Parameterization in large-scale atmospheric modelling

General parameterization problem: Evaluation of terms involving “averaged” quadratic and higher order products of (unresolved) deviations from “large-scale” variables and effects of unresolved processes in large-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)

(c) Cumulus parameterization

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

AL

Liic AtzA ),()(

Li

ie AA )1(

Large-scale variables and equations

Let an overbar denote the result of an averaging or filtering operation which suppresses fluctuations with temporal and spatial scales smaller than

pre-defined limits. e.g. for some appropriately smooth and bounded variable

after averaging:

2

2

2

2

2

2

2

2

1),,,(

t

t

Lx

Lx

Ly

Ly

Lz

Lzzyx

x

x

y

y

z

z

tdxdydzdLLL

tzyx

We refer to this as the large-scale variable and assume that our model hassufficient spatial and temporal resolution to represent the variation of this variable once we have determined the equations governing it and an appropriate solution methodology.

Typically, if the variable, has the following governing equation:

,QVtDt

D

And the mass continuity equation is:

Then applying the averaging operation gives, approximately:

QVVt

)(1

;

;0

Vt

1

z

wV

)( In cases to be considered (e.g. cumulus parameterization)

Determining this term is a goal of the parameterization

if

VkgPVkfDt

VD

2ˆ1ˆ

D TckFecLDt

Dp

Dt

DTc pradp

21

D TkecLDt

Dpgw

Dt

DS 21

TcS p; gw

Dt

D

ceDt

Dqv

Pr ecDt

Dqw

vRTP )61.1( wvv qqTT

Atmospheric Equations

0

Vt

wkvjuiV ˆˆˆ

][

VtDt

D

Motion

Mass continuity

Thermodynamic

Or:

Vapour

Condensedwater

Equation ofState (ideal gas)

2/VV

E (kinetic energy)

vp LqTch (moist static energy)

radp FTckt

ph

Dt

D

)(22EE

222

z

V

y

V

x

V

D , Molecular dynamic and kinematic viscosity

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

0])()([])([

radp FTckhVpht

EEE

For air sm /104.1 25 at 15C , 100hPa

Kolmogorov scales (for which viscosity and dissipation are independent parameters): 4/14/13 ; DD KK UL

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

Quasi-anelastic approximations for AGCM (Atmospheric GCM) parameterization

Background state:

-hydrostatically balanced - slowly varying (on the smaller, unresolved horizontal and temporal scales - e.g. that of quasi-balanced planetary scale circulation regime). - 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

Terms involving will also be neglected compared to unity. The approximate mass continuity equation which will be used is:

z

VwpVkfVV

t

VHH

H

ˆ

D RQQ

vv TTgB

Bwpwzz

hwQpV

t

phV

t

hH

)(

pVkfT

Tg

p

zkp

Dt

VDH

v

vH

ˆˆ)1(

0)1(})1({

Vt

)( qLTch vpQpVt

pp

zT

TgwpV

t

p

Dt

DhH

v

vH

)1(

Using these results leads to the following:

Terms in curly brackets:negligible for the parameterized scales but not for the resolved scales

+ other such (horizontal) terms

+ other…

0}{

Vt

Upon using this continuity equation to develop the flux-form equations and averaging:

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

1. 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).]

2. 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]

Traditional Assumptions for Cumulus Parameterization:

Starting equations (neglect terms in curly and other small terms brackets and assume implicitly that the background state is slowly varying on the parameterized scales):

z

wV

z

wV

t HH

)(

0

Parameterization of Moist Convection

v

vH T

Tg

p

zkpVV

t

V

ˆ)(

Qp

zT

TgwpV

t

pVh

t

h

v

vH

plus similar equations for vapour, condensed water, and other scalar quantities

For the traditional formulation ignore crossed-out terms

AL

Liic AtzA ),()(

Li

ie AA )1(

Spatial Averages

For a generic scalar variable, :

LAL

dAA

1Large-scale average:

Convective-scale average (for a singlecumulus 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 ,

Ensemble of cumulus clouds: i

i eii

i )1(

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 (Leibnitz rule):

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, (steady and top hat in cloud drafts: neglect crossed-out terms)

otherQDz

Mz

wV

t

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

dududucc

DDDMMM

DDDEEEMMwM

;

;;

Basic cumulus updraft equations (top-hat, traditional)

0

0

z

hMhEhD

Pcz

lMlD

cz

qMqEqD

Lcz

sMsEsD

z

MED

uuuu

uuuu

uu

uuu

uuu

uuu

uuu

uuu

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

mass conservation

dry Static Energy

moist Static Energy

condensate

vapour

uu wM

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

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

cccin

c

nn wvdlvHvE PP )](1[

cc

ccc

c wR

MwA

PE 2

c

c dlP (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 )

Kain & Fritsch (1990) (and descendants, e.g. Bretherton et al, 2004 ):- Rc is specified (constant or varying with height) for a given cumulus - 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 hh

z

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.

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.

Both may involve stochastic processes

Closure Schemes In Use (typically to determine the net mass flux at the

base of the convective layer)• Moisture convergence~ Precipitation (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., 2004)

• Stochastic closures may combine one of the above with a stochastic formulation for cumulus ensemble properties (e.g. Craig&Cohen papers, Plant&Craig)

Zonally averaged variance oflatent heating for differentconvective closures and downdraft evaporation efficiency parameters

(Scinocca & McFarlane, 2004)

Lecture 2

• Cumulus Friction and Energetics

• Parameterization of Boundary Layer Processes

0

ED

z

M ccc wM

kp

zBpVEVD

z

VM

c

ccHEccc ˆ)(

c

c

HccEccc p

zwpVBMEhDh

z

hMD

Cumulus friction and energetics

Take the dot product of with the momentum eq. It can be shown that

cc

c

Hc

Ec

Eccc BM

p

zkpV

VVEEKDK

z

KM

ˆ2

2

c

c

Hc

Ec

EDccc

p

zkpVV

VVEhKEhKDhKM

zD

ˆ)(

2)(

2

Total energy:

(usually parameterized)

cV

Assume , consistent with top-hat that

The dissipation heating term is intrinsically positive. Choose it as

2

2

Ec VVE

cD

c

Hc

p

zkpVV

ˆ

2

ˆ2

Ec

c

HcccEccc

VVE

p

zkpVBMEhDh

z

hM

is negligible

All parameterized

From the mean equations (ignoring, for simplicity, non-cumulus contributions to prime terms):

z

VVMpVkfVV

t

V HccHH

H

)(ˆ

cc

c

Hcc

H BMp

zkpVV

z

hhMQpV

t

phV

t

h

ˆ

D RQQ

Kinetic energy:

c

ccHc

cH

c

ccHccHHH

p

zwpVV

VVE

VVDBMKKM

zpVKV

t

K

22)(

22

(1)

(2)

(3)

)1(eqVH

+ cumulus k.e. eq.

parameterized

c

Hc

c

H

p

zkpVV

p

zkpVV

ˆˆ

(for top-hat profiles)

22

22)(

)()(

wwE

VVED

z

KhKhMQ

t

pKhV

t

Kh

cHc

HcccRH

H

D

Combine (2) +(3):

The R.H.S. should be in flux form. QR is the radiative flux divergence.Dissipational heating should be positive. Suggests :

Ec

cHc ww

EVV

ED DDD

2

2

22)(

cc

c

HHc

ccH

BMp

zkpVV

z

hhMQpV

t

phV

t

h

ˆ

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

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)

Parameterization of Boundary Layer Processes in AGCMs - The atmospheric boundary layer (ABL) is the region adjacent to the surface of the earth within which the exchange of momentum, heat, moisture, and other constituentsbetween the atmosphere and the surface takes place mainly by turbulent processes.

- Within a sub-layer near the surface vertical fluxes of momentum, heat, and moistureare almost independent of height.

- Within the remainder of the ABL quantities that are typically conserved under adiabatic motion are found to be nearly uniform with height (‘well mixed’)(e.g. potential temperature and specific humidity for cloud-free conditions or equivalent potential temperature and total water mixing ratio in cloudy conditions).

Cartoon of typical structurefor a cloud-free convectively Active ABL

z

VwpVkfVV

t

VHH

H

ˆ

Bwpwzz

SwQpV

t

pSV

t

SH

)(

TTgTTgB vv /

Cloud-free ABL :

- neglect effects of water vapour condensation

- ignore (for simplicity) virtual temperature effects (i.e. water vapour is passive)

Basic equations for the large (resolved) scale:

(buoyancy)

(1)

(2)

The depth of the ABL (and of turbulent regions in the free atmosphere) is typically small compared to a density scale-height (e.g. ). Thereforevertical variations in the background density are often ignored in ABL modelling.

1./ Hhabl

Potential Temperature vs Static Energy

ppT /0 )/( pcR

pVpV

t

pwB

dt

dS

p

p

dt

dp

dt

dT

p

p

dt

dH)(

11 00

TcS p

If departures from hydrostatic conditions are small:

z

S

p

p

cc

g

z

T

p

p

z pp

00 1gz

It can also be shown that

Also:

Bwz

pw

z

Sw

p

p

z

wc

wg

z

pw

z

Twc

p

pw

zc

pTc

p

pc

p

pp

pp

0

0

0

and

Turbulent Kinetic Energy Equation:

Therefore the R.H.S. of (2) is approximtely zwTcQ p

Usual current approach: combine a turbulent kinetic energy (tke) equation with an eddy diffusivity formulation. Get a tke equation by forming an equation for and averaging. VV

eE 2/2 VVVV

E

Approximate tke ( ) equation: (e.g. Stull, 1988)e

kwgpwVVwzz

VVw

tD

e

//2/

Eddy diffusion approximation for second moments:

zKpwVVw

zKw

zVKVw

e

H

m

//2/ e

Diffusivities:

eeHmeHmeHm clK ,,,,,,

Traditional approach

deHml ,,,

Dissipation:

Physical and dimensional considerations suggest

ddd lc // 2/3eeD

Specifying the lengths and coefficients is a closure issue. Large literature on this topic. Several hypotheses have been explored in recent work (e.g. Sanchez&Cuxart, 2004, Lenderink&Holtslag, 2004, and references therein)

Matching to the surface layer: Monin - Obukhov similarity requires that:

**

2*

2/122

)''(

])''()''[(/

uw

uwvwu

where

)/()()ln(

)Pr)(/(

)()()ln(

*

00*

LzLzzz

k

LzLzzz

kUu

tHLHtL

sL

mLmL

L

Boundary conditions and constraints

k : von Karmen constant, Pr : turbulent Prandtl number, UL, L: wind speed, potential temperature at reference level ( ) roughness heights (where surface values apply).tzz ,0

Lz

sv

v

wkg

uL

)''(

3*

(Monin-Obukov length)

Bulk exchange formulae (resulting from fits to non-linear solutions):

)(),/,/(

)(),/,(

),(

0**

0**

20

2*

sLLiBqLLQQ

sLLiBtLLHH

LiBLmD

qqURzzzzFCqu

URzzzzFCu

URzzFCu

])[(])()[( 2LsvsvLvLiB UgzR (Bulk Richardson Number)

The functions are derived from field campaign observations (e.g. Dyer, 1974) . Moisture and other tracers treated similarly.

Hm ,

Limitation:

-Dependence of vertical fluxes on local mean gradients does not account for heat transfer in the convectively active ABL where mean gradients are small(slightly stable) but upward heat flux is positive.

-Requires introduction of non-local effects.

For a scalar quantity, :

nlwz

Kw

Approaches:(a) Introduce prognostic equations for second moments with associated closureAssumptions to derive the nonlocal effects (e.g. Deardorf, 1966, Mellor& Yamada, 1974, Cuijpers & Holtslag , 1993, Abdella & McFarlane, 1997, Gryanik&Hartmann, 2002, ….). Simplest formulations give:

cgHnl Kw )(

(b) Represent non-local transfer effects as being associated with plume-like Eddies (e.g. Siebesma et al, 2007)

)( unl Mw

(From Siebesma et al)

)(

)(

uu

z

Mz

M