EVALUATION OF A GENERAL-CIRCULATION MODEL

PROGNOSTIC CLOUD SCHEME USING

CLOUD-RESOLVING MODEL DATA.

Cyril Morcrette∗ and Damian Wilson

Met Office, United Kingdom

1 INTRODUCTION

The purpose of a general circulation model(GCM) cloud scheme is to determine whatfraction of the grid box is covered by cloud,and what the condensed water content is inthose clouds, given the grid-box mean val-ues of temperature, pressure and humidity,These values of cloud fraction and condensateamount are then used by the GCM radiationand precipitation schemes.

A new prognostic cloud fraction and prognos-tic condensate scheme (PC2) is being devel-oped at the Met Office for use in its GCM:the Met Office Unified Model (MetUM). Eachof the physical processes in the GCM, such aslarge-scale ascent, radiation, convection andboundary-layer processes, will affect the cloudfraction and condensate amounts and theseare then advected by the dynamics.

After a brief overview of the theoretical frame-work that underlies the PC2 cloud scheme, wepresent cloud-resolving model (CRM) simula-tions of tropical deep convection which havebeen analysed using this theoretical frame-work.

The CRM temperature and humidity data isaveraged horizontally to mimic the gridbox-mean values which are the inputs to thescheme. The PC2 scheme then calculatescloud fraction and condensate tendencies inthe same way that it would when used ina GCM. These parametrized tendencies forcloud fraction and condensate amount fromthe scheme can then be compared to the evo-

∗Author address: Dr C. J. Morcrette, Met Office,

FitzRoy Road, Exeter, EX1 3PB, United Kingdom.

Email: cyril.morcrette@metoffice.gov.uk

lution of the fraction of cloudy pixels andmean condensate amount at each level foundin the CRM data.

This analysis of the behaviour of the schemewill allow to gain confidence that it is behav-ing sensibly when used in a GCM.

2 THEORETICAL

OVERVIEW

A brief summary of the PC2 cloud scheme isgiven here for convenience. Further details aregiven by Wilson et al. (2008a). The perfor-mance of the PC2 scheme in a climate modelis described by Wilson et al. (2008b).

2.1 THE ’s’ FRAMEWORK

The amount of cloud fraction and condensatedepends on:

Qc = aL(qT − qsat(TL, p)) (1)

(the difference between the grid-box meanspecific total water content and the saturationspecific humidity (Smith, 1990)) and

s = aL(q′T − αT ′

L − βp′) (2)

(the local deviation from the mean (Mellor,1977)). Where qT is the total (vapour plus liq-uid) specific humidity, TL is the “liquid tem-perature”, and p is the pressure. We use thenotation φ = φ + φ′, where φ represents thegridbox-mean value and φ′ represents the dif-ference of φ from its mean. Additionally L isthe latent heat of condensation, cp the specific

heat capacity at constant pressure, qcl the liq-uid water content, α = ∂qsat/∂T at constantpressure, β = ∂qsat/∂p at constant tempera-ture and aL = (1 + α L

cp)−1.

Within each gridbox, there is a distribution G,of s, and cloud water content will be presentwhenever s > −Qc. So the liquid cloud frac-tion, Cl, and grid-box mean liquid condensateamount, qcl, are given by

Cl =

∫

∞

s=−Qc

G(s)ds (3)

and

qcl =

∫

∞

s=−Qc

(Qc + s)G(s)ds (4)

2.2 HOMOGENEOUS FORCING

Certain physical processes, such as large-scaleascent and adiabatic cooling, can be assumedto act uniformally over the whole GCM grid-box. As a result, the underlying distribu-tion of G is unchanged (Gregory et al., 2002).By calculating the new position of Qc, whichrepresenting the grid box mean values, andknowing the values of G(−Qc), then, follow-ing Wilson and Gregory (2003), the effects ofthe cooling on cloud fraction and liquid watercontent can be quantified using:

∂qcl

∂t= C

∂Qc

∂t(5)

and

∂Cl

∂t= G(−Qc)

∂Qc

∂t(6)

This procedure is illustrated in Fig. 1. Bytaking temperature and humidity incrementsfrom say, the boundary-layer scheme, or theradiation scheme, the change in Qc can befoud using Eq. 1. The changes to qcl andCl can then be found from Eqns. 5 and 6.

Figure 1: Homogeneous forcing. The change incloud fraction, C, which is proportional to thearea under the G curve, can be estimated, byknowing the change in grid-box mean proper-ties (∆Qc), the value of the PDF at the sat-urated/unsaturated boundary G(−Qc), and as-suming that G is linear over the small interval∆Qc.

2.3 PRODUCTION OF

CONVECTIVE CLOUD

Building on the work of Tiedtke (1993), Jakob(2000) and Bushell et al. (2003) show how theevolution of cloud fraction as a result of con-vection can be written as:

∂Cl

∂t= D(1 − Cl) + M

∂Cl

∂z(7)

where D is the detrainment rate and M themass-flux. The first term on the righ-handside represents the detrainment of cloudy airout of the convective plume and into the large-scale environment, while the second term rep-resents the vertical advection of pre-existingcloud by the compensating subsidence. A sim-ilar form:

∂qcl

∂t= D(qcl,plume − qcl) + M

∂qcl

∂z(8)

is used to represent the detrainment of con-densate from the convective plume, as wellas the vertical advection. Two similar equa-tions are used to represent the detrainmentand vertical advection of ice cloud fractionand frozen condensate. The effects of adia-batic warming from compensating subsidencemay lead to the sublimation of some ice. Thisis calculated within the large-scale precipita-tion scheme, while the effects of evaporation ofliquid water by the same process are modelledusing the homogeneous forcing framework.

2.4 LARGE-SCALE

PRECIPITATION

The changes to qcl and qcf due to large-scale microphysics, are taken from the Wil-son and Ballard (1999) large-scale precipita-tion scheme which is used by the Met Of-fice GCM. This calculates the microphysicaltransfer rates between water vapour, cloud icewater content, snow and rain. In addition tothe cloud condensate tendencies provided bythe large-scale precipitation scheme, it is nec-essary to represent the effects of large-scaleprecipitation on the evolution of the cloudfractions.

2.4.1 DEPOSITION AND

SUBLIMATION

The change in cloud fraction due to deposi-tion and sublimation assume that there is auniform distribution of moisture in the partof the gridbox in which the process is acting.By knowing the change in condensate amount,it is possible to work out what the change incloud cover should be. Figure 2 shows the

Figure 2: Regions to consider, when calculatingwhere sublimation can occur.

relevant areas when working out the effectsof sublimation on ice cloud fraction. (Subli-mation does not change the liquid cloud frac-tion). Cl is the liquid cloud fraction, Cf , theice cloud fraction. Sublimation can only occurin the part of the grid box, where there is ice,but no liquid, so somewhere within Aice. Ad-ditionally, Aice, is further divided into the partabove saturation, Aice1, and below saturation,Aice2. Sublimation actually occurs only inAice2. Let C2 = C1 + dC and q2 = q1 + dqbe the end states, defined in terms of the ini-tial states C1 and q1 and difference dC and

dq. The total amount of ice is q1 = 0.5C1Q1

initially and q2 = 0.5C2Q2 at the end. Usingsimilar triangles: Q2/Q1 = C2/C1. By re-arranging and using C1 = Aice2 we find that

dCf,sublime = Aice2

√

1 +∆q

q1

− Aice2. (9)

Deposition only occurs when there is some icealready present, and leaves Cf unchanged. Asimilar derivation, shows that for depositionthe change in liquid cloud fraction is given by

dCl,deposition = Cl

√

1 −∆q

q1

− Cl. (10)

Figure 3: The amount of ice present in the sub-liming region, when assuming a “top hat” distri-bution of ice water content across the grid-box.

2.4.2 FALL OF ICE

Consider two layers with ice cloud fractions ofCf (k) and Cf (k + 1) (Fig. 4(a)). The “over-hang” between the two layers is defined asO = Max(Cf (k + 1) − Cf (k), 0), which en-sures that the overhang cannot be negative.Ice is assumed to fall out of the base of thetop layer, with the velocity, v, calculated bythe large-scale microphysical scheme. In time∆t, the ice has fallen a distance ∆h = v∆t(Fig. 4(b)), which corresponds to a fractionv∆t/∆z of the layer depth. The volume of icethat falls into the cloud-free region in the layer

below is then assumed to completely fill thegrid-box in the vertical and is redistributed,leading to an increase in cloud fraction of∆Cf = OMin(v∆t/∆z, 1) (Fig. 4(c)). Theminimum check being to limit the increase incloud fraction if the ice falls further than onelayer depth per timestep.

Figure 4: Schematic of how ice falling from onelayer increases the ice cloud fraction in the layerbelow.

2.4.3 MELTING

The melting of ice and evaporation of meltingsnow are both assumed to reduce the ice cloudfraction in proportion to how much of the icecondensate has been removed.

2.4.4 OTHER MICROPHYSICAL

PROCESSES

The remaining microphysical processes (i.e.riming, capture, evaporation of rain, accretionand autoconversion) are all assumed to haveno effect on the cloud fractions.

3 RESULTS

Results are presented from a CRM simula-tion using the Met Office Large-Eddy Model(LEM). The case studied is from TOGA-COARE between 20-26 December 1992 (Petchand Gray, 2001). The CRM was run in two-dimensions over a 256 km wide domain with ahorizontal gridlength of 500m. The model topwas at 20 km and a stretched grid was usedwith 120 levels, giving a vertical gridlength ofaround 165 m between 2 and 14 km in theregion where clouds formed.

Figure 5 shows the evolution of the liquid andice cloud fraction and of the liquid and icewater contents averaged over the large-scaleenvironment. The “large-scale environment”is the part of the domain that is not part of theconvective plume (i.e. not a bouyant cloudyupdraught).

Figure 5: a) Liquid cloud fraction and b) icecloud fraction. c) Liquid water content and d)ice water content, averaged over the environe-ment.

Following Arakawa and Schubert (1974) themass-flux can be defined as M = ρσw, whereρ is the density, σ is the fraction of the GCMgridbox occupied by the convection and w isthe vertical velocity in the convective plume.

The mass-flux from the CRM is shown inFig.6(a). In order to calculate the rate oftransport of condensate and cloud fractionfrom the convective plume into the environ-ment we need to have an estimate of the de-trainment rate. A height-dependent passivetracer is inserted into the CRM. We then com-pare the mean profiles of the tracer averagedover the convective points and averaged overthe environment. As the tracer is passive anydifference between these two profiles is due totransport from the environment into the con-vective plume (entrainment) or from the con-vective plume into the environment (detrain-ment) (Swann, 2001). Figure 6(b) shows thedetrainment rate calculated from the CRMdata.

Figure 6: a) Mass flux, b) detrainment calcu-lated using passive tracers and the equationfrom Swann (2001).

3.1 THE “TRUTH”

Figure 7 shows the time-derivative of Fig. 5.These are the “true” rates of change of liq-uid and ice condensate and cloud fraction, asmodelled in the CRM. These are the fieldsthat we want the PC2 scheme to be able toparametrize.

Although all the mechanisms that can modifythe cloud fields have been considered, for thiscase, it turn out that processes such as large-scale forcing, boundary-layer mixing and ra-diation all have minor impacts (not shown).Perhaps unsurpringly sine we are looking at acase of tropical deep convection, the dominantterms turn out to be due to the detrainmentfrom convection and the large-scale precipita-tion. We will look at the two areas of convec-tion and microphysics in more detail.

Figure 7: Rate of change of a) liquid cloudfraction, b) ice cloud fraction, c) environmen-tal liquid water content and d) environmentice water content, calculated from the LEMfields. These tendencies can be considered tobe the “truth” and are what we would want acloud scheme to be able to predict.

3.2 CONVECTION

Figure 8 shows the liquid cloud fraction ten-dencies, ∂Cl/∂t, due to convection. Theseconsist of terms due to detrainment and ver-tical advection from compensating subsidence(see Eqn. 7) as well as a term represent-ing evaporation of cloud following adiabaticwarming by compensating subsidence whichis modelled using the homogeneous forcingframework (see Eqn. 6).

Figure 8: Liquid cloud fraction tendencies dueto convection: a) detrainment of liquid cloudfraction from convective plume into the large-scale environment, b) advection of liquid cloudfraction by the compensating subsidence, c)evaporation following adiabatic warming fromcompensating subsidence.

Figure 9 shows the liquid water tendencies,∂qcl/∂t, due to convection. As for cloud frac-tion, these consist of terms due to detrain-ment and vertical advection from compensat-ing subsidence (see Eqn. 8) and evaporationfollowing adiabatic warming by compensatingsubsidence (see Eqn. 6).

Figure 9: Liquid water tendencies due to con-vection: a) detrainment of liquid water fromconvective plume into the large-scale environ-ment, b) advection of liquid water by the com-pensating subsidence, c) evaporation followingadiabatic warming from compensating subsi-dence.

Figures 10 and 11 show the ice cloud fraction,and ice water content tendencies (∂Cf/∂t and∂qcf/∂t), due to convection. As for the liquidphase, these consist of terms due to detrain-ment and vertical advection from compensat-ing subsidence, however, the effects of adia-batic warming from compensating subsidenceon the Cf and qcf are included in the sub-limation terms calculated by the large-scaleprecipitation scheme (see Section 3.3).

Figure 10: Ice cloud fraction tendencies due toconvection: a) detrainment of ice cloud fractionfrom convective plume into the large-scale envi-ronment, b) advection of ice cloud fraction bythe compensating subsidence.

Figure 11: Ice water tendencies due to con-vection: a) detrainment of ice from convectiveplume into the large-scale environment, b) ad-vection of ice water by the compensating subsi-dence.

3.3 MICROPHYSICS

Figure 12 shows the tendencies associatedwith the CRM microphysical transfers whichoccur in the large-scale environment (as op-posed to within the convective plume) andwhich we assume would be represented by theGCM large-scale precipitation scheme. Al-though microphysical processes have a directeffect on the ice cloud fraction (the dominantprocesses being sublimation, deposition andthe fall of ice all), the warm-rain microphys-ical processes (such as autoconversion, accre-tion and evaporation) are assumed to have noeffect on the liquid cloud fraction. These pro-cesses do however effect the liquid water con-tent, so changes to the liquid cloud fractionoccur indirectly.

Figure 12: The tendencies from the large-scaleprecipitation scheme for a) ice water content, b)liquid water content and c) ice cloud fraction.The large-scale precipitation is assumed to haveno effect on the liquid cloud fraction.

3.4 PC2 PARAMETRIZATION

After having calculated the tendencies fromall the physical processes we can add themup to create the parametrized tendencies thatthe PC2 scheme would produce (Fig. 13).These are what the PC2 cloud scheme wouldpredict given the horizontally averaged CRMdata as input. These should be compared tothe “truth” shown in Fig. 7.

Figure 13: Rate of change of a) liquid cloudfraction, b) ice cloud fraction, c) environmentliquid water content and d) environment icewater content, calculated by adding all thePC2 process rates calculated from the LEMfields.

4 DISCUSSION

Although Figs. 13 and 7 do not look verysimilar, the analysis has been useful at it hashighlighted the dominant terms that may re-quire further attention. The main source termis the detrainment from convection, while themain sink term comes from large-scale micro-physics.

The resolution and number of dimensions inthe domain will have an effect on the mass-flux profiles and the representation of tur-bulent mixing due to radiative cooling nearcloud top (Petch and Gray, 2001). FurtherCRM simulations, with smaller gridlengthsand a some coarse resolution 3D tests arebeing planned in order to quantify the un-certainty in the mass-flux and detrainmentprofiles which are then used to estimate thetruth for subsequent comparison with the PC2parametrization.

The other possible reason for differences be-tween Figs. 13 and 7 is because some impor-tant process has not so far been included inthe PC2 framework or because the processesthat are included could be represented morerealistically. This is an area for further re-search.

The current analysis suggests that thereis no significant sink of liquid cloud frac-tion. A mechanism known as “cloud erosion”(Tiedtke, 1993) has been included in otherGCM prognostic cloud schemes. This processaims to represent the removal of liquid cloudcondensate and liquid cloud fraction as a re-sult of sub-grid motions within the GCM grid-box mixing clear and cloudy air and leadingto evaporation of the cloud. A parametriza-tion of this effect has been included in PC2,however, in this case it does not appear toproduce sink terms which are large enough tobe significant.

4.1 FUTURE WORK

4.1.1 EROSION TRACERS

The process of “cloud erosion” is not wellunderstood. Following Stiller and Gregory(2003), PC2 represents sub-grid mixing as anarrowing of the moisture PDF. Figure 14illustrates how a narrowing of the moisturePDF (darker grey lines) leads to a reductionin cloud fraction and liquid cloud condensate(assuming that the peak of the PDF is tothe left of −Qc). Some CRM simulations areplanned, where an age tracer is used to mea-sure the time since a parcel of air was insidethe convective plume. By studying the evolu-tion of the moisture probability density func-tion we hope to gain insight into how to bet-ter parametrize the effects of “cloud erosion”in our GCM.

4.1.2 SINGLE COLUMN MODEL

It is hoped that this work will be combinedwith some simulations using the PC2 codewithin a Single Column Model (SCM) versionof the GCM. The various PC2 tendencies fromthe SCM can then be compared to their ana-logues from the CRM.

Figure 14: Schematic of how reducing the widthof the moisture PDF will reduce the amount ofcloud fraction and condensate.

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