Parameterization of the effects of Moist Convection in GCMs Mass flux schemes –Basic concepts and quantities –Quasi-steady Entraining/detraining plumes.

Parameterization of the effects of Moist Convection in GCMs

• Mass flux schemes– Basic concepts and quantities– Quasi-steady Entraining/detraining plumes (Arakawa&Schubert

and similar approaches)

– Buoyancy sorting • Raymond-Blythe, Emanuel• Kain-Fritsch

– Closure Conditions, Triggering

• Adjustment Schemes– Manabe– Betts-Miller

References

• Atmospheric Convection , Emanuel, 1994

• Arakawa and Schubert, 1974, JAS

Plus papers cited later

AL

Liic AtzA ),()( L

iie AA )1(

Spatial Averages

For a generic scalar variable, :

LAL

dAA

1Large-scale average:

Convective-scale average (for a cumulus up/downdraft) :

cAcc dA

A 1

Environment average (single convective element):

AecL

e dAAA

1

Where 1 Lc AA

ec )1( ˆ)1(*

Vertical velocity: ec www )1(

)1(/, Oec

ec www ,

Vertical flux

eeeccc wwwwwww )]ˆ)(ˆ)[(1()])([( **

But since 1 ; ec ww

)()ˆˆ)(1()()( 2** Owwww eccc

For simplicity in the following we ignore the last term (to focus on predominantlyconvective processes) . Also ignore sub-scale horizontal fluxes on the boundaries of the large-scale area (but account for exchanges between convective elements and the environment via entrainment/detrainment processes).

Time average over cloud life cycle:

2/

2/

)(1

)(L

L

t

tL

tdtt

Cumulus effects on the larger-scales

Qz

wV

t H

)(

)()(

Start with a general conservation equation for

Plus the assumption:

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

othersz

w

z

wQ

z

wV

tccc

))(())(()(

)()( **

Mass flux (positive for updrafts): cc wM

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

(similar to using anelastic assumption for convective-scale motions)

;

In practice (e.g. in a GCM) the prognostic variables are also implicitly time averages over convective cloud life-cycles

(ii) Apply cumulus scale sub-average to the general conservation equation, accounting for temporally and spatially varying boundaries:

cccc

bnc

c Qz

wwdlv

At)(

]))([()( **

Mass continuity gives:

0)(

z

awdlv

Atc

nc

; the outward directed normal flow velocity (relative to the cloud boundary)

nv

Entrainment (inflow)/detrainment (outflow):

dlvHvA

E nnc

)](1[

dlvHv

AD nn

c

)(

{ 0;10;0

)(

ff

fH

Define: dlvHvEA nb

c

nc

E )](1[ dlvHvDA nb

c

nc

D )(

Top hat: eE cD ; ;

Summary for a generic scalar, (top hat in cloud drafts):

otherQDz

Mz

wV

t

therefore

Qz

wMED

t

z

MED

t

otherQz

wM

z

wV

t

eDc

cccc

Dc

c

ccc

)1(

:

)(

0

])([

**

**

When both updrafts and downdrafts are present, both entraining environmental air:

dduucdduucc

dudududuc

DDDMMM

MMDDDEEEMMM

;

0;0;;;

Large-scale equations for dry static energy and water vapour

)(

/

..

qlqDz

qM

Dt

qD

dryingmoistening

HDsLlsDz

sM

Dt

sD

heating

uucc

uucc

Note that

..)(

..

..

HDz

hhMHDhEDh

z

hM

HDhhDz

hM

Dt

qDL

Dt

sD

uuu

u

uu

At the conv. layer top: ;0uM At c.l. base: hhu

[ Effects on horizontal momentum and associated dynamical heating: talk by Tiffany Shaw ]

Basic cumulus updraft equations (top-hat)

0

)(

)(

)(

)(

0

z

hMhEhD

t

h

Pcz

lMlD

t

l

cz

qMqEqD

t

q

Lcz

sMsEsD

t

sz

MED

t

uuuu

u

uuuu

uuu

uuu

uuuu

uuu

uuuu

uuu

{Dry static energy: s=CpT+gz; Moist static energy : h=s+Lq; }

v

vcvcuueu

u gz

PP

z

wMEwDw

t

M

])[()/)(()()(

0

mass conservation

dry SE

moist SE

condensate

vapour

vertical velocity

uu wM

Quasi-steady assumption: effects of averaging over a cumulus life-cycle can be represented in terms of steady-state convective elements .Transient (cloud life-cycle) formulations: Kuo (1964, 1974); Fraedrich(1974), Betts(1975), Cho(1977), von Salzen&McFarlane (2002).

)( ppT ovv ; )608.1( lqTTv (virtual temperature)pcR / ;

Traditional organized (e.g.plume) entrainment assumption:

cc

c

nn PwdlvHv )](1[

cc

ccc

c wR

MwA

PE 2

c

c dlP (local draft perimeter)]

Entrainment/Detrainment

Arakawa & Schubert (1974) (and descendants, e.g. RAS, Z-M): - is a constant for each updraft [saturated homogeneous (top-hat) entraining plumes]- detrainment is confined to a narrow region near the top of the updraft, which is located at the level of zero buoyancy (determines )

[ where

Kain & Fritsch (1990) (and descendants, e.g. Bretherton et al, ):- Rc is specified (constant) for a given cumulus (not consistent with varying ) - entrainment/detrainment controlled by bouyancy sorting (i.e. the effective

value of is constrained by buoyancy sorting)

Episodic Entrainment and non-homogeneous mixing (Raymond&Blythe, Emanuel, Emanuel&Zivkovic-Rothman):-Not based on organized entrainment/detrainment - entrainment at a given level gives rise to an ensemble of mixtures of undiluted and environmental air which ascend/descend to levels of neutral buoyancy and detrain

zb

zt

iii hhz

h

)()( bbi zhzh ))((),)(( *

itiiti zhzh

zdzzzhzzzhzh it

z

z

iibitibit

it

b

)])((exp[)()])((exp[)())(()(

*

Determining fractional entrainment rates (e.g. when at the top of an updraft)

2*

**

)()1)(()),(( TTOT

q

c

LTTpTqq

c

LTT

c

hhi

pii

pi

p

i

Note that since updrafts are saturated with respect to water vapour above the LCL:

This determines the updraft temperature and w.v. mixing ratio given its mse.

ec TT

Fractional entrainment rates for updraft ensembles

0;)( uutu DMzE tb zzz

(a) Single ensemble member detraining at z=zt

))((exp btbu zzzMM

Detrainment over a finite depth ttut zzMzD /)()(:tz

(b) Discrete ensemble based on a range of tops

))(();,( itii

iu zzMM ii

iuu hMhM

ii

iu ME ti i

iu z

MD

Buoyancy Sorting

Entrainment produces mixtures of a fraction, f, of environmental air and (1-f) ofcloudy (saturated cumulus updraft) air. Some of the mixtures may be positively buoyant with respect to the environment, some negegatively buoyant, some saturated with respect to water, some unsaturated

cv

v

f0 1

ev cf *f

saturated (cloudy)

positivelybuoyant

Kain-Fritsch (1990) (see also Bretherton et al, 2003):

Suppose that entrainment into a cumulus updraft in a layer of thickness z leads to mixing of Mcdz of environmental air with an equal amount of cloudy air. K-F assumed that all of the negatively buoyant mixtures (f>fc) will be rejected from theupdraft immediately while positively buoyant mixtures will be incorporated into the updraft. Let P(f) be the pdf of mixing fractions. Then:

cf

uo dfffPME0

)(2 dffPfMDcf

u )()1(21

0

This assumes that negatively buoyant air detrains back to the environment without requiring it to descend to a level of nuetral bouyancy first).

Emanuel:

Mixtures are all combinations of environement air and undiluted cloud-base air. Each mixture ascends(positively buoyant)/descends (negatively buoyant), typically without further mixing to a level of nuetral buoyancy where it detrains.

Shallow convection:

Including decent to a nuetral buoyancy level (with evaporation of cloud water) before final detrainment requires gives rise to cooling associated with evaporatively driven downdrafts in the upper levels of cumulus cloud systems – noted as a diagnostic requirement by Cho(1977)

Closure and Triggering

• Triggering:– It is frequently observed that moist convection does

not occur even when there is a positive amount of CAPE. Processes which overcome convective inhibition must also occur.

• Closure:– The simple cloud models used in mass flux schemes

do not fully determine the mass flux. Typically an additional constraint is needed to close the formulation.

– The closure problem is currently still poorly constrained by theory.

Closure Schemes In Use

• Moisture convergence (Kuo, 1974- for deep precipitating convection)

• Quasi-equilibrium [Arakawa and Schubert, 1974 and descendants (RAS, Z-M, Zhang&Mu, 2005)]

• Prognostic mass-flux closures (Pan & Randall, 1998;Scinocca&McFarlane, 2004)

• Closures based on boundary-layer forcing (Emanuel&Zivkovic-Rothman, 1998; Bretherton et al., 2003)

Emanuel& Zivkoc-Rothman(1998):

Bretherton, McCaa, & Grenier, MWR, 2003:

)/(}1)])(({exp[)](exp[max0max

bbb

bb

u zzzzzM

dzzM

M

Z-M scheme: all plumes have the same base mass flux

)()(0 *max zz *max ; zz

Closure based on CAPE depletion:

dzTTTgCAPE vvund

z

v

LNBz

b

/))(( )608.1( lqTTv

abc CAPEFMtCAPE /]/)([ )/( FCAPEM ab

Prognostic closure:

d

bb MCAPE

t

M

.0bM