a one-dimensional time dependent cloud model

Journal of the Meteorological Society of Japan, Vol. 80, No. 1, pp. 99–118, 2002 99

A One-dimensional Time Dependent Cloud Model

Shu-Hua CHEN

National Center for Atmospheric Research, Boulder, CO, USA

and

Wen-Yih SUN

Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN, USA

(Manuscript received 6 November 2000, in revised form 22 October 2001)

Abstract

A one-dimensional prognostic cloud model has been developed for possible use in a Cumulus Param-eterization Scheme (CPS). In this model, the nonhydrostatic pressure, entrainment, cloud microphysics,lateral eddy mixing and vertical eddy mixing are included, and their effects are discussed.

The inclusion of the nonhydrostatic pressure can (1) weaken vertical velocities, (2) help the cloud de-velop sooner, (3) help maintain a longer mature stage, (4) produce a stronger overshooting cooling, and(5) approximately double the precipitation amount. The pressure perturbation consists of buoyancypressure and dynamic pressure, and the simulation results show that both of them are important.

We have compared our simulation results with those from Ogura and Takahashi’s one-dimensionalcloud model, and those from the three-dimensional Weather Research and Forecast (WRF) model. Ourmodel, including detailed cloud microphysics, generates stronger maximum vertical velocity than Oguraand Takahashi’s results. Furthermore, the results illustrate that this one-dimensional model is capableof reproducing the major features of a convective cloud that are produced by the three-dimensionalmodel when there is no ambient wind shear.

1. Introduction

Because of the significant influence of moistconvection on mesoscale or synoptic weatherphenomena, Kuo (1974), Arakawa and Schu-bert (1974), Fritsch and Chappell (1980), Betts(1986), Grell (1993), Sun and Haines (1996),and many others have proposed different Cu-mulus Parameterization Schemes (CPSs) to re-present the sub-grid scale convection in a nu-merical model. A one-dimensional cloud modelincorporated into CPSs can play an importantrole in the vertical mass flux, heating profile,drying profile, and precipitation rate. Kain and

Fritsch (1990) introduced a one-dimensionalentraining/detraining plume model to improvethe Fritsch-Chappell cumulus parameteriza-tion. They assumed that any mixture resultingin negative buoyancy detrains from the cloud,while other mixtures resulting in positive buoy-ancy entrain into the cloud. The conversion ofcondensate to precipitation and the cloud glaci-ation parameterized in their model were sim-plified. For the past few years, we have workedon a diagnostic one-dimensional cloud model(Haines and Sun 1994) and a new cumulusparameterization (Sun and Haines 1996), whichcan be incorporated into the Purdue MesoscaleModel (Haines 1992; Sun and Chern 1993;Chern 1994) to study convection and severeweather. However, the resolution of mesoscalemodels has been increasing, and a time de-pendent one-dimensional cloud model becomesrequired. Therefore, this paper presents our

Corresponding author and present affiliation: Shu-Hua Chen, Department of Land, Air, and WaterResources, University of California, Davis, CA95616 USA.E-mail: [email protected]: 2002, Meteorological Society of Japan

continuous effort to develop a prognostic one-dimensional cloud model such that it can notonly produce a realistic result, but also is sim-ple enough to be incorporated into CPSs.

The effect of the nonhydrostatic pressure(pressure perturbation) on cloud developmenthas been studied by Holton (1973), Schlesinger(1978), Yau (1979), Kuo and Raymond (1980).It is also well known that cloud microphysicsplays an important role in the cloud’s develop-ment and its precipitation. In Levi and Saluzzi(1996), the effect of ice formation on convectivecloud development was discussed in detail.When the atmosphere is moderately unstable,the modifications of the freezing process inmiddle levels could remarkably influence thecloud development. However, this influence isnot significant when the atmosphere is in apronounced unstable condition.

So far, only a few one-dimensional cloudmodels have included more complete processes.Ferrier and Houze (1989) developed a quite so-phisticated one-dimensional cloud model, whichincluded pressure gradient force, cloud physics,entrainment, and vertical diffusion. However,only warm rain processes were counted inthe microphysics since their model was appliedto tropical studies. In Cheng (1989a), theentrainment/detainment was taken care ingreat detail, but the nonhydrostatic pressurewas excluded, and only warm rain was consid-ered. In addition, his model is a diagnosticcumulus ensemble model, which is more suit-able for the cumulus parameterization schemeused in coarse resolution models. In our model,nonhydrostatic pressure, microphysics, entrain-ment, lateral eddy mixing, and vertical eddymixing are all included. In particular, a moresophisticated microphysics is implemented.

A comparison between the one-dimensionalcloud model and a three-dimensional cloudmodel will be very interesting and helpful.From this comparison, we can learn the capa-bilities and deficiencies of the one-dimensionalcloud model, and then use this information tofurther improve the cumulus parameterization.The three-dimensional Weather Research andForecast (WRF) model (Michalakes et al. 2001;Skamarock et al. 2001), which can be used as acloud model or mesoscale model, is adopted forcomparison. A comparison with observationaldata, such as radar data, will certainly be in-

teresting, but we leave it for future work. It isalso noted that more work should be done inthe future to include the vertical wind shearand downdraft in this one-dimensional cloudmodel.

Section 2 describes the governing equationsand numerical methods used in this one-dimensional cloud model. Section 3 presents acomparison of results between this model andanother one-dimensional cloud model (Oguraand Takahashi 1971), and the importance ofmicrophysics and nonhydrostatic pressure isdiscussed. Section 4 gives a brief introduction ofthe three-dimensional WRF model and a com-parison between these two models’ results. Theforcing terms of vertical eddy mixing, lateraleddy mixing, entrainment, and microphysicsin the one-dimensional model are also dis-cussed. A brief summary is given in Section 5.

2. Model descriptions

2.1 Governing equationsThe one-dimensional axi-symmetric cumulus

cloud is assumed to be a function of t, r, c, andz, where t is time; r is radius; c is tangentialangle; and z is height. The radius of the cloudcan not only change with height (Ferrier andHouze 1989; Levi and Saluzzi 1996), but alsochanges with time. However, to simplify theproblem, the radius is fixed once it is deter-mined in our model. Coriolis force is ignoredbecause of a small length scale (@ a few kilo-meters) and a short time scale (@ one hour).Molecular diffusion is also negligible comparedwith eddy turbulence diffusion. In a strongconvective cumulus cloud, hydrostatic balancecan no longer be held inside the cloud. Thus,the hypotheses of nonhydrostatic balance with-in the cloud and hydrostatic balance in thesurrounding area are proposed. The governingequations of this cloud model are:

(i) Vertical momentum equation

qw

qt¼ � 1

r

qðurwÞqr

� 1

r

qðvwÞqc

� 1

r0

qðr0w2Þqz

� 1

r0

qpnh

qzþ g

ðyv � yv0Þyv0

þ gR

Cp� 1

� �pnh

p0� gQT ; ð1Þ

where u, v, w, pnh, yv, r0, R, and Cp are radial

Journal of the Meteorological Society of Japan100 Vol. 80, No. 1

velocity, tangential velocity, vertical velocity,nonhydrostatic pressure, virtual potential tem-perature, surrounding density, gas constantfor dry air, and specific heat of air at constantpressure. QT is total precipitation, and �gQT isthe drag force due to the weight of precipita-tion. Subscript zero (0) indicates environmentalvalues. The total pressure (p) within the cloudis split into the nonhydrostatic perturbationpart ( pnh) and hydrostatic part (environmentpressure, p0), and

1

r0

qp0

qz¼ �g:

(ii) Divergence equation

qd

qt¼ � 1

r

qðurdÞqr

� 1

r

qvd

qc� 1

r0

qðr0wdÞqz

� d2

� qw

qr

qu

qz� 1

r

qw

qc

qv

qzþ 2

r

qu

qr

qv

qc� 2

r

qv

qr

qu

qc

þ 1

r

qu2

qrþ 1

r

qv2

qr� 1

r

q

qr

r

r0

qpnh

qr

� �� 1

r

q

qc

1

r0r

qpnh

qc

� �; ð2Þ

where d is horizontal divergence and its for-mula is:

d ¼ 1

r

qðurÞqr

þ 1

r

qv

qc:

(iii) Thermodynamic equation

qyei

qt¼ � 1

r

qðuryeiÞqr

� 1

r

qðvyeiÞqc

� 1

r0

qðr0wyeiÞqz

þ 1

CpðLvqv � Lf qiÞ

d

dt

y

T

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð1Þ

þ microðyeiÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}ð2Þ

; ð3Þ

where qv and qi are the mixing ratios of watervapor and cloud ice, respectively. yei is calledequivalent ice potential temperature (Chern1994) and it is defined as

yei ¼ y 1 þ Lvqv

CpT� Lf qi

CpT

� �� :

Lv, Lf and T are the latent heat of vaporizationand fusion, and temperature within the cloud,

respectively. The derivation of this equation issimilar to that using the equivalent potentialtemperature, but with the consideration of icecrystals. The equivalent ice potential tempera-ture is approximately conserved if there is norain, snow, and graupel falling out of air parceland therefore, it has a better conservative prop-erty than the equivalent potential temperature.In the thermodynamic equation, term (1) is dueto the change of y=T following an air parcel andterm (2) is the contribution from cloud micro-physical processes. Because the equivalent icepotential temperature is a function of watervapor and cloud ice, the change of yei needs toaccount for the phase change between vapor/liquid and snow/graupel and for the particletransformation between ice and snow/graupel,namely to account for the changes of snowand graupel. Therefore, the formula for term (2)is:

microðyeiÞ ¼yLf

CpTðPs þ PgÞ;

where Ps and Pg are equivalent to microphysicalproduction terms of snow and graupel, respec-tively (the last term on the right-hand side ofequation (6)).

(iv) Continuity equation

1

r

qðurÞqr

þ 1

r

qv

qcþ 1

r0

qðr0wÞqz

¼ 0: ð4Þ

Density r0 is assumed only to be a function ofheight.

(v) Six equations of water substance

qqx

qt¼ � 1

r

qðurqxÞqr

� 1

r

qðvqxÞqc

� 1

r0

qðr0wqxÞqz

þ Px; ð5Þ

qqy

qt¼ � 1

r

qðurqyÞqr

� 1

r

qðvqyÞqc

� 1

r0

qðr0wqyÞqz

þ 1

r0

qðr0VtyqyÞqz

þ Py; ð6Þ

where qx is the mixing ratio of cloud water (qc),cloud ice, or non-precipitable water (qw), whichis defined as qv þ qc þ qi, and qy is the mixingratio of rain (qr), snow (qs), or graupel (qg). Vty

is the terminal velocity of precipitation (rain,

S.-H. CHEN, W.-Y. SUN 101February 2002

snow or graupel). Px and Py are the micro-physical production terms of qx and qy.

According to Asai and Kasahara (1967), wealso define

(a) the horizontal area-average value withinthe cloud

A ¼ 1

pR2

ð2p

0

ðR

0

Ar dr dc;

where A is any cloud variable and R is thecloud radius.

(b) the deviation value from the horizontalaverage of the cloud

A 0 ¼ A � A:

(c) the lateral boundary average value of thecloud

~AAðRÞ ¼ 1

2p

ð2p

0

A dc:

(d) the deviation value from the lateral bound-ary average of the cloud

A00 ¼ A � ~AA:

Similar to Holton (1973), we assume that thenonhydrostatic pressure can be expressed aspnhðr; zÞ ¼ p�ðzÞ � J0ðrÞ, where J0ðrÞ is thezeroth order of the first kind Bessel function.The radius of the cloud ðRÞ satisfies the firstroot of J0ðxÞ ¼ 0, where x ¼ a

R r and a ¼ 2:4048.It is noted that the lateral nonhydrostaticpressure perturbation ( pnh) is zero under thisassumption.

After horizontal averaging (see Appendix A),equations (1) to (6) become:

qw

qt¼ � 2

Rð~uu~ww þ gu 00w 00u 00w 00Þ � 1

r0

q½ r0ðww þ w 0w 0Þ�qz

� 2J1ðaÞr0a

qp�ðzÞqz

þ g

�yv � yv0

yv0� QT

þ R

Cp� 1

� �2J1ðaÞ

a

p�ðzÞp0

; ð7Þ

qd

qt¼ � 2

Rð~uu~ddþ gu 00d 00u 00d 00Þ � 1

r0

q½ r0ðwdþ w 0d 0Þ�qz

� d2

2þ z2

2þ 2a

r0R2J1ðaÞp�ðzÞ; ð8Þ

qyei

qt¼� 2

Rð~uu~yyei þ gu 00y 00

eiu 00y 00eiÞ�

1

r0

q½ r0ðwyei þw 0y 0eiÞ�

qz

þ 1

CpðLvqv � Lf qiÞ

d

dt

y

T

� �þ microðyeiÞ; ð9Þ

2

R~uu þ 1

r0

qðr0wÞqz

¼ 0; ð10Þ

qqx

qt¼ � 2

Rð~uu~qqx þ gu 00q 00

xu 00q 00x Þ

� 1

r0

q½r0ðwqx þ w 0q 0xÞ�

qzþ Px; ð11Þ

and

qqy

qt¼ � 2

Rð~uu~qqy þ gu 00q 00

yu 00q 00y Þ �

1

r0

q½r0ðwqy þ w 0q 0yÞ�

qz

þ 1

r0

qðr0VtyqyÞqz

þ Py; ð12Þ

where J1 is the first order of the first kind Bes-sel function. z is horizontal average of vorticity,

and it is assumed zero. For any quantity A, ~uu ~AA,gu 00A 00u 00A 00, ðq=qzÞðwAÞ, and ðq=qzÞðw 0A 0Þ are termedentrainment, lateral eddy diffusion, verticalflux, and vertical eddy diffusion, respectively.

The second order perturbation terms of (d 0d 0),(d 0z 0), and (V 0

tyq 0y), and the one from term (1) in

equation (3) are ignored. As in Holton (1973),the tilting terms are also ignored.

The continuity equation can be expressed interms of the divergence field and vertical veloc-ity field as:

dþ 1

r0

qðr0wÞqz

¼ 0; ð13Þ

Combining r0 � ð7Þ þ ðq=qzÞ½r0 � ð8Þ� withthe continuity equation (13), we can obtain thediagnostic equation for pressure perturbation( p�), which is divided into a buoyancy part ( p�

b)and a dynamic part ( p�

d). Thus,

p� ¼ p�b þ p�

d; ð14Þ

2J1

a� q2p�

bðzÞqz2

� R

Cp� 1

� �g

2J1

a

q

qzr0

p�bðzÞp0

� �� 2a

R2J1 p�

bðzÞ

¼ gq

qzr0

yv � yv0

yv0

� �� g

q

qzðr0QTÞ; ð15Þ


and

2J1

a� q2p�

dðzÞqz2

� R

Cp� 1

� �g

2J1

a

q

qzr0

p�dðzÞp0

� �� 2a

R2J1 p�

dðzÞ

¼ � 2r0

R½~uu~ddþ gu 00d 00u 00d 00� � q

qz½ r0ðwdþ w 0d 0Þ�

� r0

d2

2� 2

R

q

qz½ r0ð~uu~ww þ gu 00w 00u 00w 00Þ�: ð16Þ

Equations (15) and (16) are the diagnostic equa-tions of buoyancy pressure perturbation anddynamic pressure perturbation, respectively.Combining equations (10) and (13) obtains

d ¼ 2

R~uu: ð17Þ

In this model, d is diagnosed from equation (17)instead of being predicted from equation (8).

We further apply the eddy exchange hypoth-esis following the notations of Ogura andTakahashi (1971) and Holton (1973). For anyvariable A, the parameterization formulae are:

gu 00A 00u 00A 00 ¼ n

RðA � A0Þ;

w 0A 0 ¼ �KmqðAÞqz

;

where A represents momentum fields. How-ever, if A is a mass field, then Km is replaced byKh. n, Km and Kh are the kinematic viscosity ofair, momentum eddy coefficient, and heat eddycoefficient, respectively. The formulae of Km,Kh, and n are similar to those used in Oguraand Takahashi (1971), and are written as:

Kh ¼ n ¼ 3Km ¼ 0:1 � R � jwj:

Following the entrainment/detrainment con-cept in Asai and Kasahara (1967), we assumethat the properties of the environment will bebrought into clouds (entrainment) if conver-gence occurs. In contrast, the properties of theclouds will be carried out to the environment(detrainment) if divergence happens, and thesedetrained properties will be saved and feedback to the resolving grids (e.g. mesoscalemodel) in the future.

2.2 Cloud microphysical processesThis model includes six classes of hydro-

meteors: water vapor, cloud water, cloud ice,

rain, snow, and graupel. All parameterized mi-crophysical production terms, the last terms onthe right-hand side of equations (11) and (12),are based on Lin et al. (1983) and Rutledge andHobbs (1984). Interactions among these sixwater substances, such as evaporation/subli-mation, deposition/condensation, aggregation,accretion, Bergeron processes, freezing, melt-ing, and melting evaporation, are counted. Theformula of each process is given in AppendixB. The saturation adjustment in Tao et al.(1989) is applied. If a layer is supersaturatedand cloud water or cloud ice exists, the amountof water vapor condensed to cloud water or de-posited to cloud ice depends upon the ratio ofcloud ice and cloud water. The same principle isapplied to the undersaturated condition. If su-persaturation occurs in a layer without cloudice and cloud water, the ratio of water vapor tocloud water and cloud ice is simply a functionof temperature. Furthermore, no water existswhen the temperature is less than �40 C; liq-uid and solid phases can coexist when the tem-perature is between 0 C and �40 C; graupel orsnow can exist without ice crystals when thetemperature is above 0 C.

2.3 Numerical methodsAs shown in Fig. 1, a staggered grid is ap-

plied. The vertical velocity (w), momentumeddy coefficient (Km), heat eddy coefficient (Kh),and kinematic viscosity (n) are computed on thefull levels. The pressure perturbation ( p�),average lateral radial velocity (~uu), mixing ratios,and temperature fields are calculated on thehalf levels.

The lateral eddy exchange, entrainment, ver-tical flux, and pressure perturbation gradient

Fig. 1. The model’s vertical staggeredgrid showing full levels and half levels.


force are solved using the Forward-Backwardscheme in time (Sun 1984) and using the centraldifference scheme in space. Vertical fluxes ofrain, snow, and graupel due to downward ter-minal velocities are calculated with a forwardscheme. However, to prevent instability fromlarge terminal velocities, a smaller time incre-ment could be used in this part if necessary. Atridiagonal matrix scheme is applied to solvevertical eddy diffusion (implicit scheme), buoy-ancy pressure perturbation, and dynamic pres-sure perturbation.

The vertical velocity is zero at both the upperand lower boundaries. The pressure perturba-tion is zero at the upper boundary, and thenormal pressure perturbation gradient is zeroat the lower boundary.

3. Comparison with one-dimensionalcloud model

The importance of the non-hydrostatic pres-sure perturbation has been discussed in manypapers (List and Lozowski 1970; Holton 1973;Schlesinger 1978; Yau 1979; Kuo and Raymond1980). Ogura and Takahashi (1971) (hereafterreferred to as OT) emphasized that the role ofmicrophysical processes in cloud dynamics is asimportant as the thermodynamics in clouds.Therefore, the simulation in Section 3 is de-signed for testing the importance of pressureperturbation and cloud microphysics.

3.1 Initial conditionThe initial, boundary, and environmental

conditions of this case are from OT, we there-fore name this comparison case the OT case.The cloud radius is 3000 m and the surfacepressure is 1000 mb. The relative humidity is100 percent at the lowest level and then de-creases with height with a rate of 5% km�1.The temperature is 25 C at the lower bound-ary. The lapse rate is 6.3 C km�1 from the sur-face to 10 km and then isothermal up to 15 km.A cloud is initiated by weak upward motion inthe environment from the surface to 2 km withthe formula:

w ¼ z

zo

� �2 � z

zo

� �;

where zo ¼ 1 km. The domain is 15 km with avertical resolution of 200 m. A 5-second timestep is used and the model integrates for one

hour. Four runs as indicated in Table 1 aretested.

3.2 Results comparisonTable 1 shows that clouds reach a steady

state without cloud microphysics (Run 1 and 2)because of the lack of drag forces; therefore,their life spans are infinite. The life span of acloud is measured from the starting point of thevertical velocity reaching 2 m s�1 to the endpoint of the downward motion (or upwardmotion) reaching 2 m s�1. When cloud micro-physics is included, a cloud grows and eventu-ally dissipates as shown in Fig. 2 and Table 1.The presence of the cloud microphysics pro-duces a positive feedback with respect to thestrength of the maximum vertical velocity, ei-ther with or without nonhydrostatic pressure(Table 1). However, in OT, the maximum verti-cal velocity without cloud microphysics, 24–28 m s�1 (OT1 in Table 1), is much larger thanthat with cloud microphysics, 16–18 m s�1

(OT2 in Table 1). Besides the different numeri-cal methods used in these two models, the dif-ferences between these two models’ resultscould be from the simplification of the cloudmicrophysics in OT’s model. The maximumvalue of rain mixing ratio reaches 0.01 kg kg�1

in our model, but it was only 0.002–

Table 1. Experimental design and resultsfor OT case. The results of OT1 andOT2 can be found in Figures 2a and 3ain OT paper. ‘‘Life S.’’ is the life span ofa cloud.


0.003 kg kg�1 in OT (Fig. 3e in OT). The maxi-mum value of the graupel mixing ratio reaches0.006–0.007 kg kg�1 in our model; howevergraupel, as well as snow, was ignored in OT.Several important processes which might beinvolved with latent heat were not included inOT, such as the accretion of rain by graupel,rain accretion by cloud ice, accretion of rain bysnow, graupel melting to form rain, and accre-tion of cloud water by rain. In particular, themaximum rate of the accretion of rain by grau-pel (@ 7 � 10�4 s�1) is one order larger than theothers. At the end of the cloud life cycle, adownward motion develops close to the surface(Fig. 2) due to the drag forces of the precipita-tion, as well as low level evaporative cooling ofrain and the cooling of melting graupel. Theprecipitation reaches the ground mainly in theform of rain.

Nonhydrostatic pressure perturbation notonly reduces the maximum vertical velocity(Table 1), but also weakens the low leveldownward motion (�2.4 m s�1 in Fig. 3a vs.�5.6 m s�1 in Fig. 2) because the pressure per-turbation gradient force (Fig. 3b) is opposite tothe air parcel motion (positive value near thesurface during 32 to 40 minutes and negativevalue at the maximum updraft area). The up-ward pressure gradient force near the surfacealso elevates the location of the maximumdownward motion. Around cloud top, the up-ward pressure gradient force helps the cloudreach its mature stage sooner and also helpsmaintain a longer mature stage (Fig. 2 vs. Fig.3a). The longer mature stage allows more vig-orous microphysical processes (more rain, snow,and graupel are formed), so that the cloud can

generate more precipitation capable of reachingthe ground.

Figure 4 shows the accumulated precipita-tion amount in both Run 3 and Run 4. The rainfalls slowly at the beginning due to the smallamount of precipitation, and the upward mo-tion is approximately offset by the terminal ve-locity. The quasi-suspended situation lets therain and graupel grow quickly and results inmore precipitation, which produces a strongerdrag force with time. Besides a stronger dragforce, the evaporative cooling of rain and the

Fig. 2. Time evolution of the verticalvelocity with a contour interval of2 m s�1 from the OT case Run 3.

Fig. 3. Time evolution of (a) the verticalvelocity with a contour interval of2 m s�1 and (b) the pressure gradientforce with a contour interval of0.03 m s�2 from the OT case Run 4.

Fig. 4. Accumulated rain (mm) withoutnonhydrostatic pressure in Run 3 (lineR3) and with nonhydrostatic pressurein Run 4 (line R4) from the OT case.


melting of graupel help change the updraft to adowndraft. The downward motion occurs whenthe precipitation rate reaches its maximum.The rain starts falling earlier in Run 4 andwithin an hour, 52 mm of precipitation hadfallen, while only 28 mm fell in the run withoutthe nonhydrostatic pressure effect. The differ-ent qualitative precipitation amount due to thepressure perturbation was also briefly men-tioned in Kuo and Raymond (1980).

It is worth mentioning that, in Run 4 (Fig.3a), as soon as the old cloud dies, a new cellstarts growing. Recurrence is also found whennonhydrostatic pressure is excluded, but ittakes a much longer time. The recurrence isdue to the assumption of the undisturbed sur-rounding atmosphere during convective devel-opment. After the cloud dies, if any disturbanceor thermal bubble exists, a new cell may grow.In Run 4, except near the surface, the old cellcolumn is moister (around 40 minutes in Fig.5a) than the environment after the cell dies.The positive buoyancy (Fig. 5b) and the upwardnonhydrostatic pressure gradient force (Fig.3b) trigger a new upward motion at low levels.However, in Run 3, although the column isslightly warmer than the environment at 40–50 minutes (Fig. 6b) at low levels, a deeper and

drier layer in the lower troposphere due to astronger downward motion (dry advection) pre-vents a new cell from developing. The moistureperturbation field in Fig. 6a clearly shows thatfour oscillations occur after the cloud dies, butbefore the new cell grows. A closer look at theseoscillations (Fig. 7) shows that the weak verti-cal motion creates an opportunity for mixing ofthe dry, old cell column and the moist environ-ment. The dry layer becomes weaker andweaker and eventually the new cell is able topenetrate it and the cloud starts growing.

Fig. 5. Time evolution of (a) the watervapor perturbation contoured every0.0005 kg kg�1 and (b) the potentialtemperature perturbation contouredevery 1.0 K from the OT case Run 4.

Fig. 6. Time evolution of (a) water va-por perturbation (contour interval of0.0005 kg kg�1) and (b) potential tem-perature perturbation (contour intervalof 1.0 K) from the OT case Run 3.

Fig. 7. Time evolution of the vertical ve-locity with a contour interval of0.3 m s�1 during 40 to 100 minutes be-low 5 km. For clarity, values greaterthan 1 m s�1 are not contoured.


Figures 5b and 6b show a cold anomaly abovethe cloud top due to overshooting cooling, butthis was not found by OT. We also see that theovershooting cooling of �10.1 K with non-hydrostatic pressure is stronger than that of�5.3 K without nonhydrostatic pressure. Thevertical velocity in Run 3 is slightly strongerthan that in Run 4 and the cloud top can reachthe same height in both runs, but the longermature stage in Run 4 results in a strongerovershooting cooling. As a result, the presenceof the nonhydrostatic pressure perturbation notonly decreases the extreme vertical velocitiesbut also increases the precipitation rate, as wellas enhances the overshooting cooling above thecloud top.

Figures 8a and 8b show that the meso-highat the cloud top has both buoyancy and dynamiccomponents. Above the cloud top, the negativebuoyancy pressure perturbation is opposite tothe positive dynamic pressure perturbation.The surface high pressure and the low pressurenear 3 km between 25 and 40 minutes, aredue to the buoyancy pressure perturbation. Thepatterns show that the dynamic pressure per-turbation is as important as the buoyancypressure perturbation in this case.

4. A comparison with athree-dimensional cloud model

4.1 Three-dimensional modelThe Weather Research and Forecast (WRF)

model is a newly-developed, next-generation,fully compressible, nonhydrostatic model whichcan be applied to both idealized and realweather studies (Michalakes et al. 2001; Ska-marock et al. 2001). In this version of the WRF,the governing equations are written in fluxform for the purpose of conservation of mass,dry entropy, and scalars. The time split explicitscheme (Klemp and Wilhelmson 1978), thestaggered Arakawa C grid, a sigma-height co-ordinate, a free-slip lower boundary condition,and a rigid upper boundary condition are thefeatures of this model. Several lateral boundaryconditions, high order time and space schemes,and physics schemes (Chen and Dudhia 2000)have been implemented. Detailed informationfor this model can be found on the WRF website (www.wrf-model.org).

Since the WRF model is used for purposes ofcomparison, we have chosen from the availableschemes ones which most closely match thoseused in the one-dimensional model, such as thesecond order Runge-Kutta time scheme (Wickerand Skamarock 1998), the second (third) orderadvection scheme in the vertical (horizontal),the Smagorinsky’s subgrid turbulence scheme(1963). The cloud microphysics scheme is thesame as the one in the one-dimensional model.The horizontal resolution, the vertical resolu-tion, and the time step are 1 km, 400 m, and5 seconds, respectively. The domain size is92 km � 92 km � 20 km in x-y-z direction withopen lateral boundary conditions and a 3-kmsponge zone at the model top.

4.2 Initial conditionInitial conditions from Schlesinger (1978)

(without ambient wind) are used in this ex-periment. These conditions are typical of thesouthern Great Plains severe thunderstormenvironment. The tropopause is located at11.9 km, and the atmosphere above this levelis assumed to be isothermal up to the upperboundary. The cloud is initiated by a moistthermal bubble as was the trigger functionused in Schlesinger (1978), without the verticalvelocity perturbation. The formula of the im-posed thermal bubble (from the surface to the

Fig. 8. Time evolution of (a) the buoyancypressure perturbation and (b) the dy-namic pressure perturbation with acontour interval of 0.2 mb from the OTcase Run 4.


3.5 km) in the WRF is:

y 0 ¼ 1:0 � sinp z

3500

� � expð�r2=R2Þ

ð1 � r2=R2Þ;

where R (¼ 9 km) is the radius of the bubble,and r is the distance of any grid point to thecentral vertical axis of the bubble. The thermalperturbation used in the one-dimensional modelis from the horizontally averaged value of thethree-dimensional bubble. The relative humid-ity within the cloud (0 to 4.2 km) is 88% for bothmodels, slightly higher than the environment.The same vertical resolution and time resolu-tion are used in the one-dimensional model, butwith a smaller vertical domain (16 km).

4.3 Results comparisonWithout ambient wind shear, the three-

dimensional model produces a cylindrical axi-symmetric cloud and the updraft is surroundedby a weak downdraft as shown in Fig. 9. Whenthe updraft reaches its maximum vertical ve-locity, the maximum radius of the updraft isabout 5 km, which is used as the cloud radiusin the one-dimensional cloud model. The hori-zontal averages of the vertical velocity, potentialtemperature anomaly, and water vapor anom-aly within the 5-km radius of updraft from thethree-dimensional model are calculated and

compared with those from the one-dimensionalmodel.

Figure 10a shows the time evolution ofthe vertical velocity from the one-dimensionalmodel. The maximum value, 23.7 m s�1, occursat 9 km after a 17 minute simulation. Figure10b shows the horizontal-averaged vertical ve-locity within the radius of 5000 m cloud fromthe three-dimensional model. There exists twopeaks of local maximum values in the three-dimensional model. The larger one is 28.1 m s�1

and happens at the 24 minute mark at 11–12 km. The cloud simulated from the one-dimensional model develops earlier, with aslightly weaker intensity for both upward anddownward motion. The cloud top is also slightlylower.

The maximum potential temperature anom-aly is 7.2 K at 6–7 km around 14 minutes inthe one-dimensional model (Fig. 10c) and 7.3 Kat 10 km around 23 minutes in the three-dimensional model (Fig. 10d). Both modelsconsistently show that the altitude of the max-imum vertical velocity is higher than that ofthe maximum heating, and the maximum ver-tical velocity occurs earlier than the maximumheating. Furthermore, both models produceovershooting cooling above the top of the up-drafts and cold pool near the surface in thedissipation stage. The overshooting cooling isstronger in the three-dimensional cloud model(�29.8 K vs. �15.3 K), due to a stronger verti-cal velocity and the penetration into the strato-sphere (large potential temperature gradient).The cold pool near the surface in the three-dimensional model is also stronger.

The maximum mixing ratio anomaly ofthe water vapor in the one-dimensional modelis 6.5 g kg�1 and happens at 11 minutes at4.0 km (Fig. 10e), while the maximum is about6.0 g kg�1 at 30 minutes at the same height inthe three-dimensional model (Fig. 10f ). At theend of the cloud life, the evaporation occurs atlower levels, but it is not enough to compensatethe vertical dry advection. Therefore, a dryingprocess exists near the surface in both models.The pressure field (not shown here) indicatesthat there are two meso-highs in both modelresults: one is near the cloud top and the otheris close to the surface during the downwardmotion period. However, there is a low nearthe surface during the developing and mature

Fig. 9. The horizontal cross section of thevertical velocity (m s�1) of the WRFmodel at 10 km height after 30-minuteintegration.


stages in the three-dimensional model, but ahigh in the one-dimensional model.

Besides momentum, thermal, and moisturefields, we also calculate the vertical mass flux(Fm), heat flux (Fh), and moisture flux (Fq) ac-cording to the following formulae:

Fm ¼ rwB;

Fh ¼ CprwBðy� y0Þ;

Fq ¼ rwBðqv � qv0Þ;

where B is the horizontal area.In the mass flux field, the maximum values of

9:8 � 108 kg s�1 from the one-dimensionalmodel (Fig. 11a) and 9:0 � 108 kg s�1 from thethree-dimensional model (Fig. 11b) are veryclose. The patterns of these two models’ results

are also comparable, except there are two localmaximums and the mass profile is shifted laterin time in the three-dimensional simulation.

The maximum vertical heat flux 7:0 � 1012

J s�1 in the one-dimensional model (Fig. 11c) islarger than the value 5:7 � 1012 J s�1 in thethree-dimensional model (Fig. 11d). This is be-cause the density decreases with height. Abovethe cloud top, the downward heat flux in theone-dimensional model is much smaller thanthat in the three-dimensional model due toweaker upward motion and weaker overshoot-ing cooling. However, the patterns are stillcomparable.

The maximum vertical moisture flux in theone-dimensional model (5:2 � 106 kg s�1 in Fig.11e) is larger than that in the three-dimen-

Fig. 10. Time evolutions of the (a) vertical velocity (m s�1), (c) potential temperature anomaly (K),and (e) moisture anomaly (kg kg�1) from the one-dimensional cloud model, and the averaged (b)vertical velocity (m s�1), (d) potential temperature anomaly (K), and (f ) moisture anomaly (kg kg�1)within the radius of 5000 m cloud from the WRF model. The values in (e) and (f ) are multiplied by1 � 103.


sional model (3:3 � 106 kg s�1 in Fig. 11f ). Theheights of the maximum values and the pat-terns are quite similar. The positive anomaly,which occurs near the surface during the dissi-pation stage, is much stronger in the three-dimensional model due to stronger downwardmotion and a stronger drying process.

Figure 12 shows the accumulated precipita-tion from both model simulations. The three-dimensional result still uses the horizontallyaveraged value within the 5000 m cloud radius.The precipitation in the one-dimensional modeloccurs earlier and is concentrated between 20and 35 minutes, while the precipitation in thethree-dimensional model occurs later and at amore gradual rate. After 90 minutes, the accu-

Fig. 11. Time evolutions of the (a) mass flux (�108 kg s�1), (c) heat flux (�1012 J s�1), and (e) mois-ture flux (�106 kg s�1) from the one-dimensional cloud model, and the averaged (b) mass flux(�108 kg s�1), (d) heat flux (�1012 J s�1), and (f ) moisture flux (�106 kg s�1) within the radius of5000 m cloud from the WRF model.

Fig. 12. Accumulated rain (mm) (a) fromthe one-dimensional cloud model, and(b) from the WRF model (averagedwithin the radius of 5000 m).


mulated precipitation amount reaches 46 mmfor the three-dimensional model, while it isonly 29 mm for the one-dimensional model.

4.4 Entrainment, lateral eddy diffusion,vertical diffusion, and microphysicsin the one-dimensional cloud model

The forcing terms of the entrainment, lateraleddy diffusion, vertical diffusion, and cloud mi-crophysics in the one-dimensional cloud modelare calculated. Figure 13 shows the time evolu-tion of the maximum and minimum values ofeach term for the vertical momentum, thermo-

dynamic, and non-precipitable water equations.The results indicate that the entrainment effectis very important for all prognostic variables(the same conclusion for others). During theupdraft period, the entrainment (detrainment)occurs at lower (upper) levels (Fig. 14a). Thevertical momentum detrains out from the up-per portion of the cloud (negative value in Fig.13a), while no momentum entrains into thecloud at lower levels since the surroundings arecalm in this case. The positive entrainmentrate in the first 28 minutes matches quite wellfor both equivalent ice potential temperatureand non-precipitable water (qw) since the envi-ronment entrains high yei and qw into the cloud(Fig. 14a). The detrainment rate of qw at upperlevels is relatively smaller than the entrain-ment rate at lower levels (Fig. 13c and Fig.14a) since a lot of qw has been transformed intorain, snow, and graupel when the moist airascends. Figure 13 also shows another peak ofdetrainment rate in association with the down-ward motion (30 minutes). It is noted that thehigh entrainment rate is mostly concentratedat the lowest two to three levels, while the de-tainment rate is gradually spread out at theupper portion of the cloud. The detrainmentcould extend to 5-km depth (more than onelayer) for the equivalent ice potential tempera-ture (figure is not shown here). This entrain-ment/detrainment process is very important forthe vertical exchange of surrounding air and forthe cloud information contributed from thesubgrid scale.

For the qw, the microphysics is a sink terminside the updraft (Fig. 14b), but it becomesa source term during the downward motion(evaporation from rain). The maximum trans-formation rate from qw to rain, snow, andgraupel (microphysical processes) is compara-ble to that detrained out from upper levels ofthe updraft. For the equivalent ice potentialtemperature, the microphysics is a source(sink) term above (below) the melting level. Itis worth mentioning that the rate of change ofthe equivalent ice potential temperature due tocloud microphysics is very small (only 1% ofthat due to entrainment/detrainment) since thechange only occurs when the snow or graupelamount varies in microphysical processes. Thevertical diffusion (Fig. 14c) is a downgradientprocess, namely reducing the difference be-

Fig. 13. Time evolution of the maximumand minimum rates of several forc-ing terms, such as entrainment (— e —)(positive), lateral eddy diffusion (— l —),vertical diffusion (— d —), and cloudmicrophysics (— m —), from the one-dimensional cloud model for (a) verti-cal momentum equation (m s�2), (b)thermodynamic equation (K s�1), and(c) non-precipitable water equation(kg kg�1 s�1). In (b), the lateral eddydiffusion, vertical diffusion, and cloudmicrophysics are multiplied by 20 forvisualization.


tween the maximums and minimums, and thelateral diffusion (Fig. 14d) is a sink term for thecloud during the updraft period.

5. Summary

A one-dimensional cloud model is developedfor possible use in the cumulus parameter-ization scheme developed by Sun and Haines(1996). The effects of entrainment, cloud micro-physics, pressure perturbation, lateral eddydiffusion, and vertical eddy diffusion are takeninto account. The inclusion of nonhydrostaticpressure could (1) reduce the vertical velocities,(2) help the cloud develop sooner, (3) helpmaintain a longer mature stage, (4) produce astronger overshoot cooling, and (5) approxi-mately double the precipitation amount. Simu-

lations also indicate that both the dynamicpressure perturbation and the buoyancy pres-sure perturbation are equally important to theformation of a cloud. The analysis of the forcingterms for entrainment, cloud microphysics, lat-eral eddy diffusion, and vertical eddy diffusionshows that the entrainment effect dominatesthe other three for a cloud development. In ad-dition, the detrainment layer could extend overa 5-km depth (more than one layer) for equiva-lent ice potential temperature.

Two comparisons are examined in this study:a one-dimensional cloud model (Ogura and Ta-kahashi 1971) and a three-dimensional WeatherResearch and Forecast (WRF) model (Micha-lakes et al. 2001; Skamarock et al. 2001). Oursimulation shows that the inclusion of cloud

Fig. 14. Time evolution of the rate of change of non-precipitable water (kg kg�1 s�1) due to (a) en-trainment, (b) microphysics, (c) vertical eddy diffusion, and (d) lateral eddy. The values are multi-plied by 1 � 106 for visualization. The contour interval is 20 kg kg�1 s�1 for entrainment and2 kg kg�1 s�1 for the others.


microphysics develops a stronger cloud. How-ever, this is quite different from those pre-sented in Ogura and Takahashi (1971), whichhad simplified cloud microphysics. In our simu-lations, several microphysical processes relatedto graupel are quite important (e.g., the accre-tion of rain by graupel, rain accretion by cloudice, accretion of rain by snow, graupel meltingto form rain, and accretion of cloud water byrain), but these were not counted in Ogura andTakahashi (1971).

No ambient wind shear is considered inthe comparison with a three-dimensional cloudmodel. The horizontally averaged results with-in the three-dimensional cloud were used to dothe comparison. The one-dimensional modelis capable of reproducing the major featuresof a convective cloud generated by the three-dimensional model, such as fluxes, cold pool,overshooting cooling, and two meso-highs. Somedifferences (e.g., double maximum peaks andthe meso-low) existing between both model re-sults should not be surprising because cloudssimulated from a three-dimensional model arequite nonlinear. Hence, those features may bedifficult to be reproduced by a one-dimensionalmodel. It is also noted that the environmentremains as the initial condition in the one-dimensional model, but it changes with timein three-dimensional simulations. Definitely,there is room for improvement of this one-dimensional cloud model.

The tilting of convective clouds has oftenbeen observed and it might influence a cloud’slife, properties, and rain rate. The importanceof the downdraft, which develops besides theupdraft, has also received lots of attention. Thetilting of the cloud might be related to the sur-rounding wind shear, and it will be an in-teresting subject to study their relationship.Besides applying this model to a cumulus pa-rameterization scheme, we will keep improvingthis one-dimensional model, especially the con-sideration of the downdraft effect.

Acknowledgments

The authors thank Dr. Jiun-dar Chern forhis grateful assistance, Drs. J. Klemp and J.Bresch for their helpful discussion. We alsoacknowledge Dr. J. Bresch’s review of themanuscript. Finally, the authors thank theWRF team for their development effort of

the WRF model. This work is supported by theNSF under grant ATMS-9213730 and ATMS9631572.

Appendix A

The horizontal average of thevertical momentum equation

Since the derivations of all equations arevery similar, we only present the vertical mo-mentum equation here. The horizontal averageof equation (1) is derived term by term as fol-lows:

1

pR2

ð2p

0

ðR

0

qw

qtr dr dc ¼ qw

qt:

� 1

pR2

ð2p

0

ðR

0

1

r

qðurwÞqr

r dr dc

¼ � 1

pR2

ð2p

0

½ðurwÞ�jR0 dc

¼ � 1

pR

ð2p

0

ðuwÞdc

¼ � 2

Rfuwuw

¼ � 2

Rð~uu ~ww þ gu 00w 00u 00w 00Þ:

Here, we assume that ~uuð ~wwÞ and w 00ðu 00Þ are un-correlated.

� 1

pR2

ð2p

0

ðR

0

1

r

qðvwÞqc

r dr dc

¼ � 1

pR2

ðR

0

qðvwÞqc

� 2p0

dr

¼ 0:

� 1

pR2

ð2p

0

ðR

0

1

r0

qðr0w2Þqz

r dr dc

¼ � 1

r0

q

qzr0

1

pR2

ð2p

0

ðR

0

w2r dr dc

� ¼ � 1

r0

q

qzðr0wwÞ

¼ � 1

r0

q

qz½ r0ðww þ w 0w 0Þ�:

w and w 0 are also assumed uncorrelated. r0 isonly function of height.


� 1

pR2

ð2p

0

ðR

0

1

r0

qpnh

qzr dr dc

¼ � 1

r0

qp�

qz

1

pR2

ð2p

0

ðR

0

J0ðrÞr dr dc

¼ � 1

r0

qp�

qz

1

pR2 2p R2

a2

ð a0

J0ðxÞx dx

¼ � 2

r0

qp�

qz

1

a2½xJ1ðxÞ�ja0

¼ � 2J1ðaÞr0a

qp�

qz;

where x ¼ aR r and pnhðr; zÞ ¼ p�ðzÞ J0ðrÞ as

mentioned in Section 2.

1

pR2

ð2p

0

ðR

0

gðyv � yv0Þ

yv0r dr dc ¼ g

ðyv � yv0Þyv0

:

1

pR2

ð2p

0

ðR

0

gR

Cp� 1

� �pnh

p0r dr dc

¼ g

p0

R

Cp� 1

� �p�ðzÞ 1

pR2

ð2p

0

ðR

0

J0ðrÞr dr dc

¼ gR

Cp� 1

� �2J1ðaÞ

a

p�ðzÞp0

:

� 1

pR2

ð2p

0

ðR

0

gQTr dr dc ¼ �gQT :

Appendix B

Bulk parameterization ofcloud microphysics

The formulae for microphysical processes arebased on Lin et al. (1983) and Rutledge andHobbs (1984). The size distribution of the rain,snow and graupel are hypothesized as:

NrðDrÞ ¼ Nor expð�lrDrÞ;

NsðDsÞ ¼ Nos expð�lsDsÞ;

NgðDgÞ ¼ Nog expð�lgDgÞ;

where Nor, Nos and Nog are the interceptparameters of the rain, snow and graupel sizedistributions, respectively. Dr, Ds and Dg arediameters of the rain, snow and graupel par-ticles, respectively. The slope parameters of therain, snow and graupel size distributions (lr, ls

and lg) are written as:

lr ¼prw Nor

rqr

� �1=4

;

ls ¼prs Nos

rqs

� �1=4

;

lg ¼prg Nog

rqg

� �1=4

:

rw, rs, and rg are densities of rain, snow, andgraupel, respectively. The terminal velocities ofthe rain, snow and graupel particles are:

UrðDrÞ ¼ aDbr

ro

r

� �1=2

;

UsðDsÞ ¼ cDds

ro

r

� �1=2

;

UgðDgÞ ¼4grg

3CDr

� �1=2

D1=2g :

where b ¼ 0:8, a ¼ 2115 � 0:01ð1�bÞ, d ¼ 0:25,c ¼ 152:93 � 0:01ð1�dÞ, ro ¼ 1:29 kg m�3, andCD (a drag coefficient) ¼ 0.6. The mass-weighted mean terminal velocities of the rain,snow and graupel are:

Ur ¼aGð4 þ bÞ

6lbr

ro

r

� �1=2

;

Us ¼cGð4 þ dÞ

6lds

ro

r

� �1=2

;

Ug ¼ Gð4:5Þ6l0:5

g

4grg

3CDr

� �1=2

:

G is the gamma function.

A.1 AggregationThe aggregation rate of ice crystals to form

snow is:

a1 ¼ 1:0 � 10�3 expð0:025TcÞ;

Psaut ¼ a1ðqi � qioÞ;

where a1 is a rate coefficient (s�1), Tc isthe temperature in the units of C, and qio

(¼ 1:E �3 kg kg�1) is a threshold amount foraggregation to occur.

The collision and coalescence of cloud drop-lets to form raindrops is:

Praut ¼ 0:001 � ðqc � qcoÞ;


where qco (¼ 7:E �4 kg kg�1) is a threshold forautoconversion.

A.2 AccretionThe rate of accretion of cloud ice by snow is:

Psaci ¼pESINoscqiGð3 þ dÞ

4l3þds

ro

r

� �1=2

;

where ESI, the collection efficiency of the snowfor cloud ice, is a function of temperature andexpressed as:

ESI ¼ expð0:025TcÞ:

The accretion rate of cloud water by snow is:

Psacw ¼ pESWNoscqcGð3 þ dÞ4l3þd

s

ro

r

� �1=2

;

where ESW , the collection efficiency of the snowfor cloud water, is assumed to be 1. WhenTc < 0 C, Psacw will increase the snow content.Otherwise, it will contribute to rain content.

The accretion rate of cloud ice by rain is:

Praci ¼pERINoraqiGð3 þ bÞ

4l3þbr

ro

r

� �1=2

;

where ERI, the collection efficiency of the rainfor cloud ice, is assumed to be 1.

Piacr ¼rp2ERINoraqirwGð6 þ bÞ

24Mil6þbr

ro

r

� �1=2

;

where Mi ¼ 4:19 � 10�13 kg.The accretion rate of rain by snow, Pracs, and

snow by rain, Psacr, are:

Pracs ¼ p2ESRNor NosjUr � Usjrs

r

� �

� 5

l6s lr

þ 2

l5s l

2r

þ 0:5

l4s l

3r

!;

Psacr ¼ p2ESRNos NorjUr � Usjrw

r

� �

� 5

l6r ls

þ 2

l5r l

2s

þ 0:5

l4r l

3s

!;

where ESR ¼ 1.The accretion rate of snow by graupel, Pgacs

is:

Pgacs ¼ p2EGSNosNogjUg � Usjrs

r

� �

� 5

l6s lg

þ 2

l5s l

2g

þ 0:5

l4s l

3g

!;

where EGS ¼ expð0:09TcÞ when Tc < 0 , andEGS ¼ 1:0 when Tc >¼ 0 .

Hail grows by accretion of other water formsin either the dry (Pgdry) or wet (Pgwet) growthmode. The mode with smaller amount is chosen.

Pgdry ¼ Pgacw þ Pgaci þ Pgacr þ Pgacs:

The rate of graupel accretion cloud water(Pgacw), cloud ice (Pgaci), and rain (Pgacr) are:

Pgacw ¼ pEGWNogqcGð3:5Þ4l3:5

g

4grg

3CDr

� �1=2

;

Pgaci ¼pEGINogqiGð3:5Þ

4l3:5g

4grg

3CDr

� �1=2

;

Pgacr ¼ p2EGRNogNorrw

rjUg � Urj

� 5

l6r lg

þ 2

l5r l

2g

þ 0:5

l4r l

3g

!;

where EGW ¼ 1, EGR ¼ 1, and EGI is assumed tobe 0.1 and 1 for dry and wet growth, respec-tively.

Pgwet ¼2pNogðrLvchrs � KaTcÞ

rðLf þ CwTcÞ

��0:78l�2

g þ 0:31S1=3c Gð2:75Þ

�4grg

3CD

� �n�1=2l�2:75

g

þ ðP 0

gaci þ P 0gacsÞ 1 � CiTc

Lf þ CwTc

� �;

and

P 0gacr ¼ Pgwet � Pgacw � P 0

gaci � P 0gacs;

where the formula for P 0gaci is given as Pgaci with

EGI ¼ 1 and similarly the formula for P 0gacs is

given as Pgacs with EGS ¼ 1. c, hrs, Ka, Cw, andCi are the diffusivity of water vapor in air, wa-ter vapor mixing ratio difference between satu-ration and the environment, thermal conduc-tivity of air, and the specific heats of water andice, respectively.


The accretion rate of cloud water by rain is:

Pracw ¼ pERWNoraqcGð3 þ bÞ4l3þb

r

ro

r

� �1=2

:

where ERW ¼ 1.

A.3 Bergeron processesTwo terms, Psfw and Psfi, describe the rates at

which cloud water and cloud ice, respectively,transform to snow by deposition and rimingaccording to the growth of a 50 mm radius icecrystal.

Psfw ¼ NI50ða1ma2

I50 þ pEIWrqcR2I50UI50Þ;

Psfi ¼qi

ht1;

where a1 and a2 are temperature dependentparameters in the Bergeron process (Koenig1971), and NI50, RI50 (¼ 1:0E �4 m), mI50

ð¼ 4:8E �10 kg) and UI50 are the number con-centration (g�1), the radius, mass and terminalvelocity of a 50 mm size ice crystal (Hsie et al.1980). The collection efficiency of cloud ice forcloud water, EIW , is assumed to be 1. The timeneeded for a crystal to grow from 40 to 50 mm isgiven by

ht1 ¼ 1

a1ð1 � a2Þ½mð1�a2Þ

I50 � mð1�a2ÞI40 �;

where mI40 (¼ 2:46E �10 kg) is the mass of a40 mm size ice crystal.

A.4 Deposition (or sublimation)The depositional growth rate of snow, Psdep is:

Psdep ðor PssubÞ

¼ 2pðSi � 1ÞrðA 00 þ B 00ÞNos �

�0:78l�2

s þ 0:31S1=3c G

� d þ 5

2

� �c1=2 ro

r

� �1=4

n�1=2l�ðdþ5Þ=2s

;

where

A 00 ¼ L2s

KaRwT2;

B 00 ¼ 1

rqsic:

Rw, n, Si, and Sc are specific gas constantfor water vapor, kinematic viscosity of air, sat-uration ratio over ice, and Schmidt number

(¼ n=c), respectively. T is the temperature inthe Kelvin units.

The sublimation of hail will occur at the sub-saturated region and the formula is:

Pgdep ðor PgsubÞ

¼ 2pðSi � 1ÞrðA 00 þ B 00ÞNog

�0:78l�2

g

þ 0:31S1=3c Gð2:75Þ

4grg

3CDr

1=4

n�1=2l�2:75g

:

A.5 Melting and melting evaporationThe melting of snow is:

Psmlt ¼ � 2p

rLfðKaTc � LvcrhrsÞNos

��0:78l�2

s þ 0:31S1=3c G

d þ 5

2

� �c1=2

� ro

r

� �1=4

n�1=2l�ðdþ5Þ=2s

� CwTc

LfðPsacw þ PsacrÞ:

The rate of evaporation of melting snow is

Psmltevp ¼ 2pNosðS � 1ÞrðC 00 þ D 00Þ

� 0:78

l2s

þ0:31S

1=3c c1=2G dþ5

2

� �n1=2lðdþ5Þ=2

s

ro

r

� �1=4" #

:

The rate of evaporation of melting graupel is:

Pgmltevp ¼ 2pNogðS� 1ÞC 00 þ D 00

0:78

l2g

þ 0:31ar

n

� �1=2 poo

p

0:2� �

Gðb=2þ 5=2Þlb=2þ5=2

g

;

where

C 00 ¼ Lv

KaT

LvMw

R�T� 1

� �;

D 00 ¼ R�T

cMwesw:

b (¼ 0:37) is the fallspeed exponent for graupel;a (¼ 19:3 mð1�bÞ s�1) is a constant in the fall-speed relation for graupel; Mw (¼ 18:0160) isthe molecular weight of water; esw is a satura-tion vapor pressure for water; R� is the univer-sal gas constant.


The melting of graupel is:

Pgmlt ¼ � 2p

rLfðKaTc � LvcrhrsÞNog

��0:78l�2

g þ 0:31S1=3c Gð2:75Þ

�4grg

3CD

� �1=4

n�1=2l�2:75g

� CwTc

LfðPgacw þ PgacrÞ:

A.6 Snow crystal aggregationRimed snow crystals may collide and aggre-

gate to form graupel. The formula is:

Pgaut ¼ a2ðqs � qsoÞ;

where a2 (¼ 10�3 expð0:09TcÞ) is a rate coeffi-cient (s�1), and qso (¼ 6:E �4 kg kg�1) is a massthreshold of snow.

A.7 Raindrop freezingThe probabilistic freezing of rain to form

graupel is given as:

Pgfr ¼ 20p2B 0Norrw

r� ½expð�A 0ðTcÞÞ � 1�

� l�7

r ;

where A 0 ¼ 0:66 K�1, and B 0 ¼ 100 m�3 s�1.

A.8 EvaporationThe evaporation rate for rain is:

Prevp ¼ 2pðS � 1ÞNor

rðC 00 þ D 00Þ

� 0:78

l2r

þ0:31S

1=3c G bþ5

2

� �a1=2

n1=2l�ðbþ5Þ=2r

ro

r

� �1=4" #

� 1

r

L2v

KaRwT2þ 1

rqswc

� ��1" #

:

where S is the saturation ratio, and qsw is thesaturation mixing ratio of water vapor with re-spect to water.

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a one-dimensional time dependent cloud model

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