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Numerical Weather Prediction Parametrization of diabatic processes

Convection II

The parametrization of convection

Peter Bechtold, Christian Jakob, David Gregory

(with contributions from J. Kain (NOAA/NSLL)

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Outline

• Aims of convection parametrization

• Overview over approaches to convection parametrization

• The mass-flux approach

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Caniaux, Redelsperger, Lafore, JAS 1994

Q1c is dominated by condensation term

10

10

5z (

km

)

(K/h)-1 2

c-e

Q1-Qrtrans c-e

Q2

10

5z (

km

)

(K/h)

trans

-1-2 0 21

a b

but for Q2 the transport and condensation terms are equally important

To calculate the collective effects of an ensemble of convective clouds in a model column as a function of grid-scale variables. Hence parameterization needs to describe Condensation/Evaporation and Transport

Task of convection parametrisation total Q1 and Q2

p

secLQQQ RC

)(11

4

Task of convection parametrisation:in practice this means:

Determine vertical distribution of heating, moistening and momentum changes

Cloud model

Determine the overall amount of the energy conversion, convective precipitation=heat release

Closure

Determine occurrence/localisation of convection

Trigger

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Constraints for convection parametrisation

• Physical– remove convective instability and produce subgrid-scale convective precipitation

(heating/drying) in unsaturated model grids

– produce a realistic mean tropical climate

– maintain a realistic variability on a wide range of time-scales

– produce a realistic response to changes in boundary conditions (e.g., El Nino)

– be applicable to a wide range of scales (typical 10 – 200 km) and types of convection (deep tropical, shallow, midlatitude and front/post-frontal convection)

• Computational– be simple and efficient for different model/forecast configurations (T511, EPS,

seasonal prediction)

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Types of convection schemes

• Schemes based on moisture budgets– Kuo, 1965, 1974, J. Atmos. Sci.

• Adjustment schemes– moist convective adjustement, Manabe, 1965, Mon. Wea. Rev.– penetrative adjustment scheme, Betts and Miller, 1986, Quart. J. Roy. Met. Soc.,

Betts-Miller-Janic

• Mass-flux schemes (bulk+spectral)– entraining plume - spectral model, Arakawa and Schubert, 1974, Fraedrich

(1973,1976), Neggers et al (2002), Cheinet (2004), all J. Atmos. Sci. , – Entraining/detraining plume - bulk model, e.g., Bougeault, 1985, Mon. Wea. Rev.,

Tiedtke, 1989, Mon. Wea. Rev., Gregory and Rowntree, 1990, Mon. Wea . Rev., Kain and Fritsch, 1990, J. Atmos. Sci., Donner , 1993, J. Atmos. Sci., Bechtold et al 2001, Quart. J. Roy. Met. Soc.

– episodic mixing, Emanuel, 1991, J. Atmos. Sci.

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The “Kuo” scheme

Closure: Convective activity is linked to large-scale moisture convergence

0

(1 )ls

qP b dz

t

Vertical distribution of heating and moistening: adjust grid-mean to moist adiabat

Main problem: here convection is assumed to consume water and not energy -> …. Positive feedback loop of moisture convergence

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Adjustment schemes

e.g. Betts and Miller, 1986, QJRMS:

When atmosphere is unstable to parcel lifted from PBL and there is a deep moist layer - adjust state back to reference profile over some time-scale, i.e.,

qq

t

q ref

conv

.TT

t

T ref

conv

.

Tref is constructed from moist adiabat from cloud base but no

universal reference profiles for q exist. However, scheme is robust and produces “smooth” fields.

Procedure followed by BMJ scheme…

1) Find the most unstable air in lowest ~ 200 mb

2) Draw a moist adiabat for this air

3) Compute a first-guess temperature-adjustment profile (Tref)

4) Compute a first-guess dewpoint-adjustment profile (qref)

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Adjustment schemes:The Next Step is an Enthalpy Adjustment

First Law of Thermodynamics: vvp dqLdTCdH

With Parameterized Convection, each grid-point column is treated in isolation. Total column latent heating must be directly proportional to total column drying, or dH = 0.

dpqqLdpTTC vvref

P

P v

P

P refp

t

b

t

b

)(

Enthalpy is not conserved for first-guess profiles for this sounding!

Must shift Tref and qvref to the left…

a

b

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Imposing Enthalpy Adjustment:

a

Shift profiles to the left in order to conserve enthalpy

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Adjustment scheme:Adjusted Enthalpy Profiles:

b

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The mass-flux approach

p

secLQ C

)(1

Aim: Look for a simple expression of the eddy transport term

Condensation term Eddy transport term

?

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The mass-flux approach

Reminder:

with 0

Hence

'

)(

= =0 0

and therefore

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The mass-flux approach:Cloud – Environment decomposition

Total Area: A

Cumulus area: a

Fractional coverage with cumulus elements:

A

a

Define area average:

ec 1

ec

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The mass-flux approach:Cloud-Environment decomposition

ecec 1 1

With the above:

and

ec 1

Average over cumulus elements Average over environment

Use Reynolds averaging again for cumulus elements and environment separately:

cccc and

eeee

=

0

=

0

Neglect subplume correlations

(see also Siebesma and Cuijpers, JAS 1995 for a discussion of the validity of the top-hat assumption)

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The mass-flux approach

Then after some algebra (for your exercise) :

ecec

1

Further simplifications :

The small area approximation

ec ;1)1(1

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The mass-flux approach

Then :

c c e

Define convective mass-flux:

cc

cM wg

Then

ccgM

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The mass-flux approach

With the above we can rewrite:

p

ssMgecLQ

cc

C

)(

)(1

p

qqMLgecLQ

cc

)(

)(2

To predict the influence of convection on the large-scale with this approach we now need to describe the convective mass-flux, the values of the thermodynamic (and momentum) variables inside the convective elements and the condensation/evaporation term. This requires, as usual, a cloud model and a closure to determine the absolute (scaled) value of the massflux.

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Mass-flux entraining plume cloud models

Cumulus element i

Continuity:

0

p

Mg

ti

iii

Heat:

i

iiiii

ii Lcp

sMgss

t

s

Specific humidity:

i

iiiii

ii cp

qMgqq

t

q

Entraining plume model

i

i

ic

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Mass-flux entraining plume cloud models

Simplifying assumptions:1. Steady state plumes, i.e., 0

t

X

Most mass-flux convection parametrizations today still make that assumption, some however are prognostic

2. Bulk mass-flux approach

Sum over all cumulus elements, e.g.

p

Mg c

with i i i

iiic MM , ,

e.g., Tiedtke (1989), Gregory and Rowntree (1990), Kain and Fritsch (1990)

3. Spectral method

D

dpmpM Bc

0

),()()(

e.g., Arakawa and Schubert (1974) and derivatives

Important: No matter which simplification - we always describe a cloud ensemble, not individual clouds (even in bulk models)

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Large-scale cumulus effects deduced using mass-flux models

p

ssMgecLQ

cc

C

)(

)(1

Assume for simplicity: Bulk model, ccci

i

cMg

p

Lcss

p

sMg c

cc

Combine:

Lessp

sgMQ c

cC

)(1

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Large-scale cumulus effects deduced using mass-flux models

Lessp

sgMQ c

cC

)(1

Physical interpretation (can be dangerous after a lot of maths):

Convection affects the large scales by

Heating through compensating subsidence between cumulus elements (term 1)

The detrainment of cloud air into the environment (term 2)

Evaporation of cloud and precipitation (term 3)

Note: The condensation heating does not appear directly in Q1. It is however a crucial part of the cloud model, where this heat is transformed in kinetic energy of the updrafts.

Similar derivations are possible for Q2.

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Alternatives to the entraining plume model - Episodic mixing

Observations show that the entraining plume model might be a poor representation of individual cumulus clouds.

Therefore alternative mixing models have been proposed - most prominently the episodic (or stochastic) mixing model (Raymond and Blyth, 1986, JAS; Emanuel, 1991, JAS)

Conceptual idea: Mixing is episodic and different parts of an updraught mix differently

Basic implementation:

assume a stochastic distribution of mixing fractions for part of the updraught air - create N mixtures

Version 1: find level of neutral buoyancy of each mixture

Version 2: move mixture to next level above or below and mix again - repeat until level of neutral buoyancy is reached

Although physically appealing the model is very complex and practically difficult to use

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Closure in mass-flux parametrizations

The cloud model determines the vertical structure of convective heating and moistening (microphysics, variation of mass flux with height, entrainment/detrainment assumptions).

The determination of the overall magnitude of the heating (i.e., surface precipitation in deep convection) requires the determination of the mass-flux at cloud base. - Closure problem

Types of closures:

Deep convection: time scale ~ 1h

Equilibrium in CAPE or similar quantity (e.g., cloud work function)

Boundary-layer equilibrium

Shallow convection: time scale ? ~ 3h

idem deep convection, but also turbulent closure (Fs=surface heat flux, ZPBL=boundary-layer height)

1/3

* *; ,cb PBL

p

g HSM a w w z

c

Grant (2001)

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CAPE closure - the basic idea

large-scale processes generate CAPE

Convection consumes

CAPE

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CAPE closure - the basic idea

Downdraughts

Convection consumes

CAPE

Surface processes

also generate CAPE

b

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Boundary Layer Equilibrium closure (1) as used for shallow convection in the IFS

1[ ] 0;

( ) [ ] ;

[ ( ) ( )]

LCL LCL LCL

s

s

s s s

LCL

s

P P Phs

p v h s

P P Prad dyn

P

S LCL S conv p v

Prad dyn

u u uconv c p v l

Fh T qdp C L dp dp F h

t t t p g

T qF F P F F C L dp with

t t

F M C T T L q q q

1) Assuming boundary-layer equilibrium of moist static energy hs

2) What goes in goes out

Therefore, by integrating from the surface (s) to cloud base (LCL) including all processes that contribute to the moist static energy, one obtains the flux on top of the boundary-layer that is assumed to be the convective flux Mc (neglect downdraft contributions)

Fs is surface moist static energy flux

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Boundary Layer Equilibrium closure (2) – as suggested for deep convection

Postulate: tropical balanced temperature anomalies associated with wave activity are small (<1 K) compared to buoyant ascending parcels ….. gravity wave induced motions are short lived

Convection is controlled through boundary layer entropy balance - sub-cloud layer entropy is in quasi-equilibrium – flux out of boundary-layer must equal surface flux ….boundary-layer recovers through surface fluxes from convective drying/cooling

Raymond, 1995, JAS; Raymond, 1997, in Smith Textbook

Fs is surface heat flux

)(

);()(

0,

,

,,

,

PBLedes

uc

uc

dcPBLe

de

dcPBLe

ue

ucS

convSPBLe

FM

MMMMF

FFconst

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Summary (1)

• Convection parametrisations need to provide a physically realistic forcing/response on the resolved model scales and need to be practical

• a number of approaches to convection parametrisation exist

• basic ingredients to present convection parametrisations are a method to trigger convection, a cloud model and a closure assumption

• the mass-flux approach has been successfully applied to both interpretation of data and convection parametrisation …….

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Summary (2)

• The mass-flux approach can also be used for the parametrization of shallow convection.

It can also be directly applied to the transport of chemical species

• The parametrized effects of convection on humidity and clouds strongly depend on the assumptions about microphysics and mixing in the cloud model --> uncertain and active research area

• …………. Future we already have alternative approaches based on explicit representation (Multi-model approach) or might have approaches based on Wavelets or Neural Networks

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Trigger Functions

 

CAPECloudDepth

CIN Moist.Conv.

Sub-cloudMass conv.

Cloud-layerMoisture ∂(CAPE)/∂t

BMJ(Eta)

     

 

Grell(RUC, AVN) 

     

KF

(Research) 

 

   

Bougeault(Meteo FR)

   

     

Tiedtke(ECMWF) 

     

Bechtold(Research) 

 

   

Emanuel(NOGAPS,research)

 

       

Gregory/Rown(UKMO)

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Closure Assumptions (Intensity)

  CAPE Cloud-layermoisture

MoistureConverg.

∂(CAPE)/∂t SubcloudQuasi-equil.

BMJ(Eta)

 

     

Grell(RUC, AVN)

 

   

 

KF(Research)

 

       

Bougeault(Meteo FR)

 

   

   

Tiedtke(ECMWF)

 

       

shallow

Bechtold(Research)

 

       

Emanuel(Research)

       

Gregory/Rown

(UKMO)

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Vertical Distribution of Heat, Moisture  Entraining/Detraining

Plume 

Convective Adjustment

Buoyancy SortingCloud Model

BMJ(Eta)

 

 

 

Grell(RUC, AVN)

 

   

KF(Research)

   

Bougeault(Meteo FR)

 

   

Tiedtke(ECMWF)

 

   

Bechtold(Research)

 

   

Emanuel(Research)

   

Gregory/Rowntree

(UKMO)