1

Numerical Weather Prediction Parameterization of diabatic processes

Convection IIIThe ECMWF convection scheme

Christian Jakob and Peter Bechtold

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A bulk mass flux scheme:What needs to be considered

Entrainment/Detrainment

Downdraughts

Link to cloud parameterization

Cloud base mass flux - Closure

Type of convection shallow/deep

Where does convection occur

Generation and fallout of precipitation

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Basic Features

• Bulk mass-flux scheme• Entraining/detraining plume cloud model• 3 types of convection: deep, shallow and mid-level - mutually

exclusive• saturated downdraughts• simple microphysics scheme• closure dependent on type of convection

– deep: CAPE adjustment– shallow: PBL equilibrium

• strong link to cloud parameterization - convection provides source for cloud condensate

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Large-scale budget equations: M=ρw; Mu>0; Md<0

)()()( FMLeecLsMMsMsMp

gt

sfsubclddududduu

cu

Mass-flux transport in up- and downdraughts

condensation in updraughts

Heat (dry static energy): Prec. evaporation in downdraughts

Prec. evaporation below cloud base

Melting of precipitation

Freezing of condensate in updraughts

Humidity:

subclddududduucu

eecqMMqMqMp

gt

q

)(

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Large-scale budget equations

Cloud condensate:

u ucu

lD l

t

uMMuMuMp

gt

ududduu

cu

)(

Momentum:

vMMvMvMp

gt

vdudduu

cu

)(

uucu

lDat

a)1(

Cloud fraction:

(supposing fraction 1-a of environment is cloud free)

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Large-scale budget equations

Nota: These tendency equations have been written in flux form which by definition is conservative. It can be solved either explicitly (just apply vertical discretisation) or implicitly (see later).

Other forms of this equation can be obtained by explicitly using the derivatives (given on Page 10), so that entrainment/detrainment terms appear. The following form is particular suitable if one wants to solve the mass flux equations with a Semi-Lagrangien scheme;

note that this equation is valid for all variables T, q, u, v, and that all source terms (apart from melting term) have cancelled out

)()()(

dduuducu

DDp

MMgt

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Occurrence of convection:make a first-guess parcel ascent

Updraft Source Layer

LCL

ETL

CTL

1) Test for shallow convection: add T and q perturbation based on turbulence theory to surface parcel. Do ascent with w-equation and strong entrainment, check for LCL, continue ascent until w<0. If w(LCL)>0 and P(CTL)-P(LCL)<200 hPa : shallow convection

2) Now test for deep convection with similar procedure. Start close to surface, form a 30hPa mixed-layer, lift to LCL, do cloud ascent with small entrainment+water fallout. Deep convection when P(LCL)-P(CTL)>200 hPa. If not …. test subsequent mixed-layer, lift to LCL etc. … and so on until 700 hPa

3) If neither shallow nor deep convection is found a third type of convection – “midlevel” – is activated, originating from any model level above 500 m if large-scale ascent and RH>80%.

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Cloud model equations – updraughtsE and D are positive by definition

uuu DE

p

Mg

uuuuuu LcsDsE

p

sMg

uuuuuu cqDqE

p

qMg

uPuuuuuu GclDlE

p

lMg ,

uuuuu uDuE

p

uMg

uuuuu vDvE

p

vMg

2,1

(1 )2 , (1 ) 2

v u vu u ud u u

u v

T TK E wC K g K

z M f T

Kinetic Energy (vertical velocity) – use height coordinates

Momentum

Liquid Water/Ice

Heat Humidity

Mass (Continuity)

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Downdraughts

1. Find level of free sinking (LFS)

highest model level for which an equal saturated mixture of cloud and environmental air becomes negatively buoyant

2. Closure, , 0.3d LFS u bM M

3. Entrainment/Detrainment

turbulent and organized part similar to updraughts (but simpler)

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Cloud model equations – downdraughtsE and D are defined positive

ddd DE

p

Mg

dddddd LesDsE

p

sMg

dddddd eqDqE

p

qMg

ddddd uDuE

p

uMg

ddddd vDvE

p

vMg

Mass

Heat

Humidity

Momentum

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Entrainment/Detrainment (1)

Updraught2,1, uuu EEE 2,1, uuu DDD

“Turbulent” entrainment/detrainment

convection shallowfor 103

convection midlevel and deepfor 102.114

14

m

muu

Organized entrainment is linked to moisture convergence, but only applied in lower part of the cloud (this part of scheme is questionable)

z

qwqv

qE hu

12,

However, for shallow convection detrainment should exceed entrainment (mass flux decreases with height – this possibility is still experimental

ε and δ are generally given in units (1/m) since (Simpson 1971) defined entrainment in plume with radius R as ε=0.2/R ; for convective clouds R is of order 1500 m for deep and R=100 or 50 m for shallow

,1

,1

uu u

uu u

ME

MD

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Entrainment/Detrainment (2)

Organized detrainment:

zzK

zK

zzM

zM

u

u

u

u

)(

Only when negative buoyancy (K decreases with height), compute mass flux at level z+Δz with following relation:

with

2

2u

u

wK

and

uuDu

uu Bf

KCM

E

z

K

)1(

12)1(

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Precipitation

2

10,

crit

u

l

l

uu

uuP elw

cMG

Generation of precipitation in updraughts

Simple representation of Bergeron process included in c0 and lcrit

Liquid+solid precipitation fluxes:

gdpMelteeGpP

gdpMelteeGpP

P

Ptop

snowsubcld

snowdown

snowsnow

P

Ptop

rainsubcld

raindown

rainrain

/)()(

/)()(

Where Prain and Psnow are the fluxes of precip in form of rain and snow at pressure level p. Grain and Gsnow are the conversion rates from cloud water into rain and cloud ice into snow. The evaporation of precip in the downdraughts edown, and below cloud base esubcld, has been split

further into water and ice components. Melt denotes melting of snow.

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Precipitation

uu

precufallout r

zw

VMS

Fallout of precipitation from updraughts

2.0, 32.5 urainprec rV 2.0

, 66.2 uiceprec rV

Evaporation of precipitation

1. Precipitation evaporates to keep downdraughts saturated

2. Precipitation evaporates below cloud base

3

12

, assume a cloud fraction 0.05surf

subcld s

p p Pe RH q q

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Closure - Deep convection

Convection counteracts destabilization of the atmosphere by large-scale processes and radiation - Stability measure used: CAPE

assume that convection reduces CAPE to 0 over a given timescale, i.e.,

CAPECAPE

t

CAPE

cu

0

Originally proposed by Fritsch and Chappel, 1980, JAS

implemented at ECMWF in December 1997 by Gregory (Gregory et al., 2000, QJRMS)

The quantity required by the parametrization is the cloud base mass-flux.

How can the above assumption converted into this quantity ?

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Closure - Deep convection

cv v

vcloud

CAPE g dz

dzttgt

CAPE

cloudv

vcv

cv

v

cu

2

Assume:

1 and cloud, statesteady i.e., ,0

v

cv

cv

t

dzt

gt

CAPE

cu

v

cloudvcu

1

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Closure - Deep convection

z

M

tvc

cu

v

i.e., ignore detrainment

CAPE

dzz

Mg

t

CAPE v

cloudv

c

cu

dtdubuduc MMMMM ,, utd MM ,

, 1

1,

1 1u b nv v

u d nv u b vcloud cloud

CAPE CAPE

MM

g dz g dzz M z

where Mn-1 are the mass fluxes from a previous first guess updraft/downdraft computation

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Closure - Shallow convection

Based on PBL equilibrium : what goes in must go out - no downdraughts

0

0cbase

conv

turb dyn rad

w h h h hdz

z t t t

0

0cbase h

dzt

With ,,

u b u cbaseconv cbasew h M h h and 00, convhw

0

,

cbase

turb dyn rad

u b

u cbase

h h hdz

t t tM

h h

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Closure - Midlevel convection

Roots of clouds originate outside PBL

assume midlevel convection exists if there is large-scale ascent, RH>80% and there is a convectively unstable layer

Closure:bbu wM ,

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Vertical Discretisation

k

k+1/2

k-1/2

(Mulu)k+1/2

(Mulu)k-1/2

Eul Eul

Dulu Dulu

cu

GP,u

(Mul)k-1/2(Mul)k-1/2

(Mul)k+1/2 (Mul)k+1/2

Fluxes on half-levels, state variable and tendencies on full levels

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Numerics: solving Tendency = advection equation explicit solution

( ) ;u u

conv

g M St p

Pr

pS

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2u u u u u ukk k k k k k

conv

gM M M M S

t p

k k -1ψ ψ

2/12/1 kk PPp

if ψ = T,q

In order to obtain a better and more stable “upstream” solution (“compensating subsidence”, use shifted half-level values to obtain:

k -1/2 k -1ψ ψ

Use vertical discretisation with fluxes on half levels (k+1/2), and tendencies on full levels k, so that

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2u u u u u ukk k k k k k k

conv

gM M M M S

t p

k+1/2 k -1/2ψ ψ

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Numerics: implicit advection

( ) ;u u

conv

g M St p

Prp

S

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

n u u u u u u nk k k k k k k k

u u n u u u u nk k k -1 k k k k k

tψ -ψ g M M M M tS

p

t(1+M )ψ - M ψ g M M tS

p

n 1 n 1 n 1k k k -1

n 1k

ψ ψ

2/12/1 kk PPp

if ψ = T,q

=> Only bi-diagonal linear system, and tendency is obtained

as

For “upstream” discretisation as before one obtains: k -1/2 k -1ψ ψ

Use temporal discretisation with on RHS taken at future time and not at current time n1n

1n nk k k

convt t

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Numerics: Semi Lagrangien advection

;)( SMp

gt

uu

Pr

pS

if ψ = T,q

;)(

uuu D

p

g

pgM

tdt

d

Advection velocity

( ) ;dep u u

conv

gD

t t p

)( tgMPdep u

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Tracer transport experiments(1) Single-column simulations: Stability

Surface precipitation; continental convection during ARM

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Tracer transport experiments(1) Stability in implicit and explicit advection

instabilities

• Implicit solution is stable.

• If mass fluxes increases, mass flux scheme behaves like a diffusion scheme: well-mixed tracer in short time

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Tracer transport experiments(2) Single-column against CRM

Surface precipitation; tropical oceanic convection during TOGA-COARE

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Tracer transport experiments(2) IFS Single-column and global model against CRM

Boundary-layer Tracer

• Boundary-layer tracer is quickly transported up to tropopause

• Forced SCM and CRM simulations compare reasonably well

• In GCM tropopause higher, normal, as forcing in other runs had errors in upper troposphere

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Tracer transport experiments(2) IFS Single-column and global model against CRM

Mid-tropospheric Tracer

• Mid-tropospheric tracer is transported upward by convective draughts, but also slowly subsides due to cumulus induced environmental subsidence

• IFS SCM (convection parameterization) diffuses tracer somewhat more than CRM