AN ECMWFLECTURER

Numerical Weather Prediction Numerical Weather Prediction Parametrization of diabatic Parametrization of diabatic

processesprocessesThe ECMWF cloud schemeThe ECMWF cloud scheme

tompkins@ecmwf.int

The ECMWF Scheme ApproachThe ECMWF Scheme Approach• 2 Additional prognostic variables:

– Cloud mass (liquid + ice)– Cloud fraction

• Sources/Sinks from physical processes: – Deep and shallow convection– Turbulent mixing

• Cloud top turbulence, Horizontal Eddies

– Diabatic or Adiabatic Cooling or Warming

• Radiation, dynamics...– Precipitation processes– Advection qt

G(q

t)

qs

x

y C

Some (not all) of these are derived from a distribution approach

Basic assumptionsBasic assumptions

• Clouds fill the whole model layer in the vertical (fraction=cover)

• Clouds have the same thermal state as the environmental air

• rain water/snow falls out instantly but is subject to evaporation/sublimation and melting in lower levels

• cloud ice and water are distinguished only as a function of temperature - only one equation for condensate is necessary

Schematic of Source/Sink Schematic of Source/Sink termsterms

cloud

1. Convective Detrainment

1 2

2. (A)diabatic warming/cooling

3

3. Subgrid turbulent mixing

4

4. Precipitation generation

5

5. Precipitation evaporation/melting

ECMWF prognostic scheme - ECMWF prognostic scheme - EquationsEquations

entrlllPBLll qw

zqDqGecSqA

tq

1)()()(

)()()()( CGCDecCSCAtC

PBL

Cloud liquid water/ice ql

Cloud fraction C

A: Large Scale AdvectionS: Source/Sink due to:SCV: Source/Sink due due to shallow convection (not post-29r1)SBL: Source/Sink Boundary Layer Processes (not post-29r1)c: Source due to Condensatione: Sink due to EvaporationGp: Precipitation sinkD: Detrainment from deep convectionLast term in liquid/ice : Cloud top entrainment (not post-29r1)

Linking clouds and convectionLinking clouds and convection

Basic ideause detrained condensate as a source for cloud water/iceExamplesOse, 1993, Tiedtke 1993, DelGenio et al. 1996, Fowler et al. 1996

Source terms for cloud condensate and fraction can be derived using the mass-flux approach to convection parametrization.

Convective source terms - Convective source terms - Water/IceWater/Ice

zqlMqlqlDS u

uu

CV

k

k+1/2

k-1/2

(Muqlu)k+1/2

(Muqlu)k-1/2 (-Muql)k-1/2

(-Muql)k+1/2

DuqluStandard equation for mass

flux convection scheme

ECHAM, ECMWF and many others...

][21]1[

21

21

21 kl

kklkul

kkCV qMqMqMMS

write in upstream mass-flux form

Convective source terms - Convective source terms - Cloud FractionCloud Fraction

k

k+1/2

k-1/2(MuC)k-1/2

(MuC)k+1/2

Du

Can write similar equation for the cloud fraction

Factor (1-C) assumes that the

pre-existing cloud is randomly distributed

and thus the probability of the convective event

occurring in cloudy regions is equal to C

zCMCDCS uu

CV

1)(

Stratiform cloud formationStratiform cloud formation

Local criterion for cloud formation: q > qs(T,p)

Two ways to achieve this in an unsaturated parcel: increase q or decrease qs

Processes that can increase q in a gridbox

Convection Cloud formation dealt with separatelyTurbulent Mixing Cloud formation dealt with separately

Advection

Advection and cloud formationAdvection and cloud formation

Advection does not mix air !!! It merely moves it around conserving its properties, including clouds.but There are numerical problems in models

ut

)( 11 Tqq s

21 TT

)( 22 Tqq s

t+t

212 5.0 qqq 212 5.0 TTT

Because of the non-linearity of qs(T), q2 > qs(T2)

This is a numerical problem and should not be used as cloud producing process!

Would also be preferable to advect moist conserved quantities instead of T and q

Stratiform cloud formationStratiform cloud formationPostulate:

The main (but not only) cloud production mechanisms for stratiform clouds are due to changes in qs. Hence we will link stratiform cloud formation to dqs/dt.

diab

s

ma

s

diab

s

adiab

ss

dtdT

dTdq

dtdp

dpdq

dtdq

dtdq

dtdq

=

Stratiform cloud formationStratiform cloud formation

21 ccc

Existing clouds “New” clouds

The cloud generation term is split into two components:

And assumes a mixed ‘uniform-delta’ total water distribution

sq

1-C

qt

G(q

t)

C

Stratiform cloud formation: Stratiform cloud formation: Existing CloudsExisting Clouds

Already existing clouds are assumed to be at saturation at the grid-mean temperature. Any change in qs will directly lead to condensation.

0 1 dt

dqdt

dqCc ss

)(tqsqt

G(q

t)C

)( ttqs

Note that this term would apply to a variety of PDFs for the cloudy air (e.g. uniform distribution)

Stratiform cloud formation Stratiform cloud formation Formation of new clouds

RHcrit = 80 % is used throughout most of the troposphere

Due to lack of knowledge concerning the variance of water vapour in the clear sky regions we have to resort to the use

of a critical relative humidity

sq

RH>RHcrit

)(2

1 2

vs

s

qqqCC

sl qCq 21

1-Cqt

G(q

t)

C

sqeq

)(2 es qq )(2

1es

s

qqqCC

ese qCCqq )1( We know qe from

similarly

sq

1-Cqt

G(q

t)

C

RH<RHcrit

Term inactive if RH<RHcrit

Perhaps for large cooling this is accurate?

As we stated in the statistical scheme lecture:

1. With prognostic cloud water and here cover we can write source and sinks consistently with an underlying distribution function

2. But in overcast or clear sky conditions we have a loss of information. Hence the use of Rhcrit in clear sky conditions for cloud formation

Evaporation of cloudsEvaporation of clouds

Processes: e=e1+e2

•Large-scale descent and cumulus-induced subsidence

•Diabatic heating

•Turbulent mixing (E2)

0 1 dt

dqdt

dqCE ss

)(2 vs qqCKE

Cqql

tCqH

qeCD l

cllc

)(1)( 2

)(tqsqt

G(q

t)

C

)( ttqs

GCM grid

cell

x

Note: Incorrectly assumes mixing only reduces cloud cover and liquid!

no effect on cloud cover

Problem: Reversible Scheme?Problem: Reversible Scheme?

1-Cqt

G(q

t)

C

)(tqs)( ttqs

Cooling: Increases cloud cover

1-Cqt

G(q

t)

C

)(tqs )( ttqs

Subsequent Warming of same magnitude: No effect on cloud cover

Process not reversible

Precipitation generationPrecipitation generation

Sundqvist, 1978

2

21

1210

FFql

ql

lPcriteqFFcG

PcF 11 1

TcF 2681 22

Collection

Bergeron Process

Mixed Phase and water clouds

C0=10-4s-1

C1=100

C2=0.5

qlcrit=0.3 g kg-1

Precipitation generation – Precipitation generation – Pre 25r3Pre 25r3

Pure ice clouds (T<250 K)For IWC<100 - explicitly calculate ice settling flux

100IWCvi

16.010029.3 IWCvi

Heymsfield and Donner (1990)

Ice falling in cloud below is source for ice in that layer, ice falling into clear sky is converted into snow

Two size classes of ice, boundary at 100 m

0100 ,min

IWCIWCaIWCIWC tot

tot

McFarquhar and Heymsfield (1997)

100100 IWCIWCIWC tot

IWC>100 falls out as snow within one timestep

Precipitation generation – Post Precipitation generation – Post 25r325r3

Pure ice clouds (T<250 K)For IWC<100 - explicitly calculate ice settling flux

100IWCvi

16.010029.3 IWCvi

Heymsfield and Donner (1990)

Ice falling in cloud below is source for ice in that layer, ice falling into clear sky is converted into snow

Two size classes of ice, boundary at 100 m

0100 ,min

IWCIWCaIWCIWC tot

tot

McFarquhar and Heymsfield (1997)

100100 IWCIWCIWC tot

IWC>100 falls according to Heymsfield and Donner (1990)

Small Ice has fixed fall speed of 0.15 m/s

Numerical IntegrationNumerical IntegrationIn both prognostic equations for cloud cover and cloud water all source and sink terms are treated either explicitly or Implicitly. E.g:

ljil qKK

tq

Solve analytically and exactly as ordinary differential equations

tKj

j

itKll e

KKetqttq j 1)()(

(provided Kj>0)

Numerical IntegrationNumerical Integration

• How is this accomplished? • If a term contains a nth power for example:

nl

l Kqtq

• The first order approximation is used:

ln

ll qKq

tq 1

0

Beginning of timestep value

Numerical Aspects – Ice fall Numerical Aspects – Ice fall speedsspeeds• Remember the original ice assumptions?

“IWC>100 falls out as snow within one timestep” – V = CFL limit

Solved implicitly to prevent noise

Can calculate fall speed implied by CFL limit

Resulting Ice Fall Speed Resulting Ice Fall Speed T511: pre-25r3T511: pre-25r3

SMALL ICE

LARGE ICE

IN CLOUD ICE MIXING RATIO (KG/KG)

Resulting Ice Fall Speed Resulting Ice Fall Speed T95: pre-25r3 (corrected at T95: pre-25r3 (corrected at 25r3)25r3)

SMALL ICE

LARGE ICE

IN CLOUD ICE MIXING RATIO (KG/KG)

Real advectionPerfect advection

Numerical Aspects - Vertical Numerical Aspects - Vertical AdvectionAdvection

Problem: we neglect vertical subgrid-scales

w

Ice sedimentation: Ice sedimentation: Further Numerical ProblemsFurther Numerical Problems

Since we only allow ice to fall one layer in a timestep, and since the cloud fills the layer in the vertical, the cloud lower boundary is advected at the CFL rate of z / t

Solution (not very satisfactory): Only allow sedimentation of cloud mass into existing cloudy regions, otherwise convert to snow – Results in higher effective sedimentation rates AND vertical resolution sensitivity

Ice flux Snow flux

Numerical Aspects - Ice Numerical Aspects - Ice settlingsettling

Problem : A “bad” assumption and numerics (always imperfect) lead to a realistic looking result.

1. We implicitly assume that all particles fall with the same velocity - wrong

Start with single layer ice cloud in level k - expect that settling moves some of the ice into the next layer and that some stays in this layer due to size distribution.

Assume t = z / vice and ignore density

Perfect numerics in this case - explicit scheme:

2. Use analytical integrationz

tqtqvt

tqttq kl

kl

ice

kl

kl

)()()()( 111

)()(1 tqttq kl

kl

)(63.0)(1 tqttq kl

kl )(37.0)( tqttq k

lk

l

Correct but not as expected

as expected but incorrect

iicei qv

dzd

dtdq

1

100 vs 50layer resolution

Ice flux Snow flux

iicei qv

dzd

dtdq

1

Sink at level k depends on dz

k

k+1

This is correct, but if then converted to snow and “lost” will introduce a resolution sensivity

which is which is which?which?

Planned Revision for 29r3 (sept Planned Revision for 29r3 (sept 2005)2005)To tackle ice settling deficiencyTo tackle ice settling deficiencyOptions: (i) Implicit numerics (ii) Time splitting (iii) Semi-Lagrangian

fall speed

Constant Explicit Source/Sink

Advected Quantity (e.g. ice)

Implicit Source/Sink(not required for short timesteps)

Implicit: Upstream forward in time,

j=vertical leveln=time level

Solution

Note: For multiple X-bin bulk scheme (X=ice, graupel, snow…) with implicit

(“D” above) cross-terms leads to set of X-linear algebraic equations

what is short?

XXXX qv

dzdDqC

dtdq

1

Improved Improved Numerics in SCM Numerics in SCM Cirrus CaseCirrus Case

29r1

Sch

eme

29r3

Sch

eme

100 vs 50layer resolution

time evolution of cloud cover and ice

which is which is which?which?

time

cloud fraction

cloud fraction

cloud ice

cloud ice

Precipitation evaporation and Precipitation evaporation and meltingmelting

Melting

Occurs whenever T > 0 C

Is limited such that cooling does not lead to T < 0 C

577.0

,3

21

0

4, 109.5

11044.5

clrP

clrvsclrPP C

PppqqCE

Evaporation (Kessler 1969, Monogram)

Newtonian Relaxation tosaturated conditions

Precipitation EvaporationPrecipitation Evaporation

Jakob and Klein (QJRMS, 2000)

Mimics radiation schemes by using a ‘2-column’ approach of recording both cloud and clear precipitation fluxes

This allows a more accurate assessment of precipitation evaporation

NUMERICS: Solved Implicitly to avoid numerical problems

ttv

tts

tv

ttv qq

tqq

ts

p

v

tv

tst

vtt

v

dTdq

cLt

qqtqq

11

Precipitation EvaporationPrecipitation Evaporation

• Numerical “Limiters” have to be applied to prevent grid scale saturation

Clear sky region

t

t

Grid can notsaturate

Clear sky region

t

t

Grid slowlysaturates

Precipitation EvaporationPrecipitation Evaporation

• Threshold for rainfall evaporation based on precipitation cover

RHcrit

Clear skyPrecipitation Fraction

70%0

100%Critical relative

humidity for precipitation evaporation

• Instead can use overlap with clear sky humidity distribution• Humidity in most moist fraction used instead of clear sky mean

Clear sky

1

Cirrus Clouds, new homogeneous Cirrus Clouds, new homogeneous nucleation form for 29r3nucleation form for 29r3• Want to represent super-saturation and homogeneous

nucleation• Include simple diagnostic parameterization in existing

ECMWF cloud scheme• Desires:

– Supersaturated clear-sky states with respect to ice– Existence of ice crystals in locally subsaturated state

• Only possible with extra prognostic equation…

GCM gridbox

CClear sky Cloudy region

qvcld =?qv

env=?cloudclear

GCM gridbox

Three items of information: qv, qi, C (vapour, cloud ice and cover)

•We know: qc occurs in the cloudy part of the gridbox•We know: The mean in-cloud cloud ice•What about the water vapour? In the bad-old-days was easy: •Clouds: qv

cld=qs

•Clear sky: qvenv=(qv-Cqs)/(1-C) (rain evap and new cloud formation)

C

Unlike “parcel” models, or high resolution LES models, we have to deal with subgrid variability

I F S

GCM gridbox

The good-new-days: assume no supersaturation can exist

C

qvcldqv

env

(qv-Cqs)/(1-C) qs1.

qv+qi

qv-qi(1-C)/C3. Klaus Gierens:Humidity in clear sky part equal to the mean total water

4. (qv-Cqs)/(1-C) qsMy assumption: Hang on… Looks familiar???

qvqv2. Lohmann and Karcher: Humidity uniform across gridcell

The bad old daysNo supersaturation

GCM gridbox

qvqv2. Lohmann and Karcher: Humidity uniform across gridcell

qvcldqv

env

qvcld: reduces as

ice is formed

Artificial flux of vapour from clear sky to cloudy regions!!!

Critical Scrit reached

Assumption ignores fact that difference processes are occurring on the subgrid-scale

qvenv=qv

cld

uplifted box

From Mesoscale Model

1 timestep

1 timestep

qi: increases

GCM gridbox

qvcldqv

env

qvcld: reduces to

qs

Critical Scrit reached

qi: increases qvenv

unchanged

No artificial flux of vapour from clear sky from/to cloudy regions

Assumption seems reasonable: BUT! Does not allow nucleation or sublimation timescales to be represented, due to hard adjustment

qvenv=unchange

dqi: decreases

uplifted box

4. (qv-Cqs)/(1-C) qsMy assumption: Hang on… Looks familiar???

microphysics

Difference to standard scheme is that environmental humidity must exceed Scrit to form new cloud

From GCM perspective

Region Lat:-60./60., Lon:0./360.

0.8 1.0 1.2 1.4 1.6 1.8RH

0.001

0.010

0.100

1.000

10.000Fr

eqdefaultclipping to Koopnew parameterizationMoziac

A

C

B

RH PDFRH PDFat at

250hPa250hPaone one

month month averageaverage

Aircraft observations

A: Numerics and interpolation for default modelB: The RH=1 microphysics modeC: Drop due to GCM assumption of subgrid fluctuations in total water

Summary of Tiedtke SchemeSummary of Tiedtke Scheme• Scheme introduces two prognostic equations for cloud water and cloud

mass• Sources and sinks for each physical process - • Many derived using assumptions concerning subgrid-scale PDF for

vapour and clouds

More simple to implement than a prognostic statistical scheme since “short cuts” are possible for some terms. Also nicer for assimilation since prognostic quantities are directly observable

Loss of information (no memory) in clear sky (a=0) or overcast conditions (a=1) (critical relative humidities necessary etc). Difficult to make reversible processes.

Nothing to stop solution diverging for cloud cover and cloud water. (e.g: liquid>0 while a=0). More unphysical “safety switches” necessary.

Last Lecture: Validation!

ExampleExampleProblemProblem

1. Convection detrains into a clear-sky grid-box at a normalized rate of 1/3600 s-1. This is the highest level of convection so there is no subsidence from the layer above (see picture). How long does it take for the cloud cover to reach 50%? State the assumptions made.

2. The detrained air has a cloud liquid water mass mixing ratio of 3 g kg-1. If the conversion to precipitation is treated as a simple Newtonian relaxation (i.e. [dql / dt]precip = ql / tau) with a timescale of 2 hours, what is the equilibrium cloud mass mixing ratio?

GridboxM=1/3600 s-1

M=1/3600 s-1

M=0 s-1

Answers: (1) 42 minutes (2) 2 g kg-1.