Pier Siebesmasiebesma@knmi.nl

KNMI, The Netherlands

MotivationFundamentals, Models, Equations

Dry Convective Boundary layerShallow Moist Convection (Friday)

Parameterizations of moist and dry convection (Saturday)

Atmospheric dry and shallow moist convection

1. Motivation

o0

Equator

N30

Northoregion

windTrade

Subsidence

~0.5 cm/s

10 m/s

vE vE

inversion

Cloud base

~500m

Tropopause 10km

•Stratocumulus

•Interaction with radiation

•Shallow Convective Clouds

•No precipitation

•Vertical turbulent transport

•No net latent heat production

•Fuel Supply Hadley Circulation

•Deep Convective Clouds

•Precipitation

•Vertical turbulent transport

• Net latent heat production

•Engine Hadley Circulation

The GCSS intercomparison project on cloud representation in GCM’s in the Eastern Pacific

ECMWF IFS overestimates Tradewind cumulus cloudiness:

Siebesma et al. (2005, QJRMS)

scuShallow cuDeep cu

Water vapour is the building material for clouds

name SymbolUnits

Definition

Near surfacevalues

Atmospheric column

spec. humidity

qv [g/kg] amount of water vapour in 1kg dry air

10 g/kg 20 kg/m2

Saturation spec. hum.

qs [g/kg] Max. amount of water vapour in 1kg dry air

15 g/kg

Liquid water

ql [g/kg] amount of liquid water in 1kg dry air

1 g/kg 200 g/m2

qsat

qt

Tip of the iceberg

water vapor clouds albedo lapse rate total

(Bony and Dufresne, GRL, 2005)

High-sensitivity GCMs

Low-sensitivity GCMs

Sensitivity of the Tropical Cloud Radiative Forcingto Global Warming in 15 AR4 OAGCMs

CRFSW

SST

Fundamentals , Models and Equations

Some fundamental notions on Turbulence (1)

2

2

3j

i

ii

j

ij

i

x

u

x

pg

x

uu

t

u

Conservation of momentum: Navier Stokes equations:

storage

term

advection

term

gravity

term

pressure gradient

term

viscosity

term

In order to discuss the non-linearity consider a simpler 1d version: The Burgers Equation:

2

2

x

u

x

uu

t

u

And treat both physical processes seperately:

Some fundamental notions on Turbulence (2)

1) The diffusion equation:

2

2

x

u

t

u

dissipation gradient weakening stabilizing

0

x

uu

t

u

2) The advection equation:

General solution:

)( utxfu

Advection term gradient sharpening instability

Some fundamental notions on Turbulence (3)

2

2

x

u

x

uu

t

u

•Competition between both processes determines

the solution.

•Compare both terms by making the equation dimensionless:

L

Utt

Lxx

Uuu

~

/~/~

2

2

2

2

~

~1~

~

~

~~

~~

x

u

Rex

u

ULx

uu

t

u

UL

Re Reynolds Number measures the ratio between the 2 terms.

1Re dissipation dominates flow is stable laminar

1Re Non-linear advection term dominates flow is unstable turbulent

Turbulence made by convection in the atmospheric boundary layer

L=1000m

U=10m/s

m2s-1

95

1010

10000 UL

Re

Macrostructure dominated by non-linear advection!!

Heat flux

Potential temperature profile

Large eddy simulation of the convective boundary layer

Poor man’s artist impression of the convective boundary layer

Some fundamental notions on Turbulence (3):

Energy Cascade

Some fundamental notions on Turbulence (4):

• Energy injection through buoyancy at the macroscale

Re dominated by non-linear processes

• Hence, Large eddies break up in smaller eddies

• that have less kinetic energy:

• and a lower “local” Reynold number

• until they are so small that :

• and viscosity takes over and the eddies dissipate.

UL

Relocal

1localRe

J.L Richardson (1881-

1953) Some fundamental notions on Turbulence (4):

Richardson 1926

free after Jonathan Swift (1733):

(Sloppy) Kolmogorov

(1941) Some fundamental notions on Turbulence (4):

Kinetic Energy (per unit mass) : E

Dissipation rate :

t

E

eddy size:

eddy velocity:

eddy turnover time: )(v

)( v

Kolmogorov Assumption:

Kinetic Energy transfer is constant and equal to the dissipation rate

)(v

cstvvE

)(

)(

)(

)(

)( 323/1~)( v

Some fundamental notions on Turbulence (5):Consequences of Kolmogorov

(1941)

3/~)( ppv Structure functions:

3/53/22 ~~))(( kdlelvF ik Fourier transform of kinetic energy Famous 5/3-law!!

Kolmogorov scale : the scale at which dissipation begins to dominate:

m

vlocal

34

1

3

354

133

1010

10)(1Re

Largest eddies are the most energetic

1 hour100 hours 0.01 hour

microscaleturbulence

spectral gap

diurnal cycle

cyclones

data: van Hove 1957

Energy Spectra in the atmosphere

energy spectra at z=150m below stratocumulus

Duynkerke 1998

U Spectrum

V Spectrum

W Spectrum500m

Governing Equations for incompressible flows in the atmosphere

2

2

j

i

iu

j

ij

i

x

u

x

pF

x

uu

t

ui

2

2

jjj

xF

xu

t

2

2

jqq

jj

x

qF

x

qu

t

q

0

i

i

x

u

vdTRp

Continuity Equation

(incompressible)

NS Equations

Heat equation

Moisture equation

Gas law

jcijiu ufFi 33 with

gravity

term

coriolis

term

Condensed water eq.lq

j

lj

l Fx

qu

t

q

10 m 100 m 1 km 10 km 100 km 1000 km 10000 km

turbulence Cumulus

clouds

Cumulonimbus

clouds

Mesoscale

Convective systems

Extratropical

Cyclones

Planetary

waves

Large Eddy Simulation (LES) Model

Cloud System Resolving Model (CSRM)

Numerical Weather Prediction (NWP) Model

Global Climate Model

The Zoo of Atmospheric Models

Global Climate and NWP models (x>10km)

Qz

w

t

'

v

z

wuvvfu

t

ugc

v

z

wvuufv

t

vgc

v

vdTRp

gdz

pd

t

w

0 0

z

w

y

v

x

u

Subgrid

To be parameterized

Large Eddy Simulation (LES) Model (x<100m)

High Resolution non-hydrostatic Model: 10~50m Large eddies explicitly resolved by NS-equations inertial range partially resolved Therefore: subgrid eddies can be realistically parametrised

by using Kolmogorov theory Used for parameterization development of turbulence,

convection, clouds

ln(Energy)

ln(wave number)-11 1mm

dl-11

0 1km ~ l

Inertial Range

53

DissipationRange

Resolution LES

Dynamics of thermodynamical variables in LES

ltqforwzz

w

t

,...... terms"horizontal"

ResolvedResolved

turbulenceturbulence

subgrid subgrid

turbulenceturbulence

3 with

: with

dxdydzleclK

zKw

Subgrid turbulence:

Remark: 3/4)( llvcleclK

Richardson law!!

Simple: All or Nothing:

0),(

,1

0,0

,0

sltsltl

sltl

qqifqqq

a

qqifq

a

c

c

{

{

Cloud Scheme in LES

• Definition:

Turbulent Kinetic Energy (TKE) Equation

2225.0 wvue

pwz

ewz

wg

z

vwv

z

uwu

t

ev

o

1'''''

Reynolds Averaged budget TKE-equation:

Assume:

•No mean wind

•No horizontal flux terms

Shear production Buoyancy production Transport Dissipation

S B T D

Richardson Number: S-

BfRi

1-

1

1

f

f

f

Ri

Ri

Ri Laminar flow

Shear driven turbulence

Buoyancy driven turbulence

Mixed layer turbulent kinetic energy budget

normalizedStull 1988

dry PBL

Conditions for Atmospheric Convection

1

ULReReynolds Number Condition for fully developed turbulence

Richardson Number 1

S

BRi Condition for buoyancy drive turbulence

Atmospheric Convection = Turbulence driven by Buoyancy

Objectives

Tools

• To “understand” the various aspects of atmospheric convection

• To find closures (for the turbulent fluxes and variances)

• Observations, Large Eddy Simulation (LES) models

Methods

• Dimension analysis, Similarity theory, common sense

Application

• Climate and Numerical Weather Prediction (NWP) Models

(Simplified) Working Strategy

Large Eddy Simulation (LES) Models

Cloud Resolving Models (CRM)

Single Column Model

Versions of Climate Models

3d-Climate Models

NWP’s

Observations from

Field Campaigns

Global observational

Data sets

Development Testing Evaluation

See http://www.gewex.org/gcss.html

Dry Convective Boundary Layer

1. Phenomenology

2. Properties

3. Models and Parameterization for Convective Transport

The Place of the Atmospheric Turbulent Boundary Layer

Depth of a well mixed layer: 0~5km

Determined by:

•Turbulent mixing in the BL

•Large Scale Flow (convergence, divergence)

we

Q0

z

tropopause

Can we see the convective PBL?

PBL top

Downtown LA

10km

July 2001

(Courtesy Martin Kohler)

Typical Profiles of the convective BL

Surface layer

Mixed layer

Entrainment Zone

Stull 1988

LES View of the Dynamics: potential temperature

Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands

LES View of the Dynamics: vertical velocity

Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands

Horizontal Crosscut

Surface layer

Mixed Layer

Inversion layer

Irregular polygonal structures!

Moeng 1998

Dry Convective Boundary Layer

2. Properties

•Surface Layer

•Mixed Layer

•Inversion Layer

Monin-Obukhov Similarity

Construct dimensionless gradient terms:

*u

z

z

um

L

z

(von Karman constant) is defined such that 1 for / 0m z L

SLh

z

z *,

and evaluate this as a function of the stability parameter

stableunstable

mh

u

qwq

u

w

wuu

srfSL

srfSL

srf

,

,

Fleagle and Businger 1980

MO theory allows to formulate the turbulent fluxes in a diffusivity form:

zK

zz

w

t h

'

**

)''(

u

w o

)/(*

Lzz

zh

z

Kz

uzkw hh

1

0

Diffusion Eq.

Dry Convective Boundary Layer

2. Properties

•Surface Layer

•Mixed Layer

•Inversion Layer

Scaling Parameters for the convective mixed layer

Relevant parameters:

1) TKE production through buoyancy: srfv

buoyancybyproduction

wg

t

e

0

2) Depth of the boundary layer: z

unitsform

3

2

s

m

m

Construct a convective velocity scale:

3/1

0*

zw

gw srf

Interpretation: velocity that results if all potential energy is converted into kinetic energy in an eddy of size z*

*w

z

Typical Numbers

K

msg

mz

msKwmWwc srfsrfp

300

81.9

1000

15.0'/150'

0

2

12

smzwg

w srf /7.13/1

0*

sec600*

w

z

2/13/1

9.01

z

z

z

z

ww

Dimensionless vertical velocity variance (in the free convective limit)

Garrat 1992)

Mixed layer turbulent kinetic energy budget (LES)

pwz

ewz

wg

z

vwv

z

uwu

t

ev

o

1'''''

Shear production Buoyancy production Transport Dissipation

S B T D

Pino 2006

Dry Convective Boundary Layer

2. Properties

•Surface Layer

•Mixed Layer

•Inversion Layer

Turbulent Entrainment

quiet non-turbulent air

turbulent air

One-way entrainment: less turbulent air is entrained into more turbulent air

Mixed layer erodes into the Free atmosphere and is growing as a result of the entrainment proces

Entrainment Flux

Free Convection: Entrainment flux directly related to surface buoyancy flux

Rsrfv

invv Aw

w

,

,

Observations suggest 3.0~15.0RA

vevveinvv www ,

MLsrfv ww ,, Ri

AA

w

w RR

e

(Tennekes 1972)

Dry Convective Boundary Layer

1. Phenomenology

2. Properties

3. Models and Parameterization for Turbulent Transport

Prototype: Dry Convection PBL Case

•Initial Stable Temperature profile: qs=297 K ; = 2 10-3 K m-1

•No Moisture ; No Mean wind.

•Prescribed Surface Heat Flux : Qs = 6 10-2 K ms-1

h (km)

x(km)

0

51

Mean Characteristics of LES (virtual truth)

^Non-dimensionalise : z z/z*

w w/w*

t t/t*

Q Q/Qs

^

^

^

1700,0

04.0,0

5.1,0

*

*

z

w

Growth of the PBL

PBL height : Height where potential temperature has the largest gradient

Mixed Layer Model of PBL growth

Assume well-mixed profiles of

Use simple top-entrainment assumption.

2.0,

,

Rsrfv

invv Aw

w

Boundary layer height grows as:

)(tz

tAQtz

12)( 0

Q2.0

Simplest Model of PBL growth: Encroachment

Assume well-mixed profiles of

No top-entrainment assumed.

time

srfw )( 1t

1t

06.0

Boundary layer height grows as:

)(tz tQ

tz 02)(

Internal Structure of PBL

Rescale profiles

z

zz

ˆ min

Classic Parameterization of Turbulent Transport in de CBL

Eddy-diffusivity models, i.e.

zKw

zK

zw

zt

•Natural Extension of MO-theory

•Diffusion tends to make profiles well mixed

•Extension of mixing-length theory for shear-driven turbulence (Prandtl 1932)

K-profile: The simplest Practical Eddy Diffusivity Approach (1)

The eddy diffusivity K should forfill three constraints:

•K-profile should match surface layer similarity near zero

•K-profile should go to zero near the inversion

•Maximum value of K should be around: 1.0max zwK

z/zinv

0

1

0.1

K w* /zinv

2

0*

* 1ˆ

iih

i

hh z

z

z

z

w

uk

zw

KK

(Operational in ECMWF model)

A critique on the K-profile method (or an any eddy diffusivity method) (1)

0w

0w

0w

0

z

0

z

0

z

Diagnose the K that we would need from LES:

K>0

K<0

K>0

Forbidden area

“flux against the gradient”

Down-gradient diffusion cannot account for upward transport in the upper part of the PBL

Physical Reason!

•In the convective BL undiluted parcels can rise from the surface layer all the way to the inversion.

•Convection is an inherent non-local process.

•The local gradientof the profile in the upper half of the convective BL is irrelevant to this process.

•Theories based on the local gradient (K-diffusion) fail for the Convective BL.

“Standard “ remedy

NLwz

Kw

Add the socalled countergradient term:

zinv

Long History:

Ertel 1942

Priestley 1959

Deardorff 1966,1972

Holtslag and Moeng 1991

Holtslag and Boville 1993

B. Stevens 2003

And many more…………….

K

zK

Can we understand the characteristics of this system?

B. Stevens Monthly Weather Review (2000)

zK

ztAnd let’s find quasi-steady state solutions for

Non-dimensionalise: 1ˆ,ˆ,ˆ0

0 Q

QQ

z

zz

(leave the ^ out of the notation from now on)

0ˆ,ˆ

Q

zw

zw

KK

Quasi-Steady Solutions (1)

That is to say to find steady state solutions of:

0

z

0

z

0

z

This implies a linear flux!

Use as boundary conditions:

(Remember we work in non-dimensionalised variables)

Azzz

KQ

w

1

Which is to say solutions for which the shape of is not changing with time.

02

2

zK

zzt

Quasi-Steady Solutions (2)

Solution for the gradient

K

AzzK

z

1

Where K-profile is given by: 675.01)( zzzK and is constant

cstz

A

z

zzz

11ln

1)(

Solution of

Non-local

processesSurface fluxesTop-entrainment

AB

Surface fluxes

Top-entrainment

Quasi-Steady Solutions without countergradient (1)

cstz

A

z

zz

11ln

1)(

No countergradient:

A=0 no top-entrainment

A=-0.2 typical top entrainment value

LES K-profile without countergradient

Quasi-Steady Solutions without countergradient (2)

LES K-profile without countergradient

•The system tends to make quasi-steady solutions (in the absence of large scale forcings)

•So it allways produces linear fluxes

•It will find a quasi-steady profile that along with the K-profile provides such a linear flux

•So it is the dynamics that determines the profile (not the other way around!!!)

Quasi-Steady Solutions with countergradient term

cstz

A

z

zzz

11ln

1)(

A = -0.2, = 0.675,

=1.6, 3.2, 4.8

Height where

Countergradient: Conclusions

•Addition of a countergradient gives an improved shape of the internal structure

•But…

•How does it affect the interaction with the free atmosphere, i.e. what happens if we do not prescribe the top-entrainment anymore.?

•Can it be used in the presence of a cloud-topped boundary layer?

•Are there other ways of parameterizing the non-local flux?