Draft version January 12, 2017 PreprinttypesetusingLATEXstyleemulateapjv.08/22/09 EFFECTS OF LATENT HEATING ON ATMOSPHERES OF BROWN DWARFS AND DIRECTLY IMAGED PLANETS Xianyu Tan and Adam P. Showman DepartmentofPlanetarySciencesandLunarandPlanetaryLaboratory,UniversityofArizona,1629UniversityBoulevard,Tucson,AZ 85721,USA Draft version January 12, 2017 ABSTRACT Growingobservationsofbrowndwarfshaveprovidedevidenceforstrongatmosphericcirculationon 7 these objects. Directly imaged planets share similar observations, and can be viewed as low-gravity 1 versions of brown dwarfs. Vigorous condensate cycles of chemical species in their atmospheres are 0 inferred by observations and theoretical studies, and latent heating associated with condensation is 2 expected to be important in shaping atmospheric circulation and influencing cloud patchiness. We present a qualitative description of the mechanisms by which condensational latent heating influence n the circulation, and then illustrate them using an idealized general circulation model that includes a a condensation cycle of silicates with latent heating and molecular weight effect due to rainout of J condensate. Simulations with conditions appropriate for typical T dwarfs exhibit the development of 0 localized storms and east-west jets. The storms are spatially inhomogeneous, evolving on timescale 1 of hours to days and extending vertically from the condensation level to the tropopause. The frac- tional area of the brown dwarf covered by active storms is small. Based on a simple analytic model, ] P we quantitatively explain the area fraction of moist plumes, and show its dependence on radiative E timescale and convective available potential energy. We predict that, if latent heating dominates . cloud formation processes, the fractional coverage area by clouds decreases as the spectral type goes h through the L/T transition from high to lower effective temperature. This is a natural consequence p of the variation of radiative timescale and convective available potential energy with spectral type. - o Subject headings: brown dwarfs - planets and satellites: gaseous planets - planets and satellites: r atmospheres - hydrodynamics - methods: numerical t s a [ 1. INTRODUCTION been observed on directly imaged EGPs (Biller et al. 1 Observations of brown dwarfs (BDs) have shown in- 2015; Zhou et al. 2016). From a meteorological point of view, the directly imaged EGPs resemble low-gravity v creasing evidence of a vigorous circulation in their at- versions of BDs, for which their atmospheric dynamical 2 mospheres (Showman & Kaspi 2013). This evidence in- regimeischaracterizedbyfastrotation,vigorousconvec- 8 cludesnear-infraredbrightnessvariability(Artigauetal. tion and negligible external heating. 7 2009;Radiganetal.2012;Apaietal.2013;Buenzlietal. Motivated by the observations, several studies have 2 2014; Radigan et al. 2014; Wilson et al. 2014; Buenzli been conducted to explore the atmospheric dynamics 0 et al. 2015; Metchev et al. 2015; Yang et al. 2015, 2016; . Cushing et al. 2016), chemical disequilibrium (Fegley & of ultra cool objects (compared to stars). Local two- 1 dimensionalhydrodynamicssimulationsbyFreytagetal. Lodders1996;Saumonetal.2006,2007;Hubeny&Bur- 0 (2010) showed that interactions between the convective 7 rows 2007; Stephens et al. 2009; Visscher & Moses 2011; interior and the stratified layer can generate gravity 1 Zahnle & Marley 2014) and surface patchiness (Cross- waves that propagate upward, and the breaking of these : field et al. 2014). Cloud disruption has been proposed v waves causes vertical mixing that leads to small-scale to help explain properties of the L/T transition (Acker- i cloud patchiness. Showman & Kaspi (2013) presented X man & Marley 2001; Burgasser et al. 2002; Marley et al. the first global model of brown dwarf dynamics for the 2010), and such patchiness is also likely responsible for r convective interior, and showed that large-scale convec- a thenear-infraredbrightnessvariability(Marley&Robin- tion is dominated by the fast rotation. Using an an- son 2015). Nevertheless, the mechanism responsible for alytic theory, they proposed that atmospheric circula- cloud disruption is yet unclear. Atmospheric circulation tion can be driven by atmospheric waves in the stably isexpectedtoplayacrucialroleincontrollingcloudcov- stratified upper atmosphere. Using a two-layer shallow- eragefraction,butthedetailsremainpoorlyunderstood. water model, Zhang & Showman (2014) showed that A handful of directly imaged extrasolar giant planets weakradiativedissipationandstrongforcingfavorlarge- (EGPs) exhibit similarities with BDs: the near-infrared scalezonaljetsforbrowndwarfs,whereasstrongdissipa- colors, inference of dust and clouds, chemical disequilib- tion and weak forcing favor transient eddies and quasi- riumintheiratmospheresandfastspin(Hinzetal.2010; isotropic turbulence. Despite these studies, no global Barman et al. 2011a,b; Marley et al. 2012; Oppenheimer modelthatincludescondensatecyclesandcloudshasyet et al. 2013; Ingraham et al. 2014; Skemer et al. 2014; been published for brown dwarfs. Clouds play a signifi- Snellen et al. 2014; Macintosh et al. 2015; Wagner et al. cant role in sculpting the temperature structure, spectra 2016). Near-IR brightness variability has also recently andbrightnessvariationsofbrowndwarfs(seerecentre- Electronicaddress: [email protected] viewsofMarley&Robinson2015andHelling&Casewell 2 2014). There is a pressing need to couple condensation conclusions. cycles and clouds to global models to study how the cir- culationcontrolsglobalcloudpatchiness,andinturnhow 2. EFFECTSOFLATENTHEATINGONATMOSPHERES the condense cycle affects the circulation. 2.1. Conditional Instability In this paper, we propose the importance of latent Most atmospheres of planets and ultra cool brown heating on the atmospheric circulation and cloud patch- dwarfs have constituents that can condense. Due iness of brown dwarfs by using an idealized general cir- to atmospheric motion and diabatic heating/cooling, culation model that includes a condensation cycle of sil- air parcels containing condensable species can undergo icate vapor. Latent heating is of paramount importance change of temperature and pressure, leading to conden- inEarth’satmosphere(Emanuel1994). Forgiantplanets sation. The latent heating/cooling due to condensa- in our solar system whose atmospheres are likely analo- tion/evaporation has important effects on the stability goustobrowndwarfs’,alonghistoryofstudieshasshown of atmospheres, which we summarize here; a more de- the importance of latent heating in driving their atmo- taileddiscussioncanbefound,e.g.,inChapter7ofSalby spheric circulation (Barcilon & Gierasch 1970; Gierasch (2012). For simplicity, we begin our discussion assuming 1976; Gierasch et al. 2000; Ingersoll et al. 2000; Lian the molecular weight is constant but return to this issue & Showman 2010). Lian & Showman (2010) demon- in a later subsection. strated that large-scale latent heating from condensa- Itiswellknownthatarapidlyascendingordescending tion of water can drive patterns of zonal (east-west) jet dry air parcel follows a dry adiabatic lapse rate streams that resemble those on all four giant planets of the solar system: numerous zonal jets off the equator dlnT R = , (1) and a strong prograde equatorial jet on Jupiter and Sat- dlnp c p urn,andathree-jetpatternincludingretrogradeequato- rial flow and high-latitude prograde flow on Uranus and where T is temperature, p is pressure, R is specific Neptune. Such models also exhibit episodic storms that gas constant and cp is specific heat capacity of dry at- qualitativelyresemblethoseobservedonJupiterandSat- mosphere. Similarly for a saturated air parcel mixed urn. For brown dwarfs, the evidence for patchy clouds with condensable and non-condensable gases, it follows in controlling brightness variability and the L/T transi- a moist adiabatic lapse rate1: tion itself also suggests a strong role for an active con- dlnT R + Lmξ densate cycle, and latent heating may be similarly im- = u T , (2) pimoratgaendtEfoGrPastm. oBspechaeuriscectiermcuplaetriaotnuroefpBeDrtsuarbnadtidoinrescatlsy- dlnp cp+ RLu2mTξ2 sociated with (dry) convection at condensable pressure where Lm is the latent heat per mole, cp is the specific levelsaregenerallysmall,thelatentheatingthataccom- heat capacity per mole for the mixture, Ru is the uni- panies the condensation of relevant chemical species can versal gas constant, and ξ = pcond/pd is the molar mix- dominatethebuoyancyinthelayerswherecondensation ing ratio of condensable gas over dry gas. Under nor- occurs. mal conditions of most atmospheres, the dry adiabat is Themainpointofthispaperistoillustratehowlatent larger than the moist adiabat as long as Lm > cpT. In heatingmodifiesacirculationandinfluencescloudpatch- the presence of two adiabats, the atmosphere can have inessinthesimplestpossiblecontext,soweintentionally different stability criteria. If the atmospheric lapse rate exclude clouds, radiative transfer and detailed micro- dlnT is larger than the dry adiabatic lapse rate R/c , dlnp p physics to allow a simpler environment in which to clar- the atmosphere will be absolutely unstable; if the lapse ify the dynamical processes that are at play. Cloud mi- rate is smaller than the moist adiabatic lapse rate, the crophysics processes are highly complex (Rossow 1978), atmosphere will be absolutely stable; if the lapse rate is and significant prior work on the cloud microphysics is- in between the dry and moist adiabatic lapse rate, the sue (see a review by Helling & Casewell 2014), as well atmosphere is stable against dry convection but unsta- as parameterized cloud models (Allard et al. 2001; Ack- ble to moist convection, which is referred to as condi- erman & Marley 2001; Tsuji 2002; Cooper et al. 2003; tional instability. Examples include the tropospheres of Barman et al. 2011a) has been done for ultra cool atmo- the Earth, Titan, and probably Jupiter and Saturn. spheres. We are well aware of the important feedback of How does an air parcel behave in a conditional unsta- cloudstoatmospheres,andwillleaveitforfutureefforts. ble and unsaturated atmosphere? This is schematically Also, to resemble the vigorous convection and the dy- illustrated in Figure 1: initially starting from an arbi- namics in radiative-convective boundary caused by con- trary level below the condensation level, the ascending vective perturbation, one needs a model that can prop- air parcel follows a dry adiabat until its relative humid- erly treat both the convective interior and the overlying ity reaches 100%, and reaches the lifting condensation stably stratified layer. Therefore, we do not expect our level (LCL). Afterward it will follow a moist adiabat, current simulations to resemble the true atmospheres of and then at some point it will reach the level of free brown dwarfs and directly imaged EGPs. convection (LFC) where it has a lower density than the Thepaperisorganizedasfollows. WestartoutinSec- environment and becomes positively buoyant. The air tion 2 by describing several important effects of latent heating on the atmosphere; in Section 3, we briefly in- 1 In derivingthis formula, theClausius-Clapeyronequation for troduce our idealized model that is used to illustrate the thesaturationvaporpressureofthecondensablespecieswasused, mechanismsdescribedinSection2;inSection4,weshow assuming an ideal gas equation of state and that the condensate densityismuchgreaterthanthegasdensity. Thisformulaisappli- result of our simulations; finally in Section 5, we discuss cableforfullrangeofξ,notlimitedtoassumptionofsmallmixing our results and implications for observations, and draw ratioofcondensablegas. 3 z Neutral buoyancy level — parcel stops ascending Path followed by actual parcel Dry adiabat Reference profile Level of free convection — (LFC) parcel becomes buoyant Lifting condensation level — (LCL) condensation first occurs T v Figure 1. Aschematicplotshowinghowanairparcelbehavesinaconditionallyunstableatmosphere. Thesolidlineisthebackground referencetemperatureprofile,thedashedlineisthepathfollowedbyanactualairparcelinitiatedbelowtheliftingcondensationlevel,and thedottedlineindicatesthedryadiabaticpathiftheairparcelisdry. Horizontalaxisistemperature(or, ifmolecularweighteffectsare important,thevirtualtemperature),andtheverticalaxisisheightwithintheatmosphere. parcel then can freely convect to the top of a cumulus 2.2. Moist Convection on Controlling the Area Fraction stormwhereitsbuoyancydiminishesanditstopsascend- of Moist Plumes ing. In reality, because the atmospheric lapse rate may Vertical velocity within the moist convecting plume is be stable to dry convection, some external lifting mech- much larger than the surrounding subsidence flow, as anism is needed for the air parcel to reach the LFC, and observed in Earth’s atmosphere. The fast upwelling ve- this is why storms do not occur everywhere and in every locityisdrivenbythelargeconvectiveavailablepotential moment in Earth’s tropics even though the atmosphere energy (CAPE) in deep moist convection. CAPE is the isconditionallyunstable. Theamountofenergyrequired amount of potential energy per unit mass available for to reach the LFC is referred to as convective inhibition theconvectionofaparticularairparcel,andisessentially (CIN). To initiate moist convection, either strong initial anintegrationofbuoyancywithrespecttoheightduring diabaticheating(e.g.,inthecaseofEarth,heatingofthe the lifting of the parcel (e.g., Emanuel 1994, Chapter 6). surface by sunlight) or kinetic energy (e.g., forced lift- In contrast to the buoyant ascending convective plumes, ing by atmospheric waves or other large-scale motion) is the subsiding air is stratified and not convecting, but it neededfortheparceltoovercometheCIN.TheCINcan can gradually subside as described in Section 2.1. Be- act to limit the frequency of moist convection and pre- cause the radiative cooling time scale is generally long, serve large convective available potential energy, which the subsidence is generally slow. The asymmetry of the can be essential for the development of deep moist con- upwelling and downwelling vertical velocity results in a vection. small area fraction of moist ascending plumes by the re- One necessary condition for conditional instability is quirementofmasscontinuity. Theareacoveredbyclouds the stratification to dry convection in the troposphere. may not closely follow the area of moist plumes because One mechanism to produce the tropospheric stratifica- cloud particles will be spread by the wind field near the tion is by latent heating and moist convection. As il- cloud top, but this argument for moist plume area frac- lustrated in Figure 1, the rising air that follows a moist tion can qualitatively explain the origin of patchiness of adiabat in storms carries a higher entropy than where cumulus clouds (Lunine & Hunten 1987). For brown it is initiated. Mass continuity implies that the high- dwarfs, patchy clouds deduced from near-IR brightness entropy surrounding atmosphere at the top of the storm variability may qualitatively be explained by this mech- must subside. During the subsidence, air continues to anism. lose its entropy to space via IR radiation. Air closer to the ground has been subsiding for longer – and thus ex- 2.3. Molecular Weight Effect hibits lower entropy – than air aloft. As a result, the entropyincreaseswithheightinthebackgroundtemper- InatmospheresofgaseousgiantplanetsandBDs,con- atureprofile,whichimpliesthatthebackgroundtemper- densable species generally have a much higher molecular ature profile is stable to dry convection. On the other weight than the dominant dry constituent H2. Rainout hand, the background temperature exhibits lower tem- of these condensates can decrease the density of air, and perature than the moist adiabat, allowing moist insta- this effect can play an important role in atmospheric bility. For giant planets, the tropospheric stratification thermal structure and dynamics. For example, Guillot has been inferred from observations for Jupiter (Flasar (1995) presented the idea that moist convection in giant & Gierasch 1986; Magalh˜aes et al. 2002; Reuter et al. planets may be inhibited due to molecular weight effect 2007), and it has been demonstrated by numerical simu- if the mixing ratio of water is substantially higher than lations that the stratification in Jupiter can result from solarabundance;Li&Ingersoll(2015)proposedthatSat- latent heating of water condensation (Nakajima et al. urn’s 20-to-30-year quasi-periodic planetary-scale storm 2000; Sugiyama et al. 2014). isrelatedtothemolecularweighteffectofwater. InBDs and directly imaged EGPs, a local thin stably stratified layerassociatedwithmolecularweightgradientsmayex- 4 istrightabovethecondensationlevel,duetothefactthat where v is the horizontal velocity vector on isobars, the subsiding environmental air has experienced rain- ω =dp/dtistheverticalvelocityinpressurecoordinates, out, is thus relatively dry and has lower molecular mass. f =2Ωsinφ is the Coriolis parameter (here φ is latitude Whereas the air below the condensation level contains andΩistheplanetaryrotationrate),Φisgeopotential,kˆ significant condensable vapor and has higher molecular is the local unit vector in the vertical direction, ρ is den- mass. Therefore,asharpgradientinmolecularmassnat- sity,∇ isthehorizontalgradientinpressurecoordinate, p urally exists near the condensation level, where molecu- d/dt=∂/∂t+v·∇ +ω∂/∂p is the material derivative, p ldaurcemsasstsradteicfirceaatsioesnwtoithbointhcredarsyinagndaltmitouidste.coTnhviesctpiorno-, θ = T(pp0)R/cp is the potential temperature, p0 = 1 bar is a reference pressure, θ is the equilibrium potential contributing to CIN. Large CIN can suppress moist con- ref temperatureprofile,τ istheradiativetimescale,andL vection within (and above) the stable layer, leaving the rad islatentheatpermassforcondensate. Theidealgaslaw atmosphere gradually cooling off towards its radiative is assumed for the equation of state for the atmosphere. equilibrium by thermal radiation. The radiative equilib- Thehydrostaticassumptionusedinthestandardprim- riumtemperatureprofileinthetroposphere(forinstance, itiveequationssolvedinourmodelisagoodapproxima- on Earth and Jupiter) usually has a temperature lapse tion for large-scale flows in stratified atmospheres with rate dlnT/dlnp larger than the moist adiabatic lapse large ratio of horizontal scale to vertical scale (e.g., see rate. Thus, the existence of CIN can help contribute to Vallis 2006, Chapter 2). In the atmospheres of brown the accumulation of CAPE. This phenomenon has been dwarfs,theexpectedhorizontallengthscaleoflarge-scale shown by simulation on Jupiter as well (Nakajima et al. dynamics is 106 −107 meters, while the pressure scale 2000). By contribution to the accumulation of CAPE, heightis103−104 meters. Theaspectratiooftheatmo- the stratification from molecular weight effect also help sphere is on the order 102 −103, which is sufficient for to control the moist plume fraction via the mechanism the hydrostatic approximation to hold. discussed above. Thetracerq ismassmixingratioofcondensablevapor todryair,andq isthelocalsaturationvapormassmix- 2.4. Large-scale Latent Heating on Atmospheric s ing ratio that is determined by saturation pressure func- Dynamics tionforspecificcondensablespecies. The“on-offswitch” Moist convection provides a source of small-scale ed- function δ controls the condensation: when q > q then s dies,whichcangrowintolarge-scaleeddiesviaaninverse δ =1andvaporcondensesoveracharacteristictimescale energycascade. Theinteractionsamongtheseeddiesand τ which is generally taken as 103 sec, representative cond the mean flow in a rapidly rotating sphere can produce of a typical convective time; when q ≤ q then δ = 0. s zonallybandedstructureandvortices(Lian&Showman Latent heating is immediately applied in the thermody- 2010,alsoseeareviewbyVasavada&Showman2005for namicequation(Equation[6])oncecondensationoccurs. Jovian atmospheric dynamics). The latent heating can Forsimplicity,weincludeonlyonetracer,andchooseen- interact with the dynamics in many ways, and may pro- statite vapor (MgSiO ) to represent silicate vapor in our 3 duce organized clouds that can lead to cloud radiative brown dwarf models. Silicates are one of the most abun- feedback to the dynamics. The temperature perturba- dantcondensatesintheatmospheresofL/Tdwarfs, and tions by latent heating on isobaric surface can generate their condensation levels are closer to the photospheres a wealth of waves that propagate upward to the strato- than another dominant condensates – iron, and so sili- sphere, driving circulation by the dissipation and break- cates could have more influences on the photospheres of ing of these waves (Showman & Kaspi 2013). L/T dwarfs than iron. The saturation pressure function forMgSiO isadoptedfromAckerman&Marley(2001). 3 3. MODEL Alternativesaturationpressurefunctionsforsilicatesare Here we summarize the key aspects of our model; for available, for example, see Visscher et al. (2010). Our detailed implementation see Lian & Showman (2010). study does not aim at precisely determining where con- We solve the three-dimensional hydrostatic primitive densation occurs but rather to explore dynamics driven equationsusinganatmosphericgeneralcirculationmodel by latent heating given a plausible condensation curve (GCM), the MITgcm (Adcroft et al. 2004, see also mit- for a representative condensing species. The latent heat gcm.org). The horizontal momentum, hydrostatic equi- ofsilicatesaresimilarinAckerman&Marley(2001)and librium, continuity, thermodynamic energy and tracer Visscher et al. (2010), therefore, the detailed choice of equations in pressure coordinates are, respectively, the saturation T-P profile is not essential here. The sat- uration pressure function reads dv +fkˆ×v+∇ Φ=0, (3) dt p 58663 K e =exp(25.37− ) bar, (8) s T ∂Φ 1 =− , (4) which is shown by the dashed line in Figure 2 assuming ∂p ρ solar abundance for the mixing ratio of silicates. Here ∂ω we assume that condensate will rain out immediately. ∇ ·v+ =0, (5) p ∂p Theinfluenceofrainoutofcondensablevaporonairden- sity is properly included in the hydrostatic equilibrium dθ θ−θ Lθ q−q =− ref + (δ s), (6) equation where the density is affected by mean molecu- dt τ c T τ lar weight. The replenishment term Q crudely pa- rad p cond deep rameterizes evaporating precipitation and condensable dq q−q =−δ s +Q , (7) species mixed upward from the deeper atmosphere. It dt τ deep cond 5 takes the form Q = (q −q)/τ , where q is T Dwarf deep deep rep deep 10-2 a specified abundance of condensable species in the deep atmosphere and τ is the replenish timescale which is rep styenpictaollbyetsahkoernt c1o0m3 psaecre.dBtootdhynτcaomndicaalndtimτreespcaaleres,cahnod- 10-1 T3rreeff Saturation T-P in this limit the dynamics should be independent of the tdweeopteirmtehsacnaletsh.eTcohnedQendseeaptitoenrmleviesl.applied only at levels ure [bar]100 s The radiation effects of the system are simplified by es using the Newtonian cooling scheme (Equation [6]). For Pr simplicity, the radiative timescale τ is taken to be rad 101 constant through the atmosphere. In our application, the radiative equilibrium potential temperature θ is ref assumed spherically symmetric, and is characterized by two regimes, a nearly adiabatic deeper region and an 102500 1000 1500 2000 2500 3000 3500 4000 Temperature [K] isothermalupperregionasθ (p)=[θn (p)+θn (p)]1/n, ref adi iso where θadi represents the potential temperature of the Figure 2. Equilibrium T-P profile Tref in the model (solid line) nearly adiabatic lower layer, θ represents that of the withsquaresrepresentingthemodelpressurelevels,andthesatu- isothermal upper layer, and nisios a smoothing parame- rationT-Pprofile(dashedline)ofenstatite(MgSiO3)foratypical T dwarf temperature regime assuming 1× solar abundance. The ter that we here set to 15. The equilibrium temperature dottedlineisthecorrespondingpotentialtemperatureθ profile. ref profileisintendedtocrudelymimictheresultsfromone- current model. As stated in Lian & Showman (2010), it dimensional radiative-convection models, where the pro- is useful to first ascertain the effects of large-scale latent file of the upper atmosphere approaches nearly isother- heating associated with the hydrostatic interactions of mal and smoothly transitions to an adiabatic profile in storms with the surroundings, as we pursue here. the lower atmosphere (e.g., Marley et al. 2002; Burrows We include a weak linear damping of velocities simi- etal.2006;Morleyetal.2014). Ourequilibriumtemper- lar to that of Liu & Showman (2013) at pressure larger ature structure is based on a gray radiative-convective than 50 bars to mimic the reduction of winds due to calculationusingtheRosseland-meanopacitytablefrom the Lorentz force and Ohmic dissipation at great depths Freedman et al. (2014), and the radiative-convective wheremagneticcouplingmaybeimportant. Thisdragis boundary from our calculation is in good agreement deep and weak enough (with drag timescale of 100 days with models using realistic opacities (e.g., Tsuji 2002). atthebottom)nottoaffectthedynamicsaboveconden- Thedeepthermalstructureisgenerallyslightlyunstable sation level. rather than strictly neutral to allow dry convective mo- Wesolvetheequationsofourglobalmodelonasphere tions. Weparameterizethetemperaturestructureofthe using the cube-sphere coordinate system (Adcroft et al. adiabatic deep region by 2004; Showman et al. 2009). For most of simulations, p we assume a Jupiter radius, a five hour rotation period θadi(p)=θ0+δθlogp (9) and 500 ms−2 surface gravity. The resolution in our bot nominal simulations is C128, which is equivalent to 0.7◦ where θ0 and δθ are constants and pbot is the bottom pergridlongitudinallyandlatitudinally(i.e.,anapprox- pressure of the simulation domain. δθ is typically taken imate resolution of 512×256 in longitude and latitude). as1 Kfor1 timessolarcases, whichisqualitatively con- Thepressuredomaininourmodelisfrom0.01barto100 sistent with the argument in mixing length theory, but bars,anditisdividedinto55layerswithfinerresolution small enough to not affect the dynamics above the con- on condensation layers as shown in Figure 2. The hori- densation level. Figure 2 shows the equilibrium temper- zontal and vertical resolution is adequate to resolve the ature and saturation vapor T-P profile for a typical T Rossby deformation radius which is the typical length dwarf temperature regime, with silicates’ condensation scale of eddies expected on brown dwarfs, and the va- T-P curve. The dotted line is the corresponding poten- por partial pressure scale height above the condensation tialtemperatureθref profile, inwhichtheadiabaticlayer level, respectively. We do not include an explicit viscos- is characterized by nearly constant θ and the isothermal ity in our simulations, but a fourth-order Shapiro filter layer has an increasing θ with increasing altitude. is added to the time derivative of v and θ to maintain Real moist convection involves the formation of cumu- numerical stability. lus clouds and thunderstorms on a length scale much smaller than that can be resolved by most general cir- 4. RESULTS culation models. There has been a long history of de- 4.1. A Typical T Dwarf velopment for schemes that parameterize the effects of We begin by describing in detail a specific representa- sub-grid-scale cumulus convection on large-scale flows tive case for a typical T dwarf with a radiative timescale resolved by global models (for a review see, e.g., Ray- τ of 106 sec, solar metallicity which is typical for mond & Emanuel 1993). These schemes are often com- rad field brown dwarfs (e.g., Leggett et al. 2010), and other plex, with concepts and parameterizations constrained parameters described in Section 3 (see also Figure 2). by Earth’s atmosphere. It is not yet clear how relevant The spin-up time is about 1500 days for models with thespecificparameterizationoftheseschemesistoatmo- τ = 106 and 107 sec and about 1000 days for model spheres of brown dwarfs and giant planets, so we do not rad with τ = 105 sec. Figure 3 shows a snapshot of a include a moist convection sub-grid-scale scheme in our rad horizontal map of temperature and zonal (east-west) ve- 6 Pressure = 9.1 [bar] Pressure = 9.1 [bar] 40 1692.5 g] 50 1692 g] 50 20 e e d d e [ 0 1691.5 e [ 0 d d 0 u 1691 u atit −50 atit −50 l 1690.5 l −20 −100 0 100 K −100 0 100 m/s Pressure = 3.5 [bar] Pressure = 3.5 [bar] 1283.5 g] 50 g] 50 50 e e d d e [ 0 1283 e [ 0 d d 0 u u atit −50 1282.5 atit −50 l l −50 −100 0 100 K −100 0 100 m/s longitude [deg] longitude [deg] Figure 3. Upper row: snapshot of horizontal map of temperature (left, K) and zonal (that is, east-west) velocity (right, ms−1) as a function of longitude and latitude at pressure level of 9.1 bars at about 1736 Earth days simulation time in an atmospheric circulation model for conditions appropriate to a typical brown dwarf. Lower row: same as the upper row but at pressure level of 3.5 bar, near the tropopause. locity at 1736 Earth days simulation time at 9.1 bars thislayer,andsostrongerinteractionsofmeanflowwith (upper row) near the condensation level. The simula- turbulence and Rossby waves, generating stronger zonal tion reached a statistically equilibrium state, where la- flows. tentheatingfromthecondensatecycleisstatisticallybal- Storms occur mostly near the equator, with almost no anced by radiative cooling, and the upward transport of stormsinmid-to-highlatitudes. Noticethatinourmodel condensablevaporisbalancedbyrainoutinstorms. On setup the equilibrium potential temperature profile θ ref the temperature map (left panel), the local red regions is independent of latitude, such that we can exclude are storms with warm upwelling moist plumes, and they any latitudinal dependent forcing as a possible cause of evolve on a timescale of hours to (Earth) days. The up- banded structure seen in our simulations. Rather, any per right panel in Figure 3 shows the zonal wind map zonal banding, or latitude dependence of storms, must at the same pressure level, with yellow and red colors result from the latitudinal variation of f and β (where representingeastwardvelocity. Threeeastwardjetsform β = df/dy is the gradient of Coriolis parameter with neartheequator,withmaximumwindspeedofabout40 northwarddistance),whicharetheonlysourcesthatcan ms−1. The jets are located where storms are generated, introduce anisotropy in our simulations (Lian & Show- suggestingthatjetsarepumpedbymomentumtransport man 2010). associated with the storms. No jets form at mid-to-high Diagnosing the dynamical mechanism for the latitudi- latitudes, but velocity residuals manifest there, which nal dependent storms is difficult. Instead we offer spec- areRossbywavespropagatingnorthwardandsouthward ulation based on the fact that f and β are the only from the storm regions. possible sources of anisotropy (and therefore of any lat- Theupwellingverticalmotionsarestronglysuppressed itude dependence). A possible reason is that the hori- near the stably stratified isothermal layer at p<2 bars, zontaldivergenceofhorizontalwinds, ∇ ·v,tendstobe p causing large horizontal velocity divergence; as a result, smaller at mid-to-high latitudes, implying smaller verti- the wider spreading of the upwelling hot air produces cal velocities, which makes it more difficult to generate the larger temperature perturbation patterns near the and maintain storms; the suppression of storms in turn tropopause (lower-left panel in Figure 3), similar to sim- further weakens vertical velocities by limiting horizon- ulations for the Jupiter model (Lian & Showman 2010). tal temperature differences which are essential to drive Becausetheascendingairinsidestormsextendvertically horizontal divergence. There are two reasons that we from the storm base to the top near the tropopause, expect small horizontal divergence at mid-to-high lati- the locations of warm regions at 3.5 bars are gener- tudes. First, winds tend to be more geostrophic (the ally correlated to those at about 9 bars. The horizontal balance between Coriolis and pressure gradient forces in zonal velocity map at the tropopause exhibits the simi- the horizontal momentum equation [3]) in higher lati- lar multiple-jet configuration as that at 9 bars but with tudes where the Coriolis parameter f is larger, and this a larger maximum wind speed of about 90 ms−1. The can lead to a smaller horizontal divergence. At low lati- largerhorizontalvelocitydivergencenearthetropopause tudes,windshavelargerageostrophiccomponents,which causes more abundant turbulence and wave sources at results in a larger horizontal divergence. Thus moist in- 7 stability can be more easily triggered. The importance Temperature at Pressure = 10.3 [bar] of rotation can be characterized by the Rossby number, g] 10 de 1751.5 iRzoon=talUw/Linfd, swpheeerdeaUndanhdorLizoanrteatlhleencghtahrascctaelrei,striecsphoecr-- ude [ 0 11775501.5 tively. If Ro (cid:28) 1, winds are nearly geostrophic. We Latit −10 1750 canquantitativelyestimatethelatitudeabovewhichthe Vapor mixing ratio at Pressure = 10.3 [bar] Kx 10−4 flow tends to be geostrophic by taking U ∼ 100ms−1 eg] 10 15 d and L ∼ 106 m, which are approximately the maximum de [ 0 10 rtievlaeltyiv,eanvedlosceitttyinagndRoth∼e w1i,dathndofwaelhoacvael sφto∼rm8,◦r.esTphecis- Latitu −10 5 Vertical velocity at Pressure = 10.3 [bar] is qualitatively consistent with our simulations in which storms tend to clump inside ±10◦ latitudes (Figure 3). deg] 10 0.3 Sveercgoenndc,eeivse∇npif·flvo∼w wateavrneφgweohsetrreopvhiisc,mthereidhioorniazoln(ntaolrtdhi-- Latitude [ −100 000..12 south) velocity and a is the radius of the brown dwarf. Relative vorticity at Pressure = 10.3 [bar] m/sx 10−5 The divergence in pure geostrophic flow comes from the g] 10 gtchoramadteiesevnsetmnoafilnlfergweinoitshhtrigorehpsehprieclcatfltoittwou,dlaehtsoitrauinzddoenslta.arlIgtedriisvieneraglsoeywnectreolbsaeete-- Latitude [de −100 −−0242 itudes. However, the argument here does not mean that Relative vorticity at Pressure = 3.5 [bar] s−1x 10−5 there is no vertical motions in high latitudes in general eg] 10 10 d situations. Infact,ifoneimposesanindependentmerid- de [ 0 5 ionaltemperaturedifferencetotheatmosphere,itiseasy u 0 to generate vertical motion and overturning circulations Latit −10 −5 in high latitudes (e.g., Williams 2003; Lian & Showman 40 50 60 70 80s−1 Longitude [deg] 2008). The difficulty in generating high-latitude storms Figure 4. Snapshots of temperature, vapor mass mixing ratio, here is that the horizontal temperature differences that vertical velocity and relative vorticity in a storm active region at would be required for vertical motions are not indepen- about 1736 days for the nominal T dwarf simulation. Red in ver- dently generated but can only come from the existence ticalvelocitymaprepresentsupwarddirection. Thecondensation of vertical motion (and the associated latent heating). levelforthissimulationisatabout11bars. This additional sensitivity allows for the suppression of storms in situations where vertical motions tend to be Large-scale latent heating drives global atmospheric smaller, as at high latitudes. circulation and forms zonal jets in our simulations, as Stormsregionsarebuoyant,therebycausingascending discussed in Section 2.4. The time-averaged zonal-mean motion, which transports moist air upward from below, zonal jet configuration from simulations with three dif- leading to condensation and latent heating — thereby ferent radiative timescales (τ = 105,106 and 107 sec, rad maintaining the storms themselves. Therefore, storms with other parameters the same as the typical T dwarf are spatially well correlated with vapor mixing ratio and in Section 4.1) are shown in Figure 5. The results are vertical velocities, as shown in the horizontal maps rep- averaged over about 1000 days after the models being resenting a local storm active area in Figure 4. The ver- equilibrated. In general, two strong eastward subtropi- tical relative vorticity ζ = kˆ ·∇×v (lower two panels caljetsformatabout±12◦ latitudesandweakjetsform of Figure 4) measures the local spin of fluid in horizon- in mid-to-high latitudes, which are symmetric about the tal direction. If ζ has the same sign as the Coriolis pa- equator. At the equator, the equatorial jets are gener- rameter f, the storm is cyclonic, whereas the storm is ally westward below the condensation level, and east- anticyclonic if ζ and f have the opposite sign. In the ward equatorial jets appear just above the condensation northern hemisphere, the base of storms (near 9 bars) level. The equatorial jet speed increases with height generally have positive ζ and the top of storms (near 3.5 to the tropopause because of the baroclinic structure bar) generally have negative ζ. The dynamical picture by latent heating, with its strength depending on ra- for a single storm is that: because of the latent heating, diative timescale. Here, baroclinic means that constant the lower density of the storm column causes a greater density surfaces are not aligned with constant pressure vertical spacing of isobars, i.e., constant pressure lines surfaces, whereas barotropic means that the two sur- bow downward at the base of the storm and upward at faces are aligned. The local maximum jet speed near the top of the storm. This causes a low-pressure center the tropopause is caused by the strong dynamical per- at the base of the storm and high pressure center at the turbations from eddies generated at the tops of storms. topofthestormwithrespecttothesurroundingenviron- Jets below the condensation level are generally weak, mentalairatagivenaltitude. Asaresult,horizontalflow and the subtropical jets are presumably driven by the convergesanddivergesduetopressuregradientforcesat Coriolis force2 on the meridional circulation in the deep the base and top of the storm, respectively. Meanwhile, atmosphere that results from the circulation of the up- theflowisacceleratedbyCoriolisforce,whichdrivesthe per active layer (Haynes et al. 1991; Showman et al. flow to cyclonic at the base and anticyclonic at the top 2006; Lian & Showman 2008). The jets extend into of the storms. the upper stably stratified atmosphere. The circulation 4.2. Zonal Jets 2Noticethatbecauseofthefastrotation,theCoriolisparameter f hasalargemagnitudeofabout10−4 s−1 evenat10◦ latitude. 8 above the tropopause probably emerges from the ab- Zonal-mean zonal wind, τ =105 s rad sorption, dissipation and breaking of upward propagat- ing waves that are generated at the tropopause (Show- 40 man & Kaspi 2013). There have been extensive studies 10−1 showingthatthemechanical,wave-inducedforcingisthe bar] 20 ddoesmpiinteantthedreixviesrtefnocresotfraetqousapthoerr-itco-cpirocleultahtieornmaolnfoEracrinthg essure [100 Tropopause 0 Pr (seereviewby,forexample,Andrewsetal.1987;Haynes 101 −20 Condensation 2005). Inconditionsofoursimulatedatmosphereswhere isotropic equilibrium thermal structure is imposed, the −40 wave-induced mechanical forcing should be responsible −80 −60 −40 −20 0 20 40 60 80 m/s Latitude [deg] for the stratospheric circulation. In fact, we have ob- served upward propagating waves from levels perturbed Zonal-mean zonal wind, τ =106 s rad by latent heating in our simulations, which supports our hypothesis. The jet structure exhibits differences with 40 different radiative timescale τ , as τ can affect the 10−1 rate at which the characteristriacdhorizornadtal temperature bar] 20 dFiiffrsetr,enthceesshaonrdt-τdryadnammoicdaell psheortwusrbnaetairolnysbaarreotdroapmicpejedt. Pressure [100 0 structure, whereas relative high τrad models show baro- 101 −20 clinic structure. Second, the jet speed is generally larger ftohrerteheisrmelaotrievetliymelafrogre-jτertasdtmoopdueml,ppruepsubmeafobrleydbyencaamusie- −80 −60 −40 −20 0 20 40 60 80 m/s−40 Latitude [deg] calperturbationsaredampedout; thisrelationhasbeen formulated in Showman & Kaspi (2013) using the quasi- Zonal-mean zonal wind, τ =107 s rad geostrophic theory. The jet structure within about 2 – 3 pressure scale heights of the upper boundary for the 40 model with τ = 107 sec (the lower panel in Figure 5, 10−1 jets from 20◦ratdo 50◦ latitude) is likely affected by the bar] 20 uhpigpheerrbuopupnedrabroyucnodnadriiteison(1s0.−W3ebahravaendtelsetsesd),manoddetlhsewdityh- essure [100 0 Pr namics deeper than about 3 pressure scale heights from 101 −20 the upper boundary remains almost the same as in the original model. We conclude that despite the imperfec- −40 tionofnumericsneartheupperboundaryformodelwith −80 −60 −40 −20 0 20 40 60 80 m/s Latitude [deg] τ = 107 sec, the dynamics presented here for the at- rad Figure 5. Time-averaged zonal-mean zonal wind as a function mosphere below about 0.1 bar is physical. oflatitudeandpressureforthreesimulationswithdifferentradia- tivetimescaleτ : 105 sec(upper),106 sec(middle)and107 sec 4.3. Area Fraction of Storms rad (lower). Dashedlinesineachpanelmarkthepressuresforthecon- Thesmallareafractionofmoistplumeshasbeenvisu- densation level and the tropopause, respectively. The tropopause here is defined as the vertical level above which the lapse rate allyshowninFigure3and4foratypicalTdwarfmodel, dlnT/dlnpislessthan0.268,whichisslightlylessthantheadia- in which the discretized warm areas only occupy a small baticlapserateR/cp =0.2857butenoughtostratifyflowsinside fractionoftheareainlowlatitudeswherestormsareac- storms. tive. Not only the area occupied by storms is small, the are defined as regions outside storms. We only count ar- sizesofindividualstormsarealsosmall,havingdiameter eas within about ±9◦ latitudes since this is the primary of around 2◦ (about the length of 1700 km, or three grid regionwherestormsoccur. Wefirstcountverticalveloc- cells) above the condensation level at around 10 bars for ities as a function of pressure inside and outside storms typical storms in all our models. As shown in Figure using instantaneous snapshots of vertical velocity field 3, the storms slightly expand near the tropopause where from simulations, then define the area fraction of storms flowsexperiencestratificationandexpandlaterally. Here as3 σ (p)=|ω (p)/ω (p)|,whereω (p)andω (p)arethe wedisplayamorequantitativemeasurementofthestorm s d a d a spatiallyaveragedverticalvelocityoutsidestormsandin- fractioninmodelswithdifferentradiativetimescaleτ . rad side storms, respectively. Finally, σ (p) is averaged over Thestormsaredefinedroughlybetweenthecondensation s many snapshots at different simulation time over about level and the tropopause, and they should have both a 1000 days after the simulation equilibrates. The results saturatedmixingratioofvapor(q ≥q )andanupwelling s are shown in Figure 6 as a function of pressure for mod- velocity. Using the upward vertical velocity as an in- els with τ = 105,106 and 107 sec. In the left panel, dicator for storms is reasonable in our case because, as rad the vertical velocity is in a modified log-pressure coor- willbeshownbelow,theupwardverticalvelocitiesinside dinate −Hd(lnp)/dt, where H is pressure scale height; storms are much larger than the descending velocities thisvelocityisapproximatelyequaltotheverticalveloc- outside storms. We tested the sensitivity of this criteria ity in height coordinates. This is a standard way of rep- bychoosingdifferentnumbers,forexample,q ≥0.98q or s resenting vertical velocity in pressure coordinates (e.g., q ≥ 1.02q , and these different criteria do not affect the s results. Wedefineregionssatisfyingthesetwocriteriaas 3 This is a definition based on continuity argument, consistent beinginsidestorms. Regionsnotsatisfyingthesecriteria withthatdefinedinSection2. 9 Zonal−mean zonal wind 2 AAADDDssseeecccsss ======rrrrrraaaaaadddddd111111000000567567ssssss 2 ===rrraaaddd111000567sss 10−1 100 ar] b sure [bar] 34 sure [bar] 34 Pressure [110001 050 s s Pre 5 Pre 5 6 6 −80 −60 −40 −20 0 20 40 60 80 m/s 7 7 Latitude [deg] 8 8 9 9 Figure 7. Time-averaged zonal-mean zonal wind for a T dwarf 10 10 modelwith3timessolarabundanceandτrad=106 sec. 11 11 10-6 10-4 10-2 100 10-5 10-4 10-3 10-2 10-1 Vertical Velocity [m/s] Area Fraction Figure 6. Left: Theabsolutevaluesfortime-averagedascending verticalvelocity(ms−1)insidestorms(solidlines)anddescending 4.3 by constructing scaling relations for the governing velocityoutsidestorms(dashedlines)asafunctionofpressurefor themodelswithτ =105,106 and107 sec. Right: Areafraction equations (3) – (6). The physical picture of the model rad ofstormsσs(p)asafunctionofpressureforthesamemodels. comprisesstatisticallysteadystormsandsubsidenceout- Andrews et al. 1987). Physically, the quantity d(lnp)/dt side the storms. At its top, the storm center has high is the vertical velocity expressed in units of scale heights pressure relative to the surroundings which can drive an per second, with positive being downward. Multiplying outward divergent flow; then the high-entropy air radia- by −H converts this to the vertical velocity in m s−1, tively cools over time during the slow subsidence, reach- with positive being upward. As long as the structure of ing almost the same temperature as the environment at isobars does not change rapidly with respect to z over the condensation level by requirement of steady state. time, this quantity will be approximately equal to the Here we ignore the density variations due to rainout of vertical velocity in height coordinates. The magnitude condensate. The area fraction of storms σs is given by ofdescendingverticalvelocitiesclearlydecreasebyorder requirement of continuity oitfiemsaagrneitsuimdeilawritfhorinτcrraeda=sin1g0τ5raadn.dTτhreadas=ce1n0d6insgecv,elbouct- σs ∼|ωωd|, (10) they are a factor of ∼ 5 smaller for τ = 107 sec. As a rad a result, the area fraction of storms decreases orders of where |ω | (cid:28) |ω |. As presented in Section 4.1, storms d a magnitude as τ increases. We quantitatively explore mostly occur at low latitudes where the Rossby number rad the mechanism controlling the area fraction in Section is large ((cid:38)1). Near the top of the storm, the horizontal 5.1. forcebalanceisprimarilybetweenadvectionandpressure gradient force. In low latitudes, the Coriolis force could 4.4. Enhanced Abundance of Condensate still have a nontrivial magnitude compared to the ad- vection force. Including the Coriolis force in our scaling Giant planets tend to have metal-rich atmospheres, induces only a mild correction to our final result (Equa- having condensed out of the gas-depleted disks around tion [16]), but does not change our conclusion in this preferentiallymetal-enrichedhoststars(Gonzalez1997). section. Thereforeforthesakeofaclearerillustrationof In the context of our model, an enhanced metallicity thephysicalmechanismcontrollingthefractionalareaof means a greater abundance of silicate vapor, a higher storms, using the force balance between advection and latent heating and thus a stronger atmospheric circula- pressuregradientisreasonable. Thereforethebalancein tion. We have ran models with three times solar abun- horizontal momentum equation (3) is v·∇ v ∼ −∇ Φ, dance of silicate vapor, representing the possible con- p p which to order of magnitude reads ditions of directly imaged EGPs. Generally, the basic pattern of the condensation cycle and the zonal jet con- U2 ∆Φ figurationaresimilartothesolarabundancemodels,but ∼ , (11) L L with larger temperature perturbations (proportional to the abundance of vapor), active storms occurring up to where∆Φisthehorizontaldifferenceingravitationalpo- slightly higher latitudes and larger wind speeds. Figure tentialbetweenthetopofthestormanditssurroundings 7 shows the time-averaged zonal-mean zonal wind from on a constant pressure surface. From hydrostatic equi- a model with three times solar abundance (typical for librium (Equation 4), we can estimate the pressure dif- heavy element abundances on Jupiter) and τ = 106 ference inside and outside the storm by integrating over rad sec. Thezonaljetstructureisverysimilartothatofour the column: model with solar abundance (middle panel of Figure 5), ∆Φ∼Rδlnp∆T, (12) exceptthatthewindsareenhancedbyafactorofseveral. whereδlnpistheverticaldifferenceinlog-pressurefrom thebottomtothetopofthestormand∆T isthecharac- 5. DISCUSSIONANDCONCLUSIONS teristichorizontaltemperaturedifferenceinsideandout- 5.1. What Controls the Area Fraction of Moist Plumes? side the storm. From the continuity Equation (5), the Here we construct a simple model to quantitatively horizontaldivergenceatthestormgivenby∇ ·v∼U/L, p understand the area fraction of storms shown in Section is balanced by vertical divergence of ascent inside the 10 storms, This implies numerical results show variation as a function of pres- sure which in general are off by a factor of a few to ten U ω ∼ a, (13) compared to our analytical model. Given the simplicity L δp of the scaling theory, we can explain the order of mag- nitude decrement of area fraction with increasing τ , where δp is the difference in pressure from the bottom rad illustrating the important regulation of radiation on the to the top of the storm. Combining Equation (11), (12) moist convection. and (13), and assuming constant vertical velocity, the Real cloud formation exhibits many complexities not ascending velocity ω can be estimated by a accountedforinthissimplescalingtheory. Forexample, √ δp R∆Tδlnp theintertropicalconvergencezoneinEarth’stropicalre- ω ∼ . (14) a L gion shows organized regions of vigorous cumulus con- vection, containing transient cloud clusters rather than To estimate the descending velocity, we use the thermo- simplyregionsofsteady-stateprecipitationandmeanup- dynamic energy equation (6), and assume that vertical drafts. This is a result of interactions between local cu- advection of the potential temperature is much larger mulusconvectionandlarge-scaleatmosphericcirculation than the horizontal advection. This is reasonable near (Holton & Hakim 2012, Chapter 11). The large-scale la- thetropopausewhereverticaldifferenceofpotentialtem- tent heating scheme in our model does not represent the perature is much larger than the horizontal differences. small-scale cumulus convection, but rather the hydro- We then can obtain the balance between radiative cool- static interaction of the storms with their surroundings. ing and vertical advection, which to order of magnitude To understand the interactions between sub-grid moist reads: ωdδδpθ ∼ θ−τrθardef, where δθ is the vertical difference convectionandlarge-scaleflow, weneedabetterparam- in potential temperature outside storms between pres- eterization of moist convection in future studies. Also, sure levels corresponding to the bottom and the top of radiative feedback by cloud particles can play an impor- storms. Imaginingathermodynamicsloopwhereairrises tant role in the development of cumulus clouds. Our instormsandsubsidesinbetweenstorms,weexpectthat analysis here will be tested using more realistic models at the altitude of the storm top, the environmental air in future studies. outside storms has just been detrained from the top of the storm, and therefore that the potential temperature 5.2. Thermal Structure of the storm air and environmental air are the same at The thermal structure of the atmosphere can be af- thepressureofthestormtop. Likewise, inaclosedther- fecteddirectlybylatentheatingviaitseffectontemper- modynamic loop we expect that the potential temper- ature, and indirectly by the molecular weight effect via ature of environmental and storm air are equal at the introducing a stratification layer above the condensation storm bottom. To close the system, we assume that in a level. The upper panel of Figure 8 shows potential tem- global-mean and steady state, the higher-entropy air de- perature θ as a function of pressure for air outside and scendingfromthetopofstormsradiatesawaymostofits inside of storms from the nominal simulation with ra- entropygainedfromlatentheating,andrelaxestonearly diative timescale τ = 106 sec in Section 4, and the the reference temperature when the air reaches the bot- rad middle panel shows the corresponding virtual potential tom of storms, which implies δθ ∼ θ −θ . Assuming ref temperature θ profile. The virtual potential tempera- constant ω , we can estimate the descending velocity as v d ture θ is defined as θ =(1+q/(cid:15))θ, where (cid:15)=m /m is δp v v 1+q v d ω ∼ . (15) the ratio of molecular mass of condensable species and d τrad the dry air, and q is mass mixing ratio. It can be viewed This equation simply states that the rate of descent is as the theoretical potential temperature that a dry air bottlenecked by the efficiency of radiation: in order for parcel would have if the dry parcel has the same pres- theairoutsidestormstodescendovertheverticalheight sure and density as the moist air, so θv is a direct mea- of a storm, the air has to lose entropy (since the envi- surement of density. The solid line in the upper panel ronment is stratified), and thus this descent must occur is the equilibrium background temperature profile pre- ontimescalescomparabletotheradiativetimeconstant. scribed by Equation (9). In the deep convective region, Finally,theareafractioncanbeobtainedbysubstituting temperatures do not exactly follow the reference profile ω and ω into Equation (10): becausedrymotionstendtoneutralizethethermalstruc- d a ture by having nearly constant θ. However, due to rain- L out of condensates as shown in the lower panel of Figure σ ∼ √ . (16) s τ R∆Tδlnp 8, the layer just above the condensation level is strati- rad fied, and is stable against dry convection. As a result, Th√is is essentially a timescale comparison, where this thin layer just above the condensation level is not L/ R∆Tδlnp is the dynamical ascent timescale driven neutralized by dry motions. The stratification of this by CAPE and τrad is the timescale driven by radiative thin layer is better illustrated by looking at the virtual dissipation. According to results in Section 4, taking potential temperature θ profile in the middle panel, in La1n0∼−d311,00r7e6ssmpeec,c,∆tiwvTeel∼yh.a2vC.e5omaKrpe,aaδrflenrdapctto∼iorne1s.3uσlsatsn∼dfr1oτ0mra−d1si,∼m10u1−l0a25t,iao1nn0d6s wawchittiuhcahhlleyθivgdhietnccirmreeapaslseieessswswittrihtahtvhifiheciegaihgtihto.tnNdaeogstapeiinttsehtatdthreyinfccaorcentavsetichntagitoθnθv, in the right panel of Figure 6, our analytical model to accounting for both temperature and molecular weight order of magnitude agrees well with the maximum area gradients. The red circles represent (virtual) potential fraction for different τrad. The area fraction from our temperatures inside storms, and their profile is nearly