Adam P. Showman , Yohai Kaspi - lpl.arizona.edushowman/publications/showman-kaspi-submitted.pdfetary...

Submitted to ApJPreprint typeset using LATEX style emulateapj v. 08/13/06

ATMOSPHERIC DYNAMICS OF BROWN DWARFS AND DIRECTLY IMAGED GIANT PLANETS

Adam P. Showman1, Yohai Kaspi2

Submitted to ApJ

ABSTRACTA variety of observations now provide evidence for vigorous motion in the atmospheres of brown

dwarfs and directly imaged giant planets. Motivated by these observations, we examine the dynamicalregime of the circulation in the atmospheres and interiors of these objects. Brown dwarfs rotate rapidly,and for plausible wind speeds, the flow at large scales will be rotationally dominated, exhibitinggeostrophic balance between pressure gradient and Coriolis forces. We present three-dimensional,global, anelastic numerical simulations of convection in the interior. Fundamental theory, scalingarguments, and our anelastic simulations all demonstrate that, at large scales, the convection alignsin the direction parallel to the rotation axis. Convection occurs more efficiently at high latitudesthan low latitudes, leading to systematic equator-to-pole temperature differences that may reach ∼1K near the top of the convection zone. The rotation significantly modifies the convective properties.The interaction of convection with the overlying, stably stratified atmosphere will generate a wealthof atmospheric waves, and we argue that, just as in the stratospheres of planets in the solar system,the interaction of these waves with the mean flow will lead to a significant atmospheric circulation atregional to global scales. At scales exceeding thousands of km, this should consist of geostrophicallybalanced, stratified turbulence (possibly organizing into coherent structures such as vortices and jets)and an accompanying overturning circulation. We present a semi-quantitative, analytic theory of thiscirculation as a function of the wave-driving efficiency. For plausible efficiencies, this theory predictscharacteristic horizontal temperature variations of several to ∼50 K, horizontal wind speeds of ∼10–300 m s−1, and vertical velocities that advect air over a scale height in ∼105–106 s. The vertical mixingassociated with this large-scale circulation may help to explain the chemical quenching of CO andNH3 observed on some brown dwarfs. Moreover, the implied large-scale organization of temperatureperturbations and vertical velocities suggests that, near the L/T transition, patchy clouds can formnear the photosphere, helping to explain recent observations of brown-dwarf variability in the near-infrared.Subject headings: stars: low-mass, brown dwarfs; convection; waves; turbulence

1. INTRODUCTIONSince the discovery of brown dwarfs beginning in the

mid-1990s, our understanding of the atmospheric struc-ture of these objects has grown ever more sophisticated.Approximately 1000 brown dwarfs, and a handful of di-rectly imaged planets, have now been discovered. Obser-vational acquisition of infrared (IR) spectra for many ofthese objects have allowed the definition of the L, T, andY spectral classes (e.g., Kirkpatrick 2005; Cushing et al.2011). The theory for these objects now encompasses abroad understanding of their evolution, radii, luminosity,molecular composition, spectra, and colors, and includesprescriptions for condensate formation and rainout, sur-face patchiness, and disequilibrium chemistry. Notably,however, these theoretical advances have relied heavilyon one-dimensional (1D) models for the atmospheric ra-diative transfer and interior evolution (for an early re-view, see Burrows et al. 2001). By comparison, littleeffort has been made to understand the global, three-dimensional atmospheric dynamics of these substellarbodies.

Yet there is increasing evidence that brown dwarfs ex-hibit vigorous atmospheric circulations. This evidence

1 Department of Planetary Sciences and Lunar and PlanetaryLaboratory, The University of Arizona, 1629 University Blvd., Tuc-son, AZ 85721 USA; [email protected]

2 Center for Planetary Science, Weizmann Institute of Science,Rehovot, Israel

falls into three main classes. First, L dwarfs, particularlyof later spectral type, show a reddening of near-infrared(e.g., J −K) colors that indicate the presence of silicateclouds in the visible atmospheres (e.g., Kirkpatrick et al.1999; Kirkpatrick 2005; Chabrier et al. 2000; Tsuji 2002;Cushing et al. 2006; Knapp et al. 2004). Since cloudparticles would gravitationally settle in the absence ofdynamics, such clouds imply the presence of atmosphericvertical mixing necessary to keep the particles suspended.In the cooler T dwarfs, the condensation occurs progres-sively deeper and, for objects with sufficiently low ef-fective temperature, eventually no longer influences theinfrared spectrum. However, the L/T transition itselfremains poorly understood; it occurs over a surprisinglysmall range of effective temperature and accompanies aJ-band brightening that are not easily captured by stan-dard 1D models (Chabrier et al. 2000; Allard et al. 2001;Burrows et al. 2006b; Saumon & Marley 2008). Hypothe-ses that have been put forward to resolve this discrepancyare that, across the transition, the cloud sedimentationefficiency changes (Knapp et al. 2004) or that the cloudsbecome patchy, allowing contributions from both cloudyand cloud-free regions to affect the disk-integrated emer-gent spectrum (Burgasser et al. 2002; Marley et al. 2010).In both cases, a role for atmospheric dynamics in modu-lating the clouds is implicated.

A second line of evidence for atmospheric circulationcomes from chemical disequilibrium of CO, CH4, and

2 Showman and Kaspi

NH3 inferred for many cool brown dwarfs. Late Tdwarfs have sufficiently cool atmospheres that the pre-ferred chemical-equilibrium forms of carbon and nitrogennear the photosphere are CH4 and NH3, respectively; incontrast, CO and N2 dominate under the high-pressureand temperature regions at depth. Fitting of infrared(IR) spectra to radiative transfer models shows that, nearthe photosphere, many T dwarfs exhibit an overabun-dance of CO and an underabundance of NH3 relative tochemical equilibrium. This can be attributed to verti-cal transport of CO-rich and NH3-poor air from depthand the subsequent chemical quenching of these disequi-librium mixing ratios due to the long chemical intercon-version timescales in the low-pressure, low-temperatureregions near the photosphere. This story was first workedout for CO on Jupiter (Prinn & Barshay 1977; Bézardet al. 2002; Visscher & Moses 2011) and then for bothCO and NH3 on Gl 229b (Fegley & Lodders 1996; Nollet al. 1997; Griffith & Yelle 1999; Saumon et al. 2000).Subsequently, chemical disequilibrium and vertical mix-ing have been inferred in the atmospheres of a wide rangeof T dwarfs (Saumon et al. 2006, 2007; Hubeny & Bur-rows 2007; Leggett et al. 2007b,a, 2008, 2010; Stephenset al. 2009).

Third, recent near-IR photometric observationsdemonstrate that several brown dwarfs near the L/Ttransition exhibit large-amplitude variability over rota-tional timescales, probably due to cloudy and relativelycloud-free patches rotating in and out of view. The pos-sibility of weather on brown dwarfs has long motivatedsearches for variability. Recently, Artigau et al. (2009)observed the T2.5 dwarf SIMP0136 in J andKs band andfound peak-to-peak modulations of ∼5% (∼50 mmag)throughout the inferred 2.4-hr rotation period. Radiganet al. (2012) observed the T1.5 dwarf 2M2139 in J , H,and Ks and found peak-to-peak variations of up to ∼25%with an inferred rotation period of either 7.7 or 15.4 hr.The relative amplitudes of the variability at J , H, andKsplace strong constraints on the cloud and thermal struc-ture associated with the variability. These authors con-sidered models where the variability resulted from lateralvariations in effective temperature alone (with no vari-ations in the cloud properties), lateral variations in thecloud properties alone (with no variation in effective tem-perature), and lateral variations in both temperature andcloud properties. The observations rule out models witha uniform cloud deck and instead strongly favor modelswith significant lateral variations in both cloud opacityand effective temperature; the relatively cloud-free re-gions exhibit effective temperatures ∼100–400 K greaterthan the cloudier regions. This suggests a picture withspatially distinct regions of lower and higher condensateopacity, where radiation escapes to space from lower-pressure, cooler levels in the high-opacity regions anddeeper, warmer levels in the low-opacity regions. Theobservations of Artigau et al. (2009) and Radigan et al.(2012) both show that the lightcurves vary significantlyover intervals of several Earth days, indicating that theshape, orientation, or relative positions of the low- andhigh-condensate opacity regions evolve over timescales ofdays.

In addition to these observations of field brown dwarfs,growing numbers of young, hot extrasolar giant plan-ets (EGPs) are being imaged and characterized. Promi-

nent discoveries include planetary-mass companions toβ Pic, 2M1207, and HR 8799. Multi-band photome-try already indicates that 2M1207b and several of theHR 8799 planets exhibit clouds and probably disequilib-rium chemistry similar to that inferred on brown dwarfs(Hinz et al. 2010; Bowler et al. 2010; Currie et al. 2011;Galicher et al. 2011; Skemer et al. 2011; Madhusudhanet al. 2011; Barman et al. 2011a,b; Marley et al. 2012).With effective temperatures exceeding ∼1000 K, theseplanets radiate IR fluxes & 105 W m−2, orders of magni-tude greater than the flux received by their primary star.Stellar irradiation is therefore negligible to their dynam-ics. From a meteorology perspective, this population ofbodies will therefore resemble low-mass, low-gravity ver-sions of free-floating brown dwarfs. With next-generationtelescope facilities, including the Gemini Planet Imagerand SPHERE, significant numbers of new planets willbe discovered, greatly opening our ability to probe plan-etary meteorology at the outer edge of stellar systems.

These existing and upcoming observations providestrong motivation for investigating the global atmo-spheric dynamics of brown dwarfs and directly imagedplanets. As yet, however, no investigations of the globalatmospheric circulation of brown dwarfs have been per-formed. The only study of brown dwarf atmospheric dy-namics published to date is that of Freytag et al. (2010),who performed two-dimensional, non-rotating convectionsimulations in a box ∼400 km wide by ∼150 km tall.Their study provides valuable insights into the role ofconvectively generated small-scale gravity waves in caus-ing vertical mixing. Nevertheless, dynamics on scales oftens to hundreds of km differs substantially from that onglobal scales of 104–105 km, and thus, for understand-ing the global-scale circulation—including the implica-tions for variability—it is essential to consider global-scale models.

Here, we aim to fill this gap by presenting the firstglobal-scale models of brown dwarf atmospheric dynam-ics. Rotation periods of L and T dwarfs inferred fromspectral line broadening range from ∼2 to 12 hours (Za-patero Osorio et al. 2006; Reiners & Basri 2008), inline with the periods of SIMP0136 and 2M2139 inferredfrom lightcurve modulation (Artigau et al. 2009; Radi-gan et al. 2012). We will show that, at these periods,rotation dominates the global-scale dynamics and willconstitute an overriding factor in controlling the three-dimensional wind and temperature structure. We firstpresent theoretical arguments to highlight the fundamen-tal dynamical regime in which brown dwarfs lie, to showhow rotation organizes the large-scale wind, establishessystematic temperature differences, and shapes the con-vective velocities (Section 2). We next present global,three-dimensional numerical simulations of the interiorconvection of brown dwarfs that confirm our theoreticalarguments and provide insights into the detailed, time-evolving global wind and temperature patterns and dy-namical timescales (Section 3). We then consider the dy-namics of the stably stratified atmosphere that overliesthe convective interior, demonstrating how large-scalevortices and/or zonal3 jets are likely to emerge from in-

3 Zonal refers to the east-west (longitudinal) direction whereasmeridional refers to the north-south (latitudinal) direction. Zonaland meridional winds are winds in the eastward and northward

Atmospheric dynamics of brown dwarfs 3

teractions with the interior (Section 4). Next, we con-sider observational implications, since IR radiation tospace typically occurs from within this stratified layerand hence infrared spectra and lightcurves are stronglyshaped by its dynamics (Section 5). The final sectionconcludes (Section 6).

We emphasize that our goal is to provide a theoreticalfoundation for understanding the atmospheric dynamicsof rapidly rotating, ultracool dwarfs and young EGPs,broadly defined. As such, we emphasize dynamical con-siderations, and intentionally simplify our models by ex-cluding clouds, chemistry, and detailed representation ofradiative transfer. This provides a clean environment inwhich to identify key dynamical processes and constructa theoretical foundation for more realistic studies thatwill surely follow.

2. BACKGROUND THEORY: APPLICATION TOBROWN DWARFS

Here, we review basic concepts in atmospheric fluiddynamics and apply them to brown dwarfs to understandthe large-scale structure of the flow.

2.1. Importance of rotationBrown dwarfs rotate rapidly, and this will exert a ma-

jor influence on their atmospheric dynamics. To demon-strate, consider the momentum equation for a rotatingfluid, which is given in the rotating reference frame ofthe brown dwarf by

dvdt

+ 2Ω× v =−1ρ∇p−∇Φ, (1)

where dv/dt = ∂v/∂t + v · ∇v is the material deriva-tive, v is the 3D velocity vector, Ω is the planetary ro-tation vector, Φ is a force potential which includes boththe gravitational and centrifugal accelerations, ∇ is thethree-dimensional gradient operator, t is time, and p andρ are the pressure and density fields respectively (Ped-losky 1987). For the purpose of this discussion we willassume the dynamics are inviscid. We also for the presentneglect the Lorentz force, as appropriate for the atmo-spheres and molecular envelopes of cool brown dwarfs;nevertheless, the Lorentz force will be crucial in the deepinterior, and return to a discussion of it in Section 3.3.

The nature of the flow depends on the Rossby num-ber, given by the ratio of the advective and Coriolisaccelerations, Ro = U/ΩL, where U is a characteris-tic wind speed, L is a characteristic length scale of theflow, and Ω is the rotation rate (2π over the rotation pe-riod). If Ro � 1, the flow is rotationally dominated; ifRo ∼ 1, rotation is important but not dominant, whileif Ro � 1, rotation plays a minor role (see, e.g., Vallis2006, p. 84). Because of their fast rotation rates, Jupiter,Saturn, Uranus, and Neptune, as well as the extratropi-cal atmospheres of Earth and Mars, all exhibit Ro � 1at large scales; the only solar-system atmospheres whererotation does not dominate are Titan and Venus.

Estimating Ro for brown dwarfs requires knowledgeof wind speeds and flow length scales, which are un-known.4 Nevertheless, Artigau et al. (2009) show that

directions, respectively. Zonal jets refer to atmospheric jet streamsoriented in the east-west direction.

4 Here, we seek to understand the global-scale flow and the

Fig. 1.— Rossby numbers expected on brown dwarfs as a functionof rotation period and characteristic wind speed, assuming dynam-ical features are global in scale (length equals one Jupiter radius).Rossby numbers range from ∼10−4 to 0.1, indicating that theregional- and global-scale dynamics in brown-dwarf atmosphereswill be rotationally dominated over a wide range of parameters.

if the evolution of their light curve shapes over intervalsof days is interpreted as differential zonal advection ofquasi-static features by a latitude-dependent zonal wind,the implied differential rotation is ∼1%, which for the2.4-hour rotation period of SIMP0136 implies a zonalwind speed of ∼300–500 m s−1 depending on the latitudeof the features. A similar analysis by Radigan et al.(2012) suggests a possible zonal wind speed of ∼45 m s−1for 2M2139, although they caution that this estimate re-lies on rather tentative assumptions. Interestingly, thesevalues bracket the range of wind speeds measured forthe giant planets in the solar system, which range fromtypical speeds of ∼30 m s−1 on Jupiter to ∼300 m s−1 onNeptune (e.g., Ingersoll 1990).5 Later, we show that thelarge-scale winds in the convective interior of a browndwarf are likely to be weak (Section 3.3), but that windspotentially exceeding ∼102 m s−1 could develop in thestratified atmosphere (Section 4).

Regarding length scale, the fact that SIMP0136 and2M2139 exhibit large-amplitude variability hints that at-mospheric features could be near-global in size (partic-ularly for 2M2139 where variability reaches 25%). Thiswould imply L ∼ RJ ∼ 7× 107 m, where RJ is Jupiter’sradius. On the other hand, only a small fraction of

appropriate values are therefore not the convective velocities andlength scales but the wind speeds and length scales associated withany organized jets (i.e. zonal flows) and vortices that may exist.

5 The maximum observed speeds, expressed as a difference be-tween the peak eastward and peak westward zonal winds, are sev-eral times these typical values, reaching ∼200 m s−1 on Jupiter and∼600 m s−1 on Neptune.

4 Showman and Kaspi

brown dwarfs exhibit such large variability and it is pos-sible that length scales are typically smaller; for example,L ∼ 107 m on Jupiter, Saturn, Uranus, and Neptune.

Adopting length scales L ∼ 107–108 m, wind speedsU ∼ 10–1000 m s−1, and rotation rates of 2 to 10 hoursyields Rossby numbers ranging from 0.0001 to 0.4. Fig-ure 1 shows the Rossby number as a function of windspeed and rotation period for the case of global-scaleflows. The values are much less than one everywhereexcept for the largest wind speeds and slowest rotationperiods considered. This implies that, in general, thelarge-scale circulation on brown dwarfs will be rotation-ally dominated. The Ro� 1 condition on brown dwarfsimplies that the flow is geostrophically balanced, thatis, the primary balance in the momentum equation isbetween Coriolis and pressure-gradient forces (Pedlosky1987).

2.2. Organization of flowSignificant insight into the flow structure can be ob-

tained from the vorticity balance. Taking the curl of(1) gives a vorticity equation of the form (e.g., Pedlosky1987)

∂ω

∂t+(2Ω + ω) ·∇v− (2Ω + ω)∇·v = −∇ρ×∇p

ρ2, (2)

where ω = ∇ × v is the relative vorticity. The termon the right side, called the baroclinic term, is nonzerowhen density varies on constant-pressure surfaces. Notethat, since ω scales as U/L, the ratio ω/Ω ∼ Ro, andthe time-derivative term is generally order Ro smaller inmagnitude than the second and third terms on the leftside. Taking Ro� 1, appropriate to the flow on a browndwarf, yields a leading-order vorticity balance given by

2Ω · ∇v − 2Ω∇ · v =−∇ρ×∇pρ2

. (3)

One might expect that convection homogenizes the en-tropy within the convection zone, in which case densitydoes not vary on isobars and ∇ρ × ∇p = 0. This iscalled a barotropic flow. In this case, Eq. (3) simply be-comes the compressible-fluid generalization of the Taylor-Proudman theorem, which, expressed in a cylindrical co-ordinate system centered on the rotation axis, is

∂u

∂ẑ=∂v⊥∂ẑ

= 0 (4)

∇⊥ · v⊥ = 0. (5)where ẑ is the direction parallel to the rotation axis,u is the azimuthal (zonal) velocity, v⊥ the velocity to-ward/away from the rotation axis, v⊥ = (u, v⊥) is the ve-locity in the plane perpendicular to the rotation axis, and∇⊥ is the gradient operator in the plane perpendicular tothe rotation axis. The theorem states that if the flow hasa small Rossby number and is invicsid and barotropic thefluid motion will be completely two-dimensional, there-fore there will be no variation in the fluid velocity alongthe direction of the axis of rotation. The fluid then movesin columns aligned with the rotation axis. Note that noconstant-density assumption was made; Equations (4)–(5) hold in a barotropic, geostrophic, low-viscosity fluideven if the density varies by orders of magnitude acrossthe system.

Within a spherical planet or brown dwarf, suchcolumns can easily move in the zonal (east-west) direc-tion. However, the columns cannot easily move towardor away from the rotation axis, because this changesthe length of the columns and the local density withinthem, both of which induce non-zero ∇⊥ · v⊥ that vio-late the theorem. For such a barotropic fluid, the pre-dominant planetary-scale circulation therefore consists ofzonal (east-west) wind whose speed varies minimally inthe direction along the rotation axis; by comparison, thenorth-south flow is weak. Of course, the theorem is onlyvalid to order Ro, and so motions toward/away from therotation axis—as well as motions parallel to the fluidcolumns—can occur but only with amplitudes ∼Ro lessthan that of the primary zonal flow.6

In reality, turbulent convection results in horizontalentropy gradients and therefore the fluid is not in abarotropic state leading to a non-vanishing term on theright side of Eq. (3). As a result, shear can developalong the ẑ direction. Considering the zonal componentof Eq. (3) yields

2Ω∂u

∂ẑ= −∇ρ×∇p

ρ2· λ̂ (6)

where λ̂ is the unit vector in the longitudinal direction. Ifthe flow exhibits minimal variation in longitude, then itcan be shown that |∇ρ×∇p| = |∇p|(∂ρ/∂y)p. Since |∇p|is overwhelmingly dominated by the hydrostatic compo-nent, we have to good approximation (Showman et al.2010)

2Ω∂u

∂ẑ=

g

ρr

(∂ρ

∂φ

)p

. (7)

where φ is latitude and r is radial distance from the cen-ter of the planet. Thus, variations in the geostrophicwind along ẑ must be accompanied by variations in den-sity on isobars. This relation, well-known in atmosphericdynamics, is called the thermal-wind equation.

By itself, however, the preceding theory gives little in-sight into the spatial organization—columnar or not—ofthe internal entropy perturbations and any thermal-windshear that accompanies them. An alternative point ofview that sheds light on this issue is to consider the an-gular momentum budget. The angular momentum perunit mass about the rotation axis is given by

M ≡MΩ +Mu = Ωr2 cos2 φ+ ur cosφ, (8)where the first and second terms represent the contri-butions due to the planetary rotation and winds in thatrotating frame, respectively. Writing the zonal momen-tum equation in terms of angular momentum yields (e.g.,Peixoto & Oort 1992, Chapter 11)

ρdM

dt= − ∂p

∂λ. (9)

It is useful to decompose the pressure and density intocontributions from a static, wind-free reference state andthe deviations from that state due to dynamics. Whenwind speeds are less than the speed of sound, these dy-namical density and pressure perturbations are small,

6 Of course, at very small scales, the Rossby number exceedsunity and the convection at these small scales will not organize intocolumns (e.g., Glatzmaier et al. 2009). The columnar organizationapplies only at scales sufficiently large that Ro� 1.


leading to a continuity equation ∇ · (ρ̃v) = 0, where ρ̃ isthe reference density profile.7 Motivated by the fact thatthe convective eddies drive a mean flow, we representthe dynamical variables as the sum of their zonal means(denoted by overbars) and the deviations therefrom (de-noted by primes), such thatM = M+M ′, v = v+v′, etc.Here, we refer to these overbarred quantities as the meanflow and the primed quantities as the eddies. Substitut-ing these expressions into Eq. (9) and zonal averagingleads to the zonal-mean momentum equation (cf Kaspiet al. 2009)

ρ̃∂M

∂t+∇ · (ρ̃vM) +∇ · (ρ̃v′M ′) = 0. (10)

This equation states that temporal changes to the zonal-mean angular momentum at any given location (firstterm) can only result from advection of the zonal-meanangular momentum by the mean flow (second term) orchanges in the zonal-mean angular momentum due totorques caused by eddy motions (third term). In a sta-tistical steady state, M equilibrates and the equationsimply becomes

∇ · (ρ̃vM) +∇ · (ρ̃v′M ′) = 0. (11)Since the ratio of Mu to MΩ is essentially the Rossbynumber, it follows that for the Ro � 1 regime expectedon a brown dwarf, M ≈MΩ. Thus, for a rapidly rotatingbrown dwarf, surfaces of constant angular momentum arenearly parallel to the axis of rotation. Using this result,along with the continuity equation, Eq. (11) becomes

v · ∇MΩ = −1ρ̃∇ · (ρ̃v′M ′). (12)

This result has major implications for the circulationon rapidly rotating giant planets and brown dwarfs. Theequation implies that the mean flow, v, can only cross an-gular momentum surfaces in the presence of eddy corre-lations between v′ and M ′ (such eddy correlations causea torque that changes the zonal-mean angular momen-tum following the mean flow, as necessary for the meanflow to cross angular momentum surfaces). When sucheddy effects are small, or if the flow is axisymmetric withno variation in longitude (for which v′ and M ′ are zeroby definition), then v ·∇MΩ = 0. In such a situation, themean flow must to leading order be parallel to surfacesof constant angular momentum (Busse 1976; Kaspi et al.2009) and there can be no flow crossing these surfaces.This constraint places no limitation on the zonal-meanzonal flow, u, but requires the meridional circulation vto be small. It is important to emphasize that this con-straint differs from the Taylor-Proudman theorem, sinceit does not require the flow to be barotropic, nor does itstate that u is independent of ẑ.

But how important are the eddy torques on the right-hand side of Eq. (12)? At small Rossby number, theplanetary rotation contains so much angular momentumthat even in the presence of vigorous convection, eddytorques are unable to drive a rapid mean-meridional cir-culation; therefore, we still have v · ∇MΩ ≈ 0 at leading

7 The reference density will generally be a function of radius.Note that this approximate continuity equation (essentially theanelastic approximation) filters sound waves from the system,which is a reasonable approximation as long as wind speeds areless than the speed of sound.

order. To show this, we can estimate the timescale of themeridional circulation and compare it to the characteris-tic timescale for convection. The characteristic timescalefor convection to traverse the interior is τconv ∼ D/w,where D is a thickness of the layer in question (e.g., theplanetary radius) and w is the characteristic convectivespeed. We can estimate the timescale for the mean flowto cross angular momentum contours as follows. Theeddy correlation v′M ′ is just v′u′r⊥, where r⊥ is the dis-tance from the rotation axis. Under the assumption thatthe eddy velocities scale with the convective velocitiesw, we can write v′M ′ ∼ Cr⊥w2 where C is a correla-tion coefficient equal to one when u′ and v′ are perfectlycorrelated and equal to zero when u′ and v′ exhibit nocorrelation. To order of magnitude, Eq. (12) then be-comes

v⊥Ωr⊥ ∼ Cw2 (13)

and the meridional velocity therefore has a characteristicmagnitude

v⊥ ∼Cw2

Ωr⊥(14)

Defining a timescale for the meridional circulation,τmerid = r⊥/v⊥, implies that

τmerid ∼ τconvr⊥ΩCw

(15)

which can be expressed as

τmerid ∼τconv

C Roconv(16)

where Roconv = w/r⊥Ω is a convective Rossby numbergiving the ratio of the convective velocities to the typ-ical rotational velocity of the planet in inertial space.For typical brown dwarfs, where rotational velocities aretens of km s−1, we expect Roconv � 1; given the expectedconvective velocities (see §3), we expect Roconv ∼ 10−5–10−2. Therefore, the timescale for the meridional cir-culation is several orders of magnitude longer than thecharacteristic convection timescale.

This also means that the convective heat transport willbe more efficient along (rather than across) surfaces ofconstant angular momentum. Fig. 2 shows the onset ofconvection for an experiment driven by a constant heatflux at the bottom boundary. Two models are shown, arapidly rotating case on the left and a slowly rotating caseon the right. While for the larger Rossby number casethe dominant driving force for the turbulent plumes is thebuoyancy and therefore the plumes are driven away fromthe center of gravity, for the small Rossby number casethe convection becomes aligned along the direction of theaxis of rotating demonstrating the angular momentumconstraint (v · ∇MΩ = 0). Note that for small Rossbynumber experiment only close to the boundaries doesthe Rossby number approach one and therefore there theconvective cells can close. Thus rotation strongly mod-ulates the heat transport from the interior of the browndwarf at large scales.

6 Showman and Kaspi

3. THREE-DIMENSIONAL CIRCULATION MODELOF CONVECTION-ZONE DYNAMICS

3.1. ModelWe solve the fluid equations for a self-gravitating, con-

vecting, three-dimensional, rotating brown dwarf. Weadopt the anelastic system (e.g., Ogura & Phillips 1962;Gilman & Glatzmaier 1981; Ingersoll & Pollard 1982),which assumes that dynamics introduces only small per-turbations of the density, entropy, and pressure from aspecified reference state, which we here take to be isen-tropic. Dynamical density perturbations then enter themomentum equations in the buoyancy term but do notappear in the continuity equation; this has the effect offiltering acoustic waves from the system. The anelasticsystem is appropriate for the fluid interior of a browndwarf, where dynamical perturbations of entropy, den-sity, and pressure due to convection are expected to bemodest and convection should lead to a nearly constantentropy throughout. Although this study represents itsfirst application to brown dwarfs, the anelastic systemhas previously been used with great success for under-standing convection in Jupiter and Saturn (Kaspi et al.2009; Jones & Kuzanyan 2009; Glatzmaier et al. 2009;Showman et al. 2011) and stellar interiors (Miesch &Toomre 2009, and references therein).

Our particular implementation is that of Kaspi et al.(2009). The momentum, continuity, and energy equa-tions, respectively, are given by

∂v∂t

+(2Ω+ω)×v = −1ρ̃∇p′−ρ

′

ρ̃∇Φ−1

2∇v2+ν∇2v (17)

∇ · (ρ̃v) = 0 (18)

∂s′

∂t+

1ρ̃∇ · (ρ̃vs′)− 1

ρ̃∇ · (ρ̃κ∇s′) = Q

T̃(19)

where Q is thermodynamic heating/cooling per mass, νis the kinematic viscosity, κ is the thermal diffusivity, andother quantities are as defined previously. Here, both νand κ are taken as constants and are intended to parame-terize small-scale eddy mixing. The quantities ρ̃(r), p̃(r),and T̃ (r) are the radially varying reference profiles ofdensity, pressure, and temperature, respectively; ρ′ andp′ are the deviations of the density and pressure fromtheir local reference values, such that the total pressureand density are ρ = ρ̃+ ρ′ and p = p̃+ p′. Likewise, s′ isthe deviation of entropy from its reference state value.

The system is closed with an equation of state, whichenters through the reference profiles and through therelationship between the density, entropy, and pressureperturbations in the anelastic system

ρ′

ρ̃=

1ρ̃

(∂ρ

∂s

)p

s′ +1ρ̃

(∂ρ

∂p

)s

p′ ≡ −αss′ + βp′, (20)

where αs and β are the radially varying isobaric entropyexpansion coefficient and isentropic compressibility, re-spectively, along the model’s radially varying referenceprofile. Here, we adopt the SCVH equation of state(EOS) for hydrogen-helium mixtures (Saumon et al.1995). Given a specified internal entropy and the as-sumption that the reference state is in hydrostatic bal-

ance8, this EOS allows us to calculate the radially vary-ing reference profiles ρ̃, p̃, and T̃ (e.g., Guillot & Morel1995; Guillot et al. 2004) as well as the radial profiles ofαs and β along the reference adiabat.

Many studies of convection in rotating spherical shellsforce the system by passing a heat flux through imperme-able upper and lower boundaries, either with a constant-temperature or constant heat flux boundary condition(e.g., Christensen 2001, 2002; Aurnou & Olson 2001;Heimpel et al. 2005, and many others). However, this isunrealistic in the context of a substellar object. At highRayleigh numbers, passing a heat flux through the modelboundaries will lead to thin hot and cold boundary lay-ers at the bottom and top boundaries, respectively, whichdetach and form hot and cold convective plumes that insome cases can dominate the dynamics. Because realbrown dwarfs are fluid throughout, the bottom bound-ary layer, in particular, is unrealistic. Instead, we forcethe system by imposing a vertically distributed sourceof internal heating and cooling throughout the bottomand top portions of the domain, respectively, thus al-lowing outward convective transport of heat without thedevelopment of artificial boundary layers (for more de-tail see Kaspi et al. 2009).9 The top and bottom thermalboundary conditions correspond to zero heat flux. Thetop and bottom mechanical boundary conditions are im-permeable in radial velocity and free-slip in horizontalvelocity.

We solve the equations in spherical geometry usingthe state-of-the-art circulation model MITgcm (Adcroftet al. 2004), which Kaspi (2008) adapted for anelasticsimulations of the deep convective envelopes of giantplanets. The equations are solved using a finite-volumediscretization on a staggered Arakawa C grid (Arakawa& Lamb 1977) in longitude and latitude. Our typical res-olution is 1◦ in longitude and latitude with 120 verticallevels spaced to give enhanced resolution near the top ofthe domain where the pressure and density scale heightsare the smallest (see Kaspi et al. 2009). Most models ex-tend the full 360◦ in longitude and in latitude from 80◦Sto 80◦N. For some parameter variations, we performedsimulations in sectors 90◦ of longitude wide (using a pe-riodic boundary condition between the eastern and west-ern boundaries) with a resolution of 2◦ of longitude andlatitude and 120 vertical levels. For all models, the outerand inner boundaries are spherical surfaces with radii of1RJ and 0.5RJ respectively. This choice of inner bound-ary is sufficiently deep to minimize any artificial effect ofthe lower boundary on the surface dynamics. We gen-

8 We emphasize that the dynamical model itself is non-hydrostatic; hydrostatic balance is used only in defining the ref-erence state.

9 Real brown dwarfs of course do not have substantial internalheat sources (the burning of deuterium not playing a role exceptin the most massive objects) but rather decrease in entropy verygradually over multi-Byr timescales. Thus, ideally, one would liketo set up the problem with a heat sink near the top (i.e., cooling)and no heat source near the bottom, thereby allowing the internalentropy to decline with time in a brown-dwarf like fashion. The dif-ficulty is that, due to computational limitations, achieving steadystate requires the system to be overforced (Showman et al. 2011),and without a source of energy near the bottom the interior en-tropy would decline unrealistically rapidly. Adding a heat sourcenear the bottom, as we have done, allows the global-mean inte-rior entropy to be essentially constant over dynamical timescales,consistent with expectations for brown dwarfs.


erally use a Jupiter-like interior reference profile with apressure at the outer surface of 1 bar and 20 Mbar at thebottom boundary. The interior reference density variesby a factor of over 104 from the 1-bar level to the deepinterior (see Kaspi et al. 2009). All simulations are spunup from rest using an initial thermal profile correspond-ing to the reference profile, and are integrated until astatistical steady state is achieved.

3.2. Results: Convective and thermal structureWe perform simulations using rotation periods ranging

from 3 to 200 hours (spanning the typical range observedfor brown dwarfs) as well as an additional sequence ofparameter variations adopting rotation periods as longas 2000 hours to illustrate the effect of rotation on thedynamics.

Before presenting models for fully equilibrated browndwarfs, we first demonstrate with a pedagogical examplethe crucial importance that rotation plays in the brown-dwarf parameter regime. Figure 2 depicts the tempera-ture structure during the spin-up phase for two modelsthat are identical except for the rotation period, whichis 2000 hours in the model on the right and 10 hours forthe model on the left. The models in Figure 2 are not in-tended to be realistic brown dwarf models (for example,they are forced by a heat flux from the bottom boundary,which is not realistic in the context of a brown dwarfs)but are instead simply an illustration of the importanceof rotation in the brown dwarf parameter regime. Nev-ertheless, the models do have a realistic Jovian interiorstructure, with density increasing by a factor of ∼104from the interior to the exterior.

In the slowly rotating model (right panels of Figure 2),the Coriolis forces are sufficiently weak that the Rossbynumber is &1, so that rotation plays a negligible role inthe dynamics. Convective plumes rise from the lowerboundary and ascend quasi-radially toward the outerboundary. The plumes are equally able to radially tra-verse the domain whether they emanate from the polaror equatorial regions, and to zeroth order, the convectionappears to be isotropic.

In contrast, the rapidly rotating model exhibitsRo � 1 and is thus rotationally dominated (Fig. 2, leftpanels). As predicted by the theory in Section 2, theconvection develops a columnar structure. Plumes thatemerge in the polar regions can ascend and descend ra-dially while remaining at nearly constant distance fromthe rotation axis; they therefore easily traverse the do-main. But plumes forming at lower latitudes cannot eas-ily cross the domain because doing so requires them tochange distance significantly from the rotation axis. Ina Ro � 1 flow, angular momentum is not homogenizedand lines of constant angular momentum are nearly par-allel to the rotation axis. As a result, moving toward oraway from the rotation axis can only be achieved by sig-nificantly increasing the angular momentum of ascendingfluid parcels or decreasing the angular momentum of de-scending fluid parcels. The timescale for this angularmomentum exchange is longer than the typical convec-tion timescale for plumes to traverse the domain, andthus convection toward or away from the rotation axis isinhibited. Therefore, as expected from Section 2, rota-tion imposes on the flow a columnar structure.

Rotation strongly affects the vertical convective veloc-

Fig. 2.— A pedagogical illustration, using two anelastic mod-els, of the importance of rotation in the brown-dwarf parameterregime. The left column shows a rapidly rotating model (10 hourrotation period), and the right column shows a slowly rotatingmodel (2000 hour rotation period). A constant heat flux is ap-plied at the bottom boundary, leading to convection. Both modelsare fully three-dimensional simulations extending 360◦ in longitudeand adopt Jovian-like radial profiles of density and thermal expan-sivity from the SCVH equation of state. In each model, the toppanel shows the transient initial stage soon after convection initi-ates, and the bottom panel shows the state after the convectionis well developed. Colorscale denotes entropy perturbations at anarbitrary longitude, shown in the radius-latitude plane. Rotationvector points upward in the figure. In the slowly rotating case,rotation plays no role in the dynamics, whereas in the rapidly ro-tating case, the rotation forces the large-scale flow to align alongcolumns parallel to the rotation axis.

ities as well. To order of magnitude, convective velocitiesw and temperature perturbations δT relate to the con-vective heat flux F as

F ∼ ρwcpδT (21)

where cp is specific heat at constant pressure and ρis the local density. Convective temperature perturba-tions relate to convective density perturbations δρ viaα δT ∼ δρ/ρ, where α is thermal expansivity. The stan-dard nonrotating mixing-length scaling results from as-suming that buoyancy forces g δρ/ρ cause free accelera-tion of convective plumes over a mixing length l, yielding

8 Showman and Kaspi

10−1

100

101

101

102

αgF

ρCp

w(m

s−1)

Fig. 3.— Vertical (i.e., radial) velocities for brown dwarf con-vection models showing that rotation significantly affects the con-vective velocities. Each symbol shows the domain-averaged, mass-weighted root-mean-square vertical wind speed versus the mass-weighted mean buoyancy flux αgF/ρcp (units m2 s−3) for a givennumerical integration. Circles show slowly rotating models (rota-tion period 100 hours), while triangles show rapidly rotating mod-els (rotation period 10 hours). For each rotation period, modelswith a range of buoyancy fluxes were performed. The dotted anddashed lines show Equations (22) and (23), respectively.

(e.g., Clayton 1968; Stevenson 1979)

w ∼(αgF l

ρcp

)1/3. (22)

In contrast, in a rapidly rotating convective flow, convec-tive buoyancy forces often approximately balance verticalCoriolis forces. Assuming the turbulent motions are ap-proximately isotropic (i.e., horizontal eddy velocities arecomparable to vertical convective velocities), one insteadobtains a vertical velocity

w ∼ γ(αgF

ρcpΩ

)1/2. (23)

where we have introduced a dimensionless prefactor γthat is expected to be order unity. Laboratory experi-ments in rotating tanks demonstrate that this expressionworks well in explaining the convective velocities in therapidly rotating regime (Golitsyn 1980, 1981; Boubnov &Golitsyn 1990; Fernando et al. 1991). A similar expres-sion has also been suggested for the dynamo-generatingregion of planetary interiors where a three-way force bal-ance between buoyancy, Coriolis, and Lorentz forces mayprevail (Starchenko & Jones 2002; Stevenson 2003, 2010).Showman et al. (2011) showed that it also provides agood match for convective velocities under Jupiter con-ditions.

Our models demonstrate that, under typical browndwarf conditions, the rotating scaling (23) provides asignificantly better match to the convective velocitiesthan the non-rotating scaling (22). This is illustratedin Fig. 3, which shows vertical velocities for our fullyequilibrated brown dwarf models.10 Symbols depict the

10 These, and all subsequent, models are fully equilibrated browndwarf models forced by internally distributed heating and cooling,thereby (unlike Figure 2) avoiding the generation of unrealisticlower thermal boundary layers.

−60 −30 0 30 6020

40

60

80

Latitude

wrm

s

Fig. 4.— Characteristic vertical convective velocities versus lat-itude in a convection model with a rotation period of 10 hours,indicating that convective velocities tend to be greater at mid-to-high latitudes than at low latitudes. Velocities are calculated at agiven latitude and pressure as the root-mean-square in longitudeand time.

mass-weighted, global-mean vertical velocities for a se-quence of models that are slowly rotating (circles, rota-tion period 100 hours) and rapidly rotating (triangles,rotation period 10 hours). They are plotted against themass-weighted, global-mean buoyancy flux, αgF/ρcp, foreach model.

There are several points to note in Figure 3. First,the velocities in the rapidly rotating models are smallerthan in the slowly rotating models, indicating the rota-tional suppression of convective motions. Second, themodels show that the dependence of vertical velocity onbuoyancy flux is weaker in the slowly rotating case thanin the rapidly rotating case. The dotted line shows thenonrotating scaling (22), with a mixing length of 816 km,while the dashed line gives the rotating scaling (23), witha prefactor γ = 0.75. The agreement is good, showingthat, in our slowly rotating models, the mass-weightedmean vertical velocities scale approxiately as buoyancyflux to the one-third power, whereas in our rapidly ro-tating models, the mass-weighted mean vertical veloci-ties scale approximately as buoyancy flux to the one-halfpower—just as predicted by Equations (22) and (23), re-spectively. At a given buoyancy flux, the two scalingsshown in Figure 3 differ by only a factor of a few, butthe discrepancy becomes greater with decreasing buoy-ancy flux, and the two predictions differ significantly forconditions inside a typical L/T dwarf. Adopting pa-rameters appropriate for an L/T dwarf (α ≈ 10−5 K,cp ≈ 104J kg−1 K−1, and F ∼ 104–105 W m−2, corre-sponding to effective temperatures of ∼650 to 1150 K)yields buoyancy fluxes appropriate to the bulk interiorof ∼10−6 to 10−5 m2 s−3. For these values, the convec-tive velocities predicted by the rotating scaling are anorder of magnitude lower than those predicted by thenon-rotating scaling.

Under conditions appropriate to a typical T dwarf,Equation (23) predicts velocities of ∼0.1 m s−1 in thedeep interior, ∼10 m s−1 at 1000 bar and ∼40 m s−1 at100 bar. However, the equation likely overpredicts thevelocities near the top of the convection zone. In par-ticular, because α/ρ is large near the outer boundary,the buoyancy forces are large, and this likely implies a


breakdown of Equation (23) in the outermost part ofthe convection zone. Interactions of convection with theradiative-convective boundary may also be important inmodifying the convective velocities there, an effect notincluded in Equation (23).

The convective velocities tend to be greater at highlatitudes than at low latitudes, as expected from angularmomentum constraints. This is illustrated in Fig. 4 fora model with a rapid (10 hour) rotation period. Radialconvective motion at low latitudes can only occur if fluidparcels gain or lose significant angular momentum asthey change distance from the rotation axis; in contrast,convective motion near the poles involves comparativelymodest changes in distance from the rotation axis andcan occur more readily. The result is greater convectivevelocities near the poles than the equator. Nevertheless,rotational constraints still influence high-latitude convec-tion: the continuity equation demands that the verticalconvective motion must necessarily be accompanied byhorizontal convergence and divergence and thus motionstoward or away from the rotation axis. For convectivemotions at sufficiently large scale that Ro . 1, such ro-tational constraints will still play an inhibiting role inthe efficiency of polar convection. This may explain whythe velocities in Fig. 4 vary by only a factor of ∼2 fromequator to pole.

We now examine the large-scale flow in our fully equi-librated brown dwarf models. The temperatures developsignificant organization, as shown at the 1-bar level inFig. 5 for a typical model. Over a wide range of con-ditions, the temperatures exhibit significant latitudinalgradients, with polar temperatures exceeding equatorialtemperatures by typically a few K. This equator-to-poletemperature difference results from the greater efficiencyof convection in polar regions than equatorial regions.Cooling to space continually decreases the entropy offluid near the ∼1 bar level; in polar regions, this low-entropy fluid readily sinks and is replaced with higher-entropy material rising from below. But in equatorialregions, the suppression of radial convection inhibits thedense, low-entropy fluid at ∼1 bar from readily sinking.The characteristic hot-poles-cold-equator pattern seen inFig. 5 is the result. We emphasize that this effect emergesnaturally from the dynamics and is not the result ofany latitudinally varying forcing; indeed, our forcing andboundary conditions are independent of latitude.

The temperature patterns also develop significant vari-ations in both longitude and latitude on regional scales oftypically ∼107 m (Fig. 5). Convection produces regional-scale thermal anomalies that vary substantially in time.At high latitudes, these regional anomalies tend to ex-hibit comparable longitudinal and latitudinal scales, asmight be expected from the fact that the plumes movenearly vertically and converge or diverge horizontally ina quasi-isotropic fashion there. At low latitudes, how-ever, the structures exhibit north-south elongation. Thisis the manifestation of the columnar structure taken bythe convection at relatively large scales.

The characteristic convective temperature perturba-tions and equator-to-pole temperature differences in ourmodels decrease with increasing rotation period. Thisis illustrated in Figure 6. The top panel shows the lon-gitudinal (zonal) mean temperature versus latitude atthe 1-bar level for a model with a rotation period of 10

Fig. 5.— Temperature variations at 1 bar in a typical, rapidlyrotating brown dwarf model. Colorscale gives the temperature per-turbation T ′ at 1 bar, that is, the deviation of temperature fromits reference value. Rotation period is 10 hours.

hours, illustrating the hot poles and cold equator with adifference of ∼2 K. The bottom panel shows the equator-to-pole temperature difference (black circles), and root-mean-square (rms) temperature variations (red trian-gles), both at the 1-bar level, for a sequence of modelswith differing rotation periods. Both the equator-to-poletemperature differences and rms temperature perturba-tions are nearly constant from rotation periods of 3 hoursto ∼40 hours. At rotation periods exceeding ∼50 hours,however, the temperature perturbations decrease signif-icantly. This results from the fact that, at long rotationperiods, the Rossby number becomes large and the con-vection is no longer rotationally inhibited.

The temperature contrasts expected in the convectingregion can be understood by combining Eqs. (21) and(23) to yield a relation for the convective temperatureperturbations in a rotationally dominated flow (cf Show-man et al. 2010)

δT ∼(

FΩρcpαg

)1/2. (24)

Our models are performed for Jovian-like internal pro-files, corresponding to Ω = 1.74 × 10−4 K−1, cp =1.3 × 104 J kg−1 K−1, and gravity, density, and thermalexpansivity at the 1-bar level of 23 m s−2, 0.2 kg m−3,and 0.006 K−1, respectively. As discussed in detail by

10 Showman and Kaspi

−80 −40 0 40 80

−1

0

1

2

Latitude

Te

mp

. a

no

ma

ly (

K)

100

101

102

0

1

2

3

Rotation Period (hours)

∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)


∆T

(K

)

Fig. 6.— Top: Longitudinal-mean temperature versus latitude at1 bar for a model with a rotation period of 10 hours, illustrating theemergence of a systematic equator-to-pole temperature difference.Bottom: Black dots show longitudinally averaged pole-to-equatortemperature differences, and red triangles denote the root-mean-square horizontal temperature fluctuations, both at 1 bar, for asequence of otherwise identical models varying in rotation periodfrom 3 to 200 hours.

Showman et al. (2011), global convective models of giantplanets must, for computational reasons, be overforcedby several orders of magnitude; our model adopts a heatflux near 1-bar that is close to 107 W m−2. For thesevalues, Eq. (24) predicts δT ∼ 2 K, very similar to thevalues actually occurring in our models (e.g., Fig. 5).This indicates that Eq. (24) provides a reasonable repre-sentation of the model behavior. Extrapolating now tothe conditions of a typical L/T dwarf, we adopt a tem-perature of 1000 K, corresponding to a radiated IR fluxof F ∼ 6 × 104 W m−2. Inserting parameters appropri-ate to the 1-bar level of a brown dwarf (ρ = 0.03 kg m−2,cp = 1.3×104 J kg−1 K−1, α = 10−3 K, Ω = 3×10−4 s−1,and g = 200 m s−2), we obtain δT ∼ 0.5 K as the ex-pected convective temperature perturbation for a typicalbrown dwarf.

The convective structure exhibits significant tempo-ral variability, as can be seen in Figure 7. The figureshows the 1-bar temperature structure (at the top of theconvection zone) at 4.8-hour intervals in a brown dwarfmodel with a rotation period of 10 hours. Such convec-tive variability should cause significant variability in theoverlying atmosphere, helping to explain the variability

in lightcurve shapes observed in several L/T dwarfs (Ar-tigau et al. 2009; Radigan et al. 2012). We return to thedynamics of the stratified atmosphere in Section 4.

3.3. Large-scale flow organization in the convectionzone

Here we address the question of whether the convectionzone can develop organized, large-scale horizontal windssuch as fast east-west (zonal) jets, since these might playa role in causing differential zonal motion of cloudy andcloud-free regions in the overlying atmosphere.

At pressures & 1 Mbar, hydrogen metallizes andmagnetohydrodynamic (MHD) effects become important(e.g., Weir et al. 1996; Nellis et al. 1995, 1996; Nellis 2000,2006). Theoretical arguments and numerical simulationsof convection in electrically conducting spherical shells—as applied to Jupiter, Earth’s outer core, and relatedsystems—suggest that the Lorentz force acts to brakethe large-scale east-west (zonal) winds when the electri-cal conductivity is high, inhibiting jet formation in themetallic region (e.g., Kirk & Stevenson 1987; Grote et al.2000; Busse 2002; Liu et al. 2008). Numerical simula-tions of dynamo generation in convecting, rotating fluidsat high electrical conductivity have led to scaling lawsfor the magnetic field strength of rapidly rotating plan-ets and convective stars (Christensen & Aubert 2006;Christensen et al. 2009; Christensen 2010). Applicationof these these scaling laws to brown dwarfs predict thatbrown dwarfs will exhibit strong magnetic fields (Reiners& Christensen 2010). These dynamo experiments alsolead to scaling laws for the mean flow velocities in thedynamo-generating region (Christensen & Aubert 2006;Christensen 2010). When heat fluxes, rotation rates, anddensities appropriate to typical L/T-transition dwarfsare adopted (F ∼ 105 W m−2, Ω ∼ 10−3–10−4 s−1, andρ ∼ 1–5 × 104 kg m−3), these scaling laws predict typi-cal fluid velocites of ∼0.1–0.3 m s−1—similar to estimatesfrom Equation (23) under the same assumptions. Over-all, these results suggest that the flow speeds are weakin the metallic interiors of brown dwarfs. We for nowproceed under the assumption that the large-scale hor-izontal winds are weak in the metallic region, and askwhat happens in the overlying molecular envelope.

The emergence of large-scale, organized horizontaltemperature gradients (cf Figures 5 and 6) implies thatthe flow will develop large-scale shear of the zonal windin the direction along the rotation axis via the thermal-wind equation (7). We here write this in the form

2Ω∂u

∂ẑ≈ gkjet

δρ

ρ≈ gkjetαδT (25)

where δρ and δT are the characteristic large-scale hori-zontal density and temperature differences (on isobars)which occur over a horizontal wavenumber kjet (betweenthe equator and pole, for example). We envision thatthese horizontal density and temperature differences re-sult from large-scale organization of the convective tem-perature fluctuations, and we therefore equate δT inEquation (25) to that from Equation (24). Doing soyields a characteristic variation of the zonal wind alongẑ of

∆u ≈ kjet2

∫ (Fgα

ρcpΩ

)1/2dẑ (26)


The quantity α/ρ in the integrand of (26) varies by or-ders of magnitude from the atmosphere to the deep in-terior and must be accounted for. In contrast, F , g,and cp vary radially by a factor of two or less acrossthe molecular envelope (see Kaspi et al. 2009), and toa first approximation—here seeking simply an order-of-magnitude expression—we can treat them as constant.If we furthermore adopt the ideal-gas equation of state,which is reasonably accurate in the outermost layers, andassume that the background thermal profile is an adia-bat, we can integrate Equation (26) analytically to ob-tain the characteristic difference in zonal wind (along ẑ)between a deep pressure pbot and some low pressure p:

∆u ≈ −Rkjetθ(1− 2κ)| sinφ|pκ0

(FR

cpgΩ

)1/2 [ 1p

12−κ− 1

p12−κbot

](27)

where we have used the fact that the pressure variationalong ẑ is overwhelmingly dominated by the hydrostaticcontribution. In Equation (27), θ = T (p0/p)κ is the po-tential temperature of the adiabat (that is, a representa-tion of the entropy of the adiabat), p0 is a reference pres-sure (which we take here to be 1 bar), R is the specific gasconstant, κ = R/cp, and the region under considerationhas a characteristic latitude φ.

Adopting values appropriate to a typical brown dwarf(R = 3700 J kg−1 K−1, κ = 2/7, θ = 1000 K, Ω ≈3 × 10−4 s−1, φ ≈ 30◦, F ∼ 105 W m−2, g ≈ 500 m s−2,and kjet = 1 × 10−7 m corresponding to a wavelength ofapproximately one Jupiter radius), the equation can beexpressed

∆u ≈ 2

[(1 barp

) 12−κ

−(

1 barpbot

) 12−κ

]m s−1. (28)

We are interested in the wind shear between the deepinterior and the top of the convection zone, where thepressure is approximately p ∼ 1 bar. Interestingly, whenwe consider any deep pressure pbot � p, the second termin Equation (28) drops out and the equation becomesindependent of pbot; this is because the factor α/ρ be-comes extremely small at high pressure, so that almostall of the contribution to ∆u comes from the outermostfew scale heights of the convection zone—even if a verydeep layer is being considered. With the adopted param-eters, we then obtain ∆u ∼ 2 m s−1 for the difference inzonal wind (in the direction of ẑ) between any deep leveland 1 bar. The implication is that, given the expectedtemperature variations associated with convection, thelarge-scale wind varies by at most a few m s−1 along thedirection of the rotation axis. If the large-scale, zonal-mean horizontal wind is weak in the deep interior whereMHD effects predominate, then it will also be weak nearthe top of the convection zone. If, rather than adopt-ing a horizontal length scale 2π/kjet of a Jupiter ra-dius, we instead adopt a smaller length scale (e.g., 0.1Jupiter radius, appropriate to the regional-scale temper-ature anomalies seen in Figure 5), we then conclude thathorizontal winds of tens of m s−1 are possible at the topof the convection zone. Despite the uncertainties, theseestimates suggest that the large-scale zonal and merid-ional wind speeds in the convection zone are < 102 m s−1for typical brown dwarfs.

Fig. 7.— Snapshots at different times of the temperature per-turbations at 1 bar in a single model with rotation period of 10hours. Temperature perturbations are deviations of temperaturefrom the reference state, in K. Time separation between frames is4.8 hours.

4. DYNAMICS OF THE STRATIFIEDATMOSPHERE

We have so far emphasized the convective interior, butinfrared spectra and light curves emerge from pressuresthat for a wide range of effective temperature lie withinthe stratified atmosphere overlying the convective region(e.g., Burrows et al. 2006a). Understanding this strati-fied region is therefore crucial for understanding obser-vations. In particular, the horizontal temperature differ-ences, wind speeds, and dominant flow length scales inthis layer will control the variability in IR lightcurves,and vertical mixing rates will control cloudiness andchemical disequilibrium. Here we outline the expecteddynamics of this stratified layer.

4.1. Qualitative mechanism of atmospheric circulationAt first glance, it is not obvious that brown dwarfs

should exhibit significant large-scale circulations in theiratmospheres. Because they receive no external irradi-ation, the temperature-pressure profiles in their strat-ified atmospheres are determined primarily by absorp-tion of upwelling IR radiation from below. Since theinterior entropy of a brown dwarf varies little with lat-itude, one might therefore expect that the radiative-equilibrium temperature profile of the stratified atmo-sphere should vary little with latitude, and that—at leastat large scales—the stratified regions will be relativelyquiescent. This constrasts significantly from the tropo-spheres of most solar system planets—and hot Jupiters—


where differential stellar heating between equator andpole (or day and night) leads to a thermally driven at-mospheric circulation.

However, the interaction of convective turbulence withthe stable layer on brown dwarfs will perturb the strat-ified layer and generate a wide spectrum of atmosphericwaves, including gravity waves (e.g., Goldreich & Kumar1990; Freytag et al. 2010) and Rossby waves. In solar-system atmospheres, including that of Earth, Jupiter,Saturn, Uranus, and Neptune, such waves generated inthe troposphere by convection and various instabilitiespropagate upward into the stratosphere. The interac-tion of these waves with the mean flow—in particu-lar, the generation, absorption, breaking, and dissipationof these waves—induces a large-scale circulation in thestratosphere. Indeed, despite the existence of equator-to-pole radiative (thermal) forcing in irradiated atmo-spheres, this mechanical, wave-induced forcing is per-haps the dominant driver of the stratospheric circula-tion on the Earth and the giant planets (for reviews, see,e.g., Andrews et al. 1987; Shepherd 2000, 2003; Haynes2005). In a similar way, we envision that the break-ing, absorption, and dissipation of convectively generatedwaves will drive a large-scale circulation in the stratifiedatmospheres of brown dwarfs.

A variety of nonlinear interactions and feedbacks en-hance the ability of such wave/mean-flow interactions todrive an atmospheric circulation. For example, verticallypropagating waves are preferentially absorbed near crit-ical layers where the background flow speed matches thewave speed; such absorption causes an acceleration ofthe mean flow that is spatially coherent. In Earth’s at-mosphere, this effect allows convectively generated wavespropagating upward from the troposphere to drive zonaljets in the stratosphere, a phenomenon known as the“Quasi-Biennial Oscillation” or QBO (Baldwin et al.2001). A similar phenomenon has been observed onJupiter (Friedson 1999) and has been suggested to oc-cur in hot stars (e.g., Rogers et al. 2012). Likewise, themixing induced by breaking Rossby waves is spatiallyinhomogeneous and naturally leads to the formation ofjets and vortices (e.g., Dritschel & McIntyre 2008). Ideal-ized numerical experiments of two-dimensional and strat-ified, three-dimensional, rapidly rotating flows demon-strate that random turbulent forcing can generically leadto the generation of large-scale vortices and jets (e.g.,Nozawa & Yoden 1997; Huang & Robinson 1998; Mar-cus et al. 2000; Smith & Vallis 2001; Smith 2004; Scott& Polvani 2007; Showman 2007; Dritschel & McIntyre2008; Dritschel & Scott 2011).

Regardless of the details of this forcing, the rapid ro-tation will dominate the physical structure of such a cir-culation at large scales (defined here as say & 103 km).The small Rossby numbers expected at large scales im-ply that this circulation will be geostrophically balanced,that is, pressure-gradient forces will approximately bal-ance Coriolis forces in the horizontal momentum equa-tion. Here, we adopt the primitive equations, which arethe standard equations governing atmospheric flows instably stratified atmospheres when the horizontal dimen-sions greatly exceed the vertical dimensions (for reviews,see Pedlosky 1987; Vallis 2006; Showman et al. 2010).Using log-pressure as a vertical coordinate, geostrophy

reads

fu = −(∂Φ∂y

)p

fv =(∂Φ∂x

)p

(29)

where u and v are the east-west (zonal) and north-south(meridional) wind on isobars, Φ is the gravitational po-tential on isobars, x and y are eastward and northwarddistance, respectively, and the derivatives are taken onconstant-pressure surfaces. Here, f ≡ 2Ω sinφ is theCoriolis parameter. When combined with local hydro-static balance—valid in the stratified atmosphere at hori-zontal scales greatly exceeding vertical ones—geostrophyimplies that the vertical shears of the horizontal windrelate to the horizontal temperature gradients via thethermal-wind equation (e.g., Pedlosky 1987; Vallis 2006)

f∂u

∂z̃= −R∂T

∂yf∂v

∂z̃= R

∂T

∂x. (30)

where z̃ ≡ − ln p is the vertical coordinate. At largescales, then, the development of fast winds in theatmosphere—particularly if the large-scale winds in theconvection zone are weak—depends on the ability of theatmosphere to maintain horizontal temperature gradi-ents.

What is the nature of this wave-driven circulation?Figure 8 provides a schematic illustration of the dynam-ics. Generally, the acceleration of large-scale horizontalwinds by waves or turbulence induces deviations fromgeostrophy, leading to a mismatch between the pressure-gradient and Coriolis forces in the direction perpendic-ular to the wind. This unbalanced force drives a so-called “secondary circulation” in the plane perpendicularto the main geostrophic flow; the Coriolis forces and en-tropy advection caused by this circulation act to restoregeostrophy. This standard mechanism is well understood[see Haynes et al. (1991) for theory, and James (1994,pp. 100-107) or Holton (2004, pp. 313-327) for brief re-views] and provides the dynamical link between the hor-izontal winds, temperature perturbations, and verticalvelocities.

4.2. Quantitative model of atmospheric circulationAlthough the amplitude of the wave driving in brown-

dwarf atmospheres remains unknown, the above dynam-ical arguments allow us to determine the relationship be-tween the horizontal winds, temperature contrasts, andvertical velocities as a function of the wave-driving am-plitude. We here construct a simple analytic theory ofthis atmospheric circulation, treating the wave-drivingamplitude as a free parameter. The model is approxi-mate and makes a number of simplifying assumptions inthe spirit of exposing the dynamics in the simplest pos-sible context. As a result, the model is not expected tobe accurate in quantitative detail. Rather, the goal is tobroadly illustrate the types of physical processes govern-ing the atmospheric circulation on brown dwarfs, and toobtain order-of-magnitude estimates for the horizontaltemperature perturbations and wind speeds, quantitiesimportant in shaping the observables.

In steady state, the momentum balance in the direc-tion parallel to the geostrophic wind reads, to order ofmagnitude11

fv ∼ A (33)11 Suppose, for concreteness, that the dominant geostrophic flow


Hei

ght

~1-2 bars

Eddy accel

Coriolis

waves

Distance convection zone

stably

atmosphere stratified

Primary wind

COLD HOT

Fig. 8.— Schematic illustration of a wave-driven atmospheric cir-culation, as occurs in the stratospheres of Earth, Jupiter, Saturn,Uranus and Neptune and as we propose occurs at regional-to-globalscales in the stably stratified atmospheres of brown dwarfs. Grav-ity and Rossby waves (orange) propagate from the convective zoneinto the atmosphere, where they break or dissipate and induce anacceleration of the mean wind, here illustrated as a vector comingout of the page (black � symbol labeled “eddy accel”). This drivesa horizontal wind here also represented as a vector coming outof the page (red � symbol labeled “primary wind”). The result-ing deviation from geostrophy drives a weak secondary circulation(blue contours) in the plane perpendicular to the primary wind. Insteady state, the eddy acceleration is balanced by a Coriolis accel-eration (⊗ symbol, representing a vector pointing into the page)associated with the secondary circulation. Vertical motion asso-ciated with the secondary circulation advects entropy vertically,leading to horizontal temperature contrasts (labelled “hot” and“cold”). These temperature contrasts are precisely those neededto maintain thermal-wind balance with the vertical shear of theprimary wind. Scales are uncertain but are plausibly thousandsto tens of thousands of km horizontally and several scale heightsvertically.

consists of zonal jets, as exist on Jupiter. The significant zonalsymmetry of such jets suggests decomposing the flow into zonal-mean and deviation (eddy) components, A = A+A′. By expandingthe zonal momentum equation and zonally averaging, we obtain theEulerian-mean equation for the evolution of the zonal-mean flow,u, over time (adopting Cartesian geometry for simplicity)

∂u

∂t= −v

∂u

∂y−$

∂u

∂z̃+ fv −

∂(u′v′)

∂y− ez̃

∂(e−z̃u′$′)

∂z̃(31)

where $ = dz̃/dt = −d ln p/dt is the vertical velocity in log-pressure coordinates. Thus, the absorption, breaking, or dissipa-tion of waves can drive a mean flow, u. Scaling analysis of thisequation immediately shows that, on the righthand side, the firstand second terms are both order Ro smaller than the third term.In steady state, then, the balance in a geostrophic flow is betweenthe eddy-driven accelerations and the Coriolis force associated witha mean meridional circulation, i.e.,

fv ≈∂(u′v′)

∂y+∂(u′$′)

∂z̃. (32)

If the flow consists predominantly of large vortices rather thanzonal jets, one can alternately adopt a cylindrical coordinate sys-tem centered on a vortex, where u is the azimuthal flow aroundthe vortex, v is the radial velocity (toward/away from the vor-tex center), and the eddy-mean-flow composition denotes an az-imuthal mean around the vortex (overbars), and deviations there-from (primes). Azimuthally averaging the azimuthal momentum

where v is the horizontal flow perpendicular to the maingeostrophic flow and A is the characteristic magnitude ofthe eddy-induced acceleration of the mean flow, due tobreaking, absorption, or dissipation of gravity or Rossbywaves. What this equation implies is that the wave in-teractions with the mean flow drive a so-called secondarycirculation, v (meridional in the case of zonal jets, radi-ally toward or away from the vortex center in the case ofa large vortex). See Figure 8 for the conceptual picture.

The associated vertical velocity can be obtained fromthe continuity equation, which for the primitive equa-tions in log-pressure coordinates reads

∂v

∂y+ ez̃

∂

∂z̃(e−z̃$) = 0 (34)

which we can approximate to order of magnitude as

vl ∼ $∆z̃

, (35)

where $ is the characteristic vertical velocity (in unitsof scale heights per sec), ∆z̃ is the vertical scale of thecirculation (in units of scale heights), and l is the dom-inant horizontal wavenumber of the circulation. Equa-tions (33) and (35) imply that

$ ∼ vl∆z̃ ∼ lA∆z̃f

. (36)

Thus, wave interactions with the mean flow drive large-scale vertical motions.

These wave-driven, large-scale vertical motions will ad-vect entropy vertically, leading to the existence of hor-izontal temperature variations on isobars. These are infact exactly the temperature differences needed to main-tain the wave-driven geostrophic flow in thermal-windbalance. To quantify, consider the thermodynamic en-ergy equation, which can be expressed as

∂T

∂t+ vh · ∇hT −$

H2N2

R=

q

cp(37)

where T is temperature, vh the horizontal velocity, ∇h isthe horizontal gradient operator, H the scale height, Nthe Brunt-Vaisala frequency, q the specific heating rate,and cp the specific heat. In a statistical steady state, weenvision a primary balance between the radiative heat-ing/cooling (righthand side) and vertical advection (rightterm on left side). If isentrope slopes are sufficientlylarge, the horizontal mixing may also contribute via theterm v · ∇T . We write this balance as

−$H2N2

R=

q

cp− vh · ∇hT. (38)

We parameterize radiative heating/cooling as Newto-nian relaxation of the temperature toward the radiative-equilibrium state, expressed as q/cp = (Teq − T )/τrad,where Teq(z̃) is the radiative-equilibrium temperatureprofile and τrad is a specified radiative time constant.Since brown dwarfs receive no external irradiation, Teqis to zeroth order independent of latitude and longi-tude. To order-of-magnitude, the characteristic devia-tion of temperature from its local radiative equilibrium,

equation then leads to relationships analogous to (31) and (32). Ineither case, the equation can be expressed, to order of magnitude,as Equation (33).


T (z̃)−Teq(z̃), is comparable to the characteristic horizon-tal temperature difference on isobars, ∆Thoriz. We alsoparameterize the meridional eddy mixing as a diffusiveprocess, with eddy diffusivity D. To order-of-magnitude,we thus have

$H2N2

R=

∆Thorizτrad

+Dl2∆Thoriz. (39)

The physical interpretation is that vertical advection (leftside) attempts to increase the horizontal temperaturecontrasts, whereas radiation and meridional eddy mix-ing (right side) both attempt to decrease the horizon-tal temperature contrasts. Here, $ and ∆Thoriz refer tocharacteristic magnitudes and are defined positive. Im-portantly, the two terms on the right side have the samesign, since they both act in the same direction, namely,to damp temperature differences.12

Substituting Eq. (36) into Eq. (39) yields

lA∆z̃H2N2

fR∼ ∆Thoriz

(1τrad

+Dl2)

(40)

which can readily be solved to yield an expression for themeridional temperature difference in terms of “known”parameters:

∆Thoriz ∼lA∆z̃H2N2

fR(

1τrad

+Dl2) . (41)

We have yet to use the meridional momentum balance(or radial momentum balance in the case of a vortex),and doing so will allow us to solve for the zonal wind sev-eral scale heights above the radiative-convective bound-ary. To order-of-magnitude, the thermal-wind equationimplies

∆u ∼ Rl∆Thoriz∆z̃f

(42)

where ∆u is the characteristic difference between thewind speed at the radiative-convective boundary andsome level of interest, say at the mean IR photosphere.If the former is small as suggested in Section 3.3, ∆uwould approximately give the actual wind speed at lev-els above the radiative-convective boundary. InsertingEq. (41) into Eq. (42), we obtain

∆u ∼ l2A∆z̃2H2N2

f2(

1τrad

+Dl2) . (43)

Together, Equations (36), (41), and (43) provide theexpressions we seek for the vertical velocities, horizon-tal temperature differences, and horizontal wind speedsas a function of A, l, and parameters that are eitherknown or can be estimated. The Coriolis parameter,f , follows directly from the rotation period. For a

12 Breaking gravity waves will cause a vertical mixing that mightbe represented as a vertical diffusion of entropy, leading to an ad-ditional source term in Equation (39). Only horizontal variationsin the amplitude of this mixing will act to alter ∆Thoriz. A priori,it is not clear how such variations will correlate with the overturn-ing circulation nor how to parameterize them in the context ofEquation (39). Since our goal is to describe the dynamics of thewave-driven circulation in the simplest possible context, we there-fore forgo any inclusion of this vertical mixing term here, with theunderstanding that more realistic models of the large-scale circu-lation will probably have to account for it.

brown dwarf of a given effective temperature and grav-ity, 1D radiative-transfer models allow estimates of thevertical temperature-pressure profile (e.g., Marley et al.1996, 2002, 2010; Burrows et al. 1997, 2006b) and henceHN . Since the infrared photosphere is typically 1–3 scaleheights above the radiative-convective boundary (Bur-rows et al. 2006b), values of ∆z̃ ∼ 1–3 are most appro-priate.

What sets the dominant horizontal length scale of theflow, represented in the above theory by the wavenumberl? One possibility is the Rhines scale, given by (∆u/β)1/2where β is the derivative of the Coriolis parameter withnorthward distance y. This is generally the scale at whichRossby waves impose anisotropy on the flow, and in manysystems, it is the energy-containing scale. The Rhinesscale controls the jet widths on Jupiter, Saturn, Uranus,and Neptune (e.g., Cho & Polvani 1996) as well as in awide range of numerical simulations of stratified, rotat-ing turbulence (for a review, see Vasavada & Showman2005). For typical brown dwarf rotation rates and thewind speeds estimated in Figure 9, this yields l ∼ 3–6×10−7 m−1, corresponding to horizontal wavelengths of10,000 to 20,000 km. Another possibility is that l resultsfrom an interaction of turbulent energy transfers withthe strong radiative and/or frictional damping. Stratifiedflows forced at small scales tend to exhibit upscale energycascades, and a competition between the rate of upscaleenergy transfer and the radiative damping timescale thendetermines the dominant length scale. This possibility isat present difficult to quantify. Given the uncertainties,we simply adopt plausible values for l here, and leave adetailed investigation for future work.

It is also worthwhile expressing our solutions in termsof the power exerted by the waves in driving the large-scale circulation. The characteristic power per mass ex-erted by the waves in driving the large-scale circulation isapproximately A∆u, and the power per unit horizontalarea is Ap∆u/g, where p is the pressure at the radiative-convective boundary. Defining a dimensionless efficiencyη, corresponding to the fraction of the brown-dwarf heatflux that is used to drive the atmospheric circulation, wethen have ηF ∼ Ap∆u/g, where F is the heat flux ra-diated by the brown dwarf. Using this constraint, thesolutions become

∆Thoriz ∼(ηgF

p

)1/2HN

R(

1τrad

+Dl2)1/2 , (44)

∆u ∼(ηgF

p

)1/2l∆z̃HN

f(

1τrad

+Dl2)1/2 , (45)

and

$ ∼(ηgF

p

)1/2 1HN

(1τrad

+Dl2)1/2

. (46)

If we adopt a blackbody flux, F = σT 4, where T is thetypical photospheric temperature and σ is the Stefan-Boltzmann constant, and express the radiative time con-stant as (Showman & Guillot 2002)

τrad ∼pcp

4gσT 3(47)


then Equations (44), (45), and (46) can be expressed asfunctions of temperature and dimensionless wave-drivingefficiency (here for simplicity neglecting the horizontaldiffusion term),

∆Thoriz ∼(ηcpT

4

)1/2NH

R, (48)

∆u ∼(ηTcp

4

)1/2l∆z̃HN

f, (49)

and

$ ∼ 2η1/2gT 7/2σ

c1/2p pHN

∼ 2(ηTcp)1/2

HNτrad. (50)

Noting that HN is the approximate horizontal phasespeed of long-vertical-wavelength gravity waves, inspec-tion of (48)–(50) makes clear that, to within factors oforder unity,

• ∆Thoriz/T is η1/2 times the ratio of the gravitywave speed to the sound speed,

• ∆u over the sound speed is η1/2 times the ratio ofthe Rossby deformation radius, LD = ∆z̃HN/f ,to the dominant horizontal length scale of the flow,and

• $ is η1/2τ−1rad times the ratio of the sound speed tothe gravity wave speed (in other words, the time forthe flow to advect vertically over a scale height isη−1/2τrad times the ratio of the gravity wave speedto the sound speed).

For an isothermal, ideal-gas atmosphere, HN =R√T/cp, and the ratio of HN to the sound speed is√

κ(1− κ), which is ∼0.4 for an H2 atmosphere withκ = 2/7. Thus, the ratio of the gravity wave speed tothe sound speed is order unity. These arguments implythat, for small wave-driving efficiencies (η � 1), the frac-tional horizontal temperature differences will be small,the horizontal wind speeds will be much less than thesound speed, and the time for air to advect verticallyover a scale height will be much longer than the radia-tive time constant.

4.3. Application of the theory to giant planets andbrown dwarfs

Detailed numerical simulations of convection imping-ing on a stable layer will be necessary to quantify thevalue of η, but several previous studies provide con-straints. In a theoretical investigation of convection in-teracting with an overlying isothermal radiative zone,Goldreich & Kumar (1990) found that the fraction ofthe convective heat flux converted into gravity waves isapproximately the Mach number associated with the con-vection, which may be ∼0.01 for typical brown dwarfs.This presumably provides an upper limit on η since onlya fraction of the energy converted to waves is actuallyused to drive a large-scale circulation. Rough estimatesfor the Earth’s stratosphere13 indicate that waves drivea circulation with an efficiency η ∼ 10−3. Information is

13 In Earth’s mid and high latitudes, upwardly propagatingwaves lead to typical accelerations of the zonal-mean zonal wind

limited for Jupiter but likewise suggests η ∼ 10−3 (foot-note 13). While future work is clearly needed, these esti-mates suggest that values of η ranging from 10−4 to 10−2may be appropriate to brown dwarfs.

We first test the theory on Jupiter’s stratospheric cir-culation. Jupiter’s mean stratospheric temperature pro-file rises from the ∼110-K tropopause minimum at 150mbar to ∼170 K at 1 mbar pressure. Voyager and Cassiniobservations show that, throughout this pressure range,the temperature and zonal wind vary on characteris-tic horizontal (meridional) scales of ∼104 km. On thesescales, temperatures vary by ∼3–5 K at most latitudes,reaching 10 K at a few latitudes and pressures (Simon-Miller et al. 2006). Analysis of these observations in-dicates that, from 1–100 mbar, zonal winds are ∼20–30 m s−1 over most of the planet but reach ∼130 m s−1in specific latitude strips including the equator and anarrow jet at 23◦N (Simon-Miller et al. 2006). Verti-cal velocities are less certain but have been estimatedat ∼10−5 m s−1 throughout much of the stratosphere,reaching speeds of ∼3 × 10−4 m s−1 at high latitudes(Moreno & Sedano 1997).

To apply Equations (48), (49), and (50) to Jupiter,we adopt f = 2.4 × 10−4 s−1 (appropriate to 45◦ lati-tude), R = 3700 J kg−1 K−1, p = 0.1 bar, cp = 1.3 ×104 J kg−1 K−1, and evaluate NH using an isothermalbackground temperature profile with a temperature of165 K. Using a length scale of 104 km (implying l =6 × 10−7 m−1) and an efficiency η ∼ 10−3 (see footnote13), our theory predicts ∆Thoriz ∼ 3 K, ∆u ∼ 50 m s−1,and $ ∼ 1× 10−8 s−1, which for a scale height of 20 kmimplies a vertical velocity of $H ∼ 2× 10−4 m s−1. Thepredicted meridional temperature contrasts and horizon-tal wind speeds match the observations reasonably well.The predicted vertical velocity lies close to the upperend of the observationally inferred range, suggesting thatour theoretical estimate may be several times larger thanthe actual global-mean vertical velocity (a mismatch thatmay result from the crudity of our parameterization ofradiative heating/cooling). Over and above the spe-cific numerical comparisons, it is worth emphasizing thatobservational analysis of Jupiter’s stratospheric circula-tions supports the overall dynamical framework adoptedhere; the primary complication is that Jupiter (unlikebrown dwarfs) exhibits a latitudinal gradient of stellarirradiation, and thus its stratosphere exhibits aspects ofboth thermal and mechanically driven circulations. De-spite this complication, the comparison is encouraging,and gives us confidence in applying the theory to browndwarfs and directly imaged EGPs.

We next apply the theory to brown dwarfs. Figure 9shows the solutions (48), (49), and (50) as a function

of order A ∼ 10−5 m s−2 in the stratosphere (Andrews et al. 1987;Vallis 2006, Chapter 13). Typical zonal wind speeds in these re-gions are ∆u ∼ 20 m s−1. Adopting a pressure p ∼ 0.1 bar repre-sentative of the lower stratosphere, this implies a power per areadriving stratospheric motions of Ap∆u/g ∼ 0.2 W m−2. SinceEarth’s global-mean radiated flux is F = 240 W m−2, the impliedefficiency is η ∼ Ap∆u/(Fg) ∼ 10−3. For Jupiter, observationaldiagnosis of stratospheric heating patterns imply typical eddy ac-celerations of the zonal-mean zonal wind exceedingA ∼ 10−6m s−2in the lower stratosphere (West et al. 1992; Moreno & Sedano1997). Given a typical stratospheric wind speed of ∼20 m s−1,lower stratospheric pressure of 0.1 bar, and a radiated flux ofF ∼ 14 W m−2, this again implies an efficiency η ∼ 10−3.


Fig. 9.— Characteristic horizontal temperature differences(top), horizontal wind speeds (middle), and vertical velocities (bot-tom) from our solutions for large-scale, wave-driven circulation inthe stratified atmosphere. Plotted as a function of temperatureand dimensionless efficiency by which the waves drive the atmo-spheric circulation. Plotted values adopt a horizontal wavenumberl = 6 × 10−7 m−1 (corresponding to a horizontal wavelength of104 km) and an isothermal vertical background temperature pro-

file (for which NH = RpT/cp). Other parameter values are

R = 3700 J kg−1 K−1, ∆z̃ = 2 (implying a circulation two scaleheights tall), and f = 5 × 10−4 s−1 (corresponding to a rotationperiod of 5 hours).

of temperature and η for parameter values representa-tive of brown dwarfs. The dominant horizontal lengthscale of the flow is poorly constrained, and we hereadopt a wavelength of 104 km, similar to the jet widthson Jupiter and Saturn. For simplicity, we parameter-ize the mean atmospheric temperature profile in the ra-diative zone as vertically isothermal, which implies thatNH = R

√T/cp. The magnitude of the horizontal diffu-

sivity D is likewise poorly constrained; nevertheless, thebaroclinic eddy fluxes that D parameterizes should besmall if isentropes are nearly flat, which will be the casein strongly stratified regions if horizontal temperatureperturbations are small compared to the mean tempera-ture. We here set D = 0, yi

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