Understanding the Diversity of Planetary Atmospheresinteriors, they exhibit an impressive diversity...

Space Science Reviews special issue on “Understanding the Diversity of Planetary Atmospheres”Preprint typeset using LATEX style emulateapj v. 12/16/11

ATMOSPHERIC DYNAMICS OF HOT GIANT PLANETS AND BROWN DWARFS

Adam P. Showman1,†, Xianyu Tan2,∗ and Vivien Parmentier2,?

1Lunar and Planetary Laboratory, University of Arizona, 1629 University Boulevard, Tucson, AZ 85721, [email protected]

2Atmospheric Oceanic and Planetary Physics, Department of Physics, University of Oxford, OX1 3PU, UK

Space Science Reviews special issue on “Understanding the Diversity of Planetary Atmospheres”

ABSTRACT

Groundbased and spacecraft telescopic observations, combined with an intensive modeling effort,have greatly enhanced our understanding of hot giant planets and brown dwarfs over the past ten years.Although these objects are all fluid, hydrogen worlds with stratified atmospheres overlying convectiveinteriors, they exhibit an impressive diversity of atmospheric behavior. Hot Jupiters are stronglyirradiated, and a wealth of observations constrain the day-night temperature differences, circulation,and cloudiness. The intense stellar irradiation, presumed tidal locking and modest rotation leadsto a novel regime of strong day-night radiative forcing. Circulation models predict large day- nighttemperature differences, global-scale eddies, patchy clouds, and, in most cases, a fast eastward jetat the equator—equatorial superrotation. The warm Jupiters lie farther from their stars and arenot generally tidally locked, so they may exhibit a wide range of rotation rates, obliquities, andorbital eccentricities, which, along with the weaker irradiation, leads to circulation patterns andobservable signatures predicted to differ substantially from hot Jupiters. Brown dwarfs are typicallyisolated, rapidly rotating worlds; they radiate enormous energy fluxes into space and convect vigorouslyin their interiors. Their atmospheres exhibit patchiness in clouds and temperature on regional toglobal scales—the result of modulation by large-scale atmospheric circulation. Despite the lack ofirradiation, such circulations can be driven by interaction of the interior convection with the overlyingatmosphere, as well as self-organization of patchiness due to cloud-dynamical-radiative feedbacks.Finally, irradiated brown dwarfs help to bridge the gap between these classes of objects, experiencingintense external irradiation as well as vigorous interior convection. Collectively, these diverse objectsspan over six orders of magnitude in intrinsic heat flux and incident stellar flux, and two orders ofmagnitude in rotation rate—thereby placing strong constraints on how the circulation of giant planets(broadly defined) depend on these parameters. A hierarchy of modeling approaches have yielded majornew insights into the dynamics governing these phenomena.

1. INTRODUCTION

Giant planets, broadly defined, span an enormousrange of objects. Limiting ourselves to substellar bod-ies comprised primarily of hydrogen4, such bodies nev-ertheless encompass an impressive diversity. Jupiter andSaturn represent the canonical prototypes, and of courseare the best observed due to their proximity to Earth.Outside our solar system, hundreds of extrasolar giantplanets (EGPs) have been discovered. Hot Jupiters arethe most easily observationally characterized; they or-bit extremely close to their stars, at distances of typi-cally ∼ 0.03 − 0.1 AU, receive thousands of times morestarlight than Jupiter, and thereby achieve temperaturesof 1000 K or more (Showman & Guillot 2002). At day-side temperatures exceeding ∼ 2200 K, the molecularconstituents of their atmospheres start to dissociate andthey are called ultra hot Jupiters (Bell & Cowan 2018,

† This may be the last published work led by Adam Show-man, whose sudden death during the revision process of thismanuscript deprived the field of one of his giant. He will bemissed by all.∗ [email protected]? [email protected]

4 Thus we exclude not only terrestrial planets, but Uranus andNeptune-like planets, which have primarily fluid interiors com-prised of denser materials, such as water (e.g., Hubbard et al. 1991,Fortney & Nettelmann 2010).

Parmentier et al. 2018). Also amenable to atmosphericcharacterization are the directly imaged planets—that is,planets that are sufficiently hot and distant from theirhost stars to be imaged as distinct entities. They are hotnot because they are strongly irradiated, but becausethey are massive and young, so that they still glow fromtheir heat of formation—and therefore also have temper-atures of typically ∼1000 K Bowler (2016). Intermediatebetween these extremes are a large population of EGPsthat are irradiated, but less so than hot Jupiters, andalso which are old enough to have lost much of their in-ternal heat of formation (Guillot et al. 1996); as a result,they exhibit cooler temperatures and are harder to ob-serve. One might usefully define “warm Jupiters” to bethose objects with temperatures of 300 to 1000K (cor-responding to distances from a sunlike star of approxi-mately 1 to 0.1 AU) and “cool Jupiters” to be those ob-jects with temperatures less than 300 K (correspondingto orbital distances from a sunlike star exceeding 1 AU).Although the warm and cool Jupiters are harder to ob-serve than hot Jupiters and directly imaged planets, farmore have been discovered, and they will be increasinglyamenable to observational characterization in the futurewhen more sensitive instruments will become available..

Brown dwarfs are objects thought to have formed likestars but which have insufficient mass to fuse hydro-

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gen (Chabrier et al. 2000, Burrows et al. 2001); they aretypically defined as objects of ∼10 to 80 Jupiter masses(the stellar mass limit). Lacking strong internal ther-monuclear heat generation, they cool off over time, buttheir super-Jovian mass implies that even after billionsof years they may still exhibit atmospheric temperaturesof 1000 K or more (e.g., Burrows et al. 2001). Typically,they are isolated objects, far from any star, which makesthem easier to observe than exoplanets. Although theirformation mechanisms differ from EGPs, brown dwarfsshare many physical similarities to the currently knowndirectly imaged planets; from an atmospheric dynamicspoint of view, the former may be considered high-mass,high-gravity versions of the latter.

Although far less numerous than known EGPs and fieldbrown dwarfs, a population of brown-dwarf companionsto stars has also been discovered. Some such objectsorbit sunlike stars in tight, several-day orbits and re-semble high-mass, high-gravity versions of hot Jupiters(see a summary in Bayliss et al. 2016). Other browndwarfs orbit white dwarf stars so closely that the twoobjects nearly touch, and have orbital periods of justseveral hours (see a summary in Casewell et al. 2015).In some such systems—the cataclysmic variables—thebrown dwarf lies so close to the white dwarf that it con-tinually sheds mass onto the white dwarf, while in othersystems, the orbital separation (while still tight) is greatenough to prevent mass exchange.

Despite this broad diversity, there is merit in consider-ing these objects together as a class. They share in com-mon the fact that they are all fluid, hydrogen-dominatedobjects; they have radii similar to Jupiter to within afactor of two; their atmospheres merges continuouslyinto their interiors; and because of the large opacity andlow viscosity of hydrogen in the conditions of their inte-riors, they are generally expected to lose their internalheat by convection, implying convective well-mixed in-teriors.5 Like all planetary atmospheres, giant planetsand brown dwarfs should share in common many of thefundamental dynamical, physical, and chemical processesthat shape the structure and circulation of atmospheresgenerally. Considering these objects together thereforeprovides a unique opportunity to better understand thephysical and dynamical processes that operate in atmo-spheres over a wide range, and to understand how thesecommon processes leads to diverse outcomes for differentobjects. This “grand challenge” is far easier to completefor giant planets than for terrestrial planets because theatmospheres of giant planets are currently amenable toobservational characterization over a wide range, whereasfor terrestrial planets, such observational characteriza-tion over a comparably wide range is still decades away.

The atmospheric circulation and structure of giantplanets is shaped by a variety of factors, includingthe external irradiation (that is, the radiative flux theatmosphere receives from a nearby star), the internalconvective heat flux, the planetary mass (thereforegravity), the rotation rate, and the atmospheric com-position, including the overall bulk metallicity, as well

5 Whether the interior is indeed convective and well mixed de-pends on whether the internal heat flux is strong enough to over-come the barrier from molecular weight gradients created duringthe object formation (e.g. Leconte et al. 2017, Mankovich et al.2016).

Fig. 1.— Four distinct subfields of astronomy and planetary sci-ence are providing important constraints on atmospheric structureand circulation of giant planets—hot Jupiters; solar system giantplanets; brown dwarfs and directly imaged giant planets; and irra-diated brown dwarfs. They together span a wide range of physicalproperties.

as specific elemental ratios such as C/O. Even amongknown objects, these factors vary over ranges of ∼ 107

, 106 , 102 , 102, and 102, respectively. Thus anenormous diversity is represented. Figure 1 summarizesthe subpopulations where key observations constrain-ing the atmospheric circulation have been obtained.Fortuitously, these populations span a wide range inseveral of the above parameters. Jupiter and Saturnexhibit weak irradiation, weak interior heat flux, androtate rapidly (a factor that strongly influences thedynamical regime). Hot Jupiters are strongly irradiated,but likely have small interior heat fluxes (perhaps withinan order of magnitude of Jupiter), and due to tidallocking, they are expected to have modest rotation rates.Field brown dwarfs and directly imaged giant planetsreceive negligible irradiation from their stars, but haveenormous interior heat fluxes and are rapidly rotatinginmany cases faster than Jupiter.

To highlight the wide parameter space involved, Fig-ure 2 shows these diverse populations on a parameterspace of external irradiation and internal heat heat flux,which are two of the factors that matter most for driv-ing an atmospheric circulation, and which vary over thewidest range across the known population. Jupiter, Sat-urn, Uranus, and Neptune occupy the “weak forcing”regime near the lower left corner, with external and inte-rior heat fluxes that are small and comparable. Jupiterexhibits internal and irradiation fluxes of 7.5 W m−2and6.6 W m−2(Li et al. 2018) absorbed and internal fluxesare 0.27 and 0.70 W m−2 (Pearl & Conrath 1991). Incontrast, hot Jupiters reside in the upper left corner,with external fluxes of 104 to 107 W m−2, and inter-nal fluxes that are expected to be far smaller. The hotJupiters show us how giant planets behave when exter-nal forcing dominates. Brown dwarfs embody the oppo-site extreme, with enormous interior fluxes of 103 to 106

W m−2and typically negligible external irradiation. Ly-ing in the lower right corner of Figure 2, brown dwarfsyield insights on giant-planet behavior under extreme in-ternal fluxes when external forcing is zero. Finally, the

Atmospheric Dynamics of Hot Giant Planets and Brown Dwarfs 3

Fig. 2.— Regime diagram of various giant planet populations as a function of interior heat flux (abscissa) and global-mean energy fluxreceived from the star (ordinate). Solar system giant planets lie in the lower left corner, hot Jupiters in the upper left corner, and browndwarfs in the lower right corner. A handful of highly irradiated brown dwarfs are also providing information and reside in the upper rightcorner. Observational constraints from these diverse populations place tight constraints on how fundamental theories for the atmosphericcirculation scale with these heat fluxes over many orders of magnitude. This paper will provide a tour across this parameter space.

irradiated brown dwarfs represent the “strong forcing”regime in both internal and external fluxes; they occupythe upper right corner of Figure 2.

Our grand challenge is to understand the atmosphereand interior circulation and structure on giant planetsand brown dwarfs, broadly defined. Key questions in-clude the following:

• What is the nature of the atmospheric circulation—including the distribution and importance of zonaljets, vortices, storms, waves, and turbulence?What are the characteristic wind speeds, tempera-ture variations, length scales, and time variability?How do they depend on parameters? These ques-tions could be viewed as one of characterization,both from observations and careful numerical ex-periments.

• How does the circulation work? What are thedynamical mechanisms controlling it? How doesthe interplay of these mechanisms lead to variousoutcomes in the behavior of the global circulationacross the wide parameter space occupied by giantplanets? What is the link between these mech-anisms and those mechanisms well-known fromstudy of more familiar atmospheres like Earth andJupiter? This could be viewed as a question of un-derstanding the behavior characterized in the firstpoint.

• What is the role of condensation and clouds? Howcritical are they to driving (or influencing) the cir-culation and climate, via radiative feedbacks, la-tent heating, re-distribution of condensable chemi-cal species, or other mechanisms?

• What is the role of coupling between atmosphericdynamics, radiative transfer and chemistry?

• How do the magnetic field and the atmospheric cir-culation interact with each other, particularly inthe hottest atmospheres?

• Can we achieve a unified theory of giant planetatmospheric circulation that explains observationsof hot Jupiters, brown dwarfs, and solar systemgiant planets?

• Does this knowledge provide insights into the circu-lation and climate of (less easily observed) smallerplanets, including habitable terrestrial planets?

Our aim is to broadly survey the atmospheric circu-lation across these diverse classes of giant planets andbrown dwarfs. We place particular attention on dynami-cal mechanisms and the way they vary among these pop-ulations. This review can be viewed as a guided tourthrough Figure 2, starting with hot Jupiters in the upperleft corner (Section 2), moving downward to the warmJupiters (Section 3), hopping across to the brown dwarfsin the lower-right corner (Section 4), and then finishingwith irradiated brown dwarfs in the upper-right corner(Section 5). Given the existence of many prior reviews ofatmospheric dynamics on Jupiter and Saturn, we touchon the dynamics of solar system planets only briefly, aspoints of reference with hotter giant planets. Our re-view updates prior reviews on the atmospheric dynam-ics of exoplanets (Showman et al. 2008b, 2010, Show-man et al. 2013, Heng & Showman 2015) and comple-ments the many excellent reviews covering observationsand radiative structure of EGPs (e.g. Deming & Seager2009, 2017, Seager & Deming 2010, Burrows 2014, Mad-husudhan et al. 2014, Madhusudhan 2019, Zhang 2020)and brown dwarfs (Stevenson 1991, Burrows et al. 2001,Helling & Casewell 2014, Marley & Robinson 2015, Biller2017, Zhang 2020).


2. HOT JUPITERS

EGPs orbiting extremely close to their stars—hotJupiters—comprise the most easily characterizable typeof exoplanet, and therefore our understanding of theseexoplanets is best developed. As little as 0.030.05 AUfrom their parent star, hot Jupiters have orbital periodsof just a few Earth days. The immense stellar irradia-tion heats their atmospheres to temperatures of ∼10003000K, and they therefore radiate enormous IR heatfluxes to space, promoting direct detection of their ther-mal emission. Close-in planets such as hot Jupiters arealso more likely to transit their starsthe transit proba-bility for a planet on a 0.05 AU, randomly inclined orbitaround a sunlike star is ∼10%, versus 0.1% for a planetat Jupiters distance—and when such a transiting planetis detected, it enables the determination of the planet’sradius and allows atmospheric characterization througha wide suite of observation methods.

Hot Jupiters are too close to their stars to be distinctlyresolvable from their stars in images—what we observeis the combined light from the planet-star system—andindirect methods are therefore needed to tease apart theplanetary light from the starlight. When the planetpasses behind its star—an event known as secondaryeclipse—only the star is visible. Subtracting the to-tal system flux received during secondary eclipse fromthat received immediately before and afterward (whenboth the planet and the star contribute to the combinedlight) yields a spectrum of the planet, and in particularof the planet’s dayside. Moreover, observing the planetthroughout its orbit—as its dayside and nightside rotatein and out of view—allows the measurement of the phasevariations of the planet’s outgoing IR flux, and therebyprovides inferences on the longitudinal variation of tem-perature near the photosphere (Figure 3). In particular,such light curve observations yield day-night tempera-ture differences and offsets of the hot spot from the sub-stellar point. And observations during transit—when theplanet lies in front of its star—probe the atmosphere intransmission. Starlight passing through the planet’s at-mosphere on its way to Earth is preferentially blockedat wavelengths where the atmosphere is more absorbing;therefore, the planet essentially appears bigger at wave-lengths corresponding to atmospheric absorption, and aspectrum of the planet’s size is essentially a transmissionspectrum of the planet’s atmosphere at its terminator.For recent reviews of these and other methods for charac-terizing EGP atmospheres, see Deming & Seager (2017)and Madhusudhan (2019).

The atmospheric regime of hot Jupiters differs radi-cally from that of any planet in the solar system. HotJupiters are blasted by starlight—they typically receivestellar fluxes of 105106 W m−2(Figure 2), which is & 104

times the flux received by Jupiter and hundreds to thou-sands of times that received by Earth. Moreover, theclose-in orbital distances lead to strong tidal forces thatslow their spin; for example, a typical hot Jupiter or-biting at 0.05 AU around a Sun-like star has a synchro-nization timescale of ∼106 years, which is orders of mag-nitude shorter than the typical multi-Gyr system ages(Guillot et al. 1996). Therefore, hot Jupiters are gen-erally presumed to be synchronously rotating, with per-manent daysides and nightsides. This property, coupled

with the intense irradiation, leads to a unique climateregime of permanent day-night forcing, which is not ex-perienced by any planet in the solar system.

In what follows, we first sketch out the basic dynamicalregime of hot Jupiters before proceeding to a detailed dis-cussion from simulations and theory of the atmosphericcirculation and the processes that maintain it. We alsocover various topics of current interest including couplingof the dynamics to clouds and chemistry.

2.1. Basic dynamical arguments

The relative slowness of rotation (compared to Earth,Jupiter, and brown dwarfs) causes several important ef-fects. First, it implies that the dynamical length scaleon hot Jupiters will be relatively large, approaching theglobal scale (Showman & Guillot 2002, Menou et al. 2003,Cho et al. 2003). One measure of this effect is the equa-torial deformation radius, given by

Leq =

(NH

β

)1/2

(1)

whereN is the Brunt-Vaisala frequency, H is the pressurescale height, and β is the gradient of Coriolis parameter fwith latitude, that is β = df/dy, where f = 2Ω sinφ, Ω isthe planetary rotation rate (2π over the rotation period),φ is latitude, and y is northward distance. On a sphere,β = 2Ω/a at low latitudes, where a is the planetary ra-dius. The quantity NH can be thought of as the phasespeed of horizontally propagating gravity waves, whichfor a vertically isothermal atmosphere is just R

√T/cp,

where R and cp are the specific gas constant and specificheat at constant pressure, and T is temperature. Undertypical hot Jupiter conditions, these expressions yield anequatorial deformation radius Leq ∼ 5 × 104 km—morethan half a planetary radius.6 In contrast, for Earth andJupiter, respectively, the equatorial deformation radius isabout 30% and 11% of the planetary radius, respectively.

Thus one expects that the dynamically “tropical” con-ditions (i.e., where the Coriolis force is not dominantin the horizontal force balance) that prevail within a de-formation radius of the equator—including equatoriallytrapped waves and the effect they exert in adjusting theplanet’s atmosphere— will extend to significantly higherlatitudes than they do on Earth and especially Jupiter.Indeed, the equatorial regions comprise a waveguide, ofmeridional half-width Leq, for a wide population of trop-ical baroclinic waves (Andrews et al. 1987, Holton &Hakim 2012). These waves are confined to very lowlatitudes on Jupiter and brown dwarfs but can extendmeridionally to mid-latitudes or farther on typical hotJupiters.

Outside the equatorial waveguide, the deformation ra-dius is given by

LD =NH

f. (2)

6 For vertically isothermal conditions, R = 3700 J kg−1 K−1andcp = 1.3 × 104 J kg−1 K−1relevant to a H2-He atmosphere, andT ≈ 1500 K appropriate for a typical hot Jupiter, we obtain NH ≈1250 m s−1 . Adopting a rotation period of 3 Earth days yieldsΩ = 2.4 × 10−5s−1 and, with a radius of 8 × 107 m, implies thatβ = 6 × 10−13m−1 s−1. Together these imply Leq ≈ 5 × 107 m.(Showman & Polvani 2011).


Fig. 3.— Hot Jupiter observations and dynamical insights that can be gather from them: (a) Infrared phase curves of the hot JupiterHD209458b observed with Spitzer at 4.5 µm. (b) Day-night temperature contrast estimated from phase curve observations (Komacek et al.2017). (c) Thermal structure at different phases estimated based on the spectral phase curve of WASP-43b observed by HST/WFC3 (Steven-son 2016). (d) Brightness map of HD189733b reconstructed from the 8µm phase curve observed by Spitzer (Knutson et al. 2007). (e)Doppler measurements at the planetary limb during transit allow direct detection of atmospheric winds on the different limbs for the hotJupiter HD189733b (Louden & Wheatley 2015).

Inserting typical numbers for a hot Jupiter yields LD ≈4× 107 m, again about half the planetary radius.

An alternate measure of the role of rotation comes fromdirectly assessing the amplitude of Coriolis forces relativeto other forces in the equation of motion. This can beaccomplished by using the Rossby number, which rep-resents the ratio of the characteristic amplitude of ad-vection forces per mass, U2/L, to Coriolis accelerations,fU , in the horizontal equation of motion, where U and Lare the characteristic horizontal wind speed and lengthscale. Taking the ratio of these forces, we have that theRossby number is Ro = U/fL. Typical wind speeds inhot Jupiters are expected to be ∼13 km s−1 , and adopt-ing a rotation period of 3 days and a length scale of halfa planetary radius, we obtain Ro ≈ 0.6 − 2 at midlati-tudes. With Rossby numbers of order unity, hot Jupitersare thus transitional between regimes where the rotationplays little role (Ro 1) and where it dominates the dy-namics (Ro 1). As we will discuss later, the large-scalecirculation away from the equator on Jupiter, Earth’s at-mosphere and oceans, and probably brown dwarfs are inthe latter regime, which will lead to differing dynamicsbetween hot Jupiters and these other planets. Note thatbecause the Coriolis parameter goes to zero at the equa-tor, the Rossby number always tends to become largenear the equator, and in fact one useful dynamical mea-sure of the “tropics” corresponds to the range of lati-tudes within which Ro & 1 (e.g. Showman et al. 2013).According to this measure, some hot Jupiters— particu-larly those with faster wind speeds and/or slower rotationrateswill be “all tropics” worlds where the Rossby num-ber always exceeds one even at the poles; in contrast,

other hot Jupiters with faster rotation and/or slowerwind speeds may exhibit a transition where the low andmidlatitudes comprise the tropics but the poles exhibita more extratropical (Ro . 1) behavior.

The high temperatures of hot Jupiters imply that, inthe observable atmosphere, they will experience short ra-diative time constants. Near the photosphere, the radia-tive time constant is approximately (Showman & Guillot2002)

τrad =pcp

4gσT 3∼ 105

( p

0.3 bar

)(1200 K

T

)3

s, (3)

where σ σ is the Stefan-Boltzmann constant, and pshould be interpreted as a pressure near the IR photo-sphere. Note that, because of the T 3 dependence, theradiative time constant varies significantly across the hotJupiter population, from ∼104 s for ultra hot Jupiters,to 105 s for intermediate-temperature planets like HD189733b, to 106 s for warm Jupiters.

What are the expected wind speeds, to order of mag-nitude? A simple estimate can be obtained by balancingthe pressure-gradient force that drives the flow with thegreater of the Coriolis force, advective forces, or dragforce (if present) in the horizontal momentum equation.Generally, if the frictional drag is weak, and if Ro 1,then the balance is between the pressure-gradient andCoriolis force. This leads to the well-known thermal-wind equation (Vallis 2006, Holton & Hakim 2012):

f∂u

∂ ln p= R

∂T

∂y, (4)

where u is zonal wind, R is the specific gas constant,


and y is northward distance on the sphere. To order ofmagnitude, this equation yields

U ∼ R∆Thorizδ ln p

fL, (5)

where ∆Thoriz is the characteristic horizontal tempera-ture difference, assumed to extend vertically over a num-ber of scale heights δ ln p, L is the characteristic horizon-tal length scale, and U is the difference in characteristichorizontal wind speed between some upper level of in-terest (e.g., the photosphere) and deeper levels. Thisshould be interpreted as the characteristic eddy speed(e.g., associated with the wind flow from day to night)and not the speed of equatorial superrotation. Alter-nately, if Ro & 1, then the pressure gradient forces aretypically balanced by horizontal advection forces7. Toorder of magnitude, the former is R∆Thorizδ ln p/L, andthe latter is U2/L, so their balance implies a wind speed

U ∼ (R∆Thorizδ ln p)1/2. (6)

Since Ro ∼ 1 on typical hot Jupiters, the Coriolis andadvection forces are comparable, and we would expectthat these two expressions would yield similar estimates.Indeed, when we plug in typical numbers (e.g., R = 3700J kg−1 K−1, ∆Thoriz ≈ 400 K, δ ln p ≈ 3, f ≈ 3 × 10−5,and adopting L of a Jupiter radius), we obtain U ≈ 2km s−1 from both estimates.

2.2. GCM experiments and comparison to observations

General circulation models (GCMs) for hot Jupitershave been developed using a variety of different codesand numerical approaches. Most commonly, these solvethe primitive equations of atmospheric dynamics over thefull globe, assuming that the circulation is driven by theintense stellar irradiation gradient, under conditions ofsynchronous rotation. Consistent with the expectationsdescribed above, these models typically assume a thermalstructure that is deeply stratified throughout the atmo-sphere. The vertical domain typically extends over manypressure scale heights from a pressure of . 1 mbar at thetop, to commonly ∼100 bars at the bottom.

These GCMs predict atmospheric flows comprisingseveral key features, including (1) eddy and jet structuresof near-global scale; (2) large day-night temperature dif-ferences reaching hundreds of K; and, (3) most inter-estingly, a wide, fast eastward equatorial jet—so-calledequatorial superrotation—which straddles the equator,extends to latitudes ∼30, and achieves zonal-mean zonalwind speeds of typically 2-4 km s−1 (Figure 4). Despitethe synchronous rotation—which in radiative equilibriumwould lead to a temperature field comprising a simpleday-night temperature difference with the hottest regionsat the substellar point—the dynamics distorts the tem-perature structure in a complex manner, most promi-nently by inducing an eastward displacement of the day-side hot spot by tens of degrees longitude from the sub-stellar point. The earliest GCMs capturing these gen-eral features preceded observations (Showman & Guil-lot 2002, Cooper & Showman 2005), and predicted that

7 One can also have cyclostrophic balance, which on the sphereis between the pressure-gradient force and a metric term, but thescaling is the same.

the large day-night temperature differences and eastwardoffsets would be detectable in IR lightcurves of theseplanets. This helped motivate searches for these fea-tures in IR light curves. Observations from the SpitzerSpace Telescope first confirmed this prediction for thehot Jupiter HD 189733b (Knutson et al. 2007), and sub-sequent full-orbit IR light curve observations from theSpitzer and Hubble Space Telescopes have detected suchan eastward offset on the majority of hot Jupiters thathave been observed (Figure 3; for reviews, see Heng &Showman 2015, Parmentier et al. 2015, Parmentier &Crossfield 2018).

Over the last 15 years, many additional atmosphericdynamics models have been employed to investigate theglobal circulation of hot Jupiters; interestingly, mostof these models agree reasonably well in their quali-tative predictions, including the presence of large day-night temperature differences, equatorial superrotation,and—under appropriate conditions—eastward shiftedhot spots (e.g., Menou & Rauscher 2009, Rauscher &Menou 2010, Rauscher & Menou 2012, Rauscher &Menou 2013, Dobbs-Dixon & Lin 2008, Dobbs-Dixon &Agol 2013, Heng et al. 2011b,a, Mayne et al. 2014, 2017,Showman et al. 2008a, 2009, 2013, Showman et al. 2015,Parmentier et al. 2013, Parmentier et al. 2016, Katariaet al. 2015, 2016, Lewis et al. 2010, 2013, Cho et al. 2015,Zhang & Showman 2017, Menou 2019, Tan & Komacek2019, Mendonca 2020). Figure 4 presents representa-tive snapshots from several different groups, which, de-spite the many differences in model setup, highlight theoverall similarity in qualitative regime across these di-verse models. The earliest hot-Jupiter GCMs adoptedextremely idealized schemes to represent the day-nightthermal forcing; more recent work has implemented ra-diative transfer schemes of varying levels of realism thatrepresent the absorption of incoming starlight and radi-ation of thermal IR.

Interestingly, the qualitative properties of the circula-tion in these models—including the emergence of equato-rial superrotation—seem to be robust to model assump-tions, numerics, and the detailed formulation of the day-night thermal forcing. This contrasts with Jupiter andSaturn, for which circulation models can readily pre-dict either eastward or westward8 equatorial jets depend-ing on the detailed model assumptions (for reviews, seeVasavada et al. 2006, Del Genio et al. 2009, Showmanet al. 2018). This suggests that the mechanism thatcauses the superrotation on hot Jupiters is extremely ro-bust.

Observed IR light curves of hot Jupiters are now suf-ficiently accurate that detailed comparison to GCM ex-periments is a fruitful exercise. Performing such compar-isons self-consistently requires a model that representsthe heating/cooling with an explicit radiative transferscheme.9 Most GCMs to date represent the angular de-pendence of radiation using the two-stream approxima-tion. Approaches to treating the opacities come in sev-

8 “Eastward” refers to the same direction as the solid-body ro-tation, whereas “westward” refers to the opposite.

9 The simplest approaches to forcing a day-night temperaturedifference, such as a Newtonian cooling scheme, are ideal for un-derstanding dynamical mechanisms; however, they do not includea formal radiative energy budget, so it is difficult to formally com-pare their output to those of observed IR lightcurves.


Knutson et al. (2007) (Observation)Showman & Guillot (2002)

Heng et al. (2011) Showman et al. (2009)

Rauscher & Menou (2012)Amundsen et al. (2016)

Cho et al. (2015) Mendonca et al. (2018)

Fig. 4.— Example GCM simulations of hot Jupiters from a variety of groups in the field. Despite differences in forcing setup, numerics,and other factors, all the models exhibit similar circulation regimes, comprising significant day-night temperature differences, a fast eastwardequatorial jet, and an eastward shifted dayside hot spot. All models assume synchronously rotation and conditions appropriate for typicalhot Jupiters. Top left shows observations of HD 189733b from Knutson et al. (2007). Simulations of HD 209458b are shown from Showman& Guillot (2002), Heng et al. (2011b), Rauscher & Menou (2012), Amundsen et al. (2016), Cho et al. (2015). Simulations of HD 189733bfrom Showman et al. (2009). Simulations of WASP-43b from Mendonca et al. (2016). These seven simulations were performed with totallydistinct numerical codes, involving varying approximation of radiative forcing, and using seven independent dynamical cores. For eachimage, the substellar point is in the center of the panel, except for Mendonca et al. (2018)), where the antistellar point is in the center.

eral flavors. The simplest is a dual-band (double grey)scheme, with one opacity band in the IR and one in thevisible (e.g., Heng et al. 2011a, Rauscher & Menou 2012,Rauscher & Menou 2013, Rauscher & Kempton 2014,Perna et al. 2012, Komacek et al. 2017, Tan & Komacek2019, Mendonca et al. 2018, Flowers et al. 2019). Theadvantage of this approach is its simplicity—it is idealfor wide parameter explorations, understanding dynam-ical processes, and capturing the bulk radiative budget,which is sufficient for many applications.

On the other hand, the gaseous radiative transfer isinherently non-grey, with opacities that vary by ordersof magnitude from wavelength to wavelength. For theprice of increasing the model complexity, capturing thenon-grey behavior affords two advantages. First, it al-

lows a more realistic representation of the radiative heat-ing/cooling, so that the overall thermal structure is ac-curately simulated. Second, IR spectra and lightcurvesat different wavebands probe different pressures, wherethe temperature differ. As a result, IR spectra andlightcurves indicate that hot Jupiters are inherentlynon-grey bodies. Accurately capturing the wavelength-dependence of IR spectra and lightcurves requires non-grey radiative transfer. Several hot-Jupiter GCMshave been developed that treat the opacities using thewell-known correlated-k method: the SPARC/MITgcm(Showman et al. 2009, Lewis et al. 2010, Kataria et al.2013), and the Exeter groups hot-Jupiter implementa-tion of the UK Met Office GCM (Amundsen et al. 2014,2016). In this approach, opacities are treated by di-


viding the spectra into typically 10-40 wavelength bins,and then statistically representing the opacity informa-tion from ∼104− 105 wavelength points within each bin.This allows accuracy of typically 1% or better in heatingrates (Showman et al. 2009, Kataria et al. 2013, Amund-sen et al. 2014) while retaining much greater computa-tional efficiency than a line-by-line radiative transfer cal-culation, which is computationally prohibitive in GCMs.Intermediate approaches include bin methods, which di-vide the spectrum into several dozen wavelength bins,but treat the opacity as essentially grey within each bin(Dobbs-Dixon et al. 2012, Dobbs-Dixon & Agol 2013).A key point is that none of these differing approachesare inherently better or worse; they should be viewed asdistinct tools useful for different applications.

Detailed comparisons between IR lightcurve observa-tions (obtained with Spitzer and Hubble) and GCM sim-ulations including radiative transfer have been performedfor a wide range of hot Jupiters, including HD 189733b(Showman et al. 2009, Knutson et al. 2012, Dobbs-Dixon& Agol 2013, Drummond et al. 2018a, Steinrueck et al.2019, Flowers et al. 2019), HD 209458b (Zellem et al.2014, Amundsen et al. 2016, Drummond et al. 2018b),WASP-43b (Kataria et al. 2015, Stevenson et al. 2017,Mendonca et al. 2018), WASP-18b (Arcangeli et al.2019), WASP-19b (Wong et al. 2016), HAT-P-7b (Wonget al. 2016), WASP-103b (Kreidberg et al. 2018), WASP-121b (Parmentier et al. 2018), HAT-P-2b (Lewis et al.2014), and HD 80606b (Lewis et al. 2017). Most of thesestudies compute the radiative transfer driving the GCMassuming a solar-composition, cloud-free hydrogen atmo-sphere that is in local chemical equilibrium, and performstraight-up comparisons of the observed lightcurves to anominal model, without any special attempts to searchfor an exact match via model tuning. Nevertheless, a fewof the models explore the influence of supersolar metal-licities, nonsynchronous rotation, specified atmosphericdrag (perhaps representing the effect of Lorentz forces),disequilibrium chemistry, or specified haze distributionson the lightcurves.

Generally speaking, the lightcurves predicted fromthese models capture the qualitative features of theobservations—including large day-night flux differencesand phase offsets wherein the fluxes peak before sec-ondary eclipse, as expected due to the eastward offset ofthe dayside hotspot from the substellar point10. Giventhe lack of tuning, these aspects of qualitative agreementcould be viewed as major successes suggesting the mod-els are in approximately the correct regime. However,the GCM lightcurves also exhibit significant discrepan-cies from the observations—they tend to overpredict thehotspot offset, overpredict the nightside flux, underpre-dict the dayside flux, and therefore underpredict the day-night flux difference (i.e., the phase curve amplitude).Figure 5 shows examples for the benchmark planets HD189733b, HD 209458b, WASP-43b and WASP-18b thathighlight these successes and failures (see also Parmen-tier & Crossfield 2018, for a qualitative comaprison). Al-though the degree of discrepancy varies from planet toplanet, the persistence of these aspects of discrepancyacross the hot-Jupiter sample suggests that the models

10 We note that in one case, CoRoT-2b, the offset was claimedto be westward (Dang et al. 2018), but this remains an outlier.

may consistently be missing one or more crucial ingredi-ents.

As yet, there exists no consensus on what these missingfactors may be, although several authors have exploredthe influence of various factors in an attempt to bettermatch the observed lightcurves. Super-solar metallicities(e.g., 5 or 10 times solar) tend to increase the day-nightflux differences and decrease the hotspot offset (Show-man et al. 2009, Lewis et al. 2010, Kataria et al. 2015),which improves the agreement for certain planets, butgenerally do not solve the overall problem. For someplanets, the addition of atmospheric drag improves theagreement (e.g., Kreidberg et al. 2018 for WASP-103b,and (Arcangeli et al. 2019) for WASP-18b); however, theLorentz forces such drag parameterizes are only relevantat hot temperatures (& 2000 K), and so this effect can-not provide a solution for the cooler planets in Figure5. Chemical disequilibrium between CH4 and CO wassuggested as a possible culprit for HD 189733b and HD209458b (Knutson et al. 2012, Zellem et al. 2014), but re-cent GCM experiments rule out this explanation (Stein-rueck et al. 2019, Drummond et al. 2018a, 2020). Morelikely is that the discrepancies result from clouds and/orhazes, although this possibility remains to be quantifiedin detail.

Measurements of the Doppler shift of the atmosphericwinds on the planet’s terminator, as detected duringtransits, has been used to characterize the wind regimefor several hot Jupiters. Using terminator-averaged mea-surements, Snellen et al. (2010) showed that the high-altitude winds on HD 209458b exhibit a preferentialblueshift of 2 km s−1 , indicating that the atmosphericcirculation comprises net flow toward Earth at high al-titude (with the return flow presumably occurring at adeeper, unobserved level). Louden & Wheatley (2015,2019) used transit ingress and egress measurements ofHD 189733b and WASP-49b to discriminate between theleading and trailing limbs, and they showed that bothplanets exhibit equatorial superrotation (correspondingto a redshift on the leading limb, as the air flows from theplanet’s nightside to dayside, and a blueshift on the trail-ing limb, carrying the return flow from the dayside backto the nightside). For HD 189733b, the observations indi-cate eastward velocities on the leading and trailing limbsof about 3-4 km s−1 and 4-6 km s−1 , respectively. ForWASP-49b, their study likewise identified atmosphericsuperrotation, and they were further able to determinethat the circulation over both poles flows from day tonight.

Motivated by these measurements, several GCMs in-vestigations have explicitly determined the Doppler sig-natures that would result from the atmospheric circula-tion, and made predictions for various planets (Miller-Ricci Kempton & Rauscher 2012, Showman et al. 2013,Rauscher & Kempton 2014, Flowers et al. 2019). Thesemodels show that at high altitude (pressures of ∼0.01-1 mbar, the approximate levels sensed by the Dopplermeasurements), the circulation tends to comprise a day-night flow at high latitudes combined with a superrota-tion at lower latitudes, yielding a net signature wherethe windflow is redshifted at low latitudes on the lead-ing terminator and blue shifted everywhere else along theterminator. These studies show that the relative contri-butions of the superrotation and the day-night flow to the


HD 208458b

HD 189733b

WASP 43b

WASP 18b

Zellem et al. (2014) Amundsen et al. (2016)

Knutson et al. (2012)

Stevenson et al. (2017) Kataria et al. (2015)

Arcangeli et al. (2019)

Fig. 5.— Observed thermal phase curves of benchmark hot Jupiters exoplanets compared to 3D global circulation models outputs. ForHD209458b (top) we show the comparison of the Zellem et al. (2014) Spitzer observation at 4.5µm with the SPARC/MITgcm output (left)and with the UK Met office Unified Model (right). For WASP-43b (middle row) we show the comparison between the SPARC/MITgcmand, from left to right, the Spitzer 4.5µm phase curve, the Spitzer 3.6µm phase curve, the HST/WFC3 phase curve and the HST/WFC3dayside spectrum. For WASP-18b (bottom row left) we show the comparison between the HST/WFC3 spectral phase curve and spectrumat several phases and the SPARC/MITgcm output. Finally for HD189733b we show the four Spitzer phase curve and the related day andnightside spectrum and compare them to the SPARC/MITgcm outputs.

limb signature depend on the stellar irradiation, plane-tary rotation rate, and the exact altitude sensed (whichdepends on the atmospheric composition), among otherfactors. The addition of sufficiently strong frictional dragor Lorentz forces can damp the jet, causing the day-to-night flow regime to predominate, although the regimesunder which this can occur need further study. Generally,the GCM results agree reasonably well with the observedsignatures, lending confidence that superrotation existson hot Jupiters and that the GCMs are in approximatelythe correct regime.

2.3. Dynamical mechanisms: superrotation

We next turn to the dynamical mechanisms responsi-ble for maintaining the flow, starting with the eastwardequatorial jets prominent in hot Jupiter simulations (Fig-ure 4), as well as the mechanisms responsible for control-ling the characteristic day-night temperature differences,wind speeds, and eastward offsets of the hot spots. Sig-nificant progress has been made on these topics in thepast ten years.

The eastward equatorial jets emerging in circulationmodels of hot Jupiters are important for several reasons.

Being perhaps the dominant aspect of the atmosphericflow, they are critical in explaining observations, becausethey help to cause the eastward shift of the dayside hotspot, influence the day-night temperature distribution,and cause differing conditions on the leading and trail-ing terminators. Such eastward equatorial jets are alsodynamically interesting—they are an example of what istermed superrotation.11 The equator is the region of theplanet farthest from the rotation axis, and therefore, an

11 The term superrotation is typically defined to correspond toregions of atmospheric flow where the angular momentum per unitmass about the rotation axis, m = (Ωa cosφ + u)a cosφ exceedsthat corresponding to zero wind at the equatorial surface, Ωa2. Inthis sense, zonal jets at the equator naturally correspond to su-perrotation if they exhibit eastward wind of essentially any speed.In contrast, zonal jets at higher latitudes are superrotating only iftheir zonal wind exceeds the value u = Ωa sin2 φ/ cosφ. Generally,eastward zonal jets that are away from the equator in planetaryatmospheres (such as those on Jupiter and Earth) exhibit angularmomentum less than that of the equatorial surface and are there-fore not superrotating: despite being local maxima of zonal windas a function of latitude, they are not local maxima of angularmomentum. In this sense, eastward equatorial jets are a very dif-ferent phenomenon from either eastward or westward jets at anyother latitude.


eastward equatorial jet represents a local maximum ofangular momentum per unit mass as a function of lati-tude and height—air at higher latitudes or deeper levelswill tend to have smaller values of angular momentumper unit mass due to its closer proximity to the rotationaxis. Maintaining such a superrotating jet against fric-tion or other processes requires that the atmospheric cir-culation transport angular momentum up-gradient fromregions where it is small (outside the jet) to where it islarge (inside the jet). This is the opposite of diffusion,which causes down-gradient transport. Hide’s theorem(Hide 1969) states that such superrotating flows cannotresult from axisymmetric circulations such as angular-momentum-conserving Hadley cells, but rather that thenecessary up-gradient momentum transport can only beaccomplished by waves and/or turbulence. Indeed, inmany branches of fluid dynamics, up-gradient momen-tum transport is a fairly common outcome of turbulenceand wave propagation (indeed, wave propagation is non-local in the sense that the wave can cause a flux of angu-lar momentum that does not depend on the local gradi-ents of the flow, as would occur in a diffusive problem).The puzzle in the present context is to understand thespecific dynamical mechanism that causes the superrota-tion on hot Jupiters and to understand why it appearsto be so robust.

The dynamical mechanism for equatorial superrota-tion on tidally locked planets was first investigated byShowman & Polvani (2010, 2011), who showed that theintense day-night heating pattern on synchronously ro-tating planets induces a standing pattern of large-scaleatmospheric waves, and that these waves have a struc-ture that naturally leads to a flux of angular momentumtowards the equator, causing the superrotation.

To see how this works, it is useful to first consider thebehavior of freely propagating tropical waves. Impor-tantly, tropical waves tend to be confined to an equato-rial waveguide whose half-width is equal to the equatorialRossby deformation radius (Equation 1). Two classes oftropical wave are particularly relevant here— the Kelvinwaves and the Rossby waves (see Holton & Hakim 2012,Andrews et al. 1987). The Kelvin waves are essentiallygravity (buoyancy) waves in the east-west direction, withbroad-scale pressure anomalies that are centered on, andsymmetric about, the equator. In the north-south di-rection, Kelvin waves are geostrophically balancedmean-ing the wave-induced pressure gradients are supportedby Coriolis forces associated with the wave-induced windanomalies. This balance implies (1) that isolated Kelvinwaves exhibit very weak meridional wind relative to thezonal wind, and (2) that the zonal wind exhibits eastwardphase in the regions of pressure maxima, and westwardphase in the region of pressure minima. It turns out thattrait (2) implies that only eastward-propagating wavesare possible, and therefore Kelvin waves exhibit phaseand group propagation to the east.

In contrast, Rossby waves are vortical waves whoserestoring force relies on the planetary rotation, or morespecifically, the fact that the Coriolis parameter is a func-tion of latitude. This effect is referred to as the “β effect,”where β = df/dy is the gradient of the Coriolis parame-ter with northward distance y. Because of the tendencyof fluid parcels to conserve their potential vorticity fol-

lowing the flow12, the latitude variation of the planetaryvorticity implies that fluid parcels deflected polewardtend to generate anticyclonic vorticity, whereas those de-flected equatorward tend to generate cyclonic vorticity.This vorticity comprises both zonal and meridional mo-tions, and it turns out that the meridional motions asso-ciated with this wave-generated vorticity tend to deflectthe fluid parcel back to its original latitude. This chainof events thus acts as a restoring force that allows wavepropagation. These properties imply that Rossby wavesexhibit phase propagation to the west (Holton & Hakim2012). When equatorially trapped, Rossby waves exhibitpressure patterns that are symmetric about the equator,with maxima that peak off the equator, and velocitiesskirting the pressure maxima in a vortical pattern. Forlow-order Rossby waves of long wavelength, this struc-ture tend to fill the equatorial waveguide.

Time variable, stochastic forcing can trigger freelypropagating Rossby and Kelvin waves, which in a sys-tem with weak damping are able to propagate zonallyfor long distances within the equatorial waveguide. Forexample, in Earth’s tropics, cumulus convection tends totrigger both of these wave classes, which can be observedbecause of their influence on the cloud patterns (Wheeler& Kiladis 1999, Kiladis et al. 2009). Models of stochas-tic forcing in the tropics likewise lead to these and otherwave modes (e.g., Salby & Garcia 1987). A large heatingpulse applied at the equator, with a size comparable toa deformation radius, for example, might naturally pro-duce a Kelvin-wave packet that propagates to the east,and a Rossby-wave packet that propagates to the west.

Showman & Polvani (2010, 2011) demonstrated thatthe intense day-night thermal heating pattern on sy-chronously rotating exoplanets triggers a global-scale,standing wave pattern, with dynamics analogous to thetropical wave dynamics just described, and which ex-plains the emergence of equatorial superrotation. Thesesolutions, which built on the classic solutions of Matsuno(1966) and Gill (1980), adopted the 1.5-layer shallow-water equations, which represent the flow of an active,constant-density atmosphere (here representing the pho-tosphere levels of a giant planet) that overlies a denserinterior (representing the deep atmosphere and interior)that is assumed quiescent. Figure 6 depicts the linear,analytic, β-plane solutions for typical hot Jupiter con-ditions, and adopting radiative and drag time constantsthat are equal. In the solutions, the steady, day-nightheating pattern (Figure 6a) leads to standing eddy pat-tern known as a Matsuno-Gill pattern (Figure 6b). Atlow latitudes, it comprises predominantly east-west flowsthat diverge from a point (marked by an X) lying east ofthe substellar point. This structure can be interpreted asa standing Kelvin wave; the eastward displacement of thethermal structure at low latitudes results from the factthat Kelvin waves propagate to the east. At higher lati-

12 For a stratified atmosphere, the potential vorticity is generallydefined as (ζ + f)/h, where ζ = k · ∇ × v is the relative vorticityand h is some measure of the vertical thickness of a fluid column(e.g., the vertical spacing between constant-entropy surfaces). Inthe adiabatic, frictionless limit, the dynamical equations conservepotential vorticity following the flow, which means that changesin f caused by meridional deflections of a fluid parcel tend to becounteracted by changes in ζ. Thus, meridional motions of a fluidparcel tend to “spin up” a fluid parcel, generating relative vorticity.


tudes it comprises vortical flow, with off-equatorial pres-sure maxima (anticyclones) in the northern and southernhemispheres of the dayside, and pressure minima (cy-clones) in the northern and southern hemispheres of thenightside. This structure can be understood as an equa-torially trapped Rossby wave; the westward displace-ment results from the fact that long-wavelength Rossbywaves propagate to the west. Importantly, by distort-ing the thermal field eastward at low latitude (due tothe Kelvin wave component) and westward at higher lat-itudes (due to the Rossby wave component), this latitu-dinally varying zonal-phase shift induces a global-scalechevron-shaped eddy pattern in which the pressure con-tours and eddy velocities are tilted northwest-southeastin the northern hemisphere and southwest-northeast inthe southern hemisphere. These phase tilts are clearlyvisible in Figure 6b. See Heng & Workman (2014) fora wider range of Matsuno-Gill-type solutions, and Pier-rehumbert & Hammond (2019) for a tutorial on theMatsuno-Gill model.

This tilted pattern of eddy velocities causes an equa-torward flux of momentum and is therefore key in drivingthe equatorial superrotation. Let u′ and v′ be the eddyzonal and meridional velocities, respectively, defined asthe deviations of the zonal and meridional winds fromtheir zonal-means. Examination of Figure 6 shows that,in the northern midlatitudes, fluid parcels moving east(u′ > 0) tend to be moving south (v′ < 0), whereasthose moving west (u′ < 0) tend to be moving north(v′ > 0). Thus, the velocities are correlated such thatu′v′ < 0, where the overbar denotes a zonal average. Inthe southern midlatitudes, the correlation is reversedfluidparcels moving east tend to be moving north, whereasfluid parcels moving west tend to be moving south, suchthat u′v′ > 0. The quantity u′v′ represents a meridionalflux of eastward momentum per unit mass—a positivevalue implies northward transport of eastward momen-tum, whereas a negative value implies southward trans-port. Thus, the Matsuno-Gill pattern (Figure 6) natu-rally causes a convergence of momentum onto the equa-tor, which leads to an eastward acceleration that drivesequatorial superrotation. In this theory, the meridionalwidth of these Matsuno-Gill solutions—and the equato-rial jet that they driveis comparable to the equatorialdeformation radius.

The solutions also show that, at the equator, the stand-ing waves cause a net downward transport of zonal mo-mentum, which plays a critical role in the momentumbalance. Because of the eastward phase shift of theKelvin-wave structure, the equatorial eddy velocity onthe nightside is preferentially eastward while that on thedayside is preferentially westward (see Figure 6b). Night-side cooling causes downward transport of air, whichtakes its eastward eddy momentum with it. Daysideheating brings up air from the deeper atmosphere withonly weak zonal velocity. Thus, there is a net downwardtransport of eastward momentum out of the atmosphereinto the interior. This loss of momentum from the at-mosphere causes a westward acceleration at the equa-tor that partially—but not completely—cancels out theeastward acceleration caused by the convergence of mo-mentum onto the equator (In Figure 6c, the accelerationdue to this vertical exchange is shown in blue, that due

Fig. 6.— Linear, analytical solutions of the shallow-water equa-tions demonstrating how day-night heating can induce equato-rial superrotation on tidally locked exoplanets, from Showman &Polvani (2011). Panel (a) shows the day-night heating pattern;specifically, the colors and contours represent the radiative equi-librium state as a function of eastward distance and northwarddistance (zero northward distance represents the equator and thecenter of the plot represents the substellar point; distance is plot-ted in units of the equatorial deformation radius, which for a hotJupiter is a significant fraction of a planetary radius). The daysideis thick (hot) and the nightside is thin (cold). Panel (b) depictsthe steady-state, linear solution that emerges in response to thisheating pattern when no background flow is present. The solutioncan be interpreted as standing Kelvin and equatorially trappedRossby waves. Notice that the predominant phase tilts of the eddyvelocities are from northwest-southeast in the northern hemisphereand southwest-northeast in the southern hemisphere; this patterncauses transport of angular momentum from midlatitudes to theequator. Panel (c) shows the resulting acceleration of the zonal-mean zonal wind. The acceleration from meridional momentumconvergence is in black, the acceleration from vertical momentumexchange is in blue, and the net acceleration is shown in red. In thenet, eastward acceleration occurs at the equator, which will causesuperrotation to emerge.

to meridional momentum convergence is shown in black,and the net acceleration is shown in red.)

The solution shown in Figure 6 adopts equal radia-tive and drag timescales, but the linear behavior differswhen these timescales are unequal. In particular, Show-man & Polvani (2011) and Heng & Workman (2014)explored a wide range of differing τrad and τdrag; theyshowed analytically that, as the drag becomes weak (i.e.,the drag timescale becomes long), the equatorial waveg-


uide corresponding to these forced-damped solutions be-comes confined closer and closer to the equator. Theirsolutions show that, in the linear limit of τdrag → ∞,the meridional width of the region with prograde phasetilts shrinks to zero13, and a pattern of reverse phasetilts instead emergesnortheast-southwest in the north-ern hemisphere and southeast-northwest in the south-ern hemisphere. Such a pattern would not be expectedto converge eddy momentum onto the equator or drivestrong superrotation. The presence of these reverse phasetilts are well-understood (Showman & Polvani 2011, Ap-pendix C), but this leads to a puzzle: how can super-rotation develop in a 3D hot Jupiter GCM with noexplicit frictional drag, when the meridional width ofthe Matsuno-Gill pattern (and its prograde phase tilts)should shink to zero?

The answer is provided by nonlinearity. At the highforcing amplitudes relevant to typical hot Jupiters, thedynamics becomes nonlinear, which modifies the detailsof the wave-mean-flow interactions, both due to the non-linearity of the eddies themselves, and because a strongzonal flow develops when the forcing amplitude is large,and the zonal flow modifies the eddy structures. Show-man & Polvani (2011) showed numerically that whennonlinearity is progressively introduced back into the dy-namics, the meridional width of the Matsuno-Gill-typepattern broadens accordingly, even in the limit of weakdrag, and at very large amplitudes exhibits a meridionalwidth comparable to the equatorial deformation radius.This effect of nonlinearity on the eddies can be under-stood by appreciating that nonlinear momentum advec-tion can qualitatively play the role of drag in the multi-way force balance that controls eddy behavior, allowingprograde phase tilts to persist to high latitudes, and pre-venting the Matsuno-Gill-type behavior from collapsingto the equator (Showman & Polvani 2011; for a visualexplanation of how this effect can lead to prograde phasetilts, see Showman et al. 2013). Mathematically, this ef-fect can be appreciated from the momentum equationgoverning the 1.5-layer shallow-water system (Showman& Polvani 2011, Equation 12):

dv

dt+g∇h+fk×v = −v

[1

τdrag+

1

τrad

(heq − h

h

)H(heq − h)

](7)

where v is the horizontal velocity vector, d/dt is the ma-terial derivative, h is layer thickness, k is the local ver-tical unit vector, heq is the local radiative equilibriumthickness, and H is the Heaviside step function, whichequals one when its argument is positive and zero oth-erwise. The entire quantity in square brackets plays therole of one over an “effective drag” time constant. Atlow amplitude, the quantity (heq − h)/h 1, and thusthe second term in square brackets is not generally im-portant. At high amplitude, however, deviations fromradiative equilibrium tend to be substantial,14 implyingthat (heq − h)/h ∼ 1. This implies that, on the dayside,the second term in square brackets tends to have a mag-nitude of order 1/τrad . Thus, under typical hot-Jupiter

13 By “prograde” phase tilts, we mean northwest-southeast(southwest-northeast) in the northern (southern) hemisphere.

14 An exception occurs when the radiative time constant is muchshorter than all the dynamical timescales, in which case the solu-tion is close to radiative equilibrium even at high nonlinearity.

conditions, the “effective” drag time constant caused bythe nonlinearity has comparable magnitude to the radia-tive time constant—even when the actual drag time con-stant τdrag is infinite. The fact that the highly nonlin-ear, zero-drag regime tends to naturally exhibit radiativeand effective drag timescales that are comparable to eachother can explain why these solutions have a Matsuno-Gill-like pattern that extends to high latitudes, just likelinear solutions with equal radiative and drag timescales(Figure 6). This behavior also explains more generallywhy the standard definition of equatorial deformation ra-dius (Equation 1) provides a reasonable fit to the actualjet width in most 3D hot-Jupiter GCMs, despite the factthat these GCMs generally do not include explicit fric-tional drag in the region where the superrotation forms.Debras et al. (2020) further confirmed the above pictureby performing full 3D linear wave calculations using thenumerical package developed by Debras et al. (2019).

In sum, high-amplitude, fully nonlinear shallow-watersolutions on the sphere show that essentially the samemechanism holds even in the nonlinear regime: althoughthe details of the wave-mean-flow interactions are sensi-tive to nonlinearity, the qualitative mechanism still oc-curs even at high amplitude. As mentioned above, theeddy nonlinearity is critical for preventing the Matsuno-Gill solutions from collapsing to the equator in the weak-drag limit (Showman & Polvani 2011). Once the jetbuilds up, these planetary-scale waves can exhibit fi-nite meridional width even in the linear, weak-drag limit(Hammond & Pierrehumbert 2018, Debras et al. 2020),presumably because they are then propagating relativeto the moving airflow (even though they are standing inthe reference frame of the planet). As the jet builds up,the concomitant changes in the eddy structure lead tochanges in the amplitudes of the horizontal and verticaleddy momentum convergences. Steady state is achievedwhen they balance (along with drag, if any).

In a major study, Tsai et al. (2014) showed that adeeply stratified, 3D atmosphere exhibits global-scale,Matsuno-Gill-type wave solutions that are very similarto those in shallow water. They adopted radiative andfrictional damping times that are equal and invariantwith depth. The day-night heating was assumed to occurover a broad vertical layer peaking around 0.1 bar. Tsaiet al.’s solutions in the absence of a background flow,shown in the left column of Figure 7, yield a Matsuno-Gill pattern very similar to the shallow-water solutionsidentified by Showman & Polvani, comprising superposedequatorial Kelvin and Rossby waves whose differentialzonal propagation produces eddies tilting predominantlynorthwest-southeast (southwest-northeast) in the north-ern (southern) hemisphere. Just as in the shallow-watersystem, the eddy structure induces a meridional momen-tum convergence onto the equator in the forcing layer, aswell as a downward momentum transport at the equatorfrom the forcing levels to deeper levels. Within the forc-ing layer, the net accelerations in the meridional planeare eastward at the equator and westward at higher lat-itudes (Figure 7d), qualitatively similar to the patternof accelerations predicted by the shallow-water solutions(Figure 6c). The striking similarity between the shallow-water and 3D solutions is not surprising, because, as isstandard in tidal theory and many other wave problems(Chapman 1970), this 3D problem is mathematically sep-


arable and determined by solving a shallow-water equa-tion for the horizontal structure, coupled to a 1D vertical-structure equation for the vertical eigenfunctions.

In Tsai et al. (2014)’s solutions, below the ∼1-bar level,the waves propagate downward away from the forcinglayer. This propagation causes the Rossby gyres to shiftwestward at greater depth, while the Kelvin wave com-ponent shifts eastward with depth. The overall longi-tudinal/vertical tilts of the solution depend on the rel-ative amplitudes of the Kelvin and Rossby components,which vary with the adopted damping time constant. Inparticular, the Kelvin wave component dominates whenthe damping timescales are short, while the Rossby wavecomponent dominates when the damping timescales arelong. Despite these complications, the visual appear-ance of their solutions is that the vertical tilts appearto be modest under damping conditions relevant to hotJupiters.

Still, analysis of fully nonlinear GCMs shows that thebuild-up of the equatorial jet strongly modifies the stand-ing waves and disrupts the coherency of the progradephase tilts, leading to a wave structure in the fully equi-librated state that differs substantially from the classicalMatsuno-Gill pattern (Showman & Polvani 2011, Tsaiet al. 2014, Showman et al. 2015, Mayne et al. 2017,Mendonca 2020). Because of the disruption of the phasetilts, the net meridional momentum transports—whilestill equatorward—are weaker than would be predictedfrom the classic Matsuno-Gill model at the same waveamplitude. Quantitatively understanding how the equa-torial jet equilibrates therefore requires an understandingof how the jet modifies the eddies. Several authors haveaddressed this question.

The addition of a uniform background zonal wind tothe Matsuno-Gill model has long been known to mod-ify the standing wave modes (Phlips & Gill 1987) andthe momentum fluxes they induce (Arnold et al. 2012).To investigate this effect in the context of hot Jupiters,Tsai et al. (2014) included a uniform background zonalwind U in their analytic solutions (Figure 7, middle andright columns). The background zonal flow influences theamplitudes and relative phase offset between the Kelvinand Rossby wave components, with crucial implicationsfor the wave-induced accelerations. Because the groupvelocity for long Rossby waves is about one-third thatof the Kelvin wave, the phase offset of the Rossby-wavecomponent is much more easily influenced by the back-ground winds. In the solution in Figure 7 backgroundwinds ranging from zero to 1 km s−1 hardly change theoffset of the Kelvin-wave component, but they cause theRossby wave gyres to shift from sim20 west of the sub-stellar longitude to 50 east of it. (Compare Figure 7aand c; see Tsai et al. 2014, Figure 8, for a decomposi-tion of the Kelvin and Rossby-wave components of thesolutions.) At zonal winds of ∼1 km s−1 , the pressuremaxima of the Rossby and Kelvin components are nearlyin phase, and as can be seen from Figure 7, this destroysthe strong polewared-westward to equatorward-eastwardphase tilts. After a transient increase in the strengthof the acceleration at modest U (Figure 7e), the zonalacceleration then plummets as the zonal flow increases(Figure 7f). Still, the net acceleration remains eastwardthroughout the entire range of U shown in Figure 7, sug-gesting that the jet will continually accelerate to speeds

exceeding 1 km s−1 . Tsai et al. (2014) showed that,for the particular model parameters of Figure 7, the net,wave-induced zonal acceleration at the equator reacheszero only when the background wind reaches about 3km s−1 . This intriguing result could help to explainwhy the equilibrated zonal jet speeds in GCMs are typi-cally 3-4 km s−1 (Figure 4), although the meridional andvertical shear of the zonal wind are also likely important.

Hammond & Pierrehumbert (2018) explored the effectof meridional shear of the equatorial jet on the wavestructure and momentum fluxes in a linear shallow-watermodel. They assumed a Gaussian-shaped equatorial jet

whose zonal-mean wind is U = U0e−y2/L2

eq , qualitativelysimilar to those in GCM solutions. Their results indi-cate that, even in the presence of the sheared equatoraljet, the eddies still transport momentum from midlati-tudes onto the equator, leading to an eastward accelera-tion at the equator that can help maintain the superrota-tion. Furthermore, they showed that the inclusion of themeridionally shearing jet and its associated geopotentialanomaly triggers more forced wave modes, which resultsin a better match between the overall surface height fromthe linear shallow-water model and the GCM tempera-ture field than those without a jet.

Several authors have further investigated the dynamicsof the equatorial superrotation in full 3D GCM experi-ments. As mentioned above, the equilibrated, standingeddy structure shows significant distortion from the clas-sic Matsuno-Gill pattern. Despite this complication, di-agnostics from these GCMs still generally support a pic-ture wherein the day-night forcing leads to a large-scale,standing wave pattern that causes a meridional conver-gence of eddy angular momentum onto the equator, driv-ing the superrotation, and a downward eddy momentumtransport at the equator from above the photosphere todeeper levels (Showman & Polvani 2011, Tsai et al. 2014,Showman et al. 2015, Mayne et al. 2017, Mendonca 2020,Debras et al. 2020). This downward eddy momentumtransport helps cause the equatorial jet to slowly pene-trate more deeply over time at pressures of a few bars andgreater. At mid-to-high latitudes, all the studies agreethat the steady state zonal-momentum balance involvesnot only the eddy flux convergences (both meridionaland vertical), but also the Coriolis accelerations and mo-mentum advection associated with the mean-meridionalcirculation. At the equator, the zonal-momentum bal-ance at and above the photosphere tends to comprisea balance primarily between the (eastward) accelerationdue to meridional momentum transport and the (west-ward) acceleration due to the vertical momentum trans-port (e.g., Showman & Polvani 2011). Nevertheless, sev-eral authors have highlighted the role of vertical momen-tum transport due to the zonal-mean circulation as well(Mayne et al. 2017, Mendonca 2020)). Interestingly, al-though the primary day-night forcing would appear tocomprise a zonal wavenumber-one (diurnal) mode, Men-donca (2020) argues that the wavenumber-two (semidi-urnal) mode may be at least as important. The detailsof all these issues are likely to be sensitive to planetaryrotation rate, strength and implementation of radiativeforcing, and other modeling aspects, so it would be worthexploring these issues over a wider range of conditions.

To summarize, the mechanism predicts that the equa-


Fig. 7.— Linear, analytic, steady β-plane solutions of the Matsuno-Gill problem in 3D primitive equations, assuming equal radiative andfrictional time constants that are constant with depth, from Tsai et al. (2014). Left, middle, and right columns adopt a uniform backgroundwind of U = 0, 500, and 1000 m s−1 , respectively. Top row shows eddy geopotential and eddy wind structure; antistellar point lies in themiddle of each plot. Bottom row shows the total wave-induced acceleration; exact values are arbitrary but for plausible forcing amplitudesrange from −10−6m s−2 (westward) in dark blue to 3× 10−6m s−2 (eastward) in red.

torial jet has a meridional half-width approximatelyequal to the Rossby deformation radius. For conditionstypical of hot Jupiters, this length scale is comparableto a planetary radius, explaining the broad structure ofthe equatorial jet in hot-Jupiter circulation models (Fig-ure 4). The process that drives the jet is, in its essence,a direct wave-mean-flow interaction between the eddiesand the planetary-scale mean flow. Turbulent cascadesor other eddy-eddy interactions are possible, and wouldaffect the details, but are not essential to the basic mech-anism.

2.4. Day-night temperature differences and recirculationefficiency

Synchronously rotating exoplanets exhibit permanentdaysides and nightsides and are therefore subject to a fas-cinating climate regime of day-night thermal forcing thathas no analogy in the solar system. Lacking any direct ir-radiation, the only possible means for the nightside to ex-hibit a temperature exceeding tens of K is for heat to betransported from the dayside to the nightside by dynam-ical motions in the atmosphere or interior. This raisesthe intriguing question of what processes control thisday-night heat transport, what the expected day-nighttemperature difference is, and how it should vary withthe stellar irradiation, planetary rotation rate, atmo-spheric composition, and other parameters. This prob-lem is analogous to the important question of what con-trols the meridional temperature gradients and equator-to-pole temperature differences on planets like Earth, butin the case of hot Jupiters, the heat transport occurs notonly meridionally but zonally as well. Because the β ef-fect inhibits heat transport meridionally, whereas no suchconstraint exists in the zonal direction, the dynamics aretherefore likely to be quite different.

This problem is intriguing not only because it toucheson basic issues in geophysical fluid dynamics (GFD) butbecause it is directly amenable to observational charac-terization: on a synchronously rotating planet, energy

balance implies that the energy transported from thedayside to the nightside by the atmosphere/interior cir-culation is radiated to space on the nightside. Therefore,measurements of the total IR flux radiated to space onthe nightside—which can be directly estimated from IRlightcurves—provides an observed measure of the day-night dynamical heat transport.

Many authors have posited that the problem ofwhether the day-night temperature difference is largeor small can be cast as a comparison between twotimescales, the radiative timescale (Equation 3) and anadvection timescale for air to travel across a hemispherefrom day to night:

τadv ∼a

U, (8)

where a is the planetary radius and U is a characteristichorizontal wind speed. Showman & Guillot (2002) firstsuggested that hot Jupiters will exhibit large day-nightdifferences when τrad llτadv and small day-night temper-ature differences when τrad ggτadv. Many authors havesuggested that this timescale comparison helps to ex-plain the dependence of the day-night temperature differ-ence on pressure, opacity, and incident stellar irradiation(Cooper & Showman 2005, Showman et al. 2008a, Fort-ney et al. 2008, Rauscher & Menou 2010, Heng et al.2011b, Cowan & Agol 2011b,a, Perna et al. 2012).

The radiative timescale scales as T−3 (Equation 3), im-plying that it becomes very short for ultra hot Jupiters.In contrast, order-of-magnitude expressions for the typ-ical horizontal velocities (Equation 56) suggest that thehorizontal velocity should also be greater on strongly ir-radiated hot Jupiters, but that the dependence is weaker.Thus, one might expect that for hot Jupiters whose pho-tospheres exceed some critical temperature—perhaps oforder 2000 K—the atmosphere is close to radiative equi-librium and the day-night temperature difference is large.For hot Jupiters cooler than this value, the day-nighttemperature difference should become smaller.

Perna et al. (2012) were the first authors to systemat-


0

0.2

0.4

0.6

0.8

1

500 1000 1500 2000 2500 3000

Full redistribution

No Redistribution

(Fday-Fnight)/Fday

Equilibrium temperature

Perna 2012, γ=0.5Komacek 2017

Parmentier 2020Komacek - 1e3s drag

Parmentier - NS clouds

Fig. 8.— Fractional day-night IR flux differences in GCM simu-lations of synchronously rotating hot Jupiters as a function of theirequilibrium temperature for models with different assumptions.Perna et al. (2012) uses the FMS model with semi-grey radiativetransfer. Komacek et al. (2017) use the MITgcm model with semi-grey radiative transfer but slightly different choices of opacities androtation period than Perna et al. (2012). Parmentier et al. (2020)uses the SPARC/MITgcm with non-grey, temperature dependentopacities (assuming chemical equilibrium without TiO/VO) androtation periods similar to Perna et al. (2012). Despite their differ-ences, the three models provide similar results when used in sim-ilar conditions, showing that the decrease of heat redistributionefficiency with increasing temperature is robust. However, wheneither drag (orange triangles from Komacek et al. 2017) or theradiative feedback of nightside clouds (diamonds from Parmentieret al. 2020) are included in the models, the heat redistribution be-comes much smaller. The difference between physical ingredientsis much larger than the difference between modelling frameworks.

ically explore the influence of stellar irradiation on day-night temperature structure in GCM simulations. TheirGCM adopted a dual-band radiative transfer approachwith one broad opacity band in the visible and one inthe IR, neglecting scattering, with opacities chosen toyield a range of plausible temperature profiles. In aseries of numerical experiments, they varied the stellarirradiation over nearly three orders of magnitudecorre-sponding to planetary effective temperatures of about600 to 2500K—and examined how the circulation re-sponds. Their simulations show that, as qualitativelyexpected, the day-night temperature differences are largefor effective temperatures & 2000 K but becomes smallfor T . 1500 K (Figure 8). They showed that the tran-sition is consistent with the expectation based on thetimescale argument presented above: for planets with ef-fective temperatures less than∼1500K, the advective andradiative timescales (as post-processed from the simula-tion results) are similar, but at effective temperatures ex-ceeding 1500 K, the radiative timescale becomes shorterthan the advective timescale, and the day-night flux dif-ferences become large (Figure 8).

There are several issues with the timescale comparisonbetween τrad and τadv , however (Perez-Becker & Show-man 2013, Komacek & Showman 2016). First, it was notderived in any self-consistent manner from the governingdynamical equations, but rather represents an ad hoc, al-beit attractive, hypothesis. Second, it is not predictive—the advection timescale can only be evaluated once thewind speeds are known, yet the wind speeds themselves

depend on the temperature differences one is trying tounderstand. Third, the criterion obscures any role forother key timescales, including rotational timescale, fric-tional timescale (if any), and timescale for horizontal andvertical wave propagation. These timescales surely af-fect the circulation, including the day-night temperaturedifferences, and therefore one wound expect a timescalecriterion for day-night temperature differences to dependon them.

Perez-Becker & Showman (2013) and Komacek &Showman (2016) performed idealized numerical exper-iments and constructed analytic scaling theories forthe day-night temperature differences and characteristicwind speeds, with the aim of obtaining a self-consistentunderstanding of both the day-night temperature dif-ferences and characteristic wind speeds (hence advec-tion timescale). The former study adopted the 1.5-layershallow water model, while the latter adopted the 3Dprimitive equations, but the setup was otherwise quitesimilar across both studies, involving an idealized day-night forcing where the thermal structure was relaxedtowards a prescribed radiative equilibrium (hot on thedayside and cold on the nightside) over a prescribed ra-diative timescale τrad . In agreement with the results ofPerna et al. (2012), the simulations show that the ther-mal structure exhibits essentially no day-night variationwhen τrad is long, but that the day-night contrast be-comes large when τrad is short (Figure 9). The transi-tion occurs near τrad ∼ 105 s. Frictional drag was alsoincluded in some models to parameterize the possibleeffects of Lorentz forces, and these authors found thatsufficiently strong drag could also lead to large day-nighttemperature differences.

The analytic theory in Perez-Becker & Showman(2013) and Komacek & Showman (2016) is constructedby balancing dominant terms in the dynamical equations.Ginzburg & Sari (2016) also present a related analy-sis. In the momentum equation, the day-night pressure-gradient force, which emerges from the day-night temper-ature difference and drives the circulation, is balancedby the greater of the Coriolis, frictional drag, horizon-tal momentum advection, or vertical momentum advec-tion forces, respectively; this leads to different expres-sions for the day-night contrast in each regime, alongwith criteria for the transition between them. For ex-ample, when the dominant force balance is between theday-night pressure-gradient force and the Coriolis force,which holds when drag is weak and the rotation is suffi-ciently fast, the theory predicts that the fractional day-night contrast is

A =

(1 +

τrad

fτ2wave

)−1

, (9)

where here A is the fractional day-night thermal con-trast (i.e., the fractional day-night thickness contrast inshallow water and fractional day-night temperature dif-ference in 3D). f = 2Ωsinφ is the Coriolis parameter,Ω the planetary rotation rate and φ the latitude. Thequantity τwave = a/NH is the characteristic timescalefor gravity (or Kelvin) waves of long vertical wavelengthto propagate over a zonal distance equal to a planetaryradius. The typical horizontal wave propagation speedcan be estimated by NH, which is about 1 km s−1 for a


Fig. 9.— Dependence of circulation in synchronously rotating hot Jupiter models over a wide range of radiative time constant. Theradiative heating is implemented with an idealized Newtonian cooling scheme that relaxes the circulation toward a specified radiativeequilibrium structure. Top row presents shallow-water simulations from Showman et al. (2013) and Perez-Becker & Showman (2013); fromleft to right panels, radiative time constant is 100, 10, 1, 0.1, and 0.01 Earth days, respectively. Rotation period is 2.2 days in all models.Plot shows thickness (colorscale) and flow velocity (arrows). Bottom row: 3D primitive equation simulations from Komacek & Showman(2016) showing temperature (colorscale) and flow velocity (arrows) structure at 80 mbar; from left to right panels, radiative time constantis 107, 106, 105, 104, and 103 s, respectively. Rotation period is 3.5 days in all models. Both sets of models exhibit a transition from smallday-night thermal contrast at τrad & 105 s to large day-night contrast at τrad . 105 s.

hot Jupiter (see Section 2.1), which leads to a timescaleτwave ∼ 105 s, where we have adopted a Jupiter radiusa = 7 × 107 m. In the limit of large τrad , this equationpredicts A = 0, corresponding to no variation from day-to-night; in the limit of small τrad, it predicts A = 1, cor-responding to a temperature structure in radiative equi-librium, exhibiting large day-night temperature differ-ences. The transition between these limits occurs when

τrad ∼ fτ2wave, (10)

where f is the Coriolis parameter. Evaluating the expres-sion for a typical hot Jupiter value f = 3×10−5s−1 yieldsa critical radiative time constant τrad ∼ 3× 105 s. Thus,under typical hot Jupiter conditions, this theory predictsthat day-to-night temperature differences will be smallwhen τrad . 3×105 s and large when τrad & 3×105 s. Itcan be seen that this explains the transition occurring inFigure 9. More generally, the theory yields self-consistentpredictions of the day-night thermal contrast and windspeeds over the full parameter space of τrad , τrad , forc-ing amplitude (i.e., the day-night difference in radiative-equilibrium temperature) and other parameters, with nofree parameters and no tuning. Perez-Becker & Show-man (2013) and Komacek & Showman (2016) showedthat the theory matches the simulations reasonably wellover this wide range. In contrast, the comparison be-tween radiative and horizontal advection timescales doesnot, particularly when the forcing amplitude is weak.

Perez-Becker & Showman (2013) and Komacek &Showman (2016) showed that the dynamical mechanismfor regulating the day-night temperature differences canbe interpreted in terms of a wave-adjustment mechanism:the day-night heating contrast triggers planetary-scalewaves—including the Kelvin and Rossby wave discussedpreviously—which propagate longitudinally around theplanet. The horizontal convergence and divergence as-sociated with these waves changes the vertical thicknessof fluid columns, inducing vertical motion that pushesisentropes up or down. One can make an analogy tothrowing a rock into a pond: the perturbation to thewater surface caused by throwing the rock into the pondleads to the propagation of gravity waves whose hori-zontal convergence or divergence change the thickness;

once the waves radiate away, the end result is a flat sur-face. In the atmosphere, a similar process acts to flattensurfaces of constant entropy (for a review, see Showmanet al. 2013). If the damping is weak (i.e. the dampingtimescales are long), the waves can easily propagate fromdayside to nightside and this adjustment process is effi-cient, leading to small day-night temperature difference.If the damping is strong (i.e., the damping timescales areshort), however, then the waves become damped beforethey can propagate from day to night. This suppressesthe wave regulation mechanism, allowing the tempera-ture difference to approach radiative equilibrium.

What is the relationship of this criterion to the compar-ison between radiative and advective timescales? Perez-Becker & Showman (2013) and Komacek & Showman(2016) showed that the timescale criterion for transitionbetween large and small fractional day-night tempera-ture differences can be expressed as a comparison be-tween the radiative timescale and an appropriately de-fined vertical advection timescale τvert. This comparisonemerges naturally from the dynamical equations because,on global scales, the vertical entropy advection term gen-erally dominates over the horizontal entropy advectionterm in the thermodynamic energy equation (see Ko-macek & Showman 2016 for a discussion). This is es-sentially the “weak temperature gradient” (WTG) bal-ance well-known from Earth tropical meteorology (for re-views, see Showman et al. 2013, Pierrehumbert & Ham-mond 2019). The timescale comparison between τrad andτvert subsumes the classic comparison between τrad andτadv : they are essentially the same in the highly forcedlimit where the day-night temperature difference is orderunity, but they differ greatly in the weak forcing limit,and in that limit, the former comparison explains thesimulation results far better than the latter one.

Zhang & Showman (2017) combined these regimes intoa single compact expression for the temperature differ-ence, valid across the entire parameter space:

∆T

∆Teq= 1− 2

α+√α2 + 4γ2

(11)

where the non-dimensional, pressure-dependent con-


Fig. 10.— Percentage of water that is dissociated at the infraredphotosphere of irradiated planets as a function of their daysidetemperatures and gravity. Planets on the right hand side of the20% of water dissociation iso-contour can be considered as ultrahot Jupiters. The efficiency of thermal dissociation is pressuredependent, hence why planets with higher gravity, tend to be lessaffected by it. A stellar isochrone for an age of 4 Gyr from Baraffeet al. (2008) is shown as a dashed black line.

stants α and γ are defined as

α = 1 +

(Ω + 1

τdrag

)τ2wave

τradδ ln p(12)

and

γ =τ2wave

τradτadv,eqδ ln p. (13)

In these expressions, τadv,eq = a/Ueq is the timescalefor advection by a reference “equilibrium cyclostrophicwind”, defined as Ueq =

√∆Teqδlnp/2. The quantity

δ ln p is the difference in log-pressure between some deeppressure where the flow is assumed to decay to zero (as-sumed to be 10 bars by Komacek & Showman 2016) andsome lower pressure of interest, for example at the IRphotosphere.

Atmospheres of ultra-hot Jupiters (here loosely de-fined as having equilibrium temperature & 2000 K) aresufficiently irradiated such that all molecules, includingmolecular hydrogen and water are partially dissociatedin their dayside atmospheres (see Figure 10 and Par-mentier et al. (2018)). Heat is absorbed on the day-side to break the strong H2 bond, and then the atomichydrogen is transported by atmospheric flows towardscooler regions of the atmosphere, where hydrogen atomsrecombine to molecules and release heat. The energyper unit mass associated with the molecule-atom tran-sition of hydrogen (2.18 × 108 J kg−1) is two orders ofmagnitude higher than that associated with water phasechange, and together with the fact that hydrogen is themain atmospheric constituent, suggests that heat trans-port by hydrogen dissociation and recombination canbe enormous in ultra-hot atmospheres. Bell & Cowan(2018) first studied this effect using an semi-analytic en-ergy transport model, and they found that the heating

Fig. 11.— Predictions for day-night temperature differences fromanalytic theories and comparison to measurements from lightcurvesof various hot Jupiters, from Komacek & Tan (2018). Dotted linesshow the predictions from Komacek & Showman (2016), and solidlines show the predictions from Komacek & Tan (2018) account-ing for the latent heat caused by dissociation and recombination ofhydrogen at high temperature. Orange curve assume there is no ex-plicit frictional drag in the observable atmosphere, while the blackcurves adopt a spacially constant drag coefficient with a timescaleof 104 s.

from hydrogen recombination could significantly reducethe day-night phase-curve amplitude and increase phasecurve offsets. Building upon the theory of Komacek &Showman (2016), Komacek & Tan (2018) analyticallyinvestigated this effect on the day-night heat transportof ultra-hot Jupiters. They found that the day-nighttemperature difference increases with increasing temper-ature until reaching an equilibrium temperature of about2300K, after which the day-night temperature differencedecreases with increasing temperature. This finding canqualitatively explain the relatively small phase curve am-plitudes observed for several ultra-hot Jupiters (Figure11). Next, Tan & Komacek (2019) implemented the ef-fects of heating and cooling, change of mean molecularweight and gas thermodynamic properties caused by hy-drogen recombination and dissociation into an idealizedGCM similar to that in Komacek et al. (2017), and theysystematically investigated the influence on the atmo-spheric circulation of ultra-hot Jupiters. They likewiseconfirmed that the fractional day-night temperature dif-ference as a function of equilibrium temperature peaksat ∼2300 K, in good agreement with the prediction ofKomacek & Tan (2018). Additionally, wind speeds anddirection of the equatorial flows at relatively low pres-sures are strongly affected by including the hydrogen dis-sociation and recombination, and this influence is moreprominent with higher equilibrium temperature.

2.5. Clouds, hazes, and chemistry on hot Jupiters

There now exist several lines of evidence for cloudsand hazes in the atmospheres of hot Jupiters. First,many hot Jupiters exhibit transit (transmission) spectrathat are considerably flatter than predicted by cloud-freemodels; by causing broadband scattering, high-altitudeclouds/hazes can flatten spectral features in the observedway. Moreover, in the visible, these transmission spec-tra commonly exhibit a Rayleigh-like slope attributedto scattering by small aerosol particles (e.g., Sing et al.2011, 2016, Pont et al. 2013, Stevenson 2016, Iyer et al.2016, Heng 2016, Barstow et al. 2017, Wakeford et al.2017). Second, measurements of secondary eclipse at vis-ible wavelengths imply that several hot Jupiters exhibitsufficiently high albedo to require clouds (e.g., Evans


Fig. 12.— The optical phase curve of Kepler-41b observed withthe Kepler space telescope (Shporer & Hu 2015) compared withthe output from cloudless global circulation and models includingthe presence of silicate clouds of different particle sizes. The leftinset shows the temperature distribution on the dayside of a globalcirculation model with similar equilibrium temperature (Parmen-tier et al. 2016). The right inset shows the brightness distributionon the dayside of the same model integrated over the Kepler band-pass. The bright crescent on the left is due to the reflection byclouds and creates the positive shift of the optical phase curves.

et al. 2013, Heng & Demory 2013, Esteves et al. 2013,2015, Angerhausen et al. 2015). Energy-balance ar-guments for planets with full-orbit IR lightcurves alsoimply significant Bond albedos for several hot Jupiters(Schwartz & Cowan 2015). Most spectacularly, Keplervisible lightcurves of several hot Jupiters peak after sec-ondary eclipse, indicating that the dayside bright spotin visible wavelengths is west, rather than east, of thesubstellar point (Demory et al. 2013, Angerhausen et al.2015, Esteves et al. 2015, Shporer & Hu 2015). This isinterpreted as an inhomogeneous cloud distribution, witha cloud-dominated region west of the substellar point.

Atmospheric circulation modulates the 3D distributionof condensate clouds, but in turn the cloud distributioninfluences the radiative heating/cooling and therefore thecirculation. We consider these issues in turn, followed bybrief discussions of cloud-induced variability, hazes, andcoupling between chemistry and dynamics.

2.5.1. How does the atmospheric circulation affect the clouddistribution?

Two main processes can shape the cloud distribution inhot Jupiter atmospheres. First, the temperature affectsthe cloud distribution, leading to a day/night contrastin the cloud abundance. Second, the meridional atmo-spheric circulation leads to an equator to pole cloud vari-ation that superimposes on the day/night variation.

The temperature effect has been highlighted by theobservation of optical phase curves (Demory et al. 2013,Esteves et al. 2013, 2015) from which the distributionof clouds on the dayside atmosphere can be mapped(Shporer 2017). As shown by Shporer & Hu (2015),all the best targets for cloud mapping showed a partialcloud coverage on their dayside, indicating that most hotJupiters are likely partially cloudy, with clouds presenton the western part of the dayside and absent from thesubstellar point and the hot spot. As seen in Fig. 12,this cloud distribution is anticorrelated with the tem-perature distribution derived from infrared phase curveswhich points toward an eastward shift of the hottest part

of the atmosphere (see Parmentier et al. 2018, for a re-view). This indicates that clouds are present where theatmosphere is cold and absent where it is hot. For the ul-tra hot Jupiters this can be easily explained by the factthat no known species can condense at the high tem-peratures of their dayside (Helling et al. 2019b). Forcooler planets, however, something must prevent theclouds to form in the dayside. One possibility, proposedby Parmentier et al. (2016), is that the most refrac-tory elements that could form clouds on the dayside ofthe cooler hot Jupiters are trapped in the deep layersof the planet. If that is correct, hotter planets wouldbe expected to have silicate (MgSiO3 and Mg2SiO4),corundum (Al2O3), Iron or perovskite (CaTiO3) cloudswhereas planets with an equilibrium temperature coolerthan 1500 K would have an atmosphere dominated bysulphide clouds such as manganese sulphide (MnS), zincsulphide (ZnS) or sodium sulphide (Na2S). Such a tran-sition would be very similar to the transition betweenL and T brown dwarfs Morley et al. (2012). However,this mechanism operates only if cloud particles can besequestered below the photosphere for cooler planets,which is a balance between between settling and verticalmixing. The presence of such a deep sequestration of con-densates has been simulating for silicate clouds by Powellet al. (2018). The study shows that part, but not all, ofthe cloud material was sequestered in the deep layers ofthe atmosphere. The formation of larger particle sizes,through the inclusion of a larger number of species suchas iron could potentially help sequester a larger amountof silicates and explain the trends seen in the Kepler op-tical phase curves. Overall, determining more preciselythe cloud composition of hot Jupiters would shed lightinto the deep atmosphere, including the rate of verticalmixing and the deep thermal structure, both related tothe deep atmospheric circulation.

The atmospheric circulation can also affect the cloudcoverage by advecting cloud material from one part ofthe planet to another one. The level of coupling be-tween flow and particles depends on the pressure. For agiven particle, the settling timescale increases with de-creasing pressure and, below some pressure, always be-comes smaller than any circulation timescale. At 1 mbarin HD209458b, Parmentier et al. (2013) show that parti-cles 1µm or smaller should be coupled to the flow whereaslarger particles should be more strongly affected by thevertical settling. As a consequence, simulations oftenshow that the particles horizontal distribution is morehomogeneous for small than for large particles. Partic-ularly, for large particles, the interaction between theatmospheric circulation and the settling particles oftenlead to a peculiar behaviour at the equator. As seen inFig. 13 in three out of four models the equator gets de-pleted in clouds whereas in the Lee et al. (2016) modelsthe equator gets enhanced in clouds. Overall, more inves-tigation is needed to understand the banding of cloudsin hot Jupiters.

2.5.2. How does the cloud distribution affect theatmospheric circulation?

In hot Jupiter atmospheres, clouds affect the atmo-spheric circulation mainly through their optical proper-ties, the latent heat release during their formation beingnegligible. When present on the dayside, they can cool


Fig. 13.— Cloud particle number density at 1 mbar of four different models using different assumptions. Parmentier et al. (2013)and Komacek et al. (2019) model HD209458b using passive tracers advected by the flow with a settling velocity corresponding to 2.5µmparticles. Whereas the former uses a non-grey radiative transfer scheme including TiO/VO, the second one uses a semi-grey radiativetransfer.Lee et al. (2016) and Lines et al. (2018) use the same microphysical cloud model. The former is coupled to the GCM of Dobbs-Dixon & Agol (2013) with the parameters of HD189733b whereas the latter is coupled to the Metoffice ENDgame global circulation modelof Mayne et al. (2014) with the parameters of HD209458b without TiO. Both include the radiative feedback of the clouds. All modelsshow that the particle number density is different between the equator and the mid-latitudes, showing that banded clouds can be formedby the circulation. Models disagree, however, as to whether the equator should be depleted or enhanced in cloud particles. The dynamicalmechanism responsible for this pattern is not yet fully understood.

the planet by reflecting some of the incoming stellar ra-diation but can also warm the atmosphere by increasingthe greenhouse effect. When on the nightside, clouds al-ways tend to warm up the atmosphere. The final effect ofclouds on the atmospheric circulation therefore dependson their precise location. Roman & Rauscher (2017) showthat nightside clouds tend to warm the planet night-side, leading to smaller day/night contrast on isobars. Aglobal cloud, however, produces a temperature structuresimilar, but cooler than the one of a clear atmosphere.Although the cloud radiative feedback can strongly af-fect the day/night contrast and the hot spot offset onisobars, all models found that they do not change qual-itatively the general behaviour of the atmospheric cir-culation such as the presence of a superotating jet (seeParmentier et al. (2016), Lee et al. (2016), Lines et al.(2018) or Roman & Rauscher (2019)).

Recently, Parmentier et al. (2020) carried a system-atic study of the effect of non-grey nightside clouds onhot Jupiter atmospheres and their observational con-sequences using the SPARC/MITgcm. They confirmthat nightside clouds dramatically reduce the overallday/night heat transport (see Figure 8). Furthermore,they highlight an apparent contradiction: whereas on iso-bars nightside clouds decrease the day/night temperaturecontrast and increase the hot spot shift on isobars, theyusually increase the phase curve amplitude and decreasethe phase curve offset. This apparent contradiction is

due to the fact that phase curves probe hemisphericallyaveraged flux maps. When sharp variations in flux withlongitude are present due to sharp variation of the atmo-spheric opacities, the center of the brightest hemispheredoes not necessarily correspond to the longitude of thebrightest point. Overall, this shows that phase curve am-plitude and phase curve offsets do not necessarily probethe day/night temperature contrast and hot spot offsetson isobars. This should be kept in mind when interpret-ing the results from analytical theories that often assumeisobaric heat transport.

Importantly, for a given planet, the effect of the cloudson the atmospheric circulation can easily be over or un-derestimated. Neglecting the wavelength dependence ofthe cloud opacity (such as Roman & Rauscher (2019)),or the scattering properties of the clouds (Lee et al.2016) can strongly enhanced the heating rates and af-fect the thermal structure and modelled observation in amuch stronger way than in reality (Harada et al. 2019).Similarly, considering a species that is not present in theatmosphere can lead to a strong overestimate of the cloudradiative feedback. As shown by Lines et al. (2019), us-ing two different cloud parametrisation can lead to verydifferent radiative feedback, thermal structure and ob-servational consequences.

2.5.3. Clouds and atmospheric variability

Atmospheric variability has been long sought in hotJupiter infrared observations. Clear sky models of hot


Jupiters are predicted to vary by less than 1% in theirglobal temperature (Showman et al. 2009, Rauscher &Menou 2012, Komacek & Showman 2020), which wouldnot lead to observational evidence given current obser-vational facilities. Optical phase curves, however, havebeen shown to vary significantly for two planets (Arm-strong et al. 2016, Jackson et al. 2019). Although thesource of the time variation is still under debate, andpossibly includes magnetic coupling between the windsand the planetary magnetic field (Rogers & Komacek2014, Rogers 2017) it has been postulated that the pres-ence of clouds could significantly enhance the observablevariability (Lines et al. 2018). Indeed, a planet daysidecan rapidly evolve from barely cloudy to fully cloudywhen the global temperature changes by one to two hun-dreds degrees (Parmentier et al. 2016). Moreover, on agiven bandpass, the relative importance of the reflectedlight and thermal emission can be extremely sensitive tothe cloud abundance and the temperature. Armstronget al. (2016) postulated that the atmosphere of HAT-P-7b could oscillate between two states, one hot and rela-tively clear dominated by thermal emission and havinga bright spot at the temperature maximum, east of thesubstellar point and a colder, cloudier state dominated byreflected light and having a bright spot at the cloud max-imum, west of the substellar point. Such an oscillationwould lead to large variation from positive to negativein the phase curve offset while keeping the total daysideluminosity relatively constant.

2.5.4. Haze

Haze modelling, assuming that methane is a mainhaze precursor, have shown that hazes can form andexplain some features in the transmission spectrum ofhot Jupiters Lavvas & Koskinen (2017). However, sinceall the observed planets do not have large methane con-centrations, other pathways might be more important.Particularly, recent experiments at ≈ 1500K by (Fleuryet al. 2019) have shown that photochemical hazes canform in a pure H2-CO gas at high temperature. Hazeand condensation clouds are expected to be affected dif-ferently by the atmospheric circulation. Whereas cloudparticles form and evaporate at similar temperature andpressures, hazes can have a much higher evaporationtemperature than the temperature they formed. As aconsequence, whereas clouds are expected to track thetemperature maps of hot Jupiters, hazes might provide amore homogeneous aerosol cover. The two could poten-tially be distinguished by measuring the difference be-tween the east and west limb of the planet (Line et al.2016, Powell et al. 2019). Hazes would cover both theeast and west limbs whereas condensation clouds wouldonly be present on the cooler western limb (Kemptonet al. 2017). Recent GCM investigations are starting totest these ideas (Steinrueck et al. 2019).

2.6. Chemistry

If the atmospheres of hot Jupiters were in local chem-ical equilibrium, the large day/night temperature con-trast would naturally lead to large horizontal variationin their chemical composition. However, the consensusis that hot Jupiter atmospheres are not in chemical equi-librium. The reason is that the chemical timescale is

an exponential function of both temperature and pres-sure. As an example, the time it takes to convert car-bon monoxide to methane at 0.1 bar is of order of 10days at 2500K but of the order of millions of yearsat 1000 Kelvin (Visscher 2012). For comparison, thehorizontal advection timescale is on the order of days.As first shown by Cooper & Showman (2006), atmo-spheric mixing is expected to homogenise the atmo-spheric abundances horizontally extremely easily. Theexact value of the resulting abundances is, however, sub-ject to the details of the atmospheric circulation. To firstorder, Agundez et al. (2014) show that the dayside photo-spheric abundances set the nightside abundances of theatmosphere, with the dayside abundances being them-selves set by the vertical quenching. As a consequencethe final atmospheric abundances are determined by thevertical mixing strength on the dayside atmosphere andand the pressure and temperature at the quench point,where the vertical mixing timescale equals the chemi-cal timescale. Drummond et al. (2018a) and Drummondet al. (2020) additionally showed that the meridional cir-culation could also mix the chemical composition latitu-dinally, meaning that the 3D nature of the atmosphericmixing by the circulation should not be ignored. Finally,differences between chemical schemes (e.g. Moses et al.2011, Venot et al. 2019) and the simplifications needed tocouple the scheme to the atmospheric circulation (Tsaiet al. 2018) can also change the expected quenched abun-dances (see Fig. 13 of Drummond et al. 2018b)

Changing the chemical abundances of the atmospherethrough chemical quenching changes the optical proper-ties of the atmosphere, hence the energy balance and theatmospheric circulation. However, as shown by Stein-rueck et al. (2019), Drummond et al. (2018a,b), for themethane/carbon monoxide reaction, quenching affectsthe thermal structure of the atmosphere by ±100K,which is small compared to the day/night contrast ofthe atmosphere. Although important when comparingmodel outputs to observations, chemical quenching doesnot fundamentally alter the atmospheric circulation.

There are two situations where the chemical abun-dances of CO, CH4, and H2O are expected to departfrom being vertically and horizontally constant. First,at pressures lower than ≈ 10µ bar, photochemistry is ex-pected to significantly deplete the dayside atmospherefrom molecules such as methane, water or ammonia whileincreasing the abundances of other species such as HCNor CO2. Horizontal advection is expected to extend theeffect of photochemistry to the limb and part of the night-side atmosphere (see Fig. 14). Importantly, photochem-istry can also affect the higher pressures through verticalmixing, resulting on an inverted quenching as seen forthe case of HCN in Fig. 14.

The second situation where abundances are not con-stant through the atmosphere is extremely high temper-atures, such as occur on ultra-hot Jupiters, when molec-ular dissociation becomes important. The timescale tothermally dissociate and reform molecules is much fasterthan the timescale for other chemical reactions (Bell &Cowan 2018, Kitzmann et al. 2018). As a consequence,ultra-hot Jupiters are expected to be in close to chemi-cal equilibrium with respect to the dissociation reactions.The molecular abundances of almost all species, includ-ing H2, are expected to vary both vertically and hori-


Fig. 14.— Abundances of several molecules at the equator ofHD189733b calculated by Agundez et al. (2014) using a 2D model(longitude/pressure). The abundances are determined by a com-bination of diffusive vertical mixing, advective horizontal mix-ing and photochemistry. At pressures larger than 1-10 bars, theabundances are in chemical equilibrium, between 1-10 bars and100 − 10µbar the abundances are determined by a combinationof vertical and horizontal quenching. For pressures lower than100− 10µbar photochemistry starts being important.

zontally, CO being the only molecule to not dissociate atthese temperatures due to its higher binding energy (Par-mentier et al. 2018, Kitzmann et al. 2018). In the hottestplanets the dissociation of H2 on the dayside and its re-combination on the nightside is expected to significantlyenhance the day to night heat transport through latentheat transport (Bell & Cowan 2018, Komacek & Tan2018, Tan & Komacek 2019, Mansfield et al. 2020, Wong

et al. 2019).

2.6.1. Vertical mixing

Vertical mixing is a fundamental outcome of atmo-spheric circulation. Understanding mixing is critical tounderstanding how the atmosphere interacts with cloudsand chemistry. It affects the abundance of clouds, deter-mines their particle size distribution and whether par-ticles can sequester important chemical species deep inthe atmosphere. However, understanding vertical mix-ing in hot Jupiter is a challenge. Despite being locallystable to convection, the atmospheres of these planetscan transport and mix material with large scale atmo-spheric dynamics. Using passive tracers in a global cir-culation model of HD209458b, Parmentier et al. (2013)show that the vertical mixing in hot Jupiters is due tothe combined effect of large scale upwelling on the day-side, downwelling on the nightside and localised updraftand downdraft close to the terminators. These mixingpatterns are very different from known solar system andbrown dwarfs equivalent, and, for example, vertical mix-ing calculations based on mixing length theory cannotbe easily applied to the hot Jupiter case. As an exam-ple, for HD209458b Parmentier et al. (2013) estimatedthe vertical mixing by measuring the vertical flux of pas-sive tracers along isobars of their global circulation modeland found a vertical mixing coefficient that is 100 timessmaller than the prediction from mixing length theory.Nonetheless, the resulting vertical mixing at the photo-sphere is much larger than at the photosphere of knownsolar-system planets (e.g., eddy diffusivity of ≈ 105m2/sat 100 mbar).

Earth stratosphere is probably the best solar-systemanalogue for vertical mixing in hot Jupiters. As shownby Holton (1986), mixing in the stratosphere is deter-mined by large scale atmospheric motions. The formula-tion of Holton (1986) served as the base of the analyticalmodels developed by Zhang & Showman (2018a,b) andKomacek et al. (2019). They show that the vertical mix-ing is tightly tied to the correlation between the abun-dance on isobars of the species being mixed and the ver-tical motions. When a chemical species has an equilib-rium background abundance varying with pressure (forexample, decreasing with decreasing pressure), the at-mospheric circulation naturally causes a correlation be-tween vertical motion and abundances. Where the windsare upward they carry parcel of gas with high chemicalabundance, while where winds are downward they carryless chemical abundance. The net effect leads to a netchemical flux from regions with high chemical abundanceto regions with low abundance. Efficiency of chemicaltransport by this correlation can be modulated by twodifferent phenomena. The first one is horizontal mixing:the larger the isobaric winds, the harder it is to main-tain a horizontal perturbation due to the vertical winds.The second one is relaxation timescale. Given enoughtime a chemical species abundance will tend towards lo-cal chemical equilibrium. If this equilibrium abundanceis only pressure dependent, then the fastest the chemicaltimescale, the smaller the horizontal variations and thusthe smaller the mixing. Following these ideas, Komaceket al. (2019) derived that the vertical mixing coefficient


should scale as:

Kzz =w2

1τchem

+ 1τadv

, (14)

where w is the vertical velocity, τchem is the chemicaltimescale and τadv the horizontal advective timescale.This formula shows that a vertical mixing coefficient,Kzz, cannot be extracted from an atmospheric flow in-dependently from the chemical species. Otherwise said,different chemical species are advected following differ-ent paths in the 3D atmosphere which are equivalent todifferent 1D vertical mixing coefficients. Finally, the for-mula is valid for species such as chemical species that arenon-conservative. Cloud particles cannot be well repre-sented by this model since they do not disappear but falldown. Intriguingly, despite having investigated settlingtimescales spanning several orders of magnitude, Par-mentier et al. (2016), Zhang & Showman (2018b) andKomacek et al. (2019) all found that the vertical mix-ing coefficient derived from the global circulation modelswas rather independent of the particle size. More workis needed to understand this behaviour and derive a rela-tionship estimating vertical mixing for settling particles.

Finally, because the vertical mixing timescale inher-ently depends on the horizontal mixing, any estimate ofthe vertical mixing rely on a global averaging on isobars(e.g. Parmentier et al. (2013)). If one is lucky, the hor-izontally averaged circulation on isobars might act likea one-dimensional diffusive column. However, on localscales, mixing is not generally diffusive. As such, it isgenerally not correct to calculate local Kzz profiles from3D circulation. Doing so will likely produce spuriousvariations of mixing with height (e.g., the local verticalmixing coefficient would go to zero when the local verticalvelocity changes sign) and the sensitivity of the outputsof the model to the method to calculate the vertical mix-ing coefficient should be thoroughly tested to understandwhich conclusions are robust and which ones are not (seee.g. Helling et al. 2019b, for a discussion).

2.7. Other aspects

2.7.1. Choice of equations

All the global circulation models described in thestudies above do not solve the same set of dynamicalequations. The primitive equations (used in e.g. theSPARC/MITgcm) are the standard dynamical equationsappropriate for circulations in stratified atmosphereshaving horizontal length scales greatly exceeding theirvertical scales (see Vallis 2006 or Showman et al. 2010 fora discussion of equation sets in atmospheric dynamics).These conditions of stratification and large aspect ratioare generally met for the large-scale flow in planetaryatmospheres, including hot Jupiters, where the typicalhorizontal length scales are 104–105 km and atmosphericscale heights are ∼200–500 km, leading to aspect ratiosof ∼20–500. Large aspect ratio and stable stratificationgenerally allow the vertical momentum equation to bereplaced by local hydrostatic balance, i.e., ∂p/∂z = −ρg,where p is pressure, ρ is density, z is height, and g isgravity; this balance means that spatially and tempo-rally varying meteorological perturbations to the densityare generally in hydrostatic balance with meteorologicalperturbations to the vertical pressure gradient. Show-

man et al. (2008a,b) performed a scaling analysis of thevertical momentum equation for the large-scale flow onhot Jupiters, which suggests that, for large-scale flows,hydrostatic balance is valid locally to typically ∼1% orbetter.15 Importantly, the primitive equations make noassumptions that density variations are small, nor dothey set any explicit limits on the wind speeds, in con-trast to some other reduced equation sets in atmosphericdynamics.16 However, the fastest wind speeds in hot-Jupiter circulation models commonly exceed the speed ofsound, at least in local regions (the speed of sound in anideal-gas hydrogen atmosphere is 2.2 km s−1 at 1000 K,rising to nearly 4 km s−1 at 3000 K). This raises the ques-tion of whether acoustic shocks may form in the atmo-spheres of hot Jupiters. Shocks, by definition, are sharpfeatures with small horizontal scales across the shock,which would necessarily imply a local violation of hydro-static balance. Thus, an important question is the extentto which shocks are important and whether the primitiveequations break down in a significant way. Furthermore,in fluid planets, the smooth connection between the deepinterior and the observable atmosphere can lead to flowsspanning a large vertical extent that can get close to vio-late the traditional approximation used in the primitiveequations (Mayne et al. 2019).

To address this issue, several groups have performed3D simulations of hot Jupiters using the fully compress-ible (Euler or Navier-Stokes) equations. Dobbs-Dixonand collaborators were the first to tackle this problem(e.g., Dobbs-Dixon & Lin 2008, Dobbs-Dixon et al. 2010,2012, Dobbs-Dixon & Agol 2013). Their simulations pro-duce circulations that are qualitatively very similar tothose of primitive-equation GCMs, although differencesin radiative forcing, implementation of friction, and othermodeling details prevent a direct, head-to-head bench-mark comparison.

Mayne et al. (2014) and Mendonca et al. (2016) pre-sented new, non-hydrostatic hot Jupiter GCM codes, theformer based on the UK Met Office dynamical core, andthe latter a custom code called THOR, which both allowseveral levels of approximation to be made in the dy-namical equations. The most sophisticated system is thefully compressible, non-hydrostatic Euler equations withradially varying gravity and no thin-shell approximation(“full” in the notation of Mayne et al. 2014); these are

15 In addition to hydrostatic balance, the primitive equationsalso generally adopt the “traditional approximation,” in whichCoriolis and metric terms involving vertical velocity are droppedfrom the horizontal momentum equations and the Coriolis termis dropped from the vertical momentum equation, and a “thin-shell” or “shallow-atmosphere” approximation, in which distancefrom the planetary center is replaced with a reference planetaryradius when it does not occur inside a derivative. Conservationof angular momentum requires that the thin-shell and traditionalapproximations be taken (or not) together. Under conditions whenthe thin-shell approximation is valid, distance from the planetarycenter varies only slightly across the vertical extent of the atmo-sphere, and so gravity is also typically assumed to be constantrather than varying radially. See Vallis (2006) for a general treat-ment of these issues. Mayne et al. (2014) presents a nice summaryof the relationship among these approximations in the context ofhot-Jupiter models.

16 For example, the non-hydrostatic Boussinesq and anelasticequation sets, which are often used to study convection in planetaryinteriors, explicitly assume that dynamical perturbations to thedensity are small and that the wind speeds are small compared tothe speed of sound.


Fig. 15.— Comparisons of solutions from the primitive equations (left) versus full Navier-Stokes equations (right) in the equilibrated“shallow hot Jupiter” test case defined by Menou & Rauscher (2009), in which strong day-night heating (implemented with a Newtoniancooling scheme) occurs in an atmosphere with a surface pressure of 1 bar. Shown are zonal-mean zonal winds versus latitude and pressure.Left panel shows the primitive equation implementation of this test case from Heng et al. (2011b). Right panel shows the fully compressible,Euler-equation implementation of the test case, from Mendonca et al. (2016).

Fig. 16.— Comparisons of primitive equations (left) versus full Navier-Stokes equations (right) in spin-up models of hot Jupiters after10,000 days of integration, from Mayne et al. (2017). Planetary parameters of HD 209458b are adopted. Day-night forcing is implementedwith Newtonian cooling. Top row shows temperature (colorscale, K) and winds (arrows) at 0.2 bar; bottom row shows zonal-mean zonalwinds (m s−1) versus latitude and pressure.


followed by equation sets that successively introduce var-ious simplifications, including constant gravity (“deep”),thin-shell and traditional approximations (“shallow”),followed by the addition of hydrostatic balance (the prim-itive equations)15. The inclusion of multiple choices ofapproximation in a single code is extremely useful, be-cause it allows head-to-head comparisons between the re-sults of different equation sets, where the radiative forc-ing, domain, and other aspects of the numerics are heldfixed. As a benchmark comparison, these studies bothevaluated the “shallow hot Jupiter” test case definedin Menou & Rauscher (2009). Mendonca et al. (2016)showed that, when integrated with the “full” equations,this shallow hot-Jupiter benchmark yields very simi-lar behavior to its primitive-equation implementation(Menou & Rauscher 2009, Heng et al. 2011b, Bendinget al. 2013), as shown in the top row of Figure 16. Mayneet al. (2014)’s “full” implementation of this test casedid not show such good agreement, although subsequentwork (Mayne et al. 2017) suggested that the discrepan-cies may have resulted from their model’s implementa-tion of numerical diffusion rather than inherent differ-ences between the primitive and fully compressible equa-tions.

Mayne et al. (2017) presented further comparisons be-tween primitive and fully compressible equations sets,for a hot-Jupiter setup extending deeper into the atmo-sphere. Day-night forcing was implemented with New-tonian cooling. No frictional drag was included near thebase of the model, which implies that the winds spinup over time in these models. Comparisons betweenthe primitive equations and the “full” set after 10,000days of integration are shown in Figure 16. Qualita-tively, the two implementations produce very similar cir-culations, with roughly the same day-night temperaturedifferences, qualitatively similar spatial temperature pat-terns, and maximum zonal-mean zonal winds that differby only ∼10%. Still, a number of important quantita-tive differences occur, most notably the equatorial jet inthe primitive equation variant (Figure 16, left) extendsconsiderably deeper than in the “full” variant (Figure 16,right). Because these are not equilibriated—the zonal jetis continuing to spin up over time—it is unclear if thesedifferences reflect true differences in the equilibrium statethat would be achieved between the two equation sets,or rather simply reflect differences in the rates at whichthe zonal jet spins up in the two equation sets. Furthercomparisons and diagnostics of this type are needed toclarify the situation.

2.7.2. Hot Jupiter inflation mechanisms

Many hot Jupiters have observed radius larger thanexpectations from standard planetary evolution models(e.g., see Figure 1 in Komacek & Youdin 2017). Strongstellar incident irradiation together with the assumptionof efficient global homogenization of the incoming en-ergy significantly reduce the interior cooling and help tosustain the large radius (e.g., Guillot et al. 1996, Fort-ney et al. 2007), but is still not sufficient to explain theobserved radius. Various mechanisms involving exter-nal energy injection to the planetary interior or furtherreducing the interior cooling have been proposed. Forthorough reviews of this topic, readers are referred to,for example, Fortney et al. (2010), Baraffe et al. (2014)

and Dawson & Johnson (2018). Here we only brieflysurvey several hydrodynamic mechanisms related to theunderstanding of atmospheric dynamics of hot Jupiters.

If a small fraction (on the order of 1%) of the stellar ir-radiated energy is deposited near or below the radiative-convective boundary (RCB), the planetary contractioncan be significantly slowed down or even halted (Guillot& Showman 2002, Komacek & Youdin 2017). Showman& Guillot (2002) and Guillot & Showman (2002) pro-posed that if significant vertical wind shear develops inthe deep layers approaching the RCB where the strat-ification becomes weak, hydrodynamic instability suchas the Kelvin-Helmholtz instability could occur and dis-sipate the kinetic energy associated with the mean flowinto thermal energy via turbulent cascade near the RCB.A “Mechanical Greenhouse” mechanism has been pro-posed by Youdin & Mitchell (2010), in which forced tur-bulence (which may come from shear-wind instabilities)mix higher entropy in the outer radiative zone towardsthe interior, pushing the RCB to depth and significantlyreducing the cooling. These mechanisms are qualita-tively attractive, but the quantitative details involvingthe development and organization of small-scale turbu-lence and their interactions with the large-scale fields re-main inconclusive. So far only very few studies exist inthis direction (e.g., Fromang et al. 2016, Menou 2019),and future investigations unifying our understanding ofthe global-scale flow and small-scale (over lengthscalesmuch smaller than grid sizes typically used in globalmodels) turbulence are desired. Additional uncertaintyof the above mechanisms also arises from our lack of un-derstanding on the deep circulation of hot Jupiters (seesection 2.7.4).

Another mechanism driven by the atmospheric dynam-ics was proposed by Tremblin et al. (2017). They de-veloped a 2D radiative-advective non-grey model of hotJupiters equator with a parametrized meridional and ver-tical mass transport. The simplicity of the model allowsone to solve directly for the steady-state solution ratherthan integrating forward in time, which impedes mostglobal circulation models to solve for a converged steadystate. The model shows that for non-zero vertical trans-port, the deep atmospheric temperature pressure pro-file was converging towards a hot adiabat rather thanan isotherm as in a one dimensional radiative-convectiveequilibrium (e.g. Fortney et al. 2008, Guillot 2010, Par-mentier & Guillot 2014). When large enough verticaltransport is assumed, the downward transport of en-ergy is large enough to explain the inflated radii of hotJupiters. This behavior was further explored with a 3DGCM by Sainsbury-Martinez et al. (2019) who used aNewtonian cooling scheme and integrated their modelsfor several thousands of Earth years. They found thatfor large enough relaxation timescale for the tempera-ture (e.g. larger then 3000years at 200 bars) the modelconverges towards a hot adiabat as in the 2D model.

The large-scale circulation models proposed by Trem-blin et al. (2017) and Sainsbury-Martinez et al. (2019) isattractive in the sense that no (yet unknown) small-scaleturbulence is required, but it has not been fully demon-strated. First, the mechanism behind the results is stillunclear. To mechanically transport thermal energy fromlow to high pressure, one requires a net downward heat


flux (ωT ′ > 0, where ω is the vertical velocity in pres-sure coordinates, T ′ is the isobaric temperature variationrelative to the global mean, and the overbar denotes aglobal mean at isobar surface) through a level near theupper deep layer given that the lower boundary is im-permeable. In the deep layer, this is a thermally indirectcirculation, meaning that the circulation works againstbuoyancy17 (e.g., Holton & Hakim 2012). It cannot oc-cur spontaneously within the deep layer by its own, anddriving forces to the deep layers from the upper pho-tosphere have to be involved. It is not yet clear whatis driving the thermally indirect circulation (i.e., whereωT ′ > 0) throughout the deep layer.

Secondly, both the 2D and the 3D models parametrizea fundamental part of the problem. The formerparametrizes the vertical flow whereas the latter oneparametrizes the heat transfer through a Newtonian re-laxation scheme. Whereas both studies argue that theirmodels are efficient to transport energy downward inmost of the parameter space they explored, it is worthnoting that the radiation and the circulation are inti-mately linked. Whether the actual physical parametersof hot Jupiters allow for the aforementioned thermallyindirect circulation to develop is not clear. Particularly,a more recent study by Mendonca (2020) using a 3DGCM with semi-grey radiative transfer predicts a muchsmaller warming of the deep atmosphere then Sainsbury-Martinez et al. (2019), which would not be able to explainthe inflated radius of hot Jupiters.

Finally, one often needs to be cautious when applyingthe simulated results which are in a closed domain toreal hot Jupiters. The current simulations are often de-pendent on the choice of deep boundary conditions (seesection 2.7.4). For example, considering a more realis-tic situation wherein the GCM is connected to a coldplanetary interior (equivalent to the assumed cold initialdeep layers in Sainsbury-Martinez et al. 2019), the down-ward heat transport forced by the photospheric circula-tion could eventually trigger a sharp thermal inversionnear the interface between the upper hot adiabat and thelower-entropy interior. Hence a strong and sharp strat-ification may suppress the circulation further down andstop the heat transport. Additionally, hot Jupiters likelyhave a convective interior and it is not yet clear how thepresence of a radiative-convective boundary would affectthe deep atmospheric circulation (see e.g. Rauscher &Showman 2014).

2.7.3. Magnetic coupling

In the atmospheres of hot Jupiters, sodium, potassium,calcium and aluminium become ionized when daysidetemperatures become higher than 2000K (Batygin et al.2013, Helling et al. 2019a,b), rendering the atmospheresignificantly conductive. Interactions between the mag-netic field and the flow are therefore expected.

Magnetic fields are expected to interact with the at-mospheric circulation in two ways. First they dissipatekinetic energy through Ohmic dissipation and hence slowdown the winds. Second they affect the atmospheric cir-

17 Circulation of the whole system, including the photospherewhich is driven by the large day-night temperature contrast, is stillthermally direct—otherwise no motions will be maintained againstdissipation.

culation differentially in different directions, leading toa change in the circulation patterns. The dissipation ef-fect was recognized early on and implemented in severalGCMs by simply adopting frictional drag terms in thehydrodynamic equation sets (e.g. Perna et al. 2012). Fur-ther studies proposed that observational measurements,such as smaller than expected hot spot offsets, smallerthan expected wind speeds and and higher than expectedday/night contrast could be caused by the presence of ad-ditional magnetic drag (Komacek et al. 2017, Kreidberget al. 2018, Arcangeli et al. 2019, Koll & Komacek 2018).

In all the aforementioned studies, the additional dragterms tend to drive the atmospheric circulation from ajet-dominated towards a day-to-night flow Komacek &Showman (2016). However, as pointed out by Batyginet al. (2013) zonal flows should be much more stable thanmeridional flows when the magnetic field is aligned withthe spin axis of the planet18.

More self-consistent magnetohydrodynamic modelswere performed by (Rogers & Komacek 2014). They con-firmed that the presence of magnetic coupling leads to areduced meridional flow. Furthermore, when the conduc-tivity is allow to change due to large day-night tempera-ture difference, Rogers (2017) show that the direction ofthe zonal jet can start to oscillate, leading to a variationin the hot spot offset of up to 20 degrees. Such a largevariation in the hot spot offset may be responsible for thetime variability observed in the Kepler phase curves oftwo hot Jupiters (Armstrong et al. 2016, Jackson et al.2019).

The coupling between magnetic field and the circu-lation leads to the downward transport of stellar irra-diated energy to planetary interior through the Ohmicdissipation. Stellar energy is deposited at the photo-sphere which fuels a vigorous atmospheric circulation.This circulation produces currents that can connect todeeper atmospheric layers and dissipate energy (Baty-gin & Stevenson 2010). If more than ∼ 1% of the stel-lar energy can be deposited near the RCB, this couldexplain that many hot Jupiters have an inflated radius(e.g. Guillot & Showman 2002, Menou 2012a, Thorngrenet al. 2019). However, as shown by Rogers & Showman(2014) and Rogers & Komacek (2014), the simulationsshow that magnetic drag slows the deep winds signifi-cantly more than predicted by Menou (2012b), leadingto Ohmic heating ∼100 times too small to explain theinflated radii.

Additional magnetic effects in the atmospheres of hotJupiters that have been explored theoretically but havenot been included in GCMs include the presence of ther-mal instabilities above the photospheric layers triggeredby local heat deposition by Ohmic dissipation Menou(2012c) or the presence of an atmospheric dynamo trig-gered by the large day/night variation in conductiv-ity (Rogers & McElwaine 2017), which could change themagnetic field topography.

Finally, it is worth pointing out that no fully satisfac-tory model currently exists to compare to the observa-tions of ultra hot Jupiters. The only MHD model thathas been applied to hot Jupiter originates from stellarinterior modeling from the Rogers group. The anelas-

18 For misaligned magnetic dipole we would expect a misalignedequatorial jet (Batygin & Stanley 2014)


tic approximation used systematically leads to smallerwind speed than predicted by models solving the primi-tive or fully compressible equations. In addition, no ra-diative transfer scheme was used (only Newtonian cool-ing schemes) and therefore the model outputs cannot bedirectly compared to observations. On the other hand,models including non-grey radiative transfer are pure hy-drodynamic models that only consider the magnetic ef-fect through the inclusion of simple drag terms, whichcould lead to an incorrect atmospheric circulation pat-tern in very hot atmospheres. In the coming decades,coupling the two approaches will be a necessary step tointerpret the observations of ultra hot Jupiters.

2.7.4. Deep boundary conditions and integration times

Gaseous planets lack a distinctive boundary betweenthe atmosphere and the interior, and the choices of bot-tom boundary conditions in GCMs could affect the re-sults both at the photospheric levels and in the deeplayers. Most hot Jupiter GCMs adopted a slip-free, non-permeable bottom boundary at a pressure of around 100or 200 bars (e.g., Showman et al. 2009, Heng et al. 2011a,Rauscher & Menou 2012, Mendonca et al. 2016, Mayneet al. 2017). Compared to the shallow hot Jupiter simula-tions with a bottom pressure of about 1 bar (e.g., Henget al. 2011b, Mayne et al. 2014), flows in the “deep”simulations are typically more time-invariant. Recently,Carone et al. (2020) proposed that for rapidly rotat-ing hot Jupiters like WASP-43b, the simulated domainshould extend to a deeper pressure (∼700 bars) to prop-erly capture dynamics in the observable layers.

The radiative timescale increases rapidly with increas-ing pressure due to the increased opacity and atmo-spheric mass. Convergence of the whole simulated do-main is therefore bottlenecked by the extremely long ra-diative timescale in the deep layers. Recent GCM ex-periments by Wang & Wordsworth (2020) showed thatseveral hundred simulated years are needed for the deepflow to converge if assuming an 80-bar bottom bound-ary pressure. Mendonca (2020) showed that several tensof simulated years are required for the equatorial jet toequilibrate. Such long integration time is yet challeng-ing for GCMs coupled with non-grey radiative transfer,chemistry and cloud microphysics, and is also demand-ing on the conservation properties of dynamical cores.Proper simulating strategies should be investigated inthe near future to cop requirements of both convergenceand comprehensive physics.

In GCMs utilizing radiative transfer schemes, a hor-izontally isotropic net heat flux is typically applied atthe bottom boundary. In this case the deep thermalstructure is unconstrained and can evolve according tothe dynamics driven by the upper atmosphere. In thecase of inflated hot Jupiters, the deep model layer mayhave reached the convective zone given the likely highinterior entropy (e.g., Thorngren et al. 2019). Scalinganalysis and global convection models of planetary inte-rior with uniform surface cooling predict small (∼a fewKelvin) isobaric temperature variation in the convectivezone (e.g., Stevenson 1991, Kaspi et al. 2009, Showmanet al. 2011, Showman & Kaspi 2013). In the absenceof strong molecular gradient, the interior is expected tobe nearly adiabatic. This means that in the presence ofconvection, the zeroth-order temperature structure in the

upper convective zone could be constrained by the spe-cific interior entropy of the planet, which does not evolveover timescales relevant for GCM integration time. In ad-dition, interactions between the stratified layer and theinterior convection may trigger a wealth of turbulenceand waves that propagate upward and interact with themean flow (see Section 4). The extent of which the in-terior convection can affect the atmospheric circulationand deep thermal structure remains unexplored. In theother way around, it remains an open question whetherand how interior convection will differ from those forcedby uniform surface cooling when the hot-Jupiter-like at-mospheric circulation is applied to the surface conditionof interior convection models.

The interior of giant planets is expected to be elec-tronically conducting, and interaction of the magneticfield with the flow leads to Ohmic dissipation and re-tards the flow (e.g., Liu et al. 2008). The large-scale flowin the interior is expected to be significantly slower thanthat near the photosphere. Convection leads to a nearlybarotropic state of the interior, witch together with theslower winds (which implies a small global Rossby num-ber) may result in a TaylorProudman effects (Pedlosky2013) that tends to drag down winds in the deep GCMlayers (e.g., Schneider & Liu 2009). To crudely repre-sent this effect, some GCMs adopted a frictional dragnear the bottom boundary that relax winds towards zeroover characteristic drag timescales (e.g., Liu & Showman2013, Komacek et al. 2017, Tan & Komacek 2019, Caroneet al. 2020). Although easy to implement, the choice ofdrag timescale and the exact form of the drag are ratherloosely chosen. In addition, recent gravity measurementson Jupiter by the Juno spacecraft have revealed that thezonal jets of Jupiter could penetrate down to ∼3000 kmbelow the cloud deck (Kaspi et al. 2018, Guillot et al.2018). Similarly, the zonal jets of Saturn may extend to∼9000 km below the cloud deck which is constrained bygravity measurements from the Cassini spacecraft (Iesset al. 2019). The implication is that, for cooler planetsin which the conducting region is far below the GCMdomain, whether a deep drag that relaxes winds towardzero is questionable in general conditions.

Understanding the interactions and coupling betweenthe photospheric level dynamics and interior dynamicsis in a pressing need given the above issues. A self-consistent coupling between them is numerically chal-lenging partly because traditionally two different sets ofequations are used in different parts—primitive or fullycompressible equations are used in the upper atmosphereand anelastic approximation is used in the interior. Butmore critically, in order to achieve a statistically steadystate with realistic computational cost, global interiormodels are overly forced with non-dimensional dynami-cal parameters that are many orders of magnitude differ-ent to realistic values (e.g., Showman et al. 2011), whilemodels of the upper atmosphere do not suffer from thisissue. Proper modeling strategies are likely needed tocircumvent this challenge. For example, without a fullycoupling between the two parts, modeling of the individ-ual part can be used as idealized boundary condition ofthe other part.

3. WARM JUPITERS


A bit farther from their stars than hot Jupiters liethe warm Jupiters, which we define to be approximatelyJupiter-mass planets with effective temperatures of ∼300to 1000 K. They lie just below hot Jupiters in the upperleft corner of Figure 2, with global-mean incident stellarand radiated IR fluxes ranging from a few hundred toa few × 104 W m−2. As with hot Jupiters, their interiorfluxes are expected to be relatively weak, perhaps 10 to1000 times less than the incident stellar flux they receive(depending on the planet’s effective temperature, history,and other factors), implying that, like hot Jupiters, thewarm Jupiters should have atmospheres that in the time-mean are nearly in radiative equilibrium with their par-ent star.

As yet, there are far fewer observational constraints onthe atmospheres of warm Jupiters than hot Jupiters, butthis situation may change in the future. Over 40 transit-ing warm Jupiters have been discovered. Although mostof these are Kepler detections that are difficult to fol-low up due to their great distance from Earth, over adozen have been discovered by groundbased surveys andthe Transiting Exoplanet Survey Satellite (TESS) mis-sion around brighter, closer stars amenable to furtherobservational characterization.19

The greater orbital separations of warm Jupiters fromtheir stars will lead to key differences in behavior relativeto hot Jupiters. In particular, the tidal effects that drivehot Jupiters into a (presumed) state of synchronizationand that circularize their orbits are less dominant at thegreater orbital distances of warm Jupiters. Therefore,unlike typical hot Jupiters—where it is generally a rea-sonable assumption that the rotation rate is equal to thesynchronous value, and the obliquity and orbital eccen-tricity are zero—warm Jupiters should be expected toexhibit a range of rotation rates, obliquities, and orbitaleccentricities. This may lead to a wider range of possiblebehavior.

Let us quantify the distances at which these transi-tions occur. The tidal synchronization timescale from aprimordial rotation rate Ωp is (Guillot et al. 1996)

τ ∼ Q

(R3p

GMp

)Ωp

(M

M∗

)2(aorb

Rp

)6

, (15)

where G is the gravitational constant, M∗ is the stel-lar mass, aorb is the orbital semi-major axis, and Q, Rpand Mp are the planet’s tidal dissipation factor, radius,and mass, respectively. Evaluating this expression usingvalues appropriate for a typical hot Jupiter yields

τ ∼ 1× 106

(Q

105

)( aorb

0.05 AU

)6

yr, (16)

where we have used a solar mass for the star, a Jupitermass and rotation rate for the planet, along with a plan-etary radius of 1.2 Jupiter radii, which is typical of hotJupiters. For a canonical hot Jupiter 0.05 AU fromits star and adopting Q ∼ 105 (an appropriate time-averaged value for the giant planets in the solar system),the equation yields a spindown time of ∼106 yr—which

19 Prominent examples include WASP-69b and 84b (Andersonet al. 2014), HAT-P-17b (Howard et al. 2012), HATS-17b (Brahmet al. 2016), HATS-71b (Bakos et al. 2018), TOI-813b (Eisner et al.2020), and TOI-216b and c (Kipping et al. 2019).

underlies the common expectation that hot Jupitersshould be close to a state of synchronous rotation. How-ever, the sychronization timescale increases greatly withonly modest increases in semi-major axis: for a planet0.2 AU from its star, the spindown timescale increases to4 Byr, comparable to system ages. Thus, EGPs beyond∼0.15 AU around sunlike stars will generally not be syn-chronized. They may thus exhibit a range of rotationrates and obliquities.

We now discuss, in turn, the effect of non-synchronousrotation, non-zero obliquity, and non-zero eccentricity onthe atmospheric circulation of warm Jupiters, as cur-rently understood. All of the research done so far as-sumes the incident stellar flux greatly exceeds the inte-rior flux, which should be valid in relatively old systemsfor planets inward of ∼1 AU. We close this section byoffering a few remarks about the way these regimes maybe modified by an interior convective flux.

3.1. Non-synchronous rotation

Several GCM investigations have explored the effect ofnon-synchronous rotation under conditions of zero obliq-uity and eccentricity (Showman et al. 2009, Rauscher& Kempton 2014, Showman et al. 2015, Penn & Val-lis 2017). Non-synchronous rotation exerts two effects—first the differing rotation rate changes the strength of theCoriolis parameter and its gradient β, thereby influenc-ing the Rossby number, the Rossby deformation radius,and other factors that depend on rotation. Second, non-synchronous rotation implies that the planet no longerhas permanent daysides and nightsides, but rather thatthe dayside heating pattern sweeps in longitude acrossthe planet over time. This effect might at first seem triv-ial, but in fact it can significantly alter the planetarywave modes that arise from the day-night heating pat-tern, with consequent implications for the superrotation,hot-spot offset, and other aspects of the circulation. In-deed, GCM experiments demonstrate that both effectscritically influence the dynamical behavior.

Showman et al. (2009) performed simulations ofHD 189733b at rotation rates 0.5, 1, 1.5, and 2 timesthe synchronous value, and Rauscher & Kempton (2014)performed a similar study for both HD 189733b andHD 209458b, with the goal of understanding how thetwo effects listed above influence the circulation regimeand IR lightcurves. For HD 189733b, the rapidly ro-tating models in both studies showed the emergenceof high-latitude eastward jets in addition to the pri-mary superrotating equatorial jet. The slower rotat-ing model exhibited a robust equatorial jet flanked bystrong westward flow. In this case, the equatorial jet wasfast and—despite the slower rotation—narrower thanthe equatorial jet in the synchronously rotating model.These changes do not follow the trends that occur insychronously rotating GCM experiments when rotationrate alone is varied (e.g., Showman et al. 2008a), whichsuggests that the longitudinal migration of the daysideheating pattern is critical in setting the detailed jet prop-erties.

Strikingly, however, the slowly rotating HD 209458bexperiment in Rauscher & Kempton (2014)—corresponding to a rotation period of 6.6 days—developed a qualitatively different circulation patterncomprising a strong westward equatorial jet, a dayside


hotspot shifted westward of the substellar point, andday-night flow across most of the terminator. Theyreport that equally slowly rotating models of HD189733b (i.e., using a rotation period of 6.6 days, threetimes the synchronous value) also exhibit a similarcirculation, with a fast westward equatorial jet ratherthan superrotation. The dynamical mechanism causingthis transition remains unclear, but it appears that theslow rotation and migrating dayside heating patternsare key factors. Interestingly, Mendonca (2020) found insynchronously rotating hot-Jupiter models that rotationperiods longer than 5 days allowed equilibrated statescontaining either eastward or westward equatorial jets,which may be a related phenomenon. As expected,synthetic IR lightcurves for the models with eastwardjets reach maximum flux before the secondary eclipse,but the slowly rotating HD 209458b model exhibits anIR flux peak after secondary eclipse. Thus, this phe-nomenon presents a clear prediction for future lightcurveobservations, as well as having a distinct Doppler-shiftsignature during transit (Rauscher & Kempton 2014).

Showman et al. (2015) suggested that the circulationof non-sychronously rotating hot and warm Jupiters sys-tematically splits into two dynamical regimes, depend-ing on whether the radiative time constant is shorter orlonger than the solar day. If the radiative time constant isless than the solar day, the day-night heating pattern (di-urnal cycle) is strong, and the circulation resembles thecanonical hot-Jupiter regime already discussed: the day-night temperature differences are large, and the globalwave modes triggered by the strong day-night forcingdrive equatorial superrotation through the mechanismdescribed in Section 2.

If instead the radiative time constant is greater thanthe solar day, the amplitude of day-night heating (thediurnal cycle) is expected to be weak. Longitudinalvariations of temperature are therefore small, and thecirculation is driven by the equator-to-pole gradient inzonal-mean heating, and primarily plays the role of trans-porting heat meridionally. At low obliquities, the sun-light predominantly illuminates the low latitudes, andthe greatest temperature gradients tend to occur in mid-latitudes, with zonal-mean temperature decreasing pole-ward. Through thermal-wind balance (Equation 4), thisimplies that the strongest winds occur at midlatitudesrather than the equator, and furthermore that thosewinds are eastward at photosphere levels. The midlat-itudes become baroclinically unstable, leading to baro-clinic eddies that transport heat poleward; this trig-gers the formation of Rossby waves, which propagatemeridionally away from their latitude of generation. Themeridional propagation of the Rossby waves arising fromthese baroclinic eddies causes the convergence of angularmomentum into the instability latitudes (for a review, seeVallis 2006, Showman et al. 2013), promoting the gener-ation of midlatitude, eddy-driven jets analogous to thoseobserved on the Earth.

This transition tends to correspond approximately toa particular orbital distance inside of which the diurnalcycle is critical and beyond which it becomes less impor-tant or irrelevant. To calculate this distance, note thatthe effective temperature of a planet in energy balance

with its star (and assuming zero albedo for simplicity) is

Te =1√2

(R∗aorb

)1/2

T∗, (17)

where R∗ and T∗ are the stellar radius and effectivetemperature, respectively. Inserting this expression intoEquation (3) for the radiative time constant yields

τrad ∼pcpgσT 3

∗

(aorb

R∗

)3/2

(18)

which implies that, for a given stellar type, the radiativetime constant at the IR photosphere increases approxi-mately as semi-major axis to the 3/2 power.20 The solarday is Psolar = 1/|P−1

rot −P−1orb|, where Prot is the rotation

period and Porb is the orbital period. Equating the solarday to the radiative time constant and solving for thesemi-major axis, we obtain

aorb ∼ P 2/3rot

[1

k+gσT 3

∗pcp

R3/2∗

]2/3

(19)

where we have used Kepler’s law, P 2orb = k2a3

orb to relatethe orbital period and semi-major axis, where the con-stant k = 2π/

√G(M∗ +Mp). For a sunlike star, and

adopting typical planetary parameters (g ≈ 20 m s−2,p = 0.25 bar, and cp = 1.3×104 J kg−1 K−1), we find thatthe radiative time constant becomes longer than the so-lar day for orbital semimajor axes of 0.045 AU, 0.08 AU,0.13 AU, and 0.2 AU for rotation periods of 0.5, 1, 2, and4 days, respectively.

Thus, these arguments suggest that, around a sunlikestar, planets inward of ∼0.05–0.1 AU will exhibit strongdiurnal cycles, but outward of ∼0.15–0.2 AU, the diurnalcycle will be weak, and the circulation will be driven bythe zonal-mean stellar heating.

Figure 17 and 18 provide a test of this prediction fromShowman et al. (2015) for planets around an HD189733-like star, which is slightly dimmer than the sun. Sim-ulations were performed on a regular grid in rotationrate and orbital semimajor axis. The condition (19) isjust met for the lower left, middle, and upper right pan-els, respectively, of the two figures. The results indeedshow the expected regime transition: models in the lowerright exhibit essentially canonical hot-Jupiter-like behav-ior, while models in the upper left are in a regime wherezonal temperature variations are minimal, baroclinic in-stabilities transport heat poleward in the midlatitudes,and the fastest jets occur in midlatitudes. The transitionis broad and, not surprisingly, models near the predictedtransition exhibit hybrid behavior with fast midlatitudezonal jets superposed on weak equatorial superrotation.

Interestingly, the fastest rotating, least-irradiated sim-ulation of this set—in the upper left panel of Figures 17and 18—exhibits multiple eastward jets in each hemi-sphere, reminiscent of the pattern of multiple zonal jetson Jupiter and Saturn themselves. The zonal jets in this

20 Of course, the photosphere pressure p can depend on atmo-spheric composition and temperature, which would complicate thistrend. These effects seem to be relatively modest, however, due tothe fact that a wide variety of gas-phase species become opticallythick at pressures a bit greater than 0.1 bar (e.g., Robinson & Mar-ley 2014).


Fig. 17.— Temperature pattern over the globe (colors) and winds (arrows) for a set of 9 models straddling the transition between hotand warm Jupiters, illustrating a dynamical transition in the behavior, from Showman et al. (2015). Models in the left, right, and middlecolumns adopt rotation periods of 0.55 days, 2.2 days, and 8.8 days, respectively; the top, middle, and bottom columns adopt orbitalsemimajor axes of 0.2, 0.08, and 0.03 AU, respectively, for planets orbiting a K0 star with the properties of HD 189733. Models in thelower right have radiative time constants shorter than their solar days, and exhibit canonical hot-Jupiter circulation patterns with largeday-night temperature differences and equatorial superrotation. Models in the upper right have radiative time constants longer than theirsolar days, and instead exhibit little day-night temperature variation; baroclinic instabilities transport heat toward the poles, and thefastest wind speeds tend to occur in midlatitudes rather than the equator. Model HΩmed and WΩslow are synchronously rotating; theformer has parameters identical to HD 189733b. The solid vertical bars in each panel show the substellar longitude at the time of thesesnapshots.

simulation exhibit meridional spacings that are qualita-tively consistent with Rhines scaling, which implies thatif the jet speeds were slower (as might be expected undereven weaker stellar irradiation), then the jets would havesmaller meridional spacings and they would be more nu-merous, becoming even more similar to Jupiter and Sat-urn. The transition of the jet structure along with thechange in the rotation rate has been studied for condi-tions appropriate for terrestrial planets and qualitativelysimilar results to those in Figure 18 were obtained (e.g.,Williams 1988).

In the appropriate temperature range, a similar tran-sition can occur among planets of differing atmosphericcomposition receiving a given flux of starlight. Lewiset al. (2010) showed that the atmospheric circulationregime of the hot Neptune GJ 436b, which has an ef-fective temperature of 650K, depends on the metallic-ity. Menou (2012a), Kataria et al. (2014) and Char-nay et al. (2015) found just the same phenomenon forthe super Earth GJ 1214b, whose effective temperature

is 550K. For these planets, atmospheres with metallic-ities & 30 times solar have high opacities, and there-fore low photosphere pressures, leading to short radiativetime constants at the photosphere. This leads to strongday-night forcing at the photosphere, causing substantialday-night temperature contrasts and a fast equatorial jetwhose maximum eastward winds are at the equator. Onthe other hand, at metallicities of a few times solar orless, the gas opacities are smaller, so the photospherelevels are deeper, leading to larger radiative time con-stants at the photosphere. These models have compara-tively weaker day-night forcing, leading to temperaturesthat are nearly constant with longitude, with the fastestzonal winds occurring in midlatitude jets rather than atthe equator. These two regimes are directly analogousto those occurring in Figures 16—17, as well as to thatoccurring in Figure 9, and they occur for essentially thesame reasons. The difference is that the change in radia-tive time constant that causes the transition is broughtabout not by a difference in the incident stellar flux but


Fig. 18.— Zonal-mean zonal winds versus latitude and pressure for the same 9 models shown in Figure 17. There is a striking transitionfrom strong equatorial superrotation for models in the lower right to mid-latitude eastward jet streams for models in the upper left. Thefastest rotating, least irradiated model (upper left panel) exhibits multiple zonal jets in each hemisphere, and illuminates the dynamicalcontinuum between the warm Jupiters and Jupiter and Saturn themselves. From Showman et al. (2015).

rather by a change in the photosphere pressure, via themetallicity.

3.2. Non-zero obliquity

We next turn to consider the effects of non-zero obliq-uity. Only a few papers have so far addressed the influ-ence on obliquity on hot and warm Jupiters (Langton &Laughlin 2007, Rauscher 2017, Ohno & Zhang 2019a,b),using both shallow-water and 3D models.

Let’s first consider some basic aspects. When obliq-uity is zero, of course, there is no seasonal cycle and thestarlight predominantly irradiates low latitudes. Whenthe obliquity is non-zero, the sunlight received by eachhemisphere varies throughout the year, leading to sea-sonal effects, but additionally, the annual-mean irradia-tion changes, shifting annual-mean sunlight away fromlow latitudes toward high latitudes. For small obliqui-ties, the strongest instantaneous (daily mean) starlightreceived remains at low latitudes, even near the time ofsummer solstice. However, when the obliquity reaches

approximately 18, the daily mean insolation quicklyshifts so that, at summer solstice, the maximum starlightis received at the summer pole—even though the annual-mean insolation still maximizes at low latitudes (Ohno& Zhang 2019a). This implies that, if the radiative timeconstant is less than the planet’s year, the seasonal effectswill exhibit a large increase in amplitudes for obliquitiescrossing a threshold of ∼18. A second transition occurswhen the obliquity exceeds 54, above which even theannual mean irradiation is greater at the poles than theequator. If planets with such large obliquities have shortradiative time constants (relative to the planet’s year),they will of course have extremely strong seasons. If theirradiative time constants are long compared to the year,they will lack strong seasons, but they will exhibit theunusual situation of exhibiting warm poles and a coldequator, with the atmospheric circulation transportingheat from high latitudes toward low latitudes.

These insolation changes can trigger rich transitions indynamics, but basic aspects can be anticipated by sim-


ple arguments. At small obliquities, low latitudes tend tobe hotter than high latitudes, and through thermal-windbalance (Equation 4), this poleward-decreasing temper-ature pattern implies that the midlatitude zonal windsshould become more eastward with altitude. This is pre-cisely the way Earth behaves, and is likewise consistentwith the warm-Jupiter models shown in the upper leftcorner of Figures 17–18. On the other hand, if the polesare hotter than the equator, as expected at high obliqui-ties and long radiative time constants, then thermal-windbalance implies that the midlatitude winds become morewestward with altitude. If, during a given season, strongseasonal effects imply that one pole is warmer than theequator while the other pole is colder than the equa-tor, then thermal-wind balance implies that the warmer(colder) hemisphere will exhibit midlatitude winds be-coming more westward (eastward) with altitude.

Rauscher (2017) presented the first 3D study of howobliquity affects the circulation and observables of EGPs.They considered a rapidly rotating (10-hour period) gi-ant planet on a 10-day period orbiting a sunlike star.Under these conditions, the planet’s effective tempera-ture is 880 K; the rotation rate and irradiation level oftheir models are thus very similar to the “WΩfast” mod-els of Showman et al. (2015) (see Figures 17 and 18).Given the expectation that the diurnal cycle would likelybe unimportant, Rauscher (2017) applied diurnally aver-aged (i.e., longitudinally axisymmetric) heating and in-vestigated obliquities of 0, 3, 10, 30, and 90. Theirlow-obliquity simulations (ψ . 30) exhibit a warm equa-tor, cold poles, and eastward zonal jets maximizing in themid-to-high latitudes (Figure 19), consistent with expec-tations from thermal-wind balance, and qualitatively inagreement with the rapidly rotating, weakly irradiatedmodels of Showman et al. (2015).

Rauscher’s high-obliquity (60 and 90) cases, how-ever, exhibit a strong seasonal cycle and a very differentcirculation pattern. Near and after solstice, the summerhemisphere is much warmer than the winter hemisphere,and the hottest location on the planet is the summerpole. As expected from thermal wind balance, this signreversal in the meridional temperature gradient—withhigh latitudes warmer than the low latitudes—leads to asign reversal in the thermal-wind shear, and therefore thezonal flow is to the west. This effect is strongest at 90

obliquity, where the global flow is dominated by a single,broad westward zonal jet with speeds of ∼1 km s−1. Fig-ure 19 clearly shows that, over the majority of the sea-sonal cycle—including much of the winter—both polesare warmer than the equator. This behavior is consis-tent with the fact that, over an annual average, the polesreceive more starlight than the equator for obliquitiesgreater than 54, an effect which is modest for Rauscher’s60 case but severe for their 90 case.

Figure 20 presents IR lightcurves resulting fromRauscher (2017)’s models for the particular situation oftransiting planets (where, depending on the orientationof the planetary rotation vector relative to Earth, thesub-Earth latitude can lie anywhere between zero anda latitude equal to the obliquity). Generally, when theobliquity is small, or when the obliquity vector is largebut in the sky plane (such that neither pole preferentiallytilts toward Earth), the lightcurves are flat. However, ifthe rotation is oriented such that one pole aims toward

Earth, the lightcurve exhibits significant variations, asthe Earth-facing pole alternates between colder wintersand warmer summers. If the planet’s rotation axis istilted toward Earth (i.e., if the projection of the rota-tion vector into the planet’s equatorial plane is aimeddirectly at Earth), then the flux maxima occur aftersecondary eclipse, which is caused by the thermal lagwhereby the summer hemisphere is hottest ∼1/8 orbitafter the solstice (see Figure 19). If the rotation vec-tor tilts in other directions, the situation is complicated,and the flux maxima of the lightcurve can occur eitherbefore or after secondary eclipse. The degeneracies inFigure 20 imply that it will be hard to disentangle circu-lation behavior and obliquity for planets with unknownobliquity, but Rauscher showed that the acquisition ofdayside flux maps from ingress/egress eclipse mappingcan help break this degeneracy. A key takaway point isthat the flux peaks in Figure 19 result from a completelydifferent physical mechanism than those for close-in hotJupiters.

Ohno & Zhang (2019a) presented shallow-water mod-els of warm Jupiters that explore the diverse range ofregimes as a function of obliquity and radiative timeconstant, using an idealized model that represents theday-night forcing with a Newtonian cooling scheme inwhich the radiative time constant is an external param-eter (Figure 21). Unlike Rauscher (2017), they allowed adiurnal cycle, rather than adopting a diurnally averaged,axisymmetric forcing. They found five classes of behav-ior. As expected, when the radiative time constant isshorter than the stellar day (τrad = 0.1 day), the diur-nal cycle leads to a strong day-night thermal (height)contrast. When the radiative time constant is longerthan the stellar day but shorter than the orbital period(τrad = 5 days), the diurnal cycle is weak and the dynam-ical structure is more zonally symmetric; as expected,the seasonal cycle is weak when the obliquity is less than∼18 but strong when it is greater than 18. When theradiative time constant is longer than the orbital period(τrad = 100 days), then the seasonal cycle is weak at anyobliquity; if the obliquity is less than 54, the height isthickest at the equator and thinnest at the poles, and thepredominant winds are eastward. When the obliquity isgreater than 54, the height is thicker at the poles thanthe equator, and the predominant winds are westward.

3.3. Orbital eccentricity

EGPs with semimajor axes greater than ∼0.05 AU ex-hibit a range of orbital eccentricities from zero to nearlyone. In contrast to non-synchronous rotation and non-zero obliquities—which alter the distribution of starlightacross the globe but not the total starlight hitting theplanet—a non-zero eccentricity means that the total stel-lar power intercepted by the planet varies throughoutits orbit. Many eccentric planets are transitional ob-jects: the incident stellar flux received at periapse putsthem in the hot-Jupiter regime, whereas the orbit-meanflux is more typical of warm Jupiters. Prominent exam-ples amenable to observational characterization includethe transiting planets HAT-P-2b (e = 0.5), HD 17156b(e = 0.67), HD 80606b (e = 0.93), which have stellarfluxes that vary throughout their orbit by factors of 9, 27,and 828, respectively. The time variable forcing placeseccentric EGPs in a novel regime that is both inherently


Fig. 19.— GCM simulations from Rauscher (2017) showing the effect of obliquity on rapidly rotating warm Jupiters. Top, middle, andbottom rows show simulations with obliquities of 30, 60, and 90, respectively. Left column shows the zonal-mean temperature at theIR photosphere throughout one orbital period. Right column shows the zonal-mean zonal wind at the norther summer solstice.

Fig. 20.— IR lightcurves of oblique warm Jupiters from the GCMsimulations of Rauscher (2017). These lightcurves are calculatedfor transiting planets, such that the sub-Earth latitude can liebe-tween zero and the obliquity. Different colors represent GCMsimulations performed at different obliquities. For a given obliq-uity,different line styles represent different possible orientations ofthe planet’s rotation axis relative to the line-of-sight to Earth. Or-bital phase is defined such that transit occurs at 0 and 1, whilesecondary eclipse occurs at 0.5

interesting and may yield insights into atmospheric pro-cesses not easily obtainable from planets in circular or-bits.

Generally, eccentric planets are not expected to ex-hibit rotation periods equal to their orbital periods. Atlarge eccentricities, tidal effects are orders of magnitude

stronger near periapse than throughout the rest of the or-bit when the planet is farther from the star. Therefore, acommon expectation is that for EGPs whose periapses lievery close to the star, tides act to drive eccentric planetstoward “pseudo-synchronization,” a state in which theplanet’s rotation rate equals the instantaneous rate oforbital motion when the planet is near periapse. Debateexists about the details, but for large eccentricities, thisstate corresponds to rotation periods much shorter thanthe orbital period. The rotation period is of course con-stant throughout the orbit, and given the varying rateof orbital motion, a pseudo-synchronous rotation stateimplies that, for high eccentricities, the stellar day willbe much shorter near apoapse than near periapse.

Eccentric planets can be subject to several possi-ble thermal regimes, due to the fact that their meantemperature—and therefore characteristic radiative timeconstant at the photosphere—vary throughout the orbit.The first regime corresponds to the situation where theradiative time constant is shorter than the stellar daythroughout the orbit, so that the diurnal cycle (day-nightradiative forcing) is important at all orbital phases. Asecond regime occurs when the peripase is small enoughthat, near periapse, the radiative time constant is shorterthan the stellar day, whereas near apoapse it is longer


Fig. 21.— Regimes of behavior for warm Jupiters as a function of radiative time constant, length of day, and orbital period, fromOhno & Zhang (2019a). When the radiative time constant is less than the stellar day, the day-night contrast is large (regime I). AtProt < τrad < Porb, the diurnal cycle is weak; when obliquity is less than ∼18, the seasonal cycle is weak, whereas when obliquity isgreater than 18, it is strong (regimes II and III, respectively). When τrad > Porb, the seasonal cycle is weak and the annual-mean insolationdrives the circulation; for obliquities less than 54, the height is thicker at the equator than the poles and the winds are predominantlyeastward, whereas obliquities greater than 54, the reverse is true, and the winds are predominantly westward.

than the stellar day. In this regime, the diurnal cy-cle is important near periapse, leading to an intenseburst of day-night radiative forcing, but the diurnal cy-cle is not important throughout the rest of the orbit. Athird regime occurs when the entire orbit—including theperiapse—is sufficiently far from the star that the diurnalcycle is irrelevant throughout the orbit. So far, most ec-centric EGPs whose atmospheres are being characterizedlie in the first or second regime.

Several investigations of the atmospheric circulation ofeccentric EGPs have been performed (Langton & Laugh-lin 2008, Kataria et al. 2013, Lewis et al. 2010, 2014,2017). Langton & Laughlin (2008) adopted an idealized2D model, whereas the other studies adopted a 3D GCMwith sophisticated non-grey radiative transfer. For ec-centricities that are small—less than perhaps 0.2—the ef-fects of eccentricity on the qualitative circulation regimeare modest (Lewis et al. 2010, Kataria et al. 2013). Butfor larger eccentricities, the investigations performed todate all agree that the intense stellar irradiation near pe-riapse passage plays a dominant role in driving the circu-lation. This “flash heating” causes transient formation ofa Matsuno-Gill-like pattern of standing equatorial wavesthat can drive equatorial superrotation, which can persistthroughout the orbit even though it is not strongly forcedthroughout the more distant parts of the orbit (Katariaet al. 2013, Lewis et al. 2014, 2017). The flash heatingevent can also trigger shocks and other wave fronts thatpropagate from day to night. Still, the 3D studies per-formed to date indicate that this time-variable forcingappears to produce a time-variable version of the circu-lation regime already familiar from hot Jupiters on cir-cular orbits, as opposed to a qualitatively different, to-tally new circulation regime. As mentioned, these studiestend to be in the first or second regimes described above,and one could imagine that eccentric EGPs in the thirdregime would lack Matuno-Gill patterns and would nothave strong equatorial jets (at least not via the mecha-nism described in Section 2). Future work is needed to

explore this issue.For large orbital eccentricities and small periapse dis-

tances, the pseudo-synchronous rotation rate is fast, andthis leads to an equatorial jet much narrower than oc-curs in GCM experiments of hot Jupiters on circular or-bits. Kataria et al. (2013) performed a systematic inves-tigation across a wide range of semi-major axes and ec-centricities, exploring both pseudo-synchronous and syn-chronous rotation rates, and she showed that the merid-ional half-widths of the equatorial jet are reasonably wellmatched by the equatorial deformation radius, as pre-dicted by the analytical theory (see Section 2.3).

These studies have important implications for obser-vations of eccentric EGPs. The increase in temperaturecaused by periapse passage tends to lag the pulse of stel-lar insolation, such that the maximum temperatures oc-cur after the planet is already past periapse and recedingfarther from its star. The overall timescale for this ther-mal pulse to damp out provides an observational measureof the atmosphere’s radiative time constant, a quantitythan is difficult to infer for planets on circular orbits.The large day-night temperature difference can persistfor times considerably past periapse passage, and if therotation period is short, this can lead to a “ringing” phe-nomenon where the flux received at Earth oscillates upand down as the hotter hemisphere rotates in and out ofview (Langton & Laughlin 2008, Cowan & Agol 2011a,Kataria et al. 2013, Lewis et al. 2017).

Synthetic lightcurves show that the observational sig-natures of transiting eccentric EGPs can be subtle anddepend on the orientation of the planet’s orbit relative toEarth. Temporal flux variations in IR lightcurves reflectboth spatial effects—in which hot and cold regions onthe EGP move in and out of view as the planet rotates—and temporal effects in which the entire planet heats andthen cools as it passes through periapse. Disentanglingthese effects can be difficult. Depending on the orbitalalignment relative to Earth, the secondary eclipse canoccur either before or after periapse passage, and, partly


as a result, the peak IR flux can either lead or lag thesecondary eclipse (Kataria et al. 2013). In most cases,the peak IR flux lags the periapse passage, although thetiming depends on the orbital configuration, since theorbital alignment controls the time at which the newlyflash-heated hot hemisphere first rotates into view as seenfrom Earth (Kataria et al. 2013). As a result of theseissues, the range of possible lightcurve signatures is di-verse.

Spitzer IR lightcurves have been obtained for HAT-P-2b over its full 5-day orbit (Lewis et al. 2013) and forHD 80606b near its periapse passage (Laughlin et al.2009, de Wit et al. 2017). These observations show alarge spike in the IR flux due to flash heating, whichpeaks around the time of periapse passage and thenslowly decays. In general, this flux spike is reproducedin GCM simulations of these two planets (Lewis et al.2014, 2017), but the detailed way the flux decays in thesimulations exhibits discrepancies with the observations,especially at times & 1 day after periapse, which sug-gests possible influences of clouds, chemistry, or interiorheat flux that are not yet self-consistently included in thesimulations (Lewis et al. 2017).

4. BROWN DWARFS AND DIRECTLY IMAGED PLANETS

Brown dwarfs are fluid hydrogen objects thought toform like stars but that contain insufficient mass to fusehydrogen. They range in mass from approximately sev-eral Jupiter masses (MJ) up to the stellar mass limit of∼80MJ .21 Despite their large mass range, the hydrogenequation of state implies that these objects have radiiclose to Jupiter’s radius over nearly two orders of magni-tude in mass, from less than 1MJ to 80MJ (e.g., Guillot1999, Chabrier et al. 2000). Their lack of an internalheat source22 implies that, like Jupiter, brown dwarfscool down over time. Nevertheless, massive brown dwarfscontain so much internal energy that they cool downvery slowly and, even after billions of years, can exhibittemperatures at the IR photosphere exceeding 1000 K.Brown dwarfs were first definitively discovered in the mid1990s (Nakajima et al. 1995, Oppenheimer et al. 1995),around the same time as the first EGPs were discoveredaround main sequence stars (Mayor & Queloz 1995), andover the past 25 years the field of brown dwarf discoveryand characterization has developed in parallel with thefield of exoplanets. Because brown dwarfs are typicallyisolated—with no nearby star to drown out their light—they are generally much easier to observationally charac-terize than exoplanets. The quality and quantity of IRemission spectra for brown dwarfs is therefore exquisiteby the standards of exoplanets.

Like stars, brown dwarfs have been subdivided intospectral types, the L, T, and Y types, based on theirspectral characteristics (Kirkpatrick 2005, Cushing et al.2011, Kirkpatrick et al. 2012, Leggett et al. 2017).

21 The mode of origin of substellar objects is difficult to ascer-tain for individual objects, so for convenience brown dwarfs aresometimes defined as hydrogen objects with masses between ∼10and 80MJ . The mode of formation is not important for the presentdiscussion of atmospheric dynamics.

22 Sufficiently massive brown dwarfs can fuse deuterium, but itsabundance is sufficiently small that this process generally causesonly a modest change to the evolutionary trajectory of a browndwarf (Burrows et al. 2001).

L dwarfs are the hottest, with effective temperatures∼2100–1300K, as well as reddish colors in the near-IR and rather shallow spectral features indicating theprevalence of cloud decks of silicates and other refrac-tory condensates. T dwarfs are cooler, with effectivetemperatures of 1300 to ∼600K. Their IR spectra ex-hibit deep spectral features of water and methane, withblue near-IR colors, indicating comparatively cloud-freeatmospheres. Y dwarfs are the coldest class, with ef-fective temperatures less than ∼600 K and deep spec-tral features of water, methane, and other species. Thedisappearance of silicate clouds in T and Y dwarf spec-tra makes sense, because at their relatively cool temper-atures, refractory materials like silicates condense onlyvery deep in their atmospheres, below the optical-depthunity level, where the cloud layers cannot strongly affecttheir outgoing IR spectra. Since brown dwarfs cool downover time, this spectral sequence is also an evolutionarysequence, meaning a given object can transition from Lto T to Y over billions of years. This process happensmore quickly for lower mass objects, such that for ob-jects of a given age, a massive brown dwarf may remainan L dwarf while a lower-mass object may have alreadytransitioned to a T or Y dwarf.

A key point is that, starting in the L dwarfs, and pro-ceeding through the T and Y dwarfs, the atmospherictemperatures of brown dwarfs are sufficiently cool thatthey should magnetically decouple from any backgrounddipole magnetic field they may possess (e.g., Gelino et al.2002), making them less like stars, and strengtheningthe analogy between the atmospheric dynamics of browndwarfs and those of solar system planets like Jupiter andSaturn.

In addition to brown dwarfs, a variety of young,hot EGPs are being directly imaged and characterized.These include the planets around HR 8799, β Pic, and2M1207, among other systems. Because of the difficultyof discerning the planet from the much-brighter star,planets that can be directly imaged tend to be distantfrom their stars (10 AU or more) and have photospherictemperatures of ∼500 K or more. This implies that theyreceive negligible stellar flux compared to the interiorheat flux they radiate to space; therefore, from a mete-orology standpoint, directly imaged EGPs resemble low-mass, low-gravity versions of brown dwarfs, and they willlikely exhibit analogous dynamical processes.

Despite exhibiting similar temperature ranges, theregime of brown dwarf atmospheres differs significantlyfrom that of hot Jupiters. Most known brown dwarfsare isolated objects that receive no starlight; they arehot not due to irradiation, but because they are massiveand still losing their formidable heat of formation. Thisimplies that, unlike typical hot Jupiters, they transportan enormous internal heat flux through their interiorsand into their atmospheres to be radiated to space. Forexample, a 1500-K brown dwarf has an IR heat flux of290, 000 W m−2, while even a 1000-K object radiates anenergy flux of 60, 000 W m−2, hundreds of times greaterthan Earth’s IR flux to space of 240 W m−2. In the in-teriors, this prodigious heat flux is expected to be trans-ported by convection, but the transition to opticallythin radiative transfer near ∼1 bar pressure decreasesthe radiative equilibrium temperature gradient, ensuringthat the temperature structure transitions from nearly


Fig. 22.— Observations indicating surface patchiness on brown dwarfs. Left panel shows the IR flux versus time measured in J-band(∼1.2µm) for brown dwarf SIMP0136 by Artigau et al. (2009). The J-band flux varies by about 5% with a period of 2.4 hours, indicatingthat regional patchiness in clouds and temperature move in and out of view as the brown dwarf rotates every 2.4 hours, causing significantvariability in a disk average. As expected, the period is constant throughout the observing sequence, but the shape of the lightcurvevaries substantially on intervals of several days, which implies that the global pattern of cloud patchiness evolves on this timescale. Rightpanel shows global maps throughout a rotation period of the brown dwarf Luhman 16B obtained by Doppler imaging from Crossfieldet al. (2014). The maps indicate regional patchiness on a scale of tens of thousands of km. Note that although the patchiness is real, thepreferential north-south elongation of features seen in the maps, as well as some of the lower-amplitude features, are probably analysisartefacts; moreover, at the signal-to-noise ratio of these maps, the method may not be sufficiently reliable to retrieve a zonally bandedstructure, if any (Crossfield et al. 2014).

Fig. 23.— Modeled 1D temperature-pressure profiles of browndwarfs in radiative-convective equilibrium, from Burrows et al.(2006). Twelve models are shown, with effective temperaturesranging from 900 to 2000K in steps of 100K (solid curves). Oneach curve, red represents the convection zones and black repre-sents stably stratified layers in radiative equilibrium. Black dotsshow the radiative-convective boundaries. Stars show the approxi-mate IR photosphere level where the bulk of the radiation escapesto space. Note that in most cases, the IR photosphere occurs inthe stably stratified atmosphere. Understanding observations of IRspectra, variability, surface patchiness, and chemical disequilibriumtherefore requires an understanding of the atmospheric structure,including the ability of atmospheric dynamics to mix clouds andchemical tracers in this stratified region.

adiabatic in the deep interior to a stably stratified at-mosphere at low pressure, with the radiative-convectiveboundary expected to occur typically at a few bars pres-sure (Figure 23, Stevenson 1991, Burrows et al. 2001,Baraffe et al. 2014).

Several lines of evidence indicate that brown dwarfsexhibit vigorous atmospheric circulations. First, the IRspectra and colors of many brown dwarfs—particularly Ldwarfs—indicate the presence of silicate clouds (dust) inthe visible atmospheres (e.g., Chabrier et al. 2000, Tsuji2002, Knapp et al. 2004, Kirkpatrick 2005, Cushing et al.2006). In the absence of dynamics, cloud particles would

settle gravitationally, so the existence of clouds impliesthe presence of atmospheric motions to loft the parti-cles vertically. Second, the transition between L and Tdwarfs is a puzzling phenomenon that seemingly requiresa role for atmospheric dynamics, either in generatingcloud patchiness or in influencing how cloud microphysicsdepends on spectral type (Ackerman & Marley 2001,Burgasser et al. 2002, Knapp et al. 2004, Burrows et al.2006, Marley et al. 2010). Third, many brown dwarfs ex-hibit chemical disquilibrium of CO, CH4 and NH3, whichseems to require vertical mixing of air between deep lev-els and the atmosphere (Fegley & Lodders 1996, Saumonet al. 2000, 2006, Leggett et al. 2007, Stephens et al.2009, Miles et al. 2020). Fourth, many brown dwarfsexhibit IR variability over rotational timescales, whichprobably results from regional patches of differing tem-perature and cloud opacity—and therefore differing IRfluxes to space—moving in and out of view as the browndwarf rotates (e.g., Artigau et al. 2009; Radigan et al.2012; Apai et al. 2013; Wilson et al. 2014; Buenzli et al.2014;Buenzli et al. 2015; Metchev et al. 2015; Yang et al.2016; Lew et al. 2016; Vos et al. 2019; Vos et al. 2020;Zhou et al. 2020; Bowler et al. 2020; for reviews, seeBiller 2017 and Artigau 2018). Interestingly, the shapeof the lightcurves changes on short timescales, indicat-ing that the spatial pattern of patchiness changes overtime—presumably due to time-evolving organization ofturbulence, vortices, or other atmospheric structures overthe globe. In some situations, the lightcurves can be usedto place constraints on the size and spatial distributionof atmospheric spots on the globe (Karalidi et al. 2016,Apai et al. 2017). Fifth, Doppler imaging of the closestbrown dwarf to Earth, Luhman 16B, provides direct con-firmation of a patchy surface structure (Crossfield et al.2014), a technique that may be extended to other browndwarfs in the future.

Several authors have attempted to constrain atmo-spheric wind speeds from observations. Allers et al.(2020) recently presented the first true atmospheric


speed measurement for a brown dwarf. They indepen-dently measured the rotation period of IR variabilityand the period from radio emission for the brown dwarf2MASS J1047, showing that the former is about oneminute shorter than the latter on this ∼1.75-hour-periodobject—a 1% difference. The IR period senses atmo-spheric cloud patchiness, while the radio period presum-ably senses the magnetic field rotation rate, which isrooted in the deep interior. Thus these measurements im-ply that the atmospheric features move eastward relativeto the deep interior at a velocity of 600±300 m s−1. Thisphenomenon is most easily explained by advection of at-mospheric features by a fast eastward zonal jet, though acontribution due to zonal wave propagation is not ruledout. In fact, this method is essentially the same approachused to infer the atmospheric wind speeds on Jupiter rel-ative to those in the Jovian interior.

The number of brown dwarfs for which radio and IRperiods can both be measured is limited, however, andso several authors have attempted to infer wind speedssolely from IR lightcurves. This requires plausibility ar-guments about the nature of the atmospheric featurescausing the variability. Artigau et al. (2009) and Radiganet al. (2012) found that time evolution of their lightcurvescould be explained by the assumption that discrete, co-herent atmospheric features change in longitudinal sep-aration over time due to advection by an assumed zonalwind; the derived wind speeds are ∼300–500 and 30–50 m s−1, respectively. Similarly, Apai et al. (2017) in-ferred a speed of ∼600 m s−1 but suggested this may re-sult from differential propagation of atmospheric waves,rather than advection by zonal jets. Burgasser et al.(2014) and Karalidi et al. (2016) used lightcurves to placeconstraints on the diameter of atmospheric spots; underthe assumption that the spots may coexist with jets ofequal width and that the jets obey Rhines scaling, theinferred wind speed is ∼600 m s−2 or faster. These ap-proaches are worth exploring, but—as the above authorsare careful to admit—the assumptions they invoke arenon-unique and uncertain, so these estimates should betaken as plausibility inferences rather than tight obser-vational constraints.

Here, we survey the atmospheric dynamics of browndwarfs, first using basic arguments to sketch the expecteddynamical regime, and then summarizing current atmo-spheric circulation models of brown dwarfs, emphasizingphysical insights, basic processes, and observational im-plications derived from these studies.

4.1. Basic dynamical regime

Isolated brown dwarfs rotate rapidly, and this exerts astrong control over their atmospheric dynamics. Dopplerbroadening of atmospheric spectral emission lines (Rein-ers & Basri 2008), and rotation periods directly inferredfrom IR variability (e.g., Artigau et al. 2009, Radiganet al. 2012, Metchev et al. 2015), indicate rotation pe-riods typically ranging from 1.5 to 12 hours. Directlyimaged planets likewise rotate rapidly, with periods oforder several to 11 hours (Snellen et al. 2014, Zhou et al.2016).

Because of the fast planetary rotation, the regional-to-global circulation on brown dwarfs and directly im-aged planets exhibits Rossby number Ro 1, imply-ing that the large-scale dynamics are rotationally domi-

nated (Showman & Kaspi 2013). Adopting wind speedsof 50–1000 m s−1 and rotation periods of 1–10 hr consis-tent with the observationally inferred ranges, global-scaleflows (L ∼ 108 s) yield Rossby numbers of 0.0002 to0.04.23 If the dominant flow length scale is instead a re-gional scale (L ∼ 107 m), similar to the dominant lengthscales on Jupiter and Saturn, then the Rossby numbersrange from 0.002 to 0.4. Over almost the entire rangeconsidered, these values are much less than one, imply-ing that the dominant horizontal force balance is betweenCoriolis and pressure-gradient forces, that is, geostrophicbalance (Vallis 2006, Holton & Hakim 2012). The large-scale circulation on Earth, Jupiter, Saturn, Uranus andNeptune are also geostrophically balanced, and thereforea substantial body of literature in the dynamics of rapidlyrotating turbulence developed over the past 40 years willlikely have important insights for understanding browndwarfs. Note for a typical brown dwarf with a rotationperiod of a few hours, the “tropical regime” correspond-ing to Ro & 1 obtains only within a few degrees of theequator.

A second influence of rotation is that it leads to smallRossby deformation radii. Under brown dwarf condi-tions, the deformation radius in midlatitudes is LD =NH/f ∼ 500 to 5000 km—just a few percent of the plan-etary radius—for brown dwarfs with rotation periods of1 to 10 hours. The Rossby deformation radius is a typi-cal length scale that emerges from a variety of processesinvolving an interplay between buoyancy and rotation.For example, baroclinic instability and convection tendto produce eddies with sizes comparable to the defor-mation radius. On Jupiter and Saturn, this fact helpsto explain the large population of vortices with sizesof ∼1000–2000 km (e.g. Mac Low & Ingersoll 1986, Liet al. 2004, Vasavada et al. 2006, Choi et al. 2009): theyare probably a natural outcome of convection interactingwith the overlying, stratified atmosphere. This argumentsuggests that brown dwarfs will likewise exhibit a largepopulation of eddies with sizes of order 1000 km.

Still, the processes that drive a circulation in browndwarf atmospheres differ significantly from those on typ-ical irradiated planets. On hot Jupiters, Earth, and otherirradiated planets, the large-scale atmospheric circula-tion results primarily from the gradient in stellar irra-diation between the dayside and nightside (or betweenthe equator and pole on planets where the diurnal cycleis weak). But most brown dwarfs lack external sourceof irradiation, and so this familiar way of generating anatmospheric circulation cannot occur on a brown dwarf.Convective mixing should lead to a nearly constant en-tropy in their interiors, and if so, the temperature onan isobar near the radiative-convective boundary shouldvary little with latitude. As a result, one might ex-pect the upward IR flux emitted to space—and thereforethe atmospheric temperature-pressure profiles in radia-tive equilibrium with that upwelling flux—to be nearlyindependent of longitude or latitude. In such a situa-tion, the stratified atmosphere would exhibit essentiallyno horizontal temperature contrasts at all.

Still, as we will see, several mechanisms exist for

23 For these estimates we adopted a value of the Coriolis param-eter relevant to midlatitudes, implying that f ∼ 2 × 10−3 s−1 to2× 10−4 s−1 for rotation periods of 1 to 10 hours.


Fig. 24.— Entropy anomalies (top) and velocity and dust dis-tribution (bottom) versus horizontal distance and height in a local2D box model of convection from Freytag et al. (2010). This sim-ulation corresponds to a brown dwarf with effective temperature1858 K and g = 1000 m s−2. The convective zone roughly corre-sponds to z < 0, and convective plumes can be readily seen there.They interact with the overlying stratified layer at z > 0 to gener-ate small-scale gravity waves.

generating a circulation in the stratified atmosphere ofbrown dwarfs. Because the IR radiation that escapesto space generally originates from within the stratifiedatmosphere rather than the convection zone (see Figure23), understanding the dynamics in this stratified regionis critical to explaining observations of lightcurve vari-ability, patchy cloud structure, and chemical disequilib-rium.

4.2. Circulation models of brown dwarfs

There exist at least three independent mechanisms forgenerating an atmospheric circulation in the stably strat-ified atmosphere of an isolated brown dwarf: (1) interac-tion of the interior convection with the stratified atmo-sphere; (2) 1D cloud feedbacks between radiation, ver-tical mixing, and cloud physics; and (3) multi-D cloud-radiative-dynamical feedbacks that drive an overturningcirculation that maintains cloud patchiness. We sum-marize recent models investigating each of these threemechanisms.

4.2.1. Atmospheric circulation I: interaction of convectionwith the stably stratified atmosphere

In a pioneering study, Freytag et al. (2010) presented2D numerical models of convection in a local box, repre-senting a small patch of a brown dwarf, aimed at deter-mining how convection interacts with an overlying, sta-bly stratified atmosphere. Rotation was neglected, andthe typical domain size was 300–400 km, about 0.5% ofthe radius of a brown dwarf. They coupled the 2D dy-namics to a radiative transfer scheme and a dust cyclethat computes the transport of a condensable gas speciesand its condensate—silicate dust—as well as the inter-conversion between them. In the simulations, convectiveplumes drip off the top of the convection zone and de-scend into the interior. The convection interacts withthe overlying stratified layer, generating a wide popu-lation of small-scale gravity waves (Figure 24). Thesewaves break, triggering localized regions of vertical mix-

ing in the stratified atmosphere, which can transportthe clouds and condensable vapor upward, and gener-ate cloud patchiness on a scale of tens of km. Freytaget al. (2010) analyzed the characteristic velocities in theconvection zone and stratified layer, characterized howthey depend on effective temperature, and derived ef-fective vertical eddy diffusivities for upward mixing oftracer due to the breaking gravity waves. This helps toexplain the cloud decks on typical L dwarfs, and maybe relevant for understanding the L/T transition, but itslengthscale is too small to generate the lightcurve vari-ability shown in Figure 22. The 2D geometry and neglectof rotation also implies that the convection in their mod-els cannot generate Rossby waves and has nothing to sayabout whether zonal jets, large-scale coherent vortices,or large-scale cloud patchiness are expected on browndwarfs.

Showman & Kaspi (2013) presented the first global-scale models of the atmospheric circulation on browndwarfs. They demonstrated that rotation plays a keyrole in the dynamics, leading to behavior on large scales(& 1000 km) greatly differing from the small-scale con-vection simulated by Freytag et al. (2010). Showman& Ingersoll (1998) presented 3D spherical-shell, globalmodels of convection in the interior to understand theglobal-scale convective organization. They solved theNavier-Stokes equations subject to the anelastic approx-imation, which are valid when the dynamical perturba-tions to the background density are small and the windspeed is much less than the speed of sound, as expectedto be appropriate in the interiors of brown dwarfs. Theydemonstrated that the rotation exerts a strong controlover the structure of the interior circulation, and theycharacterized how the characteristic convective velocitiesvary with heat flux, rotation rate, and radius in the in-terior. Convection occurs more efficiently near the polesthan the equator, leading to equator-to-pole temperaturedifferences that may reach a few K in the interior.

Next, Showman & Kaspi (2013) constructed analyticmodels for the circulation in the stratified atmosphere.As mentioned previously, the absence of external stel-lar irradiation impinging on brown dwarfs suggests thatthey lack the day-night or equator-pole radiative forcingthat drives the circulation in the tropospheres of most so-lar system planets. Showman & Kaspi (2013) suggestedthat the convection will generate a wide population ofatmospheric waves, including both gravity waves (Frey-tag et al. 2010) and Rossby waves, and that these willpropagate vertically and drive a mean flow comprisingzonal jets, turbulence, and coherent structures. Fig-ure 25 illustrates the mechanism. Gravity and Rossbywaves are generated by convection near the radiative-convective boundary and propagate vertically into theatmosphere, where they break or dissipate, causing azonal acceleration, which in steady state is balanced byan equal and opposite zonal acceleration due to a merid-ional flow (blue curves). The ascending and descendingmotion associated with this circulation advects entropyvertically; because entropy increases with height in a sta-bly stratified atmosphere, this implies that, on an isobar,low entropy—hence cold—air will accompany regions ofascent, whereas high entropy—hence warm—air will ac-company regions of descent. Such a circulation is ther-mally indirect; ascending air is denser than descending


Fig. 25.— Illustration of a large-scale, wave-driven atmosphericcirculation, as occurs in the stratospheres of solar system plan-ets (including Earth, Jupiter, and Saturn) and as Showman &Kaspi (2013) proposed also occurs in the stratified atmospheresof brown dwarfs. Vertically propagating gravity and Rossby wavesare generated by convection near the radiative-convective bound-ary and propagate upward into the atmosphere (orange wavy ar-rows), where they are absorbed or break. Their damping generatesa coherent eddy acceleration (black symbol, representing a vec-tor coming toward the viewer), which in turn drives a mean wind(black and red symbols, respectively, representing vectors com-ing toward the viewer). In steady state, the eddy acceleration isbalanced by a Coriolis acceleration (black ⊗ symbol, representinga vector pointing away from the viewer) associated with a slowsecondary circulation (blue curve). The ascending and descendingbranches of this circulation advect entropy vertically, leading tohorizontal temperature contrasts.

air, and so the circulation increases rather than decreasesthe potential energy of the circulation. In steady state,this potential energy is destroyed by radiation, whichcauses cooling in the warm regions and warming in thecool regions. Of course, such a circulation cannot occurin isolation; it is driven by the absorption of the wavespropagating upward from below, and is therefore referredto as a wave-driven circulation. Such circulations arewell known in the solar system community; they occurin the stratospheres of Earth, Jupiter, Saturn, Uranus,and Neptune (e.g. Andrews et al. 1987, Conrath et al.1990, West et al. 1992, Moreno & Sedano 1997).

Showman & Kaspi (2013) constructed an analytic the-ory for such a circulation with the aim of predicting thecharacteristic horizontal temperature contrasts and windspeeds in brown dwarf atmospheres. The efficiency withwhich wave absorption drives the mean flow is difficult topredict from first principles, so Showman & Kaspi (2013)considered it a free parameter, described by η, which isa dimensionless number giving the ratio of the power perarea used to drive the circulation to the total IR heat fluxradiated to space. No fundamental theory for η in plan-etary stratospheres yet exists, but observations of Earthand Jupiter indicates η ∼ 10−3. This number is gen-erally yet unknown in the context of brown dwarfs, andhere we apply a similar number η ∼ 10−3 in the follow-ing analysis. Showman & Kaspi (2013) showed that, towithin numerical factors of order unity, the characteris-tic horizontal temperature differences ∆Thoriz, horizontal

wind speeds ∆u, and vertical wind speeds w are given by

∆Thoriz

T∼ η1/2NH

cs(20)

∆u

cs∼ η1/2 lLD (21)

w ∼ η1/2 H

τrad

csNH

(22)

where, as before, NH is the characteristic horizontalphase speed of long-wavelength gravity waves, cs is thespeed of sound, LD is the deformation radius, l is thehorizontal wavenumber of the flow, corresponding to acharacteristic horizontal length scale L ≈ π/l, and τrad

is the radiative time constant. For a vertically isother-mal, ideal-gas H2 atmosphere, NH/cs ≈ 0.4, which isorder unity. The length scale of the flow is unknown, butvalues several times the deformation radius are plausible,which suggests that lLD likewise is order unity. For smallwave-driving efficiencies (η 1), these equations there-fore predict that the fractional temperature differencesare small, the wind speeds are much less than the speed ofsound, and the timescale for air to advect vertically overa scale height, H/w, is much longer than the radiativetime constant. For η ∼ 10−3 and typical brown dwarfconditions, these equations predict that ∆Thoriz ∼ 30 K,∆u ∼ 102 m s−1, and H/w ∼ 10τrad ∼ 106 s, correspond-ing to vertical velocities of order 10−2 m s−1, given a typ-ical brown dwarf scale height of 5 to 10 km. Note thatthese velocities correspond to vertical velocities in thestably stratified atmosphere associated with the large-scale meridional, wave-driven overturning circulation;convective velocities in the underlying convection zoneare of course greater.

To further investigate whether convective perturba-tions can drive an atmospheric circulation and under-stand how the flow organizes, Zhang & Showman (2014)presented global simulations of the stratified atmosphere,using the 1.5-layer shallow-water equations. Random,small-scale mass sources and sinks were added to rep-resent convective perturbations at the base of the at-mosphere. Radiation was represented using a Newto-nian cooling scheme that damps horizontal thickness per-turbations, consistent with the expectation that the ra-diative equilibrium temperature structure is nearly in-dependent of longitude and latitude. Zhang & Show-man found that when the convective forcing was strongand/or the radiative damping was weak, the flow spon-taneously organized into a banded pattern comprisingzonal jets with superposed vortices. But when the con-vective forcing was weak and/or the radiative damp-ing was strong, the atmospheric turbulence damped be-fore it could self-organize into jets, and the flow in-stead comprised isotropic turbulence and vortices (Fig-ure 26). Zonal wind speeds ranged from tens to hundredsof m s−1, consistent with the analytical predictions byShowman & Kaspi (2013). The transition between thejet-dominated and vortex-dominated regimes is relevantto brown dwarfs; for plausible forcing amplitudes, jetsand banding tend to occur when τrad & 105–106 s. Thissuggests that some brown dwarfs may be dominated byspots, whereas others may be dominated by bands.


Fig. 26.— Shallow-water simulations of the global circulationon isolated brown dwarfs, from Zhang & Showman (2014). (A)Dynamical behavior as a function of the convective forcing rate(abscissa) and strength of damping (ordinate), here represented bythe radiative time constant. Note that short radiative time con-stant implies strong damping of the flow, and vice versa. Orangecircles represent models exhibiting a banded flow with prominentzonal jets, which occurs with strong forcing and/or weak damping.Blue circles depict models dominated by horizontally isotropic tur-bulence, which occurs with weak forcing and/or strong damping.An example of the turbulent regime is shown as the lower leftinset and the jet-dominated regime is shown in the upper right in-set. Colors in the insets represent geopotential anomaly (in unit of105 m2s−2). (B) A typical synthetic lightcurve from these simu-lations, calculated by integrating the height (thickness) field overthe Earth-facing hemisphere. The solid blue envelope in the mainpanel shows the time evolution of the overall variability amplitude,which ranges over almost an order of magnitude on timescales ofthousands of hours. The insets show the actual lightcurves overshort timescales (four rotation periods long each) at three differentintervals. These insets illustrate how the lightcurve shapes varywith time, at some times exhibiting complex multi-peaked struc-tures (left and right insets) and at other times exhibiting nearlysinusoidal behavior (middle inset).

These models have implications for explaining sur-face patchiness and lightcurve variability. Simulatedlightcurves calculated from Zhang & Showman (2014)’smodels shows that the lightcurves can exhibit a varietyof timescales—including not only the rotational modu-lation but longer, dynamical timescales as well—and al-lows for changes qualitatively similar to those seen inobserved lightcurves (Figure 26). The overall amplitudeof the variability can vary from nearly zero to ∼2%, andthe lightcurves over rotational timescales can alternatebetween sinusoidal and multi-peaked shapes, similar tothose seen in observations (Figure 22), depending on theevolving global pattern of the turbulence.

Continuing this line of inquiry, Showman et al. (2019)presented high-resolution, 3D numerical simulations ofthe atmospheric circulation on brown dwarfs forced bysmall-scale convective perturbations at the base of thestratified atmosphere. Radiative heating/cooling wasrepresented using a Newtonian cooling scheme that, likein Zhang & Showman (2014) damps thermal perturba-tions and therefore removes energy from the flow. Theconvective perturbations trigger a wide population ofwaves, which propagate upward into the atmosphere andinduce a wave-driven circulation through a mechanismsimilar to that described in Figure 25. Horizontal tem-perature variations of tens of K on isobars and zonalwinds reaching 200–400 m s−1 occur (Figure 27), quali-tatively consistent with the predictions of Showman &Kaspi (2013). Showman et al. (2019) found that strongzonal jets are confined to low latitudes when the radia-tive damping is strong (Figure 27a), but the jets occurat all latitudes when the radiative damping is weaker(Figure 27c).

In these 3D models, the equatorial jet often exhibitsan oscillation analogous to well-known stratospheric os-cillations on solar system planets. In Earths equatorialstratosphere, vertically stacked eastward and westwardjets—and associated temperature anomalies—migratedownward over time with a period of approximately twoyears. This phenomenon, dubbed the Quasi-BiennialOscillation (QBO), is driven by the upward propaga-tion of atmospheric wavesincluding equatorially trappedRossby, Kelvin and inertio-gravity wavesthat are ab-sorbed in the stratosphere (Baldwin et al. 2001). Similaroscillations have been observed on Jupiter and Saturn.On Jupiter, the oscillation has a period of four years andis called the Quasi-Quadrennial Oscillation (QQO). Theoscillation on Saturn has a period of 15 years (half aSaturn year), and is called the Saturn Semi-Annual Os-cillation (see Showman et al. 2018 for a review). Theseare the first full 3D models of a giant planet to capturethis type of oscillation. In the simulations, the oscillationtypically has periods of several to∼12 years, broadly sim-ilar to the periods of the QQO and SAO. These modelshighlight the likelihood that long-term oscillations maybe a generic feature on warm-to-cool giant planets andbrown dwarfs. A key implication is that brown dwarfswill probably be variable not only on short timescales,but also on multiannual or multidecadal timescales aswell.

There is a subtle difference in the models of Zhang& Showman (2014) and Showman et al. (2019) as towhether jets exist when damping is very strong. In Show-man et al. (2019), models of a given wind speed exhibitmore coherent zonal banding when the forcing and damp-ing are weak than when they are strong, because in theformer case, the flow has ample time to reorganize into astrong series of zonal jets, whereas in the latter case, theforcing and damping continually attempt to disrupt thebanding pattern as it forms. However, some degree ofzonal banding always occursnone of the models in Show-man et al. exhibit entirely isotropic turbulence, no mat-ter how short the radiative time constant. This occursbecause, in a 3D model, the radiation damps the horizon-tal temperature variations, which via thermal wind sup-presses the vertical shear of the zonal wind. However, thebarotropic mode—that is, the height-independent com-


Fig. 27.— Global models of atmospheric circulation on brown dwarfs forced by small-scale convective perturbations at the base of theatmosphere, with no external stellar irradiation, from Showman et al. (2019). Left column shows temperature (K) and right column showszonal wind (m s−1) in three simulations with different radiative time constants—105, 106, and 107 s in the top, middle, and bottom rows,respectively. Rotation period is 5 hours, typical for brown dwarfs.

ponent of the zonal flow—is not damped by radiativeforcing. This differs from Zhang & Showman (2014) be-cause their 1.5-layer model—which assumes a quiescentabyssal layer—lacks a barotropic mode. In the 3D formu-lation, not only strong radiative damping but also strongfrictional drag at the base of the atmosphere is requiredto suppress the jets entirely. Such frictional dampingcould be supplied by Lorentz forces that act to brake thedeep roots of the zonal jets that penetrate into the deepinterior, but this is a subtle issue that requires furtherinvestigation.

The vertical motions that cause the temperature con-trasts in these models (Figure 27) will transport conden-sate species upward, which can lead to cloud formationif the condensation level occurs in the atmosphere, as ex-pected for L and early T dwarfs. Very small cloud con-densate particles (radii less than perhaps 0.1µm) exhibitvery slow gravitational settling, so they may become wellmixed by the circulation into quasi-ubiquitous layers offine particles. Larger particles will settle more quicklyafter condensation, and since the regions of condensa-tion are confined to ascending air—which covers only afraction of the globe—this will naturally lead to cloudpatchiness on large scales. Tan (2018) confirmed this

picture with numerical simulations similar to those de-scribed here but that include coupling to passive cloudtracers that advect with the circulation and settle grav-itationally but do not influence the dynamics (i.e., nocloud radiative feedback). As expected from the qualita-tive considerations above, these models produce region-ally patchy cloud decks, which can modulate the outgo-ing IR flux and lead to lightcurve variability.

4.2.2. Atmospheric circulation II: 1D cloud-radiativefeedbacks in an atmospheric column

Recent work demonstrates that the feedback betweenradiation, dynamics, and cloud formation can lead tospontaneous emergence of an atmospheric circulation,and therefore represents a totally distinct means of gen-erating an atmospheric circulation on brown dwarfs thanthe wave-driven circulation mechanism just described.Here, we survey two distinct feedbacks that can lead toatmospheric circulation on brown dwarfs, each of whichcan separately lead to a vigorous atmospheric circulationand regional-scale cloud patchiness—even in the absenceof convective forcing that might trigger any wave-drivencirculation.

The importance of cloud radiative feedbacks on browndwarfs is easy to appreciate. Imagine an atmospheric


vertical column with an opaque cloud deck—perhaps sev-eral thousand km wide—surrounded by cloud-free air.In the cloudy region, the IR radiation to space escapesfrom top of the cloud, at relatively low pressure, whereasin the cloud-free region, the radiation to space occursfrom much deeper pressures, perhaps a few bars. Thetwo regions will therefore experience extremely differentvertical profiles of the radiative heating/cooling, whichwill lead to horizontal temperature differences on iso-bars. These horizontal temperature contrasts will drivean overturning circulation that, in turn, can advect cloudcondensate vertically, and in principle might be capableof maintaining the cloud patchiness.

To illustrate, imagine an initially quiescent, cloud-freeatmosphere in radiative equilibrium, with an effectivetemperature of Teff = 1000 K, corresponding to an IRflux to space F = σT 4

eff = 6× 104 W m−2, which is likelyradiated to space from a pressure near 1 bar (Figure 23).Because the atmosphere is in radiative equilibrium, thissame flux is supplied to the atmosphere from below, suchthat the next heating of the column is zero. Now for sakeof argument suppose that a cloud appears at a higher al-titude, where the temperature is cooler—let us say 900 K.Radiation to space from the cloudy region will occur fromthe top of the cloud deck, corresponding to a smaller flux4× 104 W m−2. The perturbation in outgoing IR flux is∆F ≈ 2 × 104 W m−2. Integrated over an atmosphericcolumn, this differential heating induces a rate of temper-ature change over time ∆Fg/cpp ∼ 10−2 K s−1, where wehave adopted g ≈ 500 m s−2, cp = 1.3 × 104 J kg−1 K−1,and p ≈ 1 bar24. Over only a few brown dwarf rota-tions, such a heating differential would cause tempera-ture changes & 100 K within the atmospheric column.

Such thermal changes due to the appearance of cloudscause two distinct processes relevant for driving an atmo-spheric circulation. First, they cause an adjustment ofthe thermal structure of the entire atmospheric column,which changes the atmospheric stratification profile andpotentially alters the altitude ranges where vertical mix-ing and clouds occur. Second, if such a cloudy atmo-spheric column is surrounded by columns of relativelycloud-free air, the horizontal differences in the verti-cal heating/cooling profile causes horizontal temperaturecontrasts, and the resulting horizontal pressure-gradientforces can potentially drive an overturning circulation.The key question is whether either of these process is self-generating and self-sustaining, that is, whether an ini-tially quiescent atmosphere will experience spontaneousemergence of atmospheric circulation due to these pro-cesses. Recent work indicates that the answer is yes.

Tan & Showman (2019) constructed a 1D, time-dependent model of the vertical structure in a browndwarf atmosphere to investigate the coupling betweenclouds, radiation, and mixing in a vertical atmosphericcolumn. This model is similar in spirit to a long history1D brown dwarf atmosphere models (e.g Burrows et al.1997, 2006, Marley et al. 2002, 2010, Saumon & Marley2008), except that most of these models assume steadystate and solve for the radiative-convective-equilibrium,

24 In a hydrostatically balanced atmosphere, the mass per unitarea in a column is p/g, so the radiative heating/cooling per unitmass (units of W kg−1) is ∆F g/p, and the cooling in units ofK s−1 is ∆F g/pcp.

whereas Tan & Showman (2019) allowed for time depen-dence associated with time-varying deviations from ra-diative equilibrium. They adopted a simple cloud schemein which clouds settle gravitationally and are verticallymixed in regions where convection should occur. Theirmodels show that the equilibrium state is generally notsteady; radiative-cloud feedbacks cause the thermal andcloud structure to vary, with growth and collapse ofcloud decks occurring episodically on timescales of ∼1to ∼30 hours (Figure 28). As foreshadowed above, tem-perature varies on isobars by ∼100 K as the cloud deckgrows and decays in a cyclical fashion. Because of thetime-varying cloud opacity, the IR photosphere pressuremoves vertically by two scale heights, which causes theeffective temperature to vary by hundreds of K over acycle. Tan & Showman (2019) showed that the existenceof variability is robust over a wide range of parametersincluding number density of cloud condensation nuclei(CCN), gravity, interior entropy, and the implementa-tion of the vertical mixing—although the exact oscilla-tion period and other details depend on these parame-ter values. A key point is that the oscillatory behavioris entirely spontaneous and does not require any exter-nally imposed perturbations. This contrasts with other1D models where variability emerged only in responseto imposed thermal perturbations (Robinson & Marley2014).

The growth and collapse of cloud decks shown in Fig-ure 28 results from several interacting feedbacks. In thegrowth phase, a positive feedback promotes upward ex-pansion of the cloud top, as follows: an upwardly per-turbed cloud top cools by adiabatic expansion, leadingto cooler cloud-top temperatures and smaller IR flux ra-diating to space from the cloud top. Since the temper-ature profile above the cloud top is determined by thestrength of this IR flux, a lessening of this OLR causesthe entire overlying T/p profile (above the cloud) to cooloff. For a given temperature on an isobar near the topof the cloud, a decrease in the overlying temperaturesacts to destabilize the air near the cloud top, allowingthe cloud to expand upward. But this growth processis eventually halted by a negative feedback that killsthe cloud: although the cloud itself remains relativelywell-mixed, a large upward heat flux against the base ofthe cloud generates stable stratification below the cloud,which shuts off the source of moisture (i.e. condensablechemical species) into the cloud. Without a source ofcondensable material to replenish it, gravitational set-tling of particles then inevitably depletes the cloud ofparticulate matter, causing the cloud to dissipate. Thecycle can then repeat.

To reiterate, this is a solely 1D feedback that does notrequire large-scale dynamics. On a brown dwarf, dis-tinct atmospheric vertical columns in different locationon the globe cloud in principle independently undergosuch oscillatory behavior; the interactions between suchcolumns will not be negligible, a topic requiring a multi-dimensional dynaical model to investigate. It is also im-portant to emphasis that the 1 to 30 hour period of theoscillation has nothing to do with rotation—it purely re-sults from the interactions between radiation, convectivemixing, and cloud physics—yet its similarity to the ro-tation timescale means that brown dwarf lightcurves canvary over short (rotational) timescales. Note that this


Fig. 28.— Results of 1D time-dependent models of brown dwarfs showing the development of cyclical oscillations in the cloud structureand thermal flux radiated to space. From Tan & Showman (2019). (A) Time-mean cloud mixing ratio (qc) as a function of pressure. (B)Time evolution of cloud mixing ratio as a function of time and pressure in logarithmic scale. The dotted red line is the pressure at which thecontribution function is at a maximum, which is where the majority of flux escape from the atmosphere. (C) The time-mean temperatureas a function of pressure is represented as the thick solid line; the dashed line represents enstatite (MgSiO3) condensation curve; and thegrey lines represent the envelope of the variation range of the temperature-pressure profile. (D) The corresponding outgoing thermal fluxas a function of time.

1D feedback occurs more easily when the cloud conden-sation occurs near the top of the convective zone. Thisis because convection below the cloud base is needed toreplenish the condensable vapor after clouds gravitation-ally settling out—otherwise without replenishment, theclouds will dissipate away and the system will be even-tually inactive. 1D model atmospheres where the clouddeck would form in the stratified atmosphere could po-tentially lack variability from this mechanism (Tan &Showman 2019).

4.2.3. Atmospheric circulation III: multi-D feedbacksbetween clouds, radiation, and dynamics

Next, we turn to consider the second cloud-radiativefeedback. Imagine an initially quiescent, cloud-free at-mosphere in radiative equilibrium, and imagine seedinga small cloud into a vertical air column that is surroundedby cloud-free air. Being at higher altitudes where tem-peratures are cooler, the IR flux to space from the topof the cloud is less than that of the surrounding, cloud-free regions. If the initial air columns were in radiativeequilibrium—receiving the same heat flux from below asthey radiate to space—then this decrease in outgoingheat flux caused by the cloud implies that, integratedvertically over an atmospheric column, the cloudy air col-umn experiences net heating. This heating promotes as-cent, causing the cloud to grow to higher altitudes. Adi-abatic expansion decreases the temperature still further,causing greater net heating of the column and furtherpromoting growth of the cloud. Gierasch et al. (1973)demonstrated this system is linearly unstable, in thatsmall perturbations will grow over time, and he suggestedits possible applicability to giant planets. The mecha-

nism is thought to promote convective self-aggregrationin Earth’s tropics (e.g., Wing et al. 2017). However, themechanism has not been previously investigated for giantplanets in the fully developed, nonlinear regime.

Tan & Showman (2020a) have shown that this mech-anism can cause the spontaneous emergence of a self-generating atmospheric circulation on brown dwarfs,even when the atmosphere would be quiescent in the ab-sence of clouds. Figure 29 shows a numerical simulationfrom Tan & Showman (2020a) demonstrating that thismechanism indeed constitutes a positive feedback thatcan generate atmospheric turbulence. Tan & Showman(2020a) solved the primitive equations for the stratifiedatmosphere on an f -plane (corresponding to a 3D Carte-sian domain where the Coriolis parameter f is assumedconstant) over a vertical domain from 10 to 0.01 bars.The initial, cloud-free atmosphere was set to be in ra-diative equilibrium. Through the positive feedback justdescribed, a small cloud seeded into the domain (top leftpanel) leads to net heating of the column, ascending mo-tion, cloud growth, and horizontal divergence of the flow.The Coriolis force deflects the horizontally diverging air,leading to the generation of a warm-core, anticyclonic,geostrophically balanced, cloud-covered vortex with a di-ameter of ∼20,000 km on a timescale of tens of hours.

Long-time integrations demonstrate that this processis self-generating and leads to an active atmospheric cir-culation that reaches a statistical, fluctuating turbulentequilibrium. Figure 30, from Tan & Showman (2020a),show snapshots of the equilibrated state for several dif-ferent rotation rates. Cloudy and cloud-free vortices,filaments, and other time-variable turbulent structuresemerge, with characteristic length scales that are smaller


Fig. 29.— Growth of a cloudy vortex on an f -plane from an initially cloud-free environment through a positive feedback betweenradiation, dynamics, and cloud development. Plotted is the cloud mixing ratio (colorscale) and winds (vectors) at 0.4 bars. In this model,f = 3× 10−4 s−1. From Tan & Showman (2017).

the faster the rotation rate. This trend matches expec-tations, since the geostrophic adjustment process shouldproduce vortices with sizes close to the deformation ra-dius, which decreases with increasing rotation rate. How-ever, vortex mergers can cause an inverse energy cas-cade that transfers energy to larger and larger scales,so the characteristic flow length scale in the fully tur-bulent state (Figure 30) can potentially become greaterthan the deformation radius, depending on the dissipa-tion rate of the kinetic energy (implemented as a bottomfrictional drag in the models). For typical brown-dwarfparameters, horizontally averaged wind speeds are typi-cally 300− 500 m s−1, and the local, outgoing IR flux tospace can vary by nearly a factor of two.

Global, fully spherical models of this type from Tan(2018) and Tan & Showman (in prep) likewise show thedevelopment of complex, time evolving turbulence in-volving cloudy and cloud-free patches. Figure 31 showsthe outgoing IR flux in the fully equilibrated state formodels with rotation periods of 2.5, 5, 10, and 20 hours.Again, as expected, the dominant turbulent length scalesare smaller when the rotation rate is faster (rotation pe-riod is shorter). The equatorial dynamics differ fromthose at higher latitudes because of the larger localRossby numbers and the presence of equatorially trappedwaves, which couple to the cloud-radiative feedbacks incomplex ways. Locally, the outgoing IR flux again variesby a factor of two (Figure 30), and integrated over thedisk, the lightcurve variability on rotational timescalescan exceed 1%, consistent with those commonly observedon brown dwarfs (e.g., Metchev et al. 2015).

4.3. Summary of brown dwarf dynamics

Studies performed to date demonstrate that at leastthree independent mechanisms can lead to a vigorouscirculation in the stratified atmospheres of brown dwarfs,despite the absence of any stellar irradiation that couldproduce day-night or equator-to-pole thermal contrasts.

These mechanisms include:

• Cloud-free dynamics: Convection impinges on thestably stratified atmosphere, triggering waves anddriving turbulence, zonal jets, and vortices. Thiswave-driven circulation can organize clouds into apatchy structure on large scales, even if cloud ra-diative feedbacks are not considered.

• 1D cloud feedback: A cloud-radiative-mixing feed-back in 1D vertical columns drives cyclical growthand collapse of cloud decks in an atmospheric col-umn. This causes IR flux variability of hundredsof K in effective temperature over periods of ∼130hours.

• Multi-D cloud feedback: A cloud-radiative-dynamical instability in 3D drives a circulationthat maintains the cloud patchiness. Wind speedsreach 300− 500 m s−1, horizontal temperature dif-ferences on isobars reaches 100-200K, and localvariations in effective temperature reach hundredsof K, corresponding to local variations in the out-going IR flux of a factor of two.

We note that the global-scale models for these threeprocesses performed to date have essentially been ideal-ized process studies that have aimed to investigate eachof the three processes in isolation. The behavior whenthe three mechanisms are combined has not yet been sim-ulated in global-scale numerical models but will likely in-volve even richer behavior than the models shown herethat have investigated them one by one. Future model-ing efforts beyond the idealized framework summarizedhere include the use of GCMs with non-grey radiativetransfer along with equilibrium or transport-induced dis-equilibrium chemistry, which are essential for straightmodel-observation comparisons. Other than clouds, pos-sible transport-induced chemical patchiness and its ra-


Fig. 30.— 3D simulations of cloudy brown dwarf atmospheres on the f -plane from simulations performed in Tan & Showman (2020a).Shown are cloud abundance at 0.2 bars (left), temperature at 0.2 bars (middle) and outgoing IR flux to space (right) for three differentrotation rates in the top, middle, and bottom rows. Cloud-radiative-dynamical feedbacks lead to an active atmospheric circulation.

diative effect may also contribute to affect the global cir-culation as well as lightcurve variability, similar to thoseexplored in the context of hot Jupiters (see Section 2.6).Utilizing convection resolving models and the inclusionof both the convective and stratified zones would help tounderstand turbulence and wave generation in the strat-ified layers and their effects on large-scale flow. Convec-tion models may also be used to examine the relativelysmall-scale cloud formation and its associated dynam-ics, as well as effects of chemical reactions in modulatingthe convective instability in brown dwarf atmospheresproposed by Tremblin et al. (2017) and Tremblin et al.(2019).

5. IRRADIATED BROWN DWARFS

Although most close-in, highly irradiated EGPs havemasses less than a few MJ , and most known browndwarfs are isolated objects receiving negligible stellarflux, there exists a population of massive brown dwarfson close-in orbits that receive intense stellar irradiation.The irradiation levels they experience are comparable totypical hot Jupiters, and yet because they are massive,they transport interior heat fluxes that rival or even ex-ceed the energy flux absorbed from their parent star. Assuch, they populate the upper right corner of Figure 2.While currently a less well-known subclass than that of

hot Jupiters or field brown dwarfs, they are equally im-portant, for they provide the link in the chain of under-standing between brown dwarfs and hot Jupiters. Onecan think of them as hot Jupiters with enormous convec-tive heat fluxes, or as brown dwarfs that are modified byintroducing external stellar forcing.

These objects fall into two distinct populations. Thefirst are brown dwarfs orbiting main-sequence stars inclose-in orbits. The rarity of these objects relative toless massive, close-in EGPs was discovered early, a phe-nomenon dubbed the “brown dwarf desert” (e.g., Marcy& Butler 2000, Sahlmann et al. 2011). Nevertheless,about two dozen transiting, close-in brown dwarfs of thistype have been discovered. Their orbital periods—andhence rotation periods if synchronously rotating—rangetypically from 3–5 days, implying that rotation plays adynamical role similar to that on hot Jupiters. The sec-ond population are brown dwarf companions to whitedwarf stars, which survived a common envelope phaseduring which the more massive companion swelled duringthe red giant phase to envelope the less-massive compan-ion. Friction caused by orbital motion of the companionthrough the red giant’s tenuous outer layers causes thebrown dwarf to slowly spiral in. Once the red-giant phaseends, the brown dwarf is left orbiting the remnant whitedwarf on an extremely tight orbit with a period of typ-


Fig. 31.— Global circulation models of isolated brown dwarfs with a dual-band radiative transfer scheme, coupled to a cloud cycle,showing the development of highly time-variable turbulence and cloud patchiness. Shown is the total outgoing IR flux, which varies locallyby a factor of two due to local variations in cloud opacity and temperature structure. The characteristic size of the turbulent structuresdecreases with decreasing rotation period (increasing rotation rate). From Tan (2018) and Tan & Showman (in prep).

ically 1–4 hours. Because they are likewise expected tobe synchrously rotating, their rotation periods should beequally short, implying that rotation plays a more con-trolling role in their atmospheric dynamics. Moreover,because the white dwarf’s radiation is primarily in the ul-traviolet, where typical atmospheric gases are much moreabsorbing than in the visible, such brown dwarfs mayexperience very different atmospheric thermal structuresand chemistry than their counterparts orbiting sunlikestars.

Lightcurves and other observations of both popula-tions have been acquired, suggesting that these objects,like hot Jupiters, experience large day-night tempera-ture differences. Examples are presented in Figure 32,which shows IR lightcurves from KELT-1b (Beatty et al.2019), a 27-MJ brown dwarf orbiting an F star on a 1.2-day orbit, and WD0137-349B, which is a 53-MJ browndwarf orbiting its primary white dwarf every 1.9 hours.Like hot Jupiters, the IR lightcurve of KELT-1b reachesa peak flux substantially before secondary eclipse (Fig-ure 32, left panel), suggesting the possibility of a simi-lar dynamical regime involving equatorial superrotationand global-scale waves driven by the day-night ther-mal forcing. In contrast, the brown dwarf orbiting awhite dwarf lacks such a phase-curve offset (Figure 32,right panel), suggesting that differing processes oper-ate at fast rotation. Several such brown-dwarf-white-

dwarf systems have been characterized, including NLTT5306 (Steele et al. 2013), WD0137-349 (Casewell et al.2015, Longstaff et al. 2017), EPIC 21223532 (Casewellet al. 2018a), WD 1202-024 (Rappaport et al. 2017)and SDSS J141126.20+200911.1 (Littlefair et al. 2014,Casewell et al. 2018b).

Note that, from the perspective of observational char-acterization, brown dwarfs around main-sequence starsface similar observational challenges as typical hotJupiters—the planetary flux is typically thousands oftimes dimmer than the parent star, so teasing out theplanetary signature from total signal is challenging. Inthe case of brown-dwarf-white-dwarf systems, however,the IR flux from the brown dwarf often dominates, aid-ing observational characterization of the brown dwarf.25

Many brown dwarfs of this type are cataclysmic vari-ables (e.g., Hellier 2001), in which the brown dwarf shedsmass onto the white dwarf. In this situation, the browndwarf’s non-spherical shape and the presence of an accre-tion disc around the white dwarf can complicate an in-terpretation of the observations (Santisteban et al. 2016).Still, some observed brown-dwarf-white-dwarf systems—including the one shown in Figure 32—are “detached,”

25 The white dwarf typically dominates the total flux, but most ofthis flux emanates in the UV. Because the white dwarf is physicallymuch smaller than the brown dwarf, its IR flux is smaller, despitehaving a higher temperature.


Fig. 32.— Spitzer IR lightcurves of irradiated brown dwarfs. Left panel shows 3.6µm (grey) and 4.5µm phase curves of the 27MJ browndwarf KELT-1b, orbiting a main-sequence (F5) star in a 1.2-day orbit. The phase curves shows two secondary eclipses (at phase 0 and1) and one transit (at phase 0.5). Modified from Beatty et al. (2019). Dashed line shows the bottom of the secondary eclipses, whichrepresents the radiation from the star alone. Right panel shows 3.6µm (black), 4.5µm (blue), 5.8µm (green) and 8.0µm phase curves ofWD0137-349, which comprises a 53MJ brown dwarf orbiting a white dwarf in a 1.9-hour orbit. Two full orbits are shown. In both phasecurves, 0 and 1 represent the times when the brown dwarf’s dayside is aimed toward Earth, whereas phase 0.5 and 1.5 represent timeswhen its nightside is aimed toward Earth. From Casewell et al. (2015).

exhibiting sufficiently great separations for these effectsto be minimal.

Motivated by these systems, Tan & Showman (2020b)performed idealized GCM experiments of sychronouslyrotating, strongly irradiated giant planets with rotationperiods ranging from 80 hours to 2.5 hours to character-ize the role of fast rotation in shaping their circulation.A subset of these models is shown in Figure 33. Underidentical forcing conditions, faster rotation leads to largerday-night temperature differences in the observable at-mosphere (pressures less than a few hundred mbar),which results from the inhibition of day-night heat trans-port caused by rapid planetary rotation. Moreover, be-cause of the decrease in the Rossby deformation radiusat fast rotation, the equatorial waveguide, Matsuno-Gillpattern, and the equatorial superrotating jet all becomeconfined within 10 of the equator. As a result, in con-trast to hot Jupiters, the overall dayside temperaturepattern does not exhibit any prominent eastward offsetwhen the rotation rate is very fast (essentially, the east-ward offset still occurs but becomes confined preferen-tially closer and closer to the equator as rotation rate in-creases). Infrared lightcurves therefore exhibit flux peaksthat coincide with the time of secondary eclipse. Thishelps to explain the fact that many brown dwarfs onclose-in orbits around white dwarfs lack prominent off-sets of the IR flux peaks, as shown in Figure 32. Interest-ingly, Figure 33 shows that deeper in the troposphere—atpressures of a few hundred mbar—the dayside tempera-ture develops a “fingering” pattern wherein meridionallynarrow, zonally elongated tongues of hot and cold air in-terleave. Outside the equatorial jet, a prominent patternof alternating eastward and westward zonal jets emerges.Tan & Showman (2020b) showed that, despite the novelforcing, these zonal jets scale well with the Rhines scale.Further work is needed to understand the implications forbrown-dwarf-white-dwarf systems and the connection tozonal jets on Jupiter and Saturn.

Lee et al. (2020) simulated the atmospheric circulationof the brown dwarf orbiting the white dwarf WD0137-349with an orbital period of about 2 hours. They adoptedthe ExoFMS GCM which utilizes a dual-grey radiativetransfer scheme assuming constant opacity in both ther-

mal and visible band. Their results are qualitativelysimilar to the rapidly rotating cases in Tan & Showman(2020b), showing a meridionally narrower equatorial su-perrotating jet compared to canonical hot Jupiter sim-ulations, nearly geostrophic flows at mid-high latitudesand large day-night temperature difference. Althoughusing different GCMs and radiative forcing setup, theagreement between Lee et al. (2020) and Tan & Show-man (2020b) suggests that strong rotation could robustlyand profoundly influence the atmospheric circulation oftidally locked objects.

Future endeavors beyond the above idealized work areneeded to obtain a completed understanding on the 3Datmospheres of this group of objects. The spectrum ofwhite dwarfs peak at UV, and significant photochemistryare expected to occur in atmospheres of the companionBDs (Longstaff et al. 2017, Casewell et al. 2018b). Ef-fects of photochemical products and radiative heating onthe thermal structures and circulation, along with theeffects of hydrogen thermal dissociation/recombinationsimilar to those found in ultra-hot Jupiter atmospheres(Bell & Cowan 2018, Komacek & Tan 2018, Tan & Ko-macek 2019), are worth to explored in a thorough man-ner. Given the insufficient day-night heat transport, vari-ous mineral clouds are expected to condense in the night-sides of these objects (Lee et al. 2020), and may haveinteresting effects on lightcurve and dynamics. Strongmagnetic field (∼kG) has been inferred from observa-tions of field brown dwarfs (e.g., Kao et al. 2016, 2018).If brown dwarfs around white dwarfs host such strongfields, their strongly irradiated atmospheres may experi-ence significant MHD effects.

6. DISCUSSION

Observations reveal a wealth of behavior in the atmo-spheres of EGPs and brown dwarfs. The intense daynightradiative forcing on hot Jupiters leads to large day-nighttemperature differences and planetary-scale waves that,according to most GCMs, drive a fast superrotating jetat the equator. Adjustment of the thermal structure byplanetary-scale waves provides a framework for under-standing the day-night temperature contrasts and theirdependence on the strength of stellar irradiation andother parameters. Analytic theories and idealized nu-


merical models have provided significant insights intothese processes. Overall, a basic picture of the dynamicalprocesses that maintain the circulation on hot Jupitersis slowly emerging. But many aspects remain poorlyunderstood; for example, the eddies that maintain thesuperrotating jet are nonlinear and interact with theshear of the jet, making their structure difficult to pre-dict in detail. The ways the circulation interacts withpatchy clouds and the detailed dynamics of vertical andhorizontal mixing of trace constituents remain less wellunderstood. Predictions for various regime transitionsacross the hot Jupiter populationand between hot andwarm Jupiters—have yet to be confronted by observa-tional data. An open frontier that has been little studiedis the extent to which (and how) the atmosphere andinterior of hot Jupiters interact, including the depth towhich the equatorial jet extends, the type of winds thatmay exist in the deep convection zone, as well as the de-gree of mixing, the nature of the thermal structure, andthe modes of energy exchange between the atmosphereand deep interior.

Warm Jupiters are a fascinating target for future obser-vations, and help to bridge the gap between hot Jupiterson the one hand and Jupiter and Saturn on the other.At present, few observational data exist to con- strainthe dynamical behavior of warm Jupiters, but a grow-ing body of models suggest a wide range of behav- iorthat organizes into a variety of regimes depending onthe rotation rate, obliquity, orbital eccentricity, and howthe photosphere-level radiative time constant comparesto the rotational and orbital timescales.

A wealth of data provide information about the ther-

mal and cloud structure of isolated brown dwarfs, indi-cating that these worlds are covered by patchy cloudsmodulated by a vigorous atmospheric circulation. Thisregime differs from hot and warm Jupiters in being fun-damentally driven by interior heat loss. In comparison tohot Jupiters, dynamical models of the global circulationon brown dwarfs are in their infancy and represents a ripefrontier for future research. Models point toward rich in-teractions between the convective interior and stratifiedatmosphere and of various subtle feedbacks between dy-namics, radiation and clouds that can lead to a sponta-neous, self-generating circulation comprising turbulence,vortices, and zonal jets, associated with patchy cloudstructures like those observed. The continuum betweenhot brown dwarf on the one hand and Jupiter and Saturnon the other deserves further study, and can face obser-vational tests from the growing quality of data on coolerT and Y dwarfs. Irradiated brown dwarfs represent aneven more open frontier, and can help bridge the gapbetween hot Jupiters and isolated brown dwarfs. Obser-vations are starting to clarify the thermal structure ofthese objects, but so far almost no modeling and theoryhas been performed to explore this regime.

We have argued that there is value in considering thesedistinct populations together as a class. The synergiesthat can result from comparisons between the diversepopulations have yet to be well exploited but can po-tentially lead to major insights across these populations.The grand challenge of understanding the behavior ofgiant planets as an integrated population across manyorders of magnitude in internal heat flux, external irra-diation, rotation rate, and other parameters beckons us.

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