Numerical Weather Prediction Parametrization of Diabatic Processes Cloud Parametrization 2: Cloud Cover Richard Forbes and Adrian Tompkins [email protected]

Numerical Weather Prediction Parametrization of Diabatic Processes

Cloud Parametrization 2:Cloud Cover

Richard Forbes

and Adrian Tompkins

[email protected]

2

Clouds in GCMs:What are the problems ?

Many of the observed clouds and especially the processes within them are of subgrid-scale size (both horizontally and vertically)

GCM Grid cell 25-400km

3

~50

0m

~100km

Macroscale Issues of Parameterization

VERTICAL COVERAGEMost models assume that this is 1

This can be a poor assumption with coarse vertical grids.Many climate models still use fewer than 30 vertical levels.

x

z

4

~50

0m

~100km


HORIZONTAL COVERAGE, aSpatial arrangement ?

x

z

5

~50

0m

~100km


Vertical Overlap of cloudImportant for Radiation and Microphysics Interaction

x

z

6

~50

0m

~100km


In-cloud inhomogeneity in terms of cloud particle size and number

7

~50

0m

~100km


Just these issues can become very complex!!!

x

z

8

Cloud Cover: Why Important?

In addition to the influence on radiation, the cloud cover is important for the representation of microphysics

Imagine a cloud with a liquid condensate mass ql

The in-cloud mass mixing ratio is ql/a

a largea small

GC

M g

rid b

ox

precipitation not equal in each case sinceautoconversion is nonlinear

Reminder: Autoconversion (Kessler, 1969)

otherwise0

if0critcrit

llllP

qqqqcG

Complex microphysics perhaps a wasted effort if assessment of a is poor

9

qv = water vapour mixing ratio

qc = cloud water (liquid/ice) mixing ratio

qs = saturation mixing ratio = F(T,p)

qt = total water (vapour+cloud) mixing ratio

RH = relative humidity = qv / qs

1. Local criterion for formation of cloud: qt > qs

This assumes that no supersaturation can exist

2. Condensation process is fast (cf. GCM timestep)

qv = qs qc= qt – qs

!!Both of these assumptions are suspect in ice clouds!!

First: Some assumptions!

10

Partial coverage of a grid-box with clouds is only possible if there is an inhomogeneous distribution of temperature

and/or humidity.

Homogeneous Distribution of water

vapour and temperature:

2,sq

q

x

q

1,sq

One Grid-cell

Note in the secondcase the relative

humidity=1 from our

assumptions

Partial cloud cover

11

Another implication of the above is that clouds must exist before the grid-mean relative humidity reaches 1.

q

x

q

sq

cloudy=

RH=1 RH<1

Heterogeneous Distribution of T and q

12

Heterogeneous Distribution of q only

• The interpretation does not change much if we only consider humidity variability

• Throughout this talk I will neglect temperature variability

• Analysis of observations and model data indicates humidity fluctuations are more important most of the time.

qt

x

tqsq

cloudy

RH=1 RH<1

13

Take a grid cell with a certain (fixed) distribution of total water.

At low mean RH, the cloud cover is zero, since even the moistest part of

the grid cell is subsaturated

qt

x

tq

sq

RH=60%

RH060 10080

C

1

Simple Diagnostic Cloud Schemes: Relative Humidity Schemes

14

Add water vapour to the gridcell, the moistest part of the cell

become saturated and cloud forms. The cloud cover is low.

qt

x

tq

sq

RH=80%

RH060 10080

C

1


15

Further increases in RH increase the cloud cover

qt

x

tqsq

RH=90%

060 10080

C

1

RH


16

The grid cell becomes overcast when RH=100%,

due to lack of supersaturation

qt

x

tqsq

RH=100%

C

0

1

60 10080RH


17

Diagnostic Relative Humidity Schemes

• Many schemes, from the 1970s onwards, based cloud cover on the relative humidity (RH)

• e.g. Sundqvist et al. MWR 1989:

critRHRHC 1

11

RHcrit = critical relative humidity at which cloud assumed to form

(function of height, typical value is 60-80%)

C

0

1

60 10080RH

18

• Since these schemes form cloud when RH<100%, they implicitly assume subgrid-scale variability for total water, qt, (and/or temperature, T)

• However, the actual PDF (the shape) for these quantities and their variance (width) are often not known

• “Given a RH of X% in nature, the mean distribution of qt is such that, on average, we expect a cloud cover of Y%”


19

• Advantages:– Better than homogeneous assumption, since

clouds can form before grids reach saturation

• Disadvantages:– Cloud cover not well coupled to other processes– In reality, different cloud types with different

coverage can exist with same relative humidity. This can not be represented

• Can we do better?


20

• Could add further predictors• E.g: Xu and Randall (1996)

sampled cloud scenes from a 2D cloud resolving model to derive an empirical relationship with two predictors:

),( cqRHFC

• More predictors, more degrees of freedom = flexible

• But still do not know the form of the PDF (is model valid?)

• Can we do better?


21


• Another example is the scheme of Slingo, operational at ECMWF until 1995.

• This scheme also adds dependence on vertical velocities• use different empirical relations for different cloud types, e.g.,

middle level clouds:

2

* 0,1

max

crit

critm RH

RHRHC

crit

crit

m

critmm

C

CC

0

0

0

*

*

Relationships seem Ad-hoc? Can we do better?

22

Statistical Schemes

• These explicitly specify the probability density function (PDF) for the total water qt (and sometimes also temperature)

sq

ttstc dqqPDFqqq )()(

qt

x

q

sq

qt

PD

F(q

t)

qs

Cloud cover is integral under

supersaturated part of PDF

sq

tt dqqPDFC )(

23

Statistical Schemes

• Others form variable ‘s’ that also takes temperature variability into account, which affects qs

)( LtL Tqas

Ls

L TT

q

1

1

L

pL C

La

LCL

L qTTp

LIQUID WATER TEMPERATUREconserved during changes of state

ss

dssGC )(

S is simply the ‘distance’ from the linearized saturation vapour pressure curve

qs

T

qt

LT

S

INCREASES COMPLEXITYOF IMPLEMENTATION

Cloud mass if Tvariation is neglected

qs

24

Statistical Schemes

• Knowing the PDF has advantages:– More accurate

calculation of radiative fluxes

– Unbiased calculation of microphysical processes

• However, location of clouds within gridcell unknown

qt

PD

F(q

t)

qs

x

y

C

e.g. microphysics

bias

25

Building a statistical cloud scheme

• Two tasks: Specification of the:

(1) PDF shape

(2) PDF moments• Shape: Unimodal? bimodal? How many

parameters?

• Moments: How do we set those parameters?

26

• Lack of observations to determine qt PDF– Aircraft data

• limited coverage

– Tethered balloon• boundary layer

– Satellite• difficulties resolving in vertical

• no qt observations

• poor horizontal resolution

– Raman Lidar• one location

• Cloud Resolving models have also been used• realism of microphysical parameterisation?

Modis image from NASA website

Building a statistical cloud schemeTASK 1: Specification of PDF shape

27

qt

PDF(qt)

Hei

ght

Aircraft Observed

PDFs

Wood and field JAS 2000

Aircraft observations low

clouds < 2km

Heymsfield and McFarquhar

JAS 96Aircraft IWC obs during CEPEX

28

Conclusion: PDFs are mostly approximated by uni or bi-modal distributions, describable by a few parameters

More examples from Larson et al.

JAS 01/02

Note significant error that can occur if PDF is

unimodal

PDF Data

29

PDF of water vapour and RH from Raman Lidar

From Franz Berger

30


Many function forms have been usedsymmetrical distributions:

Triangular:Smith QJRMS (90)

qt

qt

Gaussian:Mellor JAS (77)

qt

PD

F(

q t)

Uniform:Letreut and Li (91)

qt

s4 polynomial:Lohmann et al. J. Clim (99)

31

skewed distributions:

qt

PD

F(

q t)

Exponential:Sommeria and Deardorff JAS (77)

Lognormal: Bony & Emanuel JAS

(01)

qt qt

Gamma:Barker et al. JAS (96)

qt

Beta:Tompkins JAS (02)

qt

Double Normal/Gaussian:Lewellen and Yoh JAS (93), Golaz et al. JAS 2002


32

Need also to determine the moments of the distribution:– Variance (Symmetrical

PDFs)– Skewness (Higher

order PDFs)– Kurtosis (4-parameter

PDFs)

qt

PD

F(q

t)

e.g. HOW WIDE?

saturation

cloud forms?

Moment 1=MEANMoment 2=VARIANCEMoment 3=SKEWNESSMoment 4=KURTOSIS

Skewness Kurtosis

po

siti

ve negative

nega

tive positive

Building a statistical cloud schemeTASK 2: Specification of PDF moments

33

• Some schemes fix the moments (e.g. Smith 1990) based on critical RH at which clouds assumed to form

• If moments (variance, skewness) are fixed, then statistical schemes are identically equivalent to a RH formulation

• e.g. uniform qt distribution = Sundqvist form

esv qCCqq )1(

C

sq

1-C

tq

(1-RHcrit)qs

qt

G(q

t)

eq

2)1)(1(1 CRHq

qRH crit

s

v

critRHRHC 1

11Sundqvist formulation!!!

))1)(1(1( CRHqq critse where

Building a statistical cloud schemeTASK 2: Specification of PDF moments

34

Clouds in GCMsProcesses that can affect distribution moments

convection

turbulence

dynamics

microphysics

35

Example: Turbulence

dry air

moist air

In presence of vertical gradient of total water, turbulent mixing can increase horizontal variability

dz

qdqw

dt

qd tt

t

22

36

Example: Turbulence

dry air

moist air

In presence of vertical gradient of total water, turbulent mixing can increase horizontal variability

while mixing in the horizontal plane naturally reduces the

horizontal variability

22tt q

dt

qd

37

Specification of PDF moments

dz

qdqwq t

tt 22

Example: Ricard and Royer, Ann Geophy, (93), Lohmann et al. J. Clim (99)

• Disadvantage:– Can give good estimate in boundary layer, but above, other

processes will determine variability, that evolve on slower timescales

turbulence

22

2 ttt

t q

dz

qdqw

dt

qd

Source dissipation

localequilibrium

If a process is fastcompared to a GCM timestep, an equilibrium can be assumed, e.g. Turbulence

38

Prognostic Statistical Scheme

• Tompkins (2002) prognostic statistical scheme (implemented in ECHAM5 climate GCM)

• Prognostic equations are introduced for the variance and skewness of the total water PDF

• Some of the sources and sinks are rather ad-hoc in their derivation!

convective detrainment

precipitationgeneration

mixing

qs

39

Prognostic Statistical Scheme in action

MinimumMaximumqsat

40

MinimumMaximumqsat

Turbulence breaks up cloud


41

MinimumMaximumqsat

Turbulence breaks up cloud Turbulence creates cloud


42

Prognostic statistical schemeProduction of variance from convectionKlein et al.

Change due to difference in variance

Change due to difference in means

Transport

Also equivalent terms due to entrainment

updraught

We want this varianceDetrained

massD = Convective Detrainment Rate

43

• Change in variance

• However, the tractability depends on the PDF form for the subgrid fluctuations of q, given by G:

Where P is the precipitation generation rate, e.g:

tqP

)1()( qlcrit

ql

eKqP l

tt

q

qq

t dqqGqPt

satt

)(max_

Prognostic statistical schemeChange in variance by precipitation

44

Can quickly get untractable !

• E.g: Semi-Lagrangian ice sedimentation

• Source of variance is far from simple, also depends on overlap assumptions

• In reality of course wish also to retain the sub-flux variability too

Prognostic statistical schemeComplications - sedimentation

45

Summary of statistical schemes

• Advantages– Information concerning subgrid fluctuations of humidity

and cloud water is available– It is possible to link the sources and sinks explicitly to

physical processes– Use of underlying PDF means cloud variables are

always self-consistent

• Disadvantages– Deriving these sources and sinks rigorously is hard,

especially for higher order moments needed for more complex PDFs!

– If variance and skewness are used instead of cloud water and humidity, conservation of the latter is not ensured

46

Issues for GCMs

• If we assume a 2-parameter PDF for total water, which prognostic variables should we use ?

• How do we treat the ice phase when supersaturation is allowed ?

• How do we link other processes with the total water PDF (microphysics, radiation, convection)?

• Is there a real advantage over existing cloud schemes ?

47

Prognostic statistical scheme:Which prognostic equations?

Take a 2 parameter distribution & partially cloudy conditions

qsatqsat

Cloud

Can specify distribution with(a) Mean(b) Varianceof total water

Can specify distribution with(a) Water vapour(b) Cloud watermass mixing ratio

qv ql+i

Cloud

Variance

qt

48

qsat

(a) Water vapour(b) Cloud watermass mixing ratio

qv ql+i

qsat

qv qv+ql+i

• Cloud water budget conserved• Avoids Detrainment term

But problems arise in

Clear sky conditions

(turbulence)

Overcast conditions

(…convection +microphysics)(al la Tiedtke)

Prognostic statistical scheme:Water vapour and cloud water ?

49

qsat

(a) Mean(b) Varianceof total water

• “Cleaner solution”• But conservation of liquid water compromised due to PDF• Need to parametrize those tricky microphysics terms!

Prognostic statistical scheme:Total water mean and variance ?

50

Issues for GCMs





51

sq

ttsti dqqPDFqqq )()(

qt

PD

F(q

t)

qsqcloud

If supersaturation allowed, then the equation for cloud-ice no longer holds

Ice cloud ?

x

y

sub-saturated region

supersaturated clear region cloudy

“activated” region

52

Issues for GCMs





53

Overheard recently in the ECMWF meteorological fruit bowl….…

But a prognostic statistical scheme could provide

consistent sub-grid information for all our physical parametrizations!

We already have cloud variability in our models. Nearly all components of GCMs contain implicit/explicit assumptions concerning subgrid fluctuations,

e.g: RH-based cloud cover, thresholds for precipitation evaporation, convective triggering, plane parallel bias corrections for radiative transfer… etc.

54

Advantage of Statistical Scheme

Statistical Cloud Scheme

Radiation

Microphysics Convection Scheme

Can use information inother schemes

Boundary Layer

55

Variability in Clouds

Models typically have many independent “tunable” parameters with a limited number of “metrics” for verification.

‘M’ tuningparameters

‘N’ Metricse.g. Z500 scores, TOA radiation fluxes

With large M, task of reducing error in N metricsbecomes easier, but not necessarily for the right reasons

Solution: Increase N, or reduce M

Using a statistical cloud scheme with an underlying PDF of sub-grid variability would bring greater consistency between processes and reduce the number of independent “tunable” parameters ‘M’.

56

Issues for GCMs





57

Use of Cloud PDF in Radiation Scheme

Independent Column Approximation, e.g.

MCICA

Can treat the inhomogeneity of in-cloud condensate and overlap in a consistent way between the cloud and radiation schemes

Traditional approach

58

Result is not equal in the two cases since microphysical processes are non-linear

Example on right: Autoconversion based on KesslerGrid mean cloud less than threshold and gives zero

precipitation formation

Cloud Inhomogeneity and microphysics biases

qLqL0

cloud range

mea

n precipgeneration

Most current microphysical schemes use the grid-mean or cloud fraction cloud mass (i.e:

neglect in-cloud variability)

Ls qq

qt

G(q

t)qs

cloud

Independent column approach? – computationally expensive!

59

Summary• GCMs need to have a representation of the sub-grid scale

variability of the atmosphere (e.g. q, T)

• A prognostic statistical cloud scheme that can provide consistent sub-grid heterogeneity information across the model physics is an attractive concept (closer to the real world!)

• There are different implementations with different complexities and degrees of freedom.

• More degrees of freedom allow greater flexibility to represent the real atmosphere, but we need to have enough knowledge/information to understand and constrain the problem (form of pdf/sources/sinks)!

60

Next time: The ECMWF Scheme

• ECMWF Scheme - Considers the physical processes and derives source or sink terms for cloud fraction (cover) and cloud condensate.

• HOW? To do this, assumptions about a distribution PDF and its moments are made for many of the terms

• Thus the ECMWF scheme is essentially another expression of a prognostic statistical scheme - the approach has advantages and disadvantages...

convection

turbulence

dynamics

microphysics

Documents

Numerical Weather Prediction Parametrization of Diabatic Processes Cloud Parametrization 2: Cloud Cover Richard Forbes and Adrian Tompkins [email protected]