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Constraints on droplet growth in radiatively cooled stratocumulus clouds Austin, Philip H.; Siems, Steven T.; Wang, Yinong Apr 5, 1995

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JOURNAL OF GEOPHYSICALRESEARCH,VOL. 100,NO. D7, PAGES 14,231-14,242, JULY 20, 1995  Constraints on droplet growth in radiatively stratocumulus  cooled  clouds  P. H. Austin  Atmospheric Sciences Programme, Departmentof Geography, Universityof BritishColumbia, Vancouver,Canada  S. Siems  Depaxtmentof Mathematics,MonashUniversity,Clayton,Victoria, Australia  Y. Wang Atmospheric Sciences Programme, Department of Geography, Universityof BritishColumbia, Vancouver, Canada Abstract.  Radiative cooling near the top of a layer cloud plays a dominant role in droplet condensationgrowth. The impact of this coolingon the evolutionof small droplets and the formation of precipitation-sizeddrops is calculated using a microphysical model that includesradiatively driven condensationand coalescence.The cloud top radiative environmentused for these calculationsis determinedusing a mixedlayer model of a marine stratocumuluscloud with a subsiding,radiatively cooled inversion. Calculations of the radiatively driven equilibrium supersaturation show that net long wave emissionby cloud dropletsproducessupersaturationsbelow 0.04% for typical nocturnal conditions. While supersaturationsas low as this will force evaporationfor dropletssmaller than • 5 pm, radiatively enhancedgrowth for larger droplets can reducethe time required to produceprecipitation-sized particles by a factor of 2-4, comparedwith dropletsin a quiescentcloud without flux divergence.The impact of this radiative enhancementon the accelerationof  coalescence is equivalent to that produced in updraftsof 0.1- 0.5 ms-1, andvaries linearlywith the total emittedflux (the "radiativeexchange").  1. Introduction  persaturation, which is determined by the droplet size distribution, the net flux divergence,and the partitioning of the long wave absorptionbetweencloud droplets Long wave cooling at the top of the cloud-capped boundarylayer can exert a controllinginfluenceon layer and water vapor [Davies,1985]. The valueof this radynamics and cloud microphysics. Observationsand diatively forced equilibrium, together with the effects models show that stratocumulus clouds beneath a dry of convection and entrainment, determines the condeninversionare subjectedto coolingratesof 7-10 K hr-1 sation growth for droplets smaller than • 5 pm near [Caugheyand Kitchen,1984],with substantially larger cloud top. Droplets larger than 10 pm can grow evenunder very flux divergencespossiblecloseto the top of thick clouds low supersaturations through direct cooling to space. [Daviesand Alves,1989]. The coolingis due primarily to emissionfrom droplets, and this emissionallows The details of this radiatively driven droplet growth larger cloud dropletsto shedheat efficientlyand grow may be particularly important in 200- to 400-m-thick at rates more than an order of magnitude greater than layer clouds. Given droplet concentrationsbetween growthproduces10-14 those experienced in the center of a quiescent cloud 50 and 150 cm-s, condensation pm mean droplet radii near cloud top, with maximum [Roach,1976;Barkstrom,1978]. The growingdroplets liquid water mixing ratios of Wl • 0.40.8g kg-1. Audeplete the availablevapor and reducethe ambient sutoconversion rates (the rate of formation of embryonic persaturation as they cool the air; these two opposing precipitation particles) given these distribution paraminfluences force the cloud parcel to its equilibrium sueters are several orders of magnitude below those typical of deeper cumuli with larger liquid water contents Copyright 1995 by the American GeophysicalUnion. Paper number 95JD01268.  0148-0227 / 95/ 95JD-01268505. O0  [Austinet al., 1995],but both observations [Nicholls, 1984; Austin et al., 1995]and models[Nicholls,1987; Baker, 1993; Austin et al., 1995]indicatethat signif14,231  14,232  AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH  icant precipitation can be produced by these clouds.  dEn  dz ,i)  • (/•drops,i q-/•gas,i)4 (Bi- 0.5(El + + E•-)) One factor that can accelerate precipitation development for shallow cloudsis the condensationgrowth of = (ndrops,i q-ngas,i) 4Ed,i cloud droplets in updrafts [Kovetz and Olund, 1969; Jonasand Mason, 1974];radiativecoolingat cloudtop might be expected to have a similar impact on accelerating the onset of precipitation in layer clouds. and/•gas,i (m-•) arethe absorption coBelow we will use model-derived radiative profiles where/•drops,i  __ - (dEni k,am' )drop sq-(dEni am' )gas (1)  from typical marine stratocumulus clouds tostudy the efficients inband i; Bi,Ei+,andE?(Wm-2)arethe condensation growth ofradiatively-cooled cloud droplets, Planck function andtheupward anddownward irratheevolution oftheequilibrium supersaturation, and diances; thesubscript n denotes thenetupward flux the initiation of precipitation. The cloud droplet radiative environment will be specifiedusing the model  of $ierns et al. [1993] (referredto below as SLB); it gives profiles of temperature, vapor mixing ratio, and the upward and downward long wave fluxes for a nocturnal cloud in which a subsidinglayer of overlying air determines the radiative balance at cloud top. In Section 2 we derive an expression for the equi-  Ei+ - E•-;andz istheheight(seetheNotation listfor a full list of symbols). The thermal emissionand the upward and downward irradiancescombineto give the radiative exchangede-  finedby Roach[1976]asthe net powerleavinga droplet of radiusr per unit area per unit absorptionefficiency:  Ed,i- (Bi- 0.5(El + q-E?))  (2)  librium supersaturation,Seq, and calculatean upper Givenn(r)dr (kg-•), the mixingratio of dropletswith boundon Seqfor typicalcloudradiativeandmicrophysi- radii betweenr and r + dr, and Qa(r,i), the absorption cal conditions. We use the SLB model to obtain vertical efficiencyfor droplets in spectral band i, we can write profiles of the total net emission(termed the "radia- that portion of the flux divergencedue to dropletsin tive exchange"by Roach[1976])and the fractionof the terms of Fd(r, i) (W), the net powerradiatingfrom a total cooling due to cloud droplets and model droplet droplet of radius r in spectralinterval i: condensation growth given this radiative environment. In Section 3 we calculate droplet coalescencenear cloud top and compare it to coalescencein updrafts typical of nocturnal  stratocumulus  discussion  of these results.  2.  Condensation  clouds.  Growth  Section  4 contains  a  -- Pa i)4Ed,idr (dEn,i /0n(r)•rr2Qa(r, dz )drops --  and  the  Equilibrium  Supersaturation  2.1. Radiative  Fluxes: The Two Stream Approx-  imation  We will calculate radiative cooling due to absorption and emission by cloud droplets, water vapor, and carbon dioxide using the five-band model of Roach and  Pa  n(r)Fd,i(r)dr  (3)  where Pa is the density of dry air. 2.2. Droplet  Growth Equations  We can calculate droplet growth given Ed,i, the size and compositionof the cloud condensationnucleus,and the thermodynamicstate variables. The droplet growth  equationincludingradiationis [Roach,1976]  r•dr - G(r)(S- CK+ CRq-4•rrK,RvT2 LvFd ) (4)  5'lingo[1979].The wavelengthrangesfor the fivebands are given by Table 1; the model includes parameterizations for the transmissivity of water vapor, carbon dioxide, and cloud droplets and can be run at arbitrary  where r is the radius, S is the supersaturation, t is the  sum of separate contributionsfrom the dropletsand gas  bands.  [Bott et al., 1990]:  For a closedparcel exposedto net flux divergencethe conservationequations for energy and water are  time, and the Kelvin and Raoult terms (CK, Ca) are verticalresolution. Givenmodelvalues forEi+ andF,•- given in the Notation list. We have droppedthe band we can write the net flux divergencein band i as the subscripti from Fd to denotesummationover the five  Table  1.  Band Wavelength Ranges for Roach and  $1ingo[1979] band wavenumber(cm-• ) 1  0-400  2 3 4 5  400-560 560-800 800-1150 1150-2050  A (/•m) 25 17.9  - c• - 25 17.9 12.5 - 8.7  dT dp Pa 1 den Cpln dt= -Lv•dwv -[ RdT (5) p dt dz  dwv_dwl =_4•rp 1/o •dr c•n(r)r2 dr (6) dt  dt  where w is the vertical velocity, Wv and Wl are the vapor  and liquidmixingratios,andCpm- Cpd+ (Wv+ Wl)Cw is the heat capacity of the cloudy air.  AUSTIN  ET AL.:  CONSTRAINTS  ON DROPLET  GROWTH  14,233  Togetherwith the hydrostaticequation,(4)-(6) can section we will fix the cloud layer thickness at 300 m be solvedgiven an initial droplet numberdistribution and let the subsidinginversion evolve to steady state Wv- I gkg-1) and n(r). The supersaturation S can be diagnosed at each abovethe cloudfor dry (inversion moist (Wv5 g kg -1) conditions. Adiabaticcloudsof time step through the definition this thicknesshave large cloud top flux divergencesbeneath a dry inversion and produce a broad range of es(T) es(T)(1+ e/Wv) precipitation rates, apparently modulated by the charIf the updraft velocity and/or the flux divergenceare acter of the small droplet population. For example, steadyon timescaleslongerthan a few seconds,it can aircraft measurements during the First International CloudClimatology Project (ISCCP)RegiOnal be shownthat the supersaturationcalculatedby the in- Satellite  S-  e  1-  P  - 1  (7)  (FIRE) shownodrizzleformation in a 27gtegrationof (4) - (6) relaxesto a quasi-equilibrium value Experiment Seq[Roach,1976;Davies,1985].We usethe approach to 300-m-thick layer with a droplet number concentraof Davies[1985]to derivean expression for Seqin the tion of 150 cm-3, while in adjacentcleanair, peak rain Appendix; in a closedparcel it is determinedpredom- ratesof 5-8 mmday-1 wereobservedin a layerof the inately by the vertical velocity,the net flux divergence same thickness but with droplet concentrations below 50 cm-3 [Austinet al., 1995]. and the integral radius, I: We initialize inversion air at a fixed temperature and  Seq •  al TsW -}- CK -- Cr  a3/G/(1 Paden) dz drops a3 (1 47rpla2•CpmI Paden) dz total (8)  - 47rI• •7  +  vapor mixing ratio at a height of 3 km and let it descend with velocity Dz, where z is the height of the parcel and D, the large-scaledivergence,is set at a fixed value  D - 4 x 10-6 S--1 (a valuetypicalof the large-scale divergenceoff of the Californiacoastin July). The inversiohair warms adiabaticallyand emits and absorbs long wave radiation as it descends;the inversiontem-  whereI - f rn(r)dr, al, a2 and a3 are slowlyvary- perature above cloud top is controlled through the waing functions of the pressureand temperature and rs -  ter vapor mixing ratio, which determinesthe emissivity,  whichdetermines thetimeavailable 1/(47rpla2GI)is the relaxationtime (see,for example, andthe subsidence, Cooper[1989]). The overbarsin (8) representan av- for the parcel to experiencediabatic heating or cooling erage weighted by the integral radius, while the angle as it descends. Radiative fluxes are computed at each bracketsrepresentan averageweightedby the droplet time ste•)usingthe RoachandSlingo[1979]model. The  flux divergence(seethe Appendix).  modelrequires several hoursto evolveun.•ilthe overly-  We show in the Appendix that for typical stratocu- ing, subsidingair is in a quasi-steadystate. Following mulus droplet and aerosoldistributions, Cr is 2 orders this,:gl•eradiativeflux divergence changes very slowly; of magnitude smaller than Ck. With this approxima- for our purposes,we considerthis the steady state ration, and assumingw - 0, (8) can be written in a more diative flux profile. compact form: Figures la and lb showvertical profilesof water vapor mixing ratio, temperature, and liquid water mixing  Seq •  CK  (9)  ratio for the 300-m-thickcloudlayerandan inversion in whichWv- I gkg-1. The total numberconcentration is constantat NT -- 50 mg-1 , and the adiabaticin-  creaseof liquid water with height producesa maximum volume mean radius at cloud top of rvol -- 13.4 pm, whereF - (dEn/dZ)drops/(dEn/dz)total is the fraction where rvol is defined by  •  p-••ZZ totalpla2•CpmT  of the total flux divergencedueto the droplets[Davies, 1985]. We will follow Fukutaand Walter [1970]and use c• - 1, /• - 0.04 in (8)-(10), which produces as • 0.1 /•m, az m 4 pm. As a result, K' is not a strongfunctionof dropletradius,while G increaseswith The fluxes for these thermodynamic profilesare calcuincreasingdroplet size. lated on a grid with variable vertical spacing:Az =  rvol-(47rplNT 3Wl )1/3  2.3.  Radiative  Fluxes:  Cloud  Flux  Profiles  (10)  2.5 m in the 10 m below the inversion, followed by two  layerswith 5-m spacing,2 with 10-m spacing,and the The radiativeexchangeEd,i at cloudtop and the fractional absorption F due to droplets depend on the magnitude of the downwellingflux from the inversion,the cloud temperature, and the droplet size distribution. We will estimate these usingthe SLB model, which attempts to establish the interaction between the cloudcapped boundary layer and the overlying air. In this  remainder  set to Az = 20 m.  Figure lc showsthe corresponding valuesof the upward and downward irradiances and the total radia-  tive exchange,Ed, summedover the five bands. The warm, dry inversionproducesa total downwardflux of • 280 W m-2, while the upwardflux at cloudtop is • 320 W m-2; dropletsin this cloudexperiencevalues  14,234  AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH Ed (dashed)  wv (dashed) (gkg'1) 2  4  6  (Wm'2)  w, (dashed) (gkg'1)  8  0  0.1 0.2 0.3 0.4 0.5 0,6 i  i  i  !  i  0  10  20  30  i  b) •  •  //  ,,  o 2•t0 285 290 295 300 Temperature(solid)(K)  0  5  10  15  280  rvo • (solid)(gin)  320  360  400  E + (solid)E' (dotted)  (Wrn'2)  Figure 1. Vertical soundingsfor a 300-m-thicklayer cloudbeneatha quasi-steadystate inversion  (inversion wv - 1 gkg-1). (a) temperature T andvapormixingratiowv;(b)volumemeanradius r•oi (with total numberconcentration NT -- 50 mg-1) andadiabaticliquidwatermixingratio wi; (c) total upwardanddownward irradiances E+ andE- andthe total radiativeexchange Ed  of the radiativeexchange varyingfrom2 to 35 W m-2  •--T) drops • •drops (11)  through the upper 80 m of the cloud layer. The net flux divergencedEn/dz (not shown)variesfrom 0.1 to  (•z)total •drops q-•gas  2.6 W m -3 over the same 80-m scale.  For the subadiabatic and adiabatic clouds the droplets Increasing the emissivity of the overlying air signifi- account, respectively,for approximately 80% and 87% cantly reducesboth the flux divergenceand the radia- of the net flux divergence;these relative contributions tive exchange. This is shown in Figure 2a, which gives changelittle through the upper 100 m of the cloud. the quasi-steadystate radiative profilesfor a subsiding  inversionwith w• - 5 gkg-x atop the mixedlayerof Figure 1. The peak value of the radiative exchangeis  reducedby roughly50% for this cloud,to 18 W m-2  2.4. Constraints on Seq  At fixed temperatureand pressure,we expectSeqto be maximumin (10) for largeflux divergences and small __  __  at cloud top, while the maximum net flux divergenceis valuesof F, G, and I. Table 2 lists valuesof G and I for a now 1.6 W m-3 at cloudtop (not shown). rangeof drop sizedistributionsspecifiedby the modified Figure 2b shows the impact of droplet removal on gammafunctionof Berry and Reinhardt[1974a]' the radiative exchange. The liquid water content has been reduced by 50% at each cloud level by remov(12) n(rj) = ing droplets, consistentwith, for instance, precipitation scavenging. The lower cloud emissivity reduces the downwellingirradiance and increasesthe radiative  (l+y)(l+v) (r?) v[  NT•vo••' • • rvol _-•--exp--(1+ rvol 3]  exchange near cloudtop; Ed remainsabove5 W m-2  through the upper 100 m of the cloud. The 50% re- where rj is the radius of size classj, NT is the total duction in cloud water also halves the flux divergence number mixing ratio, rvol is the volume mean radius,  through(3), with a new cloudtop flux divergence of 1.3 and y is a parameter related to the dispersion of the Wrn -•.  number  distribution.  Table  2 also lists two other mea-  In Figure 2c, the fraction of the coolingdue to droplet suresof the distributionmoments,rg and rb, whichwill emissionF is shown for the adiabatic layer of Figure lc be discussed in Section 3. Observationsfrom the FIRE indicate that the droplet and the subadiabatic layer of Figure 2b. It is estimated spectrum below 23 /•m radius can be fit to distribuusing(1):  AUSTIN ET AL.' CONSTRAINTS Ed (dashed) 10  20  GROWTH  14,235  Ed (dashed)  (Wm'2) 0  ON DROPLET  (Wm'2) 30  0  10  20  30  subadiab.  ..........  280 320 360 400  280 320 360 4•10  E + (solid) E'(dotted)  E + (solid) E'(dotted)  (Wm'2)  (Wm'2)  0  0.2  adiab  0.4  0.6  0.8  1  F  Figure 2. (a) Asin Figurelc but fora moistinversion (Wv- 5 gkg-1). (b) Asin Figurelc but for a cloudwith awl profilethat is 50% of adiabatic. (c) FractionF of the net flux divergence due to droplet cooling.  Although I, F, and G will vary with local changes tions of the form of (13) with valuesof v rangingfrom 0.2 _<v _<2.2 [Austinet al., 1995]. The parameterF is in the droplet distribution,other coefficientsin (8) rerelatedto the dispersion(fr/Y) wherefr is the standard main approximately constant over a broad range of temperatures, pressures,and drop size distributions. deviationof n(r) and Y is the mean radiusby The thermodynamic coefficientsal, a2, and a3 vary by less than 15% in the temperature and pressure range 273 K < T < 293 K, 1000 hPa < p < 800 hPa. The  (13)  7.14(1 +  microphysicalcoefficientsCr, Ck, K • dependweakly on the drop size distribution, but can be consideredconstant for the range of distributions given by Table 2. Table 3 shows values for each of these parameters for a cloud top temperature and pressureof T=283.5 K, p=954.5 hPa. The equilibrium supersaturationis held within a com1 <_rj <_25/•m. Substituting the otherdistributions of paratively narrow rangefor the coefficientvaluesof TaTable 2 (while keepingthe dry inversion,300-m cloud bles 2 and 3. The secondterm in (10) remainsgreater thickness,and adiabatic liquid water profile) changes than zerofor all valuesof G, sothat the minimumSeqis set by Ck -- 10-4. To computean upperboundon Seq the cloudtop coolingrate by lessthan 20%.  Thus 0.2 _<• <_2.2 is equivalentto 0.2 <_fr/• • 0.35, in approximateagreementwith dispersionvaluesfound' in North Atlantic stratocumulus by $lingoet al. [1982]. The radiative profilesof Figures1-2 were computedfor distributionsof the form of (13) with • - 0.3, NT -50 mg-1, and25 dropletsizecategories with rj between  Table 2. GammaDistributionsfor Fixedwi -- 0.5 gkg-1 Case 1 2 3 4 5 6 7  8 9 10 11  Wl  NT  (/zm) (gkg-1)  rvo]  (rag-1)  v  Err/•  rg  10 10 10 11  0.5 0.5 0.5 0.5  119.36 119.36 119.36 89.68  0.45 0.30 0.00 0.45  0.31 0.33 0.37 0.31  11.91 12.09 12.60 13.10  11 11 12 12 12 14 16  0.5 0.5 0.5 0.5 0.5 0.5 0.5  89.68 89.68 69.08 69.08 69.08 43.50 29.14  0.30 0.00 0.45 0.30 0.00 0.45 0.45  0.33 0.37 0.31 0.33 0.37 0.31 0.31  13.30 13.86 14.29 14.51 15.48 16.67 19.06  G  I  rb  (•um)  (mkg-1)  (/tin2  1103.8 1093.9  76.4  9.57 10.00  1065.9  76.1  10.34 10.53 11.00  912.2 904.1  77.5 77.4  880.9  11.28  766.6 759.7 740.2  77.2 78.4  9.40  11.49 12.00 13.16 15.03  563.2 431.1  76.3  78.3 78.1 79.9 81.0  14,236  AUSTIN ET AL.- CONSTRAINTS ON DROPLET GROWTH  S (x 105)(dashed)  Table 3. CoefficientValuesfor (8) Coefficient  Value  T, K p, hPa  283.5 954.5  al, m-1  10  12 ,.  14 I .  16 I  18 I  20 I  5.2 x 10--4  a2  283.1  as, j-1 K kg pl, kgm-3  0.067 1000  Ck C--•  1.0 X 10-4 9.7 x 10-7  Cpm,J kg- 1K- 1 •, K J (msK)- 1 pa, kgm-3 r.,, s  1020 0.024 1.17 4.6  v  using the conditionssuggestedby Figures 1 and 2, we  take F - 0.87, (dEn/dz)total=10 K hr-1 (3.3 W m-3) and the drop size distribution of Case 11. This yields  Seq- 3.7 x 10-4. The equilibrium supersaturation de-  6  5  2'o  2'5  3'0  radii (solid) (gin)  creasesas NT increases'inserting (I, G) for Case 1, Table 2 halvesthe radiativecontributionto Seqand re-  Figure 3. Evolution of an adiabatic drop sizedistribution 40 m below cloud top. Solid lines showthe droplet ducesit to S•q = 2.2 x 10-4. Theselow S•qvalues radii for size classes1-25 as a function of time and parpreclude any significantradiatively-inducedaerosolac- cel saturation pressure. The dashed and dotted lines showthe equilibriumsupersaturationcalculatedby (8) tivation at the tops of the cloud layers presentedhere; (dashedline) and (10) (dottedline). roughly 97% of the CCN distribution describedin the Appendix require supersaturationslarger than 0.037% for activation.  2.5.  Condensation  Growth  Calculations  The equilibrium values of the supersaturationfor the examples given above are small enough so that in the absenceof upward motion, droplets with radii lessthan m 5/•m evaporate throughout much of the upper part of the cloudsshownin Figures 1-2. Figure 3 showsthe  with a growth rate that varieslinearly with Ed for fixed droplet size. The supersaturationdecreasesduring the cooling,as larger droplets increase the distribution mean radius  (and thereforeincreaseI) by 1.1 /•m from 12.2/•m to 13.3/•m. The flux divergencedecreasesby • 4% over this time period (not shown),as reducedthermal emission is offset by an increase in the droplet absorption  growth historiesof 25 droplet classes(solidlines)computedby an explicitintegrationof (4)-(6) for conditions taken from Figure 1. For this casewe locate the drop size distribution 40 m below cloud top, in a parcel of  coefficientndrops.The dashedline on Figure 3 shows Seqgivenby (8); it differsfrom the actual supersatu-  The droplets and vapor produce a combined net flux  saturationterm ("S") and the radiative coolingterm  ration by less than 0.1% over the courseof the integration. Also shownis the approximate value given by adiabatic cloud base air that remains at that height as (10), whichis within 2% of the moreaccuratevalue. Figure 4 showsthe contributionof individual terms it cools.The valueof Ed is 7.3 W m-2 at this height, in the droplet growthequation(4) to the growthrate while the initial drop size distribution has an adiabatic at the beginning of the integration. Both the superliquid water contentof 0.15gkg-1 and rvo1-13.3/•m.  contribute to thegrowthof divergence of 0.93 W m-3, with an initial coolingrate (CE = LvFd/(4•rrK•RvT2)) 5-/•m cloud droplets, while the aerosol mass(term R) (includingthe effectsof phasechange)of-1.6 K hr-1. Over the course of•21 minutes in this environment  the cooling decreasesthe parcel temperature by 0.25 K  has little impact on the threshold size at which drops beginto evaporate. Dropletssmallerthan 3/•m will not  and lowersthe parcel saturationlevel (the pressureat evaporateprovidedS•q > 0.03%, whichwouldbe progreaterthan • 25 W m-2 whichthe parcelwouldbe exactlysaturated)by approx- ducedby a radiativeexchange for this drop size distribution. From Table 3 and (8), imately 10 hPa. The solid lines in Figure 3 show the S•q > 0.03% would also be produced by updraftsat growth historiesof individual droplet categoriesduring cloud top larger than m 0.1 ms -1. this time. The critical radius and supersaturation for  this aerosolmassare 1.48 /•m and 5 x 10-4 respectively, so that the smallest droplets assume their un-  3. Coalescence With Radiative Cooling  activatedequilibriumradii, whiledropletssmallerthan 5/•m evaporate until they deactivate. Droplets larger  As the value of the radiative exchangeincreasesbe-  than 10/•m growat between0.10 and 0.14/•m min-1,  yond10W m-2, the radiatively drivengrowthratesof  AUSTIN  -1  0  -0.5  0.5  ET AL.- CONSTRAINTS  I  1.5  2  o_  ON DROPLET  GROWTH  14,237  sensitive to size increases in these droplet categories, because the collision efficiency increasesfrom 0.02 to 0.2 as droplets grow through this 5-/zm radius range  [Jonas,1972]. Even modestcondensationgrowth can have a rate-determining impact on the early stagesof coalescenceif it acceleratesthis droplet growth. In this sectionwe will calculate coalescence growth for a sedimentingdroplet population in a regionof constant radiative exchangeand in an updraft with a constant vertical velocity. Berry and Reinhardt[1974b]showed that a representative measure of the progressof coalescencein an evolving droplet population is given by the massmean, or "predominant"radius, rg. This is defined by  /  o-  15 volume cjrowth rate(gin3s-1)  xg= / x2n(x)dx//xn(x)dx rg -- (47rpl (Xg) (1/3) 3)(1/3)  (14) Figure 4. Volumegrowthrate 47rr2dr/dtas a function of radiusfor a radiativeexchange Ed-- 7.5 W m-2 andan equilibrium supersaturation of 1.5 x 10-4. The wherex is the dropletmass. Valuesof rg for precipi-  symbolsdesignatethe termsfrom (4): S = S, K = CK, tating stratocumulusobservedduringFIRE rangefrom R = Ca, E=cooling term, T = S + K + R + E  the largerdropletsexceedthoseproducedby typical updrafts in stratocumulusclouds. Figure 5 showsthe individual terms of (4) (givena total radiativeexchangeof  45 < rg < 75 /zm for midcloudrain ratesof 1 - 10 mmday- 1. We will useTg, the time requiredfor rg to growto 50/zm, as a measureof the rate of spectralevolutionof an initial drop size distribution. Berry and Reinhardt  [1974b]foundthat for a coalescing dropletpopulation Ed = 17.5W m-2) compared with the supersaturation(withoutcoolingor supersaturation growth),1/Tg was term assumingS=10-3 (generated,for instance,by a linearly related to the liquid water contentof the initial 0.35 m s-1 updraft actingon the initial drop sizedistri- distribution and to rb, a measure of the initial norrnalbution of Figure 3). Radiationhas a substantialimpact ized mass variance: on the growth of the larger droplets;it is the droplets in the 20-25 /zm size range that determine the initial coalescence growth rate of the condensation-produced nvar x = -droplet size distribution. Coalescenceis particularly  ((x2)(x)2) --1/(1 +.)  r} --  (15)  nvar-- rvøl 1q-F  rvøl  where the angle brackets define an averageover the mass distribution.  We calculatethe evolutionof rg for the initial distributions of Table 2, integrating the stochastic collection equation using the algorithm and the kernel of  Berry andReinhardt[1974a](with collectionefficiencies taken from Hockingand Jonas[1970]and $hafrir and Neiburger[1963]). Coalescence growthis calculatedusing 60 logarithmicallyspacedradiusbinsbetween2 and 1800/•m radius, and for each2-s time stepwe alsocompute condensationgrowth usingthe advectionequation:  Ot volume growth rate(gm3 s'1)  cond Or •  We solve(17) usingthe semi-Lagrangian advection scheme of Bott et al. [1990] and the droplet growth Figure 5. As in Figure 4 but for Ea- 17.5 Wm -2. equation (4) with S = CK = Ca = 0 (for the coolLine "S" showsthe growth rate due to a supersaturation ing cases),and S = Seq,CK : Ca = CE : 0 (for the of S- 1. x 10-3 for comparison.  14,238  AUSTIN  ET AL-  CONSTRAINTS  ON DROPLET  GROWTH  updraft cases).The pressureand temperatureare fixed a factor of 21. The initial growth rate of the predomivalues in Table 3. For this calculation we make nant radiusis independentof rg andproportionalto the  to their  radiative exchange, varying for each distribution from  the approximation that 5  Fd-- 7rr 2• Qa(r, i)4Ed,i • 7rr2•aa(r)4Ed (18) i=l  where the average absorption efficiency, Qa is taken  from Roach[1976]'  0.05 •ummin-1 for a radiativeexchange of 3.75 W m-2 to 0.3 •ummin-• for Ea=25 W m-2. For the smaller rvo•distributionsat valuesof Ea lessthan 15 W m-2, there is a transitionfor 14 < rg < 16 •um,in whichthe growth rate increasesby up to an order of magnitude. This change is due to rapidly increasing collection by droplets with radii greater than 30 •um (and collection  efficiencies approachingunity).  Qa - 1.18(1- exp(-0.28r))  (19)  In Figure7 we plot rg versus1/Tg for fourradiative  exchangesand three valuesof the updraft velocity,with with r given in microns. Although this approximation the initial distributions of Table 2. The rate of spectral overestimatesFd by 5-30% for droplets with radii be- evolution varies linearly with the value of the radiatween 5 and 15/•m, the errors decreasewith increasing tive exchangefor these initial distributions. The w = 0 drop size, and are below 4% for drops larger than 20 curve givesa baselineagainst which the impact of cooling and ascentcan be compared:a radiative exchangeof Figure 6 showsthe time evolutionof rg for four dif- 15 W m-2 or an updraft of 0.5 ms-• producesratesof ferent values of the radiative exchangeEd (3.75, 7.5, precipitationdevelopment(givenrvol----10/•m) equiva-  15, 25) W m-2 andthreedifferentinitial dropsizedis- lent to those found in a quiescentcloud with a distribu-  tributions (Cases1, 4, and 7 in Table 2). Also shown tion rvoi • 15/•m. The time required for the distribufor Case I is the evolution with neither cooling or vertion to evolveto rg = 50 /•m decreases from 81 min for tical ascent("n") and with a constantvertical velocity CaseI (no cooling)to 20.7min with Ed -- 25 W m-2.  of w - 0.5 ms-1 (initialSeq-- a•rsW- 8.3x 10-4).  Figure 6 showsthat coalescencewithout cooling or  Comparisonof the slopesof the curve families in Figure 7 shows different  sensitivities  to the initial  distribu-  ascent (line "n") proceedsvery slowlyfor this kernel, tions for coalescenceaided by supersaturation or radiawith the predominantradius rg growingat roughly tive cooling. radiatively driven droplet growth is inde6 x 10-3 /•mmin-•. Placingthe parcelin a steady0.5 pendentof dropletradiusonceQa(r) reachesits asympm s-• updraft (line "w") increases this growthrate by totic limit (at r > 15 /•m). In contrast, saturationdriven growth for 20 /•m droplets in an updraft decreasesas the dropletsgrow,becauseof the 1/r depenrg(pm) 12  14  16  18  I  I  I  I  -0  I  I  I  I  12  14  16  18  % Figure 6. Stochastic coalescencewith radiative cool-  I  ing: predominantrg versustime for threedifferentdrop sizedistributions(Cases1, 4, and 7) and four valuesof  rb (gm)  the radiativeexchange Ed = 3.75, 7.5, 15, 25 W m-2. Solid lines are rvoi =  10/•m.  Dotted lines are rvol =  11/•m. Dashedlinesare rvoi= 12/•m. Alsoshownare growth curveswith neither adiabatic ascentor radiative  cooling ("n") andin a constant updraftofw = 0.5ms-• ("w") without radiation.  Figure 7. Time Tg requiredto reach rg -- 50 /•m for the initial  distributions  of Table  2. Lines are least  square fits through the cases,labeled by the appropriate value of the total radiative exchangeEd or updraft velocity w.  AUSTIN ET AL'  CONSTRAINTS  ON DROPLET  GROWTH  14,239  dencein (4) and becauseof the decrease in Seqas the by • 1 •um. This is equivalent to a similar period spent integralradius increasesin (8). in an updraft of 0.2 - 0.25 m s-•. For the initial distribution with rvo• - 12 •um in Figure 6, this I •um increaserepresentsa significantportion of the growth neededto move to the more rapid stage of coalescence We have modeled radiatively driven droplet growth at rg > 16 •um. An equivalenteffectcan be produced near cloud top for a 300-m-thick stratocumulus cloud by higher values of the radiative exchangeand proporbeneath moist and dry inversions. Our results indicate tionally shorter residencetimes in the upper part of the 4.  Discussion  an upper bound on the radiatively-induced supersatu-  ration of lessthan 0.04%. Equation (10) suggests that this maximum might increase in thicker clouds with a larger total flux divergence,in cloudswith lower values of I, or in layers in which vapor absorption constituted a larger fraction of the total layer absorption. Countering such an increaseis the coupling of the integral radius  layer. A lower bound on the cloud top residencetime can  be found by assumingthat the mixed layer circulation is organized, consistent with balloon and aircraft mea-  surementsof nocturnalclouds[Caugheyand Kitchen, 1984;Nicholls,1989]. Theseobservations indicatethat air parcels rise to cloud top, cool by 0.1-0.2 K, and then  and the flux divergence through(3); in the absenceof descendin downdrafts spaced100-150 m apart. Given a precipitation, an increasein dEn/dz due to increased convective velocityscalew. • 0.5 ms-• (inferredfrom cloudliquid water content(and the resultingincrease either the observations or our mixed layer model), this in •;drops) will be offsetin (10) by the correspondingimplies a residencetime at cloud top of 3-5 min, a flux increase in I. divergence of 1.6 W m-3 (or Ed • 15 W m-2), and an If precipitation is considered,observationssuggest rg increaseof • I •umin a parcelcompletingthis circua second limit, as clean marine clouds with low total droplet concentrationsproduce drizzle, which removes  lation.  value of the accommodation  mixed parcels, with reduced number concentrations, will also experiencecomparatively large equilibrium supersaturations in cloud-top updrafts. A parcel with an  That residencetime could be significantly extended liquid water and reducesthe emissivity(and flux diver- by the entrainment of inversion air, since the cloudgence)of the cloud. Measurements from FIRE suggest top entrainmentinstabilitycriterion[Randall,1980]is precipitation scavengingsufficient to halve the cloud not met for either the moist or dry inversionsof Figliquid water path in less than 25 min in 250- to 300- ure I and Figure 2. Mixtures of inversionand cloud air m-thickcloudswith I m 650 m kg-• (NT -- 50 mg-•, will be more buoyant than surrounding cloud and will rvo•-- 13 •um)[Austinet al., 1995].This leavesa reduc- require more cooling to produce the 0.1-0.2 K tempertion in F as the third possibility for increasedvalues of ature deficit of the descendingplumes. As Figure 2b Seq,but the resultsof Section2.3 indicate that lower indicates, removal of cloud water has little impact on values of F occur in subadiabatic layers, with reduced the value of the radiative exchange, and droplets that liquid water paths and correspondinglylower valuesof survive mixing with inversion air will grow as rapidly dEn/dz. as their neighborsin adiabatic cloud, while the reduced One sourceof uncertainty in thesecalculationsis the emissivity will lower the parcel cooling rate. These and condensation  coeffi-  cients. Growth measurementsof water droplets held in  electrodynamic balance[Sageevet al., 1986]seemto indicate a • I as used above. There is less support for the usual choice of/3 - 0.04; recent measurementsof  integralradiusof I - 200 m kg-•, experiencing a ver-  tical velocityof 0.1 - 0.2 ms-• will produceSeqfrom  /3 [Hagenet al., 1989]showvaluesthat vary logarith- 0.1% - 0.2% from (8); from Figure 5 this would more mically with droplet size for droplets grown in a fast expansion chamber, with /• decreasingfrom I to 0.01 as the droplet radius increasesfrom I to 15 •um. Hagen et al. suggestthat this variation is related to increasing concentrationsof surfacecontaminantson the older  than double the growth rate for a 20-•umdrop in a ra-  diativeexchangeof 17.5 W m-2. A more complete treatment of the radiative contribution to spectral broadeningshould considercorrelations between perturbations in the radiative exchange, the  (and larger) drops. If we usetheir average/3for aged verticalvelocity,and the integralradius. Cooper[1989] droplets(/3 - 0.01 + 7%) and recomputeG we find a hasshownthat givenhigh (and uncorrelated)variability 30% reduction in G for the conditions of Table 3. This  in I and w, mixing betweenparcelswith different histories could have a substantial broadeningeffect on mean 30-40%. droplet spectra. Fluctuations in Ed could act as another The stochastic coalescencecalculations presented in sourceof variability in I, as droplets residing in the upSection 3 show a linear relationship between the radia- per 10-20 m of the cloud are exposed to significantly tive exchange,Ea, and the growth rate of the predomi- different values of the radiative exchange.These radianant radius for fixed liquid water content. The growth tive exchangefluctuations would be particularly large rates shownin Figure 6 suggestthat 10 min exposureto for entrained parcels with low I and a few large, sura radiativeexchangeof Ea - 7.5 W m• wouldincrease viving droplets, becausethe removal of overlying cloud the predominant radius of the distributions of Table 2 would reduce the downward flux and increase Ed. We  would increasethe radiative contributionto Seqby •  14,240  AUSTIN ET AL.' CONSTRAINTS ON DROPLET GROWTH  have also neglected the impact of supersaturationfluctuations at the scale of individual cloud droplets. Our  When we substitute (5) into (A1) we obtain for the supersaturation dS  Seqis the traditional far-field supersaturation,which may differ significantlyfrom the value near the droplet surface. Srivastava[1989]has shownthat this kind of supersaturation variability also has the potential to significantly broaden the drop size distribution. We plan to calculate the cumulative impact of turbulence, entrainment, and radiative cooling on large droplets cycling through cloud using large eddy simulations. Recent work with a simple one-dimensional turbulence model, however, does suggestthat condensation growth may play a significantrole in the initia-  •  dt  alw-- a24•rplI (•S - GCk+ GCr)  (G>(1 den) + a2pla3 •7 Pa dz drops  (A5)  + Cpm a3(Pa1den) dz total Solving(A5)'using a multiplyingfactor yields(8). We require an aerosol distribution to evaluate Cr and  will make the simplifying assumptionthat each droplet has formed on an identical aerosolparticle, consisting  tion of stratocumulusprecipitation[Austinet at., 1995]. of ammonium bisulphate with a dry aerosol diameter of The results presented here indicate that radiative cooling has a similar potential, given a dry inversionand a cloud with peak rvol > 10/•m.  0.2 •m. Observationsindicate that sulphate is the principal constituent of the remote marine aerosol;observed massdistributionscan be fit to a lognormaldistribution with a geometric mean diameter of 0.2/•m and a geo-  Appendix- The Equilibrium Supersaturation  metric standard deviation of 1.7 [Ctarke et at., 1987; Twohy et at., 1989]. Particleswith dry diametersless than 0.2 •m constitute half of the total aerosol mass  A prognosticequation for the supersaturationcan be  and 95% of the total aerosol number available as cloud  found by differentiatingits definition[Pruppacherand condensation nuclei for this choice of distribution parameters. As the Cr values of Table 3 and Figure 4 Ktett, 1978]' indicate, our results are not sensitive to the choice of  dS [gw LvdT-}  pRw dwv  aerosol  size.  (A1)  The result givenby (8) differsslightlyfrom that presentedin equation (20) of Davies [1985]. To permit a  where we have made the approximation(l+S)=l. To obtain (8), we beginby inserting(4) into (6) and integrate over the drop size distribution:  term by term comparisonwith his expression,we define  dt  RdT  RwT 2 dt  esRd dt  a new averagefor the ratio {Kt/D t} that satisfies  ø•n(r)Fd (r)dr  dwv  -- 4•pl [ISm- IGCk-}-IGCrJ  dt  (A6)  [TR,• K' + L2• ]  Lv G/ /•I den] (h2) - PlRvT 2/ •7 dz7drops  dr  where we have used (3) to define the flux divergence With this definition and using the expressionfor F  due to droplets.  The overbarsin (A2) denotean averageweightedby the integral radius, while the angle bracketsdenote an average weighted by the droplet cooling:  xn(r)rdr X--  oo  givenin (10), (8) becornes  Seq •  xn(r)Fd(r)dr , <x)-  oo  (A3)  /0 n(r)rdr /0n(r)Fd(r)dr For the distributions of Table 2 the use of separate  integral radius-weightedaveragesfor the coefficients in  (A2) introduceserrorsin eachterm of lessthan 2%: GCk  •  GCk  GCr  •  G Cr  •7  •  Kt  (A4)  -  Ck--Cr  ([(4•rI•) a4•7  (A7)  Padz total  2 {Kt X [Fpacpm - (1- F)R•T L2vps  Writing Davies (20) in our notation  Seq,Davies • Ck -- Cr  ( X [Fpacpm-a5(1F)psLv-  AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH  There are two offsettingapproximations in (A8). The F extra 1/T term that appearsin both the numeratorand Fd,i denominatorof (A8) (and is alsofound in the droplet growthequationusedby Roach[1976])arisesfrom the G(r) assumption that the vapor flux toward the droplet can be written in terms of the vapor density gradient V ps. This is strictly true only under isothermal conditions.  If the gradient V(nv/n) (where nv and n are the vapor number density and the total number density of  14,241  fraction of the flux divergencedue to droplets. net power emitted by droplet in band i. TRy  L•  p•-i D_r•e, _[ - K•R• T2  ß  K  thermal conductivity of air.  K•  f•K [Fukutaand Walter,1970].  I  integralradius,I- f n(r)rdr.  Lv  latent heat of vaporization.  M  molecular weight of aerosol.  moist air) is usedinstead,the droplet growth equation takesthe form of (4), whichis the sameas that usedby Q•(i, r) absorption eificiencyin band i for droplet of radius r. Cooper[1989]or Srivastava[1989](W. A. Cooper,per- m aerosol mass. sonal communication,1993). The 5% decreasein the  bracketedterm in (A8/ causedby the additionof the 1/T term is verynearlyoffsetby the approximationthat K•/D • • •K•/D•). As a result, (A7) and (A8) agreeto within 1-2% for the droplet distributions of Table 2. Notation a•  accommodationlength,  a•  K(2•rRMdT)•/2/(ap(Cv + R/2)). condensation length,(2•r/(RvT))i/2D//•.  al a2 a3  T  cpRvT  Ra  '  (pa/ps)+ (L•/(RvcpmT•)). Lv/(evT2pa).  a4  Lv/(CpmPaRv T2 + L•ps).  a5  (Lv/(RvT•)) - (l/T). a•/(Cp•pa + Lvpsa•).  a6  Bi  Planck  function  in band i.  CE CK  coolingterm, LvFd(r)/(4•rK•RvT2). Kelvin term, 2fs/(p•rRvT).  Ca  Raoultterm,vmM/(W((4•/3)p•r3- m)).  Cv  molar heat capacity for water vapor.  Cpd  specific heat at constant pressurefor dry air.  Cw  specific heat for liquid water.  Cpm  specific heat at constant pressure for cloud parcel.  D  diffusivity of water vapor in air.  D•  f•D [Fukutaand Walter,1970].  e  vapor  es  saturation vapor pressure.  Ed,i  radiative exchangein band i.  En,i  net upward irradiance in band i.  pressure.  molecular weight of dry air. NT  total number mixing ratio.  nv  vapor number density.  n  total number density of moist air.  p  total pressure.  r  droplet radius.  rb  normalized  rg  predominant radius.  rvol  volume  mass variance  mean  radius.  radius.  R  universal gas constant.  Rv  gas constant for water vapor.  S  supersaturatione/es- 1.  Seq  equilibrium supersaturation.  Sv  dry virtual static energy.  T  temperature.  T•  time requiredfor coalescence to producerg -50 •m. Van't  Hoff factor  w  vertical velocity.  w1  liquid water mixing ratio.  Wv  water vapor mixing ratio.  molecular weight of water. Acknowledgments.  We would like to thank W. A.  Cooperfor pointingout the consequences of isothermalvapor diffusiondiscussedin the Appendix, and two anonymous reviewers for helpful suggestions. This article was typeset in DTF_• using the AGUTEX JGR style and Patrick  Daly's AGU++ article class. The researchwassupported by grants from the National Research Council of Canada, the Atmospheric Environment Service, and NOAA Grant NA37RJ0203.  upward irradiance in band i. downward  irradiance  in band i.  References  r/(r + a•).  r/(r + a•). surface  tension.  Austin, P., Y. Wang, R. Pincus, and V. Kujala, Precipitation in stratocumulus clouds: observationaland modeling results, J. Atmos. $ci., 52, 2329-2352, 1995.  14,242  AUSTIN ET AL.: CONSTRAINTS  ON DROPLET GROWTH  Baker, M. B., Variability in concentrations of cloud condensation nuclei in the marine cloudtopped boundary layer, Tellus, Set. B, J5, 458-472, 1993.  Nicholls, S., A model of drizzle growth in warm, turbulent, stratiform clouds, Q. J. R. Meteorol. $oc., 113, 1141-  Barkstrom, B. R., Some effectsof 8-12 /•m radiant energy transfer on the mass and heat budgets of cloud droplets, J. Atmos. $ci., 35, 665-673, 1978.  Nicholls, S., The structure of radiatively driven convection in stratocumulus, Q. J. R. Meteorol. $oc., 115, 487-512,  1170, 1987.  1989.  Berry,E. X., andR. L. Reinhardt,An analysisof clouddrop Pruppacher, H. R., and J. D. Klett, Microphysics of Clouds and Precipitation. D. Reidel, p. 355, Norwell, Mass., 1978. growth by collection,I, Double distributions,J. Atmos. Randall, D. A., Conditional instability of the first kind $ci., 31, 1814-1824, 1974a. upside-down, J. Atmos. $ci., J1, 402-413, 1980. Berry, E. X., and R. L. Reinhardt, An analysisof cloud drop growthby collection,II, Singleinitial distributions, Roach, W. T., On the effect of radiative exchange on the growth by condensationof a cloud or fog droplet, Q. J. J. Atmos. $ci., 31, 1825-1831, 1974b. R. Meteorol. $oc., 102, 361-372, 1976. Bott, A., U. Sievers,and W. Zdunkowski, A radiationfog Roach, W. T., and A. Slingo, A high resolution infrared model with a detailed treatment of the interaction beradiative transfer schemeto study the interaction of raditween radiative transfer and fog microphysics,J. Atmos. ation with cloud, Q. J. R. Meteorol. $oc., 105, 603-614, Sci., J7, 2153-2166, 1990. 1979. Caughey,S. J., and C. K. Kitchen, Simultaneousmeasurementsof the turbulent and microphysicalstructureof noc- Sageev, G., R. E. Flagan, J. H. Seinfield, and S. Arnold, Condensation rate of water on aqueous droplets in the turnal stratocumuluscloud, Q. J. R. Meteorol.$oc., 110, 13-34, 1984. transition regime, J. Colloid Interface $ci., 113, 421-429, 1986. Clarke, A.D., N. C. Alquist, and D. S. Covert, The Pacificmarineaerosol:Evidencefor naturalacidsulfates,J. Shafrir, U., and M. Neiburger, Collision efficienciesof two spheresfalling in a viscousmedium, J. Geophys.Res., 68, Geophys.Res., 92, 4179-4190, 1987.  4141-4147, 1963. Cooper,W. A., Effectsof variabledropletgrowthhistories on dropletsizedistributions,I, theory,J. Atmos.$ci., J6, Siems, S. T., D. H. Lenschow,and C. S. Bretherton, A nu-  1301-1311, 1989.  Davies, R., Responseof cloud supersaturationto radiative forcing, J. Atmos. $ci., J2, 2820-2825, 1985. Davies, R., and A. R. Alves, Flux divergenceof thermal radiation within stratifiormclouds,J. Geophys.Res., 9•,  merical study of the interaction between stratocumulus and the air overlying it, J. Atmos. $ci., 50, 3663-3676, 1993.  Slingo, A., S. Nicholls, and J. Schmetz, Aircraft observations of marine straocumulusduring JASIN, Q. J. R. Meteorol. $oc., 108, 833-838, 1982. 16,277-16,286, 1989. Fukuta, N., and L. A. Walter, Kinetics of hydrometeor Srivastava, R. C., Growth of cloud drops by condensation, J. Atmos. $ci., j6, 869-887, 1989. growthfrom a vapor-spherical model,J. Atmos.$ci., 27, Twohy, C. H., P. H. Austin, and R. J. Charlson, Chemi1160-1172, 1970. cal consequencesof the initial diffusional growth of cloud Hagen, D. E., J. Schmitt, M. Trueblood,J. Carstens,D. R. droplets I: Clean marine case., Tellus, Set. B, J1, 51-60, White, and D. J. Alofs, Condensationcoefficientmeasure-  ment for water in the UMR cloudsimulationchamber,J. Atmos. $ci., J6, 803-816, 1989. Hocking,L. M., and P. R. Jonas,The collisionefficiency of small drops, Q. J. R. Meteorol.$oc., 96, 722-729, 1970. Jonas,P., and B. J. Mason,The evolutionof dropletspectra  by conde•-•ktion andcoalescence in cumulus clouds,Q. J.  Roy.Meal$oc.,100,286-295, 1974.  Jonas,P. R., The collisionefficiencyof small drops, Q. J. R. Meteorol. $oc., 98, 681-682, 1972. Kovetz, A., and B. Olund, The effect of coalescenceand  1989.  P. H. Austin and Y. Wang, Atmospheric SciencesProgramme, •217 Geography, 1984 West Mall, University of British Columbia, Vancouver, B.C. V6T 1Z2 CANADA  (email: phil@geog.ubc.ca; yinong@geog.ubc.ca) S.  Siems,  Department  of  Mathematics,  condenastion on rain formation in a cloud of finite vertical  extent, J. Atmos. $ci., 26, 1060-1065, 1969. Nicholls, S., The dynamics of stratocumulus: Aircraft ob-  servationsand comparisonswith mixed layer models, Q. J. R. Meteorol. $oc., 110, 783-820, 1984.  Monash  University, Clayton, Victoria, 3168 Australia (email: siems@cyclone.maths.monash.edu.au)  (ReceivedJuly 12, 1994;revisedApril 5, 1995; acceptedApril 5, 1995.)  


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