Parameterization of convective momentum transport and energy

conservation in GCMs

N. McFarlane

Canadian Centre for Climate Modelling and

Analysis (CCCma)

Parameterization in larger-scale atmospheric modeling

General parameterization problem: Evaluation of terms involving averaged products of (unresolved) deviations from (resolved) larger-scale variables and effects of unresolved processes in larger-scale models (e.g. NWP models and GCMs).

Examples:

(a) Turbulent transfer in the boundary layer

(b) Effects of unresolved wave motions (e.g. gravity-wave drag) on larger scales

(c) Cumulus parameterization

Other kinds of parameterization problems: radiative transfer, cloud microphysical processes, chemical processes

** The parameterization scheme(s) should preserve integral constraints of the basic equations – e.g. conservation of energy and momentum

kgPVkfDt

VD ˆ1ˆ

radp FecLDt

Dp

Dt

DTc

1

radFecLDt

Dpgw

Dt

DS

1 TcS p; gw

Dt

D

ceDt

Dqv

P ecDt

Dqw

vRTP )61.1( wvv qqTT

Basic Equations

0

Vt

wkvjuiV ˆˆˆ

][

VtDt

D

Motion

Mass continuity

Thermodynamic

Or:

Vapour

Condensedwater

Equation ofState (ideal gas)

2/VV

K (kinetic energy)

vvvp LqpTcLqTch (moist static energy)

222

z

V

y

V

x

V

D

Energy Conservation (e.g., Gill, 1982, ch. 4)

0])([])([

radFhVpht

KK

For air sm /105.1 25 at 15C , 100hPa

Kolmogorov scales 4/14/13 ; KK UL

These are small for the atmosphere (~ 1mm, .1 m/s) => Permissibleto neglect viscous terms for modeling/parameterization purposes but not to ignore effects/processes that lead to dissipation and associated heating

(k.e. dissipation rate)

Energy conservation:

Hydrostatic primitive equations and parameterization

(GCM resolved) Background state:

-hydrostatically balanced - slowly varying (on the smaller, unresolved horizontal and temporal scales). - deviations from it are small enough to allow linearization of the equation of state (ideal gas law) to determine relationships between key thermodynamic variables:

vTRp

z

T

Tzpg

z

p v

v

11

=>v

v

v TR

g

z

T

g

R

TR

g

z

11

v

v

T

T

p

p

=>

v

v

T

Tgg

z

p

v

v

T

Tg

p

zg

z

p

z

VwpVkfVV

t

VHH

H

ˆ

Bwpwzz

hwQpV

t

phV

t

hRH

)(D

/phqLTc HvvH KKEvv TTgB

GCM equations [hydrostatic p.e + parameterized terms]

0

Vt

2HHH VV

K

z

VVwVVw

zpVV

tH

HHHH

)()()(

KK

Bwpwzz

VVw

z

hwVVw

zQp-V

tH

HR

)()()]([(

DEE )

(+ other terms)

Boville and Bretherton (2003) in the context of turbulent transfer (PBL) in CAM2:

2

z

VK

z

VVw H

mH

D

Neglect the terms Bwpwz

)(

Include a tke equation (Stull,1988; Lendrick & Holtslag,2004)

D

Bwz

VVwpwVVw

zDt

D

2/e

dd lc /e 2/3D

2e VV

emm lK

AL

Liic AtzA ),()(

Li

ie AA )1(

Cumulus effects on the larger-scales

Qz

wV

t H

)(

)()(

Start with basic equations, e.g. for a quantity

(i) Average over the larger-scale area (assuming fixed boundaries):

othersz

w

z

wQ

z

wV

tccc

))(())(()()(

)( **

Mass flux (positive for updrafts): cc wM

ec QQQ ))(1()( “Top hat” assumption: 0)( ** cw

Quasi-steady assumption: effects of averaging over a cumulus life-cycle can be represented in terms of steady-state convective elements .

Pressure perturbations and effects on momentum ignored[Some of these effects have been reintroduced in more recent work, but not necessarily in an energetically consistent manner]

ec )1( 1 cww

0

ED

z

M ccc wM

kp

zBpVEVD

z

VM

c

ccHEDc ˆ)()(

c

c

HcEDc p

zwpVMBEhDh

z

MhD

)(

(usually parameterized)

(2) Average over convective areas (updrafts/downdrafts). Note:

cc

c

HEDc wB

p

zwpVE)D

z

MD

KKK

()(

(K.E.1)

wM

cccc VwVMVM

)( cccccccc MVMVMMM KKKK

)(

2/ccc VV

K 2/ VV

K 2/)(2/ DDDD VVVV

K

top-hat: neglect 2

2

Ec VVE

cD

2

ˆ2

Ec

c

HcccEccc

VVE

p

zkpVBMEhDh

z

hM

K.E. equation from the cumulus momentum equation:

22

22

EcDcED

ccccVV

EVV

DEKDKz

VVw

z

VVwKM

cDc

ccc BMKDz

VVMKMKM

z

c

HcDcEc p

zkpV

VVD

VVE

ˆ*22

22 D

cc

c

Hc BMp

zkpV

ˆ

*()

(K.E.2)

(K.E.1 – K.E.2):

and cD VV

=> (top hat)

From the mean equations:

z

VVMpVkfVV

t

V HccHH

H

)(ˆ

cc

c

Hcc

RH BMp

zkpVV

z

hhMQpV

t

phV

t

h

ˆ

D

Kinetic energy:

c

ccHc

cH

c

ccHccHHH

p

zwpVV

VVE

VVDBMKKM

zpVKV

t

K

22)(

22

)1(eqVH

+ cumulus k.e. eq.

c

Hc

c

H

p

zkpVV

p

zkpVV

ˆˆ

(for top-hat profiles)

(1)

22

22)(

)()(

wwE

VVED

z

KhKhMQKhV

t

pKh

cHc

HcccRH

H

D

Total energy::

The R.H.S. should be in flux form. QR is the radiative flux divergence.

2

2

22)(

wwE

VVED cH

c

D

cc

c

HHc

ccRH

BMp

zkpVV

z

hhMQpV

t

phV

t

h

ˆ

D

In summary, assuming top-hat cumulus profiles

2

2

Ec

c

HcccEccc

VVE

p

zpVBMEhDh

z

hM

22

ˆ

22

HcH

c

cc

c

HHccc

RH

VVE

VVDBM

p

zkpVV

z

hhMQpV

t

phV

t

h

0

ED

z

M c

kBp

zkpVEVD

z

VMc

c

HEccc ˆˆ)(

z

VVMpVkfVV

t

V HccHH

H

)(ˆ

(a)

(b)

(c)

(d)

(e)

)( VVE

)( hhE

Parameterization of convective (horizontal) momentum transport in GCMs

Schneider & Lindzen, 1976 (SL76) - ignored the PGF

Helfand, 1979 – Implemented SL76 in the GLAS GCM

Zhang & Cho, 1991 (ZC91), Wu&Yanai, 1993– parameterized PGF using an idealized cloud model

Zhang & McFarlane, 1995 (JGR) – ZC91 in CCC GCMII

Gregory et al., 1997 (G97) – parameterized the PGF on the basis of CRM results (implemented in the UKMO GCM)

Grubisic&Moncrieff, 2000; Zhang & Wu, 2003 – evaluated G97 and other possibilities from CRM results

Richter & Rasch, 2008 – effects of SL76 and G97 using NCAR CAM3

The cumulus pgf term must be parameterized, e.g. Gregory et al, 1997propose the following for the horizontal component associated with updrafts:

z

VMp uuH

7.

For the vertical component, the pgf is often assumed to partially offsetthe buoyancy and enhance the drag effect of entrainment. Since cE ww

c

c

ccc Bp

zDwwM

z

c

c

cc

c Bp

zEw

z

ww

Let

1/cc

c

BaEwp

z

Typical choice: 1a 20 (Siebesma et al, 2003)

(H)

(V)

Need to ensure that (H) and (V) are consistent with eachother.

Gregory et al. parameterization for updraft PGF in

)()1( VVDz

VM

t

Vuuu

u

(Zhang&Wu, 2003: )

Grubisic & Moncrieff alternative - adddependence on detrainment for PGF:

z

VuMVuVD

z

VuVuM

u

)1()()(

<=>

Zhang & McFarlane, 1995

(JJA)

ZM95

(Richter&Rasch, 2008; CAM3, Z-M) SL: = 0; GR: = .7

ZM95

Control(no CMF)

=.2

Summary

1. Momentum transport by convective clouds has important effects in GCM simulations of the large-scale atmospheric circulation

2. Introduction of parameterized momentum transport by cumulus clouds may result in an unbalanced total energy budget if associated dissipational heating effects areignored