circulation Models but it has sometimes been substantially

simplified when used in this application. A running canopy water balance is

still made, but evaporation is set equal to lEI whenever C > 0, and D = 0 whenever

C < S, but D = p-S whenever p > S.

Recently there have been developments of the original Rutter model described

above to allow its use in sparse canopies (e.g., Valente et al., 1997). In essence the

approach used is to separate the landscape into two portions, a fraction c without

vegetation cover for which it is assumed there is no interception loss, and a frac-

tion (1-c) with vegetation cover for which evaporation is assumed calculated using

a version of the original Rutter model, perhaps with a simplified description of

drainage similar to that decribed in the last paragraph. This revised sparse canopy

version of the Rutter model is now accepted as being the preferred form because

as the fractional vegetation cover changes, it has appropriate asymptotic limits.

For a recent comprehensive review of models of wet canopy interception loss the

reader is referred to Muzylo et al. (2009).

Equilibrium evaporation

Consider the enclosed volume, V, of air above a water surface of area, A, inside a

thermally insulating box, shown in Fig. 22.9. Assume the air is well-mixed and has

reached saturation. Now assume that for a period δt, a radiant energy input flux,

Rn, is applied to the water surface but the air remains well-mixed. The incoming

energy will be used to both evaporate some of the water and to raise the temperature



of the air so that it remains saturated. To do this there is a sensible heat flux, H, and

a latent heat flux, λE, from the surface to the air above.

While the air is heating and moistening over the period δt, energy conservation

requires that:

nR H E= + l

(22.19)

Over this same period, heat conservation in the air requires that the heat input

from the surface equals the change in heat stored in the air as its temperature rises

by δT, i.e., that:

( ). ( ).a pHA t V c T= ρd d

(22.20)

Mass conservation in the air also requires that the flux of moisture from the surface

is equal to the change in water vapor in the volume V, i.e. that the increase in the

specific humidity, δq, is such that:

( ) ( )aEA t V q= ρd d

(22.21)

Because δq = 0.622(δe/P), see Equation (2.9), this means:

0.622( )( ) aV eEA t

Pρ

=d

d

(22.22)

However, by definition, Δ = (δe/δT), hence:

0.622( )( ) aV TEA t

Pρ Δ

=d

d

(22.23)

Dividing Equation (22.20) by Equation (22.23), then dividing both sides of the

resulting equation by λ and identifying the psychrometric constant among the

terms gives:

Well Mixed, Saturated Air

Radiant Energy Input

Thermallyinsulating

box

H

Rn

lE

Water

Figure 22.9 Equilibrium

evaporation into a saturated

atmosphere in response to a

radiant energy input to the

evaporating surface.



HE

=Δg

l (22.24)

Finally, combining Equations (22.19) with Equation (22.24) gives:

nE RΔ=Δ +

lg

(22.25)

This evaporation rate is called the ‘Equilibrium evaporation rate’. It is the rate of

evaporation that would occur from natural surfaces in response to incoming energy

if the overlying atmosphere was saturated and remained saturated. As such, it is a

useful concept because it defines a lower limit on natural evaporation rates. However,

in the real world, evaporation rates are higher than this because the atmosphere is

usually not saturated because water is removed aloft by precipitation.

Evaporation into an unsaturated atmosphere

Most often natural evaporation occurs during the day into an unsaturated ABL

which is partly (but not wholly) contained by a stable inversion. The surface

evaporation rate is determined by the surface resistance and aerodynamic

resistance but also by atmospheric variables (which may be measured) in the

surface layer, specifically by net radiation, VPD, wind speed, and temperature.

However, because the ABL is partly contained, the values of these atmospheric

variables are themselves influenced by the surface energy inputs, and the potential

for surface–ABL feedbacks therefore exists.

If daytime containment of the ABL were totally effective, evaporation rate

would presumably be close to the equilibrium evaporation rate. But this is not the

case. Precipitation processes (and the associated loss of water vapor and release of

latent heat aloft) mean that the air in the free atmosphere above the daytime

inversion is on average drier and (in terms of potential temperature) warmer than

that in the ABL. As discussed in Chapter 18, intermittent breakdown of the

inversion layer during the day allows some entrainment of this drier, warmer air

from the free atmosphere into the ABL, and the ABL grows as a result. Again as

discussed in Chapter 18, some of the moisture evaporated from the ground may

remain in the ABL, but most is used to moisten the incoming drier air from the

free atmosphere. As a result the change in absolute moisture content in the ABL

can be small. On the other hand, during the day the air in the ABL is warmed both

by sensible heat from the ground and by the incoming warmer air from above.

Consequently, the temperature of the ABL rises and the VPD remains finite.

Because the VPD in the ABL is finite, natural evaporation rates from moist surfaces

are greater (typically 25% greater) than the equilibrium evaporation rate.

The extent to which dry, warm air from the free atmosphere is entrained

depends on the strength of the stable inversion. The stronger the inversion the



more difficult it is to entrain air downward through it, and vice versa.

Entrainment also increases when the surface sensible heat flux (strictly the

surface buoyancy flux) increases because the intermittent breakdowns in the

inversion layer are related to the strength of turbulence in the ABL. For a given

strength of stable inversion, feedback processes can come into play to moderate

the VPD in the ABL. If, for example, the VPD tended to increase, then (assuming

the surface resistance and the available energy do not alter) surface evaporation

will increase and surface sensible heat flux (and also entrainment of drier air)

will decrease. The net result is, therefore, to counteract the increase in VPD. The

converse is true if VPD decreases. Because the ABL grows beneath a partially

contained inversion layer during the day when most evaporation occurs, and

because surface-atmosphere feedback constrains the magnitude of the VPD in

the ABL, natural evaporation rates are restricted to being about 25% greater

than the equilibrium evaporation rate. This applies in moderate humid

atmospheres when the surface resistance is reasonably small and is the reason

why hydrologists and meteorologists have been able to postulate the hypothetical

existence of potential rates of evaporation, a point discussed further below and

in Chapter 23.

McNaughton and Spriggs (1989) explored the effect of surface–atmosphere

interactions on evaporation rate using a simple ‘slab model’ (see the section

on low order closure schemes in Chapter 19). The evolution with time of

the ABL represented in the model is illustrated in Fig. 22.10. This slab model

Modeled ABL(at time t )

Free atmosphere

Mixed layer

Specific humidity = qm(t)

Potential temperature = qm(t )

VPD = D(t )

lE(t ) Rn(t ) H(t)

Entrainment layer

Modeled ABL(at time t + dt )

Free atmosphere

Mixed layer

Specific humidity = qm (t+δt )

Potential temperature = qm(t+δt )

VPD = D(t+δt)

lE(t+dt ) Rn(t+dt )) H(t+dt ))

Entrainment layer

h h

qq

qm (t )

qm (t+dt))qm (t )

qm (t+dt ))

Figure 22.10 The simulated growth of the ABL over the time interval dt in the McNaughton and Spriggs (1989) slab

model simulation.



used the Penman-Monteith equation to calculate surface energy balance for

different prescribed values of surface resistance, with the aerodynamic resist-

ance set to:

0*

1lna

Lr

ku z⎛ ⎞

= ⎜ ⎟⎝ ⎠

(22.26)

in which u* and zo are prescribed and the absolute value of the Obukov Length,

L , was calculated at each model time step. The time evolution of the mean

potential temperature, qm

, and specific humidity, qm

, in the simulated ABL were

determined by the energy and humidity conservation, as follows:

( ) ( )ma p a p s f

d dhc h H t cdt dt

= + −q

r r q q

(22.27)

= + −( ) ( )ma a s f

dq dhh E t q qdt dt

r r

(22.28)

where h is the depth of the ABL, H(t) and E(t) are the time-dependent modeled

surface sensible heat and evaporation fluxes, respectively, and qf and q

f are the

potential temperature and specific humidity of the free atmosphere, respectively.

The rate of growth of the ABL was assumed to be directly related to the surface

buoyancy flux and inversely to ( )v hz∂ ∂q , the rate of change of virtual potential

temperature at the top of the ABL, as follows:

( ) 0.07 ( )

va p

h

dh H t E tdt c h z

+=∂⎛ ⎞

⎜ ⎟∂⎝ ⎠

lqr

(22.29)

McNaughton and Spriggs initiated this model run using nine days of data from a

tower site at Cabauw in the humid climate of the Netherlands (Driedonks, 1981;

1982). These nine days included some with weak and some with strong inversions.

Observed profiles of potential temperature and specific humidity measured at

05:45 am were used to initiate the model profiles, and the measured time series of

net radiation minus soil heat flux through the day was used to force surface energy

balance (the growth of boundary layer cloud was not simulated). The surface

evaporation calculated by the model with different prescribed values of average

surface resistance was averaged over the daytime hours when the ABL was grow-

ing. For each value of prescribed area-average surface resistance, the effective

value of the parameter α in the equation:

E AΔ=Δ +

l ag

(22.30)



was calculated, where A is the energy available at the surface. Shuttleworth et al.

(2009) subsequently re-normalized McNaughton and Spriggs’ daytime average

values of α to give the equivalent all-day average values.

Figure 22.11 shows (as thin lines) the original results of the McNaughton and

Spriggs model study for the nine days on which simulations were made and reveals

the substantial day-to-day variability given by the different initiations, much of

this variability being related to differences in the strength of the inversion. The

thick line in Figure 21.10 is a polynomial fit to the average values of α over these

nine days (Shuttleworth et al., 2009), which has the form:

2

&

3 4 5

1.26 0.24141 ln 0.07199 ln70 70

0.0099 ln 0.00504 ln 0.00083 ln70 70 70

s sM S

s s s

r r

r r r

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

a

(22.31)

Figure 22.12 compares the predictions of Equation (22.31) to experimental

measurements of the relationship between α and daily average surface resistance

made in the semi-arid climate of southern Arizona, over three different land

covers (Shuttleworth et al., 2009). The results suggest that the model relationship

1.4

1.2

1.6

1.0

0.8

0.6

0.4

0.2

0.010 100 1000 10000

Natural logarithm of surface resistance (in s/m)

Alp

ha (

dim

ensi

onle

ss)

Figure 22.11 Model-simulated

values of a in Equation (22.27)

given by a ‘slab model’

simulation of surface-ABL

coupling with different

prescribed values of surface

resistance initialized using field

data on nine days. Also shown is

the polynomial fit to the average

values of these different curves

described in the text. (Redrawn

from Shuttleworth et al., 2009,




although originally developed using initiation fields taken in the humid

climate of the Netherlands may have more general applicability.

Arguably the most significant aspect of Figs 22.11 and 22.12 is that they sug-

gest that for values of surface resistance less than about 100 s m−1 (a value typical

of many unstressed natural surfaces), the ABL feedback processes described

earlier in this section seem reasonably effective at constraining the evaporation

rate to be approximately 25% greater than the equilibrium evaporation rate.

This result is broadly consistent with the proposal of Priestley and Taylor (1972)

that Equation (22.30) is an ‘appropriate framework’ for apportioning surface

1.0

0.5

0.0

1.5

10 100 1000 10000

M&S

Woodland

Surface resistance (s m−1)

(a)

α(a)

1.0

0.5

0.0

1.5M&S

Grassland

(c)

10 100 1000 10000


α

(c)

1.0

0.5

0.0

1.5M&S

Shrubland

(b)

10 100 1000 10000


α

(b)

Figure 22.12 Comparison between measured values of a and daily average surface resistance over (a) woodland,

(b) shrubland, and (c) grassland covers in southern Arizona, (Redrawn from Shuttleworth et al., 2009, published

with permission.)



energy between sensible heat and evaporation, and that for ‘saturated surfaces’

a reasonable estimate of evaporation can be made from:

1.26E AΔ=

Δ +l

g (22.32)

However, Fig. 22.12 also shows that as surface resistance rises and water

availability at the surface decreases, evaporation rate necessarily falls and

ultimately becomes zero.


● Use of in-canopy results: results from in-canopy studies of individual leaf

exchange and near-canopy turbulent transfer can improve representation of

whole canopy exchanges by single source, ‘big leaf ’ Penman-Monteith models.

● LAI dependency of aerodynamic properties: taking 0.2 as the average drag

coefficient for an individual leaf inside a canopy, Shaw and Periera (1982)

used a second order closure model of canopy exchange to give simulation

of zero plane displacement and aerodynamic roughness consistent with

field observations.

● Excess resistance: in one-dimensional models the fact that the exchange

coefficient for individual leaves is significantly less for momentum than for

other exchanges is often accommodated by assuming other exchanges act at

a source/sink deeper in the canopy and have 10 times less aerodynamic

roughness length than momentum.

● Enhanced efficiency of near surface turbulence: enhanced efficiency of

turbulent transfer near vegetation canopies has been accommodated in K

Theory by a height dependent re-definition of similarity relations and

typically reduces aerodynamic resistance for short crops by ∼10% but for tall

(forest) crops by about a factor of two.

● Rutter model: the most successful representation of evaporation from

canopies wet or partially wet during and after rain is using the Rutter model

or derivatives thereof, which make a running balance of rain water stored on

vegetation and assume evaporation of intercepted water is proportional to

the fractional fill of a canopy water store.

● Equilibrium evaporation: when incoming radiant energy is incident on a

water surface evaporating into an isolated volume of air, the energy is shared

as latent and sensible heat inputs to the air such as to keep the air saturated

and evaporation occurs at a well-defined equilibrium evaporation rate given

by Equation (22.25).

● Greater than equilibrium evaporation: evaporation mainly occurs during

the day into an ABL partly (but not wholly) constrained by a stable inversion,



consequently the ABL usually remains unsaturated and the evaporation rate

from an underlying moist surface is therefore higher than the equilibrium

evaporation rate.

● Modeling ABL feedbacks: modeling feedback processes in a partly

constrained ABL gives an approximate relationship between land surface

evaporation rate and surface resistance that is consistent with observations

within (albeit large) experimental errors, see Fig. 22.12.

● Origin of potential evaporation hypothesis: surface-atmosphere feedbacks

constrain the magnitude of the VPD in the ABL such that for land surfaces

with fairly small surface resistance, evaporation into a humid atmosphere is

typically ∼25% greater than the equilibrium evaporation rate. The presence

of ABL feedbacks is why it has been possible for hydrologists to postulate the

hypothetical existence of potential rates of evaporation.

References

Driedonks, A.G.M. (1981) Dynamics of the well-mixed atmospheric boundary layer.

Scientific Report W.R. 81-2 K.N.M.I., De Bilt, The Netherlands.

Driedonks, A.G.M. (1982) Models and observations of the growth of the atmospheric

boundary layer. Boundary-Layer Meteorology, 23, 283–306.

Garratt, J.R. (1992) The Atmospheric Boundary Layer. Cambridge University Press,

Cambridge, UK.

Gash, J.H.C. (1979) An analytical model of rainfall interception by forests. Quarterly Journal

of the Royal Meteorological Society. 105, 43–55.

Muzylo. A., Llorens, P., Valente, F., Keizer, J. J., Domingo, F. and Gash, J.H.C. (2009)

A review of rainfall interception modeling. Journal of Hydrology, 370, 191–206.

McNaughton, K.G. and Spriggs, T.W. (1989) An evaluation of the Priestley-Taylor equation.

In Black, T.A., Spittlehouse, D.L., Novak, M.D. and Price, D.T. (eds) Estimation of Areal

Evaporation. IAHS Publication No.177, IAHS Press, Wallingford, UK.

Priestley, C.H.B. and Taylor, R.J. (1972) On the assessment of surface heat flux and evapora-

tion using large scale parameters. Monthly Weather Review, 100, 81–92.

Rutter, A.J., Kershaw, K.A., Robins, P.C. and Morton, A.J. (1971) A predictive model of

rainfall interception in forests, 1. Derivation of the model from observations in a planta-

tion of Corsican pine. Agricultural Meteorology, 9, 367–384.

Rutter, A.J., Morton, A.J. and Robins, P.C. (1975) A predictive model of rainfall interception

in forests. II. Generalization of the model and comparison with observations in some

coniferous and hardwood stands. Journal of Applied Ecology, 12 (1), 367–380.

Shaw, R.H. and Pereira, A.R. (1982) Aerodynamic roughness of a plant canopy: A numeri-

cal experiment. Agricultural Meteorology, 26, 51–65.

Shuttleworth, W.J., Serrat-Capdevila, A., Roderick, M.L. and Scott, R.L. (2009) On the

theory relating changes in area-average and pan evaporation. Quarterly Journal of the

Royal Meteorological Society, 135, 1230–1247.

Valente, F., David, J.S. and Gash, J.H.C. (1997) Modelling interception loss for two sparse

eucalypt and pine forests in central Portugal using reformulated Rutter and Gash analytic

models. Journal of Hydrology, 190, 141–162.


Introduction

Broadly speaking, the understanding of processes and phenomena involved in

the interaction between the (often vegetation-covered) land surface and the

atmospheric boundary layer that were discussed in previous chapters is used in

two ways. The first way is to combine this understanding in the form of computer

sub-models, which then become important components of hydrological or

meteorological models. Such sub-models are often called soil-vegetation-

atmosphere schemes (SVATS) and they are described in the next chapter. This

chapter covers a second broad application of understanding of surface-atmosphere

interactions, which is to provide daily estimates of evaporation for use in other

hydrological and agricultural applications. This use of evaporation estimates

predates the creation of SVATS and, although the methods and formulae used

have the same basic origin, the applications are generally simpler and the methods

used involve making more assumptions.

Thus, the distinction between these two applications is in part related to

complexity. However, it is more fundamentally associated with the availability of

the meteorological data needed to calculate estimates of surface-atmosphere

exchange. In the case of SVATS, the meteorological data needed are generally

readily obtainable and they are usually available for sampling intervals of an hour

or less, perhaps from automatic weather stations in the case of hydrological

models, or as a byproduct of calculations made in other model components in the

case of meteorological models. The availability of relevant meteorological data is

usually a much more problematic issue when making the simple daily estimates

of evaporation discussed in this chapter. Moreover, data limitations can affect

the likely reliability of the method adopted to make an estimate. At best, the avail-

able data is that which might be provided by a standard agro-meteorological

climate station reporting daily. Hence, the meteorological variables used to

23 Daily Estimates of Evaporation




Daily Estimates of Evaporation 335

estimate evaporation are themselves the estimates of daily average or daily total

values that can be derived from the limited measurements taken at such climate

stations, as discussed in the next section. In many cases the recommended

approach used to estimate daily values is obvious, or versions of the equations used

have been defined earlier; to reduce the scope for misunderstanding these methods

are also given explicitly below.

Daily average values of weather variables

Temperature, humidity, and wind speed

Daily maximum and minimum air temperature, Tmax

and Tmin

, respectively, are

usually measured at climate stations. These can be used to estimate the daily

average air temperature, T, and the daily average saturated vapor pressure, es, from:

max min

2

T TT

+=

(23.1)

and:

max min( ) ( )

2sat sat

se T e Te +

=

(23.2)

where, for example, esat

(Tmax

) represents the function of Tmax

given as Equation

(2.17). Because the relationship between saturated vapor pressure and tempera-

ture is not linear, using Equation (23.2) is preferable to estimating es from e

sat(T).

A measurement of humidity may be available at least once (but often only once)

each day, perhaps in the form of a measurement of the wet bulb and dry bulb tem-

perature, Tdry

and Twet

, respectively, or as a measurement of relative humidity, RH,

or as a measurement of dew point temperature, Tdew

. If Tdry

and Twet

(in °C) are

available, e is calculated from Equation (2.24) which is here re-written in the form:

( ) *( - )sat wet dry wete e T T T= − g (23.3)

to emphasize that the effective value of the psychrometric constant required may well

not be γ = (cpP)/(0.6622λ) as used elsewhere, but rather a different empirically deter-

mined value, γ *. This is very likely the case if the wet and dry bulb thermometers are

not aspirated and the wet bulb depression is then smaller than the true value.

If RH (%) is available at a particular time (e.g., 09:00 local time) when the dry

bulb temperature is Tdry

′, the daily average value of e is best calculated from:

( )100

sat dryRHe e T⎛ ⎞= ⎜ ⎟⎝ ⎠

¢

(23.4)



If RH is considered to be an all day average value, es should be used instead of

esat

(Tdry

′) in Equation (23.4).

If Tdew

, (in °C) is available, e is calculated from:

( )sat dewe e T= (23.5)

The value of the daily average vapor pressure deficit, D, required in many of the

equations used to estimate evaporation then follows from:

sD e e= − (23.6)

It may also be necessary to adjust the value of wind speed, uz, if this is measured at

a height, zu, other than 2 m. To do this, it is necessary to assume the logarithmic

wind speed profile given in Equation (19.22) applies, and to prescribe values for

the aerodynamic roughness length, z0, and zero plane displacement, d, at the loca-

tion where wind speed is measured. When this is done, the wind speed, u2, that

would have been measured at 2 m is given by:

0

2

0

(2 )ln

( )ln

zu

dz

u uz dz

⎡ ⎤−⎢ ⎥⎣ ⎦=⎡ ⎤−⎢ ⎥⎣ ⎦

(23.7)

In practice, when making this correction it is often assumed that the required

values of z0 and d are those relevant for a ‘reference crop’ of short grass (described

later), in which case z0 = 0. 0148 m and d = 0.08 m are used.

Table 23.1 gives examples of the above calculations at three example sites, at

all of which albedo is assumed to be 0.23, but where humidity is measured in

different ways, as follows:

SITE A A site near Oxford, England, at latitude 51.7°N, elevation 129 m, on

July 15 (i.e., day of the year 196), where humidity is measured using

an aspirated wet and dry bulb thermometer, where cloud cover

(but not bright sunshine hours) is measured, and where wind speed

is measured at 10 m. At this site, the general climate is designated as

humid and measured pan evaporation is 5.5 mm per day.

SITE B A site near Tucson, USA, at latitude 32.2°N, elevation 720 m on

May 15 (i.e., day of the year 135), where humidity is measured as

relative humidity, where the number of bright sunshine hours (but

not cloud cover) is measured, and where wind speed is measured

at 2 m. At this site, the general climate is designated as arid and

measured pan evaporation is 14 mm per day.

SITE C A site near Manaus, Brazil at latitude 3.1°S, elevation 80 m, on

February 15 (i.e., day of the year 46), where humidity is measured by



Table 23.1 Demonstration of example calculations of the daily average values of air

temperature, saturated vapor pressure, vapor pressure, vapor pressure deficit, and wind

speed at 2 m at the three sites A, B, C specified in the text where humidity is measured

in different ways.

Origin Variable Units Site A Site B Site C

(Data) Maximum air temperature (°C) 22.00 37.00 30.00(Data) Minimum air temperature. (°C) 13.00 17.00 23.00Equ. (23.1) Average temperature (°C) 17.50 27.00 26.50Equ. (2.17) Sat. vapor pressure (Max. temp) (kPa) 2.644 6.275 4.243Equ. (2.17) Sat. vapor pressure (Min. temp) (kPa) 1.498 1.938 2.809Equ. (23.2) Average sat. vapor pressure (kPa) 2.071 4.106 3.526Assumed Wet bulb psychrometric constant (kPa °C−1) 0.066 – –(Data) Dry bulb temperature (°C) 17.50 – –(Data) Wet bulb temperature (°C) 15.00 – –Equ. (23.3) Vapor pressure (kPa) 1.538 – –(Data) Relative humidity (%) – 20.00 –Equ. (23.4) Vapor pressure (kPa) – 0.821 –(Data) Dew point (°C) – – 23.00Equ. (23.5) Vapor pressure (kPa) – – 2.809Equ. (23.6) Vapor pressure deficit (kPa) 0.533 3.285 0.717(Data) Wind measurement height (m) 10.00 2.00 5.00(Data) Wind speed (m s−1) 7.00 4.00 5.00Equ. (23.7) Modified wind speed (m s−1) 5.23 4.00 4.19

a dew point hygrometer, where cloud cover (but not bright sunshine

hours) is measured, and where wind speed is measured at 5 m. At

this site, the general climate is designated as humid and measured

pan evaporation is 5.1 mm per day.

Net radiation

When making daily estimates of evaporation the availability of a measured

(or more likely) estimated value for daily average net radiation is arguably the

most valuable meteorological variable because (providing water is not limit-

ing) available energy is the major control on evaporation rate. Also, as a daily

average value, net radiation estimates tend to be more ‘transferable’ from one

location to another within a region than other weather variables. Thus, as a

general rule, it is generally best to base an estimate of daily evaporation on

measured or estimated daily average net radiation if this is available or can

be obtained.

The equations needed to make an estimate of daily average net radiation

from measurements of sunshine hours or cloud cover, daily average temperature,



and daily average humidity were given in Chapter 5. An overview of the steps

required to make the calculation is as follows:

1. Use the day of the year, Dy, to calculate: (a) the eccentricity factor, d

r, from

Equation (5.5); and (b) the solar declination, d, from Equation (5.8).

2. Use the calculated value of d and the latitude of the site, f (in radians)

to calculate the sunset hour angle, ws (also in radians), from

Equation (5.12).

3. From the calculated values of dr, d, and w

s, calculate the solar radiation

incident at the top of the atmosphere at the site, So

d in mm of evaporated

water per day, from Equation (5.15).

4. If estimated fractional cloud cover, c, and locally derived values of the con-

stants as and b

s are available, use these values in Equation (5.16) to calculate

the daily total solar radiation, Sd, reaching the ground. If local values of as

and bs are not available, assume a

s = 0.25 and b

s = 0.5.

5. Alternatively, if a measure of bright sunshine hours, n, is available (rather

than cloud cover), first calculate the day length, N, in hours from ws using

Equation (5.13), then calculate Sd from Equation (5.17) using locally

derived values of as and b

s if available, but assume a

s = 0.25 and b

s = 0.5

otherwise.

6. Select a value for the albedo, a, (see Table 5.1, for example) then use this

with the calculated value of Sd to calculate the net daily solar radiation, Sn

d,

from Equation (5.18).

7. From a measurement or estimate of ed, the daily average vapor pressure in

kPa, calculate an estimate of the effective emissivity, e’, from Equation

(5.23).

8. Calculate the daily total solar radiation that would have reached the ground

had the sky been clear, Sdclear

, from Equation (5.16) with c = 0, using the

same values of as and b

s as were used in step 4 (or step 5).

9. On the basis of available measurements or otherwise, categorize the site as

being either ‘humid’ or ‘arid’, and on this basis select either Equation (5.24)

or (5.25) to calculate empirical cloud factor, f, using Sd from step 4 (or step 5)

and Sd

clear from step 8.

10. Use the value of the Stefan-Boltzmann constant re-expressed in units of

mm of evaporated water per day (i.e., σ = 2 × 10−9 mm d−1 m−2 K−4), with e’

from step 7, and f from step 9, and Tair

, the measured daily average air tem-

perature (in °K), to calculate the daily average net longwave radiation, Ln

d,

in mm of evaporated water per day using Equation (5.22).

11. Finally, use the values of Sn

d from step 5 and Ln

d from step 10 to calculate

the daily average net radiation flux, Rn

d, in mm of evaporated water per day

from Equation (5.26).

Table 23.2 demonstrates examples of the steps in calculation of net radiation for

three example sites A, B, C specified earlier with cloud cover measured at A and C

and the number of bright sunshine hours measured at B.



Table 23.2 Demonstration of the sequence of steps undertaken to calculate daily average net radiation as described in the

text applied in the three cases A, B, C specified previously. In two cases (A and C) cloud cover is measured and in the third

case (B) the number of bright sunshine hours is measured.


(Data) Day of year (none) 196 135 46Equ. (5.5) Eccentricity fctor (none) 0.9679 0.9774 1.0232Equ. (5.8) Solar declination (radians) 0.3773 0.3254 −0.2355Equ. (5.12) Sunset hour angle (radians) 2.0964 1.7849 1.5838(Data) Latitude (deg) 51.7 32.2 −3.1Latitude × π/180 Latitude in radians (radians) 0.9023 0.5620 −0.0541Equ. (5.15) Extraterrestrial solar radiation (mm day−1) 16.45 16.36 15.61(Data) Cloud fraction (none) 0.50 – 0.70Equ. (5.16) Solar at ground (cloudy sky) (mm day−1) 8.23 – 6.24(Data) Number of bright sunshine hours (hours) – 13.00 –Equ. (5.13) Maximum daylight hour (hours) – 13.64 –Equ. (5.17) Solar at ground (cloudy sky) (mm day−1) – 11.89 –Selected from above Solar at ground (cloudy sky) (mm day−1) 8.23 11.89 6.24(Data) Selected value for albedo (none) 0.23 0.23 0.23Equ. (5.18) Net solar radiation (mm day−1) 6.33 9.16 4.81Table 23.1 Vapor pressure (k Pa) 1.538 0.821 2.809Equ. (5.23) Effective emissivity (none) 0.166 0.213 0.105Equ. (5.16) (with c=0) Solar at ground (clear sky) (mm day−1) 12.34 12.27 11.70(Assigned) Assigned site humidity (none) Humid Arid HumidEqu. (5.24) or (5.25) Cloud factor (none) 0.667 0.958 0.533Table 23.1 Average temperature (°C) 17.50 27.00 26.50Equ. (5.22) Net longwave (mm day−1) −1.58 −3.30 −0.90Equ. (5.26) Net radiation (mm day−1) 4.76 5.86 3.90

Open water evaporation

All the methods recommended for estimating daily evaporation in this text are in

some way derived from the Penman-Monteith equation. In the case of open water

evaporation, the equation is used with surface resistance set equal to zero because

air is assumed saturated at the evaporating water surface. When calculating open

water evaporation the available energy, Aw, used in the Penman-Monteith equation

must be calculated with an albedo appropriate for a water surface (often 8% is

assumed), and should allow for any change in the energy stored in the water body

as a result of heat advection. Heat advection might result if water enters and leaves

the water body with temperatures that differ.

The most appropriate form of expression to be used for the aerodynamic resist-

ance for open water, ra

ow, has been debated over the years, but the empirical form

originally (implicitly) defined by Penman (1948) is considered appropriate and is

selected here. It takes the form:



( )24.72ln

(1 0.536 )

owm oow

am

z zr

u=

+

(23.8)

where um

is the wind speed measured at height zm

and z0

ow = 0.00137 m is the

effective value for the aerodynamic roughness for an open water surface that is

implicit in Penman’s original equation (Thom and Oliver, 1977). With this value of

z0

ow, the Penman-Monteith equation relevant for estimating open water evapora-

tion in mm d−1 when the daily average wind speed and vapor pressure deficit are

both measured at 2 m becomes:

126.43 (1 0.536 )

( ) (mm d )owOW n h h

u DE R A S −+Δ

= − − +Δ + Δ +

gg g l

(23.9)

where Rn

ow is the net radiation relevant to an open water surface (preferably

measured over the water surface) in mm d−1, u2 is the daily average wind speed

in m s−1 and D is the daily average vapor pressure deficit in kPa, both measured

at 2 m, and Sh and A

h are the estimated changes over the period for which the

evaporation estimate is made in the energy stored in the water body and the energy

advected to the evaporating water body, respectively, both in mm d−1. For example,

for a lake:

−+ += 11 1 0 0

( ) (mmd )P

h w w

q T q T pTcA r

l (23.10)

where rw and c

w are the density and specific heat of water, respectively, q

I and q

O

are respectively the inflow and outflow per unit area of lake in mm d−1, p is the

precipitation in mm d−1, and TI, T

0 and T

P are respectively the temperatures of the

inflow, outflow and precipitation. The term Sh has often been neglected and it is

probably reasonable to do so in tropical regions where the rate of change in water

temperature is low. However, at high latitudes this can be a dominant large term

in the energy balance, which is months out of phase with the solar cycle. For exam-

ple, Blanken et al. (2000) report that the water in Great Slave Lake in Canada

provided a substantial energy sink throughout most of the spring and summer

before switching to an energy source in the fall and early winter. To overcome the

lack of water temperature measurements Finch and Gash (2002) applied a simple

numerical, finite difference scheme to calculate a running balance of lake energy-

storage which gave good agreement between modeled evaporation loss and

mass-balance measurements of the water loss.

Calculations made using Equation (23.9) require prior calculation of l and Δ

from the daily average temperature using Equation (2.1) and Equation (2.18),

respectively, and also γ from Equation (2.25), and the value of cp = 0.001013

MJ kg−1 K−1. If air pressure is not available as a measurement, it usually is adequate

to estimate it from, El, the elevation of the site above sea level using:



5.256293 0.0065

101.3293

lEP−⎛ ⎞

= ⎜ ⎟⎝ ⎠ (23.11)

This equation is derived from Equation (3.13) assuming the pressure and temperature

at sea level are 101.3 kPa and 293 K, respectively, and that there is an environmental

lapse rate equal to that of the US Standard Atmosphere (6.5 K km−1).

Table 23.3 demonstrates examples of the calculation of open water evaporation

for the meteorological conditions in the three example cases A, B, C. In these

calculations the albedo of the water surface is assumed to be 0.08 and the advected

energy Ah and stored energy S

h is assumed negligible.

Reference crop evapotranspiration

Early researchers correctly believed that the rate of evaporation from terrestrial

surfaces was primarily meteorologically determined. This led to the hypothetical

concept of ‘potential’ rates of evaporation, and empirical relationships were

sought to estimate these rates from available weather data (e.g., Thornthwaite,

1948; Blaney and Criddle, 1950; Hargreaves, 1975). Penman (1948) (see previous

section) was the first to formulate the basic physics of evaporation using two

terms, an energy term related to radiation and an aerodynamic term related to the

vapor pressure deficit of the air and wind speed. At that stage, Penman suggested

Table 23.3 Demonstration of the sequence of steps undertaken to calculate daily average net radiation as described in the

text applied in the three cases A, B, C specified previously. In two cases (A and C) cloud cover is measured and in the third

case (B) the number of bright sunshine hours is measured.


Table 23.1 Average temperature (°C) 17.50 27.00 26.50Table 23.1 Vapor pressure deficit (kPa) 0.533 3.285 0.717Table 23.1 Modified wind speed (m s−1) 5.23 4.00 4.19Table 23.2 Solar at ground (cloudy sky) (mm day−1) 8.23 11.89 6.24Table 23.2 Net longwave (mm day−1) −1.58 −3.30 −0.90(Data) Elevation (m) 129.00 720.00 80.00Equ. (23.11) Air pressure (kPa) 99.79 93.08 100.36Equ. (2.1) Latent heat (MJ kg−1) 2.460 2.437 2.438Equ. (2.18) Delta (kPa °C−1) 0.1260 0.2086 0.2033Equ. (2.25) Psychrometric constant (kPa °C−1) 0.0659 0.0622 0.0670(Data) Selected value for Albedo (water) (none) 0.08 0.08 0.08Equ. (5.18) Net solar radiation (mm day−1) 7.57 10.94 5.74Equ. (5.26) Net radiation (mm day−1) 5.99 7.64 4.84Equ. (23.9) Open water evaporation (mm day−1) 5.76 12.14 5.16



the potential rates of evaporation for well-watered grass and moist bare soil might

be related to that from open water using multiplicative factors.

By the mid-1970s this by then long-established way of thinking about the evaporation

process determined the United Nations’ Food and Agriculture Organization (FAO)

recommended method for estimating the water requirements for irrigated crops

(Doorenbos and Pruitt, 1977), i.e., vegetation for which stomatal resistance is not

subject to water stress. FAO followed Penman’s approach by first defining a potential

rate called ‘reference crop evapotranspiration’, ETRC

, which was defined to be the evapo-

transpiration rate for short green grass plentifully supplied with water. This rate, it was

recommended, was to be estimated by one of several alternative equations depending

on available weather data. Evapotranspiration from any other well-watered crop, ETc,

was then assumed to be calculated using a crop specific coefficient, Kc, thus:

C C RCET K ET= (23.12)

FAO provided a table of Kc values for a range of well-watered vegetation stands the

values of which (although not individually traceable by reference) are assumed

to have been derived from field studies where the well-watered crop evapotran-

spiration rate ETc and the weather variables needed to calculate ET

0 were also

measured, so that Kc could be derived. Many years of application followed and

refinements to this approach were introduced, including experiments to evaluate

and/or validate Kc for different crops in different climates (e.g., Howell et al., 2002;

Inman-Bamber and McGlinchey, 2003; Barton and Meyer, 2008).

However, the agricultural community’s adoption of the original Penman (1948)

approach as recommended by FAO failed to recognize important subsequent advances

in the specification of evapotranspiration. Penman (1963) himself determined that the

‘two stage process’ using the ‘factor’ approach wasn’t needed, and computed ‘potential

evaporation’ from any natural surface using an aerodynamic term and an energy term

specific to that surface. Shortly afterwards, Monteith generalized Penman’s ‘one step’

approach and derived the Penman-Monteith equation (introduced in Chapter 21).

Notwithstanding the publication and widespread adoption of the Penman-Monteith

equation, agriculturalists continued to use the original two-step approach. But there

were signs that the approach might be problematic when Kc values derived in one

place were used for the same crop in another place with different weather conditions.

The reason for this was demonstrated by Wallace (1995) who showed that crop

coefficients are inherently a complex mixture of both the physiology of the crop they

represent and the climate within which Kc values are derived and/or used. They also

depend on the method used to calculate reference crop evaporation.

Penman-Monteith equation estimation of ERC

With this last point in mind, and recognizing the greater realism of the

Penman-Monteith equation, FAO subsequently modified their guidelines (Allen

et al., 1998) by adopting the Penman-Monteith equation to calculate reference



crop evapotranspiration (ET0). However, this required retention of the two step

approach because it was still necessary to multiply ET0 by the relevant crop

coefficient to obtain actual evapotranspiration (ETc). Later in this chapter a recent

approach for estimating daily average evaporation is described that is more realis-

tic because it uses the Penman-Monteith equation to calculate estimates using

crop-specific values for unstressed surface resistance, and an aerodynamic

resistance that reflects the height of the crop. However, the FAO crop factor

method currently remains in widespread use.

The FAO definition of reference crop evaporation rate is the standard

Penman-Monteith equation used to calculate a daily average evaporation rate

assuming a specific value of surface resistance and a specific expression for

aerodynamic resistance. On the basis of field calibrations for short, well-watered

grass, rs is set equal to 70 s m−1 in Equation (21.33), while r

a (which is assumed the

same for both latent and sensible heat transfer) is calculated from Equation (22.8),

assuming the wind speed, temperature, and specific humidity and therefore, by

implication, vapor pressure deficit are all measured 2 m above 0.12 m high grass.

On the basis of field studies, the zero plane displacement, d, and aerodynamic

roughness, zo, for the grass crop are assumed given by:

0.123oz h= (23.13)

and

0.67d h= (23.14)

where h is the height of the grass crop. Substituting these values into Equation

(22.8) gives ra = 208/u

2. With these assumptions, the short grass crop-specific

version of the Penman-Monteith equation that can be used to estimate reference

crop evaporation, ERC

, in mm d−1 is:

2 2

2

900

275RC

m m

E A u DT

⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Δ + Δ + +⎝ ⎠⎝ ⎠ ⎝ ⎠g

g g

(23.15)

where A is the energy available to evaporate water in mm d−1, T2, u

2, D

2 are respec-

tively the temperature in °C, wind speed in m s−1, and vapor pressure deficit in kPa

measured at 2 m, and gm is the ‘modified psychrometric constant’ which is given by;

2(1 0.33 )m u= +g g

(23.16)

Commonly A = (Rn – G), where R

n is the net radiation and G is the soil heat flux

both in mm d−1 for the short grass crop.

Equation (23.15) is strongly recommended as the preferred method for

estimating reference crop evaporation whenever daily average values of all the

required weather variables are on hand. But sometimes not all the required varia-

bles are available, and less reliable estimates of ERC

then have to be made which are

described in the next sections.



Radiation-based estimation of ERC

As previously stated, when estimating evaporation rates, it is recommended that

the starting point should be a measurement or estimate of the energy available to

support evaporation; because this has a strong influence on evaporation rate

when water is not limiting. Equation (22.30) can be calculated if a measurement

of the available energy (Rn-G), and temperature and pressure are available.

Temperature is needed to calculate Δ from Equation (2.18), and pressure is also

needed to calculate γ from Equation (2.25). As Fig. 22.11 shows, for a range of

surface resistances around 70 s m−1, the value of surface resistance selected for the

reference crop, Equation (22.30) can provide a reasonable estimate of evaporation

with α = 1.26.

However, it is important to recognize that this result only applies if there is

surface-ABL coupling over an extensive area with a similar surface resistance.

It could be that the area of reference crop for which an estimate of evaporation

is sought is not extensive. It could be a limited area of irrigated reference crop in

an otherwise arid landscape, for example. Therefore, some means of quantify-

ing the aridity of the advected atmosphere is required. In this context, it is

relevant that the Penman-Monteith equation (Equation (21.33) can be

re-written in the form:

lim

( )a c

a s

r rE Ar r

+= Δ

Δ + +l

g g (23.17)

where rclim

is the climatological resistance which is defined by:

2

lim (if is in W m )

a pc

c Dr A

A−⎛ ⎞

= ⎜ ⎟Δ⎝ ⎠

r

(23.18)

or:

( )1

lim

187219 187219 (if is in mm d )

(275 )1 0.622 (273.15 )c CC

D Dr AT AA e P T

−⎛ ⎞= ≈ ⎜ ⎟⎝ ⎠+ ΔΔ + +g g

(23.19)

The value of rclim

provides a measure of the relative influence of the (advected)

vapor pressure deficit and available energy on reference crop evaporation rate.

Equating the estimated evaporation rates given by Equations (22.30) and

(23.15), rearranging, and substituting rs = 70 s m−1 and r

a = 208/u

2 gives a

effective, the

value of α required to give a good estimate of reference crop evaporation, thus:

2 lim

2

( ) (208 / )

( )(208 / ) 70

ceffective

u ru

Δ + +⎡ ⎤⎣ ⎦=Δ + +

ga

g g

(23.20)



Figure 23.1 shows the calculated variation of aeffective

as a function of rclim

when

T = 20°C, P = 100 kPa and u2 = 2 m s−1 and reveals that, in these conditions, a

effective

is within 10% of the value 1.26 used in the Priestley-Taylor equation when rclim

is

between about 40 to 70 s m−1. When the atmosphere is more arid, the value of

aeffective

must be higher to give a reasonable estimate of reference crop evaporation.

Jensen et al. (1990) proposed α = 1.74 as being the value required for a reasonable

estimate of reference crop evaporation in arid conditions and Fig. 23.1 shows that

when T = 20°C, P = 100 kPa and u2 = 2 m s−1, a

effective is within 10% of 1.74 when r

clim

is between about 90 to 140 s m−1.

Thus, if measurements or estimates of either vapor pressure deficit, or wind

speed (or both) are not available, but an estimate of daily average net radiation

can be made, the best available estimate of reference crop evaporation is from

the equation:

RC effectiveE AΔ=Δ +

ag (23.21)

with aeffective

set to 1.26 if the climate of the area is considered to be generally humid,

or to 1.74 if the climate of the area is generally arid.

Temperature-based estimation of ERC

A temperature-based estimate of reference crop evaporation should only be made

when the available data is limited to measurements of maximum and minimum

temperature. There have been several empirical equations proposed for relating

2.50

2.25

2.00

1.75

1.50

1.25

1.00

0.75

0.50

0.25

0.000 25 50 75 100

rclim

125 150 175 200

α effective

Humid conditions

Arid conditions

Figure 23.1 Variation in the

value of aeffective

required for

Equation (22.28) to give an

estimate of reference crop

evaporation rate consistent

with an FAO estimate as a

function of rclim

.



ERC

to temperature but here the Hargreaves equation (Hargreaves, 1975) is

recommended on the grounds that its performance is at least as good as alternatives

and it is simple to use. However, the estimate given by this equation is only rele-

vant for monthly average estimates of reference crop evaporation at humid sites.

The Hargreaves equation has the form:

0.5 0.0023 ( ) ( 17.8)dRC T oE S T≈ +d

(23.22)

where So

d is the solar radiation incident at the top of the atmosphere in mm of

evaporated water per day, dT is the difference between mean monthly maximum

temperature (Tmax

) and mean monthly minimum temperature (Tmin

) in °C, and

T is the temperature in °C. Arguably the predictive ability of this empirical

equation is based on the fact that it bears some relationship to Equation (23.21).

The temperature variation of the term (T+17.8) approximates that of Δ/(Δ+γ),

the equation has an explicit link to maximum solar radiation via So

d, and through

dT, it also includes some implicit measure of the extent to which the radiation

at the top of the atmosphere reaches the surface to warm the atmosphere near

the ground.

Evaporation pan-based estimation of ERC

The measurement of weather variables requires the use of fairly expensive sensors.

For this reason evaporation pans (see Chapter 7) were often preferred in many

agricultural applications, and many pans remain in operation today (see

Chapter 7). The required estimate of reference crop evaporation is assumed to be

directly related to the measured rate of evaporation from the evaporation pan

using an equation similar to Equation (23.12), thus:

RC p panET K ET=

(23.23)

The ‘constant’ in this equation is called a ‘pan factor’. In the past the value of the

pan factor has been defined empirically by comparing reference crop evaporation

rate, lErc, with measured pan evaporation rate, lE

pan, at one location and in one

climate, and then applying this ratio elsewhere. On this basis approximate values

of pan factor were tabulated in different weather conditions (e.g. Doorenbos and

Pruitt, 1977; Shuttleworth, 1993), but such tabulation was made without proper

theoretical understanding of the origins of such variations.

In recent years there has been research into the physics that controls evapora-

tion from the Class A evaporation pan. Rotstayn et al. (2006) developed the

‘Penpan’ equation which is based on the work of Thom et al. (1981) and Linacre

(1994), and which is a physically-based description of pan evaporation in terms

of ambient climate variables. The Penpan equation is an implementation of the

Penman-Monteith equation in which the effective aerodynamic resistance for a

Class A evaporation pan is prescribed to be:



2

( )1 1.35

pa pan

Cr

u=

+

(23.24)

where u2 is the wind speed (in m s−1) measured at 2 m, C

p has units of s m−1 and the

factor 1.35 (implicitly) has units of s m−1. Rotstayn et al. (2006) prescribed the

effective value of surface resistance for a Class A evaporation pan when evapora-

tion is calculated using the Penman-Monteith Equation as:

( ) 1.4( )s pan a panr r=

(23.25)

with the albedo of the pan set to 14%. Roderick et al. (2007) experimentally verified

the Penman equation against Class A pan data from pan sites in Australia where the

measured meteorological variables required in the equation were also available.

They showed that, on average, the equation gave a reasonable description of

monthly-average measured pan evaporation rate, albeit with systematic site-to-site

discrepancies in the order of 10–20% when Cp is set to an average value 224 s m−1.

Thus, both the Penpan equation and FAO’s recommended equation for

calculating reference crop are implementations of the Penman-Monteith equation

with different values of aerodynamic and surface resistance, i.e., (208/u2) and 70

for a reference crop and (ra)

pan = C

p/(1+ 1.35u

2) and (r

s)

pan = 1.4r

a for a pan, respec-

tively. By substituting these pairs of values into Equation (23.23) and taking the

ratio of the two calculated rates, it follows that:

[ ]2lim 2

2 lim 2

( 2.4 ) / (1 1.35 )[ 208/ ]

70 ( )208/ ( / ) / (1 1.35 )

pcp

c p pan rc

C ur uK

u r C A A u

⎡ ⎤Δ + ++ ⎣ ⎦=+ Δ + ⎡ ⎤+ +⎣ ⎦

g

g g

(23.26)

where (Apan

/Arc) is the ratio of the energy available to support evaporation from a

reference crop to that available for an evaporation pan. Shuttleworth (2010) dem-

onstrated that the value of Kp is estimated to within an accuracy of a few percent

by setting (Apan

/Arc) = 1.15 for a wide range of short wave and longwave radiation

values, and recommended that in the absence of better information the value of

rclim

in Equation (23.26) should be calculated from:

[ ]2

clim

2

(1 0.377 )2081

ur

u⎛ ⎞Δ + +

= −⎜ ⎟Δ +⎝ ⎠

a gg

(23.27)

where u2, the wind speed measured at 2 m, is in m s−1 and with α set to 1.26 and

1.74 at humid and arid sites, respectively. Shuttleworth also showed that the value

of Kp has limited sensitivity to temperature (via the value of Δ), but has a strong

sensitivity to wind speed. In the absence of measurements, he recommended that

a default temperature of 20°C and default wind speed of 2 m s−1 are used in

Equation (23.26). When the temperature is 20°C and pressure is 100 kPa, in humid

conditions Kp has the wind speed dependent form:



22

2 2

0.303 / (1 1.35 )[58+208/ ]

[4.63 43.8/ ] 58 1.15 / (1 1.35 )

pp

p

C uuK

u C u

⎡ ⎤+⎣ ⎦=+ ⎡ ⎤+ +⎣ ⎦ (23.28)

while in arid conditions, it has the form:

22

2 2

0.303 / (1 1.35 )[120+208/ ]

[4.63 43.8/ ] 120 1.15 / (1 1.35 )

pp

p

C uuKu C u

⎡ ⎤+⎣ ⎦=+ ⎡ ⎤+ +⎣ ⎦ (23.29)

The implicit dimensions of the constants that appear in Equations (23.28) and

(23.29) require that Cp in s m−1 and u

2 in m s−1. Assuming C

p is the average value

224 s m−1 found by Roderick et al. (2007), then, when u2 = 2 m s−1, the default values

of Kp are 0.88 and 0.82 in humid and arid conditions, respectively. Were a calibration

of Cpan

at the pan site to be made (perhaps by temporarily deploying the sensors

needed to gather the weather data required by the Penpan equation), then the

subsequently sustained collection of wind speed measurement would give improved

accuracy for pan-based estimates of reference crop evaporation at the site.

Table 23.4 demonstrates example calculations of reference crop evaporation using

the Penman-Monteith-based FAO method and the less reliable radiation-based,

temperature-based and pan-based estimates described above for the three example

sites A, B, C. In the case of the radiation-based estimate, the value is calculated using

the most appropriate value, either 1.26 or 1.74, in Equation (23.21) depending on

whether the site is considered to have a generally humid or generally arid climate.

At the sites with humid climates A and C (sites near Oxford, England and Manaus,

Brazil), the radiation-based and temperature-based estimates of reference crop

evaporation give values that are roughly comparable to those given using the FAO

method. However, at site B near Tucson, Arizona, the radiation-based and tempera-

ture-based methods both give estimates of ERC

which are much lower than the values

given when using the FAO method, despite the fact that aeffective

= 1.74 is selected

when making the radiation-based estimate of ERC

. Pan-based estimates are made

both using relevant default values of Kp and relevant wind speed dependent estimates,

i.e., using Equation (23.27) to calculate rclim

in humid and arid conditions as

appropriate, then calculating a wind-corrected value of Kp from Equation (23.26).

The default values of Kp give overestimates because in each case the wind speed is

significantly greater than 2 m s−1.

Evaporation from unstressed vegetation: the Matt-Shuttleworth approach

Daily estimates of evaporation from vegetation cover that is plentifully supplied

with water are still usually made using Equation (23.12) with assumed values for

KC, taken from the tables provided by FAO (Allen et al., 1998) for irrigated crops.



Table 23.4 Demonstration of the calculation of reference crop evaporation using the FAO, radiation-based, and

temperature-based methods as described in the text for the three cases A, B, C specified previously.


Table 23.1 Maximum air temperature (°C) 22.00 37.00 30.00Table 23.1 Minimum air temperature (°C) 13.00 17.00 23.00Table 23.1 Average temperature (°C) 17.50 27.00 26.50Table 23.1 Vapor pressure deficit (kPa) 0.533 3.285 0.717Table 23.1 Modified wind speed (m s−1) 5.23 4.00 4.19Table 23.2 Extraterrestrial solar radiation (mm day−1) 16.45 16.36 15.61Table 23.2 Net radiation (mm day−1) 4.76 5.86 3.90Table 23.2 Assigned site humidity (none) Humid Arid HumidTable 23.3 Latent heat (MJ kg−1) 2.460 2.437 2.438Table 23.3 Delta (kPa °C−1) 0.1260 0.2086 0.2033Table 23.3 Psychrometric constant (kPa °C−1) 0.0659 0.0622 0.0670Data Measured pan evaportion (mm) 5.5 14 5.1Assumed Value of Cp in Equ. (23.24) (s m−1) 224 224 224Assumed Value of (Apan/Arc) (none) 1.15 1.15 1.15Equ. (23.16) Modified psychrometric constant (kPa °C−1) 0.1797 0.1443 0.1597Equ. (23.27) rclim assigned in Equ. (23.26) (s m−1) 44 70 37Selected Default pan coefficient (none) 0.88 0.82 0.88Equ (23.26) Wind corrected pan factor (none) 0.71 0.75 0.77Equ. (23.15) Ref. crop evap. (FAO) (mm day−1) 3.81 10.36 3.84Equ. (23.21) Ref. crop evap. (radiation based) (mm day−1) 3.94 7.85 3.70Equ. (23.22) Ref. crop evap. (temperature based) (mm day−1) 4.01 7.54 4.21Equ. (23.23) Ref. crop evap. (pan: default Kp) (mm day−1) 4.84 11.48 4.49Equ. (23.23) Ref. crop evap. (pan: wind corr. Kp) (mm day−1) 3.91 10.53 3.95

Because the FAO tables provide estimates of evaporation for irrigated crops, they

describe the seasonal evolution in the value of KC in terms of four growth stages

with crop-specific duration as shown for a hypothetical crop in Fig. 23.2.

The reluctance of the agricultural irrigation community to change practice and

adopt estimates based on using the Penman-Monteith equation with crop- specific

surface resistances may partly be due to a lack of appreciation of the basic flaws in

using Equation (23.12). However, a more fundamental inhibition on change is that

specified values of aerodynamic resistance and surface resistance are required for

non-stressed, well-watered, irrigated crops. Hitherto these have not been readily

available. Shuttleworth (2006) addressed this need by combining modern thinking

in surface energy exchange and boundary layer meteorology to derive a means of:

● specifying aerodynamic resistance of any crop from readily available (2 m)

climate station data; and

● converting existing Kc values to their equivalent values of surface resistance.



The resulting method is called the Matt-Shuttleworth approach.

To address the need to allow readily available climate data measured at 2 m to be

realistically applied to calculate aerodynamic resistance from tall crops when

using the Penman-Monteith equation, Shuttleworth derived a version of the

equation that is indexed to a common blending height arbitrarily selected to be at

50 m. This means the reference height and value of D are the same when calculat-

ing both the evaporation rate for any well-watered crop and reference crop

evapotranspiration rates. With A in units of W m−2, this version of the Penman-

Monteith equation has the form:

2 2 5050

2

250

1 ( )

a p

cRC

s cc

c u D DADR

ETurR

⎛ ⎞Δ + ⎜ ⎟⎝ ⎠

=⎛ ⎞

Δ + +⎜ ⎟⎝ ⎠

r

g (23.30)

where (rs)

c is the crop-specific surface resistance, D

50 and D

2 are the vapor pressure

deficit at 50 m and 2 m, respectively. The ratio of these two values, (D50

/ D2) is

given by:

1.0

0.5

0

Planting dateTotal growing season

TimeStage 4Stage 3Stage 2Stage 1

Late-seasonMid-seasonCropdevelopment

Initial

Kc (stage 3)

Kc (stage 4)

Kc (stage 1)

Kc App

rox.

10%

Gro

und

cove

r

App

rox.

70%

Gro

und

cove

r

Figure 23.2 Simplified seasonal

pattern used by FAO to specify

the time dependence of Kc

values for agricultural crops in

terms of four growth stages with

period lengths defined for each

crop, and with crop-specific

values of Kc defined to apply

during a stage, or as limiting

values with linear interpolation

during the stage.

50 2 2

2 2 clim 2 2 2

( )302 70 ( )302 701 208 302

( )208 70 ( )208 70

D u uD u r u u u

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ + + Δ + += + −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟Δ + + Δ + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

g g g gg g g g

(23.31)

and Rc50 the aerodynamic coefficient for a crop of height h

c given by:



50

2

(50 0.67 ) (50 0.67 ) (2 0.08)ln ln ln

(0.123 ) (0.0123 ) 0.0148

(50 0.08)(0.41) ln

0.0148

c c

c cc

h hh h

R

⎡ ⎤ ⎡ ⎤− − −⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦=

−⎡ ⎤⎢ ⎥⎣ ⎦

(23.32)

Shuttleworth (2006) also derived a method for converting the existing values of Kc

into the required values of (rs)

c. This involves specifying ‘preferred conditions’ in

which the reference crop evapotranspiration rate calculated by the FAO method and

the equivalent rate calculated by the Priestley-Taylor equation are equal. The justifi-

cation for forcing this equivalence is that in the original FAO recommendations

(Doorenbos and Pruitt, 1977), the FAO crop coefficients were considered applicable

to a range of estimates of potential evapotranspiration based on different formulae,

including the Priestley-Taylor equation. Such preferred conditions are therefore

those described earlier as ‘humid’ conditions, and the corresponding ‘preferred’

value of climatological resistance, (rclim

)pref, is therefore given by rearranging Equation

(23.20) and substituting aeffective

= 1.26 and u2 = 2 m s−1. Substitution of u

2 = 2 m s−1 is

because FAO states the wind speed for which their tabulated crop coefficients apply

best is 2 m s−1. The resulting equation for (rclim

)pref is then:

clim

1.67( ) 104 1.26 1

prefpref

prefr⎛ ⎞Δ +

= −⎜ ⎟Δ +⎝ ⎠g

g

(23.33)

where Δpref is the value of Δ calculated at the ‘preferred’ temperature Tpref.

In these preferred atmospheric conditions (rs)

c, the value of the surface resistance

for a well-watered crop that is equivalent to the FAO crop coefficient, can be

calculated (see Shuttleworth, 2006 for details) from:

1

2( ) ss c s

c

rr r

K= −

(23.34)

where:

50

50clim

21

50clim

2

( )2 151( ) 70

151 ( )

prefprefc

pref

s prefpref

R Dr

Dr

Dr

D

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟ ⎛ ⎞⎝ ⎠ Δ + +⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎝ ⎠⎛ ⎞⎜ ⎟+ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

g gg

(23.35)

and:

502 ( )

2

prefc

sRr Δ += g

g

(23.36)

In Equation (23.35), (D50

/ D2)pref is given by Equation (23.31) with Δ = Δpref and

u2 = 2 m s−1. The values of r

s1 and r

s2 are therefore solely functions of the crop height



and the temperature (known or assumed) at which the original crop coefficient Kc

was calibrated. Consequently the remaining need is to specify the value of Tpref.

Shuttleworth and Wallace (2010) investigated the sensitivity of the calculated

value of surface resistance to the assumed value of Tpref for a range of irrigated

crops in Australia and found a lack of sensitivity for many. On the basis of their

study, they concluded that, pending field studies to better define the crop-specific

values of surface resistance, Tpref = 20°C should be used in Equations (23.35) and

(22.36) when estimating (rs)

c from the tabulated values of K

c given by FAO

(Doorenbos and Pruitt, 1977). On this basis, Shuttleworth and Wallace then calcu-

lated the values of Rc50 from Equation (23.32) and (r

s)

c from Equation (23.34) for

the selection of crops given in Table 23.5.

Table 23.5 Values of crop factor, Kc, and crop height, h

c, for a selection of irrigated crops

together with the equivalent derived values of Rc50 and (rs)

c required for in Equation (12.21)

when calculating daily average evaporation using the Matt-Shuttleworth approach.

Irrigated crop Kc (dimensionless) hc (m) Rc50 (dimensionless) (rs)c (s m−1)

Reference crop 1.00 0.12 302 70Alfalfa (average) 0.95 0.70 196 127Bermuda 1.00 0.35 235 92Clover (average) 0.90 0.60 204 149Rye (average) 1.05 0.30 244 66Pasture (rotation) 0.95 0.23 260 109Pasture (extensive) 0.75 0.10 314 254Small vegetables 1.05 0.38 230 72Solanum family 1.15 0.70 196 50Cucurbitaceae 1.00 0.34 237 91Roots and tubers 1.10 0.68 198 66Legumes 1.15 0.55 209 44Cereals 1.15 1.00 177 60Cotton 1.18 1.35 162 60Maize (grain) 1.20 2.00 143 64Sorgum (grain) 1.05 1.50 157 100Rice 1.20 1.00 177 46Millet 1.00 1.50 157 118Sugar cane 1.25 3.00 124 63Cacao 1.05 3.00 124 113Coffee 0.95 2.50 133 143Tea 1.00 1.50 157 118Grape (table) 0.85 2.00 143 184Grape (wine) 0.70 1.75 149 273Almonds 0.90 5.00 102 169Avocado 0.85 3.00 124 186Citrus (50% canopy) 0.60 3.00 124 345Kiwi 1.05 3.00 124 113Walnut 1.10 4.50 107 106Olives 0.70 4.00 112 265



To enhance comparability with the FAO method for estimating reference crop

evaporation, Equation (23. 30) can be re-written to give crop evaporation, EC, as:

50 2 2

50

2 2

187219

* * 275C

m m c

D u DE AT D R

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟Δ + Δ + +⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠g

g g

(23.37)

where A is the energy available to evaporate water from the crop in mm d−1, T2, u

2

and D2 are respectively the temperature in °C, wind speed in m s−1, and vapor

pressure deficit in kPa measured at 2 m, and γm

* is the ‘re-modified’ psychrometric

constant, given by;

2

50

( )* 1 s c

mc

r uR

⎛ ⎞= +⎜ ⎟⎝ ⎠

g g

(23.38)

with the values of (rs)

c and R

c50 taken from tables such as Table 23.5.

Table 23.6 demonstrates example calculations of daily average evaporation from

unstressed crops calculated using the Matt-Shuttleworth approach and the FAO

crop factor method at the three example sites A, B and C specified previously, with

values of Kc, (r

s)

c and R

c50 taken from Table 23.5 for three example crops, i.e., cereal

crops, an alfalfa crop and small vegetables. These results show there are differences

between the estimated daily evaporation rates when calculated using the more real-

istic Matt-Shuttleworth approach relative to those calculated using the FAO crop

factor method. At the two humid sites, the estimated daily average evaporation rates

given by the Matt-Shuttleworth approach are slightly less than those given by the

FAO crop factor method. However, at the arid site the estimated daily average evapo-

ration rates given using the Matt-Shuttleworth approach are significantly greater

than those given by the FAO crop factor method. This is to be expected and is real-

istic: the higher evaporation rate reflects the fact that for crops with crop height

greater than the reference crop, the aerodynamic resistance is less. Consequently the

evaporation rate will necessarily be more sensitive to atmospheric aridity, in general,

and will be greater in arid conditions because the advection term in the Penman-

Monteith equation becomes more significant in comparison with the radiation term.

Evaporation from water stressed vegetation

The approach used to represent the effect of soil water restrictions on evaporation

rate which applied in conjunction with the simple models of daily average evapo-

ration described in this chapter is usually not complex. Typically, a volume of soil

is defined that is considered to be accessible to the atmosphere via plants, the

depth of which may be related to the nature of the overlying vegetation through an

assumed rooting depth. A running water balance is then made for this soil sample.

The primary input to this water balance is precipitation, and the primary output is



Table 23.6 Example calculations of daily average evaporation from unstressed crops calculated using the Matt-Shuttleworth

approach and the FAO crop factor method at the three example sites A, B, and C specified previously with values of Kc, (rs)

c

and Rc50 for cereal crops, an alfalfa crop, and small vegetables.


Table 23.1 Average temperature (°C) 17.50 27.00 26.50Table 23.1 Vapor pressure deficit (kPa) 0.533 3.285 0.717Table 23.1 Modified wind speed (m s−1) 5.23 4.00 4.19Table 23.2 Extraterrestrial solar radiation (mm day−1) 16.45 16.36 15.61Table 23.2 Net radiation (mm day−1) 4.76 5.86 3.90Table 23.2 Assigned site humidity (none) Humid Arid HumidTable 23.3 Air pressure (kPa) 99.79 93.08 100.36Table 23.3 Latent heat (MJ kg−1) 2.460 2.437 2.438Table 23.3 Delta (kPa °C−1) 0.1260 0.2086 0.2033Table 23.3 Psychrometric constant (kPa °C−1) 0.0659 0.0622 0.0670Table 23.4 Modified psychrometric constant (kPa °C−1) 0.1797 0.1443 0.1597Table 23.4 Ref. crop evaporation (FAO) (mm day−1) 3.81 10.36 3.84Equ. (23.18) rclim (s m−1) 37 104 38Equ. (23.31) (D50 / D2) (none) 1.10 1.29 1.18Cereal crops

Table 23.5 Crop factor (none) 1.15 1.15 1.15Table 23.5 Rc

50 (none) 177 177 177Table 23.5 (rs)c (s m−1) 60 60 60Equ. (23.38) Re-modified psychrometric constant (kPa °C−1) 0.1828 0.1465 0.1622Equ. (23.37) Matt-Shuttleworth estimate (mm day−1) 4.31 13.85 4.45Equ. (23.12) FAO estimate (mm day−1) 4.38 11.92 4.42Alfalfa cropTable 23.5 Crop factor (none) 0.95 0.95 0.95Table 23.5 Rc

50 (none) 196 196 196Table 23.5 (rs)c (s/m) 127 127 127Equ. (23.38) Re-modified psychrometric constant (kPa °C−1) 0.2893 0.2234 0.2490Equ. (23.37) Matt-Shuttleworth estimate (mm day−1) 3.04 10.56 3.42Equ. (23.12) FAO estimate (mm day−1) 3.62 9.85 3.65Small vegetablesTable 23.5 Crop factor (none) 1.05 1.05 1.05Table 23.5 Rc

50 (none) 230 230 230Table 23.5 (rs)c (s/m) 72 72 72Equ. (23.38) Re-modified psychrometric constant (kPa °C−1) 0.1738 0.1401 0.1549Equ. (23.37) Matt-Shuttleworth estimate (mm day−1) 3.88 11.66 4.01Equ. (23.12) FAO stimate (mm day−1) 4.00 10.88 4.03

evaporation. In practice, there may also be a modeled drainage loss to depth from

this soil sample, but for crops that are irrigated with just enough water to replace

evaporative loss, this term may be negligible. The maximum volumetric water

holding capacity of the sample of soil is specified in terms of the depth and nature

of the soil, and any excess water above this is assumed to drain.



The restriction on evaporation rate is then parameterized in terms of the

ambient volumetric soil moisture content of the soil sample, qs. Often a simple

moisture stress function f(qs) with the general form shown in Fig. 23.3 is assumed,

with values of soil moisture defined when the soil is saturated, qssat, when it is at

‘field capacity’ (after draining from saturation), qsfc, and at a ‘wilting point’, q

swilt.

At wilting point, it is assumed the stomata are closed and evaporation ceases.

In Fig. 23.3, typically the value of (qssat/q

sfc) is 0.5 to 0.8. If the evaporation estimate

is made using a crop factor approach, the value of the ambient water-stressed crop

factor, Kc’, at any point in time is assumed to be given by:

( )c s cK f K=¢ q (23.39)

If the evaporation estimation is parameterized in terms of the surface resistance of

the crop, (rs)

c, the value of the ambient water-stressed surface resistance, (r

s)

c’, at

any point in time is given by:

( ) ( ) ( )s c s c sr r f=¢ q (23.40)

In such models, the daily running water balance takes the general form:

( )1n n n n n ns s sM M P E D−= + − −q

(23.41)

where Msn and M

sn−1 are the depth of water in mm in the soil store on days n and

(n−1), respectively, and Pn, En(qsn) and Dn are the precipitation, evaporation, and

drainage on day n, with En(qsn) either calculated from Equation (23.12) using the

soil moisture weighted crop factor Kc′ given by Equation (23.39), or from Equation

(23.30) or Equation (23.37) with the soil moisture weighted surface resistance

given by Equation (23.40).


● Daily average meteorological variables: daily estimates of evaporation are

usually made from meteorological variables that are themselves estimates of

daily average or daily total values derived from the limited measurements taken

at agro-meteorological climate stations using the procedures given in the text:

– for temperature, humidity and wind speed, by Equations (23.1) to (23.7)

with calculation illustrated by example in Table 23.1;

– for net radiation, using the equations defined in the relevant section with

calculation illustrated by example in Table 23.2.

● Use of the Penman-Monteith equation: all the preferred methods for esti-

mating daily evaporation recommended in this text are derived from the

Penman-Monteith equation, including:



– Open water evaporation, which is given by Equation (23.9) and associated

equations with calculation illustrated by example in Table 23.3.

– Reference crop evaporation, which is given by Equation (23.15) and asso-

ciated equations with calculation illustrated by example in Table 23.4.

● Compromise estimates of evaporation: when not all the required weather

variables are available to estimate Reference Crop Evaporation from Equation

(23.15), a compromise estimate using less/other variables is required, includ-

ing making a radiation-based estimate using Equation (23.21); a temperature-

based estimate using Equation (23.22); or a pan-based estimate using

Equation (23.23) with either Equation (23.28) or (23.29), calculations of all

also being illustrated by example in Table 23.4.

● Matt-Shuttleworth approach: estimates of crop evaporation are calculated

more realistically from standard weather variables measured at 2 m using the

Matt-Shuttleworth approach which involves using a version of the Penman-

Monteith equation indexed to a (50 m) blending height (Equation (23.37),

and associated equations) with crop-specific values of surface resistance

either calculated by Equation (23.34) for partial crop cover or taken from

Table 23.5 for full crop cover. These calculations are illustrated by example in

Table 23.5.

● Evaporation in water-stressed conditions: often evaporation from water-

stressed vegetation is calculated using Equation (23.39) with the value of the

water-stress factor given as a function of ambient soil water content in the

plant rooting zone modeled by a daily running water balance.

References

Allen, R. G., Pereira, L. S., Raes, D. and Smith, M. (1998) Crop evapotranspiration. Irrigation

and Drainage Paper 56. UN Food and Agriculture Organization, Rome, Italy.

Barton, A. B. and Mayer, W. S. (2008) An analysis of method and meteorological

measurement of evapotranspiration estimation. Part 2: Results using measured

evapotranspiration and weather data from Ayr (Qld), Kununurra (WA) and Griffith

0

1

f(qs)

θswilt θsd θsfc θssat

Figure 23.3 Typical variation

in moisture stress function f(qs)

with average volumetric soil

moisture content, qs, in a

volume of soil considered

accessible to the atmosphere via

plants. The soil is saturated

when q is qssat, at ‘field capacity’

when q is qsfc, and at ‘wilting

point’ when q is qswilt.



(NSW). In: CRC for Irrigation Futures Technical Report No. 09−2/08, November 2008.

18pp.

Blaney, H. F. and Criddle, W. D. (1950) Determining water requirements in irrigated areas

from climatological and irrigation data. US Department of Agriculture Soil Conservation

Service TP−96, p. 48.

Blanken, P. D., Rouse, W. R., Culf, A. D., Spence, C., Boudreau, L. D., Jasper, J. N.,

Kochtubajda, R., Schertzer, W. M., Marsh, P. and Verseghy, D. (2000) Eddy covariance

measurements of evaporation from Great Slave Lake, Northwest Territories, Canada.

Water Resources Research, 36, 1069–1077.

Doorenbos, J. and Pruitt, W. O. (1977) Crop water requirements. Irrigation and Drainage

Paper, 24. Rome, Italy: United Nations Food and Agriculture Organization.

Finch, J. W. and Gash, J. H. C. (2002) Application of a simple finite difference model for

estimating evaporation from open water. Journal of Hydrology, 255, 253–259.

Hargreaves, G. H. (1975) Moisture availability and crop production. Transcripts of American

Society of Agricultural Engineers, 18 (5), 980–984.

Howell, T. A. and Evett, S. R. (2002) The Penman-Monteith Method. USDA-Agricultural

Research Service Conservation & Production Laboratory, Bushland, Texas, USA. http://

www.cprl.ars.usda.gov/wmru/pdfs/PM%20COLO%20Bar%202004%20corrected%20

9apr04.pdf

Inman-Bamber, N. G. and McGlinchey, M. G. (2003) Crop coefficients and water use esti-

mates for sugarcane based on long-term Bowen ratio energy balance measurements.

Field Crops Research, 83, 125–138.

Jensen, M. E., Burman, R. D. and Allen, R. G. (eds) (1990) Evapotranspiration and Irrigation

Water Requirements. ASCE Manuals and Reports on Engineering Practice No. 70, ASCE,

New York, p. 332.

Linacre, E. T. (1994) Estimating US Class A pan evaporation data from few climate data.

Water International, 19, 5–14.

Penman, H. L. (1948) Natural evaporation from open water, bare soil, and grass. Proceedings

of the Royal Society of London, Ser. A 193, 120−145.

Penman, H. L. (1963) Vegetation and hydrology, Technical Communication 53,

Commonwealth Bureau of Soils: Harpenden, England.

Roderick, M. L., Rotstayn, L. D., Farquhar, G. D. and Hobbins, M. T. (2007) On the attribu-

tion of changing pan evaporation. Geophysical Research Letters, 34, L17403.

doi:10.1029/2007GL031166.

Rotstayn, L. D., Roderick, M. L. and Farquhar, G. D. (2006) A simple pan-evaporation

model for analysis of climate simulations: Evaluation over Australia. Geophysical Research

Letters, 33, L17715. doi:17710.11029/12006GL027114.

Shuttleworth, W. J. (1993) Evaporation. In Maidment, D. (ed.) Handbook of Hydrology.


Shuttleworth, W. J. (2006) Towards one-step estimation of crop water requirement.

Transactions of the ASABE, 49 (4), 925–935.

Shuttleworth, W. J. (2010) Back to the basics of understanding ET. In: Khan, S., Savenije, H.,

Demoth, S. and Hubert, P. (eds) Hydrocomplexity: New Tools for Solving Wicked Water

Problems: Kovacs Colloquium, July 2010. IAHS Publication No. 338, IAHS Press,

Wallingford, UK.

Shuttleworth, W. J. and Wallace, J. S. (2010) Calculating the water requirements of irrigated

crops in Australia using the Matt-Shuttleworth approach. Transactions of the ASABE, 52

(6),1895–1906.



Thom, A. S. and Oliver, H. R. (1977) On Penman’s equation for estimating evaporation.

Quarterly Journal of the Royal Meteorological Society, 103, 345–357.

Thom, A. S., Thony, J. L. and Vauclin, M. (1981) On the proper employment of evaporation

pans and atmometers in estimating potential transpiration. Quarterly Journal of the Royal

Meteorological Society, 107, 255–278.

Thornthwaite, C. W. (1948) An approach toward a rational classification of climate.

Geographical Review, 38, 55–94.

Wallace, J.S. (1995) Calculating evaporation: Resistance to factors. Agricultural and Forest

Meteorology, 73, 353–366.


Introduction

As previously mentioned, one important way that understanding of the interaction

between land surfaces and the ABL is used is in computer sub-models that become

important sub-components of meteorological, hydrological and/or coupled

atmospheric-hydro-ecological models. These sub-models are often called ‘soil

vegetation atmosphere schemes’ (SVATS) or, when used in meteorological models,

‘land-surface parameterization schemes’ (LSPs). They are important in atmospheric

models to specify the lower boundary condition, in hydrological models to specify

the upper boundary condition, and in coupled models as the essential interface

between model components.

Again, as previously mentioned, the availability of frequently sampled weather

variables from which SVATS can calculate surface exchanges is not usually an issue

because, in the case of meteorological models and coupled surface-atmosphere

models, the required variables are themselves regularly calculated by the remain-

der of the model and, in the case of hydrological models, the use of SVATS would

likely not be attempted unless the required data are available from measurements,

perhaps made with one or more automatic weather stations.

Basis and origin of land-surface sub-models

Putting aside for the moment the important indirect influence of terrestrial

vegetation on the concentration of atmospheric CO2, continental surfaces

influence the atmosphere in two main ways, (i) via the surface energy balance and

(ii) through the efficiency with which momentum is transferred to the ground

from the moving air. In the case of the surface energy balance, this influence is

exerted both through land surface characteristics that control the net capture of

24 Soil Vegetation Atmosphere Transfer Schemes





radiant energy, and through the properties and processes that control how this

energy, once captured, is returned to the atmosphere either as latent or sensible

heat. There are also interactions between the two surface exchanges of energy and

momentum. The aerodynamic roughness of the surface controls not only the

transfer of momentum to the surface but also the efficiency of the exchange of

latent and sensible heat. Similarly, the surface energy balance affects atmospheric

buoyancy and, through this, the efficiency of both momentum and energy transfers.

Over the past four decades, the complexity and realism of the SVATS used in

meteorological and hydrological models has increased dramatically, as described

below. However, when seeking to understand the purpose of this development and

to classify the diversity of the models that have resulted from it, it is helpful to

recognize that SVATS strictly speaking only need to meet a limited number of the

basic requirements at each point in time, albeit these are required as area-averages

over the grid area used in the model. When estimating primary variables, present

day SVATS often also calculate several secondary variables, and are distinguished

by differences in the number of secondary variables calculated and the complexity

used in their calculation. Nonetheless, the motivating purpose for calculating

these additional values remains to define the time evolution of the primary set of

area-average requirements listed in Table 24.1.

Table 24.1 Requirements in a Soil-Vegetation-Atmosphere Transfer (SVAT) scheme: (A) Basic variables that must be

calculated at each model time step by a SVAT if it is used in a meteorological model; (B) Additional required calculations to

allow representation of the hydrological impacts of climate; (C) Additional required calculations to allow representation of

changes in CO2 (and perhaps other trace gases) in the atmosphere.

A. Basic requirements in meteorological models

1. Momentum absorbed from the atmosphere by the land surface – requires the effective area-average aerodynamic roughness length.

2. Proportion of incoming solar radiation captured by the land surface – requires the effective area-average, wavelength average solar reflection coefficient or albedo.

3. Outgoing longwave radiation (calculated from area-average land surface temperature) – requires the effective area-average, wavelength average emissivity of the land surface.

4. Effective area-average surface temperature of the soil-vegetation-atmosphere interface - required to calculate longwave emission and perhaps energy storage terms.

5. Area-average fraction of surface energy leaving as latent heat (with the remainder leaving as sensible heat) - to calculate this other variables such as soil moisture and/or measures of vegetation status are often required, these either being prescribed or calculated as state variables in the model.

6. Area-average of energy entering or leaving storage in the soil-vegetation-atmosphere interface (required to calculate the instantaneous energy balance).

B. Required in hydro-meteorological models to better estimate area-average latent heat and to describe the hydrological impacts of weather and climate

7. Area-average partitioning of surface water into evapotranspiration, soil moisture, surface runoff, interflow, and baseflow.

C. Required in meteorological models to describe indirect effect of land surfaces on climate through their contribution to changes in atmospheric composition

8. Area-average exchange of carbon dioxide (and possibly other trace gases).


Soil Vegetation Atmosphere Transfer Schemes 361

Originally, land surface models (e.g., Manabe et al., 1965; Shukla and Mintz,

1982) were as simple as they could be. They ignored energy storage in the soil-

vegetation-atmosphere interface (Table 24.1, value 6), and did not calculate a

detailed surface water partition (Table 24.1, value 7) or carbon dioxide exchange

(Table 24.1, value 8). They simply assumed typical fixed values of aerodynamic

roughness length, albedo, and surface emissivity and applied these to all continental

surfaces to calculate values 1, 2, 3, 4 and 5 in Table 24.1.

In early models, energy sharing between latent and sensible heat and (implicitly)

the surface temperature (requirements 4 and 5) was calculated using a simple

‘Budyko Bucket’ model (Budyko, 1948; 1956), see Fig. 24.1. A form of potential

evaporation rate was assumed and calculated using the Penman-Monteith

equation assuming zero surface resistance and an aerodynamic resistance equal to

that for momentum transfer in neutral conditions with the assigned aerodynamic

roughness length. The actual evaporation rate was then calculated at any point in

time by making a running water balance of the water level in a hypothetical

‘bucket’ located at the land surface, which is filled by precipitation and emptied by

evaporation and, when the water depth, d, stored in the bucket exceeds a critical

value, dmax

, also by runoff. The actual evaporation rate was assumed to vary linearly

with the fractional fill of this bucket between zero and the calculated potential

evaporation rate.

Early evidence for land-surface influences on climate came through model

experiments made with General Circulation Models (GCMs) which used the

simple model just described. These experiments involved making imposed

changes in the values of one of the (few) parameters used in these simple SVATS

and comparing the model-simulated climates before and after such a change.

The first such study was made by Charney et al. (1975) and was motivated by

the hypothesis (Fig. 24.2a) that expansion of desert regions, especially those in

the Sahel region of Africa, might result from a human-induced, land-surface

Precipitation, P

FieldCapacity

WaterDepth

dmax

d

Evaporation, E = bEp

Runoff, R

1

b

0

∂d

If d′ > dmax d → dmax and R = (d ′ − dmax)

= P − E∂t

d0 dmax

Figure 24.1 Schematic diagram of the SVATS used in early studies of the effect of land surfaces on weather and climate

based on the ‘Budyko Bucket’.



driven positive feedback process. The hypothesis was that overgrazing reduced

vegetation cover and increased the surface albedo, and this in turn reduced the

energy entering the atmosphere and consequently atmospheric ascent, causing

precipitation and vegetation cover to be further reduced. In this model

experiment, Charney increased the albedo in selected regions of the globe

including the Sahel, (Fig. 24.2b) and modeled a reduction in precipitation of

about a factor two (Fig. 24.2c).

A second important early modeling study also used such a simple land surface sub-

model to represent the land surface. By making GCM runs with land surface evapo-

ration across the globe fixed first to zero and then to the potential evaporation rate,

Shukla and Minz (1982) demonstrated that water evaporated from continental

surfaces recycles in the atmosphere and can contribute significantly to modeled

precipitation (Fig. 24.3).

Developing realism in SVATS

There have been rapid developments in the realism of land-surface sub-models

in meteorological and hydrological models over the past decades, motivated

partly by the sensitivity of climate to the land surface as demonstrated by early

experiments such as those just described, and stimulated by the need for better

predictions of human influence on the atmosphere resulting from land use

160°W

80°N

60°N

40°N

20°N

0°

20°S

40°S

60°S

(c)

(b)

Reducedradiation

Lessascent

Higheralbedo

Lack ofvegetation

Lack ofrainfall

(a)

80°S Time (weeks)

Humid area (0.14) Ice cover (0.7)

Idealised permanentdesert (0.35)

ALBEDOES PRESCRIBEDIdealised margionalarea (0.14 or 0.35)

Pre

cipi

tatio

n (m

m d

−1)

SAHEL

10

2

4

6

2 3 4

120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

Figure 24.2 The first GCM modeling study demonstrating the effect of land surfaces on climate: (a) the

‘desertification’ hypothesis that motivated the study; (b) areas of the globe where different albedo values were used in

the model, including the Sahelian region; and (c) the modeled difference in precipitation in the Sahel. (Data from

Charney et al., 1975.)



change and global warming. At this point in time, the developments in SVATS

can be conveniently considered as falling into three groups distinguished by

the sequence in which the associated research and development occurred

(Liang, 2005). These are described below.

44

4

36

1

8

31

1

(b)

45

3

1

6

106

B

54

44

8 6

445

3

6

33

810

64

3

4

46 6

6

B

H

3

6

(a)

18060S

40S

20S

0

20N

40N

60N

80N

60S

40S

20S

0

20N

40N

60N

80N

120W 60W 0 60E 120E 180

L

H

2

2

2

2

2

22

2

2

1 H

H

H

L

H 3

H2

H

HH

H

HH

L1

HH

H

B

H

HH

H

H

LL

L

L

1

L

H

H

2

2

6H

H

H

L

H

L

1

1

H

LL1

1

1

3

2

L

L

4

4

43

568

1216

20N

116

6

1

11

1

2L

12

2

2

422

2

23

2

2

2

2

2

3

Precipitation contours in (mm d−1)

Figure 24.3 The first GCM study to demonstrate moisture recycling over continental surfaces contributes to precipitation,

(a) and (b) respectively show modeled global precipitation for July with land-surface evaporation fixed at potential rate and

zero. (From Shukla and Mintz, 1982, published with permission.)



Plot-scale, one-dimensional ‘micrometeorological’ models

Following the use of the Budyko Bucket model, the next generation of SVATS can be

characterized by the developments of Deardorff (1978), who made simplifying

improvements in the representation of soil heat fluxes through the ‘force-restore’

scheme, and Dickinson et al. (1986) and Sellers et al. (1986), who made substantial

improvements in the representation of vegetation controls on evaporation. The

essential essence of this group of SVATS which is illustrated schematically in

Fig. 24.4, is that they were one-dimensional representations of the micrometeoro-

logical interaction of uniform plots of vegetation, but applied at the (much larger)

grid scale used in GCMs.

Notably among the improvements in vegetation-related features was the

introduction of the effect of leaf stomatal resistance through development of the

Monteith (1965) ‘big leaf ’ assumption and of a canopy water balance to calculate

interception loss using a (usually simplified) version of the Rutter model, both of

which were described in Chapter 22. In this group of SVATS, the temperature of

the vegetation was also sometimes calculated explicitly as a state variable, and dry

canopy surface resistance was usually either assumed constant for a particular

vegetation cover, or parameterized in terms of a series of stress factors in a version

of the Jarvis-Stewart model (see, for example, Shuttleworth, 1989) in which surface

resistance is expressed in the form:

0

1 s c R D T M

s

g g g g g g gr

= =

(24.1)

lE

lE

lEH

H

H Sr

Lu

Lu

LuSr

Sr lEH

Lu

SrlE

H

Lu

Sr

P S Ld

Runoff

Runoff

RunoffRunoff

Bare soil

Snow pack


Deep drainage

Deep drainage


diagram of second

generation one-dimensional

SVATs in which a plot-scale

micrometeorological model

with an explicit vegetation

canopy was applied at grid

scale. See Plate 3 for a colour




where g0 is a constant; g

c is a canopy cover factor (to allow for gaps in the canopy

and/or patches of soil in the landscape); gR is a radiation stress function, typically

parameterized in terms of the incoming solar radiation using the functional form:

(1000 )( )

1000( )R

RR

S Kg SS K

+=

+

(24.2)

where S is the incoming solar radiation (in W m-2); gD is a vapor pressure deficit

stress function, typically parameterized in terms of the VPD using the functional

form:

1 2 2( ) 1D D Dg D K D K D= + +

(24.3)

where D is the vapor pressure deficit of the air (in kPa); gT is a temperature stress

function, typically expressed in terms of the air temperature and specified tem-

perature parameters using the functional form:

0

( )( )( )

( )( )

T

T

L HT

o L H

T T T Tg TT T T T

α

α

− −=

− −

(24.4)

where T is the temperature of the air in K, and aT is given by

( )

( )H o

To L

T TT T

−=

−a

(24.5)

and gM

is a soil moisture stress function, variously parameterized in terms of the

available soil moisture in the plant rooting zone, including in the functional form:

1 2( ) 1 exp( [ ])SM M M og SM K K SM SM= − −

(24.6)

SM is the soil moisture in a depth of soil accessible to the atmosphere via plant

roots, with maximum soil moisture holding capacity, SMo. The several constants

that appear in equations (24.1) to (24.6) (i.e., KR, K

D1, etc.) are specific to the type

of plant represented and their values were sometimes calibrated against plot-

scale field data. Using observations made over several forest canopies, for

example, Shuttleworth (1989) demonstrated the functional forms for gT , g

R, and

gD shown in Fig. 24.5.

In some SVATS in this group, the aerodynamic roughness and zero plane

displacement of the vegetation were prescribed depending on the vegetation

represented, but in others (e.g., Sellers et al., 1986) it was calculated from more basic

canopy characteristics, and in several, the difference in solar reflection coefficient

above and below 0.7 μm was allowed for via the ‘two stream’ approximation.

Seasonal changes in the vegetation (especially leaf area index) were also often

recognized and prescribed through ‘look up’ tables.



One very important development, which was introduced with this group

of models, was the feasibility of representing GCM grid squares as being

covered with one of several ‘biomes’ of vegetation and one of several soil

classes. Such biomes and soil classes were defined by specifying groups of

the parameters used in the sub-model to calculate requirements 1–6 in Table

24.1. Introducing this feature was important because it enabled model

experiments to investigate the effect on climate of potential large-scale veg-

etation changes such as large-scale Amazonian deforestation, for example,

see Chapter 25.

Subsequently, a large number of novel land-atmosphere sub-models were devel-

oped that were essentially similar in concept (e.g., Noilhan and Planton, 1989;

Xue et al., 1991; Koster and Suarez, 1992; Ducoudré et al., 1993; Verseghy et al.,

1993; Viterbo and Beljaars, 1995; Wetzel and Boone, 1995; Desborough and

Pitman, 1998). Although the land-atmosphere sub-models in this group were all

essentially one-dimensional plot-scale models of vertical energy and water move-

ment with limited attention paid to hydrological processes or recognition of the

(a)

403020

Temperature (°C)

100

0.5 TL

A

g T(T

)

1.0(b)

Solar Radiation (W m−2)

g S(S

)

00 500 1000

0.5

1.0

A

L

T

J

(c)g D

(D)

00 10

VPD (g kg)

20

0.5

J

1.0

T

A

L

Figure 24.5 Form of the

stress factors in the

Jarvis-Stewart model for

several forest stands:

A - Amazonia (tropical

rain forest), J - Jadraas,

Sweden (Scots pine),

L - Les Landes, France

(maritime pine), and

T - Thetford, UK

(Scots-Corsican pine).

(From Shuttleworth, 1989,

published with

permission.)



effects of horizontal heterogeneity, they nonetheless differed substantially in

significant detail with respect to the level of complexity adopted.

Improving representation of hydrological processes

The next generation of land surface sub-models is characterized by the work of

Famiglietti and Wood (1994), Liang et al. (1994; 1996a, 1996b), Peters-Lidard

et al. (1997), Ducharne et al. (1999), Schaake et al. (1996) and Chen et al. (1996).

Development of this group of models, which are illustrated schematically in

Fig. 24.6, was motivated by the fact that over-simplification of hydrologic

processes in earlier models was recognized as having the potential to lead to

significant errors in water and energy budget related calculations and this could

limit the ability to project future climate change, e.g., Chen et al. (1997);

Crossley et al. (2000); and Gedney and Cox (2003).

This group of models therefore attempted to address the effects of subgrid spatial

variability on water and energy budgets due to heterogeneity of soil properties,

Mixedvegetation

grid squares

S

lE

lElE

lE

H

HH

H

Sr

SrSr

Sr

Snow pack

Water table

Bare soil

P

m

1-m


Topography

Lu

Lu

Lu

Lu

Ld

Figure 24.6 Schematic diagram of SVATS with improved representation of hydrologic processes. See Plate 4 for a colour




topography, vegetation, and precipitation using statistical-dynamical approaches.

Models in this group also sought to improve the basic parameterizations of

hydrologic processes, such as infiltration, surface runoff, subsurface runoff, and

snow processes. The representation of these processes was considered to be overly

simplified in the earlier generation of micrometeorological models.

Although models in this group sought greater realism with the aim of improv-

ing the area-average calculations of the values 1–7 in Table 24.1, they did not give

priority to including description of CO2 exchange. Improvements in the represen-

tation of hydrological processes in land surface models continues (e.g., Koster

et al., 2000; Liang and Xie, 2001; Milly and Shmakin, 2002; Cherkauer and

Lettenmaier, 2003; Huang et al., 2008), and there is now interest in introducing

improved hydrological representation into the group of models described in the

next section, including the impacts of subgrid variability in precipitation based on

the schemes of Shuttleworth (1988) and Liang et al. (1996b), for instance.

Improving representation of carbon dioxide exchange

Motivated by the need for a more comprehensive representation of the carbon

cycle in GCMs to address climate change issues, the development of a further

generation of land surface sub-models was fostered by the work of Bonan (1995),

Sellers et al. (1996), Dickinson et al. (1998), Cox et al. (1998), and Dai et al. (2003).

The characteristic feature of this group of models, which are illustrated

schematically in Fig. 24.7, is that they seek to include representation of the plant

physiological processes and vegetation dynamics, in an attempt to account for

carbon uptake by plants and the feedbacks between climate and vegetation. Pitman

(2003) provides a comprehensive discussion of land surface models designed for

coupling to climate models. Two main approaches are used in predicting seasonal

variations in vegetation dynamics, i.e., a plant physiological process-based

approach (e.g., Lu et al., 2001), and a rule-based approach (e.g., Foly et al., 1996;

Levis and Bonan, 2004; Kim and Wang, 2005).

Studies of plant biochemistry had suggested a different approach to modeling

stomatal control that is somewhat less empirical than the Jarvis-Stewart model,

and therefore hopefully more transferable from one plant species to the next and

(perhaps) only dependent on whether species are C3 or C4 plants in terms of their

photosynthetic function. In such models, the assimilation of carbon is viewed as

the controlling factor, and stomatal conductance is described by (sometimes a

derivative of) the so-called Ball-Berry equation (Ball et al., 1987), i.e.:

min( )s n s l eg m A C P F g= +

(24.7)

where gmin

is a prescribed minimum stomatal conductance; m is a slope parameter

(~9 for C3 plants); An is the net carbon assimilation; C

s, is the partial pressure of

carbon dioxide; Pl is atmospheric pressure adjacent to the leaf; and F

e is a humidity

dependent stress factor, which in the original Ball-Berry equation was set as



numerically equal to the relative humidity, but which in some SVATS (e.g.,

Dickinson et al., 1998), is assumed to be a function of vapor-pressure deficit. The

introduction of this alternative means of describing the behavior of stomata has

sometimes been referred to as the ‘greening of SVATS’. In this formula, the simplest

estimate of An, is given (Farquhar and Sharkey, 1982) by:

min( , , )n c e sA J J J=

(24.8)

where Jc , J

e , J

s are functions expressing the assimilation rates when limited by the

Rubisco enzyme, light, and transport capacity, respectively, for C3 and C4 plants;

see Collatz et al. (1991; 1992), Sellers et al. (1996) and Cox et al. (2001). In practice,

it has been observed that the transition between these three limiting rates is not

abrupt but gradual, and some SVATS (e.g., Collatz et al., 1991; Cox et al., 2001)

have devised mathematical ways to simulate this smooth transition.

Thus, a key difference between the Ball-Berry formula for stomatal conductance

(resistance) and the Jarvis-Stewart formula is that stress factors (apart from that

for humidity) are no longer combined as a product; rather, one factor is considered

to be the dominant limitation on carbon assimilation and hence on stomatal

conductance. SVATS continue to make progress in describing vegetation and are

Vegetationdynamics Vegetation

growth cycle

CO2

N2

lE

lE

lE

H

H

H Sr

Lu

Lu

Sr

Sr lEH

Lu

SrlE

H

Lu

Sr

PS Ld

Runoff

Snow packRunoff

RunoffRunoff

Bare soil


Deep drainage

Deep drainage

Figure 24.7 Schematic diagram of SVATS with improved representation of vegetation related processes, including CO2

exchange and ecosystem evolution. See Plate 5 for a colour version of this image.



now seeking to describe the evolution of the biome (i.e., vegetation cover)

represented in the GCM at a particular place in response to long-term changes in

modeled climate (Foley et al., 1996; 2000; Cox et al., 2000; 2001; Kucharik et al.,

2000; Oyama and Nobre, 2003; 2004). Ultimately this capability may also become

relevant in long-term, large-scale hydrological modeling studies.

As yet, this group of sub-models does not consider interactions between the

biogeochemical cycles of carbon, nitrogen and phosphorus. In fact, such

interactions are not yet considered in many terrestrial biogeochemical

models, including the CENTURY model (Wang et al., 2007). However, including

such interactions is likely necessary for a more comprehensive and realistic

representation of the effect of land surface processes on the carbon cycle. It is also

true that most of the land surface models in this group of sub-models have simpler

treatments of sub-grid spatial variability and hydrological processes than those

described in the last section. Consequently, there is a need to combine the model

improvements in describing hydrological processes with those describing carbon

dioxide exchange and vegetation dynamics in a new generation of land surface

sub-models. Fortunately, ecologists, climatologists, and hydrologists have begun

to work together to improve the realism and functionality of SVATS.

Ongoing developments in land surface sub-models

The impacts of surface and groundwater interactions on the land-atmosphere sys-

tem have hitherto received little attention, but recognizing the role roots (especially

deep roots) may play in the plant-soil-land continuum, researchers have now begun

to investigate the dynamic interactions of surface water and groundwater, and

whether such interactions can affect vegetation and, via vegetation, land-atmos-

phere interactions (e.g., Winter, 2001; Gutowski et al., 2002; York et al., 2002; Liang

et al., 2003; 2006; Maxwell and Miller, 2005; Yeh and Eltahir, 2005; Fan et al., 2007;

Niu et al., 2007). Several different approaches are under investigation, including:

(a) A ‘TOP model’ type based approach (e.g., Walko et al., 2000).

(b) Solving for soil moisture in unsaturated zones and pressure head profiles in

saturated zones separately by applying (variations of) the Richards equa-

tion to each zone respectively (e.g., Gutowski et al., 2002; York et al., 2002;

Yeh and Eltahir, 2005; Fan et al., 2007; Niu et al., 2007). In this approach,

the coupling is essentially one-way rather than two-way.

(c) Solving for the hydraulic pressure profile for the unsaturated and saturated

zones together based on a mixed form of the Richards equation (e.g.,

Maxwell and Miller, 2005). This approach involves two-way coupling.

(d) Solving for soil moisture profile by applying the Richards equation to the

unsaturated zone only, with the groundwater table treated as a moving

boundary (e.g., Liang et al., 2003). This relatively simple approach does not

introduce any additional parameters other than those already used by a

typical land surface model and also involves two-way coupling.



Among these four approaches (a) arguably has least accuracy in representing the

dynamic movement of the groundwater table, and the two-way coupling in (c) and

(d) is likely preferable. Of these last two, (d) is favored on the grounds of

computational efficiency unless the groundwater table is close to the bedrock.

Despite the limitations involved in all these approaches, these several studies have

together suggested the importance and potential impact of interactions between

groundwater and surface water on surface water and energy budgets, surface runoff

generation, soil moisture, groundwater recharge and subsurface flow characteristics

at different spatial scales (e.g., Jiang et al., 2009), see Fig. 24.8. However, work in

this area is ongoing and the most effective way forward is yet to be defined.

Hydraulic redistribution (or hydraulic lift) is the passive movement of water from

roots into soil layers or from soil layers into roots. It can occur upwards, downwards

or laterally depending on the water potential gradients in the soil (e.g., Burgess et al.,

1998; Hultine et al., 2003a; 2003b; Leffler et al., 2005). It is now believed that

hydraulic redistribution may have significant impact on weather and climate

through its impact on evapotranspiration. Hasler and Avissar (2007) showed, for

example, that global and regional meteorological models both currently tend to

overestimate the dry season water stress in the Amazon basin relative to eddy

covariance flux measurements from eight towers, probably due to their

misrepresentations of the soil water and plant processes. While Lee et al. (2005)

incorporated a simple empirical formulation of hydraulic redistribution into version

2 of the NCAR Community Climate Model coupled to the Community Land Model

Figure 24.8 Cumulative observed average precipitation for the region 30-40°N 92.5-107.5°W compared with ensemble-

average cumulative precipitation calculated by the WRF model with the NOAA land surface scheme (labeled ‘control’),

a NOAA scheme including simulation of dynamic vegetation and a NOAA scheme including simulation of both dynamic

vegetation and ground water interactions. (Adapted from Jiang et al., 2009.)



(CLM) and estimated its impact on the climate of Amazonia and elsewhere. Based

on two simulations, Lee et al. showed that photosynthesis and evapotranspiration

increase significantly in the Amazon during the dry season if the hydraulic

redistribution process is included, resulting in less water stress and greater

evapotranspiration. Other important new findings from field observations, such as

the functioning of the plant roots in arid and semi-arid climates (Seyfried et al.,

2005) may also merit representation in SVATS, in conjunction with better

representation of the dynamics of surface water and groundwater interactions.

All the models in the groups of land surface sub-models discussed in the

previous three sections implicitly apply the one-dimensional Richards equation

when describing the soil moisture movement within the soil column, while the

lateral flows and/or interactions are represented through parameterizations and

flow routing. However, the routing scheme of Guo et al. (2004) is significantly

different in four respects:

1. it allows grid-based runoff to exit modeled grid squares in multiple direc-

tions simultaneously instead of just one of the eight discrete directions

employed in most routing schemes, a feature which is likely to be most use-

ful for models with coarse grid resolution;

2. it introduces a ‘tortuosity coefficient’ which adjusts some geomorphology-

related parameters such as channel slope and length, to reduce the impacts

of different spatial resolutions on flow routing;

3. it uses a flow network which is reasonably realistic; and

4. it explicitly differentiates between overland and river flow in the flow

network.

Choi et al. (2007) also propose applying a three-dimensional Richards equation to

more accurately represent both vertical and lateral flow interactions. There are

issues still to be resolved before three-dimensional approaches can be applied

effectively in SVATS, but this new direction of investigation merits attention.

On the basis of the above review of SVATS, it is clear that there has already been

substantial progress and that this field of interest remains active, with progress still

being made. It is anticipated that land surface sub-models will continue to evolve to

include, for example, better representation of surface-groundwater interactions,

sediment transport, biogeochemical processes, and sub-grid spatial variability

associated with the integrated atmosphere-vegetation-land-soil system, see, for

example, Fig. 24.9. However, given the problems associated with defining the

growing number of parameters that will need to be specified globally in such

models, it is not yet clear that further development and associated sub-model

complexity will necessarily have a major positive impact on the accuracy and

reliability with which predictions of weather and climate can be made.

What is arguably more certain is that further development may enhance the

capability to interpret predicted weather and climate in terms of their impact on

human welfare and ecological status because representation of features relevant to

such impact are included in the models themselves, with parameters that can be



locally calibrated. Further development may thus be a mechanism through which

meteorologists, hydrologists, biologists and ecologists can be brought together

with applications specialists concerned with the management of water, agricul-

tural and industrial resources and public health.


● Purpose of SVATS: is to specify the lower boundary conditions in atmospheric

models, the upper boundary conditions in hydrological models, or provide

the interface between land surfaces and the atmosphere in coupled models.

● Function of SVATS: is to calculate the exchange of energy, water, and carbon

dioxide and perhaps also other trace gases between land surfaces and the

atmosphere, and sometimes also components of the surface water balance

from ‘forcing data’ that comprises frequently sampled (either modeled or

measured) values of near-surface weather variables.

Mixed vegetationwith vegetation

dynamics

SLd

(CO2, N2,...)

Snowpack

Routing

Dataassimilation

m

1-m


Routing

(CO2, N2,...)

LuLu

SrSr

HH

lElE

Figure 24.9 Schematic diagram of potential future developments in SVATS. See Plate 6 for a colour version of this image.



● Requirements from SVATS: modern SVATS can be complex and may calculate

many variables but the motivating purpose for calculating these additional

values remains to calculate the time evolution of the limited set of area-

average requirements listed in Table 24.1.

● Early SVATS: originally SVATS prescribed fixed, globally-applied surface

parameters (e.g., surface roughness, albedo, emissivity, etc.) and used a sim-

ple ‘bucket model’ for surface energy partition, but using these in GCMs

demonstrated the sensitivity of modeled climate to changes in these

parameters.

● Micrometeorological SVATS: subsequent development resulted in a genera-

tion of SVATS, here called ‘micrometeorological’ SVATS, that were fairly

detailed one-dimensional models of the interactions of different uniform

vegetation canopies that strictly only apply at the scale of a few hundred

meters.

● Hydrological improvements in SVATS: further development sought

improvements in SVATS’ ability to describe hydrological processes,

including:

— improving basic parameterizations of hydrologic processes such as

infiltration, surface runoff, subsurface runoff and snow processes;

— representing the effects of subgrid spatial variability due to heteroge-

neity of soil properties, topography, vegetation and precipitation using

statistical-dynamical approaches.

● ‘Greening’ of SVATS: recognition of potential climate change caused a major

shift in direction in SVAT development toward providing improved capabil-

ity to simulate CO2 exchange, and this shift resulted in a substantial revision

in the preferred representation of plant stomatal behavior in SVATS (e.g.,

from the Jarvis-Stewart to the Ball-Berry parameterization).

● Ongoing development of SVATS: current development is concerned with

addressing more difficult aspects of surface-atmosphere interactions includ-

ing the impacts of surface-groundwater interactions on the land-atmosphere

system, and hydraulic redistribution (or hydraulic lift).

● Future value of SVAT development: further developments of SVATS that

add complexity will not necessarily improve their performance in climate

and weather prediction, more likely it will improve ability to interpret pre-

dicted weather and climate in terms of their impact on human welfare and

ecological status.

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Introduction

As mentioned in Chapter 9, on average across the globe about half the energy

that drives atmospheric circulation enters from the surface of the Earth. Because

two-thirds of the Earth’s surface is covered with oceans they provide an important

control on atmospheric circulation. The remaining one-third of the Earth’s

surface is continents, and the interaction between land surfaces and the overlying

atmosphere is also a substantial influence in the coupled atmosphere-ocean-land

climate system.

Even at the scale of large-scale atmospheric circulation the differences between

the ocean-atmosphere interactions and land-atmosphere interactions are apparent.

Continents are aerodynamically rougher than the oceans so large-scale circulations

around high pressure at 30°N and 30°S are less strong and persistent over

continents than over the smoother oceans. The seasonal north-south shift in the

pattern of atmospheric pressure and circulation is also influenced by the difference

between ocean-atmosphere interactions and land-atmosphere interactions. As

Fig. 9.5 shows, in the northern hemisphere the oceanic subtropical highs

are farther north and more intense in the northern hemisphere summer than

they are in the southern hemisphere in the southern hemisphere summer. In

the northern hemisphere winter there is also a more marked reversal in the

pressure difference between oceans and the continents. The seasonal differences

in the surface air temperatures over continents relative to those over oceans result

in the Asian-Australian monsoon system, while the presence of the Rocky

Mountains influences the penetration into North America of moist maritime air

in the mid-latitude westerly winds.

In the context of this chapter it is important that the overwhelming majority

of humankind lives on land, and as the human population grows the relative

25 Sensitivity to Land Surface Exchanges




Atmospheric Sensitivity to Land Surface Exchanges 381

importance of what they do has an increasing influence on climate. The indirect

impact of human activity that results from changing the chemical composition of

the atmosphere is well-recognized and increasingly well-predicted (IPCC, 2007).

However, it is the direct impact of human-induced changes on land surface–

atmosphere exchange processes that concerns us in this chapter. Land surface

exchange processes depend on the nature of the vegetation present, but humans

increasingly change the type of vegetation cover and heterogeneity of land

surfaces, create large urban complexes that replace natural surfaces with artificial

surfaces, and make water more readily available to the atmosphere by irrigating

crops and building dams. As humans do this, the extent to which water, energy,

and momentum is transferred between the atmosphere and the ground

necessarily alters.

In the past, when human population was low, the proportion of the Earth’s land

surfaces that we altered was not large, so its consequence when averaged to the

regional or global scales was less important. But even when conditions higher

in the ABL are little-changed, meteorological variables measured (say) 2 m above

the ground are altered if there is a change in local surface roughness and/or local

surface energy balance (e.g., Bastable et al., 1993). The proportion of land subject

to human influence is already extensive (Fig. 25.1a), and for many of the world’s

important biomes it will continue to increase (Fig. 25.1b) as human population,

which is currently around 6.5 billion, increases by about 50% by the year 2050.

Influence of land surfaces on weather and climate

The available scientific literature on studies of the influence of land surfaces on

weather and climate is large and diverse. To give structure this overview chapter

is divided into three main sub-sections that consider research into, and evidence

for, the influence on weather and climate of (a) existing land-atmosphere interac-

tions; (b) transient changes in land surfaces; and (c) imposed persistent changes

in land cover. When considering influential mechanisms within these three

classifications, in each case we consider first whether the mechanism has a

plausible physical basis. Then we review the evidence that it actually exists by

discussing relevant analyses of weather and climate data (including reanalysis

data), computer model experiments and any other related experimental and

observational studies. This leads to an assessment of the credibility of the

evidence that the mechanism considered is indeed a way in which land surfaces

influence climate and weather using ‘IPCC-like’ categories (IPCC, 2007) of

credibility. Finally we make an assessment of how well this influential mechanism

is currently quantified and represented in models. For convenience these

judgments on the evidence that a particular influential mechanism exists and

the adequacy of our current understanding and modeling of each influence are

tabulated at the end of the chapter. The goal of this table is to give guidance on

priorities by identifying where future research is needed.


Cultivated Systems:Areas in which at least30% of the landscapeis cultivated

(a)

MEDITERRANEAN FORESTS,WOODLANDS, AND SCRUB

TEMPERATE FORESTSTEPPE AND WOODLAND

TEMPERATE BROADLEAFAND MIXED FORESTS

TROPICAL ANDSUB-TROPICAL DRY

BROADLEAF FORESTS

FLOODED GRASSLANDSAND SAVANNAS

TROPICAL AND SUB-TROPICALGRASSLANDS, SAVANNAS,

AND SHRUBLANDS

TROPICAL AND SUB-TROPICALCONIFEROUS FORESTS

DESERTS

MONTANE GRASSLANDSAND SHRUBLANDS

TROPICAL AND SUB-TROPICALMOIST BROADLEAF FORESTS

TEMPERATECONIFEROUS FORESTS

BOREALFORESTS

TUNDRA

Conversion of original biomesLoss by1950

Fraction of potential area converted−10 0 10 20 30 40 50 60 70 80 90 100%

Loss between1950 and 1990

Projected lossby 2050*

(b)

Figure 25.1 (a) Land areas which were more than 30% cultivated in 2000. (b) Projected change in original land cover by

2050 given by biome according to the four Millennium Ecosystem scenarios. (Redrawn from MEA, 2005, published with

permission.) See Plate 7 for a colour version of these image.



A. The influence of existing land-atmosphere interactions

1. Effect of topography on convection and precipitation

Evidence that topography affects weather and climate is irrefutable. When moist

air moving across the Earth encounters topography the resulting orographic flow

gives updrafts and cooling of the air that can generate cloud and so enhance the

probability of rain. Heterogeneous surface heating in mountainous terrain can

also generate mesoscale circulations that impact local atmospheric convection

and result in variations in the spatial distribution of precipitation related to topo-

graphic features such as slope, aspect and elevation. Some effects of topography

can be remote. The atmospheric convection initially generated by topography

can, for example, result in organized mesoscale complexes that give sustained

convective activity that propagates downwind (e.g., Tucker and Crook, 1999).

There is also evidence that topography has a substantial influence on the large-scale

atmospheric circulations involved in the Asian-Australian Monsoon (e.g., Ueda

and Yasunari, 1998). The North American Monsoon Experiment (NAME; Higgins

and Gochis, 2007) is a recent example of an observational study framed around

providing better understanding of monsoon flows including the topographic

influences on precipitation. Figure 25.2 shows the rain gauge transects used

during the NAME and examples of the observed spatial distribution of total

precipitation across the mountainous Sierra Madre Occidental of northwest

Mexico, together with the variation in frequency and maximum intensity of

precipitation as a function of time of day and elevation.

There is a substantial body of literature describing region-specific, observation-

based statistical models of the variations in precipitation (and sometimes

temperature) as a function of location and elevation (e.g., Brown and Comrie,

2002). The quantification of the effect of topography is reasonably good in such

statistical models but they only describe time-average values and, because they are

empirical, they are strictly only relevant in the region in which they are calibrated.

However some, notably the PRISM methodology (Daly et al., 1994; 2008; Johnson

et al., 2000), have been used to extrapolate distributions more generally. There

have been some experimental attempts to understand and describe the influence

of topography on convection using very fine-scale meteorological models (e.g.,

Gopalakrishnan et al., 2000), but the spatial scale of many of the topography-

related processes involved means describing them in mesoscale models operating

at grid-scales of a few kilometers is problematic. Consequently, from the physical

modeling perspective, our present ability to understand and quantify the influence

of topography on weather at the mesoscale remains limited.

Because air temperature falls with height, not only can the amount but also

nature of precipitation change with altitude, especially at mid-latitudes. Much of

the water used by humans originally falls and is often then temporarily stored as

frozen precipitation (Barnett et al., 2005), including most of the water resources of

the western USA for example. The buildup of, and sublimation loss from, snow and

ice in mountainous terrain and the evolution and ultimate melting of snowpack to


384 Atmospheric Sensitivity to Land Surface Exchanges

generate hydrological flows remains poorly understood and modeled. Redressing

this shortcoming remains a major challenge for terrestrial hydrometeorology.

Because the physical basis for an influence of topography on weather and

climate is plausible and there is much evidence of its existence in climate records

and from experiments and models, the credibility of this mechanism providing

a means for land surfaces to affect climate and weather is assessed as being

‘extremely likely’ in Table 25.1. Quantification and modeling of the mechanism is

assessed as being of ‘medium’ quality for long-term time averages but is assessed

as still ‘poor’ at short time scales. There is a clear need for more research in this

last area.

0.40

0.35

0.30

0.25

0.20

0.15

Pre

cipi

tatio

n fr

eque

ncy

0.10

0.05

0.000 2 4 6 8 10

Time of day (LST)12 14 16 18 20 22 24

(c)0-500500-10001000-15001500-20002000-25002500-3000Network mean

28 N

24 N

(a)

110 W 106 W

30�00�N

28�00�N

26�00�N

24�00�N

28 N

(b)

24 N

110 W 106 W

30�00�N

28�00�N

26�00�N

24�00�N

10.0

0-5009.0

8.0

7.0

6.0

Pre

cipi

tatio

n in

tens

ity (

mm

/hr)

5.0

4.0

3.0

2.0

1.0

0.00 2 4 6 8 10

Time of day (LST)12 14 16 18 20 22 24

(d)500-10001000-15001500-20002000-25002500-3000Network mean

Figure 25.2 Contours of total precipitation for (a) Aug 2002 and (b) July and Aug 2003 with contour interval 40 mm

derived from the transects of rain gauges located at the positions shown by stars along transects across the Sierra Madre

Mountain Range. Diumal cycles of (c) hourly precipitation frequency and (d) hourly precipitation intensity derived from

these rain gauges separated into elevation-band averages. (Redrawn from Gochis et al., 2004, published with permission.)



2. Contribution by land surfaces to atmospheric water availability

The hypothesis that land surfaces influence climate and weather by contribut-

ing to the water vapor in the atmosphere is entirely plausible. It requires only

that the amount of water vapor contributed regionally by evaporation from

land surfaces is significant in comparison with that contributed by oceans.

There are now numerous observations and model analyses that indicate the

recycling of water over land surfaces is significant. In fact this significance has

been apparent for decades in global water balance studies (e.g., Baumgartner

and Reichel, 1975; Korzun, 1978) and has been confirmed by more recent

studies (e.g., Oki and Kanae, 2006). As a global average only about 35–45% of

precipitation falling over land leaves as runoff, which implies that the remain-

der is re-evaporated. There is also long-established evidence from global mod-

eling studies (e.g., Shukla and Mintz, 1982) that the atmospheric water vapor

resulting from land surface evaporation has a large effect on modeled

continental precipitation.

Early isotope studies (e.g., Salati et al., 1979) further demonstrated that in

certain areas such as the Amazon River basin about 30% of area-average

precipitation originates from evapotranspiration, and later studies using

reanalysis data sets (e.g. Brubaker et al., 1993; Eltahir and Bras, 1996; Costa and

Foley, 1999; Bosilovich et al., 2005) confirm recycled evaporation accounts for

20–27% of precipitation in this region. The analysis of Makarieva and Gorshkov

(2007) suggests that the efficiency of recycling by forests is greater than for other

land covers so their extensive presence may help maintain precipitation amounts

for greater distances away from coasts. As reanalysis data sets have become

increasingly available there have been numerous studies of atmospheric cycling

including demonstrations of its importance in the context of ecoclimatological

stability (Dominguez and Kumar, 2008) and in monsoon systems (Dominguez

et al., 2008).

Thus, the phenomenon of precipitation recycling is now a well-studied and

well-established facet of terrestrial hydrometeorology and its quantification

and consequences are increasingly well-defined. For this reason in Table 25.1 the

credibility of this mechanism providing the basis of a land surface influence on

climate and weather is assessed as being ‘extremely likely’. But quantification and

modeling of the mechanism is assessed as being of ‘medium’ quality because

further research is justified, with focus on achieving greater realism and accuracy

when modeling surface evaporation and especially the atmospheric mechanisms

involved in releasing precipitation.

B. The influence of transient changes in land surfaces

1. Effect of transient changes in soil moisture

It is physically plausible that moisture stored in the soil which entered from

precipitation can later become re-accessable to the atmosphere, often via plants.



It is also plausible that transient changes in the amount of water available for

release from the soil can influence climate and weather in different ways,

specifically by:

1. contributing to changes in atmospheric water concentration (see previous

section);

2. modifying the downwind structure of the atmosphere and in this way

modifying the probability of precipitation; and/or

3. generating mesoscale circulations in the atmosphere in response to spatial

differences in the surface energy balance if the spatial pattern of soil moisture

is heterogeneous.

Hydrometeorological records provide some observational evidence that transient

soil moisture status influences precipitation during the summer months. Findell

and Eltahir (1997), for example, plotted the coefficient of determination between

the average soil moisture measured at sites in the Illinois Climate Network on a

particular day of the year and the state-wide average precipitation during the sub-

sequent 21 days, see Fig. 25.3. They suggested that during summer a significant

amount of the variation in precipitation could be explained by antecedent soil

moisture status. Early convincing evidence for the influence of soil moisture on

precipitation was also provided in a modeling study made with the European

Centre for Medium-term Weather Forecasting (ECMWF) model (Beljaars et al.,

1996) which showed that introducing improved modeling of the seasonal evolu-

tion of soil moisture resulted in enhanced skill in predicting the major Mississippi

floods in the summer of 1993. Better modeling of soil moisture gave improved

simulation of the vertical profile of atmospheric temperature downwind, with

warmer air aloft and greater opportunity for deep cloud and heavier convective

Figure 25.3 Between 1981 and 1994, the 21 day running average of the coefficient of

determination between the average soil moisture measured to a depth of 10 cm at sites in

the Illinois Climate Network on each day of the year and the statewide average

precipitation for Illinois measured over the subsequent 21 days. The horizontal lines show

5% and 10% levels of significance in the coefficient of determination. (Adapted from

Findell and Eltahir, 1997.)



precipitation (c.f., Case 3 in Fig. 10.4 and the associated text). Internationally

coordinated GCM studies involving several models have identified regions where

soil moisture control is most important (GLACE; Koster et al., 2006), see Fig. 25.4,

although there are currently some shortcomings in the quality of the representa-

tion of soil moisture evolution in the GCM models used (Teuling et al., 2006).

There is also modeling and some observational evidence that, at least in regions

such as the Sahel (where there are few other surface features such as topography or

heterogeneous land cover) for atmospheric turbulence to respond to, transient soil

moisture heterogeneity can influence mesoscale atmospheric circulations and the

likely location of future rain (e.g., Taylor et al., 2007).

In summary, it is physically plausible that transient changes in soil moisture can

have an impact on weather and climate, and there is a growing body of convincing

evidence that such an impact does occur especially at regional scales, although

currently there is less evidence for the existence of mesoscale influences.

Consequently in Table 25.1 the credibility of this influential mechanism is given as

‘extremely likely’ at regional scale and ‘likely’ at mesoscale. However, hitherto

measurement of area-average soil moisture at scales appropriate for accurately

quantifying, building and testing models of the influence of transient changes in

soil moisture has inhibited progress in this area, and the assessment of quantifica-

tion and modeling is given as ‘medium’ for both regional scale and mesoscale in

Table 25.1 for this reason. Fortunately, new methods for measuring mesoscale

area-average soil moisture are emerging by observing the field of aboveground

60N

30N

EQ

30S

60S180 120W 60W 0 60E 120E 180

−0.03

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Figure 25.4 Geographical distribution of land-atmosphere ‘coupling strength’ (i.e., the degree to which anomalies in soil

moisture can affect rainfall generation and other atmospheric processes) averaged for eight GCMs in the GLACE study

(Redrawn from Koster et al., 2006, in which paper ‘coupling strength’ is defined, published with permission.) See Plate 8 for

a colour version of this image.



neutrons produced by cosmic rays (Zreda et al., 2008; Shuttleworth et al., 2010), and

at regional scales using satellite systems (Kerr et al., 2001; Entekhabi et al., 2010).

Future research that exploits these new measurement techniques is, therefore,

a priority, and will likely give improved understanding and modeling of this

land-atmosphere feedback mechanism, leading to an assessment of ‘good’.

2. Effect of transient changes in vegetation cover

Because water often leaves soil and enters the atmosphere via transpiration from

plant leaves, the status of the vegetation in terms of leaf cover and plant vigor can

influence the surface energy balance and, through this, weather and climate.

Moreover, living plants can intervene in subterranean flow processes by modifying

soil characteristics or by redistributing water vertically in the soil through the

body of the plant, thus changing the ease and extent to which soil water is

accessible to the atmosphere. The influence of transient changes in vegetation

vigor on weather and climate is therefore physically plausible.

As discussed in detail in Chapter 23, there is long-established evidence of

seasonal changes in evaporation (and therefore in the surface energy balance)

through the growth cycle of annual crops. A common approximate representation

(Allen et al., 1998) has been to assume a seasonal pattern in the crop factor used in

Equation (23.12), see Fig. 23.2. However, most of the evidence for there being

an effect of vegetation cover on climate and weather is derived from modeling

studies, and it takes the form of a simulated difference in model-calculated climate

with and without representation of seasonal changes in leaf area index of the

vegetation covering the ground. The early one-dimensional ‘micrometeorological’

land surface models (e.g., Dickinson et al., 1986; Sellers et al., 1986) used in GCMs

(see Chapter 24 for description) took it as self-evident that transient changes in

vegetation cover will influence climate, and they had seasonal variations in the leaf

area index of the vegetation prescribed using look-up tables to accommodate

this. More recently (again see Chapter 24) SVATS have emerged with improved

representation of carbon dioxide exchange (e.g., Sellers et al., 1996; Dickinson

et al., 1998; Cox et al., 1998) and this allows simulation of seasonal changes in

simulated leaf area index of vegetation through the year. Such SVATS are some-

times referred to as having interactive vegetation. Incorporating interactive

vegetation can result in significant changes in the modeled surface evaporation

and precipitation, see Fig. 25.5. Because remote sensing can be used to provide an

indirect estimate of the amount and vigor of vegetation (see Fig. 5.6 and associated

text), some modeling studies have also considered the effect of introducing

remotely sensed estimates of changing vegetation on modeled climate and

have found that doing so can give significant effects (e.g., Matsui et al., 2005).

It is widely accepted among the scientific community that transient (generally

seasonal) changes in vegetation cover will affect weather and climate and,

consistent with this, this influential mechanism is assessed as being ‘extremely

likely’ in Table 25.1. Motivated by a desire to include the feedback effects of

vegetation change in response to atmospheric carbon dioxide concentration, over



the last few decades there has been major research investment in developing

interactive vegetation models for inclusion in GCMs. Progress has been good and

classification of the quantification and modeling of this mechanism is currently

assessed as being of ‘medium’ quality in Table 25.1. Validation of such models is

a research priority, and when the predictive performance of such interactive

vegetation models has been fully validated (perhaps against remote sensing data)

this assessment will become ‘good’.

3. Effect of transient changes in frozen precipitation cover

There are several physically plausible ways in which changes in frozen precipitation

cover can alter surface energy exchanges and through this modify weather and

climate. For example, the seasonal presence of snow and ice on the ground

generally causes an associated marked change in albedo and radiant energy cap-

ture (see Chapter 5), and the magnitude of this change varies both with time and

with the nature of the vegetation covering the ground. Also, while soils are still

frozen plants cannot extract water from the soil, so there is a resulting inhibition

on transpiration until the soil water melts, and this inhibition may persist into

60

(W m−2) (mm d−1)

4

3

2

1

0.5

−0.5

−1

−2

−3

−4

June June

Latent Heat Flux Precipitation

July July

August August

40

30

20

10

−10

−20

−30

−40

−60

Figure 25.5 Map of differences in monthly average latent heat flux (W m-2) and precipitation (mm d-1) given when a

description of interactive vegetation cover was introduced into the Weather Research and Forecasting (WRF) model

coupled with the Noah land surface model to substitute for prescribed changes in vegetation cover. The modeled domain

covers the contiguous US between 21°N–50°N and 125°W–68°W. (Redrawn from Jiang et al., 2009, published with

permission.) See Plate 9 for a colour version of this image.



spring and early summer. Snow and ice also provide long-term storage of water

above the ground through winter and spring, and when they melt the resulting

water supplements soil moisture and this modifies the surface energy balance in

the subsequent summer season.

There is some direct observational evidence for the above processes.

Figure 25.6a, for example, showed the measured seasonal changes in albedo for

four different vegetation covers in Canada and clearly demonstrates the marked

vegetation-related differences in the effect of snow cover during the boreal winter.

Coniferous trees shed snow from their branches during the winter months,

consequently the albedo for forest is generally much less than that for snow-

covered grassland. Figure 25.6b shows that when the value of albedo for snow was

Figure 25.6 (a) Seasonal changes in albedo for different vegetation covers measured at study sites in the BOREAS

experiment in Canada during 1994. (Adapted from Betts et al., 1996.) (b) Net radiation measured over boreal forest in

Canada in 1996 and modeled net radiation calculated in the ECMWF model assuming an albedo value appropriate for

snow. (Adapted from Betts et al., 1998.) (c) Time series of spring interannual variability of snow cover over a broad region

of the western United States in March, April, and May together with average rainfall measured in the subsequent July and

August at 47 NOAA cooperative reporting stations distributed across the state of New Mexico. (Adapted from Gutzler

and Preston, 1997.)



(wrongly) assigned to the boreal forest in Canada during the snow-covered winter

months in the ECMWF model it calculated values of net radiation that were

substantially less than observed values. Figure 25.6c shows that the snow cover

over a broad region of the western United States in March, April, and May appears

to be anti-correlated with area-average monsoon rainfall measured in the

subsequent July and August in New Mexico, a feature which has been ascribed

to the effect of snow melt on summer soil moisture.

Because the influences of transient changes in frozen precipitation cover are

physically plausible and consistent with observations, this influential mechanism

is assessed as being ‘extremely likely’ in Table 25.1. Over the last decade there has

been significant effort deployed toward improving the representation of frozen

precipitation in SVATS (e.g., Bowling et al., 2003; Luo et al., 2003; Nijssen et al.,

2003; Etchevers et al., 2004; Niu and Yang, 2004; 2006; 2007; Fassnacht et al.,

2006). Because progress has been reasonably good, the classification of the

quantification and modeling of this mechanism is currently assessed as being of

‘medium’ quality in Table 25.1. Adequate representation of area-average behavior

of heterogeneous frozen precipitation cover and its influence on meteorological

feedbacks and hydrological flows remains elusive and merits further research.

4. Combined effect of transient changes

For simplicity and clarity the mechanisms through which land surfaces can

influence climate and weather associated with soil moisture, vegetation cover and

frozen precipitation have been considered separately above. But it is important to

recognize that in practice they are intimately interrelated. This is true in the real

world and should also be true in any well-conceived model of surface processes.

The atmosphere’s ability to access soil moisture is related to vegetation cover

because plants’ roots, stems and leaves serve as an important conduit between

soil and the overlying air. Plants can also intercept precipitation and return it to

the atmosphere before it enters the soil, thus changing the amount of moisture

present in the soil that is available to the atmosphere. Coverings of snow and ice

and associated frozen soil influence the effect of vegetation cover by altering

surface albedo and by inhibiting transpiration from plants when soil is frozen.

Water temporarily stored in frozen form on the surface of soil and vegetation

during winter usually melts several months later and supplements the soil mois-

ture available to transpiring vegetation in the subsequent summer. The evolution

of the snow cover with time depends on what vegetation is present because tall

forests may shade snow, while snow on short grassland remains exposed to the

sun. Similarly, the seasonal cycle in vegetation growth and the extent to which

plants’ controls act to moderate transpiration rate are related to the amount of

moisture in the soil.

Because of the complexity of the regional scale interaction between the

three influential mechanisms associated with soil moisture, vegetation cover

and frozen precipitation, it is perhaps not surprising that all are assessed as

‘extremely likely’ in Table 25.1, and that the present status of understanding and



modeling these mechanisms are all assessed as ‘medium’. In practice it is likely

that future improvements in understanding and modeling will most effectively

be made by comprehensive studies in which models are challenged using a suite

of sustained observations made over several seasons that includes measure-

ments of the variables associated with all three of these influential mechanisms

simultaneously.

C. The influence of imposed persistent changes in land cover

As mentioned earlier in this chapter, the land cover and hydrological behavior of

large areas of the globe have now been altered greatly by human intervention and

continued changes seem inevitable as human population continues to grow.

Because land surface exchange helps determine how the overlying atmosphere

behaves, persistent changes in the nature of land surfaces inevitably have an effect

on weather and climate. Small-scale changes have some effect on weather variables

that are measured near the ground (typically at 2 m) because local surface

influences in part determine these. Larger regional-scale changes in land surface

features can have influence higher in the atmosphere and give rise to regional

or even global scale modifications to atmospheric processes and flows. Such

modifications arise in two main ways, either because the area-average surface

exchanges are modified over a large area, or because development introduces

heterogeneity into land surface exchanges at a spatial scale such as to generate

mesoscale circulations in the atmosphere that may alter the probability and

whereabouts of cloud and precipitation. For simplicity, the influence of persistent

changes in land cover are considered separately below, although in practice all

three influences may well act simultaneously.

1. Effect of imposed land cover change on near surface observations

The values of climate variables measured at (say) 2 m above the ground depend on

vertical profiles that are determined by the land-cover dependent turbulent

transfer of energy, water and momentum fluxes between the surface and the mixed

layer in the Atmospheric Boundary Layer (see Chapter 19 and Fig. 19.6). This fact

is recognized explicitly in the derivation of the Matt-Shuttleworth approach

(Shuttleworth, 2006; Shuttleworth and Wallace, 2009), see the relevant section

in Chapter 23. Observed differences in near-surface climate are easily observed

and have, for example, been reported over small clearings and natural forest in

Amazonia (Figs 25.7a,b,c), and in the form of the ‘urban heat island’ phenomenon

when regions undergo extensive urbanization (Fig. 25.7d).

Because the changes in near-surface climate that are associated with imposed

land cover change are readily understood in terms of present day turbulent

transport theory and are readily observed, the assessment of likelihood for

this influential mechanism is assessed as ‘extremely likely’, and the level of

understanding and modeling as ‘good’ in Table 25.1.



2. Effect of imposed land-cover change on regional-scale climate

The physical basis for expecting modification of regional-scale and perhaps

global-scale climate when there is extensive imposed land-cover change is that

the key properties of the land surface that determine surface exchanges with the

overlying atmosphere such as albedo, surface roughness and vegetation-related

moisture stores and controls will be altered. Since these key properties control

the influence of the land surface on the overlying atmosphere, it is plausible that

there will be some impact on weather and climate if the spatial scale of imposed

land-cover change is sufficiently large. Presumably the impact will be greatest

for dramatic land-use change, such as from forest to agricultural cover or

pastureland.

Collecting ‘before’ and ‘after’ observations of large-scale land-cover change is

problematic so most of the evidence for regional and global-scale modification

of climate in response to large-scale land-use change comes from model studies

using mesoscale meteorological models and GCMs. There is a huge body of

scientific literature in this area of research; example results from a study using a

mesoscale model are shown in Fig. 25.8 and using a GCM in Fig. 25.9. Narisma and

Pitman (2003) used the fifth generation Pennsylvania State University–National

Figure 25.7 Daily variations in near surface (a) air temperature, (b) specific humidity deficit, and (c) wind speed measured

over a small grassland clearing and undisturbed tropical rainforest near Manaus in Amazonia. (Adapted from Bastable

et al., 1993.) (d) Measured change in near-surface temperature in Japanese cities between 1907 and 2007 associated with

the urban heat island. (Adapted from Wikimedia Commons, available at http://commons.wikimedia.org.)



Center for Atmospheric Research Mesoscale Model (MM5) operating with a 50-km

grid spacing in an ensemble simulation of January and July climate using natural

(1788) and current (1988) vegetation cover in Australia. Figure 25.8a shows the areas

where vegetation cover was changed in their model experiment and Figures 25.8b, c

the simulated changes in total rainfall and temperature obtained in January

simulations. Werth and Avissar (2002) quantified the effects of land-cover changes

in Amazonia on local and global climate using the Goddard Institute for Space

Studies Model II global climate model in an ensemble approach with six control

Temperature (January)

(c)

−1 −0.8−0.6−0.4−0.2 0.2 0.4 0.6 0.8 1

115E39S

36S

33S

30S

27S

24S

21S

18S

15S

120E 125E 130E 135E 140E 145E 150E

(a)

42S

39S

36S

33S

30S

27S

24S

21S

18S

15S

12S

115E 120E 125E 130E 135E 140E 145E 150E −3 −1 1 3−0.5 0.5

(b)

39S

36S

33S

30S

27S

24S

21S

18S

15S

115E 120E 125E 130E 135E 140E 145E 150E

Total Rainfall (January)

Figure 25.8 Simulated changes in climate made with the MM5 mesoscale model using natural (1788) and current (1988)

vegetation cover in Australia: (a) areas where vegetation cover was changed; (b) simulated change in total rainfall in January

(in mm); and (c) simulated change in average temperature in January (in °C). (Redrawn from Narisma and Pitman, 2003,

published with permission.) See Plate 10 for a colour version of these image.



simulations compared with six deforested simulations. They found the effect of

deforestation was strong in Amazonia itself, with reductions in precipitation,

evapotranspiration and cloudiness. But they also detected a noticeable impact in

several other regions of the world, some of which show a reduction in rainy season

precipitation: the 8-year average modeled precipitation at other sites around the

world is shown in Fig. 25.9 for a forested and deforested Amazon region.

Because it is physically plausible that extensive imposed land-cover change will

influence weather and climate at regional scale and because there is an extensive

literature of modeling studies that suggest this influence does occur, the mecha-

nism is assessed as ‘very likely’ in Table 25.1. There is some consistency in these

numerous modeling studies, but there is also significant numerical variability in

the magnitude and to some extent the nature of the predicted changes. The level of

understanding and modeling of the influence is assessed as being ‘medium’.

3. Effect of imposed heterogeneity in land cover

It is plausible that imposed land-cover heterogeneity creates spatial differences in

land surface energy balance. If present at an appropriate scale (i.e., in patches of a

few 100s of square kilometers or with linear dimensions of a few 10s of kilometers,

rather than patch-scale), such heterogeneity can generate organized patterns of

buoyancy in the ABL giving mesoscale atmospheric circulations that influence

Figure 25.9 8-year average modelled precipitation at sites in North and South America (shown as rectangles on the map)

calculated for a forested and deforested region in Amazonia by the Goddard Institute for Space Studies Model II GCM

when used in an ensemble approach in which six control simulations were compared with six deforested simulations.

(Adapted from Werth and Avissar, 2002.)



cloud generation and potentially precipitation. Sometimes (over flat regions) the

effect of organized circulations linked to patterns in surface vegetation becomes

visible in the form of boundary layer cloud cover, thus confirming that such a

mechanism does indeed exist. However, most of the evidence for the existence

of mesoscale circulation stimulated by heterogeneity in surface cover is derived

from mesoscale modeling studies. Early numerical studies (e.g., Avissar and Liu, 1996)

Uniform fluxes(average of observed)

Heterogeneous fluxes(observed)

Vertical ascent (m s−1) in RAMS model

GOES-8 Visible Image

Atmospheric responseImposed surface heterogeneity

60 km(a)

(c)

(b)

30 km

30 km

15 km

−0.25 0 0.25 0.75 1 1.25 1.50.5

Figure 25.10 (a) Patterns of imposed heterogeneity of dry and wet surfaces imposed in the Regional Atmospheric

Modeling System (RAMS) during a 12 hour simulation beginning at 06:00 local time on July 28, 1989 and (b) accumulated

precipitation in millimeters calculated by RAMS between 06:00 and 18:00 on this day with these patterns of surface

wetness. (Redrawn from Avissar and Liu, 1996, published with permission.) (c) Simulated horizontal distribution in vertical

wind speed at 1117 m across the ARM-CART site calculated by RAMS at 15:30 on July 13, 1995 when surface sensible and

latent heat fluxes are set to the average values across the site (left figure), and when these are set to the measured

distribution of surface sensible and latent heat fluxes (center figure), and the cloud cover shown in the GOES-8 satellite

visible image at 15:15 on this day. (Redrawn from Weaver and Avissar, 2001, published with permission.)


Tabl

e 25

.1 A

sses

smen

t o

f th

e cr

edib

ilit

y th

at s

pec

ifie

d l

and

-atm

osp

her

e co

up

lin

g m

ech

anis

ms

act

to i

nfl

uen

ce w

eath

er a

nd

cli

mat

e m

ade

on

th

e b

asis

of

ph

ysic

al p

lau

sib

ilit

y an

d t

he

avai

lab

ilit

y o

f o

bse

rvat

ion

al a

nd

mo

del

ing

evid

ence

(o

verv

iew

ed i

n t

he

text

) to

geth

er w

ith

an

ass

essm

ent

of

the

curr

ent

leve

l of

un

der

stan

din

g o

f ea

ch m

ech

anis

m a

nd

pre

sen

t d

ay a

bil

ity

to r

epre

sen

t it

in

mo

del

s.

Land

-sur

face

infl

uenc

e on

clim

ate

or w

eath

er

Plau

sibl

e ph

ysic

al b

asis

fo

r th

e in

flue

nce

Obs

erva

tion

al

evid

ence

for

th

e in

flue

nce

Mod

elin

g ev

iden

ce

for

the

infl

uenc

e

Cred

ibili

ty

of in

flue

ntia

l m

echa

nism

*

Qua

ntif

icat

ion

and

mod

elin

g of

infl

uenc

e (G

ood,

Med

ium

, or

Poo

r)

A.

Infl

uen

ce o

f ex

isti

ng

la

nd

-atm

osp

her

e in

tera

ctio

ns

1. I

nflu

ence

of t

opog

raph

yYe

sYe

sYe

sEx

trem

ely

likel

yLo

ng-t

erm

: Med

ium

2. C

ontr

ibut

ion

to a

tmos

pher

ic w

ater

av

aila

bilit

y (“

recy

clin

g”)

Yes

Yes

Yes

Extr

emel

y lik

ely

Shor

t-te

rm: P

oor

Med

ium

B.

Infl

uen

ce o

f tr

ansi

ent

chan

ges

in

lan

d s

urf

aces

1. T

rans

ient

cha

nges

in s

oil m

oist

ure:

a.

Reg

iona

l sca

le in

fluen

ce o

n cl

imat

eYe

sYe

sYe

sEx

trem

ely

likel

yM

ediu

m

b.

Mes

osca

le in

fluen

ce o

n w

eath

erYe

sYe

sYe

sLi

kely

Med

ium

2. T

rans

ient

cha

nges

in v

eget

atio

nYe

sYe

sYe

sEx

trem

ely

likel

yM

ediu

m3.

Tra

nsie

nt c

hang

es in

froz

en p

reci

pita

tion

cove

rYe

sYe

sYe

sEx

trem

ely

likel

yM

ediu

m

(N

ote:

in p

ract

ice

thes

e in

fluen

ces

are

stro

ngly

cou

pled

)C

. In

flu

ence

of

imp

ose

d c

han

ges

in

lan

d c

ove

r1.

Loc

al e

ffect

on

2 m

clim

ate

Yes

Yes

Yes

Extr

emel

y lik

ely

Goo

d2.

Effe

ct o

f reg

iona

l-sca

le c

hang

es

in la

nd c

over

Yes

Som

eYe

sVe

ry li

kely

Med

ium

3. E

ffect

of i

mpo

sed

land

-cov

er

hete

roge

neity

Ye

s

Yes

Ye

s

Extr

emel

y lik

ely

Med

ium

* Ex

trem

ely

likel

y >

95%

; Ver

y lik

ely

> 9

0%; L

ikel

y >

66%

; Mor

e lik

ely

than

not

> 5

0%; U

nlik

ely

< 3

3%; V

ery

Unl

ikel

y <

10%

; Ext

rem

ely

unlik

ely

< 5

%



investigated the phenomenon with artificially imposed patterns of surface heating

and demonstrated modeled mesoscale circulation linked to precipitation, see

Fig. 25.10. Later studies (e.g., Weaver and Avissar, 2001) sought validation of such

circulations and their consequences with reference to observations, see Fig. 25.10.

Some attempts have been made to parameterize the effect of land surface hetero-

geneity on ABL turbulence (e.g., Liu et al., 1999) but, at this writing, this is rarely

if ever done in GCMs, perhaps because other surface features such as topography

and soil moisture heterogeneity can also give rise to mesoscale circulations.

Given the plausibility that mesoscale circulations will arise in response to land

surface heterogeneity circulations, and that these can be simulated in mesoscale

meteorological models and associated boundary layer cloud patterns have some-

times been observed, this mechanism is considered ‘extremely likely’ in Table 25.1.

Assessing the level of understanding and modeling of this influential mechanism

is complicated by the fact that the phenomenon is readily represented in mes-

oscale meteorological models suggesting an assessment of ‘good’, but they are not

yet represented in GCMs suggesting an assessment of ‘poor’. As a compromise, in

Table 25.1 the assessment given is ‘medium’.


● Land surfaces do matter: because a continental influence is evident in

global-scale atmospheric general circulation, see Chapter 9.

● Review: a critical review of the available literature in three general areas,

i.e.,(a) the influence of existing land-atmosphere interactions; (b) the

influence of transient changes in land surfaces; and (c) the influence of

imposed persistent changes in land cover indicates the following conclusions

which are summarized in Table 25.1:

— The credibility of all the land surface influences on the atmosphere

considered is assessed as being ‘extremely likely’ or ‘very likely’, except

in one case when it is considered ‘likely’.

— Present ability to quantify the magnitude and model all these

influences is assessed as ‘medium’, except in one case it is assessed as

‘high’ and in one case as ‘poor’.

— The influence of soil moisture, vegetation cover, and frozen

precipitation cannot be separately modeled. Future improvements will

most effectively be made by studies in which sustained observations

are made over several seasons and include measurements associated

with all three of these influential mechanisms simultaneously.

● Recommendations: areas that priorities in future research are as follows:

— Quantification and modeling of the effect of topography on weather

and climate at short time scales is assessed as being poor and there is a

clear need for more research in this area.



— Further research is justified in the area of precipitation recycling, but

with focus on achieving greater realism and accuracy when modeling

surface evaporation and, in particular, the atmospheric mechanisms

involved in releasing precipitation.

— To give improved understanding and modeling of the soil moisture

feedback mechanism, research that exploits the emerging capability to

make area-average soil moisture measurement techniques is a priority.

— Research is required to fully validate recent interactive vegetation

models, perhaps using remote sensing data.

— Further research is needed to provide adequate representation of the

area-average behavior of heterogeneous frozen precipitation cover

because modeling its influence on meteorological feedbacks and

hydrological flows remains elusive.

— Research is needed to resolve uncertainty in the nature of, and to

reduce the numerical variability in the magnitude of, the predicted

changes associated with large-scale changes in land cover.

— Investigation is required to assess the importance of mesoscale circula-

tions generated in response to land cover heterogeneity relative to those

generated in response to topography and soil moisture heterogeneity,

and mechanisms sought to include their parameterization in GCMs.

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Introduction

This chapter gives example questions based on the text, together with example

answers. The purpose is to allow students opportunity to gain greater insight and

experience in applying their understanding. Some questions seek numerical

answers while others provide opportunity to express opinions. In the latter case,

example opinions are given in the answers, but these are not necessarily unique

and students are encouraged to propose alternative or additional opinions and to

discuss these with their instructor. In these questions the time within a day is given

in local time expressed in military-time format, i.e., 6:00 AM is written as 06:00

and 3:15 PM as 15:15.

Example questions

Question 1 (Uses understanding and equations from Chapters 2 and 3.)

At 14:00 on June 25 just above the ground near the desert floor about 60 miles west

of Tucson, at an altitude of 3700 ft, the temperature and pressure of the air are

114°F and 29.8 inches (of mercury), respectively, and the relative humidity is 25%.

(a) What are the air temperature in °C and K, the air pressure in mb and in

kPa, the saturated vapor pressure at air temperature in kPa, the vapor

pressure in kPa, the specific humidity in kg kg−1 and Ra, the gas constant,

for the moist air in J kg−1 K?

26 Example Questions and Answers




Example Questions and Answers 405

(b) A hydrometeorologist is making measurements at 7080 ft at the nearby Kitt

Peak Observatory. Neglecting any small changes in specific humidity

between the desert floor and the top of Kitt Peak (and hence in the gas

constant for moist air) and assuming the lapse rate in the lower atmosphere

is that of the US Standard Atmosphere, estimate what she measures for the

air temperature in K and the air pressure in kPa.

(c) She decides to boil water to make coffee. Water boils when its saturated

vapor pressure equals air pressure. Calculate the temperature in °C at which

she finds her water boils. (Hint: compare with the calculation of dew point.)

Assume parcels of air that are warmed by the surface are 5°C warmer than the

surrounding ambient air but have the same vapor pressure.

(d) At what temperature will these parcels saturate? Assuming the air parcels

rise and cool at the adiabatic lapse rate, at what height above the desert floor

will they saturate? At approximately what height do the warmed parcels

lose relative buoyancy? Was there convective cloud on this day? Why?

(Assume 0°C = 273.15 K; 1 inch = 2.54 cm; 1013.3 mb = 30.006 inches of mercury;

cp = 1010 J kg−1 K−1; and the gas constant for moist air R

a = 286.5(1+0.61q) J kg−1 K−1).

Question 2 (Uses understanding and equations from Chapters 2 and 5.)

Assume that at the top of the atmosphere the instantaneous incoming flux of solar

radiation, Stop, can be computed in W m−2 from:

( ). .cos( ) . . sin sin cos cos costopo r o rS S d S d= = +q f d f d w (26.1)

where So is the solar constant ( = 1367 W m−2) ; d

r is eccentricity factor of the

Earth’s orbit (no units); f is the latitude of the site in radians; d is the solar

declination in radians; and w is the hour angle in radians. This equation is implicit

in Equations (5.14) and (5.15). When Equation (26.1) computes a negative value

for Stop the Sun is below the horizon and the true value is zero. The variables dr and

d are functions of the day of the year, and w is a function of the hour, t, within the

day in local time. (Definitions of dr, d and w are given in Chapter 5). Equation

(5.16), which is called the Brunt Equation, is normally used to estimate the all-day

average solar radiation reaching the ground from the all-day average value at the

top of the atmosphere. However, for the purpose of this question the Brunt

Equation is also assumed to apply when calculating Sgrnd, the instantaneous flux of

solar radiation reaching the ground, hence Sgrnd is given by:

Sgrnd=[as+(1-c).b

s]Stop (26.2)



where c is the fractional cloud cover and as and b

s empirical constants here assumed

equal to their typical all-day average values, i.e., as = 0.25 and b

s = 0.5.

Develop a spreadsheet to make the calculations in sections 2(a) to 2(e) below

and sections (g) and (h) then reduce to applying this spreadsheet in different

conditions. The spreadsheet should set the value of Stop to zero for hours when

Equation (26.1) gives a negative value and the Sun is below the horizon. Set up the

spreadsheet to also calculate the daily average values of solar, net solar, net

longwave and net radiation.

On July 13 at an arid site 32.5°N of the equator the measured all-day average

values of air temperature and relative humidity are 71.6 °F and 50 %, respectively.

(a) What are the equivalent all-day average values of air temperature in K, the

saturated vapor pressure in kPa, and the vapor pressure in kPa on this day.

(b) Assuming the cloud factor c = 0.7 all day, estimate the all-day average net

longwave radiation in W m−2 (giving results of intermediate calculations) at

this arid site and recalling that the Stefan-Boltzmann constant is 5.67 × 10−8

W m−2 K−4.

(c) Still assuming the cloud factor c = 0.7 all day, now estimate the all-day

average net longwave radiation in W m−2 (giving results of intermediate

calculations) had this been assumed to be a humid site.

(d) Still assuming the cloud factor c = 0.7 all day and also that the albedo at this

site is equal to 0.23 and is constant through the day and that the net

longwave radiation flux is also constant all day. At hourly intervals between

05:00 and 23.30 hours, calculate and plot the incoming solar radiation, net

solar radiation, and the net radiation fluxes assuming first that this is an

arid site, and second, a humid site.

(e) Calculate the all-day average values of the incoming solar radiation, net

solar radiation and net radiation assuming first that this is an arid site, and

second, a humid site.

You have now created a spreadsheet which you can use to make estimates of solar,

net solar, longwave and net radiation at any latitude, for any day of the year, in dif-

ferent cloud cover conditions and for different types of land cover, as characterized

by their albedo. Using this spreadsheet make the following investigations. You will

need to make appropriate selections for albedo from Table 5.1.

(f) Explore the effect of seasonality by making calculations and plotting graphs

for a humid grassland site near Saskatoon, Canada at 55°N on January 15

when the air temperature is 33°F, and on July 15 when the air temperature

is 90°F. For simplicity, assume c = 0.6 all day and the relative humidity is

80% on both days.

(g) Explore the effect of deforestation on the surface radiation balance by

making calculations and plotting graphs for a humid site near Manaus,

Brazil on March 23 with forest cover and pasture cover. Assume a

temperature of 90°F, a relative humidity of 85%, and a cloud cover of 70%.



Question 3 (Uses understanding and equations from Chapters 1, 4, 6, and 7.)

A farmer has a copy of Terrestrial Hydrometeorology and therefore has

wide-ranging knowledge of the subject. He has a field that is currently bare soil

near Casa Grande, Arizona which is 32.5°N of the equator. This is the arid site for

which you calculated values of net radiation in 2(d). He recently irrigated the field

prior to planting and the sandy soil is close to saturated. He decides to measure the

evaporation loss from the field but he only has three thermometers and a set of tall

stepladders with which to do so. He installs one thermometer in the soil to measure

the temperature very close to the surface of the soil. He wraps the mercury bulb of

a second thermometer in a small piece of cloth that he is careful to keep moist so

that it measures wet bulb temperature. The third thermometer he uses as a dry

bulb thermometer to measure air temperature.

Starting at midnight on July 13 he measures the wet bulb and dry bulb

temperature 0.5 m above the ground and then quickly runs up the stepladder and

makes the same measurements at 3.0 m from the ground. Every 5 minutes he

repeats this operation throughout the next 24 hours. In his spare time he monitors

the thermometer in the soil and notices that the minimum temperature of 20°C

occurs at 01:00 and the maximum temperature of 24°C occurs at 13:00. He also

monitors the sky and decides that the fractional cloud cover is 0.7 and fairly

constant all day. He computes the hourly-average values of wet and dry bulb

temperature at the top and bottom of the stepladder given in Table 26.1.

Having read Chapter 6 in Terrestrial Hydrometeorology, the farmer realizes that if

he assumes the soil is uniform with depth and the diurnal cycle in soil surface

Table 26.1 Values of hourly average dry and wet bulb temperatures measured by the farmer in question 3.

Bottom Top Bottom Top

Time (hour)

Dry bulb (°C)

Wet bulb (°C)

Dry bulb (°C)

Wet bulb (°C)

Time (hour)

Dry bulb (°C)

Wet bulb (°C)

Dry bulb (°C)

Wet bulb (°C)

0.5 16.786 11.714 15.654 11.177 12.5 33.139 21.096 28.347 16.7031.5 14.337 11.026 14.609 11.117 13.5 34.790 21.547 29.391 17.0432.5 12.482 10.157 14.069 10.948 14.5 34.820 21.365 29.932 17.0933.5 9.963 9.146 14.068 10.948 15.5 34.600 20.918 29.931 16.8884.5 8.838 8.482 14.609 11.052 16.5 33.255 20.360 29.391 16.7335.5 13.222 10.473 15.653 11.563 17.5 30.856 18.978 28.346 16.2826.5 17.093 13.081 17.130 11.996 18.5 27.105 17.012 26.870 15.7697.5 20.394 15.325 18.938 12.808 19.5 25.255 15.567 25.061 15.1258.5 23.451 17.181 20.956 13.978 20.5 24.786 15.274 23.043 14.3949.5 26.654 18.610 23.044 14.851 21.5 22.795 14.562 20.955 13.44510.5 29.122 19.718 25.062 15.733 22.5 20.439 13.644 18.938 12.68511.5 31.598 20.625 26.870 16.306 23.5 18.632 12.692 17.129 11.870



temperature is sinusoidal, and if he chooses the form of the sinusoidal wave to agree

with the timing and magnitude of the minimum and maximum soil temperatures

that he measured and also selects values of soil properties appropriate for the moist

sandy soil, he can calculate the soil heat flux at any time during the day.

(a) What were the values of the mean soil surface temperature, and the

amplitude and the time slip of the cycle in soil surface temperature that he

selected?

(b) What were the values of soil thermal conductivity, ks, and thermal

diffusivity, αs, that he selected, what was the value of damping depth, D, for

the daily time period (expressed in seconds) that he calculated. Which

equation from Terrestrial Hydrometeorology did he use to calculate the

instantaneous surface soil heat flux?

The farmer assumes that the psychrometric constant is 0.0667 kPa K−1 when

applying the wet-bulb equation and when calculating the Bowen ratio. For

simplicity he also assumes that the difference in virtual potential temperature is

equal to the difference in measured air temperature between the two levels. (This

is a common assumption when calculating Bowen ratio). In the course of his

calculations he found that the all-day average air temperature and vapor pressure

at the bottom level were the same as those you calculated and used in question

2(a). Using these values with the day of the year and latitude of the site he was able

to calculate the same estimates of net radiation for this arid site that you calculated

in question 2(d). You can therefore adopt those values of hourly net radiation for

use in this question.

(c) Develop a spreadsheet to tabulate the values of vapor pressure at the bot-

tom level, vapor pressure at the top level, Bowen ratio, net radiation [copied

from 2(e)], soil heat flux, available energy, latent heat flux and sensible heat

flux at hourly intervals between 0.5 and 23.5 hours.

(d) Plot the calculated net radiation, soil heat flux, available energy, latent heat

flux and sensible heat flux as a function of time through the day.

(e) What were the all-day average values of the Bowen ratio and Evaporative

Fraction at his site on this day?

(f) Suppose the farmer had chosen to neglect soil heat flux in his calculation of

available energy. Without recalculating all the rates, can you suggest

whether he would have overestimated or underestimated the all-day

average evaporative fraction and explain why?

Question 4 (Uses understanding from Chapters 1, 2, and 8.)

(a) Shuttleworth says, ‘As an annual-average, the value is about 1.2 m. However

we, as land dwellers, see only about 10% of this, and we lose almost two-

thirds back to the atmosphere. We keep an even smaller proportion



in Arizona.’ In your opinion, what is Shuttleworth talking about? The

annual-average of what is about 1.2 m? How do we lose two-thirds back to

the atmosphere? Approximately what proportion do we keep in Arizona?

(b) Shuttleworth says, ‘These two components of these models have jargon

names and are run alternately. The first applies the conservation laws while

the second, by representing relevant processes, changes the divergence

terms in these laws prior to their next application.’ In your opinion, what

components of what models is he talking about and what are their jargon

names. Which ‘laws’ does he refer to? Can you suggest some of the relevant

processes that change the divergence terms in these laws?

(c) Shuttleworth says, ‘Of course, if all the continents were constrained to be in

the tropics then, as a global average, the proportion of the Sun’s radiant

energy reflected at the Earth’s surface would vary less between summer and

winter.’ In your opinion, what is at least one reason why Shuttleworth might

be correct?

(d) Shuttleworth says, ‘Most of the time the temperature gradient in the lower

atmosphere is less than the dry adiabatic lapse rate. Water vapor is also

strongly concentrated at the bottom of the atmosphere. Presumably, the

same processes are responsible for both of these phenomena.’ In your

opinion, could Shuttleworth’s presumption be correct? What process or

processes might simultaneously reduce the actual lapse rate below the

adiabatic rate and also reduce the vapor content of the atmosphere at levels

well above the ground?

(e) Shuttleworth says, ‘These models are used in three main ways, each with a

different objective. However, in fact, one application was a by-product of

the original model application. “Initiation” is a keyword in all of these

applications.’ In your opinion, now what is Shuttleworth talking about?

What models? What are the three different objectives? Can you suggest

why he puts emphasis on model initiation?

(f) Shuttleworth says, ‘The specific heat is 4 times bigger and the density is

nearly 1000 times bigger. If this wasn’t true, we might have http://www.

weather.gov/ but we probably would not have http://www.cpc.ncep.noaa.

gov/’ In your opinion, what has a specific heat and density respectively 4

and 1000 times bigger than what? If this were not the case, can you explain

why in your opinion this might mean that http://www.cpc.ncep.noaa.gov/

would not be needed but http://www.weather.gov/ likely still would be?

(g) Shuttleworth says, ‘One important potential consequence of ‘greenhouse

warming’ is that it will enhance the hydrological cycle. It is interesting that

non-linearity in the basic relationship that would cause this enhancement

tends to compensate for the projected warming being twice as large at the

poles than at the equator.’ In your opinion, what does Shuttleworth mean

by this? Can you suggest what basic relationship might allow greenhouse

warming to enhance the hydrological cycle? Why might this relationship

be more effective at the equator, thus compensating for the potentially

enhanced warming at the poles?



Question 5 (Uses understanding from Chapter 9.)

The planet Malleable is fascinating. In many respects it is identical to the Earth. It

has identical dimensions and is located in a solar system identical to ours. It rotates

around an identical sun, in an identical orbit, and its solar declination changes

seasonally as does the Earth’s. Moreover, on average, the relative area of oceans and

continents is the same as that on Earth. The planet is the adopted home to an

advanced civilization that can manipulate the location of the continents on

Malleable’s surface. When it was settled, the ‘Founding Fathers’ of Malleable chose

to distribute these continents as shown in Fig. 26.1.

The planet Malleable is governed by a single planetary government. It is election

year and three main parties are seeking election. They are as follows.

The Reduce Warm Deserts (RWD) Party, whose platform is to redefine the

continents so as to reduce the non-productive continental areas that are deserts

on planet Malleable.

The Reduce Tropical Storms (RTS) Party, whose platform is to redefine the conti-

nents so as to reduce the ‘seed areas’ for tropical storms on planet Malleable.

The Maximize Monsoons (MM) Party, the central theme of whose platform is to

enhance seasonal precipitation in subtropical regions and thus allow the

production of additional seasonal crops on planet Malleable.

Each political party has devised a banner that expresses their ideas symbolically by

approximately representing the new continental distributions they are respectively

suggesting. As the most effective printer on Malleable, you have been awarded the

Current continental arrangement

Banner A

Banner B

Banner C

Figure 26.1 Current

continental arrangement on

planet Malleable and the three

banners used by three political

parties in the election.



contract to print these banners. Unfortunately the three party symbols have

arrived at your printing works without you knowing which symbol belongs to

which party. On the basis of your understanding derived from Terrestrial

Hydrometeorology, you must choose the most appropriate symbol for each party’s

banner. These symbols are also shown in Fig. 26.1.

(a) In your opinion, which banner (A, B, or C) most likely represents the

continental distribution advocated by the RWD Party and briefly explain

what you think is the basis for them suggesting this particular continental

distribution.

(b) In your opinion, which banner (A, B, or C) most likely represents the

continental distribution advocated by the RTS Party and briefly explain


distribution.

(c) In your opinion, which banner (A, B, or C) most likely represents the

continental distribution advocated by the MM Party and briefly explain


distribution.

Late in the election campaign, a fourth party, the Reduce El Niño (REN) Party,

emerges. Their objective is to seek to reduce the severity of fluctuations in climate

associated with building unstable ‘warm pools’ in Malleable’s tropical oceans.

Their hope is to gain a share of power by forming a coalition with one of the other

parties after the election. They see most opportunity of making a deal with either

the RWD or the RTS parties and have opened secret discussions with these two

parties before the election.

(d) In your opinion, how might the RWD Party be arguing for the support of

the REN Party after the election?

(e) In your opinion, how might the RTS Party be arguing for the support of the

REN Party after the election?

Question 6 (Uses understanding from Chapters 9, 10 and 11.)

Briefly answer the following.

(a) In your opinion, what is the fundamental cause of the difference between

the thermal structure of the oceans and the atmosphere? What is your

opinion on the consequence of the above phenomenon on the vertical

structure of the oceans, and say how this changes with latitude and

season.

(b) A student said, ‘Ocean currents tend to go north on the eastern sides of

continents, and south on the western side of continents’. In your opinion, is

the student correct? Why?



(c) In your opinion, why do ocean currents tend to behave this way?

(d) Discuss the statement, ‘The geographical distribution of land masses

influences the effect of ocean circulation on tropical Sea Surface

Temperature’ in the context of the Atlantic Ocean, and give your opinion

on any consequences on the relative frequency of tropical storms in Cuba

and in northeast Brazil.

(e) A student said, ‘The hydroclimatic mechanism which most influences the

food supply of half the world’s population is related to difference in the way

surface radiation is shared for continents and oceans.’ In your opinion,

what did she mean?

(f) For clouds to occur in the atmosphere a mechanism which gives rising air

and therefore cooling air is required. In your opinion, what are two other

requirements and which of them is most likely to be the limiting

requirement?

(g) If a parcel of air is moister than its surroundings but it has the same

temperature and pressure, in your opinion will it tend to rise or will it tend

to fall? Briefly explain why.

(h) In convective conditions parcels of air are heated and start to rise because

they are warmer and lighter. As soon as a parcel rises the air cools. In your

opinion why does this cooling not necessarily stop the air parcel rising to

the cloud condensation level?

(i) Once the cloud condensation level is reached, cloud formation begins. Give

your opinion on what effect the condensation process will have on the

buoyancy of the parcel of air and its further ascent in the cloud.

(j) In a particular mid-latitude cloud, the air temperature is -25°C. In your

opinion, which phases of water (solid, liquid or vapor) are likely to be

present in the cloud, and what is likely to be the most important physical

mechanism giving ice particle growth in the cloud.

Question 7 (Uses understanding from Chapters 12, 13 and 14.)

(a) Draw a diagram of what in your opinion is the ideal site and mounting for

a rain gauge. It should show its relation to surrounding objects and to the

ground. Give a brief explanation of why you consider your design to be

good. Discuss this design with your instructor.

(b) Obtain values of mean monthly precipitation for a site that interests you

(e.g., your home town, state, or country). Compute the Seasonality Index

from these data using the formulae given in your class notes (or an

alternative measure of seasonality if you prefer; there are alternative

measures.) Comment on what this index implies about the seasonality of

the precipitation climate that you choose.

(c) Using the same data you used in question 7(b), draw a ‘pie’ diagram to

illustrate the seasonal behavior of the rainfall showing the percentage

contributions to the annual rainfall in each month.



(d) During a storm the chart from a siphon rain gauge produced the (simu-

lated) form illustrated in Fig. 26.2. This was digitized to give the numerical

time sequence in Table 26.2. Plot the mass curve for this particular storm

and, on the basis of this mass curve, speculate as to whether the chart was

most likely to be for a frontal or convective storm and explain your

reasoning.

(e) The July rainfall amounts in Tanzania over the period 1931–1960 are given

in Table 13.1. Calculate the mean value and estimate the median value of

the July rainfall in Tanzania between 1931 and 1960. If you find they differ,

say why.

10

8

6

4

2

00 20 40 60

Time (minutes)

80 100 120

Rai

nfal

l (m

m)Figure 26.2 A (simulated) chart

of precipitation for a storm

measured using a siphon rain

gauge. Note that once the

chamber reaches a storage that is

equivalent to 10 mm of rainfall,

the chamber is siphoned empty

and then continues to refill as the

storm proceeds.

Table 26.2 Digitized form of a chart measured using a siphon rain gauge illustrated in

Figure 12.2 and used in question 7(d).

Time (minutes)

Gauge reading (mm)

Time (minutes)

Gauge reading (mm)

Time (minutes)

Gauge reading (mm)

0.00 2.62 31.40 10.00 60.00 7.495.00 5.25 31.40 0.00 65.00 9.27

10.00 7.76 35.00 9.48 68.69 10.0011.85 10.00 35.19 10.00 68.69 0.0011.85 0.00 35.19 0.00 70.00 0.2615.00 3.82 39.02 10.00 75.00 1.2118.38 10.00 39.02 0.00 80.00 1.8218.38 0.00 40.00 2.50 85.00 2.5520.00 2.97 43.14 10.00 90.00 2.6823.45 10.00 43.14 0.00 95.00 3.3223.45 0.00 45.00 4.02 100.00 3.8625.00 3.16 48.70 10.00 105.00 4.8327.60 10.00 48.70 0.00 110.00 5.9627.60 0.00 50.00 2.11 115.00 6.2930.00 6.30 55.00 4.52 120.00 6.72



(f) Compute and plot the time variations in the 7-year running mean for July

Tanzanian rainfall data between 1934 and 1957.

(g) Compute and plot the mass curve for July Tanzanian rainfall data between

1931 and 1960.

(h) Compute and plot the cumulative deviation for July Tanzanian rainfall data

between 1931 and 1960.

(i) A farmer owns the field illustrated in Fig. 26.3 which is 6 km by 4 km. He

has access to the data from three rain gauges which are located at P1, P

2, and

P3 in this diagram. In April these three gauges measure 16, 8, and 7 mm of

rainfall, respectively, in May they measure 26, 34, and 43 mm of rainfall,

respectively, and in June they measure 51, 44, and 37 mm of rainfall,

respectively. He decides to estimate the area-average rainfall for his field by

using the Reciprocal-Distance-Squared to estimate rainfall estimates at the

center of each square kilometer of his field (i.e., at the points shown), and

then averaging these values. What were the area-average precipitation

values he calculated for April, May, and June?

Question 8 (Uses understanding and equations from Chapters 16, 17, and 18.)

(a) Starting from Equation (16.46), i.e., the basic equation for conservation of

water vapor in the atmosphere, by analogy with the derivation given in

Chapter 17 for vertical velocity in your class notes or otherwise, derive

2−2

−2

4

4

(Distance in km)

6 8

2

P1

P2

P3

(−1.0, −1.5)

(2.5, 5.0)

(7.0, 2.5)

Field

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

(Dis

tanc

e in

km

)

Figure 26.3 The field for which

area-average precipitation is to

be calculated, and the three rain

gauge positions P1, P2, and P3 at

which the gauges are located

from which calculations are to be

made in question 7(i).



Equation (26.3), the prognostic equation describing for water vapor

fluctuations in the atmosphere:

( ) ( ) ( )

2

' ' ' ' ' '

qq

a a

Sq q q q Eu v w qt x y z

u q v q w qx y z

∂ ∂ ∂ ∂+ + + = + + ∇

∂ ∂ ∂ ∂⎛ ⎞∂ ∂ ∂

− + +⎜ ⎟∂ ∂ ∂⎝ ⎠

ur r

(26.3)

explaining each step in the derivation as you do so.

The prognostic equations describing the conservation of mean potential tem-

perature has a similar form, thus:

( ) ( ) ( )

2

' ' ' ' ' '

n

a p a p

R Eu v wt x y z c c

u v wx y z

θ

∇∂θ ∂θ ∂θ ∂θ+ + + = − − + υ ∇ θ

∂ ∂ ∂ ∂⎛ ⎞∂ θ ∂ θ ∂ θ

− + +⎜ ⎟∂ ∂ ∂⎝ ⎠

r r

(26.4)

Give the terms that become negligible in Equations (26.3) and (26.4) when the

following assumptions are made.

(b) There is no mechanism for creating water vapor chemically in the

atmosphere.

(c) There is no phase change between water vapor and liquid/solid water.

(d) There is no horizontal or vertical change in the net radiation flux in the

ABL.

(e) Molecular diffusion can be neglected.

(f) There is no ascent or subsidence (i.e. no persistent rising or sinking of the

air).

(g) There is no horizontal divergence of turbulent fluxes.

(h) There is no horizontal advection of humidity or potential temperature.

(i) After making all of the above simplifying assumptions, write down the

(much simpler) versions of Equations (26.3) and (26.4) which then apply.

Figure 26.4 sketches the simplified average height dependence of the sensible heat

flux H(z) and the moisture flux E(z) through the daytime atmospheric boundary

layer over uniform terrain in cloudless conditions when there is no subsidence. It

labels five levels different levels (i), (ii), (iii), (iv), and (v). You are only allowed to

choose between the following three options for changes in air temperature – be

warmer, be cooler, or change little; and you are only allowed to choose between the

following three options for changes in atmospheric humidity – be wetter, be drier,

or change little. On the basis of the answer to question (i), say how the temperature

and humidity will change over the next few minutes at the levels given below.



(j) How will the temperature and humidity change at level (i)?

(k) How will the temperature and humidity change at level (ii)?

(l) How will the temperature and humidity change at level (iii)?

(m) How will the temperature and humidity change at level (iv)?

(n) How will the temperature and humidity change at level (v)?

Question 9 (Uses understanding and equations from Chapters 2, 21, 22 and 24.)

The molecular diffusion coefficients are υ = 1.33 × 10−5 (1+0.007T) m2 s−1,

DH = 1.89 × 10−5 (1+0.007T) m2 s−1, D

V = 2.12 × 10−5 (1+0.007T) m2 s−1, and D

C =

1.29 × 10−5 (1+0.007T) m2 s−1, see Chapter 21.

Assume that the aerodynamic interactions of the leaves on deciduous trees can

be approximated by those of a circular flat plate with a diameter of 5 cm while

those of evergreen conifers can be represented by the aerodynamic interactions of

cylinders of diameter 2.5 mm. The boundary-layer resistance to heat transfer per

unit surface area of each vegetation element (i.e., leaf or needle) is estimated by

Equation (21.9). The in-canopy wind speed, U, is 0.5 m s−1 and the in-canopy

temperature is 20°C. By first calculating the Reynolds number from Re = (Ud)/n,

where d is a characteristic dimension of the leaf or needle (in this case the

diameter), and then by selecting the relevant empirical equation for the Nusselt

number, Nu, from Table (21.1), use Equation (21.9) to estimate the boundary-

layer resistance per unit area for heat transfer for:

(a) individual deciduous leaves

(b) individual coniferous needles

Assume the transfer from individual vegetative elements is always by forced con-

vection so that the relative transfer resistances for other exchanges is determined

Mixed layer

Surface layer

Entrainment layer

Free atmosphere(v)

(iv)

(iii)

(ii)

(i)

H(z) E(z)

Figure 26.4 The simplified

average height dependence of the

sensible heat flux H(z) and the

moisture flux E(z) through the

daytime atmospheric boundary

layer over uniform terrain in

cloudless conditions when there

is no subsidence.



only by their relative diffusion coefficients, see Equations (21.10) and (21.11).

From the answer to Question 9(b), estimate the boundary-layer resistance for:

(c) vapor transfer for coniferous needles;

(d) carbon dioxide transfer for coniferous needles.

(e) Equations (22.2), (22.3) and (22.4), approximately describe how the zero

plane displacement, d, and aerodynamic roughness length, zo, of a vegeta-

tion stand vary relative to the crop height, h, as a function of the leaf area

index, LAI, for a canopy with maximum vegetation density approximately

halfway through canopy depth. Assuming the aerodynamic roughness

length for bare soil, z0’, can be neglected, plot the values of (d/h) and (z

o/h)

as a function of leaf area index in the range LAI = 0 to 5 and comment on

why these two ratios vary with LAI in this way. Calculate the values of (d/h)

and (zo/h) when LAI = 4 for use in 9(f).

(f) Assume the aerodynamic resistance for latent and sensible heat transfer

to a vegetation stand in neutral conditions, ra, is given by Equation (22.9).

If both wind speed and vapor pressure deficit are measured 2 m above

the top of a 10 cm high grass stand, at 2 m above the top of a 1 m high

cereal crop stand, and at 2 m above the top of a 30 m high forest stand,

and all these stands have LAI = 4, plot the aerodynamic resistance of

these three vegetation stands as a function of wind speed from 0.25 m s−1

to 8 m s−1.

(g) Some SVAT represent the behavior of the surface resistance using the

Jarvis-Stewart model. Assume the surface conductance for the forest stand

considered in (f) is given by Equation (24.1) with g0 = 40 mm s−1 and g

M = 1

(i.e., there is no soil moisture stress); and with gR, g

D , and g

T , given

by Equations (24.2), (24.3) and (24.4), and (24.5), with KR = 200 W m−2,

KD

1 = –0.307 kPa−1, KD

2 = 0.019 kPa−2, TL =273 K, T

0 = 293 K, and T

H = 313 K.

Plot the variation in the individual stress functions gR, g

D , and g

T over the

solar radiation ranges 0–1000 Wm−2, VPD range 0–4 kPa, and temperature

range 0–40°C, respectively. If any stress function is calculated to be less

than zero, it should be set to zero. (Hint: do your calculations look plausible

in comparison with Fig. 24.5?)

In the following, assume the meteorological data given in Table 26.3 were measured

2 m above the top of the 30 m high forest and that it is acceptable to use the

aerodynamic resistance ra calculated in 9(f) and the formulae for surface

conductance specified in 9(g) to calculate the surface resistance rs (= g

s−1).

(h) Plot the variation through the day of the individual stress functions gR, g

D ,

and gT , and also the total stress function, i.e., the product (g

R g

D g

T g

M).

(i) Using the Penman-Monteith equation (Equation 21.33) and the surface

energy balance, plot the variation through the day of available energy,

latent heat and sensible heat fluxes.



Table 26.3 Meteorological data measured 2 m above the 30 m high forest for use in question 9(h) and 9(i).

Time (hour) Wind speed (m s−1)

Solar radiation (W m−2)

Available energy (W m−2) Temperature (°C)

Vapor pressure deficit (kPa)

0.5 1.39 0 17 26.74 0.561.5 1.11 0 1 26.13 0.482.5 1.30 0 9 25.47 0.413.5 1.36 0 −2 25.08 0.334.5 0.70 0 2 24.83 0.285.5 0.87 0 0 24.32 0.206.5 1.84 0 2 23.72 0.117.5 1.34 84 7 24.74 0.248.5 0.36 333 30 26.02 0.429.5 0.77 602 412 27.56 0.69

10.5 1.46 832 564 28.89 0.9311.5 2.36 965 697 30.00 1.2512.5 1.75 981 638 30.93 1.5713.5 3.16 1075 755 31.75 1.9714.5 2.77 994 618 32.11 2.0515.5 2.68 732 374 32.03 2.0916.5 2.85 617 321 32.66 2.3317.5 1.90 346 131 32.48 2.2918.5 1.97 85 26 31.75 2.0819.5 0.88 0 −3 30.39 1.7220.5 1.18 0 −16 28.72 1.1221.5 0.98 0 −9 27.71 0.9022.5 2.42 0 −2 27.58 0.9223.5 1.90 0 16 27.36 0.91

Question 10 (Uses understanding and equations from Chapters 2, 5, and 23.)

Create spreadsheets to make the calculations that are demonstrated in Tables 23.1,

23.2, 23.3 23.4 using the data for three sites in Australia given in Table 26.4. Then

create a spreadsheet to make calculations of crop evaporation in a table similar to

Table 23.6 but in this case for Alfalfa, Cotton and Sugar Cane. In this way you will

create a spreadsheet that you can use to give daily estimates of evaporation

wherever relevant data are available.

Example Answers

Answer 1

(a) Near the desert floor where the altitude is 3700 ft, or 1128 m, air tempera-

ture is 45.56°C, or 318.71 K, air pressure is 1006 mb, or 100.6 kPa, saturated



vapor pressure is 9.86 kPa [from Equation (2.17)], vapor pressure is 2.46

kPa [from Equation( 2.19)], specific humidity is 0.0152 kg kg−1 [from

Equation (2.9], and the gas constant for moist air is 289.2 J kg−1 K [from the

equation given in the question].

(b) At Kitt Peak where the altitude is 7080 ft, or 2158 m, air temperature is

312.01 K assuming the local lapse rate is 0.0065 Km−1, and air pressure is

90.1 kPa [from Equation (3.13)].

(c) From Equation (2.21), at the ambient air pressure on Kitt Peak water boils

at 96.5°C.

(d) The warmed parcels of air near the desert floor have a temperature of

50.56°C or 323.7 K, and their vapor pressure is still 2.46 kPa. They will

saturate at a dew point temperature of 20.85°C [from Equation (2.21)], or

at 294.0 K. If the ascending parcels cool at the adiabatic lapse rate of

0.00968 Km−1, they would need to reach a height of 3068 m above the

desert floor before their temperature falls from 323.71 K to 294.0 K and

they saturate. The warmed air parcels will lose buoyancy at a height h at

which their temperature has fallen such that it is equal to that of the

surrounding air, i.e., when (323.7 – 0.0097h) = (318.17 – 0.0065h), thus at

about 1500 m above the desert floor. Consequently there is unlikely to be

any convective cloud on this day because the warmed air parcels lose

buoyancy at ∼1500 m above the desert floor before they can saturate at

about 3000 m.

Table 26.4 Site characteristics and meteorological variables for the three Australian

sites considered in question 10.

Variable Units Site 1 Site 2 Site 3

Maximum air temperature (°C) 29.10 35.00 23.00Minimum air temperature (°C) 17.90 21.40 11.50Dry bulb temperature (°C) 24.00 n/a n/aWet bulb temperature (°C) 19.00 n/a n/aRelative humidity (%) n/a 39 n/aDew point (°C) n/a n/a 11.00Wind measurement height (m) 10.00 10.00 2.00Wind speed (m s−1) 5.60 4.70 3.70Day of year (none) 40 46 52Latitude (deg) −19.62 −15.78 −33.13Cloud fraction (none) 0.40 0.10 n/aNumber of bright sunshine hours (hours) n/a n/a 4.000Elevation (m) 12 44 30Assigned site humidity (none) Humid Arid HumidMeasured pan evaporation (mm) 7.6 11.2 4.9Selected value for Albedo (none) 0.23 0.23 0.23Albedo of open water (none) 0.08 0.08 0.08



Answer 2

(a) At this site on this day, the all-day average air temperature is 22°C, or

295.13 K, the saturated vapor pressure is 2.644 kPa, and the vapor pressure

is 1.322 kPa.

(b) When calculating net longwave radiation, the effective emissivity e’ = 0.194

[from Equation (5.23)] and assuming c = 0.7 all through the day, for an arid

site the empirical cloud factor f = 0.37 [from Equation (5..25)]. The esti-

mated all-day average net longwave radiation is therefore –28 W m−2 [from

Equation (5..22)].

(c) Were this a humid site the empirical cloud factor, f, would then be 0.533

[from Equation (5..24)] and the estimated all-day average net longwave

radiation would be -41 W m−2 [from Equation (5..22)].

(d) July 12 is Day Of Year 193, hence the eccentricity factor dl = 0.968 [from

Equation (5.5)] and the solar declination δ = 0.385 radians [from Equation

(5.8)]. The latitude of the site is 32.5°N, which is +0.567 radians, and the

hour angle, w, can be calculated in radians from the time of day, t, in hours

[using Equation (5.10)], with t running in hourly increments from 0.5 to

23.5. Consequently, the solar radiation at the top of the atmosphere, Stop, can

be calculated [from Equation (26.1)] and solar radiation at the ground, Sgrnd,

calculated [from Equation (26.2)]. Values of net solar radiation can then

be calculated for each value of t by allowing for the albedo of 0.23 [by

comparison with Equation (5.18)]. Values of net radiation can then be cal-

culated for each value of t by adding the relevant values of longwave radiation

for arid and humid conditions calculated in sections (b) and (c), respectively.

The resulting values of solar radiation, net solar radiation, and net radiation

for an arid and a humid site are given in Table 26.5 and plotted in Fig. 26.5.

(e) The required all-day average values are 190 W m−2 for (incoming) solar

radiation, 146 W m−2 for net solar radiation, 118 W m−2 for net radiation at

this arid site, and 105 W m−2 for net radiation were it assumed to be a

humid site.

(f) The required all-day average fluxes and plots of the diurnal variation in

solar, net solar, and net radiation for the Saskatoon site with fresh snow

cover on January 15 and grassland cover on July 15 are given in Fig. 26.6.

(g) The required all-day average fluxes and plots of the diurnal variation in

solar, net solar, and net radiation for the Manaus site on March 23 with for-

est and pastureland cover given in Fig. 26.7.

Answer 3

(a) The mean soil surface temperature, Tm

, is (24+20)/2 = 22°C, the amplitude

of the daily cycle, Ta, is (24–20)/2 = 2°C, and the time slip, t

o, which gives a

minimum 01.00 am and a maximum at 13.00 in Equation (6.11) is (7 ×

60 × 60) = 25,200 seconds.



(b) For saturated sandy soil the thermal conductivity, ks, is 2.20 W m−2 K−1 and

the thermal diffusivity, αs, is 0.74 × 10−6 m2 s−1. The damping depth, D, for

the daily time period P = (24 × 60 × 60) = 86,400 seconds is 0.143 m [from

Equation (6.13)]. The farmer used Equation (6.14) to estimate the instanta-

neous surface soil heat flux.

(c) The vapor pressure at the bottom level, vapor pressure at top level, Bowen

ratio, net radiation, soil heat flux, available energy, latent heat flux and sen-

sible heat flux are tabulated in Table 26.6.

(d) The net radiation, soil heat flux, available energy, latent heat flux and the sen-

sible heat flux are plotted as a function of time through the day in Fig. 26.8.

60 12Time (hrs)

18 24

Solar

Net (arid)Net (humid)

600

500

400

300

200

100

0

−100

Ene

rgy

flux

(W m

−2)

Figure 26.5 The diurnal cycle

of (incoming) solar radiation, net

solar radiation, longwave

radiation, and net radiation

calculated in 2(d) assuming it is

an arid and a humid site.

Table 26.5 Values of (incoming) solar radiation, net solar radiation, longwave radiation, and net radiation calculated in 2(d)

assuming it is both an arid and a humid site.

Hour (local)

Solar (W m−2)

Net solar (W m−2)

Net (arid) (W m−2)

Net (humid) (W m−2)

Hour (local)

Solar (W m−2)

Net solar (W m−2)

Net (arid) (W m−2)

Net (humid) (W m−2)

0.5 0 0 −28 −41 12.5 517 398 369 3571.5 0 0 −28 −41 13.5 489 376 348 3352.5 0 0 −28 −41 14.5 435 335 306 2943.5 0 0 −28 −41 15.5 358 276 248 2354.5 0 0 −28 −41 16.5 265 204 176 1635.5 53 41 12 0 17.5 161 124 95 836.5 161 124 95 83 18.5 53 41 12 07.5 265 204 176 163 19.5 0 0 −28 −418.5 358 276 248 235 20.5 0 0 −28 −419.5 435 335 306 294 21.5 0 0 −28 −41

10.5 489 376 348 335 22.5 0 0 −28 −4111.5 517 398 369 357 23.5 0 0 −28 −41



All-day average values

30

6

−46

−40

Solar (W m−2)

Net solar (W m−2)

Longwave (W m−2)

Net (W m−2)


208

160

−19

141

Solar (W m−2)

Net solar (W m−2)

Longwave (W m−2)

Net (W m−2)

200

150

100

50

0

−50

−100

0 6 12

Time (hrs)

18 24Ene

rgy

flux

(W m

−2)

Solar

Net solar

Net

Saskatoon Jan 15 (fresh snow)

−100

0

100

200

300

400

500

600

Time (hrs)

0 6 12 18 24

Solar

Net solar

Net

Saskatoon Jan 15 (grassland)

Ene

rgy

flux

(W m

−2)

Figure 26.6 Diurnal variation

in all-day average radiation

fluxes calculated in 2(f).

(e) The all-day average values of the Bowen ratio and Evaporative Fraction at

his site are calculated from the all-day average values of latent and sensible

(not by averaging the hourly average values) and are 0.486 and 0.673,

respectively.




175

154

−15

139

Solar (W m−2)

Net Solar (W m−2)

Longwave (W m−2)

Net (W m−2)


175

135

−15

120

Solar (W m−2)

Net solar (W m−2)

Longwave (W m−2)

Net (W m−2)

600

500

400

300

200

0

100

−1000 6 12

Time (hrs)18 24

En

erg

y fl

ux

(W m

-2) Solar

Net solar

Net

Manaus March 23 (forest)

−100

0

100

200

300

400

500

600

Time (hrs)

0 6 12 18 24

Solar

Net solar

Net

Manaus March 23 (pasture)

En

erg

y fl

ux

(W m

-2)

Figure 26.7 Diurnal variation

in all-day average radiation

fluxes calculated in 2(g).

(f) Had the farmer neglected soil heat flux he would have estimated greater

available energy during the day when evaporation is the dominant flux,

and less at night when sensible heat is the dominant flux. The net effect

would have been to overestimate the all-day average evaporation flux.


424 Example Questions and AnswersTable 26.6 The vapor pressure at the bottom level, vapor pressure at top level, Bowen ratio, net radiation, soil heat flux,

available energy, latent heat flux and sensible heat flux at hourly intervals calculated in 3(c).

Vapor pressure

Time (hour)

Bottom (k Pa)

Top (k Pa)

Bowen ratio

Net radiation (W m−2)

Soil heat (W m−2)

Available energy (W m−2)

Latent heat (W m−2)

Sensible heat (W m−2)

0.5 1.036 1.028 9.129 −28 −35 6 1 61.5 1.093 1.088 −4.402 −28 −27 −2 1 −32.5 1.084 1.098 7.579 −28 −17 −12 −1 −103.5 1.104 1.098 −52.074 −28 −6 −23 0 −234.5 1.084 1.078 −72.617 −28 6 −34 0 −355.5 1.083 1.088 29.388 12 17 −5 0 −46.5 1.236 1.058 −0.014 95 27 69 70 −17.5 1.400 1.068 0.292 176 35 141 109 328.5 1.538 1.128 0.406 248 40 207 147 609.5 1.604 1.140 0.519 306 43 263 173 90

10.5 1.666 1.162 0.538 348 43 305 198 10711.5 1.693 1.146 0.577 369 40 329 209 12012.5 1.693 1.122 0.559 369 35 335 215 12013.5 1.683 1.116 0.635 348 27 321 197 12514.5 1.640 1.089 0.592 306 17 290 182 10815.5 1.557 1.051 0.615 248 6 242 150 9216.5 1.526 1.057 0.550 176 −6 181 117 6417.5 1.398 1.043 0.472 95 −17 112 76 3618.5 1.262 1.048 0.073 12 −27 39 36 319.5 1.119 1.054 0.196 −28 −35 6 5 120.5 1.098 1.060 3.070 −28 −40 12 3 921.5 1.106 1.039 1.822 −28 −43 15 5 1022.5 1.106 1.048 1.714 −28 −43 15 5 923.5 1.070 1.038 3.170 −28 −40 12 3 9

400

350

300

250

200

150

100

50

0

−50

−100

0 6 12 18 24

Net

Soil

Available

Latent

Sensible

Ene

rgy

flux

(W m

−2)

Figure 26.8 The net radiation,

soil heat flux, available energy,

latent heat flux and the sensible

heat flux calculated in 3(d)

plotted as a function of time

through the day.



Answer 4

(a) Shuttleworth is speaking about the annual average evaporation from the

oceanic surfaces of the globe which is about 1.2 m per year. Around 90% of

this water falls back to the ocean while 10% moves and falls over land. When

averaged over all continental surfaces, about 65% of the precipitation falling

over land re-evaporates back to the atmosphere but in semi-arid areas the

proportion is higher. In Arizona, for instance, around 95% re-evaporates.

(b) Shuttleworth is talking about General Circulation models (GCMs). The

two components of these models he is referring to are the dynamics and the

physics. The dynamics applies conservation laws to calculate the fields of

atmospheric variables such as temperature, humidity and wind speed using

prescribed values for the divergence terms in these laws, see Chapters 16

and 17. The physics re-calculates the values of the divergence terms using

the (now modified) fields of atmospheric variables. Some of the processes

represented in the physics include radiation absorption, convection, and

precipitation processes in the atmosphere, and boundary layer and surface

exchange processes

(c) If all the Earth’s continents were clustered at the equator the seasonality of

the global average surface reflection coefficient for solar radiation, i.e. the

global average albedo, might well be less because, being on average warmer

than at present, they would presumably experience less snowfall. The

change in albedo associated with seasonal variations in snow and ice cover

is large because the albedo of fresh snow is around 80% while that for most

natural surfaces is around 20%. Alternative reasons for reduced seasonality

in global albedo include the possibility of reduced seasonal changes in the

vigor of the vegetation covering the continents.

(d) Shuttleworth is referring to the fact that the processes giving rise to

precipitation above, but comparatively close to the Earth’s surface, release

water vapor from the atmosphere and return it to the ground as precipitation,

which at the same time releases latent heat in the atmosphere. On average,

they therefore have the dual effect of reducing the lapse rate in the atmospheric

boundary layer so it is less than the adiabatic lapse rate while simultaneously

ensuring that atmospheric water vapor largely remains fairly near the surface.

(e) Presumably Shuttleworth is talking about GCMs again, because GCMs

have three main applications, namely (i) weather forecasting, (ii) climate

forecasting, and (iii) the synthesis of model-calculated fields of atmos-

pheric and surface variables across the entire globe as a by-product of

application (i). Weather forecasting seeks to predict actual weather a few

days ahead from well-defined initial conditions that may become the data

product (iii). In the case of (ii), initiation is less important because in this

application it is not actual weather but rather the statistics of weather (i.e.,

climate) that is the objective.

(f) Shuttleworth is respectively referring to the specific heat and density of

water relative to that of air. This difference means that the water in oceans



absorbs and releases energy more slowly than air and this is the fundamental

basis for the seasonal predictions that are the focus of interest of the Climate

Prediction Center whose website is at http://www.cpc.noaa.gov. Even if

seasonal predictions were not possible, presumably short-term prediction

would still be possible, and weather forecast centers such as the National

Weather Service at http://www.cpc.noaa.gov would still exist.

(g) ‘Greenhouse warming’ is predicted to increase near-surface air tempera-

tures. Since 70% of the Earth is covered with ocean, this likely will increase

surface evaporation rates because the saturated vapor pressure is a strong

function of temperature. Putting more water into the air as water vapor will

likely enhance the Earth’s overall hydrologic cycle by increasing the moisture

available for release by precipitation processes. At first sight, the effect will be

greatest at the poles because the projected temperature increases are greatest

there. However, saturated vapor pressure is a non-linear function of tem-

perature and the rate of change in saturated vapor pressure with temperature

is more than twice as large at 29°C (typical of sea surface temperature at the

equator) than it is at 0°C (typical of sea surface temperature at the poles).

Answer 5

The answers below give one opinion but, as is generally the case in politics,

different people have different opinions. If your opinions differ, discuss them with

your instructor.

(a) The RWD Party is probably using the banner C. Malleable has Hadley

Circulation similar to on Earth. Having looked through a telescope at their

sister planet, RWD Party followers notice that the resulting falling air

currents at approximately 30°N and 30°S of the equator suppress the

formation of precipitation and give rise to warm deserts when this occurs

over continents, e.g. the Sahara Desert. Their proposal is to remove the

continents at this band of latitudes.

(b) The RTS Party is probably using the banner A. The oceans on Malleable are

currently arranged to inhibit the inclusion of cold polar water in oceanic

circulation towards the equator. Looking through a telescope, followers of

the RTS Party notice that on their sister planet Earth, there is a marked

difference in the general shape of continental areas between the two

hemispheres. Those in Earth’s Northern Hemisphere are similar to those

on Malleable and inhibit inclusion of polar water in oceanic circulation.

However, the more open nature of the continents in the Earth’s Southern

Hemisphere allows cold polar water to penetrate towards the equator in the

western Pacific and Atlantic oceans. On average, this reduces the sea

surface temperature of equatorial western oceans in the Earth’s Southern

Hemisphere, and this in turn inhibits the production of tropical storms in

these regions. The RTS Party argues for opening the channels that link



polar and tropical oceans on Malleable so as to encourage the inclusion of

colder water towards their planet’s tropical oceans via oceanic circulation.

(c) The MM Party is probably using the banner B. Looking through a telescope,

followers of the MM Party notice that on their sister planet Earth there is

strong hydroclimatic feature that involves a marked seasonal reversal in

wind direction between areas on land and ocean. By listening in to the

radio broadcasts of weather services on Earth they learn this is called a

Monsoon. They notice that when the land surface is preferentially heated

by the shifting axis of rotation of the Sun there is a seasonal flow which

brings moisture over land that falls as precipitation in some months and

this is useful for growing agricultural crops. They also notice the effect is

greater when the flow is between a warm tropical ocean and large areas of

land that have a boundary which lies roughly parallel to the equator, e.g.,

the Indian Ocean and the continent of Asia. They are therefore suggesting

the continents on Malleable are arranged to favor such Monsoon flows.

(d) The RWD Party’s argument for the REN Party forming a coalition with

them is quite strong. They point out that relative to the existing continental

distribution, their proposed redistribution will significantly reduce the

portion of Malleable’s tropical ocean in which the Trade Winds can estab-

lish ‘warm pools’. The REN Party is negotiating for a bigger proportion of

the available land area at the equator but is facing opposition from the more

conservative faction of the RWD Party who have grown up in cooler cli-

mates with marked seasons.

(e) The RTS Party’s argument for the REN Party forming a coalition with them

is also reasonably convincing. They point out that their proposal will much

lessen the distance the Trade Winds have to establish ‘warm pools’ because of

the reduced distance between their four (as opposed to two) continents, and

because each proposed continent has more land area at the equator relative to

the existing continents. The REN party is negotiating for yet more continents

with more of their land area at the equator if they agree to form a coalition.

Answer 6

(a) Solar radiation heats the atmosphere from below, while it heats the oceans

from above. This results in a buoyant mixed layer on the surface of the

oceans which is typically 100–1000 m deep, and separated from the lower

ocean by the thermocline. The oceanic structure is fairly constant in time

in tropical regions, but the mixing layer changes depth with season at mid-

latitudes, being shallower and warmer in summer months when surface

heating is greater and the ensuing buoyancy of the surface water is greater.

(b) The student was a ‘Northo-centralist’ and was wrong. She should have said

‘ocean currents tend to go away from the equator on the eastern sides of

continents, and towards the equator on the western side of continents.’

(c) The near-surface mixed layer circulation of the oceans is primarily influ-

enced by the prevailing low level wind fields. The ocean, being massive, in



effect acts like a filter, picking out and following the average wind flow. Sea

water tends to be blown easterly near the equator and westerly at mid-lati-

tudes, and the near-continent currents are formed as part of this circula-

tion. But nothing is quite that simple in the global system and thermocline

circulation caused by changes in density, mainly salt concentration, and

Coriolis acceleration also play a role.

(d) This is actually a very complex problem, but shape of the land masses sur-

rounding oceans clearly influence the surface currents of oceans. In par-

ticular, the presence of substantial land at high latitude tends to inhibit the

linkage between (say) the Atlantic Ocean and cold polar waters, see answer

7(b). South America and Africa tend to taper towards the south pole

(unlike in the north Atlantic) and there is little land south of 40 degrees

south. The Benguela current in the South Atlantic (like the Peru current in

the Pacific) can access cold polar water more easily than the Canary current

in the North Atlantic, and this results in a lower SST at about 10-20 degrees

from the equator in the eastern portion of the ocean.

Tropical storms and hurricanes are formed in a region about 10–20 degrees

north and south of the equator (because there is not enough Coriolis force

at the very low latitude), and initially tend to move east to west in the

prevailing Trade Winds. The SST in central and eastern portions of 10–20

degrees north of the equator in the Atlantic is warmer than the 26.5 degrees

required for formation of tropical cyclones for a substantial portion of the

year, and the islands of the Caribbean suffer in consequence. But south of

the equator the equivalent phenomenon is suppressed by the cooler SST in

the eastern tropical Atlantic, and partly because of this and partly because

it is nearer to the equator, the climate of northeastern Brazil, though still

subject to oceanic influence, is spared.

(e) The student realized that the relative proportion of surface radiation used

to evaporate water over the oceans is greater than over land where more is

used to warm the lower atmosphere. The yearly cycle in near-surface tem-

perature is therefore greater over land surfaces than it is over oceans, and

this seasonal temperature differential (between the Asian continent and the

Indian ocean) is the driving mechanism behind the South East Asian mon-

soon, which is a major hydroclimatological influence on that region of the

world where a large portion of human population is concentrated.

(f) For clouds to occur not only must there be a mechanism that gives ascent

and cooling, but there must also be (i) sufficient moisture available in the

atmosphere, and (ii) cloud condensation nuclei (CCN) for cloud droplets

to form around. However, there are usually enough CCN available in the

air hence, choosing between these two, moisture availability is probably the

limiting requirement. However, quite often the atmosphere has both

enough CCN and enough moisture and the absence of an atmospheric

ascent mechanism is then the limiting criterion.

(g) The molecular weight of water molecules is less than the average molecular

weight of the mixture of oxygen and nitrogen molecules that make up dry



air. At the same temperature and pressure, the moister air has the same

number of molecules as the drier air, but some of the heavier air molecules

are replaced by lighter vapor molecules. Rearranging the Ideal Gas Law for

moist air, gives:

ra = [P/(R

d T)] / [1 + 0.61(r

v / r

a )] = [P/(R

d T)] / [1 + 0.61q]

Rd is a constant so assuming the temperature and pressure are the same, if

a parcel of air is moister than its surroundings (i.e. if q is greater) its density

is less than the surrounding air and it will tend to rise. It is the effect of

this greater buoyancy which is allowed for by re-expressing potential

temperature as virtual potential temperature.

(h) In convective conditions parcels of air heated to a temperature above that

of the surrounding atmosphere near the ground can often keep on rising to

the cloud condensation level because both the air in the parcel and the sur-

rounding air cool with height. However, the ascending air cools at the dry

adiabatic lapse rate and the surrounding air cools less quickly so there can

be situations where ascent is suppressed prior to reaching the level at which

water vapor in the rising air parcels saturates, see answer 1(d).

(i) Once at the cloud condensation level, condensation and cloud formation

begins. This releases latent heat which further warms the air thus tending

to make the air more buoyant and enhancing its further ascent within the

cloud.

(j) In mid-latitude clouds with a temperature of –25°C all the phases of water

(solid, liquid or vapor) are likely to be present. In such clouds the Bergeron-

Findeison process is likely to be the most important process responsible for

cloud particle growth.

Answer 7

(a) Very open situations are not necessarily always the best rain gauge sites

because near-ground wind speeds tend to be higher and wind-related

blow-in/blow-out gauge errors possibly higher. Consequently an optimum

site might be surrounded by obstructions but should be located in a flat

open area of short mown grass and should be sufficiently far from up-wind

of obstructions that they all subtend a vertical angle of less than 30°. Ideally,

the gauge would be placed at the center of a pit in the ground (say) 1–2 m

across such that the top of the gauge is level with the ground. This avoids

splash-in errors. The top of the pit should be covered with an open mesh

(plastic mesh is cheap and easy to find) so that it has a similar aerody-

namic roughness to that of the surrounding grass. If this is done the near-

surface wind flow is essentially parallel to the ground and the effect of

wind on the gauge is minimized. Such a site might look like that shown

in Fig. 26.9.



(b) Over the period 1961–1990 the monthly average precipitation for the

months January through December for Tucson were 0.87, 0.7, 0.72, 0.30,

0.18, 0.20, 2.35, 2.19, 0.67, and 1.07 inches, respectively. The seasonality

index calculated using Equation (13.1) from these values is 0.55, which

implies a rainfall regime that is fairly seasonal.

(c) The monthly average precipitation for the months January through

December for Tucson given in (b) are plotted as a pie diagram in Fig. 26.10.

(d) The percentage mass curve for the rainstorm is given in Fig. 26.11. About

50% of the rain during this storm falls in the first quarter of the storm and

about 90% in the first half of the storm which suggests the storm is of con-

vective origin.

(e) The mean value of July Tanzanian rainfall for the years 1931 to 1960 is

24.57 mm while the median value is 6.5 mm. The large difference between

these two is because the probability distribution is so heavily skewed, see

Figure 13.3.

(f), (g), and (h) The required plots of 7-year running mean, mass curve, and

cumulative deviation for Tanzanian July rainfall data are given in Fig. 26.12

(i) The area-average precipitation values the farmer calculated for his field in

April, May, and June were 9.26, 36.18, and 42.47 mm, respectively.

Surrounding obstructions subtendan angle of less than 30� with

respect to the ground.

Gauge set in pit with top at ground level,surrounded by a plastic grid to simulate the

aerodynamic roughness of surrounding area.

30�

Figure 26.9 Preferred

arrangement for a rain gauge site.

Figure 26.10 Monthly-average

precipitation for the months

January through December

for Tucson over the period

1961–1990 plotted as a pie

diagram.

2%3%

6%

6%

8%10%

6%

10%

6%

20%

21%

2%

Percentage precipitation per month for tucson

Jan.Feb.March.Apr.May.June.July.Aug.Sept.Oct.Nov.Dec.



00

20

40

60

80

100

20 40 60

Percentage of time during storm

Percentage mass curve

80 100

Per

cent

age

of r

ainf

all d

urin

g st

orm

Figure 26.11 Mass curve for

the precipitation during a

storm measured using a siphon

rain gauge, see Fig. 26.2 and

Table 26.2.

7-year running mean80(a) (b)

(c)

70

60

50

40

30

20

10

0

Pre

cipi

tatio

n (m

m)

Mass curve800

700

600

500

400

300

200

100

0

Cum

mul

ativ

e pr

ecip

itatio

n (m

m)

Cummulative deviation100

50

0

−50

−100

−150

−200

−250

−300

1 6 11 16 21 26

Cum

mul

ativ

e de

viat

ion

(mm

)

Figure 26.12 The required (a) 7-year running mean values, (b) mass curve, and (c) cumulative deviation for Tanzanian

July rainfall data given Table 13.3 (a).



Answer 8

(a). Start from Equation (16. 46), the basic equation for conservation of water

vapor in the atmosphere, i.e.

∂ ∂ ∂ ∂ ∂ ∂ν∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤∂+ + + = + + + +⎢ ⎥

⎣ ⎦

2 2 2

2 2 2

qq

a a

Sq q q q q q q Eu v wt x y z x y z r r

and expand the variables u, q and ra as mean and fluctuating part, thus:

( ) ( )

∂ ∂ ∂ ∂∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂

+ ′ + ′ + ′ + ′+ + ′ + + ′ + + ′

∂⎡ ⎤+ ′ + ′ + ′

= + + + +⎢ ⎥+ ′ + ′⎣ ⎦

2 2 2

2 2 2

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) qq

a a a a

q q q q q q q qu u v v w w wt x y z

Sq q q q q q Ex y z

nr r r r

Multiply out and separate the factors, thus:

( ) ( )2 2 2 2 2 2

2 2 2 2 2 2

'

qq

a a a a

q q q q q qq q q qu u u u v v vt t x x x x y y y y

Sq q q q q q q q q q Ew w w wz z z z x x y y z z

′ ′⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ′ ′+ + + + + ′ + + + ′ + ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦′ ′ ⎡ ⎤⎡ ⎤ ′ ′ ′

+ + + ′ + ′ = + + + + + + +⎢ ⎥⎢ ⎥+ ′ + ′⎣ ⎦ ⎣ ⎦

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

n

nr r r r

Average this equation and apply the Boussinesq approximation (in this case this just

means using average values for density because there is no buoyancy term), thus:

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′⎢ ⎥+ + + + ′ + ′ + + + ′ + ′⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥+ + + ′ + ′ = + + + + + + +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2 2 2 2 2 2

2 2 2 2 2 2

qq

a a

q q q q q q q q q qu u u u v v vt t x x x x y y y y

Sq q q q q q q q q q Ew w w wz z z z x x y y z z

n

nr r

Apply Reynolds averaging to eliminate terms 2, 4, 5, 8, 9, 12, 13, 16, 18, and 20 and

to eliminate overbars on already averaged terms, thus:

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ′ ′⎡ ⎤ + + ′ + + ′ + + ′ = + + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2 2 2

2 2 2

qq

a a

Sq q q q q q q q q q Eu u v w wt x x y y z z x y z

n nr r

Re-order the terms and rewrite the viscosity term in vector format, thus:

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤′ ′ ′+ + + = ∇ + + − ′ + ′ + ′⎢ ⎥

⎢ ⎥⎣ ⎦2.

qq

a a

Sq q q q q q qEu v w q u v wt x y z x y z

nr r



Multiply Equation (17.17) (the divergence equation for turbulent fluctuations in

the Atmospheric Boundary Layer) by q′, take the time average, then substitute the

resulting equation into the final term in the last equation to give:

( ) ( ) ( )∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤′ ′ ′ ′ ′ ′⎢ ⎥+ + + = ∇ + + − + +⎢ ⎥⎣ ⎦

2.q

qa a

S u q v q w qq q q q Eu v w qt x y z x y z

nr r

In Equations (26.1) and Equations (26.2) the terms that become negligible with

the assumptions given in the question are as follows.

(b) q

a

S

r

(c) a

Er

and −a p

Ecr

(d) ∇

− n

a p

Rcr

(e) ∇2.q qn and ∇2

qu q

(f) ∂∂q

wz

and ∂∂

wzq

(g) ∂

∂′ ′( )u qx

, ∂

∂′ ′( )qy

n,

∂∂

′ ′( )uxq

and ∂

∂′ ′( )

yn q

(h) ∂∂q

ux

, ∂∂qy

n , ∂∂

uxq

and ∂∂ yq

n

(i) After making all of the above simplifying assumptions Equations (26.1)

and (26.2) become:

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

′ ′ ′ ′= − = −

( ) ( )and

q w q wt z t z

q q

(j) At level (i) the temperature will be warmer and the humidity will change little.

Recalling that the chain rule gives:

∂ ∂ ∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂

′ ′ ′ ′ ′ ′′ ′= ′ + ′ = ′ + ′

′ ′ ′ ′= ′ + ′

( ) ( ). . ; . . ;

( ). .

u q q v q qu uu q v qx x x y y y

q w q ww qz z z

The final term in Equation (26.5) can be re-written to give the required prognostic

equation for mean humidity in the atmosphere, thus:

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤′ ′ ′′ ′ ′+ + + = ∇ + + − ′ + ′ + ′ + ′ + ′ + ′⎢ ⎥

⎢ ⎥⎣ ⎦2.

qq

a a

Sq q q q q q qE u v wu v w q u q v q w qt x y z x x y y z z

nr r

(26.5)



(k) At level (ii) the temperature will be warmer and the humidity will change

little.

(l) At level (iii) the temperature will be warmer and the humidity will change

little.

(m) At level (iv) the temperature will be cooler and the humidity will be wetter.

(n) At level (v) the temperature will change little and the humidity will change

little.

Answer 9

(a) At 20°C the molecular diffusion coefficients are υ = 1.52 × 10−5 m2 s−1,

DH = 2.15 × 10−5 m2 s−1, D

V = 2.42 × 10−5 m2 s−1, and D

C = 1.47 × 10−5 m2 s−1.

If the in-canopy wind speed, U, is 0.5 m s−1 for (spherical plate) leaves 0.05 m in

diameter the Reynolds number, Re, is 1649. Selecting the relevant empirical equa-

tion from Table 21.1, the Nusselt number, Nu ≈ 0.62 × Re 0.5 ≈ 0.62 × 41 ≈ 25. From

Equation (21.9), the boundary-layer resistance for heat transfer for (spherical

plate) leaves 0.05 m in diameter is RH (flat leaf) ≈ 0.05/(2.15 × 10 −5 × 24) ≈ 92 s m−1.

(b) For (cylindrical) needles leaves, the Reynolds number is 82 and the Nusselt

number, Nu ≈ 0.62 × 9.1 ≈ 5.6. From Equation (21.9) the boundary-layer

resistance for heat transfer for (cylindrical) conifer needles is RH (needle) ≈

0.0025/(2.15 × 10−5 × 5.6) ≈ 21 s m−1.

Assuming the transfer from individual vegetative elements is always by forced

convection and the relative transfer resistances for other exchanges is determined

only by their relative diffusion coefficients, see Equations (21.10) and (21.11), the

boundary-layer resistance for:

(c) vapor transfer for coniferous needles is RV (needle) ≈ 0.93 × 21 ≈ 19 s m−1.

(d) carbon dioxide transfer for coniferous needles is RC (needle) ≈ 1.32 × 21 ≈

27 s m−1.

(e) The required plots of the ratio of zero plane displacement to vegetation

height versus leaf area index and of aerodynamic roughness to vegetation

height versus leaf area index are shown in Fig. 26.13.

As additional leaf area is included in a canopy (of fixed height) a progressively

greater proportion of the momentum is lost higher in the canopy – the limit of infinite

LAI it is equivalent to raising the ground to the level by h. Initially this additional leaf

area raises the aerodynamic roughness above that of the bare soil by putting taller

roughness elements into the air stream. However, after the canopy begins to ‘close’

(when LAI is around one) and becomes denser and denser, depressions in the top of

the canopy become less significant and the aerodynamic roughness progressively falls.

When LAI = 4 the values of (d/h) and (zo/h) required for use in (f) are 0.73 and

0.08, respectively.

(f) The required plot of the aerodynamic resistance for a 10 cm high grass

stand, a 1 m high crop stand, and a 30 m high forest stand (all with LAI = 4)

is given in Fig. 26.14. Notice the large difference between these values of



2

LAI (dimensionless)

3 4 5

1.0(a)

(b)

0.8

0.6

0.4

0.2

0.00 1

d/h

(dim

ensi

onle

ss)

2

LAI (dimensionless)

3 4 5

0.15

0.10

0.05

0.000 1

z0/h

(di

men

sion

less

)

Figure 26.13 (a) Ratio of zero

plane displacement to vegetation

height versus leaf area index, and

(b) aerodynamic roughness to

vegetation height versus leaf area

index calculated in question 9(e).

0 1 2 3 4

Wind speed (m s−1)

5 6 7 81

10

100

1000

Aer

odyn

amic

res

iste

nce

(s/m

)

Grass Crop Forest


aerodynamic resistance for a

10 cm high grass stand, a 1 m

high crop stand, and a 30 m high

forest stand all with LAI = 4

calculated in question 9(f).



aerodynamic resistance as the height and roughness of the vegetation

stands increase.

(g) The required plots of gR, g

D, and g

T are given in Fig. 26.15.

(h) The required plots of gR, g

D , and g

T and the total stress function ( g

R g

D g

T g

M ,)

through the day are given in Fig. 26.16.

(i) The required plots of available energy, latent heat and sensible heat fluxes

are given in Fig. 26.17.

1.0

0.8

0.6

0.4

0.2

0.0

g s (

dim

ensi

onle

ss)

0 200 400 600 800 1000

1.0

0.8

0.6

0.4

0.2

0.0

g D (

dim

ensi

onle

ss)

0 1 2

VPD (k Pa)

3 4

1.0

0.8

0.6

0.4

0.2

0.0

g T (

dim

ensi

onle

ss)

0 105 20 2515Temperature ( �C)

30 4035

Solar Radiation (W m−2)

(c)

(b)

(a)


stomatal conductance stress

factor associated with

(a) radiation, (b) vapor pressure

deficit, and (c) temperature

calculated in question 9(g).



Answer 10

The spreadsheet calculations (equivalent to Tables 23.1, 23.2, 23.3 23.4 and 23.6)

made using the data for the three sites in Australia are given in Tables 26.7(a),

26.7(b), 26.7(c), 26.7(d), and 26.7(e), respectively.

1.0

0.8

0.6

0.4

0.2

0.00 6 12

Time (hrs)18 24

Radiation

Temperature

VPD

Total

Str

ess

Fac

tor

(dim

ensi

onle

ss)

Figure 26.16 Variation in gR,

gD and g

T and the total stress

function (gR, g

D g

T g

M,)

through the day calculated

in question 9(h).

800

700

600

500

400

300

200

100

0

−1000 6 12

Time (hrs)

18 24

Available EnergyLatent Heat

Sensible Heat

Ene

rgy

Flu

x (W

m−2

)


available energy, latent heat and

sensible heat fluxes through the

day calculated in question 9(i).


Table 26.7(b) Daily average net radiation for a crop calculated at three Australian sites

in question 10.


Day of year (none) 40 46 52Eccentricity factor (none) 1.0255 1.0232 1.0206Solar declination (radians) −0.2688 −0.2355 −0.1998Sunset hour angle (radians) 1.6691 1.6387 1.7033Latitude (deg) −19.62 −15.78 −33.13Latitude in radians (radians) −0.3424 −0.2754 −0.5782Extraterrestrial solar radiation (mm day−1) 16.61 16.34 15.68Cloud fraction (none) 0.40 0.10 -Solar at ground (cloudy sky) (mm day−1) 9.14 11.44 -Number of bright sunshine hours (hours) - - 4.00Maximum daylight hour (hours) - - 13.01Solar at ground (cloudy sky) (mm day−1) - - 6.33Solar at ground (cloudy sky) (mm day−1) 9.14 11.44 6.33Selected value for Albedo (none) 0.23 0.23 0.23Net solar radiation (mm day−1) 7.04 8.81 4.88Vapor pressure (k Pa) 1.863 1.593 1.313Effective emissivity (none) 0.149 0.163 0.180Solar at ground (clear sky) (mm day−1) 12.46 12.26 11.76Assigned site humidity (none) Humid Arid HumidCloud factor (none) 0.733 0.910 0.538Average temperature (°C) 23.50 28.20 17.25Net longwave (mm day−1) −1.68 −2.44 −1.37Net radiation (mm day−1) 5.35 6.37 3.51

Table 26.7(a) Daily average air temperature, saturated vapor pressure, vapor pressure,

vapor pressure deficit, and wind speed at 2 m calculated at three Australian sites in

question 10.


Maximum air temperature (°C) 29.10 35.00 23.00Minimum air temperature (°C) 17.90 21.40 11.50Average temperature (°C) 23.50 28.20 17.25Sat. vapor pressure (Max. temp) (kPa) 4.029 5.623 2.809Sat. vapor pressure (Min. temp) (kPa) 2.051 2.549 1.357Average sat. vapor pressure (kPa) 3.040 4.086 2.083Wet bulb psychrometric constant (kPa °C −1) 0.066 - -Dry bulb temperature (°C) 24.00 - -Wet bulb temperature (°C) 19.00 - -Vapor pressure (kPa) 1.863 - -Relative humidity (%) - 39 -Vapor pressure (kPa) - 1.593 -Dew point (°C) - - 11.00Vapor pressure (kPa) - - 1.313Vapor pressure deficit (kPa) 1.177 2.492 0.770Wind measurement height (m) 10.00 10.00 2.00Wind speed (m s−1) 5.60 4.70 3.70Modified wind speed (m s−1) 4.19 3.51 3.70


Table 26.7(c) Daily average open water evaporation calculated at three Australian sites

in question 10.


Average temperature (°C) 23.50 28.20 17.25Vapor pressure deficit (kPa) 1.177 2.492 0.770Modified wind speed (m/s) 4.19 3.51 3.70Solar at ground (cloudy sky) (mm day−1) 9.14 11.44 6.33Net longwave (mm day−1) −1.68 −2.44 −1.37Elevation (m) 12 44 30Air pressure (kPa) 101.16 100.78 100.95Latent heat (MJ kg−1) 2.446 2.434 2.460Delta (kPa °C −1) 0.1740 0.2217 0.1243Psychrometric constant (kPa °C −1) 0.0672 0.0674 0.0668Selected value for Albedo (none) 0.08 0.08 0.08Net solar radiation (mm day−1) 8.41 10.52 5.82Net radiation (mm day−1) 6.72 8.09 4.46Open water evaporation (mm day−1) 7.65 10.63 5.00

Table 26.7(d) Reference crop evaporation using the FAO, radiation-based, temperature-

based and pan-based methods calculated at three Australian sites in question 10.


Maximum air temperature (°C) 29.10 35.00 23.00Minimum air temperature (°C) 17.90 21.40 11.50Average temperature (°C) 23.50 28.20 17.25Vapor pressure deficit (kPa) 1.177 2.492 0.770Modified wind speed (m s−1) 4.19 3.51 3.70Extraterrestrial solar radiation (mm day−1) 16.61 16.34 15.68Net radiation (mm day−1) 5.35 6.37 3.51Assigned site humidity (none) Humid Arid HumidLatent heat (MJ kg−1) 2.446 2.434 2.460Delta (kPa °C−1) 0.1740 0.2217 0.1243Psychrometric constant (kPa °C−1) 0.0672 0.0674 0.0668Measured pan evaportion (mm) 7.6 11.2 4.9Value of Cp in Equ. (23.24) (s m−1) 224 224 224Value of (Apan/Arc) (none) 1.15 1.15 1.15Modified psychrometric const. (kPa °C−1) 0.1600 0.1456 0.1484rclim assigned in Equ. (23.26) (s m−1) 40 76 49Default pan coefficient (none) 0.88 0.82 0.88Wind corrected pan factor (none) 0.77 0.77 0.79Ref. Crop Evap. (FAO) (mm day−1) 5.78 8.62 3.75Ref. Crop Evap. (radiation based) (mm day−1) 4.87 8.50 2.87Ref. Crop Evap. (temperature based) (mm day−1) 5.28 6.38 4.29Ref. Crop Evap. (pan: default Kp) (mm day−1) 6.69 9.18 4.31Ref. Crop Evap. (pan: wind corr. Kp) (mm day−1) 5.85 8.59 3.87



Table 26.7(e) Daily average evaporation from unstressed crops calculated using the

Matt-Shuttleworth approach and the FAO crop factor method for an alfalfa crop, a

cotton crop, and a sugar cane crop calculated at three Australian sites in question 10.


Average temperature (°C) 23.50 28.20 17.25Vapor pressure deficit (kPa) 1.177 2.492 0.770Modified wind speed (m s−1) 4.19 3.51 3.70Extraterrestrial solar radiation (mm day−1) 16.61 16.34 15.68Net radiation (mm day−1) 5.35 6.37 4.20Assigned site humidity (none) Humid Arid HumidAir pressure (kPa) 101.16 100.78 100.95Latent heat (MJ kg−1) 2.446 2.434 2.460Delta (kPa °C−1) 0.1740 0.2217 0.1243Psychrometric constant (kPa °C−1) 0.0672 0.0674 0.0668Modified psychrometric constant (kPa °C−1) 0.1600 0.1456 0.1484Ref. Crop Evap. (FAO) (mm day−1) 5.78 8.62 3.75rclim (s m−1) 53 73 76(D50 / D2) (none) 1.21 1.28 1.19

Alfalfa cropCrop factor (none) 0.95 0.95 0.95Rc

50 (none) 196 196 196(rs)c (s/m) 127 127 127Re-modified psychrometric constant (kPa °C−1) 0.2494 0.2210 0.2270Matt-Shuttleworth estimate (mm day−1) 5.22 8.55 3.39FAO estimate (mm day−1) 5.49 8.19 3.56

CottonCrop factor (none) 1.18 1.18 1.18Rc

50 (none) 162 162 162(rs)c (s/m) 60 60 60Re-modified psychrometric constant (kPa °C−1) 0.1713 0.1552 0.1584Matt-Shuttleworth estimate (mm day−1) 7.17 11.36 4.77FAO estimate (mm day−1) 6.82 10.17 4.42

Sugar CaneCrop factor (none) 1.25 1.25 1.25Rc

50 (none) 124 124 124(rs)c (s/m) 63 63 63Re-modified psychrometric constant (kPa °C−1) 0.1673 0.1518 0.1924Matt-Shuttleworth estimate (mm day−1) 6.87 10.82 5.15FAO Estimate (mm day−1) 7.22 10.77 4.69




Index

absorptivity 51

AIRS instrument 172

atmospheric boundary layer (ABL)

atmospheric variables in 223

daytime profiles 261–3, 264, 265

diurnal evolution, over land,

under clear sky

conditions 260–1, 261

entrainment layer 260

equations of atmospheric flow

in 231–47

equations of turbulent flow

in 248–58

exchange processes 301–6, 302, 304

feedback processes, modeling 316,

327, 328, 331

higher order moments 265–75

inversion layer 260

mixed layer 260

nature and evolution of 259–61,

260, 261

nighttime profiles 263–5

resistance 302–6

structure 259–60

surface layer 259

turbulent flux at 223


in 31–3

accretion 143

actual daily total solar radiation 59

adiabatic atmosphere 33

adiabatic lapse rates 27–9

dry 27–8

environmental 28–9

moist 28

advected energy 41

advective (mean) flux 225–9

aerodynamic resistance 289, 296–9

aerosols 131

sources 131aggregation 143

AIRS precipitation product 172

Aitken Nuclei 131

albedo of natural surface 52, 52for different surfaces 53variation with solar altitude 52,

53, 54

Amazonian deforestation 366

Andres Mountains 125

anemometer 214

annual precipitation 177

Antarctica 9

anthroposphere, features of 10–11

Archimedes principle 32

areal mean precipitation,

calculation 200–4

areal reduction factors (ARFs)

205–7, 207

Asian-Australian monsoon

system 118, 119, 383

atmosphere

circulation 7, 7

composition 5, 6

features of 5–7, 7

variance 7

water vapor in 14–24

atmospheric circulation, global scale

influences on 107–25, 108

atmospheric stability 32–4, 34

static stability parameter 32–3

atmospheric water vapor 14–24

conservation 244–5

content 15

residence time 3, 6–7

autocorrelation 187

available energy 41

averaging rules for time-dependent

variables 218Avogadro’s number 16

Azores High 123

Ball-Berry equation 368, 369

Bayesian inversion method 173

Bergeron-Findeisen process 134, 143,

144, 146, 147, 147

big-leaf assumption

of canopy resistance 312–13

of plant–atmosphere

interactions 316, 364

biochemical energy storage 40–1

biosphere, features of 10

blackbody 49

blackbody radiation laws 49–51, 50

blending height 37

bluff body transfer 302, 303, 319

‘boil off ’ rate 19

BOREAS experiment 390

Boussinesq approximation 249–50,

252, 265

Bowen ratio/energy budget

measuring method 45–6,

83–5, 85, 86

Boyle’s Law 16

Budyko Bucket model 361, 361, 364

buoyant acceleration 32–3

Campbell-Stokes recorder 78, 78

canopy capacity 324

Note: Page references in italics refer to Figures; those in bold refer to Tables

Shuttleworth_bindex.indd 441Shuttleworth_bindex.indd 441 11/3/2011 5:39:12 PM11/3/2011 5:39:12 PM

442 Index

canopy processes 300–14

canopy resistances 300–14

big leaf approach 312–13

energy budget of dry canopy

311–14, 312, 313

energy budget of dry leaf 310–11

shelter factors 306–8, 306

stomatal resistance 308–10, 309

capacitance probes 91

carbon cycle, impact of change to 11

carbon dioxide, hydroclimate changes

and 110

Cartesian Grid 102, 102

CENTURY model 370

Charles’s law 16

chlorinated fluorocarbons (CFCs) 11

cirrus 140

climate prediction 100

cloud condensation nuclei (CCN) 128,

131–2

cloud cover, daily estimates 77–8

cloud droplets

collection efficiency of 144, 145

collision efficiency of 144

collision of 144–5, 145

size, concentration and terminal

velocity 133, 133cloud factor 61

in arid conditions 62

in humid conditions 62

cloud formation 128–41

cloud droplet size, concentration

and terminal velocity 133, 133extratropical fronts and

cyclones 138–40, 139

Foehn effect 136–8, 137, 138

ice in 134–5

mechanisms 129–30, 129

processes 135–40

requirements 128

saturated vapor pressure of curved

surfaces 132–3, 132

temperature ranges and

constituents 134

thermal convection 135–6, 136

cloud genera 140–1, 141cloud indices 171

cloud type, rain production by 148–9

coalescence 143

coalescence efficiency 144

cold clouds 134

cold front 138

collection efficiency of cloud

droplets 144, 145

collision efficiency of cloud

droplets 144

collision of cloud droplets 144–5, 145

Community Land Model

(CLM) 371–2

conditional probabilities 195, 196

conservation laws 101, 231

conservation of atmospheric

moisture 244–5

conservation of energy 231, 245–6,

245, 257conservation of mass 231, 243–4,

245, 257conservation of moisture 245, 254,

255, 257conservation of momentum 231,

234–42, 245, 257axis-specific forces 239–42, 240

combined momentum forces 242

pressure forces 235–6, 235

in turbulent ABL 252–4

viscous flow in fluids 236–9, 238

conservation of scalar quantity 245,

246, 254, 255, 257contact nucleation 134

continental land cover 110

continental topography 109

continental water balance,

estimated 5Continuity Equation for mass of

air 243

convective cloud 130

conventional frequency

distribution 193

Coriolis force 113, 114, 240

correlation of variables 221–3, 223

correlogram 187

counter gradient flow 282

cryosphere, features of 9

cumulative percentage

deviations 183–4, 185

cumulonimbus cloud 140, 147

cumulus 128, 140

cyclones 119–20, 120, 139–40, 139

daily average values of weather

variables 335–8

net radiation 337–8, 339

temperature, humidity and wind

speed 335–7, 337daily estimates of evaporation 334–55

Matt-Shuttleworth approach

348–53, 352, 354, 392

open water evaporation 339–41,

341reference crop

evapotranspiration 341–2, 348evaporation pan-based estimation

of 346–8

Penman-Monteith equation

estimation of 342–3

radiation-based estimation

of 344–5, 345

temperature-based

estimation 345–6

vs SVATS 334

unstressed vegetation 348–53, 352,

354, 392

water stressed vegetation 353–5

daily precipitation 180, 181

Dalton’s law of partial pressures 16

damping depth 73, 74, 75

day length 57

decomposed variables, averaging

of 217–19

deforestation 110

density of soil 69–70, 69design storms 205–7

dew 152–3

dew point hygrometer 21

dew point of air 21

dimensionless gradients 290–2

of specific humidity 291, 292, 295

of virtual potential

temperature 291–2

of wind speed 290

dimensionless measure

of atmospheric stability 289

of buoyant production 291,

292, 293

dimensionless prognostic equation for

TKE 290

divergence equation for turbulent

fluctuations 250

divergence of net radiation flux 246

Doldrums, the 112

Doppler effect 169

drag coefficient 302, 304

drizzle 149


Index 443

dry adiabatic lapse rate 27–8, 135

dry air 15

dry bulb temperature 22, 23

dry growth 147

dynamics 101

Earth, elliptical orbit, distance from

Sun and 55, 55

Earth Observing System Aqua

polar-orbiting satellite 172

eccentricity factor 55

eddies, turbulent see turbulent eddies

eddy correlation method 85–7

eddy diffusion of momentum

flux 295–6

eddy diffusivities 281, 283, 285, 296,

298–9

effective depth of soil heat flow 74

El Niño Southern Oscillation (ENSO)

110, 120–2, 121

electromagnetic radiation 48

elliptical storms 203–4, 204

emissivity 51

energy budget measuring method

(Bowen ratio) 45–6, 83–5,

85, 86

energy budget of open water 46

energy, conservation of 231, 245–6,

245, 257enhanced efficiency of near surface

turbulence 316

ensemble 100

ENSO 177

environmental lapse rate 28–9, 135

equilibrium evaporation rate 325–7, 326

equivalent flux of latent heat 295

European Centre for Medium-term

Weather Forecasting

(ECMWF) model 386,

390, 391

evaporation measurement from

integrated water loss 87–91

comparison of methods 91, 92–3evaporation pans 88–9, 89

evaporative fraction 45

evapotranspiration 342

excess resistance approach to

boundary layer

resistance 319–21, 320

extratropical fronts 138–40, 139

extremal distributions 193

far infrared waveband 48

fetch 83

Fick’s law 281

fixed area analysis 206–7

flood control systems 206

flux-gradient relationships 293–4,

293flux sign convention 41

difference values of fluxes 41–5,

42–5

Foehn effect 136–8, 137, 138

fog drip 153

force–restore scheme 364

form drag 302, 303

fossil water 2

four-dimensional data assimilation

(4DDA) 98

Fourier analysis 187, 215

Fourier series 103

Fourier’s law 281

freshwater, as reservoir of water 3

friction velocity 286

frontal cloud 130

frost 152–3

frozen precipitation cover 389–91,

390

gauges, in precipitation

mapping 199–200

General Circulation Models

(GCMs) 96–106, 107, 325, 361,

362, 363, 364

in climate prediction 100

definition 96–7

grid scale 97, 98, 99

operational sequence 100–2, 101

partitioning in Cartesian

coordinates 97, 98

physics, calculation of 103–4

boundary-layer scheme 103

convection scheme 104

large-scale precipitation

scheme 104

radiation transfer scheme 103

surface-parameterization

scheme 103

solving dynamics 102–3

use of 98–100

in weather prediction 98–100

geostrophic wind 251, 251

Giant Aerosols 131

GLAC 387

glaciers, as water reservoir 3

Global Atmospheric Research

Programme (GARP) Atlantic

Tropical Experiment

(GATE) 172

Global Precipitation Climatology

Project (GPCP) 173

Global Precipitation Index (GPI)

172

Global Precipitation Measurement

(GPM) mission 173

Goddard Institute for Space

Studies Model II 394–5,

395

Goddard Profiling Algorithm

(GPROF) fractional

occurrence of

precipitation 173

gradient Richardson number

278

graupel 147

gray surfaces 51

greenhouse effect 60

Greenland 9

ground-based radar 168–71, 169

Gumbel distributions 193, 194

Hadley circulation 112–13, 113

hail 147

Hargreaves equation 346

harmonic analysis see Fourier

analysis

heat capacity per unit volume of

soil 69, 70

heterogeneous nucleation 134

horse latitudes 112

hour angle 57

hurricanes 120

hydroclimate system, global

components of 4–9

hydroclimatology 1–2

hydrological cycle, global annual

average 3, 4

hydrometeors, measurement of

168–70, 169

hydrometerology vs

hydroclimatology 2

hydrosphere, features of 8

hydrostatic pressure law 26–7, 26

hydrostatic vertical gradients 25


444 Index

ice in cloud formation 134–5

ice particles in cloud

accretion onto ice particle 146–7

aggregation of 146

growth, by vapor transfer 147

ice sheets, impact of melting 9

icebergs, as fractional runoff 4

Icelandic Low 123

Ideal Gas Law 16–17, 25, 29, 101, 231,

245, 247, 257fluctuations in 248–9

ideal surfaces 37

Illinois Climate Network 386

in-canopy processes 316, 317

insolation 56

instantaneous radiation balance 62

interactive vegetation 388, 389

Intergovernmental Panel on Climate

Change (IPCC) 104–5

intra-annual precipitation

177–80, 178

inversion, atmospheric 33

ishyets 199–200, 200

isohyetal method 200–1, 201

isomers 179

isothermal atmosphere 33

Jarvis-Stewart model 364, 366,

368, 369

jet streams 7

Julian day 56

K Theory 282, 283, 285, 296,

301, 316

Kevin-Boltzmann statistics 19

kinematic flux 223–4, 225kinematic units, returning fluxes from,

to actual fluxes to 294–6

Kipp pyranometer 78–9, 79

Kirchoff ’s Principle 51, 60

kriging techniques 200

La Niña 110, 122

lakes, water balance of 89–90

Land Surface Parameterization

Schemes (LSPs) 10

land-atmosphere ‘coupling

strength’ 387

land-atmosphere interactions,

influence of 383–5

land surface exchanges 380–99

contribution of, to atmospheric

water availability 385

cultivated land areas 381, 382

influence of imposed persistent

changes in land cover 392–8

imposed heterogeneity

395–8, 396

near surface observations

392–3, 393

regional-scale climate 393–5

influence of land surfaces on

weather and climate 381–3,

382influence of transient changes in

land surfaces 385–92

combined effect 391–2

frozen precipitation cover

389–91, 390

soil moisture 385–8, 386, 387

vegetation cover 388–9, 389

Large Nuclei 131

latent heat flux 37, 39, 295

latent heat measurement 82–91

latent heat of fusion 14

latent heat of vaporization 15

leaf area index (LAI) 312

dependency of aerodynamic

properties 318, 319

lifting condensation level 136

Linear Correlation Coefficient 221–3,

223

lithosphere, features of 9–10

longwave radiation 38, 48, 49,

59–62, 61

lower atmosphere circulation

111–16

Hadley circulation 112–13, 113

latitudinal bands of pressure and

wind 111–12, 112

mean low-level circulation 113–15,

114

mean upper level circulation

115–16, 115

lysimeters 90–1

mapping precipitation 199–200

Markov chain model 195

Marshall-Palmer equation 149

mass, conservation of 231, 243–4,

245, 257mass curve 184–6, 186, 189

Matt-Shuttleworth approach 348–53,

352, 354, 392

McCullum model 190

mean flow of atmospheric entities 216

mean flux 225–9

Mean Kinetic Energy (MKE) 220–1

merged products 170

mesosphere 5

micrometerological measurement of

surface energy fluxes 83–7, 83

mixed clouds 134

mixing length theory 283–8, 283, 285,

287, 292

mixing ratio 15

moist adiabatic lapse rate 28–9, 135

moist air 15

moisture, conservation of 245, 254, 255, 257

moisture flux 295

molecular diffusion coefficient 305–6

momentum, conservation of see

conservation of momentum

momentum flux 224

momentum transfer 303

by bluff body transfer 303

by skin friction 303

monsoon oceanic flow 118¸ 119

National Operational Hydrologic

Remote Sensing Center 167

natural siphon rainfall recorders

160–1, 161

natural surfaces, integrated radiation

parameters for 52–4

NCAR Community Climate

Model 371

near infrared waveband 48

neglecting subsidence 250

neglecting molecular diffusion 255–8

net radiation 38, 39

daily average 62

flux 63

height dependence of 63–4

instantaneous radiation balance 62

measurement of 80–1

net radiometers 80–1, 81

neutron probes 91

Newton’s law for molecular

viscosity 281

Newton’s second law of motion 234

NEXRAD system 169, 170


Index 445

Nipher gauge 159

nitrogen, atmospheric 6

Noah land surface model 389

North American Monsoon experiment

(NAME) 383

North American Monsoon System

(NAMS) 118

North Atlantic Oscilllation

(NAO) 123, 123

numeric filters 187

Numerical Weather Prediction

(NWP) 98, 99–100

Nusselt number 303–4, 304

relationship with Reynolds

number 305

Obukov length 292–3, 329

ocean mixed layer 116

ocean to continent surface

exchanges 109

oceanic influences on continental

hydroclimate 118–23

El Nino Southern Oscillation

(ENSO) 110, 120–2, 121

monsoon flow 118¸ 119

North Atlantic Oscilllation

(NAO) 123, 123


(PDO) 110, 122, 122

tropical cyclones 119–20, 120

oceanic circulation 110, 116–18, 117

oceanic movement, response time 7

Ohm’s Law 297

open water, energy budget of 46

open water evaporation 46

daily estimates of 339–41, 341outward longwave radiation

(OLR) 172

oxygen, atmospheric 6

ozone 6


(PDO) 110, 122, 122

paired catchments 89–90, 90

pan coefficients 89

pan factor 346

parameterizations 216

pauses in atmospheric temperature, 5

Pearson distributions 193, 194

Penman-Monteith equation 316, 324,

329, 339–40, 349, 350, 353, 361

calculation of evaporation 346–7

estimation of reference crop

evapotranspiration 342–3

single leaf 311

whole-canopy 312, 313–14

Pennsylvania State-National Center

for Atmospheric Research

Mesoscale Model (MM5)

393–4, 394

Penpan equation 346–8

permafrost as water reservoir 3

PERSIANN precipitation product 172

photoelectric pyranometers 79, 80

Photosynthetically Active Radiation

(PAR) 54

physical energy storage 40

pie diagrams 179

Planck’s Law 50

planetary interrelationships 109

pluviometric coefficients 179

Plynlimon paired catchments 90, 90

point area precipitation

relationships 206

by duration 205, 206

by return period 205, 205

polar diagrams 179

potential rate of evaporation 328, 341

potential temperature 25, 30

precipitable water 124, 124

precipitation

cloud type and 148–9

extreme, statistics of 190–5

forms 149, 149frozen, types 151

rates and kinetic energy 151

seasonal, time of onset 180

precipitation analysis in

space 198–211

precipitation analysis in time 176–97

precipitation climatology 176,

177–80

annual variations 177

daily variations 180, 181

intra-annual variations 177–80, 178

precipitation, formation of 143–53,

148

in cold clouds 146

in mixed clouds 146–7

in warm clouds 144–6, 145

precipitation frequency

distribution 192–3, 192

precipitation intensity-duration

relationships 189–90, 190, 191

precipitation measurement and

observation 155–75

precipitation oscillations 186–7

precipitation recycling 385

precipitation trends 181–6, 182, 183

cumulative percentage

deviations 183–4, 185

mass curve 184–6, 186, 189

running means 182–3, 184

Priestley-Taylor equation 345, 351

PRISM methodology 383

probability distributions 193–5

conventional frequency 193

Gumbel distributions 193, 194

Pearson distributions 193, 194

extremal distributions 193

transformal distributions 193

probable maximum precipitation

(PMP) 207–9, 209

prognostic equations 247, 258

for turbulent departures 265–9

for turbulent kinetic energy

269–71, 270, 271, 272

for variance of moisture and

heat 271–5, 274, 275

of velocity components 279, 280

psychrometric constant 23, 335

pyrgeometers 81, 81

quantum sensors 79

radar

ground-based, precipitation

estimation 155

spaceborne 173

radar reflectivity factor 170

radiant energy, latitudinal imbalance

in 110–11, 111

radiation spectrum 48, 49

radiation exchange 51

radiation properties 51

radioactively active gases 58

radioactively active gases, absorption

spectra of 60

rain gauges 155, 156–65

areal representativeness of

measurements 162–4, 163

design specifications 156, 156designs 160–2


446 Index

rain gauges (cont’d )

instrumental errors 157

inter-gauge correlations 164, 165minimum gauge densities 164, 165site and location errors

157–60, 158

tipping bucket design 157

turbulence minimisation

158–9, 159

turf wall construction 158–9, 159

raindrop

shape 150, 150

size distribution 149, 149

rainfall see entries under precipitation

Rayleigh scattering 58, 170

reanalysis data 100

reciprocal-distance-squared

methods 200, 201, 202

reference crop evapotranspiration

341–2, 348

evaporation pan-based estimation

of 346–8

Penman-Monteith equation

estimation of 342–3

radiation-based estimation

of 344–5, 345

temperature-based

estimation 345–6

reference level 38

reflectivity 51

relative humidity 20

remote sensing, precipitation

estimation 155–5

resistance analogues 296–9

Reynolds averaging 219, 231, 248, 249,

252, 253, 255, 267

Reynolds number 300, 301

relationship with Nusselt

number 304, 305Richards equation 370, 372

Richardson number 277–9, 279,

292, 293

riming 143

Rocky Mountains 125

running means 182–3, 184

runoff ratios 4

Rutter model of wet canopy

evaporation 323–5, 324

salt water, as reservoir of water 2

saturated vapor pressure 18–20, 18, 20

saturation, measures of 20–1

scalar quantity, conservation of 245,

246, 254, 255, 257sea-surface temperature (SST)

116, 117

Seasonality Index 179

sensible heat flux 37–8, 39, 224

sensible heat measurement 82–91

shelter factors 306–8, 306, 317

Sherwood number 305

Simple Biosphere Model (SiB) 10, 11

siphon and chart recorders 160

skin friction 301, 319

Slab Model 280–1, 281, 328

sleet, formation 152

SNOTEL network 168, 168

snow board 166

snow courses 167, 167

snow formation 152

snow pads 167–8

snow pillows 167–8

snowflake formation 146

snowfall measurement 165–8

inverted funnel method 166

radioactive methods 167

SNOTEL network 168, 168

snow courses 167, 167

snow pads or snow pillows 167–8

using gauges 165–6

using snow board 166

satellite systems 171–3

cloud mapping and

characterization 171–2

passive measurement of cloud

properties 172–3

spaceborne radar 173

soil density 69–70, 69soil heat flow, formal

description 71–2, 72

damping depth and 74

soil heat flux 39–40

measurement 81–2, 82

soil heat flux plates 81–2, 82

soil moisture 385–8, 386, 387

depletion 91, 91

soil surface temperature 66–7

surface energy balance and 67

thermal waves in 72–5, 73

Soil Vegetation Atmosphere Transfer

Schemes (SVATs) 10, 11, 334,

359–74, 388, 391

basis and origin of land-surface

sub-models 359–62

developing realism in 362–73

greening of 369

ongoing developments of land

surface sub-models 370–3,

371, 373

plot scale, one-dimensional

‘micrometerological’

models 364–7, 364

‘two stream’ approximation 365

representation

of carbon dioxide exchange

368–70, 369

of hydrological processes

367–70, 367

requirements 360soil

homogeneous, thermal waves

in 72–5, 73

thermal properties 68–71, 69solar (shortwave) radiation 48, 49

Solar Constant 36, 54

solar declination 56

solar energy impact, latitudinal

differences in 109

solar radiation 38

actual, at the ground 59

atmospheric attenuation of 58–9, 58

maximum at ground 56–7

maximum at top of

atmosphere 54–6

measurement of 77–80

solar zenith angle 56

South America

Amazon River, ‘river breeze’

effects 8

Andes 9

spaceborne radar 173

spatial correlation of

precipitation 209–11, 210

spatial organization of

precipitation 203–4

specific heat of soil 69, 70

specific humidity of air 15

spectral gap 12, 216

spectral grid 102, 102

spontaneous nucleation 134

stability corrections 289

standard deviation of atmospheric

variable 219–20


Index 447

state variables 101

static stability parameter 32–3

statistics of extremes 190–5

Stefan-Boltzmann constant 338

Stefan-Boltzmann Law 50, 59

stomatal resistance 308–10, 309

per unit area of leaf 309, 310

reciprocal of area-weighted

average 309–10

storm centered analysis 206

stratocumulus clouds 128

stratosphere 5

stratus 140

subadiabatic atmosphere 33

subsidence 250

substratum heat flux 40

subsurface soil temperatures 67–8,

68, 69

summation convention 232

Sun—Earth distance 55, 55

sunset hour angle 57

superadiabatic atmosphere 33

surface emissivity of natural

surface 52

for different surfaces 53surface energy fluxes 36–47

units 36

energy balance of ideal

surface 38–45, 38

surface layer scaling 289–99

surface mixing cloud 130

surface, net radiation at 62–3

system signature

of precipitation 187–8, 189

of storm 176

Taylor expansion 235

Television Infrared Operational

Satellite (TIROS) Operational

Vertical Sounder (TOVS)

instruments 172

terrestrial radiation 48–65

net radiation at surface 62–3

Theissen method 200, 202–3,

204, 207

thermal conductivity of soil 69, 70

thermal diffusivity of soil 69, 71

thermocline 116

thermoelectric pyranometers 78–9, 79

thermohaline circulation 118

thermosphere 5

time-average of weather, climate

as 1–2

time-domain reflectometer sensors 91

time rate of change in a fluid 232–4, 233

tipping bucket rain gauge 157,

161–2, 162

TOP model 370

topography, effect of, on convection

and precipitations 383–5

TOVS precipitation estimate 172

Trade Winds 112

transformal distributions 193

transmissivity 51

Triangle method 200, 202, 203

Tropical Rainfall Measuring mission

(TRMM) 173

tropical storms 119–20, 120

troposphere 5

turbulence, atmospheric

advective and turbulent fluxes

225–9, 227–9

averaging of decomposed

variables 217–19

kinematic flux 223–4, 225linear correlation coefficient

221–3, 223

mean and fluctuating

components 216–17, 217, 217measures of strength of 220–1

signature and spectrum of 213–16,

214, 215

turbulent flux 223

turbulent closure 279–80

local 280

local, first order 281–2, 282, 287–8

low order 280–1

nonlocal 280

turbulent eddies 85–7, 87, 214, 214,

223, 225–9

divergence 250

turbulent flux see turbulent eddies

turbulent intensity 220

turbulent kinematic flux of water

vapor 284

Turbulent Kinetic Energy (TKE) 221

prognostic equations for 269–71,

270, 271, 272

typhoons 120

ultraviolet radiation 6

ultraviolet waveband 48

United Nations Environment

Programme (UNEP) 104

United Nations Framework

Convention on Climate

Change 104

universal gas constant 16

USA

Great Lakes, ‘lake effect’ 8

Rocky Mountains 9

vapor pressure deficit 20, 336

vapor pressure of air, measuring

21–3

vapor pressure of moist air 17

variance of atmospheric

variable 219–20

vector algebra 232

vegetation cover 388–9, 389

vertical gradients in

atmosphere 25–34

vertical pressure and temperature

gradients, atmospheric

29–30, 30


26, 224

atmospheric 31–2, 31

virtual temperature 17–18, 26

visible waveband 48

volcanic pollution 110

von Kármán constant 286

warm clouds 134

warm front 138, 139

water balance equation 88, 88

water cycle, global 1–13

water molecule capture rate by water

surface 19

water reservoirs, estimated sizes 2, 3water vapor flux 224

water vapor in the atmosphere 123–5,

124, 125

watersheds, water balance of 89–90

wavelength, separation of 48, 49

weather prediction 98–100

Weather Research and Forecasting

(WRF) model 389

Wein’s Law 50

wet bulb depression 23

wet bulb equation 23

wet bulb temperature 22, 23

wetting errors 157


448 Index

whole-canopy interactions 316–33

aerodynamics and canopy

structure 317–19, 318

equilibrium evaporation 325–7, 326

evaporation into an unsaturated

atmosphere 327–32, 328,

330, 331

roughness sublayer 321–3

wet canopies 323–5

World Meterological Organization

(WMO) 98, 104

recommended rain gauge

densities 164, 165World Weather Watch (WWW) 98

zero flux plane 91

zero identity 252

zero order closure 280

zero plane displacement 286

Z-R relationship 170


TERRESTRIAL HYDROMETEOROLOGY - Universidad ZamoranoTERRESTRIAL HYDROMETEOROLOGY W. JAMES...

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Transcript of TERRESTRIAL HYDROMETEOROLOGY - Universidad ZamoranoTERRESTRIAL HYDROMETEOROLOGY W. JAMES...