Insights into cloud parametrization provided by a prognostic approach

Q. J. R. Meteorol. Soc. (2002), 128, pp. 1485–1504

Insights into cloud parametrization provided by a prognostic approach

By DAVID GREGORY, DAMIAN WILSON and ANDREW BUSHELL∗Met Office, UK

(Received 15 January 2001; revised 21 January 2002)

SUMMARY

Starting from an assumption that within an area there is a distribution of temperature and humidity, equationsare derived for the prognostic evolution of cloud fraction and area-mean condensed water under the action ofuniform forcing. These are contrasted to alternative equation sets which are based upon consideration of humidityvariations alone, and approximations involved in the derivations of these are highlighted. The properties of,and relationships between, current diagnostic and prognostic cloud schemes are examined using the prognosticformulation, and assumptions concerning the nature of the subgrid distribution of thermodynamic variables arediscussed. While the approaches of Smith and Tiedtke provide a consistent description of cloud formation, theSundqvist approach is found to implicitly include changes in the character of the distribution as cloud coverincreases, inconsistent with an assumption of uniform forcing. Additional properties of the Tiedtke scheme areexplored and extensions suggested to improve the treatment of cloud dissipation.

KEYWORDS: Condensation Prognostic treatment

1. INTRODUCTION

The representation of clouds within the context of large-scale models has receiveda great deal of attention. This is in part due to the importance of clouds in determiningthe current and possible future state of the Earth’s climate. Moist processes alsohave important consequences upon a range of weather-forecasting issues, from severeprecipitation events to maximum daytime temperatures. A range of techniques havebeen suggested to represent clouds in models which are unable to resolve the physicalprocesses associated with cloud formation and dissipation. This study considers thetheoretical basis of a prognostic approach to cloud parametrization.

Over the past twenty years the trend in cloud parametrization has been towardsrepresenting cloud condensate explicitly, so as to provide a better specification of cloudwater path for the purposes of radiative calculations. Explicit cloud water also allowsmore realistic representations of precipitation microphysics. The scheme of Sundqvist(1978, 1988) was pioneering in this regard, considering the prognostic evolution ofcloud condensate while retaining a diagnostic approach to predicting cloud fraction usedin earlier schemes. The scheme of Smith (1990) transfers the ideas of Sommeria andDeardorff (1977) from large-eddy simulation (LES) to large-scale models. Althoughdiagnostic, Smith’s scheme provides a clearer description of the humidity distributionwithin an area which previous diagnostic cloud schemes inherently assumed withoutclearly defining its nature. The Smith scheme and its derivatives (LeTreut and Li 1991;Ricard and Royer 1993) provide a consistent approach to the specification of cloudcondensate and fraction.

Both the Sundqvist and Smith-type approaches assume that the evolution of cloudis related to the evolution of large-scale variables. Some applications of the Sundqvistapproach have attempted to break this indirect link between forcing processes andcloud production by allowing some processes to affect the cloud field directly. Thishas been considered primarily in the context of convectively produced cloud waterwhich is treated as a source within the cloud scheme. However, these extensions stilltypically retain a diagnosed cloud fraction. Tiedtke (1993) introduced a treatment where

∗ Corresponding author: Met Office, Hadley Centre for Climate Prediction and Research, London Road,Bracknell, Berkshire RG12 2SY, UK. e-mail: [email protected]© Crown copyright, 2002.

1485

1486 D. GREGORY et al.

both cloud condensate and fraction are prognostic variables, evolving under the actionof various forcings such as convection, radiation and vertical motion. This allowed aconsistent treatment of the evolution of cloud amount and condensate. The Tiedtkescheme has been used successfully in the European Centre for Medium-Range WeatherForecasts (ECMWF) forecast model and the first ECMWF re-analysis (ERA-15) (Jakob1999), but the original derivation of the scheme has little direct connection to thedistribution concepts used by other schemes. The analysis of this paper shows that aprognostic approach to cloud parametrization is possible while still retaining the conceptof a subgrid moisture and temperature distribution, and that such an approach resemblesthat of Tiedtke with certain modifications.

Starting from a generalized description of the thermodynamic state of an area,prognostic equations for the evolution of cloud condensate and fraction are derived.Wang and Wang (1999) have carried out a similar analysis for the evolution of con-densate in cloudy boundary layers, assuming a Gaussian distribution of thermodynamicvariables. Here, no particular distribution is assumed, the focus being the formation ofclouds by processes which are uniform, that is having little spatial variation. This isakin to many current cloud parametrizations which predict condensation in responseto the evolution of model mean variables. The resulting framework is used to evaluateseveral types of cloud schemes, both diagnostic and prognostic. Specifically, the Smith(1990), Sundqvist (1978, 1988) and Tiedtke (1993) schemes are considered, these beingarchetypal of the variety of cloud schemes used in atmospheric models. The relationshipbetween the schemes is explored and also their specification of the subgrid distributionof thermodynamic variables. In the case of the latter, the study highlights inconsistenciesbetween the assumed distributions and the uniform-forcing assumption.

2. PROGNOSTIC EQUATIONS FOR CLOUD CONDENSATE AND FRACTION

(a) Cloud condensateEquations for the evolution of temperature (T ), specific humidity (q) and conden-

sate (l) at a point, and averaged over an area are:

∂T

∂t= AT + L

cp

Q − L

cp

E;∂T

∂t= AT + L

cp

Q − ∂w′T ′∂z

− L

cp

E (1)

∂q

∂t= Aq − Q + E;

∂q

∂t= Aq − Q − ∂w′q ′

∂z+ E (2)

∂l

∂t= Al + Q − P − E;

∂l

∂t= Al + Q − ∂w′l′

∂z− P − E (3)

where Aφ is the forcing of a quantity φ due to dynamical processes and radiation,Q is the net condensation/evaporation due to reversible processes, P is the loss dueto precipitation, E is evaporative loss due to irreversible processes (such as turbulentmixing) and (∂w′φ′/∂z) is the contribution due to turbulent transports not representedby the area-mean flow. Area-mean values of quantities are denoted by φ and departuresfrom this by φ′. L and cp are the latent heat of condensation and the specific heatcapacity, respectively.

Here it is assumed that forcing processes are uniform across the area being con-sidered. Turbulent effects are neglected, as is the loss of water due to precipitationand evaporation. Examples of uniform forcing include large-scale ascent or radiation,although the presence of clouds will introduce some inhomogeneity here. Non-uniform

INSIGHTS INTO CLOUD PARAMETRIZATION 1487

processes are considered to be those which are sparsely distributed through the area,such as convection. Although the distinction between these two types of forcing maybe artificial, it does relate to the way in which many cloud schemes are formulatedassuming forcing to be uniformly distributed.

With these assumptions Eq. (3) becomes,

∂l

∂t= (Al + Q) (4)

with similar simplification of the temperature and humidity equations.Following work by Sommeria and Deardorff (1977) and Mellor (1977), a use-

ful starting point is to consider the impact of forcing applied across an area whosethermodynamic state is characterized by a distribution of liquid-water temperature(TL = T − (L/cp)l) and total water (qT = q + l). Specifically, the distribution is definedby the mean values of liquid-water temperature and total water together with appropri-ate measures of spread and skewness. At any point above saturation within an area A,assuming no supersaturation, the condensate is given by

l = qT − qs(T ) = q ′T − {qs(T ) − qT} (5)

where qs is the saturation humidity. The area-averaged value of condensed water is

l =∫ ∞

−∞

∫ ∞

qs(T )−qT

[q ′T − {qs(T ) − qT}]G(q ′

T, T ′L) dq ′

T dT ′L (6)

where q ′T = qT − qT, T ′

L = TL − TL and G(q ′T, T ′

L) is the probability distribution func-tion of the thermodynamic variables.

The local value of condensed water is thus expressed in terms of total waterand temperature. To evaluate this integral for a given distribution the local value ofcondensed water must be expressed in terms of total water and liquid-water temperature,as in Sommeria and Deardorff (1977). However, here only the rate of change of liquidwater with time is required and such a transformation is not necessary.

Differentiating Eq. (6) with respect to time and applying the Leibnitz rule gives

∂l

∂t=∫ ∞

−∞

∫ ∞

qs(T )−qT

{∂q ′

T

∂t−(

∂qs(T )

∂t− ∂qT

∂t

)}G(q ′

T, T ′L) dq ′

T dT ′L

+∫ ∞

−∞

∫ ∞

qs(T )−qT

[q ′T − {qs(T ) − qT}]∂G

∂tdq ′

T dT ′L (7)

where the change of the boundary term is zero.With the assumption of uniform forcing, all parts of the distribution are affected

equally and so the rate of change of G with time is zero, eliminating the second integralin Eq. (7). In a parallel analysis focusing upon cloud-topped boundary layers, Wangand Wang (1999) retain this term assuming the distribution to be Gaussian with varyingspread. Changes in the spread of the distribution due to turbulent mixing between cloudand clear air only contribute significantly to the condensate budget near cloud top.Through much of the cloud layer, processes were well characterized as being uniform.More generally, changes to the shape of the distribution, such as increasing skewnesstogether with variations in spread, also contribute to the second integral of Eq. (7).

Substituting Eqs. (1) to (3) into Eq. (7) for the rates of change of temperature andhumidity (neglecting eddy transports, precipitation and evaporation) gives

∂l

∂t=∫ ∞

−∞

∫ ∞

qs(T )−qT

{(Aq + Al) − αT

(AT + L

cp

Q

)}G(q ′

T, T ′) dq ′T dT ′

L (8)


where αT = (∂qs/∂T )T and for simplicity the variation of qs with pressure has beenneglected∗.

Under the assumption of uniform forcing, Aq , AT and Al are independent of q ′T

and T ′L and can be taken out of the integral in Eq. (8) to give

∂l

∂t= C(Aq + Al) − AT

∫ ∞

−∞

∫ ∞

qs(T )−qT

αT G(q ′T, T ′) dq ′

T dT ′L

− L

cp

∫ ∞

−∞

∫ ∞

qs(T )−qT

αT QG(q ′T, T ′) dq ′

T dT ′L (9)

where C is the cloud fraction defined as

C =∫ ∞

−∞

∫ ∞

qs(T )−qT

G(q ′T, T ′

L) dq ′T dT ′

L. (10)

An expression for the local condensation (Q) is required in terms of the forcing.Differentiating Eq. (5) with respect to time and substituting from Eqs. (1), (2) and (3)leads to an expression for the condensation ‘at each point’ within the saturated part ofthe distribution;

Q = aTL (Aq − αT AT ) (11)

where aTL = {1 + (L/cp)αT }−1.

Substituting for Q in Eq. (9) and using T to determine αT and aTL rather than the

local temperature gives, after further rearrangement,

∂l

∂t= aT

L C(Aq − αT AT ) + CAL (12)

where αT = (∂qs/∂T )T and aTL = {1 + (L/cp)αT }−1. Using Eq. (4) gives an expression

for the area-average condensation;

Q = aTL C(Aq − αT AT ) − Al(1 − C). (13)

This provides a physical understanding of how different forcing processes affect thearea-mean condensed water. Changes to mean T and q due to various forcing processescause condensation/evaporation within the cloudy part of the area. Forcing also affectsthe condensed water but within the clear part of the area added condensed water isevaporated, as represented by the second term on the right-hand side (r.h.s.) of Eq. (13).This is a consequence of applying average forcing tendencies across the whole areawithout preserving the correlation between saturation and condensed water.

(b) Cloud fractionDifferentiating Eq. (10) with respect to time gives an equation for the rate of change

of cloud fraction with time;

∂C

∂t=∫ ∞

−∞

∫ ∞

qs(T )−qT

∂G

∂tdq ′

T dT ′L −

∫ ∞

−∞GS

(∂qs(T )

∂t− ∂qT

∂t

)dT ′

L (14)

∗ The variation of qs with pressure is neglected throughout the derivations to simplify discussion. Extension toinclude this term is straightforward.


where GS = G{qs(T ) − qT, T ′L}, the amplitude of the distribution function at the

saturation point for a given temperature T .With uniform forcing the distribution is invariant with time and so the first integral

is zero. On physical grounds if this were not the case it would imply a mass transferfrom clear to cloudy parts of the area even if the saturation temperature did not change.Such a change in the distribution might result from mixing across the cloud boundarydue to the internal turbulence of the system.

Expanding the rates of change of saturation specific humidity and total water in thesecond integral on the r.h.s. of Eq. (14) and applying the uniform-forcing assumptiongives

∂C

∂t= G̃S(Aq + Al) − G̃SαT AT − αT

L

cp

∫ ∞

−∞QGS dT ′

L (15)

where G̃S = ∫∞−∞GS dT ′. Again αT has been approximated by αT .

Substituting for Q from Eq. (11) gives, after rearrangement, a prognostic equationfor cloud fraction

∂C

∂t= G̃S{aT

L (Aq − αT AT ) + Al}. (16)

This has a similar form to the area-average condensate equation (Eq. (12)). Thequantity G̃S is the amplitude of the distribution at the saturation boundary summedover the range of temperatures present within the area considered. The presence of thisquantity is reasonable in the context of a prognostic equation for cloud fraction in thatonly changes in T and qT at points close to saturation will produce a change in C.Equation (16) for the rate of change of cloud fraction together with Eq. (12) for theevolution of condensate provide a viable cloud scheme provided a suitable G̃S is definedto provide closure for the scheme. This will be considered in a later study.

(c) Alternative frameworksThe above analysis of a general distribution of thermodynamic variables within

an area gives prognostic equations for the evolution of cloud condensate and fraction.Several previous attempts to represent cloud in large-scale and mesoscale models haveused a similar approach, although not always in a prognostic framework. These havetended to use simplifications to reduce the general two-dimensional distribution to onedimension. Here the implications of such simplifications upon the development of aprognostic approach are considered.

(i) ‘s’-distribution. Following Mellor (1977), the variables of the distribution aretransformed from q ′

T and T ′L to s and t defined as

s = aT LL (q ′

T − αT LT ′

L) (17)

and

t = aT LL (q ′

T + αT LT ′

L) (18)

where αTL= (∂qs/∂T )TL

and aT LL = {1 + (L/cp)αT L

}−1.From Eq. (5) the condensed water at any point within the saturated part of the grid

box can then approximately be written as

l ≈ aT LL {qT − qs(TL)} + s = QC + s (19)


which is only dependent upon s. Hence the definition of grid-box mean condensate water(Eq. (6)) and cloud fraction (Eq. (10)) become single integrals over the range of s;

l =∫ ∞

−QC

(QC + s)G̃(s) ds (20)

C =∫ ∞

−QC

G̃(s) ds (21)

where

G̃(s) =∫ ∞

−∞G(s, t) dt. (22)

Differentiation of Eqs. (20) and (21) with respect to time leads to the same equationset for the rates of change of condensed water and cloud fraction as derived in section 2above with G̃S = G̃(−QC).

(ii) Constant-temperature approximation. The ‘s-distribution’ allows the formationof cloud to be discussed in terms of a one-dimensional distribution of a thermodynamicvariable even though a distribution of total water and liquid-water temperature exists.A similar one-dimensional treatment can be achieved by assuming a constant liquid-water temperature across the area, resulting in the same equations as in the previoussection.

However, it is more common to achieve a one-dimensional distribution by assumingthat the temperature at each point in the area equals the area mean. The cloud schemesof Sundqvist (1978) and Tiedtke (1993) follow this approach. Equivalent to Eqs. (6) and(10) the area-averaged cloud condensate and cloud fraction then become:

l =∫ ∞

qs(T )−qT

[q ′T − {qs(T ) − qT}]G̃(q ′

T) dq ′T (23)

C =∫ ∞

qs(T )−qT

G̃(q ′T) dq ′

T (24)

where G̃(q ′T) = ∫∞

−∞G(q ′T, T ′

L) dT ′L.

Note that Eq. (23) implies that the liquid-water content at any point within thesaturated part of the area is defined relative to T giving a local condensate value of

l = qT − qs(T ), (25)

an approximation to the true value given by Eq. (5).Differentiating Eq. (23) with respect to time gives

∂l

∂t=∫ ∞

qs(T )−qT

{(Aq + Al) − αT

(AT + L

cp

Q

)}G̃(q ′

T) dq ′T

+∫ ∞

qs(T )−qT

[q ′T − {qs(T ) − qT}]∂G̃

∂tdq ′

T. (26)

Again, as the distribution is defined in terms of a conserved variable, its rate ofchange with time under uniform forcing is zero.


Comparison with Eq. (8) reveals that AT and Q are replaced by AT and Q. Whilethe use of area-averaged temperature forcing in Eq. (26) is consistent with the uniform-forcing approximation, using the area-mean condensational heating implies a spuriousheat flux between the clear and cloudy parts of the area. From Eq. (13) the meancondensation is composed of a contribution due to processes within the cloud and theevaporation of condensed water within the clear air. The presence of Q in Eq. (26)implies that the contribution from evaporative processes within the clear air has an‘unphysical’ influence upon the cloud part of the area.

Applying the uniform-forcing assumption to Eq. (26) and following a similaranalysis to that in section 2(a) above, gives the area-average condensation rate, which issimilar to Eq. (13):

Q = a∗LTC(Aq − αT AT ) − a∗

LTAl(1 − C) (27)

where a∗LT = {1 + C(L/cp)αT }−1.

Considering the evolution of cloud fraction within this approximate framework,differentiating Eq. (24) with respect to time gives

∂C

∂t=∫ ∞

qs(T )−qT

∂G̃

∂tdq ′

T + G̃S

(∂qT

∂t− ∂qs(T )

∂t

)(28)

where G̃S = G̃{qs(T ) − qT}.Again, the time dependence of the distribution is zero, so following section 2(b) the

rate of change of cloud fraction with time is

∂C

∂t= G̃S

a∗L

T

aTL

{aTL (Aq − αT AT ) + Al}. (29)

This has a similar form to the rate of change of cloud fraction derived previously(Eq. (16)) but the value of the distribution at saturation is multiplied by a factorwhich varies with cloud fraction. This is an effect of using the area-mean condensationthrough the assumption that the local temperature is equal to the area-mean temperatureeverywhere. This might be termed the uniform-condensation assumption. The analysisof section 2 above suggests that such an approximation is not necessary in the context ofa prognostic approach to cloud parametrization. The implications of such an assumptionare discussed in the following section.

3. CRITIQUE OF CURRENT APPROACHES TO CLOUD PARAMETRIZATION

Cloud schemes used in large-scale atmospheric models can be categorized broadlyinto three types; (a) diagnostic cloud water and fraction, (b) prognostic cloud waterand diagnostic cloud fraction and (c) prognostic cloud water and fraction schemes.The above prognostic framework can in part be applied to each of these categories,providing insight into the hierarchy of the schemes together with the nature and conse-quences of the assumptions made concerning the distribution of thermodynamic vari-ables.


(a) Diagnostic cloud water and fractionIn this scheme l and C are diagnosed directly from Eqs. (6) and (10) (or their

equivalent) in response to changes in qT and TL associated with a variety of forcingprocesses. The shape and spread of the distribution function G̃ must be specified.The spread is usually related to a critical relative humidity (RHc), which definesthe onset of condensation. Although no tendencies are calculated, Eqs. (12) and (16)describe the trajectory of cloud formation in l and C space between successive equilibriaunder uniform forcing.

The cloud scheme of Smith (1990) is archetypal of this category, couched in termsof the ‘s-distribution’ framework. Smith (1990) used a triangular distribution for G̃which is contrasted in Smith (1994) with a top-hat distribution. The top-hat distributionis often used in other cloud schemes (e.g. Tiedtke 1993) and is considered here. Sucha distribution implies that between the upper and lower boundaries of the distribution,s = (−bS, bS) each value of s has the same probability:

G̃(s) = G̃S = 1

2bS

so that∫ bS

−bS

G̃ ds = 1. (30)

Following Smith (1990, 1994), bS is defined in terms of a critical relative humidity(RHc);

bS = aT LL (1 − RHc)qs(TL) (31)

this form arising from consideration of the values of qT and TL when condensation

first occurs (Smith 1990). Properly, the width should be defined using q s(TC=0L ), where

TC=0L is the area-mean liquid-water temperature when cloud first forms. However, Smith

(1990) replaces this by TL, which is time varying leading to bS changing with time.Differentiating Eq. (30) gives the rate of change of G(s) as

∂G̃(s)

∂t= −G̃S

bS

∂bS

∂t= − G̃S

qs(TL)

∂qs(TL)

∂t. (32)

The change in the character of the distribution with time introduces additional termspreviously neglected. Differentiating Eq. (21) with respect to time, and including the rateof change of the distribution with time, leads to a modified equation for the evolution ofcloud fraction:

∂C

∂t= a

T LL G̃S

(∂qT

∂t− ∂qs(TL)

∂t

)+∫ ∞

−QC

∂G̃

∂tds. (33)

Now ∫ ∞

−QC

∂G̃

∂tds =

∫ bS

−QC

∂G̃

∂tds + G̃(bS)

∂bS

∂t(34)

which on substitution from Eq. (32) and using Eqs. (30) and (31) gives, after rearrange-ment, the rate of change of cloud fraction as,

∂C

∂t= a

T LL G̃S

(∂qT

∂t− ∂qs(TL)

∂t

)+ a

T LL G̃S(1 − 2C)(1 − RHc)

∂qs(TL)

∂t. (35)

A similar term arises in the prognostic condensate equation if the width of thedistribution is related to qs(TL). For a cooling, below 50% cloud cover the change


Figure 1. Evolution of (a) distribution width, (b) cloud fraction, (c) area-mean condensate (l), (d) condensationalheating and (e) relation between in-cloud condensate and cloud fraction, for uniform forcing and a top-hat

distribution of differing widths; Smith (1990)—solid; fixed—short dash.

in the distribution retards the increase in cloud cover compared with a time-invariantone. Above 50% cloud cover the converse is true. The magnitude of the additionalterms depends upon the critical relative humidity. The effect of the variation in thedistribution using the Smith (1990) definition of spread is illustrated in Fig. 1 whichshows results from a simple layer model of cloud formation. From an initial mean liquid-water temperature of 10 ◦C, a pressure of 850 hPa and a relative humidity (definedrelative to the saturation mixing ratio estimated from the liquid-water temperature) of85%, the layer is cooled at a rate of 10 K day−1. A top-hat distribution is assumed with acritical relative humidity for cloud formation of 85%. A time step of 10 minutes is used.The sensitivity of the cloud evolution to different specifications of width is examined.Cloud fraction and area-mean condensate are derived diagnostically from Eqs. (20) and(21), although as discussed above their evolution is identical to that derived from aprognostic framework for the same distribution (if processes which cause the loss ofcloud condensate are neglected).

Simulations using the Smith (1990) definition of distribution width (Eq. (31)) anda fixed width, again based upon Eq. (31) but using the initial value of TL are compared(Fig. 1). As the critical and initial relative humidities are identical, the latter is equivalentto defining the width using the temperature when condensation first occurs. Figure 1(a)


shows that the distribution width of the Smith scheme decreases as the layer cools.The behaviour of the cloud fractions is as described above, although the variation ofthe distribution width has a relatively small effect compared with the growth of cloudcover in response to the imposed forcing. This is also true of the area-mean condensate(Fig. 1(c)), although the fixed-width distribution gives slightly larger values. However, alarger impact is seen on in-cloud condensate (l/C) (Fig. 1(e)), important for determiningthe radiative properties of cloud layers and rates of microphysical transformations.Using a fixed width leads to a 10% greater value at full cloud cover than for the Smithformulation. Wood and Field (2000), comparing the Smith scheme with observationaldata, suggest that the scheme under predicts cloud amount, especially above half cover,leading to an over prediction of in-cloud condensate (Wood, personal communication).Use of a fixed width would seem to exacerbate both of these signals.

It is important to note that the effect of the changing distribution with time uponcloud fraction and condensate are implicitly included in a diagnostic approach. Whilesuch changes in the distribution are inconsistent with the assumption of uniform forcing,the spread and amplitude remain finite for all reasonable temperatures and the effect ofthe additional terms is small.

(b) Prognostic l, diagnostic C

In theory, this approach to cloud parametrization can be related to the theorydiscussed previously. The use of a diagnostic relation for C is not inconsistent withthe subgrid distribution but requires that Eq. (16) becomes a diagnostic equation forG̃S. The scheme described by Sundqvist (1978, 1988) is archetypal for this category,being derived in the ‘constant temperature approximation’ framework described above.Rasch and Kristjansson (1998) is a more recent example of this approach, which usesa different diagnostic cloud formulation from that originally used by Sundqvist, butprovides a clear description of the principles behind this type of scheme. However, whilethe diagnostic cloud fraction of these schemes defines the distribution of humidity withinan area, as discussed below their treatment of condensation and cloud formation appearsto be different from that derived above.

In the Sundqvist scheme (Sundqvist 1988), the clear-sky humidity (q e) is specifiedto vary with cloud fraction as,

qe = RHcqs(T ) + C(1 − RHc)qs(T ). (36)

As the area-mean humidity (q) can be written as

q = Cqc + (1 − C)qe = Cqs(T ) + (1 − C)qe, (37)

substitution for qe from Eq. (37) leads to the variation of C with area-mean relativehumidity as

C = 1 −√

1 − RH − RHc

1 − RHc(38)

where RH = q/qs(T ).This variation of cloud cover with area-averaged relative humidity corresponds to a

top-hat distribution of qT (see appendix (a)) whose amplitude is given by

G̃S = 1

2qs(T )(1 − RHc). (39)


This is similar to the top-hat distribution used to demonstrate the properties of theSmith scheme previously. A consistent condensational heating, and so rate of changeof condensate, can be obtained by using Eq. (38) in Eq. (27). However, as outlinedbelow, the Sundqvist approach departs from the previous analysis in its treatment ofcondensation and is inconsistent with the implied distribution suggested by the cloudfraction.

Partially following the analysis laid out in Rasch and Kristjansson (1998), differen-tiating Eq. (37) with respect to time gives

∂q

∂t= {qs(T ) − qe}∂C

∂t+ (1 − C)

∂qe

∂t+ C

∂qs(T )

∂t. (40)

Expressing the rate of change of saturation humidity with time in terms of the rateof change of temperature, and using the area-mean versions of Eqs. (1) and (2) gives,after rearrangement, the area-mean condensation

Q = −a∗LT

({qs(T ) − qe}∂C

∂t+ (1 − C)

∂qe

∂t− M

)(41)

where M = (Aq − CαT AT ). Sundqvist (1978, 1988) and others have interpreted M asthe forcing of the area which leads to cloud formation.

Sundqvist partitions the area-mean condensation between condensation in existingclouds Q

cand that due to new cloud formation Q

n, i.e.

Q = CQc + Q

n. (42)

Assuming that the forcing ‘M’ is uniformly distributed across the area, the portionwithin the cloudy region must result in the formation of new condensate. ConsideringEq. (41) and choosing appropriate terms for the cloudy region

CQc = a∗

LTCM. (43)

Sundqvist assumed that new cloud has the same in-cloud condensate (l/C) asexisting cloud giving

Qn = a∗

LT l

C

∂C

∂t, (44)

and so from Eq. (42) the total condensational heating is

Q = a∗LTC

((Aq − CαT AT ) + l

C2

∂C

∂t

). (45)

Further details of the Sundqvist approach are to be found in appendix (b). For thepresent it is sufficient to point out that the condensation equation derived by Sundqvist’smethodology is different from Eq. (27), derived directly from the distribution of qT.Rather than a linear dependency upon C in the term involving temperature forcing, aquadratic dependency is found. An additional term involving the rate of change of cloudwith time is also introduced. This implies there is an inconsistency in the way that theSundqvist approach treats condensation and cloud fraction, the latter being consistentwith a top-hat distribution of qT under the action of uniform forcing while the former isnot.

This difference in the treatment of condensation causes significant differences in theprediction of cloud fraction, illustrated in Fig. 2 for initial conditions identical to those


Figure 2. Evolution of (a) cloud fraction, (b) area-mean condensate (l), (c) condensational heating, (d) relativehumidity (RH) of the clear part of the area, and (e) relation between in-cloud condensate and cloud fraction, foruniform forcing and the Sundqvist scheme (long dash) and scheme with diagnostic cloud fraction and prognostic

condensate based on a top-hat distribution of qT (solid).

used to discuss the properties of the Smith scheme. The scheme based upon the theoryin section 2(c) uses the same diagnostic cloud fraction as Sundqvist (Eq. (38)) withcondensation given by Eq. (27) and is referred to hereafter as the ‘consistent scheme’.The contribution due to condensation that results from changes in the amplitude of thedistribution as temperature reduces is assumed small, a reasonable assumption fromthe discussion of the Smith scheme previously. As before, a timestep of 10 minutes isused.

For an RHc of 85%, the cloud fraction predicted by the Sundqvist scheme(Fig. 2(a)) grows more rapidly than for the ‘consistent’ scheme, with slower ini-tial growth of area-mean cloud condensate (Fig. 2(b)). The area-mean condensation(Fig. 2(c)) is lower until after 400 minutes, with the relative humidity of the clear partof the area (Fig. 2(d)) increasing more rapidly than for the consistent scheme. Althoughthe growth of cloud fraction is more rapid with the Sundqvist scheme, the in-cloudcloud condensate is lower (Fig. 2(e)), i.e. the Sundqvist scheme predicts clouds whichare more extensive but more tenuous than the consistent scheme. This can be viewedas resulting from a non-uniform change in the initial distribution, with the total waterof the clear part approaching saturation more rapidly than the rate at which that of the


cloudy part of the distribution departs from it. Such changes cannot result directly fromthe imposed forcing, which is only applied to temperature. They might be consideredthe result of turbulence associated with the condensation processes, although this is notexplicitly accounted for in the derivation of the Sundqvist formulation.

After 400 minutes, the rate of growth of cloud cover slows slightly and the area-mean cloud condensate approaches levels similar to those predicted by the consistentscheme. The relative humidity of the clear sky also approaches saturation at a slowerrate. Again, this can be seen as being due to non-uniform changes in the underlyingdistribution of qT, being the converse of the situation described previously. The totalwater of the cloudy end of the distribution departs more rapidly from saturation thansuggested by the uniform forcing, while that of the clear part approaches saturation at aslower rate. As both schemes approach full cloud cover their condensational heatingrates converge, implying that the condensation treatment of the Sundqvist schemebecomes consistent with a top-hat distribution of total water.

These results emphasize that the approach to condensation used by Sundqvist isinconsistent with the distribution-based theory discussed here under the action of uni-form forcing. Non-uniform changes in the evolution of the distribution are introduced,and while such changes may be physically plausible, perhaps resulting from turbulenceassociated with the condensation processes, they are inconsistent with the assumed di-agnostic cloud fraction of the Sundqvist scheme, namely, a top-hat distribution.

Comparing the performance of the Smith-type scheme of Fig. 1 and the consistentscheme of Fig. 2 gives insight into the relative merits of the s-distribution/constant-TL and constant-temperature frameworks. As both schemes here assume a top-hatdistribution the growth of cloud fraction and in-cloud condensate is relatively linearin time, but the growth to full cloud cover is slower for the scheme couched in theconstant-temperature framework (Fig. 2(a) cf. Fig. 1(b)) while in-cloud condensate islarger (Fig. 2(e) cf. Fig. 1(e)). The a∗T

L dependence on cloud fraction in Eq. (27) leadsthe condensational heating predicted in the constant-temperature framework to increasemore nonlinearly with time than for the Smith scheme (Fig. 2(c) cf. Fig. 1(d)). Thesedifferences result directly from the assumption that the temperature field of both theclear and cloudy parts of the area are described by the area-mean temperature. Asdiscussed previously, this is equivalent to assuming that condensational heating, whichin reality only affects temperature in the cloudy part of the area, applies to the clearair also (there being a spurious transfer of heat from the cloudy to the clear part of thearea). Hence, under the action of the imposed temperature forcing, the temperature ofthe clear air falls more slowly than it should if it did not feel the effects of condensationalheating. This leads to slower growth in cloud fraction as the rate at which the clear airapproaches saturation is reduced. Conversely, the temperature of the cloud falls morerapidly resulting in a faster growth of condensate within cloud.

Use of a larger critical relative humidity compensates somewhat for the physicallyinaccurate description provided by the constant-temperature framework but results in adelay in the onset of condensation. For example if RHc is increased to 91%, keepingthe same initial temperature and total water (with a relative humidity of 85%) onset ofcondensation is delayed until 150 minutes but full cloud cover is reached over the sametime interval as the Smith scheme with an RHc of 85%. The in-cloud condensate whenfull cloud cover is first reached is also similar to that of the Smith scheme. However, it isinteresting to note that the behaviour of the Sundqvist approach is similar to that of theSmith scheme (Fig. 1) for the same critical relative humidity. Hence, the condensationtreatment of Sundqvist compensates the consequences of inaccurate treatment of thecondensation process provided by the ‘constant temperature’ framework.


(c) Prognostic l and C

For this category of scheme, Eqs. (10) and (16) describe changes in l and C directlyfrom defined forcing terms. As noted previously, to estimate the rate of change of cloudfraction with time it is not necessary to define the nature of the whole distribution butonly its value at the saturation boundary of the distribution, G̃S. The scheme of Tiedtke(1993) is archetypal of this category. Comparison of the whole scheme with the aboveanalysis is not possible as it considers both non-uniform and uniform forcing terms.However, a subset of the processes (large-scale ascent, radiation) are considered uniformand are compared here.

As for the Sundqvist scheme, the Tiedtke scheme uses a ‘constant-temperatureassumption’ and so the equation set of section 2(c)(ii) is appropriate. However, Tiedtkeargues that forcing of moisture variables cannot lead to cloud formation, i.e. onlychanges in the saturation humidity due to temperature lead to additional condensationand changes in cloud fraction. Equivalent cloud fraction and condensate equations tothose of Tiedtke can be written by setting Aq = Al = 0 in Eqs. (26) and (29) andassuming the distribution does not change with time:

∂C

∂t= −G̃Sa

∗TL αT AT (46)

∂l

∂t= −Ca∗T

L αT AT . (47)

Tiedtke (1993), and a subsequent corrective analysis briefly described in Jakob et al.(1999), derived an expression for G̃S assuming a top-hat distribution of humidity withinthe clear area. From Eq. (37) the mean clear-sky environmental mixing ratio is

qe = q − Cqs(T )

1 − C. (48)

An upper bound on the distribution of humidity within the clear air is qs(T ) and soto satisfy Eq. (48) the lower bound of the distribution of humidity within the clear partof the area must be 2qe − qs(T ).

The fractional area of clear sky within the area is given by

1 − C =∫ qs(T )−qT

2qe−qs(T )−qT

G̃(q ′T) dq ′

T (49)

which for a top-hat function gives

G̃ = (1 − C)

2{qs(T ) − qe} = (1 − C)2

2{qs(T ) − q} . (50)

For a top-hat humidity distribution G̃ = G̃S the value at the saturation boundary.Note that there appears to be no explicit dependence upon a critical relative humidity asin the Smith and Sundqvist schemes. However, further consideration shows that this isnot the case. Differentiating Eq. (50) with respect to time shows G̃ to be constant withtime, consistent with the derivation of Eqs. (46) and (47) (see appendix (c)). Hence, theabsolute value of G̃ is given by setting C = 0 in Eq. (50):

G̃ = 1

2{qs(TC=0) − q} = 1

2qs(TC=0)(1 − RHc)(51)


where TC=0 is the temperature at which condensation first occurs and RHc is a criticalrelative humidity. The amplitude of the distribution defined by Eq. (51) has similardependencies to that of the Smith (Eqs. (30) and (31)) and Sundqvist (Eq. (39)) schemes,although the saturation specific humidity is a fixed value. Equation (51) shows thatthe critical humidity determines the amplitude of the humidity distribution and so theclosure of the scheme. The need for a critical humidity is noted by Tiedtke (1993)but its relation to the distribution assumption was not discussed. The behaviour of thecondensation process described by the scheme is the same as for the consistent schemewhich was compared with the Sundqvist scheme in Fig. 2.

The scheme makes no statement concerning the distribution within the cloudy area;only the mean in-cloud water content is known. However, by applying the same modelas used for the cloud environment, a top-hat distribution of total water between qs(T )

and qs(T ) + 2(l/C), from the definition of cloud fraction (Eq. (24)) the amplitude ofthe cloud distribution is given by

G̃ = C2

2l. (52)

Using Eqs. (46) and (47) it can be easily shown that the rate of change of G̃ withtime is zero (see appendix (c)). Hence, the absolute magnitude of Eq. (52) must be givenby Eq. (51) for a top-hat distribution.

Equations (50) and (52) provide two possible closures for the scheme, both equiv-alent but useful in different forcing situations. Equation (50), derived from consideringthe clear-sky humidity distribution, provides an estimate of the amplitude of the distri-bution at the onset of condensation (C = 0) and is best used in the case of increasingcloud cover. The Tiedtke (1993) scheme and subsequent extensions described by Jakobet al. (1999) also use this clear-sky closure for cloud dissipation. However, the closuredescribed by Eq. (52), related to the properties of the cloudy region, may be more ap-propriate in this instance. This alternative cloudy closure also better accounts for clouddissipation in the case when non-homogeneous processes, such as precipitation, removecloud water without changing cloud fraction. This reduces the amplitude of the distribu-tion within the cloudy part of the grid box but not in the clear sky. In this instance, useof Eq. (50) as the closure would overestimate the decrease in cloud cover.

4. FURTHER DISCUSSION

Consistent prognostic equations for cloud fraction and cloud condensate have beenderived under the assumption that an area containing a humidity and temperature distri-bution is subject to uniform forcing. These are similar to terms used in the pioneeringprognostic cloud scheme described by Tiedtke (1993), although the derivation is moregeneral than previous ones and avoids approximations introduced by assuming con-stant temperature across the area. Terms involving changes in the distribution withinthe cloudy part of the area are introduced, although with uniform forcing these shouldbe zero in the absence of turbulence or precipitation. Key to the use of the prognosticapproach to cloud parametrization is the specification of the distribution’s amplitude atthe saturation boundary. This has not been considered in detail here and will form partof a later study.

In light of the prognostic framework several approaches to cloud parametrizationhave been reviewed and their properties and relationships discussed. Common to themall is a distribution of thermodynamic variables, although in some schemes the nature of


this and its variation with cloud fraction is poorly stated. The scheme of Smith (1990)explicitly assumes a distribution which changes with time, its width narrowing andamplitude increasing as temperature decreases. This is inconsistent with uniform forcingbut, assuming a top-hat distribution for the thermodynamic variables, the effects of thesechanges upon the prediction of cloud fraction are small.

Although the distribution of total water is unspecified in the Sundqvist (1978, 1988)scheme, the diagnostic cloud fraction is consistent with a top-hat distribution. However,the treatment of condensation in the scheme is inconsistent with such a distributionunder the action of uniform forcing. The total-water amounts at the upper and lower endsof the distribution change at rates which are different from that of the imposed uniformforcing, even in the case when the total-water forcing is zero and cloud formation resultsfrom cooling. These changes in the distribution result in a more rapid growth in cloudfraction than for a fully consistent treatment of the distribution such as that provided bythe Tiedtke scheme.

The Tiedtke scheme (Tiedtke 1993; Jakob et al. 1999) also uses a top-hat distri-bution but its specification is invariant in the case of uniform forcing. While originallythe distribution of the Tiedtke scheme was only specified in the clear part of the area,extension to the cloudy part is straightforward. A closure suggested by this componentof the distribution may be more suitable for treatment of cloud dissipation than thatused in the original Tiedtke scheme. However, a potential drawback of the scheme isits assumption that the temperature of each part of the area is characterized by the area-mean value, which causes significant errors in estimating condensational heating andcloud condensate for less than full cloud cover. Some compensation can be achievedthrough the use of higher critical relative humidity although this delays the onset ofcondensation.

The analysis here has assumed that forcing processes are uniformly distributedacross the area and so do not change the character of the distribution. Although idealisticthis does at least provide a framework in which the basic condensation processes of acloud scheme can be evaluated. The description of the condensation process providedby the Smith and Tiedtke approaches appears reasonable, although as noted above theuse of the ‘constant temperature’ assumption in the latter does cause a spurious transferof condensational heating from the cloudy to the clear-air part of the area. In the caseof the Sundqvist scheme inconsistencies have been highlighted in the description of thecondensation processes and uniform forcing.

The idealized framework which is used here to explore the behaviour of paramet-rizations for condensation should be seen in the wider context of interpreting cloud-related errors in general-circulation models (GCMs). A full GCM used for operationalforecast or climate prediction contains a high degree of complexity due to the inter-actions within and between its component physical parametrizations. When comparingbasic model fields with observed datasets there are significant difficulties in attributingmodel errors to specific processes, still less to particular inadequacies in their represen-tation.

Development of techniques for identifying GCM errors related to cloud radiativeproperties is an active research area. For example, Webb et al. (2001) in a recent studyused a combination of satellite datasets to highlight compensating errors in GCMs usingdifferent cloud schemes. Webb et al. (2001) note, however, that even where they believeerrors in the radiation fields are cloud related, more work is needed to distinguishbetween cloud errors that arise because the model and observation environments differ,and errors which stem directly from problems in the individual cloud parametrizationschemes.


One approach adopted to isolate the cloud-specific problems is to set up idealizedtest cases for cloud-resolving models (CRMs) before using the tested CRMs themselvesas intermediaries in comparisons between observed data and regional GCMs, whichoften operate at much coarser grid scales. Even idealized CRM tests, however, intro-duce complexity that may make direct comparison with a GCM cloud parametrizationdifficult. For instance, the condensation processes can drive turbulence which needs tobe modelled in the GCM, or at least included as a narrowing or skewing of the subgridmoisture distribution, to make the two models comparable.

When developing prognostic cloud and condensation scheme parametrizations,clear identification of such components is most important, especially when trying tocompare the behaviour of different prognostic schemes. This leads naturally to aneven deeper level of idealized test, where specified forcings are applied directly tokey components of the parametrization schemes in order to check their behaviourfor consistency without the distraction of competing processes that a full GCM runentails. This is the kind of test (akin to dynamical core testing) that was carried outabove by concentrating upon the large-scale condensation response to homogeneousforcing of temperature or moisture variables. Although necessarily limited, it has proveduseful because the assumptions underlying the schemes that were examined have strongsimilarities. The theoretical analysis in this paper has shown, though, that differentapproaches to cloud parametrization provide different descriptions of clouds and thein-cloud variables important for determining cloud radiative properties.

Clearly as a scheme is developed, the methods used to test it will change. How-ever, similar caveats to those mentioned for CRM tests apply to comparisons withobservational studies such as that of Wood and Field (2000); their derived relationshipsapply to a condensate distribution accumulated as a result of competing growth anddissipation processes. In comparing the data with the equilibrium (diagnostic) Smithscheme, the assumption is made that the observed cloud field is measured in instanta-neous quasi-equilibrium, with a distribution which implies cloud fraction increases tofull cloud cover faster than it should for a purely symmetric distribution. The impliedasymmetry could arise because the distribution of thermodynamic variables is skewedor narrows as the cloud fraction increases, possibly under the action of turbulence asmentioned above. The behaviour of the Sundqvist scheme in such a way is thus fortu-itous, despite its introduction in an apparently ad hoc way that does not clearly identifya turbulence parametrization. Terms in the full Tiedtke scheme linking convection andboundary-layer turbulence to cloud formation can be seen as an attempt to accountfor these effects. However, they may also be required in the case where clouds formunder the action of radiative cooling or vertical motion, effects which are currentlyneglected in the Tiedtke approach. Large-eddy model simulations of how distributionschange as cloud fraction increases, as well as observational studies, will prove use-ful in clarifying these issues and should assist in the development of improved cloudparametrizations.

ACKNOWLEDGEMENTS

D. Gregory acknowledges the stimulation of working with a prognostic cloudscheme while working at ECMWF and discussions with C. Jakob. The authors aregrateful to Tony Slingo for providing comments upon the work.


APPENDIX

(a) Diagnostic cloud equation for a top-hat distribution of qT

For a top-hat distribution defined by Eq. (39), evaluation of Eqs. (23) and (24) givethe area-averaged condensate and cloud fraction as

l = bsC2 (A.1)

C = {qT − RHcqs(T )}2bs

= {q + l − RHcqs(T )}2bs

(A.2)

where bs = qs(T )(1 − RHc).Substituting for l from Eq. (A.1) into Eq. (A.2) gives, after rearrangement, a

quadratic for the cloud fraction:

C2 − 2C +(

RH − RHc

1 − RHc

)= 0. (A.3)

The physical solution of Eq. (A.3) leads to cloud-fraction diagnostic equation (38).

(b) Further details of the Sundqvist schemeThe Sundqvist scheme is partially derived in section 3(b), although a final expres-

sion for area-mean condensation heating was not stated. Equations (41) and (45) providetwo different expressions for the area-averaged condensation. Equating these gives, afterrearrangement, the rate of change of cloud fraction with time:

∂C

∂t= (1 − C)

2{qs(T ) − qe} + l/C

{M −

(qe

qs(T )

) (∂qs

∂t

)T

}. (A.4)

Also, from the definition of the clear-sky humidity (Eq. (36)), differentiating withrespect to time,

∂qe

∂t= (1 − C)

(qe

qs(T )

) (∂qs

∂t

)T

+ {qs(T ) − qe}∂C

∂t. (A.5)

Substitution of Eqs. (A.4) and (A.5) into Eq. (41), expanding (∂qs/∂t)T asαT (∂T /∂t) together with the use of Eqs. (1) and (2) (again neglecting turbulence andevaporation due to irreversible processes) leads, after considerable rearrangement, to anexpression for the area-averaged condensation in terms of known variables:

Q = Aq − K1αT AT − K2M

{1 + αT (L/cp)K1} (A.6)

where

K1 = (1 − C − K2)

(qe

qs(T )

)+ C (A.7)

and

K2 = 2{qs(T ) − qe}(1 − C)

2{qs(T ) − qe} + l/C. (A.8)


Equation (A.6) together with the diagnostic equation for cloud fraction (Eq. (38))and Eqs. (1), (2) and (3) form the basis of the model used to generate the results for theSunqdvist scheme presented in Fig. 2.

As discussed in section 3(b) the Sundqvist approach assumes that the new cloudhas the same in-cloud condensate as existing cloud. When cloud first forms there isno cloud to act as such a reference and so an initial in-cloud condensate, l ci, mustbe specified. Equations (A.4) and (A.6), (A.7) and (A.8) remain valid at C = 0 withl/C being replaced by lci. With C provided by Eq. (38), an initial value for in-cloudcondensate can be derived from Eq. (A.1), i.e.

lci = l

C= bsC (A.9)

which is consistent with a top-hat distribution of total water.

(c) Invariance of distribution in ‘Tiedtke’ cloud schemeThe amplitude of the distribution within the clear part of the area is given by

Eq. (50). Differentiating with respect to time gives,

∂G̃

∂t= − (1 − C)

{qs(T ) − q}∂C

∂t− (1 − C)2

2{qs(T ) − q}2

(∂qs

∂t− ∂q

∂t

). (A.10)

From Eqs. (46) and (47)

∂C

∂t= G̃S

C

∂l

∂t(A.11)

while from Eq. (28)

∂C

∂t= −G̃S

∂qs(T )

∂t. (A.12)

Note both Eqs. (A.11) and (A.12) assume that the distribution is unchanging intime.

Also, if Aq = Al = 0, then

∂q

∂t= −∂l

∂t. (A.13)

Substituting for ∂qs(T )/∂t and ∂q/∂t using Eqs. (A.12) and (A.13), together withEq. (A.11), it can be shown that (∂G̃/∂t) = 0, consistent with the assumptions usedin Eqs. (A.11) and (A.12). Extension to the case of non-zero moisture or condensateadvection is straightforward.

Similarly for the cloudy component of the distribution, differentiating Eq. (52) withrespect to time gives,

∂G̃

∂t=(

C

l

)∂C

∂t−(

C2

2l2

)∂l

∂t. (A.14)

Substitution of ∂l/∂t from Eq. (A.11), and using Eq. (52), shows that again(∂G̃/∂t) = 0 .


REFERENCES

Jakob, C. 1999 Cloud cover in the ECMWF reanalysis. J. Climate, 12, 947–959Jakob, C., Gregory, D. and

Teixeria, J.1999 ‘A package of cloud and convection changes for CY21R3’.

Research Department Memorandum, ECMWF, ShinfieldPark, Reading RG2 9AX, UK

LeTreut, H. and Li, Z. X. 1991 Sensitivity of an atmospheric general circulation model toprescribed SST changes: Feedback effects associated withthe simulation of cloud optical properties. Clim. Dyn., 5,175–187

Mellor, G. L. 1977 The Gaussian cloud model relations. J. Atmos. Sci., 34, 356–358Rasch, P. J. and Kristjansson, J. E. 1998 A comparison of the CCM3 model climate using diagnosed and

predicted condensate parametrization. J. Climate, 11, 1587–1614

Ricard, J. L. and Royer, J. F. 1993 A statistical cloud scheme for use in a AGCM. Ann. Geophys., 11,11–12

Smith, R. N. B. 1990 A scheme for predicting layer clouds and their water contentsin a general circulation model. Q. J. R. Meteorol. Soc., 116,435–460

1994 ‘Experience and developments with layer cloud and boundarylayer mixing schemes in the UK Meteorological Office Uni-fied model’. In proceedings of the ECMWF workshop onParametrization of the cloud topped boundary layer, 8–11June 1993, Shinfield Park, Reading, Berks, UK

Sommeria, G. and Deardorff, J. W. 1977 Subgrid-scale condensation in models of non-precipitatingclouds. J. Atmos. Sci., 34, 344–355

Sundqvist, H. 1978 A parametrization scheme for non-convective condensation in-cluding prediction of cloud water content. Q. J. R. Meteorol.Soc., 104, 677–690

1988 ‘Parametrization of condensation and associated clouds in modelsfor weather prediction and general circulation simulation’.Pp. 433–461 in Physically based modelling and simulationof climate and climate change. Vol 1. Ed. M. E. Schlesinger,Kluwer Academic

Tiedtke, M. 1993 Representation of clouds in large-scale models. Mon. WeatherRev., 121, 3040–3061

Wang, S. and Wang, Q. 1999 On condensation and evaporation in turbulence cloudparametrization. J. Atmos. Sci., 56, 3338–3344

Webb, M., Senior, C., Bony, S. andMorcrette, J.-J.

2001 Combining ERBE and ISCCP data to assess clouds in the HadleyCentre, ECMWF and LMD atmospheric climate models.Clim. Dyn., 17, 905–922

Wood, R. and Field, P. R. 2000 Relationships between total water, condensed water, and cloudfraction in stratiform clouds examined using aircraft data.J. Atmos. Sci., 57, 1888–1905

Documents

Insights into cloud parametrization provided by a prognostic approach