—a - ‘a- —________

ENERGY ANALYSIS OF CLIMATIC INPUTS TO AGRICULTURE

BY

DENNIS PETER SWANEY

•1

I

A THESIS PRESENTED TO THE GRADUATE COUNCIL OFThE UEIVERSTTY OF FLORIDA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR ThEDEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

1978

‘, 444.7

ACKNOWLEDGEMENTS

To my committee members, I am grateful for suggestions and criticism,

Professor H. T. Odum, my supervisory professor, inspiredme to undertake

the study in the first place, and taught me not to fear large—scale research

problems. Professor Richard Fluck initially inspired my interest in

agricultural energetics. Professor Wayne Huber and Professor Flora Wang

read and critiqued some calculations and clarified my understanding of some

climatic processes.

Neil Sipe taught me much of the computer graphics and Bruce Darby

helped with data.

I owe my thanks to John Alexander for his advice and encouragement,

oan Breeze for her ability to smile in the face of revision, and to

my associates for their support and criticism.

Work was supported by Department of Energy contract #EY—76—5—05—4398,

“Energy Bases of the United States,” H. T. Odum, principal investigator.

to .1

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ITABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS

LIST OF TABLES v

LIST OF FIGURES

ABSTRACT xiii

INTRODUCTION 3

METhODS 25

Summary of Energy Evaluation Procedures Used 25Methods of Calculation of Climatic Energy Flows 27Mapping of Energy Flows 36

RESULTS 38

Quality Factors 38Maps of United States Climate in Energy Units 43Maps of United States Agricultural Productivity 105

DISCUSSION 119

Comparison of Production Functions with AgriculturalProductivity 130General Effects of the Environment on United StatesAgriculture 132Limitations to Results 133Investment Ratio for U.S. Agriculture 133Summary and Conclusions 134

APPENDIX I CALCULATION OF SOLAR INSOLATION 140

APPENDIX II DERIVATION OF EDDY DIFFUSION COEFFICIENTS 156

APPENDIX III CALCULATION OF PRODUCTION OF TURBULENT KINETIC ENERGY 159

APPENDIX IV AThOSPHERIC HEAT FLUX AND RATE—OF—CHANOF 160

APPENDIX V ATMOSPHERIC RATES—OF—CHANCE OF VAPOR PRESSURE . . . 166

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APPENDIX VI FLOW OF CHEMICAL POTENTIAL IN HUMIDITY GRADIENTS 170

APPENDIX VII FREE ENERGY VALUE OF VERTICAL DIFFUSION OF WATER

VAPOR173

APPENDIX VIII FREE ENERGY VALUE OF WATER VAPOR ADVECTION 174

APPENDIX IX FREQUENCY DISTRIBUTIONS OF CLIMATIC VARIABLES 181

REFERENCES194

BIOGRAPHICAL SKETCH198

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iv

V

Table

LIST OF TABLES

1

2

3

4

Page

Values of quality factors

Energy values of crops used to calculate qualityfactors for agricultural productioi

Mean values of climatic variables for the United States

Values of correlation coefficients befl.zeen productionfunctions and agricultural production variables

39

100

117

131

.‘: t ç 1

.;. . . 1. .:.!

— :

H... .H

fAST OF Ft CIJRCS

Key to symhols used in energy circuit diagrams(odtim et a 1. 1976) 5

2 Energy circuit model of agricultural interactionsillustrating complexity of the probi em 7

3 a. Aggregated model of agricultural interactions;b. Aggregated model illustrating climatic interactionsonly 10

4 Partial production function No. 1: Sum of heat—equivalentvalues of inputs (J1) as a measure of contribution toagricultural prodtictiviLy (J2) 12

5 Partial production function No. 2: Sum of solar—equivalentvalues of inputs (J1) as a measure of inputs to productivity (J2) 14

6 Partial prodiicti on function No. 3: Sum of solar energyplus solar equivalent values of those inputs greaterthan die local value of solar insolat ion (J1) as a measure

2of inputs to productivity (12) 16

7 Partial production function No. 4: Product of heat—equivalent values of inputs (J1) as a measure of inputto agricultural productivity (J2) 18

8 Partial production function No 5: Percent availableinsolation at surface level as a measure of productivity (j2) 20

9 Incoming—minus—reflected solar Insolation at,tlie groundfor the United Slates, January 1975. (Kc/C/day.) Datavalue extremes are 136.06 Kc/r/day and 3316.39 Kc/m2/day.Absolute value range applying to each level Kc/m2/day ... 45

10 tucoming—miritis—refiected solar insolation at the ground forthe tin I ted States, April , 1975. (Kc/nt’/day . ) Data vu Lueextremes are 890. 20 Kc/m2/duy and 6715.67 Ke/m2/&Iny.Absolute value range apply i iig to cacti leve I Kc/m2/day ... 67

11 incoming—udnus—reflected solar insolation at the ground[or the Un! ted States, .Tu] y, 1975. (Kc/m2/diiy. ) Data valueextremes are I S/ 3.11 Kc/m2/tlay and 8056.98 Kc/m2/dnyAbsolute v;Lltle range apply lug to each level Kc/m2/day . . . 69

vi

Figure

_

12 Incoming—minus—reflected solar insolation at the groundfor the United States, 0ctober 1975. (Kc/m2/day.) 9Datavalue extremes are 359.86 Kc/rr/day and 5018.54 Kc/m/day.Absolute value range applying to each level KcIm2/day 51

13 Average atmospheric boundary layer production of mechanicalturbulence over the United States, January, 1975. (Heat

2equivalents Kc/m2/day.) Data value extremes are 0.0 Kcfm Iday and 3.76 Kc/m2/day Absolute value range applying toeach level Kclm2lday 53

16 Average atmospheric boundary layer production of mechanicalturbulence over the United States, April, 1975. (Heat

2equivalents Kc/m2day.) Data value extremes are 0.0 Kc/m /day and 8.50 KcJm /day Absolute value range applying toeach level Kc/m2/day 55

15 Average atmospheric boundary layer production of mechanicalturbulence over the United States, July, 1975. (Heat

2equivalents Kc/m2/day.) Data value extremes are 0.0 Kc/m /day and 3.70 Kc/m2/day Absolute value range applying toeach level Kc/m2/day 57

16 Average atmospheric boundary layer production of mechanicalturbulence over the United States, October, 1975. (Heatequivalents 1Cc/rn /day.) Data value extremes are 0.0 1Cc/rn /day and 3.58 Kc/m2/day Absolute value range applying toeach level Kc/m2/day 59

17 Rate of change of heat in the atmospheric boundary layerdue to (vertical) convection over the United States,January, 1975. (Heat equivalents Kc/m2/day.) Data valueextremes are —312.60 Kc/m2/day and 27.02 Kc/m2/day.Absolute value range applying to each level Kc!m2/day . 61

18 Rate of change of heat in the atmospheric boundary layerdue to (vertical) convection over the United States,April, 1975. (Heat equivalents Kc/m2/day.) Data valueextremes are —1905.79 Kc/m2/day and 17.09 Kc/m2/day.Absolute value range applying to each level Kc/m2fday . 63

19 Rate of change of heat in the atmospheric boundary layerdue to (vertical) convection over the United States,July, 1975. (Heat equivalents Kc/m2/day.) Data valueextremes are —2890.05 Kc/m2/day and 2.24 Kc/m2/day.Absolute value range applying to each level Kc/m2/day . 65

20 Rate of change of heat in the atmospheric boundary layerdue to (vertical) convection over the United States,October, 1975. (Heat equivalents Kc/m2/day.) Data valueextremes are —1605.78 Kc/m2/day and 10.55 Kc/m2/day.Absolute value range applying to each level Kc/m2/day . . 67

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21 Rate of change of heat in the atmospheric boundary layerdue to (horizontal) advection over the United States,January, 1975. (Heat equivalents, Kc/m2/day.) Data valueextremes are —9692.02 Kcfm2/day and 11353.82 Kc/m2lday.Absolute value range applying to each level Kc/m2/day . . 69

22 Rate of change of heat in the atmospheric boundary layerdue to (horizontal) advection over the United States,April, 1975. (Heat equivalents, Kc/m2/day.) Data valueextremes are —876.97 Kc/m2/day and 1187.13. Kc/m2/day.Absolute value range applying to each level Kc/m2/day 71

23 Rate of change of heat in the atmospheric boundary layerdue to (horizontal) advection over the United States,July, 1975. (Heat equivalents Kc/m2/day.) Data valueextremes are —4531.14 Kc/m2/day and 11842.31 Kc/m2/day.Absolute value range applying to each level Kc/m2/day . 73

24 Rate of change of heat in the atmospheric boundary layerdue to (horizontal) advecton over the United States,October, 1975. (Heat equivalents, Kc/m2/day.)’ Data valueextremes are —1909.63 Kc/m2/day and 3217.49 Kcfm2/day.Absolute value range applying to each level Kc/m2/day . 75

25 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (vertical) convectionover the United States, January, 1975. (Heat equivalentsKc/m2/day.)2 Data value extremes are 0.04 Kc/nr/day and

a 715.84 Kc/m /day. Absolute value range applying to eachlevel Kc/m2/day 77

26 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (vertical) convectionover the United States, April, 1975. (Heat equivalentsKcIm2/day.) 2ta value extremes are 0.11 I(c/m2/day and1663.70 Kc/m /day. Absolute value range applying to eachlevel Kc/m2/day 79

27 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (vertical) convectionover the United States, July, 1975. (Heat equivalentsKc/m2/day.) Data value extremes are 0.03 Kc/m2/day and12665.96 Kc/m2lday. Absolute value range applying to eachlevel Kc/m2/day 81

28 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (vertical) convectionover the United States, October, 1975. (Heat equivalentsKc/m2/day.) 9ata value extremes are 0.0 Kc/m2/day and1871.82 Kc/m /day. Absolute value range applying to eachlevel Kc/m2/day 83

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29 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (horizontal)advection over the United States, January, 1975. (Heatequivalents Kc/m2!day.) Data value extremes are —976.58Kc/m2/day and 948.11 Kc/m2/day. Absolute value rangeapplying to each level Kc/m2lday 85

30 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (horizontal)advection over the United States, April, 1975. (Heatequivalents Kc/m2!day.) Data value extremes are —656.29Kc/m2/day and 1283.01 Kc/m2/day. Absolute va)ue rangeapplying to each level Kc/m2/day 87

31 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (horizontal)advection over the United States, July, 1975. (Heatequivalents Kc/r/day.) Data value extremes are —7323.61Kc/m2/day and 78980.50 Kc/m2/day. Absolute value rangeapplying to each level Kc/m2/day 89

32 Rate of change of chemical potential of water vapor inthe atmospheric boundary layer due to (horizontal)advection over the United States, October, 1975. (Heatequivalents Kc/m2/day.) Data value extremes are —514.80Kc/n2/day and 897.41 Kc/m2/day. Absolute value rangeapplying to each level Kc/m2/day 91

33 Chemical potential of rainfall/runoff for the United States,January, 1975. (Heat equivalents Kcal/m2/day times 100.)Data value extremes are 0.0 Kc/m2/day and 16.06 Kc/m2/day.Absolute value range applying to each level Kc/m2/day . . . . 93

34 Chemical potential of rainfall/runoff for the United States,April, 1975. (Heat equivalents Kcal/m2/day times 100.)Data value extremes are 0.0 Kc/m2/day and 9.52 Kc/m2/day.Absolute value range applying to each level Kc/m2/day . . . . 95

35 Chemical potential of rainfall/runoff for the United States,July, 1975. (Heat equivalents Kcal/m2/day times 100.)Data value extremes are 0.0 Kc/m2/day and 13.60 Kc/m2Jday.Absolute value range applying to each level Kc/m2/day . . . . 97

36 Chemical potential of rainfall/runoff for the United States,October, 1975. (Heat equivalents Kcal/rn/day times 100.)Data value extremes are 0.0 Kc/m2/day and 19.15 Kc/m2/day.Absolute value range applying to each level Kc/m2/day . . . . 99

37 Energy value of corn harvest for the United States, 1975.(heat equivalents Kc/m2/day.) Data value extremes are 0.0Kc/m2/day and 6.94 Kc/m2/day. Absolute value range applyingto each level Kc/m2/day 107

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38 Energy value of wheat harvest for the United States, 1975.

(Heat equivalents Kc/m2/day.) Data value extremes are 0.0

Kc/m2/day and 4.33 Kc/m2/day. Absolute value range applying

to each level Kc/m2/day109

39 Energy value of soybean harvest for the United States, 1975.

(Heat equivalents Kc/m2/day.) Data value extremes are 0.0

Kc/m2/day and 0.89 Kc/m2/day. Absolute value range applying

to each level Kc/m2/dayIll

40 Energy value of vegetable harvest for the United States,

1975. (Heat equivalents KcJm2/day.) Data value extremes

are 0.0 Kc/m2lday and 2.23 Kc/m2/day. Absolute value range

applying to each level Kc/m2/day 113

41 Energy value of four sectors of harvest for the United

States, 1975. Area—weighted average (Kc/sq. meter—day)

heat equivalents. Data value extremes are 0.0 Kc/m2/day

and 4.36 Kc/m2/day. Absolute value range applying to

each level Kc/m2/day115

62 Energy value of Partial Production Function Ill for the

United States, 1975. Sum of all climatic inputs. (Heat

equivalents Kc/m2/day.) Data value extremes are 3059.27

9 2Kc/m/day and 19834.69 Kc/m /day. Absolute value range

applying to each level Kc!m2/day 121

43 Energy value of Partial Production Function 112 for the

United States, 1975. Sum of all climatic inputs. (Solar

equivalents KcJm2/day.) Data value extremes are 21927.53

Kc/m2/day and 233069.56 Kc/m2/day. Absolute value range

applying to each level Kc/m2/day 123

44 Energy value of Partial Production Function 113 for the

United States, 1975. Local solar input plus climatic

inputs with values greater than local solar input. (Solar

equivalents Kc/m2/day.) Data value extremes are 20578.46

Kc/m2/day and 232975.06 KcJm2/day. Absolute value range

applying to each level Kc/m2/day 125

45 Energy value of Partial Production Function 114 for the

United States, 1975. Multiplicative interqction of climatic

inputs times .001. (heat equivalents Kc!m’/day.) 2flata value

extremes are 1466.00 Kc/m2/day and 3335643.00 Kc4m /day.

Absolute value range applying to each level Kc/m’/day . . . . 127

46 Energy value of Partial Production Function //5 for the

United States, 1975. Percent insolation need at the surface.

(From surface albedo values of Kung, Bryson, and Lenschow,

1964.) Data value extremes are 76.67 and 87.Q0. Absolute

value range applying to each level percent 129

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47 Energy signature—heat equivalents. Log (energy) versuslog (quality factor). Straight line represents world—average heat equivalent energy flows. See Table 3 136

48 Energy signature—solar equivalents. Log (energy) versuslog (quality factor). Upper horizontal line representsworld—average solar—equivalent energy flows 138

Al Summer surface albedoes for North America (Kung, Bryson,and Lenschow, 1964)143

AZ Spring and Fall surface albedoes for North America (Kung,Bryson, and Lenschow, 1964) 145

A3 Winter surface ?lbedoes for North America (Kung, Bryson,and Lenschow, 1964) 147

A4 Distribution of eddy diffusion coefficients for the UnitedStates2 January, 1975. Data value extremes are 0.0 m2/s to2.75 m’/s. Absolute value range applying to each level . . . 149

A5 Distribution of eddy diffusion coefficients for the UnitedStates, April, 1975. Data value extremes are 0.01 m2!s and6.07 m2/s. Absolute value range applying to each level . . . 151

A6 Distribution of eddy diffusion coefficients for the UnitedStates, July, 1975. Data value extremes are 0.0 m2/s and5.85 m2/s. Absolute value range applying to each level . . 153

A7 Distribution of eddy diffusion coefficients for the UnitedStates October, 1975. Data value extremes are 0.0 m2/s and4.56 vr’/s. Absolute value range applying to each level . . . 155

A8 Locations of 66 data—gathering stations used in the calculation and mapping of climatic data 176

A9 Locations of 235 data—gathering stations used in the calculation and mapping of climatic variables 178

AlO Frequency distribution of solar insolation, 1975. A) January;B) April; C) July; 0) October 181

All Frequency distribution for mechanical production of turbulence. A) January; B) April; C) July; n) October . . 183

A12 Frequency distribution for vertical component of rate—of—change of heat. A) January; B) April; C) July;0) October

185

A13 Frequency distribution of horizontal component of rate—of—change of heat. A) January; B) April; C) July;1)) October187

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A14 Frequency distribution for vertical component of rate—of—change of chemical potential of water vapor. A) January;B) April; C) July; D) October 189

AU Frequency distribution for horizontal component of rate—of—change of chemical potential of water vapor. A) January;B) April; C) July; 0) October 191

A16 Frequency distribution for rate—of—change of chemicalpotential of rainfall. A) January; B) April; C) July;0) October 193

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Abstract of Thesis Presented to the Graduate Councilof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

ENERGY ANALYSIS OF CLThL&TIC INPUTS TO AGRICULTURE

DENNIS PETER SWANEY

December, 1978

Chairman: H.T. OdumMajor Department: Environmental Engineering

An evaluation was made of the potential energy content of climatic

inputs to productivity in the continental United States using energy

analysis concepts involving embodied energy. Spatial distribution of

energy flows of climatic variables for four months of the year 1975 were

calculated and computer—drawn maps of the variables were prepared using a

computer—graphics package (SYMAP). The climatic variables calculated

include:

1) net solar insolation (incoming minus reflected)

2) rate—of—change of heat in the atmospheric boundary layer due to

advection

3) rate—of—change of heat in the atmospheric boundary layer due to

convection

4) rate—of—change of chemical potential of water vapor in the

atmospheric boundary layer due to advection

5) rate—of—change of chemical potential of water vapor in the

atmospheric boundary layer due to convection

6) rate of mechanical production of turbulent kinetic energy in the

atmospheric boundary layer

xiii

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7) rate—of—change of chemical potential energy from rainfall to

surface water.

Solar quality factors for all variables were calculated from world—

average data, in order to transform all inputs .qnd outputs into energy

equivalents of the same type for purposes of comparison. In solar equi—

valents kilocalories/heat equivalent kilocalories these are: horizontal

heat change — 5.25; vertical heat change — 12.6; change of chemical

potential of water vapor (horizontal and vertical) — 13.7; mechanical pro

duction of turbulence — 56.4.

Average energy values of the inputs and outputs for the United States

Were plotted against their quality factors to obtain an “energy signature”

for agriculture, which was compared to a theoretical world—average energy

signature. Most climatic inputs to the U.S. agriculture were potentially

much larger than those from the economy when expressed in energetic terms.

Energy flows were combined in five production functions and added for

the United States to estimate impact on the economy. The ratio of pur—

chased inputs of agriculture to climatic inputs was calculated in solar

equivalents and shown to be smaller (0.02) than when calculated using solar

insolation only (0.5), since the value of sunlight is being counted more

than once in climatic flows.

Energy values of production for four agricultural’ sectors, as well as

their area—weighted average were also calculated and mapped over the United

States by state for the same year. Overall, the u.s. apparently receives

more climatic energy generated from elsewhere and delivered to the U.S.

than vice versa.

4 ‘fQLtttChairman

xiv

INTRODUCTION

This thesis investigates the energetic inputs from climate to

agricultural production in the United States. Solar insolation, rainfall,

wind action, temperature and humidity are well—known factors in crop

growth. Energy budgets of crops and forests have been studied inten

sively for many years) but contributions of climatic factors to the output

and economics of agriculture have not been clear because climatic energies

are of different quality. By representing all energy flows in embodied

Calories of sunlight, each energy source can be compared. Production

also depends on energy flow from important stored constituents in soil

and water reservoirs, which are not included in this study.

By mapping the spatial distribution of climatic variables over the

United States (in terms of their heat—equivalent energy flaws) and applying

energy transformation factors (energy quality factors), indices of flows

of potential energy were developed and related to observed agricultural

production. The average embodied climatic energies for the U.S. were then

compared to the economic energies that can be attracted in the forms of

labor, fuel, machinery and fertilizer, by calculating the embodied energy

of their dollar value, and determining the investment ratio of economic

inputs to climatic inputs.

The analysis of the interaction between climatic energy flows and

the agricultural system was described in three conceptual parts:

1. Description and evaluation of relevant climatic energy flows.

2. Summary of production of corn, wheat, soybeans, and vegetables

I

2

and the sum of those four sectors of United States agricultural

production.

3. Evaluation of the energetic effects of climate on the pro

duction functions of the agricultural system.

The climatic energy values evaluated ware the following:

1. available solar insolation

2. production of turbulent kinetic energy of wind

3. atmospheric rate of change of heat (horizontally and vertically)

4. chemical free—energy change of water vapor flows

5. chemical free—energy change of rain to runoff.

These flows vary spatially over the climatic regions of the United

States, as well as seasonally. For this reason, these energy flows were

evaluated at many locations over the United States for four different months

of the base year, 1975.

Maps of the climatic energy flows for each case were constructed using

the SYMAP (Dougenik and Sheehan, 1975) computer graphics routine, to identify

zones of climatic energy flow.

Similarly, zones of agricultural energy transformation were identified

using SYMAP, by obtaining energy values of the harvests of the following

crops for the year 1975, by state:

1. corn

2. wheat

3. soybeans

4. vegetables

Interactions of the energy variables were originally conceptualized

Using energy circuit models as a basis. An energy circuit model is a

Conceptual model of a system in the Energy Circuit Language of II. T. Odum

3

(Odum, 1972). Models of this type can be described mathematically as

systems of non—linear, deterministic differential equations which are

often analytically intractable. Explanations of the Energy Circuit

Language symbols appear in Fig. 1. An Energy Circuit Model which reveals

some of the complexity of agricultural systems is shown in Fig. 2. Included

are energetic effects of climatic variables, soil, water, and feedbacks from

human systems.

Input flows from the environment as well as feedback flows from labor

and industry interact to produce a storage of agricultural goods, which,

in turn, are harvested as food. Money flows in the opposite direction of

energy for the economy, but none flows to the atmosphere or the lithosphere

directly. These provide an energy subsidy to the economy because they are

external to the market system.

Figure 2 is an illustration of the energetic interaction between

agriculture, environment and economy. This model was constructed by con

sidering actual mechanisms of energy and mass flow in the environment, as

well as the components of human activities necessary to bring a crop to its

maturity and harvest. A yet more detailed model could be constructed to

include interaction between climatic variables and human inputs (e.g., the

effect of varying weather conditions on mechanical harvest efficiency).

Although quantitatively more precise than the production functions

used in this analysis, a model with the complexity of Fig. 2 is difficult to

evaluate, and even so, only approximates the actual relationships between

the variables shown. The value of such a model is often to provide a con

ceptual basis for developing more simplistic “mini—models,” which can he

evaluated, tested, and computer—simulated. An example of a simplified

model is shown in Fig. 3a.

_,..tfl

Fig. 1. Key to symbols used in energy circuit diagrams(Odum et al., 1976).

5

Energy source (forcing function),

( ) source of external cause.

S.———.’

_L.. Heat sink, outflow of used energy.

Energy interaction, one type of energyamplifies energy of a different quality(usually a multiplier).

—-cs—.!* Economic transaction and price function.

-

Storage (state variable).

}Depreciotion

—Circulating energy transformer with

I ) Michaelis—Menton kinetics (diminishingreturns transfer function).

On—off control work (digital actions).

.-J- {j- 1 13)}

and (3) general purpose box for miscel—-r laneous subsystems.

- -- -- r_..r-- ,-,,—

Fig. 2. Energy circuit model of agricultural interactionsillustrating complexity of the problem.

7

*t

8

The model of Fig. 3a has aggregated and simplified several flows in

•-,rder to accentuate the major relationships of the agricultural system.

Most of the original components remain in the model, but some of the

Inaii5 of their interaction have been omitted. Aggregated models are

r;eful in conceptualizing and explaining the “whole—system” features and

Eroperties of systems without a mass of detail.

The variables of climatic energy change were combined according to

five ways of calculating effect on agricultural production. These numerical

:uices are partial production functions since they do not include all the

Lifl factors in production. The indices were correlated with the energy

value of agricultural production (by state) to determine their contribution

to prediction of productivity.

The five partial production functions as represented in energy circuit

language are shown in Figs. 1—3.

1. Summation of climatic flows in heat equivalent energy units

2. Summation of climatic flows in solar equivalent energy units

(i.e., including quality factors for each flow)

3. Summation of climatic flows whose inputs are greater than local

solar insolation (solar equivalent energy units)

6. Product of climatic flows (heat equivalent energy units)

5. Percent solar insolation available at the surface.

The first partial production function is actually equivalent to the

anount of energy used for agricultural production in Fig. 4) plus that

amount which is used and dispersed as heat in the process (J1 in Fig. 4).

If We assume that the unused portion is proportional to that which is used,

th0 the stun of the two (which is most easily measured) should be an index

of production

______

___

p

Fig. 3a. Aggregated model of agricultural interactions.

b. Aggregated model illustrating climatic interactions only.

I0

AGRICULTURAL

PRODUCTION

NI A N

ECONOMY

(a)

W ND

AGRICULTURAL

PRODUCTION

(b)

Fig. 4. Partial production function No. 1: Sum of heat—equivalent

values of inputs (J1) as a measure of contribution toagricultural productivity (J2).

12

Ja

JI

Fig. 5. Partial production function No. 2: Sum of solar—equivalentvalues of inputs (J1) as a measure of inputs to productivity(J2).

14

PRODUCTION

Fig. 6. Partial production function No. 3: Sum of solar energyplus solar equivalent values of those inputs greater thanthe local value of solar insolation (J1) as a measure ofinputs to productivity

16

JI

Ja

AL SEDO

Fig. 7. Partial production function No. 6: Product of heat—equivalentvalues of inputs (J1) as a measure of input to agriculturalproductivity (J2).

st

18r

‘I

F ROD U CT 0 NJa

AL8EDO

-a

Fig. 8. Partial production function No. 5: Percent availableisolation at surface level (J1) as a measure of productivity

20

J2

•

— _‘Etr.’......

21

A different interpretation of the sum of all inputs evaluated in

heat equivalents could be visualized as the input to a heat engine (a

highly simplistic view) with agricultural production and waste heat as

output. The ratio of agricultural production to the sum of all inputs

(both in heat equivalent energy units) can be considered a Carnot

efficiency for this simple heat—engine model.

The second alternative (summation of inputs in solar equivalents)

weights the inputs of small magnitude higher than those of large magni

tude, recognizing the significance of the role of “high quality” energy

in specialized functions (e.g., turbulence is necessary for fast—response

mixing in plant canopies and must be generated from lower quality energy

sources for this to occur). Yet, the input of turbulent kinetic energy

(heat calories) relative to some other forms is small.

This alternative permits a comparison between the embodied energies

of all inputs (i.e., the total energy necessary to product the inputs)

and the embodied energies of the agricultural productivity, as measured

by the wind.

The third alternative (summation of inputs of greater magnitude in

solar equivalents than solar insolation) assumes that if the energy of

each climatic sector were distributed uniformly over the globe, the value

in each sector would, by definition, be equal to that of solar insolation

(in solar equivalents). If one or more of the climatic sectors inputs

energy greater than this value, it represents a local excess of energy in

the sector above the global average, and is thus, locall.y significant,

i.e., local systems adapted to this source may have advantages over com

Peting systems. Climatic encrgy sectors with values less than the value

Of solar insolation are assumed to be locally less than the global average,

22

and thus, perhaps less significant. Therefore, this partial production

fijnction alternative includes only those energetic inputs that are deemed

(theoretically) to be locally significant.

The fourth alternative of the five partial production functions

evaluated (products of all inputs) recognizes the mechanism of the

‘liniting factor.” In the previous three production functions, production

varies directly with the magnitudes of the inputs, but falls to zero only

if the magnitudes of all inputs fall to zero. This partial production

function is the simplest of a class of functions whose outputs go to zero

if any one of the inputs goes to zero, a behavior often attributed to many

natural systems.

The fifth and final partial production function to be used is percent

incoming solar insolation available, which is defined as:

F = 100—A

where: F = percent incoming solar insolation available at the surface

A = local surface albedo, expressed as a percent

The basis for including this function as an index of production is the

theory that natural systems adapt to maximize the use of local available

energy. Consequently, a well—adapted system might therefore cause a rela

tively low surface albedo and consequently use a high percentage of the

available solar insolation. Assuming that agriculture in the United States

iS well—adapted to the environment, this function could be a valid index of

productivity at the end of the growing season.

Agricultural productivity data were obtained by state for the year

1975 for four principal growing sectors:

1) corn for grain

2) wheat (all types)

23

they

3) soybeans

6) vegetables

5) sum of the above four sectors

The first four variables were obtained both as mass of harvest per unit

area (yield) and as a dollar value per unit area (U.S.D.A., 1977). The mass

of harvest was converted to heat equivalents by multiplying by the kilocalorie

content of each sector. Solar equivalents were calculated from economic value

of each sector

The method of analysis used herein, i.e., energy analysis, is a rela

tively new tool in the study of environmental and social systems, although

the energy budget of the atmosphere has been studied by climatologists and

meteorologists for many years (Budyko, 1974; Sellers, 1965; Trewartha,

1968). As long ago as 1735, the interrelationship between plant life and

heat flow was scientifically observed by Reaumur (Thornthwaite and liather,

1954). Several models of the general circulation of atmosphere are in

existence (Smagorinsky, 1963; Arakawa, Katayama, and Mintz, 1968), but

have not been coupled with models of agriculture to investigate their

interaction, possibly because of the incredible level of complexity of the

models. Many studies of the relationships between single meteorological

variables (such as temperature) and response of individual species have

been made (Andrewartha and Birch, 1974); and between climatic variables

and crops (Monteith, 1965; Thornthwaite and Mather, 1954; Thompson, 1974;

Jackson, 1977). Holdridge, in 1947, proposed an index of ecosystem

classification based on rainfall and temperature, thus emphasizing climatic

dependence of vegetation (Holdridge, 1947). The interrelationships between

temperature, relative humidity, evapotranspiration, potential evapotrans—

piration, and productivity have been examined by several authors (Rosenzweig,

_______-

24

1968; Penman, 1948; Holdridge, 1959; Thornthwaite and Nather, 1957; Van

Bavel, 1966). Simple regression models of net primary productivity with

temperature and rainfall, with maps of their global distribution were

generated by Lieth and Whittaker (1974).R

Energy analyses of agricultural processes were done by Odum (1967,

1970) and Steinhart and Steinhart (1974); Hurst (1974); and Pimentel at al.

(1973); Leach (1976) and Slesser (1973). Energy inputs from the economy

were included but the climatic energy inputs were not calculated and

represented in a comparable way.

—

F METHODSt

Energy Circuit Language was used to conceptualize models of climatic

input to production, as discussed in the introduction.

Next, WATFIV (Merchant and Sturgul, 1977) computer programs were

developed to calculate heat equivalent values of climatic energy van—

Ifables and production functions from spatially distributed climatic data.

Energy quality factors for each variable were calculated from world—

average data to obtain solar equivalent energy values. Energy equivalents

of crops were also calculated from state average values of yield. After

the values of each variable had been calculated, another computer package,

SYMAP (Dougenik and Sheehan, 1975), was used to map the distribution of the

calculated energy values over the continental United States.

Finally, SAS (Barr, et al. , 1976) statistical programs were used to

compute the correlation between models of each production function, and

calculated agricultural productivity, as both vary over the continental

United States.

Summary of Energy Evaluation Procedures Used

The energetic value of environmental flows, as well as those of

feedbacks from the human economy, may be assessed in at least two ways:

as the physically measurable value of the energy of the specific flows,

or as the “embodied energy” of the flows. There are reasons for doing

both.

The physically measurable quantity of energy in each flow (heat

calories) is likely to be universally agreed upon, once the flow has been

25

_! v..: —

IlL,26

clearly specified and precisely measured. However, all inputs to a complex

system can never be estimated directly because of the ranges of scale in the

flows, their temporal range, and the sheer number and complexity of inputs.

Main flows, however, can be estimated so as to evaluate the model in Fig.

3a.

The “embodied energy” of a flow is that amount of energy required to

create the flow from a standard form of energy by way of a stated process

or series of processes. The physically measurable quantity of the energy

of a flow tells us nothing of the amount of energy required to initiate

and support the flow (embodied energy), which is the energy value required

to replace the flow, and hence, important in considering alternate flows.

For example, the embodied energy of an automobile in solar equivalent energy

units is equal to the amount of sunlight consumed in all support processes

needed to construct the automobile. To obtain a true value of the embodied

energy of a flow or storage, we must take the analysis beyond the human

sector to the subsidies of environmental parameters. Ultimately, all pro

cesses on the earth have the sun as a source of energy except for some

geological and radioactive heat sources. (Insofar as the earth was a by—

product of the birth of the sun, even these are of solor origin.) The most

important assumption in this work is that of the method of energy analysis

developed by H. T. Odum, et al. (1977). Following this method, all pro—

cesses are evaluated in terms of the equivalent of the solar energy needed

to produce them globally (often a complex calculation). A solar quality

factor, i.e., the ratio o the solar energy embodied in the production of the

flow to the flow itself, can then be estimated, and flows thus described can

then be used to calculate the embodied energies and solar quality factors of -1.hierarchical flows. For example, the ratio of the annual global rate of

H

27

solar insolation to the annual global rate of production of kinetic energy

of wind is an energy quality factor for wind. Energy quality factors for

climatic flows used in this thesis were calculated from global data, and

are shown in the results section.

Methods of Calculation of Climatic Energy Flows

In the next step of the analysis, the flows and storages of the Energy

Circuit Model for the system were evaluated from actual data for the year

1975. Raw data on climate variables were obtained in monthly—averaged form

from the Climatological Data and Rawinsonde Data sections of the Climatological

Data for the United States (National Climatic Center, 1975).

Average monthly values of the following data were obtained (for 235

stations of the continental United States——Climatological data — see

Appendix):

1. surface temperature (°C)

2. surface dew point temperature (°C)

3. percent solar insolation

4. rainfall (mm)

5. resultant wind vector velocity and direction (mis)

(for 64 stations of the continental United States — Rawinsonde data — see

Appendix X):

1. vertical gradient of temperature (°C/m)

2. vertical gradient of dew—point temperature (°C/m)

3. vertical gradient of wind velocity (mis)

These data were obtained for four months (ropresenLing seasonal

variations): January, April, July, and October, and stored as data in a

computer. Five climatic energy flows were calculated from this data, using

the methods of Appendices I — IX, and the WATFIV programming language

t

28

(Merchant and Sturgul, 1977). The climatic energy flows evaluated were

the following:

1. net solar insolation

2. turbulent kinetic energy of wind

3. atmospheric rate—of—change of heat (horizontally and vertically)

4. chemical free—energy change of water vapor (horizontally and

vertically)

5. chemical free—energy change of rain

These flows vary spatially over the climatic regions of the United

States, as well as seasonally.

Solar Insolation

Net solar insolation over the continental U.S. was evaluated for each.

of the four months by equation (1) — (see Appendix I):

Rnet=Rsc•Kp [adsin4)+0.Sbd cos 4) J[ 1—al (1)

where: Rt = solar insolation absorbed at the surface (Kc/m2day)

Rsc = solar constant C = 2.0 Langleys/min)

= % possible sunshine

adt bd = functions of solar declination which vary by month

(see Appendix I)

4) latitude of radiation

a surface albedo

The values of K, percent possible sunshine, are determined by local

atmospheric conditions, and were taken for each location and month from the

tables of climatological data. This value does not Include attenuation of

radiation due to atmospheric scattering and absorption, and thus slightly

overestimated insolation.

29

Values for albedo were obtained From those calculated for the U.S. in

winter, summer, and transition months in Kung, Bryson and Lenschow (1964).

Turbulent Kinetic Energy

The mechanical production of turbulent energy in the planetary boundary

layer by the vertical gradient of horizontal wind vector is given by

equation (2) (see Appendices II and III).

P = 20.56 ZbPK(BU)2

.

(2)

where: P = rate of production of turbulent kinetic energy (Kc/m2/day)

= 1000 = height of planetary boundary layer

p = density of dry air = 1. 23 Kg/rn3

Km = eddy diffusion coefficient (m2/s)

-

= vertical wind velocity gradient(el)

Eddy diffusion coefficients vary with wind velocity and the thermal

stability of the atmosphere. They were estimated for all locations by the

KEYPS equation (Rosenberg, 1974 and Panofsky, 1963; see Appendix II).

The values of were computed by taking the vector difference between

surface wind and wind near the top of the boundary layer, and dividing by

the difference in height [all values from (Rawinsonde Data) — Climatological

Data for the U.S. — 1975J:

(z2-z1)[(u1 - u2 cos ÷ Cu2 sin

fl)2JU2 (3)

where:

subscript (1) refers to the surface value

subscript (2) refers to upper value

0 = angle of difference between the two wind vectors.

30

Turbulent eddies in the boundary layer are responsible for mixing the

armosphere in plant canopies with the drier and sometimes cleaner air aloft,

as well as transporting heat, and thus play a significant but perhaps under

rated role in agricultural production.

Atmospheric flow of heat

Vertical flux of heat in the atmosphere was estimated by the equation:

=—8.64 l0 CPK (4)

where: = vertical heat flux due to turbulent diffusion (Kc/m2 day)

C = specific heat of dry air = 0.24 Kc/Kg —

p = density of dry air = 1.23 Kg/m3

K, = eddy diffusion coefficient for heat (m2/s)

= vertical gradient (lapse rate) of potential temperature

= + F) (°K/m)

F = dry adiabatic lapse rate (—.0098 °K/m)

T = ambient atmospheric temperature (°K)

The values of are related to those of Km (see Appendix II) and vary

with atmospheric conditions. They were calculated separately for each

location, as with Km

The gradient of potential temperatures was calculated by taking the

difference in temperature near the top of the boundary layer and the surface,

dividing by the difference in height; and adding the value of F:

(5)

(All values from “Rawinsonde Data” — Climatological Data — 1975.)

Because most of the rawinsonde data was collected during runs at

1200 OCT (700 EST — early morning), the lapse rate values were

strongly stable, probably due to nocturnal radiational cooling at

the surface. In order to compensate for this, surface values of

the monthly mean maximum temperatures at each location were obtained

from the Climatological Data (corresponding to late afternoon

conditions), heat fluxes were computed using these as T1, and the

average of the two fluxes, early morning and late afternoon, was

determined.

The rate of change of heat in the boundary layer due to vertical heat

flux on a per unit area basis is:

F— H

1 Z3z1b ôZ

where: F1 = rate—of—change of heat in the boundary layer due to

convection (Kc/m2lday)

Zb = height of the boundary layer

j =l000m

Thus, in this analysis, F1 was taken to be numerically equal to J11.

In addition to vertical heat flux, the rate of change of the column of

air ft the boundary layer due to heat advection, i.e. the horizontal

transport of heat by wind, was determined from equation (6):

F2 8.64 104Z5 (6)

where: F2 = rate—of—change of heat in a column of air of the height

of the boundary layer (Kc/m2/day)

31

32

• Zb = height of boundary layer = 1000 m

C = specific heat of dry air = 0.24 KcIKg °K

p = density of dry air = 1.23 Kg/rn3

-r = horizontal temperature gradient C K/rn)

= horizontal velocity vector (mis)

In order to estimate mean monthly surface temperatures and wind

velocities from each of 235 stations of the continental U.S., as well as

x, y coordinates for each station were obtained, in order to generate a

map of surface isotherms using the SYMAP computer graphics package. A

SYMAP program fit a polynomial equation in x and y (the coordinates of

• the map) to the temperature surface for each month. Correlation co

efficients of the fits were all > 0.85.

The x and y derivatives of the least—squares equations for temperature

were obtained by hand, and the resulting equations for - and the x

and y components of the horizontal temperature gradients were multiplied

• by the x and y components of velocity (by a WATFIV computer program) at

each point:

F = 8.64 lO6Zb c u, + - Uyl (Kcim2/day) (7)

The value of F represents the difference between horizontal heat

fluxes into and out of a unit air column of boundary layer height. Some

of this heat serves to increase local temperature. The remainder flows

vertically, or serves as a local source of energy.

Because the heat content and flux of the local environment is known

to affect metabolism of all life forms, these values were included as

important components of the total climatic input to agriculture.

Chemical Free Energy of Water Vapor

The rate—of—change of Gibbs free energy in the planetary boundary layer

due to vertical and horizontal flux of water vapor was estimated by

equations (8) and (9)(see Appendices V and VI).

For the vertical case:

0 = 2.056 103K [i_ + + 2e()2]

[ 1 ÷ ln 1 (8)

where: K = eddy diffusion coefficient for water vapor (rn2/s)

e = water vapor pressure (rub)

p = density of dry air = 1.23 Kg/rn3

g = gravitational acceleration = 9.8 rn/s2

p = atmospheric pressure (rub)

G = rate of change of Gibbs free energy due to vertical vapor

flux (Kc/m2day)

For the horizontal case:4-

3 e 4- eGw = 2.056 10 . u [ 1 + in (9)

where: G = rate of change of Gibbs free energy due to horizontal

vapor flux (Kc/m2’day)

= horizontal vapor pressure gradient vector (mb/n)

u = velocity vector (mis)

Equation 8 was evaluated using the “Rawinsonde Data” section of

Clirnatological Data — 1975. Vapor pressure was computed from surface and

Upper values of dew—point temperature using the empirical equation of

Andersen (Andersen

e = 7.749 l0 exp I— Td+405:0265

(10)

Where: e vapor pressure (mb)

Td = dew point temperature (°F)

34

First and second derivatives of e were calculated in a computer

program as:

e—e 2 e—ee 2 l,_ 2 1

— ZfZ1 2— (ZZ)

2

where: subscript 2 refers to upper value

subscript 1 refers to surface value.

The value of the second derivative calculated in this fashion is an

approximation which holds true only for e(z) = constant. To obtain a more

accurate value, at least three levels of values are needed, and, conse

quently, more data and computer storage. For this reason, this simple

approximation was used.

Calculation of p was accomplished by using the hydrostatic approxi—

mation:

P(Z) = 1018 — .01 pgZ (tub) (12)

p = 1.23 Kg/rn3

g=9.8 rn/s

Z = height (m)

The values for the horizontal component of the Gibbs free energy

change were computed analogously to the horizontal components of rate of

change of heat. A least—squares—fit was obtained for the horizontal

variation in x and y for each month, the gradients computed from the least—

squares equation, and multiplied by the velocity components at each point.

The rate of change of Gibbs free energy of water vapor reflects the

ability of air to take up water from the surface. The rate—of—change

has a higher (negative) value for steeper vapor pressure gradients than

for slight ones.

35

Transpiration and water uptake in plants are also affected by the

water content of the air. Transpiration, like perspiration, functions

better in lower humidities, and is aided by the wind—transport of vapor.

The value of the transport of vapor by the atmosphere can perhaps

best be seen if we consider the case of a closed system. A completely

sealed chamber containing a perspiring human, a transpiring plant, and

water at constant temperature and pressure would eventually reach equilib

rium — i.e., the water in the chamber would become saturated due to

evapotranspiration. At this point, the organisms in the chamber would

either stop transpiring water via this mechanism or supply the energy

needed to overcome saturation. Hence, there is a value implicit in the

atmosphere flushing of water vapor.

Chemical Potential of Rainfall

The Gibbs free energy of rainfall per day was estimated by equation

106_CC ds1

J =dRT[nln—]=dRT[nln IC r w C r w 6w 10—C

where; Jg = average rate of Gibbs free energy in rainfall (Kc/m2 day)

dr = average daily rainfall (mm)

R = universal gas constant = 1.986 l0 Kc/mole—°K

= number of moles/liter of pure water in rainfall

°dsnumber of moles/liter of total dissolved solids in rainfall

T = mean temperature, °K

C = average concentration of pure water in rainwater (ppm)1

13:

2 ds2

(13)

C = average concentration of pure water in runoff water (ppm)w2

a..

36

Cd = average concentration of total dissolved solids in

rainwater (ppm)9 it

Cd = average concentration of total dissolved solids in2

runoff water (ppm)

Values of dr and T were obtained for 235 stations from Climatological j’I.

data — 1975 The values of concentrations were obtained for rainfall from

annually averaged data for 32 stations over the continental U.S. (Costanza,

1978), and for surface water, from average data from 47 stations (EPA, 1975)

and interpolated between stations. Thus, the variation in this value over

time is a function of the time variation of rainfall only: we assume no

time variation in the concentrations of the chemical species.

Energy Value of Crops

The heat—equivalent energy value of crops was determined from bomb—

calorimetry values for each case (Watt and Merrill, 1974). For vegetables,

as weighted average (by percent edibility) was used (see Table 2).

To obtain solar—equivalent energy values for the variables, their

dollar value per acre harvested was multiplied by a factor of 3.6 l0

solar equivalent ICc/dollar, obtained from a fossil fuel equivalent/dollar

ratio of 18,000 Kc/dollar, multiplied by a value of 2,000 solar equivalent

Me/fossil fuel equivalent Ke (Odum, 1976). hMapping of Energy Flows

Contour maps of the climatic energy inputs as they vary over the U.S.

for each of the four months were prepared using the SYIIAP computer graphics

package. Climatic energy values were input as data, and the computer pro

duced contours of the values by interpolating linearly over the entire

continental u.s. Maps of agricultural productivity by state were also

37

prepared. This provided a visual correlation between climatic inputs and

agricultural outputs, as well as a grouping of the data appropriate to

further analysis.

Evaluation of Simplified Aggregate Energy Models

For the final part of the analysis, the calculated values of the energy

inputs and outputs were spatially averaged by state to calibrate partial

production models using simplified production functions (Figure 3) that

include the atciospiteric factors evaluated.

The averages for each state were calculated in a computer program in

which the map coordinates data values of each energy flow were read, as well

as the map coordinates of the centroid of each state. The computer program

selected the five closest data coordinates to each state centroid, averaged

the values of the data, and assigned the average value to each state. Thus,

a production function of the averaged variables could be computed for each

state and compared with actual production.

Calculation of the Investment Ratio

The investment ratio (i.e. , ratio of feedback energy flows to primary

energy flows in units of identical energy quality (Odum, 1975) was calcu—

lated for U.S. agriculture as a whole, using the sum of state averaged

values of climatic variables as the primary energy value, and the economic

(dollar) value of agricultural inputs from the economy as the feedback

energy value, by using the fossil fuel/dollar ratio of 18,000 Kc/dollar

and the value of 2,000 solar equivalent 1Cc/fossil—fuel Kc (Odum, 1976).

Thus, each dollar spent by farmers was assumed to return 3.6 lO 1Cc

(solar equivalents) as input from the economy. This value, divided by the

farm area of the U.S. in 1975 (1,084,046,000 acres) gives 8.21 . 106

Kc/m2—year—dollar or 2.25 . 10 Kc/m2—day—dollar.

RESULTS

In order to calculate solar—equivalent values for each of the variables

studied, quality factors were determined from global averaged data from the

literature of climatology and meteorology, and are tabulated below.

These factors along with the heat equivalent values of each of the

variables (maps of which follow) , were used to calculate various indices of

the climatic effects on production and the agricultural economy. The invest—

sent ratio of United States agriculture as a whole, was calculated from

averages of the climatic variables in solar equivalents, and economic data

on purchased inputs to agriculture in 1975.

Quality Factors

Quality factors were evaluated for seven climatic energy variables

(including the horizontal and vertical components of heat change and vapor

pressure potential change) by determining the ratios of annual solar

radiation for the globe to annual rates of flow (or production) of each

Variable for the globe. Results appear in Table I, with details of the

calculations in the accompanying footnotes.

The factors range from an identity value of 1.0 for sunlight itself,

to 8.3 . lO for the value of chemical potential of rainfall. Thus suggests

an energetic hierarchy in the atmosphere with flows of lower quality driving

the production of flows of higher quality, (e.g., atmospheric heat gradients

drive wind), although feedbacks from high to low quality may also occur, as

in turbulent transport of heat.

38

r

Table 1

Quality Factors

Number of Fig. 2 Name Quality Factor FootnoteCalories of sun/Calorie

1 Sun 1 Defined

4 Horizontal heat advection 5.25 3

3 Vertical heat exchange 12.6 2

5 Vertical vapor exchange(Gibbs free energy) 13.7 4

6 Horizontal vapor advection(Gibbs free energy) 13.7 4

2 Wind 56.4 1

Fossil fuel 2000 6

Average of 4 crops 2332 7

7 . Rain relative to runoff(Gibbs free energy) 8.31x10 5

1 Rate of production of atmospheric kinetic energy, according to Palmen,

in MonTh, Weather Forecasting as a Problem in Physics 2 in’2 KW for the

19entire atmosphere, or 1.51 . 10 1Cc/yr.

Rate of solar insolation (Odum, 1977)

= 8.5 ,o20 1Cc/yr

- solar insolation 8.5.1020Quality factor . = = 56.4

K.E. production 1.51.1019

2 Global average turbulent sensible—heat transfer, in Trewartha, An

Introduction to Climate, and Budyko, Climate and Life, = l.310 Kc/m2/yr.

14 2Surface area of earth = 5.1667 10 m

Thus, total average turbulent sensible—heat transfer =

6.7l7l0 1Cc/yr.

39

i

40

solar insolation s.s.io20Quality factor = = = 12.6

turbulent heat transfer6 717• o19

3 Average meridianal advection of heat fron the equator to 40°N latitude

= +11.12.1016 Kc/day. From 40°N latitude to the pole, the flux is of the

same magnitude, but opposite sign.

If we sum the absolute values of the fluxes, we double the total

magnitude. (If we neglect to take the absolute value, the sum equals zero.)

Thus, for the northern hemisphere, the absolute value of heat flux

= 22.24.1016 Kc/day. For a global rate, we double this figure, yielding

total global heat advection = 44.48.1016 ICc/day

or l.6210 Kc/yr.

solar insolation 8.5.1020Quality factor = . = = 5.25

heat advection 1.62.1020

4 Total mass of atmosphere = 5.3l021g

Total mass of water in atmosphere = 1.241019g 1120

average specific humidity= 1.24.1019

= 2.34’103g H20/g air5.310

= turnover time for water in the atmosphere = 11.23 days (Monin)

average vapor pressure e = 1.60 8 p q = 3.39 m b

(P is taken to be a boundary layer average pressure = 900 m b)

average flux of vapor pressure ==

3.49106mb/sec

equation for flow of Gibbs free energy due to vapor—pressure flux =

(Flux) x (1+ln)

= (3.49106mg/s) (-4.58) = 1.59810 mb/s

= 1.598l0 watts/m

Multiplying by the effective height of the atmosphere (10,000 m)(Monin)

= 15.98 watts/rn2

= (15.98) (5.1667.1014) = 8.26.1015 watts = 6.22.1019 Kc/m2/yr

41

solar insolation 8.5.1020Quality factor = = = 13.7

g 6.22-10

S Global rate—of—change of chemical potential or rainfall to surface water:

[nWRT ln ()] (Kc/yr)

where: = global average rainfall/year = dA

d = annual global average rainfall (cm) = 86.4 (Reichel, 1952)

A = surface area of the earth = 5.101.1018 cm2 (UNR Scientific)

Encyclopedia)

= average number of moles of pure water/liter of rainC

wl

=x .001 = 55.5

= average concentration of pure water in rain (ppm)

106_CdS ppm1

C = average concentration of pure water in surface water (ppm)w2

106_Cd ppm2

t1ds= average number of moles of total dissolved solids in rainwater

Cds1

Hx.00l

ds

CdS= average concentration of total dissolved solids in rainwater

(ppm) = 16.4 (Kern, 1970; and Carroll, 1962)

CdS= average concentration of total dissolved solids in river

water (ppm) = 90 (Livingstone, 1962)

R = universal gas constant = l.986l0 Kc/mole—°K

T = average global surface temperature (°K) = 286’K (Sellers, 1971)

Md = average molecular weight of chemical species of dissolved

42

solids = 42 g (Hem, 1970)

H = molecular weight of water (18 gm)

Plugging all values into the equation, we obtain:

= 1.0225.1015 Kc/yr

The energy quality factor for this value equals the average annual

global solar radiation flux divided by

EQF= solar insolation (CIyr) — 8.5.1020

AG —

(C/yr) 1.023.1015

7 Assuming an average value of 18,000 FFE Kc/dollar in 1975, and a solar

quality factor of 2,000 for fossil fuel (Odum, 1976), then the average

value of solar—equivalent kilocalories/dollar — 3.610.

We multiply this value by the price of a commodity per pound to find

its solar—equivalent value per pound, and divide this value by the actual

bomb—calorimeter heat value to obtain a quality factor.

‘I

sLs’) a$r

6 Odum, 1976

= 8.3l’lO Kc solar radiation/Kc chemical potential.

rr T’7)

y -)€1 J q

t —

• it,,.

h

43

Distribution of Climatic Energy Values

Figures 9—36 show the spatial and seasonal variation in the seven

climatic inputs evaluated in heat—equivalent Kc/m2/day. As can be seen,

the values show significant variation in both space and time, factors

which must be included when discussing “typical” (average) values of a

parameter for the United States as a whole. Values of solar insolation

ware the largest input, on the average, and values of chemical potential

change in rainfall the smallest (see Table 2) . However, values of the

advective changes in heat and water vapor potential were significantly

higher than solar values in the summer season. In general, values of

the parameters are higher in the South than then North, with the Northwest

having the lowest values. Details of the maps are mentioned below.

Naps of United States Climate in Energy Units

Solar Insolation

Figures 9—12 show the distribution of net solar insolation (Kc/m2—day)

(direct—reflected) at the surface for the United States.

The distribution varies most noticeably with latitude, with the

Southwest generally receiving the greatest amount.

The range over the year is from 136 Kc/m2—day for the Northwest in

January to 8,057 Kcfm2—day for parts of California in July.

IYrulent Kinetic Energy

Figures 13—16 show the distribution of average turbulent kinetic

energy due to the mechanical effects of wind in the planetary boundary

layer of the atmosphere (Kc/m2—day) . The distribution is patchy, with

some of the higher values corresponding to mountainous regions. The

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Values for calculating energy quality of crops

100

Sector Annual Heat Solar Solar Referenceaverage equivalent equivalent qualityprice/lb kilocalories/ calories/lb factor

(1975) lb (3.6l0 xprice)

Corn .0882 1579 3.175.106 2011 1

Wheat .0587 1500 2.ll3l0 1409 2

Soybeans .0820 1828 2.952.106 1615 3

Vegetables .0626 580.0 2.256.106 3879 6

Weighted averageof 4 sectors 2331.6

1 $2.47/bushel shelled corn x 1/56 bushel shelled/lb shelled (Handbook of:

Ag. Statictics, 1977).

2 $3.52/bushel wheat x 1/60 bushels wheat/lb wheat (Handbook of Ag.

Statistics, 1977).

3 $4.92/bushel soybeans x 1/60 bushels/lb soybeans (Handbook of Ag.

Statistics, 1975).

4 $3,185,179,000.00/50,854,200,000 lbs vegetables harvested (Handbook of

Ag. Statistics, 1975) over the U.S. for 1975.

4 U.Q.5 Average = Z

i=l a

a1 = 67,222,000

a3 = 53,761,000

1

a. = harvested average of each sector

a2 = 69,641,000

a4 = 3,401,590

6 from (Watt and Merrill, 1974)

7 weighted average value for vegetables, calculated by

ePE= Z

1

101

Iiej = bomb—calorimeter value of vegetable; (Kc)(Watt and Merrill, 1974)

= percent edible portion of vegetable

P = EP. = sum of percent edible portions of all vegetables

(Watt and Merrill, 1974).1: fI

Vegetables included are: artichokes, asparagus, lima beans, snap beans, beets,

brussel sprouts, cabbage, cantaloupes, cauliflower, celery, sweet corn,

cucumbers, eggplant, escarole, garlic, honeydew melon, lettuce, onions, green

peas, green peppers, spinach, tomatoes, and watermelons.

3‘it

102

values range from approximately 0.0 in many areas to 8.5 Kc/m2—day in the NSouthwest in April.

Vertical Heat Change

Figures 17—20 show the distributions of average vertical heat change

2 2for the United States (Kc/m —day). Values range from 2,890 Kc/m —day upward

in the heat of the Southwest in July, to 27 Kc/m2—day downward in the Mid

west in January. The average heat change is upward for virtually all

locations except in January, when it is downward for most locations,

suggesting strongly stable atmospheric conditions and a cold surface

temperature. The pattern of heat change is generally one of strong upward

fluxes in the Southwest, with values decreasing with distance North and East,

corresponding to a similar variation in solar insolation. This pattern

varies significantly over the seasons, with a secondary region of upward

flux centering around Tennessee (Figure 17).

Advective Heat Change

The tate—of—change of heat in the planetary boundary layer due to

horizontal flows are shown in Figures 21—24. These maps exhibit several

interesting patterns in distribution. The January map (Figure 21) shows

heat being lost in the Far West and South and converging in the Midwest

and Northeast, suggesting the influence of the Westerlies, as well as

circulation from the comparatively warm Gulf of Mexico.

The map for April (Figure 22) is similar to that of January, but

the zone of heating now runs across the breadth of the United States from Vthe Southwest to the Northeast, while above and below the band, there

exist regions of advective heat loss. i; tiVtIll

It.it.

103

July (Figure 23) appears to be dominated by Westerlies, revealing

a gradual increase in advective heating from West to East, with an

anomabus region of cooling in the Nidwest

October (Figure 24) shows advective heat losses in the South and

West, gradually giving way to heat gains as one moves North and East.

Change of Free Energy of Water Vapor (Vertical)

The distribution of the rate—of—change of the Gibb’s free energy of

water vapor in the planetary boundary layer due to vertical flux of water

vapor is shown in Figures 25—2S. This value is an index of energy sub

sidy of dry air to evapotranspiration. The higher the value, the greater

the tendency toward evapotranspiration. The annual range is from 0.0

Kc/m2—day

in the cooler months over a widespread area, up to an extreme of.

12,666 Kc/m2—day in a hot dry area of the Southeast in July (Figure 27).

The values for January (Figure 25) are generally very small (between

0.0 and 10.0 Kc/m2—day), but seasonally increase two to three orders of

magnitude by July. The January distribution of values is quite patchy,

but the pattern evolves seasonally to a high energy band in the South

west, decreasing North and East.

Change of Free Energy of Water Vapor (Horizontal)

The distribution of rate—of—change of Cibb’s free energy of water

vapor in the planetary boundary layer due to advection of water vapor is

shown in Figures 29—32. The extremes in range for this value are

anomalously large, from —7323 to +78,980 Kc/m2—day (bath in July).

The values for January (Figure 29) are much smaller (between ± 1,000.0

Kc/m2—day), with high values in the South, and no strongly defined pattern.

106

The April distribution (Figure 30) of values has roughly the same

range as January, with most values clustered around zero. Negative values

appear in the South and Midwest, and high positive values in the Northeast.

The July distribution (flgure 31) is marked by a strong North—South

variation from high values in the South to small and negative values in

the North.

The October pattern (Figure 32) is less extreme, with a large cluster—

ing of values near zero, and generally higher values in the Northeast.

The values for most of Florida are negative.

Change of Free Energy of Rainfall

This parameter includes one component only: the change in the chemical

potential of pure water component in rainfall to surface water. Inter

actions with soil are not included.

The distribution of this value over the U.S. for January 1975 (Figure

33) shows a high value for Northern California/Oregon, and a fairly high

region extending from the Great Lake states down to Oklahoma. Most of the

South and West have relatively low values.

The April distribution (Figure 34) shows much the same distribution

with slightly higher values, and the high region of the Great Lakes

extending into the Dakotas. The general increase in values reflects

greater rainfall for April than in January.

The July distribution (Figure 35) is markedly changed from the pre

vious months, with a strong band of high values appearing in Texas,

Oklahoma and Kansas. The Great Lakes states and North Dakota, as well as

the Central Atlantic coast also exhibit high values.

a_

105

The October distribution (Figure 36) is relatively uniform, with a

region of high values on the west coast, extending into Utah, and some

patches of middle values in the Nidwes tern states and Texas.

Maps of United States Agricultural Productivity

Corn Productivity

Corn productivity (Figure 37) shows little significant spatial pattern.

Low values occur in the Southern and Northeastern states. Highest values

occur in California, Indiana, and Utah, with medium to high values occur

ring throughout the country.

Wheat Productivity

Highest whest productivity (Figure 38) occurs in the states of the Far

West, followed by the Great Lakes states. Lowest values occur in the

Northeast.

pybean Productivity

Highest soybean productivity (Figure 39) is found in the states of the

Cornbelt and in the states immediately surrounding it. Virtually no soy

bean production occurs in the Northeast, nor in many Western states.

geLable Productivity

Vegetable productivity (Figure 40) is highest in the Southwest and

California, Florida, and Ohio. Low values occur in the Northwest, North

east, and in some South—Central states.

g4ghted Average of the Four Sectors

The average productivity (Figure 41) for the above four sectors is

highest in Wisconsin, Illinois, and Pennsylvania, with lesser peaks in

Kansas, Arizona, Nevada and New York. This distribution reveals a high

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productivity region around the Great Lakes, and a smaller one in the

Southwest.

Investment Ratio

Table 3 shows the average values of the climatic inputs as calculated

by averaging the values of the contiguous 48 states. Also shown is the

dollar value of counoodities purchased by the agricultural sector in 1975

(Agricultural Statistics, 1975), and the solar equivalent values of both

purchased inputs and climatic variables. Dividing the total solar equi

valent value of the purchased inputs by the sum of the climatic inputs

yields the investment ratio: 0.02 for the entire U.S., and 0.05 for the

four crop component of agriculture of corn, wheat, soybeans, and vegetables.

Sun

Table

3

Avera

ge

valu

es

of

cli

mati

cvari

able

s

4600.

3360

6

wate

rvapor

AH

ori

zonta

l

BV

ert

ical

-336.

336.

4

3045

709

r51

13.7

41717

13.7

!

9713

Rate

of

change

of

heat

AH

ori

zonta

l876.

BV

ert

ical

-!

365.

Pro

ducti

on

of

turb

ule

nt

kin

eti

cenerg

y—

81.6

1345

5.2

5

-.

279

12.6

0.9

656.4

7061

3515 54

I-k

I-.

-J

Input

Worl

dA

ver

age

U.S

.A

vera

ge

Energ

yE

ner

gy

Val

ue

(nati

onal

Ener

gy

Valu

eE

ner

gy

Valu

equa1it

(sola

reqyiv

ale

nts

)av

erag

e)(h

eat

equiv

ale

nts

)(h

eat

equiv

ale

nts

)fa

cto

rK

c/m

2/d

ay!

Kc/

m2/

da34

.K

c/m

2/da

y2-

Chem

ical

pote

nti

al

of

3350

ct-

-—

Chem

ical

pote

nti

al

of

rain

fall

/runoff

-;4

-

4.6

10

0.1

89.9

2•1

O17

856

•.--.--.

..---n

.-.

Table

3—

Co

nti

nu

ed

Input

Worl

dA

vera

ge

U.S

.A

ver

age

Ener

gy

Ener

gy

Valu

e(n

ati

onal

En

erg

yV

alu

eE

ner

gy

Valu

eq

ua1

it(s

ola

requiv

ale

nts

)avera

ge)

(heat

eq

uiv

ale

nts

)(h

eat

eq

uiv

ale

nts

)fa

cto

r-’

Kc/

m2l

day4

Kc/m

2/d

ay

lK

c/m

2/d

ay2

Fan

pro

du

cti

on

75

27

ex

pen

se(t

ota

lan

nu

al)

4.7

48’l

O$/

m3

.61

01706

Far

om

pro

du

cti

on

ex

pen

se

—4

secto

rs—

52

1(a

nnual)

8ll

.23’l

O$

/m3

.6’l

O’

60

44

av

era

ge

wo

rld

insola

tion

div

ided

byquali

tyfa

cto

r3

av

era

ge

of

state

—avera

ge

valu

es

of

cli

mati

cvari

able

s4

fro

mT

ab

le1

5pro

du

ct

of

colu

mns

1an

d2

60du

m(1

977)

7in

cid

ent

min

us

refl

ecte

din

sola

tion

10calc

ula

ted

fro

mto

tal

farm

pro

ducti

on

expense

sfo

rth

eU

.S.,

1975

($7.5

85

81

0)

inclu

din

gfe

ed

,li

vesto

ck,

seed,

ferti

lizer

and

lim

e,

repair

san

dopera

tion

of

cap

ital

item

s,depre

cia

tion,

hir

ed

12

labor,

pro

pert

yta

xes,

mo

rtg

age

inte

rest,

and

land

rent;

div

ided

byto

tal

land

infa

rms

(4.3

87

10

&.

(Agri

cult

ura

lsta

tisti

cs,

1977).

calc

ula

ted

fro

mcro

pp

rod

ucti

on

ex

pen

ses

for

the

U.S

.,1975,

of

corn

,w

heat,

soybeans,

and

anar

ea—

weig

hte

d—

avera

ge

vegeta

ble

pro

ducti

on

inF

lori

da,

1975

—76

.C

ost

sin

clu

de

seed

,ferti

lizer,

lim

e,

herb

icid

es,

insecti

cid

es,

cu

sto

mo

pera

tio

ns,

all

lab

or,

fuel

and

lub

ricati

on,

repair

s,

inte

rest,

and

land

rent.

(Cost

sof

Pro

ducin

gS

ele

cte

dC

rops

inth

eU

.S.—

l975,

1976,

and

Pro

jecti

on

sfo

r1977,

UR

San

dU

SDA

,1977.)

(Flo

rid

adata

fro

mB

rooke,.1977.)

9p

lan

ted

—are

aw

eig

hte

davera

ge

of

pro

ducti

on

co

stof

corn

($1

91

.33

/acre

),w

hea

t($

91

.48

/acre

),soy

beans

(l2

5.0

0/a

cre

)an

dF

lori

da

vegeta

ble

s($

1461.0

0/a

cre

);div

ided

bysu

nof

pla

nte

dare

as

over

the

U.S

.(m

)of

corn

(3.1

6.1

011),

wh

eat

(2.2

7.1

011),

soy

bean

s(2

.21.1

011),

and

veg

eta

ble

s(1

.38.1

010).

I.-.

2-

—-—

z’

‘ra..1

DISCUSSION

Discussion of Climatic Energy Values

The climatic energy values calculated were found to vary significantly

over the United States, geographically as well as seasonally.

The heat—equivalent value of the change of chemical potential of rain

fall was the smallest of the climatic sectors, on a yearly average, and

solar insolation was the largest. However, values of the horizontal com

ponent of chemical potential change of water vapor, and rate—of—change of

atmospheric heat were occasionally significantly higher than solar insola

tion, in July and October. In terms of the amount of work that these flows

can accomplish, these high values are misleading. The work available from

a gradient of heat is limited by the Carnot efficiency of the process.

For vertical gradients in the atmosphere, the temperature gradient is of

the order of 10 °K/Km. For Z = 1 Km (the height of the boundary layer) the

Carnot ratio is l0°K/300°K = .03, which would place the value of work

available for a temperature gradient at 3% of the actual heat available.

Horizontal temperature gradients are even smaller, and hence less efficient.

Finally with regard to a variable such as the chemical potential

change of rainfall, the value of the input depends on the local scarcity

of the intput. Rain is valuable in the desert because of the relative

lack of the substance. Consequently, some caution must be exercised in

treating these variables as predictors of any particular system behavior.

Spatial Distribution of the Partial Production Functions

The state average values of the five production functions evaluated are

shown in Figures 42—46. Despite the differences in actual range of the

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values between each of the functions, the first four exhibit considerable

similarity of trends. This is true because they are each positive functions

(sums or products) of the climatic variables, weighted in different ways.

Texas has the highest value in all cases, with Florida also generally

appearing high, followed by a Northeast region and states of the West and

Midwest. The Northwestern states are generally relatively low—valued.

The fifth production function (percent available insolation at the

surface), however, has a marked dissimilar distribution. The net avail

able insolation at the surface seems to be highest in the Southeast, and

lowest in the Southwest, with variations scattered throughout the North.

This distribution is not particularly surprising, although comparing it J

with those of the other functions illustrates the fact that climatic

distribution is not solely dependent on surface insolation, despite the

fact that climate is ultimately driven by the sun.

•1

Comparison of Production Functions with Agricultural Productivity

The distribution of agricultural productivity for the four sectors

studied (Figures 37—40) can be qualitatively compared with those of the

five evaluated production functions (Figures 42—46). Coefficients of

correlation between agricultural productivity and each production function‘1’

were computed, using the SAS statistical package, and are shown in Table 4.

As shown in the table, each of the four production functions evaluated

explain less than 20% of the variance in agricultural yield over the U.S.

for the four sectors studied. In an effort to improve the correlation, a

stepwise correlation of the input variables as heAt equivalents, solar

equivalents, and the product of heat equivalents was computed, with the

computer optimizing r—square by weighting each input separately. This

approach resulted in r—squares of 0.33 to 0.36.

IN

31.

H!7

Values of correlation

Table 4

coefficients of production

131

Production function

functions

Correlation coefficient Significance level

0.008

0.018

r

ii

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‘htic.

‘frtiIl0.139

0.112

0.107

0.004

0.04

0.36

0.33

0.33

Sum of heat equivalents

Sum of solar equivalents

Sum of solar equivalents4 >solar value

Product of heat equivalents

Percent available insolation

Suit of heat equivalents(computer optimized)

Sum of solar equivalents>solar value

F (computer optimized)

Product of heat equivalents(computer optimizedCobb—Douglas function)

0.021

0.67

0.166

6.006

0.001

0.013

-. .

.

I’.

.,;—....‘

.

General Effects of the Environment on United States Agriculture

The computer—weighted production function of the sum of solar—equivalent

inputs greater than the sun was found to be most highly correlated with

United States agrict’ltural output over the range of the data, and consequently

is the logical function to study more carefully to determine general effects

of climate on agriculture. This production function explained only 36% of

the spatial variance of yield for the four sectors. Neither long wave

radiation, nor latent heat transport was considered in the analysis, be

cause a) these parameters are somewhat analogous to sensible heat transport,

which was considered, and b) because the potential of these parameters to

generate net energy was considered negligible. However, other parameters

fr must surely be significant: a geographic distribution of soil type, fossil

fuel and fertilizer consumption, ground water and surface water storages.

In addition, secondary climatic effects, such as temperature and humidity

relationships with fungi or insects may significantly affect yield in some

cases. Clearly, agricultural yield is not a function of a selection of

W .monthly averaged meteorological variables alone. Perhaps the productivity

of the natural systems of the United States are more closely linked to

these variables, as they are presumably comparatively independent of human

energy sources. Total landscape production includes forest, range, and

pasture production as well as crops. The fifth production index (Figure 46)

ratios can be calculated using sunlight as the only primary energy source.

Values of 2.5 for the U.S. economy and 1.2 for the entire U.S. food pro—

duction system, including processing, distribution, and domestic energy

use have been calculated (Burnett, 1978). However, relative to the system

of agriculture, climatic flows other than insolation are also primary and

can be included in the calculation.

1324:

• 1

ti

iii

• I

133

Limitations to Results

The number of values used in this analysis was fairly small (64

to 235 for most parameters — less for others). Even assuming a smooth !&

spatial variation of the data, a linear interpolation of the data (upon

which the maps are based) may not be appropriate. Host of the data con

sists of monthly averaged variables (some of even longer averages, e.g.

constituents of rainfall), which may not be appropriate in a production

function when a week—long perturbation (intense frost, for example) could

have a significant effect on the actual output.

Since rates—of—change of climatic variables were based on calcu

lated gradients of parameters using the difference of only two data values

for the vertical case, approximations were involved. Also analytical

derivatives of a polynomial function were fitted to the data field for the

horizontal case.

Finally, the production functions developed to test against actual

, yield are probably not entirely adequate; they indicate potentials avail—

. able, not actual relationships. The computer was able to generate several

functions by individually weighting each climatic variable which explained

more in terms of climatic variables than those originally selected. Nowever,

the computer—generated functions were produced to optimize the correlation

coefficient, which may not be the same as producing & realistic model. What

is needed nextis a sound model that included other important factors (as in

Figures 2 and 3) which can be tested for several locations and evaluated.

Investment Ratio f or U.S. Agriculture

The investment ratio for the system is defined as the ratio of feed—

back inputs to primary inputs, in energy units of equal quality. Investment

ratios can be calculated using sunlight as the only primary energy source.

134

As shown in Table 2, a ratio has been evaluated by summing the solar

equivalent values of climatic inputs, and dividing the solar equivalent

energy value of purchased inputs by this sum. Whereas the investment ratio

as calculated with solar insolation only, and including economic feedbacks

to farm production only, is equal to 0.5, the new method yeilds a value of

0.02 to 0.05.

Significance of Climatic Variables in Energy Analysis

As shown in Table 3 and Figures 47 and 48, the embodied energy input

of climatic variables to the farm is greater than those input from the economy.

Even if only 2% of the solar equivalent value of all climatic inputs are trans

formed, this value still matches the efforts spent by the economy in purchased

inputs to the farm, the entire value of which does not benefit agriculture.

It would seem appropriate (even wise) for mankind to invest some of its

economic potential in further studying the variables of climate, in hope of

using more of the immense potential of the energy sources to which the bio

sphere is attuned.

Sunmiary and Conclusions

Everyone is familiar with the weather, and knows its power. The farmer,

probably more than anyone else, realizes how much he depends on the weather

for his livelihood. This sutdy has shown the magnitude of some of the vari

ables of climate, and that the amount of energy invested by farmers in their

crops is small compared to most of them. Even with a crude production

function, it can be shown that climatic variables can explain 36% of the

variance in agricultural energy yeidl as it varies over the nation.

Several significant factors (e.g. soil parameters, fossil fuel use,

fertilizer use) were omitted from the analysis, and it is believed that

these variables, together with those of climate, should be studied to

Fig. 67. Energy signature—heat equivalents. Log (energy) versus log(quality factor) . Straight line represents world—averageheat equivalent energy flows. See Table 3.

L

136

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Fig. 48. Energy signature—solar equivalents. Log (energy) versus log(quality factor). Upper horizontal line represents world—average solar—equivalent energy flows.

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obtain a more precise understanding of the coupling between natural systems

and human systems of agriculture.

iCh L,ç4

•:1• •

• 2;

APPENDIX I

CALCULATION OF SOLAR INSOLATION

Solar insolation was estimated by the equation (WA LAB, 1972)

11= K9 sina (1)

where: R = net incoming solar radiation (Langleys/min)

R solar constant (=2.0 Langleys/min or Kc/m2 — radianof hour angle

= % possible sunshine

a = average solar altitudet.i

and:

sin ci = sin d sin + cos d cos cos h (2)

d 23.45 cos (172 - D)1 (3)

where: d solar declination (radians)

• latitude of observing station (radians)—

h = average solar hour angle (radians). 1r

Integrating (2) over the length of a day (—7T/2<h<m/2), we have:I.’

J... IiR 1(9 [irsin d sin • + 2.0 cos d cos (4)

where:

d 23.45 cos (172 - D).

1)Thus, R is a function of K1,, latitude and day of the year. For each

month of interest, average values of sin d and ens d were computed:

d+n.

ad = sin d = sin125n

(172 - D)] (5)0

and similarly for = cos d

140

141

Thus:

R = KR [iTad sin c1 + 2.0 bd cos (6)

From this value, coefficients of albedo were subtracted to obtain

net absorbed solar radiation:

R = K [ 1 — a 3 = KR [Trad sin + 2.0 hd cos 4’] [ 1 — a J (7)

The distribution of albedo coefficients are shown in Figs. Al — A3

(Kung, Bryson and Lenschow, 1964).

It should be noted that this value of net insolation assumes that

atmospheric attenuating effects (i.e., absorption and scattering) are

negligible. If atmospheric attenuation were included, the values of net

insolation would be somewhat smaller. This value remains a valid index

of solar insolation, however.

‘Ui..

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APPENDIX II

DERIVATION OF EDDY DIFFUSION COEFFICIENTS

The turbulent diffusion of heat, momentum, and passive substances

such as water vapor in the atmosphere is often characterized by equations

of the form:

J = PK- (1)5 sz

where: = the vertical flux of parameter s

(s per unit area per time)

s = the concentration of the parameter of interest (Kg per Kg of

dry air)

30 = air density = 1.23 kg/rn

K5 = the eddy diffusion (exchange) coefficient for s (m2Is)

- the vertical gradient of s

Thus, can be determined from the gradient of s if K5 is known.

Unfortunately, the apparent simplicity of equation (1) belies the fact that

K is a complex function of local meteorological conditions, and, in

particular, the local stability of the atmosphere. A region with a steep

lapse rate (vertical temperature gradient) and light winds is liable

to possess K51s of different magnitude than a region with high winds and

an adiabatic lapse rate.

The eddy diffusion coefficient for momentum can be written as:

(2)

where: k = von Karman’s constant = 0.41

156

157

Z = height above surface Cm)

= vertical gradient of horizontal wind CS)

= a function of atmospheric stability.

The exact functional form of has been investigated by several

authors. We follow Panofsky et al.

Cl + 18 Ri)’4 for stable conditions ( >

= I. for neutral conditions = 0) (3)

(1 — 18 Ri)’4 for unstable conditions < 0)

where R1 = Richardson number (dimensionless)

potential temperature lapse rate (°K/m)

Richardson number is an index of atmospheric stability. It is

3defined by the equation:

aeR.

=

(4)a 2

where: g = acceleration of gravity = 9.8

-

= lapse rate of potential temperature °K/m

o = potential temperature °K at the geometric—mean height, (Z1Z2)2

-

= vertical gradient of velocity (us)

Plugging in equation (4) into (3) for stable conditions, we obtain:

3°hi

• =[l+18( .,z II

m

and from equation (2);

158

K2Z2 -

[1+18( )}Du 2

1/2where: Z = geometric—mean height = (Z1Z9) Cm)

To obtain representative values for K at various locations in theiii

United States, rawinsonde data for twenty—five stations (as published

in the monthly climatological data for the United States) were obtained

for the months of January and July, 1915. Estimates of and 4- for

the planetary boundary layer were obtained by calculating the gradients

using the monthly averaged values of 0 and u at the surface and at 1000 in

above the surface at each location.

The eddy diffusion constants forheat and water vapor, !c and Kw and

related to Km by the equation (Pruitt et al., cited in Rosenberg, 1974):

a2K = = l.13K (1 + a1R.)

+ 95. for stable conditions > (3)where: 21=

— 60. for unstable conditions (-5 < 0)

—0.11 for stable conditions (ft > 0)

20.074 for unstable conditions < 0)

APPENDIX III

CALCULATION OF PRODUCTION OF TURBULENT KINETIC ENERGY

The rate of production of turbulent kinetic energy per unit mass of

air by mechanical effects (wind shear) is (Webb, 1965):

1!(1)

where J = vertical momentum flux (often referred to as t) (Kg/rn—s

p = air density = 1.23 kg/rn3

= vertical gradient of wind (s)

On a per—unit—area basis for the boundary layer, we have.

= ZbØ(E) = ZbJ? (2)

where Zb = the average height of the atmospheric boundary layer = l000m

= rate of production of turbulent kinetic energy per unit area

in the PBL

The momentum flux is given by the following equation (Webb, 1965):

J =pK. (3)in rn3z

where K = eddy diffusion coefficient.UI

Thus, the rate of production of turbulent kinetic energy per unit

area in the planetary boundary layer is given by:

P =Z PlC(3132

(4)UI b in

This represents a flow of energy from the average wind energy acting

to organize the turbulent structure in the boundary layer.

159

APPENDIX IV

ATMOSPHERIC HEAT FLUX AND RATE—OF—CHANGE

The equation for the flux of heat in the atmosphere, including terms

for advection and turbulent diffusion, is:

Cl)

where:-

= heat flux (C/m2—s)

= specific heat of dry air at constant pressure = 0.24 C/Kg

p = dry air density = 1.23 Kg/n3

= turbulent eddy diffusion coefficient for heat (m2/s)

vs + 4 + - gradient of potential temperature (O)(°K/m)

U = potential temperature (°K)

= 3 dimensional velocity vector + ft + (n/s)

For the one—dimensional case, the X component of equation (1) is:

JHXCP [_K1d+ ft] (2)

To find the rate—of—change of heat in an infinitesimal volume element,

we take the divergence of equation 1:

—i —

________________

= flux in = flux out

AX IC

160

F 161

h=i/V=_VJH

and-

A

H.= —v

-; where: h = specific rate—of—change of heat (C/m3—s)

H = rate—of—change of heat (C/s)

3V = volume of element (m )

7.J = divergence of heat flux (C/m2—s)

Substituting from equation (1) into (3) yields:

:1 h = c, 0 — 70 (4)

where: 720 = + -4 ÷ = divergence of gradient of potentialDx Dy Dz temperature

VO = gradient of potential temperature

If we look at the one—dimensional case:

hXCpP[K.dL9_ *where: h = the specific rate of change of heat due to advection and

k diffusion in the x—direction.

For the atmospheric case, we asse that:

(6)

÷and W- 0 (7)

Thus, from 5, 6, and 7:

(8)

Fr. .

(9)Dz

162

and:

= Cp& (10)

=— CK11 (-g) (11)

where:

h = specific rate of change of heat due to J11 (C/m3—s)

= specific rate of change of heat due to J (C/m3—s)

= horizontal heat flux (C/m2—s)

= vertical heat flux (C/rn2—s)

i.e., horizontal heat flows are primarily advective, on the average,

and vertical flows, on the average, are those of turbulent diffusion.

A) Vertical transport of heat in the atmosphere.

The equation of turbulent transport of heat in the atmosphere is

similar to the equation for turbulent transport of momentum (Webb, 1965):

JHCPQ (12)

where: = the vertical flux of heat due to turbulent transport (C/m2—s)

specific heat of dry air = 0.24 C/kg — °K

p = air density 1.23 kg/rn3

= eddy diffusion coefficient for heat (m2/s)

= potential temperature lapse rate (.l°K/m)

The coefficient l11 was determined from rawinsonde data as shown in

Appendix II.

The rate—of—change equation (9) is equal to the divergence of (11),

and expresses the rate—of—change of atmospheric heat due to vertical flux

on a per—unit—volume basis. If we wish to consider the rate—of—change

in a column of air in the planetary boundary layer, per unit area, we

multiply equation (9) by the boundary layer height

aFM. = ZbP(h) = ZbCP K1ç—j

. (13)z az

and, for purposes of calculation,

= ZbcPPKlg_4 = CpPICia cplc (14)z b

where:

FM= rate of change of heat in the boundary layer of the atmosphere

(due to vertical flux) per unit area (C1m2—s)

= height of the boundary layer

( = 1000 in)

= specific heat of dry air

C = 0.24 C/Kg)

p = air density = 1.23 Kg/rn3

= eddy diffusion coefficient for heat (m2/s)

AG = potential temperature difference over the height of the

boundary layer (°K)

B) Energy Value of Thermal Advection

Thermal advection is the transport of heat by wind from one location

to another. This transport occurs in all directions, but we consider here

only advection by the average wind velocity in the horizontal plane.

163

164

The rate of change of heat due to horizontal advection is:

—h = cj

x0 0

where F= rate of change of heat/unit volume into the atmosphere at

x (kc/m3—s)

p = density of dry air (kg/rn3) = 1.23

a1, = specific heat of dry air (Ci°K—kg) = .240

U = resultant wind velocity vector (mis)

temperature gradient vector in the direction of (°K/m)

h is the rate of change of heat in a unit volume of air at location

z due to the horizontal advection of heat (see Fig. 1). In order to

transform h into units of rate of energy flow/unit area, we integrate

over an appropriate height (the height of the atmospheric boundary

layer Z 1000 in)

Assuming that 3(h)0

3z

thus:F Zh

x0 0

where Z = height of boundary layer C = 1000 m)

F = rate of change of heat in the boundary layer per area (C/m2—s)

Plugging in all constants yields:

F (per area) = 295.2 4 i (KC/m2—s)

÷(note: U is evaluated as some boundary—layer average velocity)

165

Note on calculation of vertical changes.

The rate of change equation for a parameter s due to vertical fluxes is:

2?s ,9s3tt’ 2

az

in difference equation notation, this can be represented as:

(s —s ) Cs —s )As 2 1 1 0 1AZ — AZ

K= E2 (sz_25l+s0)

In order to simplify the equation from 3 to 2 levels, and because the

vertical data were sometimes limited, the rate of change equation for the

vertical components of parameters in this study is:

= KAt

AZ2

This is a simplification which is equivalent to assuming a constant

vertical distribution of s (i.e., zero gradient) in the lower boundary

layer. In fact, the values of data used to calculate gradients generally

appeared to be uniform, but errors may result, depending upon the varia

tions of vertical gradients at some location.

— IM

APPENDIX V

ATNOSPHERIC BATES—OF—CHANGE OF VAPOR PRESSURE

The total flux of water vapor including turbulent diffusion and

advection, is given by an equation analogous to that for heat flux:

= q (1)

where:

flux of water vapor (Kg/m2—s)

p = air density = 1.23 KgJm3

= eddy diffusion coefficient for water vapor (m2/s)mass of water vaporq = specific humidity ( . )mass of dry air

u wind velocity (mis)

= the 3—d gradient of specific humidity (m)

The flux of specific humidity (Jq) and rate of change of specific

humidity (q) are obtained from the following equations.

Jq = (—K.Vq+ qt) (2)

q= (icV2q-Vq t) (3)

where: = flux of specific humidity (mis)

q = rate of change of specific humidity (us)

In order to obtain the flux of water vapor pressure and rate—

of—change of water vapor pressure, we use the approximation

q =O.622 (4)

where e = water vapor pressure (nib)

P = atmospheric pressure (nib)

166

167

and thus, = .622 — e 8!’(5)

where:

= q rate of change of specific humidity (el)

= rate of change of vapor pressure (mbfs)

-

= rate of change of atmospheric pressure (mb/a)

Assuming = 0 (i.e., constant atmospheric pressure), we have:

g .622 Bep

and

Be P(6)at 0.622 Bt

Equation (6) can be considered as the sum of rates—of—change of

vertical and horizontal effects. To determine the component of

for each dimension, we examine the vertical and horizontal components

of equation (3).

Considering only the vertical dimension:

2- ÷=v IS_?&.h.

(7)q 14 2 Bz BtBa

+If we consider = 0, then

2-

(8)Ba

and, from (6)

= .622 ‘S4 -

168

If we apply equation 5 to the right hand side of (9):

a2q — a .622 8e .6223 ar2az P 3z 2 azaz

= .622 +-ei q+3cj!) 2(10)

Making the hydrostatic approximation:

a2p= —pg and —j = 0

we now have:

= .622 1 .L + g p + 0 + 2e()2

(11)

and, from equations (1) and (11):

e=Kw[ij+f& +2e(&) 2]

(12):

where: = rate of change of vapor pressure due to vertical flux (mg/s)

Kw = eddy diffusion coefficient for water vapor (m2/s)

p = air density = 1.23 Kg/rn

P = atmospheric pressure (nib)

2g = 9.8 m/s

If we consider only the horizontal dimension, we have from equation

(4):

• (13)

If we assume that Kiq

we have

I

169

Again, from equation (5):

= .622 [ — £. f (15)

eAssuming —j y 0, then:

B_ .622 ae16ax i’ ax

Combining equation (14) and (16) yields:

• .622 aa—

p (1’)

and, plugging into equation (6):4. 4.

• 3e K(18)

where:

e = rate of change of vapor pressure due to horizontal flows ov

vapor.

= horizontal gradient of vapor

= horizontal velocity vector.at

APPENDIX VI

FLOW OF CHEMICAL POTENTIAL IN lllThtIDIfl GRADIENTS

4The value of the specific Gibbs function for any component in a

mixture of gases is: (1)

RT in (P. + 4)

where = the specific Gibbs functions of the th component (Kc/mole)

R = universal gas constant

T = absolute temperature of the system (°K)

= the partial pressure of the th component (mb)

= a function of temperature (not of iuunediate concern)

This is equivalent to the chemical potential, of component I

where:

g. .g+RT1nX. (2)

g :RT1n (P+p) (3)

P.

x

2 — = mole fraction of component i (4)i P

P = total pressure = Z P. (5)ii

g = specific Gibbs function at total pressure P

To

obtain the total chemical potential of water vapor, we multiply

equation (2) by the number of moles of water vapor (note the change in

subscript I to w for the water vapor component):

C =Ng =N g+11 RT1nX (6)w ww w w w

Using the equation of state for water vapor, we can put G on a

per—unit—volume basis:

#1 170

171

PV=NRT (7)w

Pvthus G,=1— g+PV1nX (8)

or, per unit volume:-

PgG mx (9)w liT w w

and from (4)

Pg PG—+P1n () (10)

To obtain the rate of change of chemical potenia1 due to humidity

changes, we take the time derivative of (10), assuming constant T and P.

wt= friFw[F ê+1n(-)

BP Pc =— 1 1 + in + ](KC/m3— 5) (11)

where: = vapor pressure of water vapor

P = atmospheric pressure

We note from 3 that g is a function of p and 4). Sears and Salinger

define • as a function of T, as:

4) = [C,(T—t) — CT in liT in — S (T—T)]

where: the subscript denotes a reference state

T = the temperature of that state (°K)

the pressure of that state (tub)

S0 = specific entropy of the state (an arbitrary value)

g = specific Gibbs free energy of the state (also arbitrary)

172

Thus, from (3) we have:

(C—S )(T—T) C gp ° °

RI? P RT R T RT0 0

If we set the arbitrary constants S and g and use our previous

assumption of constant T and P, we have:

(14)

Equation (11) simplifies to:

P1+1nJ (15)

To obtain C on a unit—area basis for the planetary boundary layer,

we multiply the right hand side of (11) by Zb ( = 1000 nO, the height of

the boundary layer:

Fg= ZbEr [1+ 1n] KC/m2 — S (16)

(Note: the vapor pressure of water, is often denoted by e.)

If the pressures are measured in millibars, a conversion factor

of 23.8 x 10 is needed to convert J to units of KG/rn2 — 5,

thus:

23.8 x 10 Zb ir 1 + in ] (1CC/rn2 — 5) (17)

where e, P are measured in rnb

= 1000 m

APPENDIX VII

FREE ENERGY VALUE OF VERTICAL DIFFUSION OF WATER VAPOR

From Appendix IV, we have an expression for the rate of change of

water vapor pressure due to turbulent flux vertically:

* = K+

4 2 (ip)2e1 (1)

where: K = coefficient of eddy diffusion

= boundary layer height — 1000 rn (rn2/s)

p = air density = 1.23 Kg/rn

2& = 9.8 mIs

P = atmospheric pressure (mb)2

(100 p (mb) = pressure in newtons/rn

e = water vapor pressure (nib)

= rate of change of vapor pressure (rnb/s) due to verticaleddy diffusion of water vapor

Frorn Appendix VI, the expression for rate of change of Gibbs free energy

due to water vapor change is:

WKlObt[P] (2)

And thus, for the component of free energy flange due to vertical

diffusion of vapor:

= 23.8 x 1O ZbKW( + ft ÷ 2 c1g)2eHi + in ](C/m2s)

173

APPENDIX VIII

FREE ENERGY VALUE OF WATER VAPOR ADVECTION

From Appendix IV we have an expression for the rate of change of

water vapor pressure due to advective flux:

— 4. +ae De 3x a ÷

. u (1)

where:

= rate of change of vapor pressure (mb/s)

= horizontal gradient of vapor pressure (mb/rn)

+

u = horizontal velocity vector (m/s)

From Appendix VI, we have an expression for rate of change of chemical

potential due to the rate of change of vapor pressure:

= 23.8 x lO Zb - [ 1 + in J (KC/m2-s) (2)

combining (1) and (2) yields:

Ô = 23.8 x lO Zb ][ 1 + in ] (KC/m2—s)

which is the expression of the rate of change of the Gibbs free energy of

air due to the advection of water vapor.

176

.4

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Fig. AL Locations of 64 data—gathering stations used in thecalculation and mapping of climatic data.

aI—

II

4

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I

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-

I

I

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IIII

*

Fig. A9. Locations of 235 data—gathering stations used in thecalculation and mapping of climatic variables.

I 7I a I I I 4. e I 4 I

I-.

b

APPENDIX IX

FREQUENCY DISTRIBUTIONS OF CLIMATIC VARIABLES

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__

— - —

BIOGRAPHICAL SKETCH

Dennis Peter Swaney was born in Ashtabula, Ohio, on May 15, 1953.

He lived in Ashtabula and Kingsville, Ohio, until his graduation from

Edgewood High School. During the summer of his senior year, he parti

cipated in a National Science Foundation program in space science for

high school students. After graduation from high school in 1971, he

attended Prescott College in Prescott, Arizona, for two years, followed

by six months working as a U. S. Forest Service firefighter on the

Prescott National Forest.

In September, 1974, he returned to school, this time to New College

in Sarasota, Florida, where he majored in physics. During the summer

term of his senior year, he pursued a student research fellowship in

atmospheric physics at Argonne National Laboratory. He continued his

work there during the fall quarter, developing his research for his

senior thesis.

After completing his studies at New College in 1976, he entered the

Department of Environmental Engineering Sciences at the University of

Florida to pursue a Master of Science degree under the supervision of

Dr. 11. T. Odum.

Dennis Swaney is single and is currently living in Gainesville,

Florida.

198

I certify that I have read this study and that in my opinion itconforms to acceptable standards of acholarly presentation and is fullyadequate, in scope and quality, as a thesis for the degree of Master ofScience.

Howard T. Odum, ChairmanGraduate Research Professor of

Environmental EngineeringSciences

I certify that I have read this study and that in my opinion itconforms to acceptable standards of scholarly presentation and is fullyadequate, in scope and quality, as a thesis for the degree of Master ofScience.

( )) /

Wayne C/ [TuberAssociate Professor of Environmental

Engineering Sciences

I certify that I have read this study and that in my opinion it• conforms to acceptable standards of scholarly presentation and is fully

adequate, in scope and quality, as a thesis for the degree of Master ofScience.

çRichard C. FluckProfessor of Agricultural

Engineering

This thesis was submitted to the Graduate Faculty of the College ofEngineering and to the Graduate Council, and was accepted as partialfulfillment of the requirements for the degree of Master of Science.

December, 1978

Dean, College of Engineering

Dean, Graduate School

• •. :

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