A Condensation-Coalescence Cloud Model for Exoplanetary Atmospheres: Formulation and Test Applications to Terrestrial and Jovian Clouds

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by  Kazumasa Ohno

DraftversionJanuary5,2017
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ACONDENSATION–COALESCENCECLOUDMODELFOREXOPLANETARYATMOSPHERES:FORMULATION
ANDTESTAPPLICATIONSTOTERRESTRIALANDJOVIANCLOUDS
KazumasaOhnoandSatoshiOkuzumi
DepartmentofEarthandPlanetarySciences,TokyoInstituteofTechnology,Meguro,Tokyo,152-8551,Japan
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1 ABSTRACT
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A number of transiting exoplanets have featureless transmission spectra that might suggest the presence of
2
cloudsathighaltitudes. Arealisticcloudmodelisnecessarytounderstandtheatmosphericconditionsunder
n
whichsuchhigh-altitudecloudscanform. Inthisstudy,wepresentanewcloudmodelthattakesintoaccount
a
J the microphysicsof bothcondensationand coalescence. Our modelprovidesthe verticalprofilesof the size

4 anddensityofcloudandrainparticlesinanupdraftforagivensetofphysicalparameters,includingtheupdraft
velocityandthe numberdensity of cloudcondensationnuclei(CCN). We testour modelby comparingwith

]
observations of trade-wind cumuli on the Earth and ammonia ice clouds in Jupiter. For trade-wind cumuli,
P
E themodelincludingbothcondensationandcoalescencegivespredictionsthatareconsistentwithobservations,
. whilethemodelincludingonlycondensationoverestimatesthemassdensityofclouddropletsbyuptoanorder
h
of magnitude. For Jovian ammonia clouds, the condensation–coalescencemodel simultaneously reproduces
p
- the effective particle radius, cloud optical thickness, and cloud geometric thickness inferred from Voyager
o
observations if the updraft velocity and CCN number density are taken to be consistent with the results of
r
t moist convection simulations and Galileo probe measurements, respectively. These results suggest that the
s
a coalescence of condensate particles is important not only in terrestrial water clouds but also in Jovian ice
[ clouds. Our model will be useful to understand how the dynamics, compositions, and nucleation processes

1 inexoplanetaryatmospheresaffectstheverticalextentandopticalthicknessofexoplanetarycloudsviacloud
v microphysics.
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Keywords:Earth–planetsandsatellites: atmospheres–planetsandsatellites: individual(Jupiter)
1
9
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1. INTRODUCTION A realistic cloud model that predicts the size and spatial
0
1. Recent observations of the transmission spectra of exo- distributions of condensation particles for arbitrary atmo-
sphericconditionsisnecessarytounderstandtheatmospheric
0 planets revealed that some hot Jupiters (e.g., Pontetal.
7 2013; Singetal. 2015, 2016), hot Neptunes (e.g., propertiesofbothexoplanetsandbrowndwarfs.
1 The microphysicsthat governsthe formationof cloudsis
Crossfieldetal. 2013; Ehrenreichetal. 2014; Knutsonetal.
:
v 2014a; Dragomiretal. 2015; Stevensonetal. 2016), and highlycomplex,andthereareatleasttwoprocessesbywhich
i cloud particlescangrow. The firstprocessis the condensa-
X super-Earth (e.g., Beanetal. 2010; Kreidbergetal. 2014;
tion of vapor onto particles in an adiabatically cooling up-
Knutsonetal. 2014b) have featureless spectra. One inter-
r
a pretation of the featureless spectra is that these exoplanets draft. In terrestrialwater clouds and Jovian ice clouds, this
processisresponsibleforthegrowthofsmallparticlesto10
have dust clouds that block starlight at high altitudes (e.g.,
µminradius(e.g.,Rossow1978). Furthergrowthofthepar-
Seager&Sasselov 2000; Fortney 2005). Dust clouds are
ticles proceedsthroughthe second process, the coalescence
also believed to have the crucial impacts on the observed
drivenbythedifferentialsettlingundergravity. Thissecond
spectra of brown dwarfs whose effective temperature fall
process is essential for the initiation of precipitation in ter-
into the same range of exoplanets (e.g., Saumon&Marley
restrialwaterclouds(Pruppacher&Klett1997).
2008). For example, the observations of brown dwarfs
However,previousmodelsofcloudsinexoplanetsaswell
showabluewardshiftofspectralenergydistributionsduring
asinbrowndwarfsneglectedoratleastparametrizedcoales-
L/T transition that might suggest the sinking of condensate
cence. TheconvectivecloudmodelbyAckerman&Marley
particles (Ackerman&Marley 2001; Burgasseretal. 2002;
(2001), which has been used by Morleyetal. (2013, 2015)
Marleyetal. 2002; Saumon&Marley 2008). Observations
for modeling the transmission spectrum of super-Earth GJ
also suggest spectral variability that might imply the effect
1214b,encapsulatestheeffectsofparticlegrowthduetocon-
ofcloudspatialdistributions(Buenzlietal.2012;Yangetal.
densation and coalescence in a single free parameter f .
2015,2016). sed
This parameter is given by the ratio of the particle termi-
2 Ohno&Okuzumi
nal velocityto the atmosphericconvectivevelocity, and de-
pendsontheparticlesizethroughtheterminalvelocity. Itis
commonlyassumed that f is constantthroughouta cloud
sed
(Ackerman&Marley 2001; Morleyetal. 2013, 2015), but at w–v(r) =0,
t
thereisnoguaranteethatitmustbeforarbitraryconvective Cloud to Rain Conversion
clouds. The recent cloud model for Earth-like exoplanets
by Zsometal. (2012) treats the microphysics of condensa-
Cloud Particle Rain Particle
tion, butgreatly simplifies the coalescence processes by in-
troducingtheefficiencyofprecipitationasafreeparameter. w–v(r) >0 w–v(r) <0
t t
ThedustcloudmodeldevelopedbyWoitke&Helling(2003, Coalescence Coalescence
2004), Helling&Woitke (2006), and Hellingetal. (2008),
which has recently been applied to clouds in hot Jupiters
HD209458bandHD189733b(Leeetal.2015;Hellingetal.
Condensation Sweepout
2016), takes into accountcondensationand evaporationbut
not coalescence. Woitke&Helling (2003) neglected coa-
lescencebecause,accordingto Cooperetal.(2003), coales-
cence takes place much slower than condensationin brown
dwarfs. However,asnotedbyWoitke&Helling(2003),co-
Updraft Velocity w
alescencebecomestheonlyparticlegrowthmechanismeven
inbrowndwarfsifthesupersaturations,definedbythefrac-
tionalexcessofthepartialpressurefromsaturation,issignif-
icantlylow, e.g., s 1%. WhileCooperetal.(2003)fixed CCN Condensing Gas
≪
the supersaturation to 1%, the actual supersaturation in an
updraftdepends on the number density of initial condensa-
Figure1. Schematic illustration of our condensation–coalescence
tionnuclei,whichishighlyuncertainforexoplanetsaswell cloud model. We consider small “cloud particles” (light blue
asforbrowndwarfs. spheres) and large“rain particles”(dark blue spheres) whose ver-
In this paper, we presenta simple one-dimensionalcloud tical velocity relative to the ground, w−vt, is positive and nega-
tive, respectively. Themodel includesparticlegrowthduetocon-
modelthatissimplebuttakesintoaccountthemicrophysics
densation, coalescence, and sweepout (for the definitions of these
of coalescenceas well as of condensation. We describe the processes,seeSection2.1),andalsoverticaltransportduetogravi-
basicequationsandcloudmicrophysicsinourmodelinSec- tationalsettlingandtheupdraftmotionofthegas.
tion 2. We test our model by comparing with the observa-
tionsofthecloudsontheEarthandJupiterinSection3. We We denote the total number density and mass density of
discussthethresholdvelocityofstickingandtheoutlookon thecloud(rain)particlesbyN (N)andρ (ρ),respectively.
c r c r
applicationto exoplanetarycloudsinSection4. We present Weassumethatthecloudandrainparticleshavecharacteris-
asummaryinSection5. ticradiir andr andcharacteristicmassesm =(4π/3)ρ r3
c r c int c
andm = (4π/3)ρ r3,respectively,whereρ istheinternal
r int r int
density of the particles. For liquid particles, ρ is equalto
2. MODELDESCRIPTION int
thematerialdensityofthecondensate,ρ ,whileforsolidpar-
p
2.1. Outline
ticles,ρ canbelowerthanρ becauseanaggregateofsolid
int p
Our modelprovidesthe vertical distributions of the mass particlescanbeporous.Theporosityofcondensateparticles
andnumberdensitiesofcondensateparticles(Figure1). We canpotentiallyaffectthegrowthandmotionoftheparticles
adopt a one-dimensional Eulerian framework in which the as demonstrated by theoretical studies on dust evolution in
gasascendsataverticalvelocityw. Eachcondensateparticle protoplanetarydisks(e.g.,Ormeletal.2007;Okuzumietal.
isassumedtofallrelativethetheupwellinggasataterminal 2009).Wehereneglectthiseffectbyassumingconstantbulk
speed v, which is given by the balance between gas drag density,ρ =ρ ,butweplantotakeitintoaccountinfuture
t int s
andgravityandisthereforeafunctionofthe particleradius work.
r (seeEquation(A5)fortheexpressionofv adoptedinthis Our model determines the vertical distribution of N , ρ ,
t c c
study).Thus,eachparticlehasanetverticalvelocityw v(r). N,ρ, andthecondensatevapormassdensityρ bynumer-
t r r v
−
We divide the populationof condensate particles into small icallysolvingthesetofverticallyone-dimensionaltransport
“cloud particles” whose net vertical motion is upward, w equationswith terms representingcondensationand coales-
−
v >0,andlarge“rainparticles”whosenetverticalmotionis cence (Sections 2.2 and 2.4–2.6). Condensation refers to
t
downward,w v < 0. Inthispaper,we assumethatinitial particle growth through the accretion of supersaturated va-
t
−
cloud particles form through heterogeneousnucleation (see por,whilecoalescencereferstothegrowththroughcollisions
Section2.3fordetails). with othercondensateparticlesundergravity. In this study,
Acondensation–coalescencecloudmodelforexoplanets 3
werefertothecoalescencebetweencloudandrainparticles 2.3. Nucleation
as sweepout, in order to distinguish the coalescence of two
We assume that cloud particles form at the cloud base
cloudparticlesoroftworainparticles.1 Weneglectthecon-
through the condensation of vapor onto small refractory
densationof vaporontorain particlesbecausethe timescale
grains that already exist in the atmosphere. This process
ofcondensationgrowthislongerthanthetimescaleofsedi-
is known as heterogeneous nucleation, and such refractory
mentationfortheselargeparticles. Thecharacteristicmasses
grains are termed cloud condensation nuclei (CCN). On
andradiioftwoparticlespecies(cloudandrain)areautomat-
EarthCCNincludeseasalt,ash,anddustfromtheland(e.g.,
ically determinedby the mass and numberdensities via the
Rogers&Yau 1989; Seinfeld&Pandis 2006). Lacking in-
relations m = ρ /N and r = (3m /4πρ )1/3, respectively,
j j j j j p formationaboutCCN inotherplanetsincludingexoplanets,
where j = c for cloudparticles and j = r for rain particles.
weparametrizethemwiththeirnumberdensityN andra-
CCN
Such a framework is called a double-moment bulk scheme
diusr . TheCCNnumberdensityisaparticularlyimpor-
CCN
in meteorology (e.g., Ziegler 1985; Ferrier 1994), and a
tantparameterbecauseitdeterminesthenumberdensityand
characteristic size method in the planet formation commu-
maximumreachablesizeofcloudparticlesgrowingthrough
nity(e.g.,Birnstieletal.2012;Ormeletal.2014;Satoetal.
condensation.
2016). Thischaracteristicsizemethodallowsustosimulate
Inprinciple,cloudparticlesmayalsoformthroughhomo-
the growth of particles with much less computational time
geneous nucleation, where molecules in supersaturated va-
than that required with spectral bin schemes where the full
por spontaneously collide to form initial nuclei. Although
sizedistributionofparticlesisevolved(Birnstieletal.2012;
homogeneousnucleationis the simplestnucleationprocess,
Satoetal.2016).Ourmethodisparticularlyusefulforstudy-
this occurs only at a supersaturation ratio much larger than
ingcloudformationoverawideparameterspace.
unitybecauseofthefreeenergybarrierarisingfromthesur-
Because the terminal velocity of particles generally in-
face tension (e.g., Rogers&Yau 1989; Marleyetal. 2013).
creases as they grow, there is a height z = z at which
top Bycontrast,heterogeneousnucleationgenerallyoccurswhen
thenetupwardvelocityofcloudparticlesreacheszero,i.e.,
the supersaturationratio is slightly above unity because the
w [v(r )] = 0. Atthisheight,whichwecallthecloud
− t c z=ztop CCNs lower the free energy barrier (Rogers&Yau 1989).
top,weconvertthecloudparticlesintorainparticlesandal-
However,ifthereareonlyafewCCNsavailableintheatmo-
low them to fall and continue growing (see Figure 1). Our
sphere, homogeneous nucleation would dominate over het-
implementationofthe cloud-to-rainconversionis described
erogeneous nucleation (Woitke&Helling 2004). For sim-
inSection2.7.
plicity, we ignore homogeneous nucleation in the present
study, but we plan to include this effect in our future mod-
2.2. TransportEquations
eling.
Thetransportequationsusedinourmodelaregivenby Thecloudbaseisdefinedbythelocationabovewhichthe
∂Nc = ∂ [(w v(r ))N ] ∂Nc ∂Nc , (1) saturationvaporpressurePsofcondensinggasunderconsid-
∂t −∂z − t c c − ∂t − ∂t erationexceedsthepartialpressureinthegasphase.Thislo-
coal sweep
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) cationismainlydeterminedbytheverticaltemperaturepro-
(cid:12) (cid:12) (cid:12) (cid:12)
∂ρc = ∂ [(w v(r ))ρ ]+ ∂ρ(cid:12)(cid:12)c (cid:12)(cid:12) m(cid:12)(cid:12) ∂Nc(cid:12)(cid:12) , (2) file of the atmosphere, and weakly depends on the mixing
∂t −∂z − t c c ∂t !cond− c(cid:12) ∂t (cid:12)sweep ratioofthecondensinggasunderthecloudbase.
(cid:12) (cid:12)
∂Nr = ∂ [(w v(r))N] ∂Nr(cid:12)(cid:12)(cid:12) ,(cid:12)(cid:12)(cid:12) (3) 2.4. Condensation
∂t −∂z − t r r − ∂t
coal
∂ρ ∂ (cid:12)(cid:12)(cid:12)∂N(cid:12)(cid:12)(cid:12) If the cloud particle size rc is much larger than the mean
r = [(w v(r))ρ]+m (cid:12)(cid:12) c(cid:12)(cid:12) , (4) freepathofcondensingvapormolecules,therateofincrease
∂t −∂z − t r r c ∂t
(cid:12)(cid:12) (cid:12)(cid:12)sweep inρc duetocondensationisgivenby(Rogers&Yau1989)
(cid:12) (cid:12)
∂ρ ∂ ∂ρ (cid:12) (cid:12)
∂tv =−∂z(wρv)− ∂tc (cid:12) ,(cid:12) (5) ∂ρc = 4πrcNcD(ρv−ρs), (6)
where (∂ρ /∂t) is the rate of in crea!sceonidn ρ due to con- ∂t !cond RLvT −1 LKDTρs +1
c cond c
where Lis the specific late(cid:16)ntheato(cid:17)f vaporization, N isthe
densation, ∂N /∂t (∂N/∂t )istherateofdecreasein c
c coal r coal
| | | | number density of the cloud particles, K is the coefficient
N (N)duetothecoalescenceofcloud(rain)particlesthem-
c r
of thermal conductivity of the atmosphere, and D, R , and
selves, and ∂N /∂t isthe rate ofdecrease in N due to v
c sweep c
| | ρ = ρ (T) are the diffusion coefficient, specific gas con-
sweepout. Theexpressionsfor(∂ρ /∂t) , ∂N /∂t ,and s s
c cond j coal
| | stant,andsaturationvapordensityofthevapor,respectively.
∂N /∂t aregiveninSections2.4–2.6.
c sweep
| | The saturation vapor density is related to the saturation va-
por pressure P by ρ = P /R T. Equation (6) neglects
s s s v
1 Sweepoutisoftentermed“accretion”intheliteratureofcloudmicro- the effects of surface tension and nonvolatile solutes on the
physicalmodels(e.g.,Ziegler1985). saturation vapor pressure over the particles’ surfaces. This
4 Ohno&Okuzumi
is a good approximation for activated condensate particles betweentwoparticlesofradiirandr (r>r ),thecollection
′ ′
(Rogers&Yau1989). efficiencycanbeexpressedintermsoftheStokesnumber
The assumption that the particle sizes are larger than the
v(r )v(r) v(r )
atmosphericmeanfreepathisjustifiedforthecloudsonthe Stk= t ′ | t − t ′ |, (9)
gr
Earth and Jupiter we considered in Section 3. The atmo-
sphericmeanfreepathintheterrestrialwaterclouds(where which is approximatelythe ratio between the stoppingtime
T 300 K and P 1 bar) and in Jovian ammonia clouds =v(r )/gandcrossingtime r/v(r) v(r ) ofthesmaller
t ′ t t ′
≈ ≈ ∼ | − |
(whereT 130KandP 0.5bar)is 0.1µm,whichissig- particle. WhenStk 1,thesmallerparticleisstronglycou-
≈ ≈ ∼ ≪
nificantlysmallerthanthetypicalparticleradiusintheclouds pled to the flow around the large particle. To zeroth order,
(seeFigures3and5inSection3),However,thisassumption E behaves as E 0 at Stk 1 and as E 1 at Stk 1
≈ ≪ ≈ ≫
is notnecessarily validforcloudsin exoplanets. For exam- (Rossow1978).Inthisstudy,weevaluateEusingasmoother
ple,ZnSandKClcloudsonsuper-EarthGJ1214bcouldform analyticfunction(Guillotetal.2014,theirEquation(99)).
at P 0.01 bar (Morleyetal. 2013) , where the mean free
∼ E =max[0,1 0.42Stk 0.75], (10)
pathis 100µm. Furthermore,theobservationofGJ1214b − −
∼
suggeststhepresenceofhigh-altitudecloudsatP 10−5bar whichvanishesatStk.0.3andapproachesunityatStk 1
∼
(Kreidbergetal. 2014; Morleyetal. 2015), where themean ≫
(see Figure 12 of Guillotetal. 2014). This expression as-
freepathisaslongasl 1cm. Whenoneappliesourcloud
g sumes that flow around the particle is laminar. If the gas
∼
modeltosuchhigh-altitudeclouds,oneshouldreplaceEqua-
flowisturbulent,thecollectionefficiencycanbehigherthan
tion (6) by the expressionin the free molecular regime (for
assumed here (Homannetal. 2016). We approximate Stk
details,see Woitke&Helling2003),
in E as Stk v(r )ǫv /(gr ). Because Stk generally in-
t j j j
≈
∂ρc =4πr2N C (ρ ρ ), (7) creases with rj, coalescence occurs only after the particle
∂t c c s v− vs size exceeds a threshold above which Stk > 1. Therefore,
!cond
theproductionofprecipitatingraindropletsthroughcoales-
whereC isthemeanvelocityofgasmolecules.
s cencerequiresgrowthbeyondthisthresholdbycondensation
(Pruppacher&Klett1997).
2.5. Coalescence
Equation(10)applieswhenthebackgroundgasbehavesas
Undertheassumptionthatthecloudandrainparticlesize
a fluid,i.e.,theparticleradiusismuchlargerthanthemean
distributionsarenarrow,therateofdecreasein N (j =cor
j free path of the gas molecules. In the opposite case where
r’)duetocoalescenceisapproximatelygivenby
the gas behaves as a free molecular flow, one may assume
E =1. Thefree-molecularflowregimeisexpectedtogovern
∂Nj 1π(r +r )2N2∆v(r )E thecollisionalgrowthofparticlesathighaltitudeswherethe
∂t ≈2 j j j j gasdensityislow.
(cid:12) (cid:12)coal
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) =2πr2jN2j∆v(rj)E, (8)
(cid:12) (cid:12) 2.6. Sweepout
where ∆v is the relative velocity between the particles in-
Similar to Equation(8), the rate of decrease in N due to
ducedbygravitationalsettling, E isthecollectionefficiency c
sweepoutisgivenby
definedby theratioof theeffectivecollisionalcrosssection
to the geometric one (see below). The factor 1/2 prevents dN
c =π(r +r )2v(r) v(r )NN E. (11)
doublecountingofthecollisions. Wehaveusedthatthege- dt r c | t r − t c | r c
sweep
(cid:12) (cid:12)
ometric collisional cross section between two similar-sized (cid:12) (cid:12)
particles is approximatelygiven by π(rj +rj)2. We express where(cid:12)(cid:12)(cid:12)the c(cid:12)(cid:12)(cid:12)ollection efficiency E is given by Equation (10)
withStk=v(r )v(r) v(r )/(gr)(seeEquation(9)).
the differential settling velocity as ∆v = ǫvt(rj), where the t c | t r − t c | r
factorǫ(<1)encapsulatestheeffectofnon-zeroparticlesize
2.7. Cloud-to-RainConversion
dispersion.Satoetal.(2016)andKrijtetal.(2016)showthat
whenthedifferentialdriftvelocityisproportionaltothepar- Cloud-to-rain conversion occurs at the height z = z
top
ticle size, bulkschemeswith ǫ 0.5bestreproducethe re- where the terminal velocity of cloud particles v(r ) equals
t c
≈
sults of spectral bin schemes that take into account the full the updraftvelocityw. Once the terminalvelocity of cloud
sizedistribution.Inthisstudy,weadoptǫ =0.5forarbitrary particles exceeds the updraft velocity, we numerically fix
valuesofr . the net vertical velocity, w v(r ), to zero, and let the
j t c
−
The collection efficiency E accounts for the effect of the cloud particles evolve into rain particles at the rate given
gasflowaroundalargeparticlemovingrelativetotheback- by t 1 = β(t 1 + t 1 ), where t 1 (∂ρ /∂t)/ρ and
c−onv c−ond c−oal c−ond ≡ c c
groundgas:thegasflowsweepsasideparticlesthatareaero- t 1 ∂N /∂t/N are theratesof growthdueto condensa-
c−oal ≡ | c | c
dynamically well coupled to the gas (see, e.g., Slinn 1974, tionandcoalescence,respectively,andβisanumericalfactor
Pruppacher&Klett1997, their Chapter14). Fora collision (see also below). The equations that describe the cloud-to-
Acondensation–coalescencecloudmodelforexoplanets 5
rainconversionarethengivenby (10 m for terrestrial water clouds and 20 m for Jovian am-
moniaclouds),andintegratetheequationsintimeusingthe
∂N N
c = c , (12) first-orderexplicitscheme.
(cid:12) ∂t (cid:12)z=ztop −tconv
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)∂ρc(cid:12)(cid:12)(cid:12) = ρc , (13) 3.1. WaterCloudsontheEarth
(cid:12) ∂t (cid:12)z=ztop −tconv Wefocusontrade-windcumuli,whicharerelativelyshal-
(cid:12)(cid:12) (cid:12)(cid:12) lowwatercloudsbutareyetdeepenoughtodevelopprecip-
(cid:12)(cid:12) ∂Nr(cid:12)(cid:12) = Nc , (14) itation. vanZantenetal.(2011)conductedcomparativestud-
∂t t
(cid:12) (cid:12)z=ztop conv ies of terrestrial cloud models using the data of trade-wind
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12)∂ρr(cid:12)(cid:12) = ρc . (15) cumuliinthenorthwesternAtlanticoceanobtainedfromthe
∂t t Rain in Cumulus over the Ocean (RICO) Field Campaign
(cid:12) (cid:12)z=ztop conv
(cid:12) (cid:12) (Rauberetal. 2007). FollowingvanZantenetal. (2011), we
Strictly speaking, o(cid:12)(cid:12)ur c(cid:12)(cid:12)loud–rain two-population model
(cid:12) (cid:12) examinehowaccuratelyourcloudmodelreproducesthever-
breaks down near the cloud top, where the true size distri-
ticalclouddistributionfromtheRICOcampaign.
butionof condensateparticles (whichis notresolvedin our
Following Weidenscilling&Lewis (1973), we construct
model)cannotbeapproximatedwithtwopeaksthatarewell
the vertical temperature profile by using the dry adiabatic
separatedfromeachother. Thenumericalfactorβweintro-
lapserate(g/c =9.8Kkm 1,wherec isthespecificheatat
duced above arises from the lack of information about the p − p
constantpressure)belowthecloudbaseandthewetadiabatic
continuousparticle size distribution from cloud to rain par-
lapserate(e.g.,theEquation(7)ofAtreyaetal.2005)above
ticles. We findthatβ & 1–100causesanoscillatingmotion
thecloudbase. Theverticalpressureprofileiscalculatedby
ofthecloudtopthatpreventsusfromobtainingasteadyso-
assuminghydrostaticequilibrium(whichisagoodapproxi-
lution. Bycontrast,β . 1providesastablesteadysolution,
mationaslongastheupdraftmotionisslow). Inaccordance
and we find that the vertical distribution of cloud and rain
withvanZantenetal.(2011),wefixthesurfacetemperature
wellbelowthe cloudtop is insensitiveto the choiceofβ as
andCCNnumberdensityto298Kand70cm 3,respectively,
longasβ.1(seeSection3.1,3.2). Wewilladoptβ=0.1in −
and adjust the water vapor mixing ratio (or the specific hu-
thetestsimulationspresentedinthefollowingsection.
midity)belowthecloudbasesothatthecloudbaseislocated
at 500 m from the ground. The updraftvelocity is taken to
2.8. BoundaryConditions
be either 0.9 ms 1 or 2.0 ms 1, which represent the typ-
− −
Thelowerboundaryofthecomputationaldomainissetto
ical updraft velocities in the dense and diffuse parts of the
the cloud base, which refers to the height above which the
clouds, respectively, in the simulations by vanZantenetal.
saturation ratio S ρ /ρ exceedsunity. To determine the
v s (2011,theirFigure5).
≡
locationof thecouldbase, we introducethemixingratioof
We approximate the vapor pressure for liquid water with
the cloud-forming vapor below the cloud base as an input
theArrheniusfunction
parameter. Atthecloudbase,wesetthenumberdensityand
L 1 1
radiusofcloudparticlestobeequaltothoseofCCN, NCCN ps,H2O =611exp R 273K − T Pa, (16)
and rCCN, respectively. For rain particles, we simply allow " v !#
themtoleavethecomputationaldomainwiththe(downward) where the temperature is in K and L = 2.5 106 Jkg 1
−
×
fluxdeterminedjustabovethecloudbase. (Rogers&Yau 1989). The thermal conductivity K and dy-
The upper boundary is taken to be the cloud top, and namicviscosityηofairaresetto2.4 10 2Wm 1K 1and
− − −
the boundaryconditionsat that locationare givenby Equa- 1.7 10 5 Pas, respectively,andthe×diffusioncoefficientD
−
×
tions(12)–(15). Neithercloudparticlesnorrainparticlesare ofwatervaporinairissetto2.2 10 5m2s 1. Thesearethe
− −
×
allowedtogoabovethecloudtopbecause,bydefinition,the values at 273.0 K and 100 kPa (Rogers&Yau 1989). The
netverticalvelocitiesoftheparticlesvanishatthecloudtop. CCN radius r is assumed to be 0.5 µm; the results pre-
CCN
sentedbelowareinsensitivetothechoiceofr aslongas
CCN
3. TESTAPPLICATIONS
itistakentobesmallerthan1µm.
Now we test our cloud modelby comparingthe observa- Figures2and3presenttheresultsofourtestcalculations.
tionaldataofwatercloudsontheEarthandofammoniaice InFigures2,weshowthesteady-stateverticalprofilesofthe
cloudsonJupiter. Forsimplicity,weassumethatthevertical number and mass densities of the cloud and rain particles,
structureand updraftmotionof the backgroundatmosphere for w = 0.9 and 2.0 ms 1. To highlight the effects of co-
−
is independent of the presence of clouds. We numerically alescence, we also show the results of simulations without
solveEquations(1)–(5)withEquations(12)–(15),underthe the coagulation and sweepout terms (see the dotted lines).
fixedboundaryconditionsatthecloudbase,untilsteadypro- Thesizesofthecloudandrainparticlesasafunctionofthe
filesofN ,ρ ,N,andρ areobtained.Todoso,wediscretize height z from the ground are shown in Figure 3. In these
c c r r
theverticalcoordinatezintolinearlyspacedbinsofwidth∆z simulations, the size of the cloud particles reach 100 µm
∼
6 Ohno&Okuzumi
2500
2000
m] 1500
ht [
g
ei
H 1000
500
0
0 20 40 60 80 100 0 1.0 2.0 3.0 4.0 5.0 10-5 10-4 10-3 10-2 10-1 100 10-3 10-2 10-1 100 101
N [cm-3] ρ [g/m3] N [cm-3] ρ [g/m3]
c c r r
Figure2. VerticalstructureoftradecumuluscloudsontheEarthderivedfromourmodelcalculationsaswellasfromtheRICOobservations.
Thefourpanelsshow, fromlefttoright, thenumber andmassdensitiesofcloudparticles, N ,ρ , andthoseofrainparticles, N, andρ, as
c c r r
afunctionoftheheightzfromtheground. Theblueandredsolidlinesshowthesteady-statedistributionsobtainedfromourcondensation–
coalescence model for w = 0.9and 2.0ms 1, respectively. Thedottedlinesshow theresultsfromthemodel neglecting coalescence. The
−
lightanddarkgray-shadedareasspanthe5to95%and25to75%rangesoftheRICOobservationdata,respectively,takenfromFigure8of
vanZantenetal.(2011).
3000
andraindensitiesfallwithinthe5–95%rangeoftheobserva-
tions,exceptathighaltitudesz&1700mwheretheresultfor
2500 w = 0.9 ms 1 considerablyunderestimatesthe cloud num-
−
] berdensity.Bycontrast,themodelneglectingcoalescenceis
m
2000
t [ foundtooverestimatethemassdensityofcloudparticlesby
h
g uptoanorderofmagnitude.Furthermore,thiscondensation-
Hei 1500 onlymodelfailstoreproduceprecipitationbecausethemax-
imumparticlesizereachablewithcondensationistoosmall
1000 to fall against an updraft of w 1 ms 1 (see the dashed
−
∼
linesinFigure3). Withcoalescence,cloudparticlesdogrow
500
100 101 102 103 104 large enough to start falling as rain particles as we already
Particle Radius [μm] describedabove.
Althoughcoalescenceresolvestheorder-of-magnitudedis-
Figure3. Vertical distributions of the cloud particle radius rc crepancy between the model predictions and observations,
(solid lines) and rain particle radius r (dotted line) from our
c
condensation–coalescence cloud calculations shown in Figure 2. the cloud and rain mass densities are still systematically
Theblueandredlinesshowtheresultsforw = 0.9and2.0ms−1, higher than the averagesof the observedvalues. This indi-
respectively. Thedashedlinesshowtheverticaldistributionofthe
cates that the updraftsthatproducethe observedcloudsen-
cloudparticleradiusneglectingcoalescence.
train dry ambient air (e.g., Pruppacher&Klett 1997). En-
trainmentreducesthetemperatureandhumiditytheupdraft,
both of which act to suppress the condensation growth of
at z 1500 m for w = 0.9 ms 1 and at z 2200 m for
−
≈ ≈ cloud particles. The suppressed growth in turn leads to a
w = 2.0 ms 1. At these heights, the net vertical velocity
−
slowerdecreaseinthecloudnumberdensitywithheight,be-
w v(r )ofthecloudparticlesbecomeszero,andweconvert
t c
− cause coalescence takes place only after the particles grow
themintorainparticlesthatareallowedtocontinuegrowing
sufficiently large (see Section 2.5). Therefore, a model in-
astheyfalltowardtheground.
cludingentrainmentmightbetterreproduceboththenumber
Now we compare our simulation results with the RICO
and mass density of cloud particles. However, the model-
flight observations. The light and dark gray-shaded ar-
ing of entrainment within a 1D framework necessarily in-
eas in Figure 2 indicate the 5–95% and 25–75% ranges
troduces additional poorly constrained free parameters (see
of the observed values, respectively, taken from Figure
e.g., Pruppacher&Klett 1997, Chapter 12.7, 12.8), which
8 of vanZantenetal. (2011). Overall, we find that our
weavoidinthisstudy.
condensation–coalescence model reproduces the observa-
Figures 2 and 3 show that there is a jump in the cloud
tionstoorderofmagnitude.Thepredictedvaluesofthecloud
Acondensation–coalescencecloudmodelforexoplanets 7
3000 1000
β=0.1
β=1.0 CCN=107 m-3
β=10.0 CCN=106 m-3
2500 ]
m CCN=105 m-3
µ
ght [m]2000 dius [
ei1500 a100
H R
e
1000 tiv
c
e
500 ff
10-2 10-1 100 101 100 101 102 103 104 E
Mass Density [g m-3] Particle Radius [µm]
10
Figure4. Influencesofnumericalfactorβontheverticaldistribu- 1
tionsofthecloudparticleradiusr andmassdensityρ (solidlines),
c c
andtherainparticleradiusrrandmassdensityρr(dashedline).The ht
red,blue, andgreenlinesshow thesimulationresultsforβ = 0.1, g
ei
1.0,and10.0,respectively. TheCCNnumberdensitiesandupdraft H
velocityaresettoNCCN =100cm−3andw=2.0m/s. ale
Sc 0.1
particle radius and a peak in the cloud mass density at the s /
cloud top. This is an artifact arising from our treatment of s
e
thecloud–rainconversionatthislocation. Asemphasizedin n
k
c
Section 2.7, our cloud–rain two-population model does not hi
T
perfectlytreatthecloudtopwherethetwopopulationsinre-
0.01
alitymergeintoasingledistributionofparticles. Infact,we 100
findthatthe jumpandpeakfeaturesinr andρ , aswellas
c c
theheightofthe cloudtop, weaklydependson thevalueof
the numericalfactor β introducedin ourcloud–rainconver- th 10
sioncalculations. Thisis shownin Figure4, wherewe plot p
e
the vertical distributions of the sizes and mass densities of D
al
cloudandrainparticlesforβ= 0.1,1.0,and10.0. However, c
we confirm that this artifact has little effect on the vertical pti 1
O
distributionswellbelowthecloudtop.
3.2. AmmoniaIceCloudsonJupiter
0.1
0.1 1 10
Wealsoattempttoreproducetheobservationsofammonia Updraft Velocity [m/s]
clouds on Jupiter. Following Ackerman&Marley (2001),
we focus on ammonia ice clouds that cover Jupter’s upper Figure5. Effective radius reff (top panel), cloud geometric thick-
ness H normalized by pressure scale height H , (middle panel),
troposphere where P 0.7 bar and T 130–140 K (e.g., andclopudopticaldepthτatvisiblewavelengths(gbottompanel)for
∼ ∼
Westetal. 1986). In accordancewith the measurementsby Jovianammoniaclouds. Thesolidlinesshowthepredictionsfrom
the Galileo probe, we model the vertical temperature pro- ourcondensation–coalescencemodelfordifferentvaluesoftheup-
draft velocity w (x-axis) and CCN density N (dashed lines for
file as T = 166 K + Γ(z z ), where Γ = 2 Kkm 1 is CCN
− 0 − − 107m−3,solidlinesfor106m−3,anddottedlinesfor105m−3).The
the lapse rate and z0 is the height at which P = 1.0 bar orangehorizontalbarsindicatetheretrievalsfromtheVoyagerIRIS
(Seiffetal. 1998), and set the mixingratio of ammonia gas observations by Carlsonetal.(1994), whilethevertical bars indi-
underthecloudbasetobe6.64 10 4 kgkg 1 (Wongetal. cate the updraft velocity inferred from the 2D simulations of Jo-
× − − vian moist convection by Sugiyamaetal. (2014, their Section3.1
2004). Theverticalpressureprofileisdeterminedunderthe
andFigure9). Thepredictionfromthecondensation–coalescence
assumptionof hydrostaticequilibrium. We take the updraft modelsatisfiesalltheseconstraintswhenwand N aretakento
CCN
velocity and CCN number density as free parameters rang- be2–3ms−1andNCCN ≈106m−3,respectively.
ingfrom0.1ms 1to10ms 1andfrom103m 3to108m 3,
− − − −
respectively. Forthethevaporpressureofammoniaice,we
use the expression by Ackerman&Marley (2001, see also which is the value for Jovian atmosphere at T = 134.3 K
Ackerman&Marley2013), (Hansen1979). Thedynamicviscosityoftheatmosphereis
givenbyη=6.7 10 6Pasbasedontheformulaforthemix-
−
2161.0 86596.0 ×
ps,NH3 =exp 10.53− T − T2 bar, (17) turegasofH2andHe(Woitke&Helling2003)togetherwith
! theassumptionT 130K. Thediffusioncoefficientofam-
≈
where the temperatureis in K. The thermal conductivityof moniagasisgivenbytheformulaD =2η/(3ρf),whereρis
the atmosphere is taken to be K = 9.0 10 2 Wm 1K 1, theatmosphericdensityand f =5forammoniavapor(Equa-
− − −
×
8 Ohno&Okuzumi
1000
tion (14)of Rossow 1978). Following Ackerman&Marley
β=0.1
(2001), we take the bulk density of ammonia ice ρp to be m] ββ==11.00
0.84gcm 3. Asintheprevioussubsection,theCCNradius µ
− [
r isassumedtobe0.5µm. s
CCN u
We comparethesteady-stateverticalprofilesofthecloud di
a 100
andrainparticlesobtainedfromourmodelwiththeretrievals R
by Carlsonetal. (1994) based on the infrared observations e
v
ofJovianammoniacloudsintheEquatorialZoneandNorth ti
c
TropicalZonefromVoyagerinstrumentIRIS. Carlsonetal. e
f
(1994) retrieved the effective particle radius reff, geomet- Ef 10
ric thickness H , and optical depth τ at the wavelength of
p
0.5µm,wheretheeffectiveradiusreferstothearea-weighted ht 1
g
averageoftheradiusofvisiblecondensateparticles(see,e.g., ei
Kokhanovsky2004). Weevaluatetheeffectiveradiusinour H
simulatedcloudsas e
al
reff = R((rrcc32NNcc++rrrr32NNrr))eexxpp((−−ττzz))ddzz. (18) ss / Sc 0.1
e
whereτz is the optiRcalthicknessaboveheightz. Thefactor n
k
exp( τz)accountsforthefactthatonecanonlyobservepar- c
ticles−residing at τ . 1. In the calculation of τ and τ , we hi
z z T0.01
apply the geometric optics approximation to the extinction
100
crosssectionatvisiblewavelengths.Thegeometricthickness
ofthesimulatedcloudistakentobethedistancebetweenthe
h
cloudbaseandcloudtopinthisstudy. t
p
The results are summarized in Figure 5. Here, the e 10
D
solid lines show the values of reff, Hp, and τ from our al
condensation–coalescencemodelfordifferentsetsoftheup- c
ti
draft velocity w and CCN density N . The orange hor- p 1
CCN O
izontal bars indicate the retrievals by Carlsonetal. (1994):
r = 70–100 µm, H 0.3H , and τ = 1.2–2.0, where
eff p g
≤
Hg = 20 km is the pressure scale height and the range for 0.1
r is based on the interpretation by Ackerman&Marley 0.1 1.0 10
eff Updraft Velocity [m/s]
(2001). Wefindthatthepredictionsfromthecondensation–
coalescence model satisfies all these observational con- Figure6. Influence of β on the results for Jovian ammonia ice
straints when the updraftvelocity w and CCN numberden- clouds. The value of N is fixed to 106 m 3. The dotted red
CCN −
sityN areassumedtobe 2–3ms 1and 106m 3(see lines show the simulations for β = 0.1, the solid blue lines show
CCN ≈ − ≈ − for β = 1.0, and the dashed dark-green lines show for β = 10.0,
thesolid blacklines), respectively. If N & 107 m 3 (see
CCN − respectively.
thedashedlines),thepredictedopticaldepthfallswithinthe
retrievedrangeonlywhenw 0.2–0.5ms 1; however,for
−
≈
thisrangeofw, thepredictedeffectiveradiusistoosmallto inglengththeoryformulatedby(Ackerman&Marley2001)
beconsistentwiththeretrieval. If N . 105 m 3 (seethe suggestedtheupdraftvelocityatammoniacloudregionisap-
CCN −
dottedlines),ourpredictionreproducestheretrievedoptical proximately1–3ms 1,whichcorrespondstotheeddydiffu-
−
depthonlywhenw 3–7ms 1, butthenoverestimatesthe sioncoefficientK =2–6 108cm2s 1andthemixinglength
− z −
≈ ×
effectiveradiusfromtheretrieval. equal to H = 20 km. Furthermore, the 2D simulations of
g
TheresultspresentedinFigure5are little affectedbythe theJovianmoistconvectionshowthattheupdraftvelocityat
choiceofthe βparameterintroducedin ourcloud-toptreat- ammoniacloudregionis0.5–3ms 1(Sugiyamaetal.2014)
−
ment,asshowninFigure6. Onecanseethatthepredictions shownintheverticalorangebarofFigure5. Therefore,our
convergeforβ & 1. Forβ = 0.1, thepredictedeffectivera- bestfitvalueofupdraftvelocityof2–3ms 1 isarealistic
−
dius and optical depth at w & 0.5 m/s are higher than the valuefortheJovianammoniacloudregion.
convergedvalues,butthedeviationfromtheconvergedvalue Unfortunately, there is no direct observational constraint
isassmallasafactoroflessthan2. on the CCN number density for Jovian ammonia clouds.
We comparethe updraftvelocityof 2–3 ms 1 with other However,animportanthintforthe CCN densitycan beob-
−
calculations to validate our best fit value. First, the mix- tainedfromthein-situobservationoftheJovianatmosphere
Acondensation–coalescencecloudmodelforexoplanets 9
inarelativelyparticle-freehotspotbytheGalileoprobe.The interrestrialwatercloudsbutalsoinJovianiceclouds.
observationshowedthataconcentrationofsmallparticlesof Equation (8) assumes that two particles stick whenever
ameanradiusof0.5–5µmandanumberdensityof1.9 105– they collide. However, this assumption breaks down if the
×
7.5 106m 3waspresentataheightwhereammoniaclouds collisionvelocityissohighthatthecollisionresultsinbounc-
−
×
form(Ragentetal.1998).Itisunlikelythattheobservedpar- ingorfragmentationoftheparticles. Inprinciple,solidpar-
ticleshadexperiencedcoalescence,becausetheparticlesare ticles are less sticky than liquid particles because a harder
toosmalltocollidewitheachother(i.e.,forparticlessmaller particlehasasmallercontactareaandhenceasmallbinding
thanamicroninradius,Stk < 0.5andhenceE < 0.3inthe energyassociatedwithintermolecularforces(Rossow1978).
ammoniacloudformingregion).Therefore,wecaninferthat Our future modeling will take into account the potentially
thenumberdensityofCCN,whichisapproximatelyequalto lowstickingefficiencyofsoliddustparticles.
thenumberdensityofsmallparticlesbeforecoalescencesets We have also assumed that the internal density of con-
in,was 105–107 m 3 inthehotspottheGalileoprobeen- densate particles is constant. However, the internal den-
−
∼
tered. Interestingly,thisvalueisconsistentwith ourpredic- sity can decrease as the particles grow into porous aggre-
tion for the CCN density in the Equatorial Zone and North gates through coalescence (e.g., Blum&Wurm 2000). Be-
TropicalZone(seeabove).Wedonotthinkthatthiscompar- cause porous aggregates have a larger aerodynamical cross
ison justifies our cloud model, because the CCN density in sectionthancompactparticlesofthesamemass,porousag-
relativelycloud-freehotspotsisnotnecessarilyequaltothat gregatesmightascendtohigheraltitudesthancompactones
incloudyEquatorialZoneandNorthTropicalZone. Rather, and provide featureless transmission spectra of exoplane-
thiscomparisonsuggeststhattheCCNconcentrationsinthe tary atmospheres. On the other hand, the coalescence of
tworegionscouldbesimilar. porous aggregates can be faster than that of compact parti-
cles(Okuzumietal.2012).Ifthisisthecaseinexoplanetary
4. SUMMARYANDOUTLOOK atmospheres,theporosityevolutionmightpreventtheforma-
Wehavedevelopedanewcloudmodelforexoplanetsand tionofhigh-altitudeclouds.Recenttheoreticalstudiesofthe
browndwarfsthatissimplebuttakesintoaccountthemicro- graingrowthinprotoplanetarydiskshaveyieldedadetailed
physics of both condensation and coalescence. Our model modelfortheporosityevolutionofgrainaggregatesbasedon
produces the vertical distributions of the mass and number graincontactmechanics(e.g.,Kataokaetal.2013). Wewill
densitiesofcloudandrainparticlesasafunctionofphysical studytheimpactofporosityevolutiononcloudformationby
parameters,includingtheupdraftvelocity,themixingratioof usingthesetheoriesinfuturestudies.
thecondensinggasatthecloudbase,andthenumberdensity
of CCN. Therefore, our model will be useful to understand
Theauthorsthanktheanonymousrefereeforusefulcom-
how the dynamics, compositions, and nucleation processes
ments. We thank Hanii Takahashi and Yuka Fujii for use-
in exoplanetaryatmosphereswouldaffecttheverticalstruc-
ful discussions about observationsof terrestrial clouds, and
tureofexoplanetarycloudsviacloudmicrophysics.
Chris Ormelfor manyhelpfulcommentson the manuscript
We have tested our model by comparing with the obser-
of this paper, and Neal Turner, Shigeru Ida, Masahiro
vationsoftheterrestrialwatercloudsandtheJovianammo-
Ikoma, Mark Marley, Yasuto Takahashi, and Xi Zhang
niaclouds. Forterrestrialwaterclouds,ourmodelplausibly
for useful comments and discussions. This work is sup-
reproduces the observed vertical distributions of the cloud
portedbyGrants-in-AidforScientificResearch(#23103005,
mass and number densities from in situ observations when
15H02065)fromMEXTofJapan.
we assume the terrestrial typical updraft velocity, height of
cloud base, and CCN number density. For Jovian ammo-
APPENDIX
nia clouds, our model simultaneously reproduces the cloud
opticaldepth,thegeometricthickness,andtheparticleeffec-
A. TERMINALVELOCITY
tive radius in the EquatorialZone and North Tropical Zone
For particles much larger than the mean free path of the
retrievedfrom Voyager measurementswhen we assume the
updraftvelocityofw 2–3ms 1andtheCCNnumberden- molecules in air, the terminal velocity of a particle under
−
sity of N 106≈m 3. Our best-fit updraft velocity is gravitygisgivenby
CCN −
≈
consistentwithestimatesfrommixingtheoryandfromcloud
8grρ
convectionsimulations. Thebest-fitCCNdensityiscloseto v(r)= p, (A1)
thenumberdensityofsmallparticlesinahotspotmeasured t s3CDρa
bytheGalileoprobe,suggestingthattheCCNdensityinthe
whereρ isthemassdensityoftheatmosphere,andC isthe
a D
EquatorialZoneandNorthTropicalZoneissimilartothatin
dragcoefficientdeterminedbythelocalReynoldsnumberof
hotspots. Thegoodagreementbetweenourpredictionsand
theparticle,
theobservationsindicatesthatthecoalescenceofcondensate
2rv(r)ρ
particlesisanimportantprocessofcloudformation,notonly N t a, (A2)
Re ≡ η
10 Ohno&Okuzumi
whichdoesnotinvolvev, andanalyticallyexpressN asa
t Re
functionofX. WerequirethefunctiontoreproduceStokes’
and Newton’s drag laws, C = 24/N (N = X/24) and
D Re Re
C = 0.45 (N = √X/0.45), in the limits of N 1
D Re Re
≪
and N 1000, respectively. To determinethe functional
Re
≫
formintheintermediateregime1. N .1000,weusethe
Re
dataforrigidspheresfromTable10.1ofPruppacher&Klett
(1997). Wefindthatthefunction
N = X −0.8+ X −0.4 −1/0.8 (A4)
Re 24 0.45
"(cid:18) (cid:19) (cid:18) (cid:19) #
well reproduces the data points in the intermediate regime
and the correct limiting behaviors at N 1 and N
Re Re
≪ ≫
1000(seeFigureA1).SubstitutingthisexpressionandEqua-
FigureA1. ParticleReynoldsnumber NRe versusX ≡ CDNR2e. The tion(A3)intoEquation(A2),weobtaintheanalyticexpres-
dashedanddottedlinesindicatetheStokesandNewtondraglaws,
respectively, and the solid circles show the data for rigid spheres sionforv asafunctionofr,
t
fromTable10.1ofPruppacher&Klett(1997).Thesolidlineshows
ourfit,Equation(A4). 2gr2ρ 0.45gr3ρ ρ 0.4 −1.25
p a p
v = 1+ . (A5)
t 9η 54η2
whereηisthedynamicviscosityoftheatmosphere.  ! 
Because NRe depends on vt, one needs to solve Equa- If the particles are small the atmosphericmean free path,
tions (A1) and (A2), together with the relation between oneshouldusetheEpsteinlaw
C and N , to obtain v as a function of r. Following gρ
D Re t p
v(r)= r (A6)
Ackerman&Marley (2001), we do so by introducing the t 3C ρ
s a
quantity
instead of Equation (A5). This is the case for high-altitude
32gr3ρ ρ
X C N2 = a p, (A3) cloudsinexoplanets(seeSection2.4).
≡ D Re 3η2
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