The Mechanical Greenhouse: Burial of Heat by Turbulence in Hot Jupiter Atmospheres
aa r X i v : . [ a s t r o - ph . E P ] A ug Draft version October 21, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
THE MECHANICAL GREENHOUSE: BURIAL OF HEAT BY TURBULENCE IN HOT JUPITERATMOSPHERES
Andrew N. Youdin
Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada
Jonathan L. Mitchell
Dept. of Earth & Space Sciences and Dept. of Atmospheric and Oceanic Sciences, University of California Los Angeles, 595 CharlesYoung East Drive, Los Angeles, CA 90095-1567, USA
Draft version October 21, 2018
ABSTRACTThe intense irradiation received by hot Jupiters suppresses convection in the outer layers of theiratmospheres and lowers their cooling rates. “Inflated” hot Jupiters, i.e., those with anomalously largetransit radii, require additional sources of heat or suppressed cooling. We consider the effect of forcedturbulent mixing in the radiative layer, which could be driven by atmospheric circulation or by anothermechanism. Due to stable stratification in the atmosphere, forced turbulence drives a downward fluxof heat. Weak turbulent mixing slows the cooling rate by this process, as if the planet was irradiatedmore intensely. Stronger turbulent mixing buries heat into the convective interior, provided theturbulence extends to the radiative-convective boundary. This inflates the planet until a balanceis reached between the heat buried into and radiated from the interior. We also include the directinjection of heat due to the dissipation of turbulence or other effects. Such heating is already knownto slow planetary cooling. We find that dissipation also enhances heat burial from mixing by loweringthe threshold for turbulent mixing to drive heat into the interior. Strong turbulent mixing of heavymolecular species such as TiO may be necessary to explain stratospheric thermal inversions. We showthat the amount of mixing required to loft TiO may overinflate the planet by our mechanism. Thispossible refutation of the TiO hypothesis deserves further study. Our inflation mechanism requiresa deep stratified layer that only exists when the absorbed stellar flux greatly exceeds the intrinsicemitted flux. Thus it would be less effective for more luminous brown dwarfs and for longer periodgas giants, including Jupiter and Saturn.
Subject headings: diffusion – opacity – planet-star interactions – planets and satellites: atmospheres– radiative transfer – turbulence INTRODUCTIONHot Jupiters — giant planets receiving intense irradi-ation from their host stars — are the best characterizedclass of exoplanets. Their proximity to their host staryields frequent transits of and occultations by their hoststar that are observed over a range of wavelengths. Tworemarkable features of hot Jupiters are their inflated radiiand the variety of infrared emission signatures, some ofwhich have been interpreted to reveal stratospheric ther-mal inversions.Many hot Jupiters have larger radii than standard cool-ing models predict, even with the intense irradiation fromthe host stars included in the radiative transfer. Thisimplies a mechanism that injects heat into and/or re-tards the loss of heat from the planets’ interiors. SeeFortney et al. (2009) for a review of proposed mecha-nisms.Guillot & Showman (2002, hereafter GS02) arguedthat atmospheric winds, driven by intense irradiation,could explain inflated radii. In their model, the kineticenergy of the winds dissipates as heat below the pen-etration depth of starlight. However the energy neednot be deposited into the convective interior (a commonmisconception). Dissipating energy in outer radiativelayers suffices to delay planetary contraction. Turbu-lence, which the winds can trigger via Kelvin-Helmholz instabilities, is an efficient mechanism to dissipate kineticenergy (Li & Goodman 2010). MHD drag is an alterna-tive dissipation mechanism provided weather-layer windsextend to the high-pressure, metallic zone of hydrogen(Perna et al. 2010; Batygin & Stevenson 2010).Thermal inversions, i.e. regions where the atmospherictemperature rises with height, may also implicate tur-bulent mixing in hot Jupiter atmospheres. Transitspectra of several hot Jupiters have been interpretedas being thermally inverted (Richardson et al. 2007;Burrows et al. 2007; Knutson et al. 2010). These obser-vations appear to confirm the predicion of Hubeny et al.(2003) that molecular absorbers, mainly TiO, in thestratosphere could generate inversion layers. Howevervapor phase TiO could rain out of the upper atmosphereif it condenses in cold traps. Turbulent mixing can coun-teract this settling.Spiegel et al. (2009, hereafter S09) showed that eddydiffusion coefficients of K zz ≈ — 10 cm / s areneeded to maintain sufficient stratospheric TiO for ther-mal inversions. The range of K zz values accounts forthe varying extent of cold traps in planets with differentthermal profiles and the size of grains that condense. S09 Eddy diffusion models small-scale turbulent processes in anal-ogy to molecular diffusion, with tracers fluxed down their meangradients.
Youdin & Mitchellargued that the need for strong mixing renders the TiOhypothesis “problematic,” pending improved estimatesof K zz . One goal of our study is to determine if large K zz values are energetically problematic.Sulfur has also been proposed as a high-altitude ab-sorber. Photochemical models of sulfur abundances(Zahnle et al. 2009) also include turbulent mixing at astrength K zz ≈ cm / s, though the dependance on K zz is unclear. Eddy diffusion is also used in browndwarf models to explain disequilibrium chemical abun-dances (Griffith & Yelle 1999; Hubeny & Burrows 2007).Turbulence not only mixes chemical species, it alsotransports heat. This paper develops a model that in-cludes this turbulent heat flux in the radiative layers ofhot Jupiters. While convection drives an outward fluxof energy, forced turbulence in stably stratified regionsdrives a downward flux of energy. This effect is distinctfrom — though it accompanies — the dissipation of tur-bulence as heat, which we also include.By altering the flow of energy, we change the coolingand contraction rates of hot Jupiters. For modest lev-els of turbulent diffusion, the outward radiative flux ispartially offset by the downward flux of mechanical en-ergy. This reduces the net cooling flux from the convec-tive interior, which self-consistently pushes the radiativeconvective boundary (RCB) to higher pressure. For sufficiently strong eddy diffusion, the downwardflux of energy exceeds the outward radiative flux thata planet of fixed entropy can provide. In this case theturbulent heat flux flows into the convective interior, in-creasing the internal entropy and inflating the planet. Aschematic of this mechanism is shown in Fig. 1. Becausehigher entropy planets are more intrinsically luminous,inflation leads to an equilibrium between turbulent heatburial and radiative losses.Our mechanism bears some resemblance to the run-away greenhouse. In the latter, the atmosphere is com-posed of a greenhouse gas in vapor pressure equilibriumwith a large, surface volatile reservoir. The cooling emis-sion to space emanates from a pressure ∼ g/κ , with sur-face gravity g and (Rosseland mean) opacity κ . Theemission is independent of the surface temperature foroptically thick atmospheres. Vapor pressure equilibriumdetermines the temperature at the emission level, thuslimiting the cooling radiation that the atmosphere canachieve (Kombayashi 1967; Ingersoll 1969). If the ab-sorbed sunlight exceeds this limiting cooling emission,the surface temperature increases until either the volatilereservoir is depleted or the atmosphere becomes suffi-ciently transparent to the surface blackbody emission.The role of limiting cooling flux in the runaway green-house is played in our mechanism by the cooling fluxof the core. The role of absorbed sunlight is played bythe downward, mechanical flux of energy. If the latterexceeds the former, the planet heats up by increasingthe core entropy until energy balance can be achieved.The analogy is somewhat incomplete, however, in thatthe traditional runaway greenhouse involves a radiative-thermodynamic feedback which does not exist in ourmechanism. For simplicity we will describe convectively stable regions as“radiative,” even when we include the transport of heat by bothturbulence and radiation. h e a t IR photosphereMixinglayer deep interiorwinds
Figure 1.
Schematic of the mechanical greenhouse effect to in-flate hot Jupiters. A downward flux of heat ( large black arrow ) isdriven by turbulence in the convectively stable “mixing layer” anddeposited in the deep interior. This downward flux can balanceor even exceed the convective losses ( gray overturning arrows ).Atmospheric circulation ( “winds” ) launched near the photospheredrive turbulence in the mixing layer. Other mechanisms, such asnon-linear gravity wave interactions, could also drive the turbulentflux.
This paper is organized as follows. In § §
3. We derive our mixing length formulae for the tur-bulent transport of heat in § § § §
4. We first treat constantdiffusion (partly to connect with S09) and ignore energydissipation in § K zz in § § § § § § STANDARD ATMOSPHERIC MODELSWe start with a review of the standard radiativetransfer approximations used in this work ( § § Radiative Transfer
Our goal is to understand energy balance. We focuson the deep atmosphere which is optically thick both toincoming stellar irradiation and the planet’s emitted flux.Here the equation of radiative diffusion dTdP = F rad k rad (1)ot Jupiter Atmospheres 3relates the outgoing radiative flux F rad to the variation oftemperature T with pressure P via the radiative diffusioncoefficient k rad = 16 σT g κ , (2)where σ is the Stephan-Boltzmann constant. Hydrostaticbalance, dP/dz = − ρg allows pressure to replace height z as the vertical corrdinate, with ρ the atmospheric density.We hold gravity g constant, invoking the plane-parallelapproximation for thin atmospheres.For the Rosseland mean opacity, our calculations willuse a power law, κ = κ P α T β ≡ κ o (cid:18) PP kb (cid:19) α (cid:18) TT (cid:19) β . (3)The two forms are equivalent, with the constant κ beingmore compact, while κ o has units of opacity and is nor-malized to P kb = 1 kbar and T = 2000 K. Unless statedotherwise, calculations will use κ ∝ P , i.e. α = 1 , β = 0as a rough approximation to collision induced molecularopacity. Our normalization choice of κ o = 0 .
18 cm / gwill be justified below ( § § T deep , an approach used in AB06 and advocated byIro et al. (2005). This approach is valid when the inci-dent stellar flux exceeds the emitted radiation, resultingin a deep isothermal region at the top of the opticallythick atmosphere. In this physical situation, the pre-cise location of the upper boundary is not important,as we explain further in § F irr ≡ σT ∗ gives a characteristic tempera-ture T ∗ ≈ L / ∗ , ⊙ M / ∗ , ⊙ P / (4)where stellar mass and luminosity, M ∗ , ⊙ and L ∗ , ⊙ arenormalized to solar values, and the orbital period P day is normalized to a (24 hour) day. Horizontal tem-perature gradients are important for driving winds inthe weather layer. However these winds efficientlysmooth temperature gradients at pressures & bar, wheretimescales for advection are shorter than for radiativelosses (Showman et al. 2009). Thus 1D models are ap-propriate for basic considerations of energy balance.Because the planet is not a perfect blackbody, T deep may not match T ∗ . Greenhouse or anti-greenhouse effectsdepend on the relative transparency of the atmosphereto stellar and emitted longwave radiation. (To be clear, we are now referring to standard radiative effects, notthe mechanical greenhouse.) If incoming starlight pen-etrates below the infrared photosphere, then the green-house effect gives T deep > T ∗ . If, however, significantincoming radiation is absorbed above the photosphere,a stratospheric thermal inversion gives T deep < T ∗ . SeeHubeny et al. (2003) for a more quantitative analysis. Inmost of our examples we adopt T deep = 1500 K, as appro-priate for short ( ∼ ∇ ≡ d ln Td ln P = 3 κP g F rad σT (5)characterizes its stability, and the final equality followsfrom equation (1). Convection occurs where ∇ > ∇ ad .We set ∇ ad = 2 /
7, the adiabatic index of an ideal di-atomic gas. In reality, non-ideal interactions lower ∇ ad at the high pressures of exoplanet atmospheres, and pro-mote convection. At the top the atmosphere, where theoptical depth τ = κP/g ≈ F rad ≪ σT , equa-tion (5) shows that ∇ ≪ ∇ increases with depth, and gives a transitionto convection (even under the ideal gas approximation).In convective regions we set ∇ = ∇ ad , i.e. an adiabaticprofile with T ∝ P ∇ ad . The efficiency of convective en-ergy transport makes the modest super-adiabaticity neg-ligible. The level of the adiabat is determined by the in-ternal entropy. A global calculation of entropy is beyondour illustrative scope. Instead, motivated by Hubbard(1977), we label our adiabats by T , the temperature itwould have at P = 1 bar pressure, even though the adia-bat likely does not extend to such low pressure. We definea reference entropy (per unit mass) S ref , correspondingto T = 250 K. Relative entropy values for different T are then computed as∆ S ≡ S − S ref = C P ln( T /
250 K) (6)where the specific heat (at constant pressure) C P = R/ ∇ ad is assumed constant. For the gas constant R = k B / ( µm p ) we use a mean molecular weight µ = 2 . m p .The stable atmosphere matches smoothly onto theconvective adiabat at the radiative-convective boundary(hereafter RCB). Since the temperature T c and pressure P c at the RCB lie on the interior adaibat we require T = T c (cid:18) P P c (cid:19) ∇ ad . (7)The location of the RCB is crucial for global evolu-tion. The secular cooling of the convective interior isdetermined by the radiative flux, F c , leaving the RCB.Combining equations (3), (5) and then (7) at the RCB Youdin & Mitchellgives F c = ∇ ad gσT − β c κ P α c (8)= ∇ ad gσ κ T P ∇ ad P ∇ ∞ −∇ ad c ! − β ∝ T P / , (9)with ∇ ∞ ≡ (1 + α ) / (4 − β ) = 1 / . (10)Core flux increases with the interior entropy. Pushingthe RCB to higher pressures decreases the core flux if ∇ ∞ > ∇ ad . This condition is satisfied for our opac-ity choice, and is generally required for a transition toconvection (as we show shortly). We emphasize that thedependance of F c on P c is independent of the mechanismthat changes P c , though previous works have mostly con-sidered irradiation. These basic considerations are use-ful in interpreting numerical studies of planetary cool-ing histories and radii evolution (Burrows et al. 2000;Baraffe et al. 2003; Chabrier et al. 2004).2.2. Radiative Equilibrium Solutions
We now apply the two approximations of radiativeequilibrium (RE) to the stable layer. First, radiationis the only relevant energy transport mechanism. Thus F rad in equation (1) is the total flux of energy. Sec-ond the flux is constant through the radiative layer with F rad = F c , the flux from the convective interior. Thisassumes that local (thermal) energy release is negligible.Fig. 2 plots radiative equilbrium atmospheres for κ ∝ P , with two values of T deep matched on to interior adia-bats labeled by T . We obtain analytic RE solutions byintegrating equation (1) with F rad = F c and T = T deep at P = 0 to get T = T deep " ∇ ad ∇ ∞ − ∇ ad (cid:18) PP c (cid:19) α / (4 − β ) . (11)This solution uses equation (8) and we have imposed therequirement T ( P c ) = T c to find T c = T deep (cid:18) ∇ ∞ ∇ ∞ − ∇ ad (cid:19) / (4 − β ) (12)A valid solution — one that transitions to convection —thus requires ∇ ∞ > ∇ ad and α > − β < T c increases with T deep but is independendent of the interior entropy.The RCB sinks to larger pressure as entropy decreasesor as T deep increases, P c = k ∇ P (cid:18) T deep T (cid:19) / ∇ ad , (13)which follows from equations (7) and (12) with the con-stant k ∇ ≡ (cid:18) ∇ ∞ ∇ ∞ − ∇ ad (cid:19) ∇ ∞ / [ ∇ ad (1+ α )] ≈ . . The numerical scaling in equation (9) ignores the effect thathigher entropy would lower gravity by inflating the planet. Thiseffect cancels when computing the total luminosity, which is ulti-mately more important. T = K T = K T = K !!! """ !!! """ Radiative Equilibrium1 10 100 1000 10
10 00050002000300015007000 P ! bar " T ! K " T deep = 2000 K1500 K Figure 2.
Radiative equilibrium (RE) atmospheres with deepisotherms of T deep = 1500 K( blue curves ) and 2000 K ( dotted greencurves ) matched onto internal adiabats ( dashed red curves ) withentropy increasing from bottom to top. Grey dots mark the loca-tion ( T c and P c ) of the radiative-convective boundary (RCB), and squares show P deep , where ∇ = ∇ ad / The core flux for RE atmospheres follows from equa-tions (8), (12) and (13) as F c = k F gκ T T −∇ ad / ∇ ∞ deep ! (1+ α ) / ∇ ad ∝ T T . (14)This gives the well known result that increased irradia-tion reduces the cooling of the planet, while higher en-topy planets are more luminous. The constant k F ≡ σ ∇ ad P α (cid:18) − ∇ ad ∇ ∞ (cid:19) ∇ ∞ / ∇ ad − . Equation (14) is consistent with, but more specific than,equation (9) in assuming that RE sets the location of P c .We chose the parameters for Fig. 2 — used throughoutthis work — by roughly matching the analytic solutionsto more detailed hot Jupiter models, as in AB06.We constrain the entropy parameter T by appealingto the typical P c ≈ T =260 K, we reproduce a 1 kbar RCB for T deep = 1500K. We also consider larger values of T to describe moreinflated planets, but keep P c ≫ Requiring F c = σ (100 K) for the standard pa-rameters and P c = 1 kbar, gives κ o = 0 .
18 cm / g for g = 10 cm /s. We emphasize that this is not a realisticopacity law (in particular it is too low at small pressures).We are merely choosing parameters that allow the simpleanalytic model to mimic properties of more detailed hotJupiter models.The lapse rate for RE solutions is (from eq. [11]) ∇ = ∇ ad ( P/P c ) α (cid:16) − ∇ ad ∇ ∞ (cid:17) + ∇ ad ∇ ∞ ( P/P c ) α , (15) Remarkably, κ o does not affect the location of the RCB alonga given adiabat, only T deep and the power laws are required. Overlong times though the opacity at the RCB affects entropy evolutionand thereby RCB location. ot Jupiter Atmospheres 5demonstrating that ∇ = ∇ ad at P = P c and that thesolution becomes isothermal ∇ → ∇ with P . Opacity windows give more complicated profiles of ∇ ,including multiple zones of convection (see § P deep , the effective depth of the isothermallayer, as the location where ∇ = ∇ ad /
2. This occurs at P deep = (cid:18) ∇ ∞ − ∇ ad ∇ ∞ − ∇ ad (cid:19) / (1+ α ) P c ≈ . P c . (16)Our definition of P deep differs from AB06, who define P deep as a characteristic scale that might exceed P c .We now revisit the validity of applying the boundarycondition T = T deep at P = 0. Due to the isother-mal layer at low pressures, applying the boundary con-dition at any P ≪ P deep gives indistinguishable solu-tions. However solutions are only physically valid inoptically thick regions, for P ≫ P thick ∼ g/κ min ∼
10 mbar[ κ min / (0 . / g)] − . The relevant opacity κ min is the smaller of the opacities to starlight and emit-ted radiation near P thick . Indeed the penetration ofstarlight below the infrared photosphere can push thetop of the isotherm to ∼ P thick ≪ P deep , solutions are physically consistent below P thick . ENERGETICS OF TURBULENT RADIATIVELAYERSWe now generalize the radiative equilibrium model toinclude two effects. First we allow for turbulent eddiesto drive an advective heat flux F eddy . The total flux F = F rad + F eddy . (17)includes the radiative and eddy contributions. Second,we allow for the release of energy at a rate ǫ . In steadystate this heating is balanced by cooling from the diver-gence of the total flux, dFdP = − ǫg . (18)Sources of ǫ include the viscous dissipation of turbulence( § F rad is no longer constant with height. The fractionalcontribution of F rad to the total flux can vary, and thetotal flux itself can vary. To proceed further we requirea model for F eddy and ǫ .3.1. Turbulent Heat Transport
We derive F eddy using basic elements of mixing lengththeory. This theory is usually applied to convectively un-stable regions, but instead we apply it to forced turbu-lence in convectively stable regions. We will show that inthis case the energy flux is inwards. We leave the forcingmechanism of turbulent motions unspecified, and theirstrength a free parameter.We consider parcels of gas which conserve entropy andmaintain pressure equilibrium with their surroundings asthey exchange position over a vertical distance ℓ and thendissolve. These parcels contain excess heat δq = ρC P δT with: δT = (cid:18) dTdz (cid:12)(cid:12)(cid:12)(cid:12) ad − dTdz (cid:19) ℓ = − ℓTC P dSdz (19)where δf gives the difference of any quantity f betweenthe parcel and its surroundings. For stable stratifica-tion ( dS/dz >
0) rising parcels ( ℓ >
0) cool and sinkingparcels heat. We express the heat flux, F eddy = wδq with w the characteristic eddy speed, in terms of the turbulentdiffusion, K zz = wℓ , as F eddy = − K zz ρT dSdz = − K zz ρg (cid:18) − ∇∇ ad (cid:19) . (20)The flux is always negative for stable stratification. Itvanishes at the RCB where dS/dz = 0 (and ∇ = ∇ ad ).We do not model overshoot, which could allow energyexchange (in either direction) between convectively sta-ble and unstable zones. In the upper isothermal regions F eddy ∝ − K zz P , declines in magnitude with height, un-less K zz increases with height to compensate, as we willconsider in § F eddy , we analyze itsdivergence, which describes cooling and (when negative)heating dF eddy dz = K zz ρgR dSdz − K zz ρT d Sdz − dK zz dz ρT dSdz . (21)The terms on the right hand side represent cooling rates, − δ ˙ q , which we relate to rates of work, δ ˙ w , using the firstlaw, δq = δe − δw . Thus − δ ˙ q = δ ˙ w/ ∇ ad since the internalenergy, δe = ρC V δT = (1 − ∇ ad ) δq with C V = C P − R . The first term in equation (21) arises from buoyantwork δ ˙ w B = δρgw = K zz ρg ( dS/dz ) /C P , where δρ/ρ = − δT /T by pressure equilibrium. The first term on theRHS of equation (21) is thus − δ ˙ q B = δ ˙ w B / ∇ ad . Thisbuoyant cooling will be evident in the stratified regions( P < P deep ) of Fig. 6.The second term represents the tendency of mixingto heat by filling in entropy minima (for d S/dz > dS/dz consider two parcels that arrive at z , onefrom above and one from below. Compressional work isdone on the parcels at a rate P ∇ · v ± = − Pρ δρ ± δt = ∓ P ℓC P dSdz (cid:12)(cid:12)(cid:12)(cid:12) z ± ℓ wℓ (22)where the top (bottom) sign refers to sinking (rising) Youdin & Mitchellparcels, ∇ · v is a velocity divergence, and δt = ℓ/w givesthe expansion rate. The net work is the sum of theseterms, δ ˙ w C = −∇ ad K zz ρT d S/dz . We identify the sec-ond RHS term in equation (21) as − δ ˙ q C = δ ˙ w C / ∇ ad .This compressional heating dominates in deeper regions( P > P deep ) of Fig. 6. The third and final term repre-sents a flux imbalance that arises from non-uniform eddydiffusion as in § Model Equations and Self-Similar SolutionTechnique
Our atmospheric model is described by equations (1),(17), (18) and (20) which reduce to the following pair ofcoupled ODEs dTdP = F + F iso k rad + F iso P/ ( ∇ ad T ) (23) dFdP = − ǫg (24)where F iso ≡ K zz ρg (25)is (minus one times) the isothermal limit of the eddyflux. The thermal profile in equation (23) describes thecombined effects of radiative and eddy fluxes. Ignoringmixing ( F iso →
0) recovers standard radiative diffusionof equation (1). Strong mixing ( F iso → ∞ ) creates anisentropic profile ( ∇ → ∇ ad ).Solving the coupled ODEs for T and F requires pre-scriptions for K zz and ǫ and boundary conditions. Aswith the RE solutions, we fix T deep at the top of the at-mosphere, and match onto an adiabat with a given T atthe bottom. This matching generally requires iterativetechniques. We avoid this complication by finding self-similar solutions. This technique is only possible becauseof the idealizations — notably power law opacities andideal gas EOS — described in § T and F are unity.We set the strength of diffusion not with a physicalvalue for K zz , but by the parameter ψ c = K zz ρ c gF c , (26)the ratio of F iso to F = F rad at the RCB. For dissipa-tion we must similarly specify ǫP/ ( gF ) at the RCB. Seeappendix B for details.To get a physical solution, we scale a dimensionlesssolution to any desired value of T deep and T . Setting T = T deep at P = 0, the solution gives T c at the RCB.Specification of T then fixes P c via equation (7). Wethen determine F c and K zz via equations (8) and (26).3.3. Relating Diffusion and Dissipation
Turbulence that gives rise to diffusion, K zz , will alsodissipate at some rate ǫ . We now consider a prescriptionthat sets a lower bound on ǫ from turbulence. We also allow for stronger heating, perhaps from non-turbulentsources.In a Kolmogorov cascade, the dissipation rate ǫ = w /ℓ and the diffusion K zz = wℓ give a simple relation ǫ = K zz /t o , with t o = ℓ/w the turnover time of thedominant eddies. Unfortunately we lack a reliable modelfor eddy timescales. Moreover turbulence in a stratifiedatmosphere is likely anisotropic and not well describedby Kolmogorov scalings. Fortunately, our results arenot very sensitive to this limitation, as we will show insubsequent sections. Eddies with long turnover times, t o > /N organize into horizontally extended pancakes,where the squared buoyancy frequency is N = g RT [ ∇ ad − ∇ ] . (27)Assuming that the buoyancy frequency sets the rele-vant timescale, t o = 1 /N , gives a dissipation rate ǫ buoy ≈ K zz N . (28)In strongly stratified, isothermal regions the buoyancy is N deep = g/ p C P T deep ∼ c s /H (29)with c s the sound speed and H the scale height. Forsub-sonic turbulence with w ≪ c s , our prescription gives ℓ ∼ w/N ≪ H . This is consistent with the expectationthat stratification limits the vertical extent of turbulentstructures to less than a scale height. Forced turbulencewith t o < /N is also possible. Since this would giveeven stronger dissipation, we consider ǫ buoy a reasonablelower bound on dissipation for stratified turbulence.Near the RCB, as N → ǫ o = f ǫ K zz N (30)with the dimensionless normalization f ǫ giving the ratioof ǫ o to ǫ buoy in isothermal regions. If we use rotationas the other relevant timescale, then the floor would bequite low with f ǫ ∼ Ω /N ∼ − P − / . Our fullprescription considers both terms, ǫ = ǫ o + ǫ buoy , (31)as discussed in § RESULTS4.1.
Constant Diffusion, No Dissipation
We describe solutions to the atmopheric model of § K zz and no dissipation, i.e. ǫ = 0. Thisrequires integration of equation (23). While the decay ofturbulence always gives some dissipation, the effect onenergetics can be small (as we will show in § F eddy <
0, in convectively stable regions, weot Jupiter Atmospheres 7rearrange flux balance as − F eddy = F rad − F . For alarge downward eddy flux we need the total flux F to besmall compared to the outgoing radiative flux. Ignoringdissipation helps in this regard by preventing F fromincreasing through the layer. Pushing the RCB to highpressure also helps by lowering the constant F = F c fromthe RCB. The solutions below show that strong mixingdoes indeed push the RCB to high pressure.4.1.1. An Upper Limit to K zz Fig. 3 shows the effect of varying K zz while hold-ing irradiation ( T deep = 1500 K) and internal entropy( T = 250 K) fixed. As K zz increases, the downwellingeddy flux pushes the RCB to higher pressures, P c , andlowers the flux from the interior, F c . These effects arecoupled since T c ∝ P − / along a fixed adiabat (eq. [9]).Turbulent mixing — like strong stellar irradiation — re-duces the planet’s cooling rate. P c F c T deep ! T !
250 K0 500 1000 15000.010.1110100 K zz ! cm " s R e l a t i v e t o K zz ! Figure 3. As K zz increases the RCB moves to high pressure ( P c , dashed green curve ) while the core flux ( F c , blue curve ) drops.Both diverge at at finite K zz = K zz, crit ≈ / s, whendissipation is ignored ( ǫ = 0). This upper limit varies with in-ternal entropy and and T deep as shown in Fig. 4. Quantities areplotted relative to the K zz = 0 case where P c = 1 . F c = σ (100 K) . T d e e p = K T d e e p = K T d e e p = K
250 350 500 75010 T ! K " K zz , c r i t ! c m s " ! S ! k B m p " Figure 4.
Maximum eddy diffusion in the stable layer ( K zz, crit ),vs. internal entropy for T deep = 1000 , dotted red,blue and dashed green curves, respectively ). See equation (32) foranalytic fits. Entropy is given in terms of T [eq. 7] and relative tothe T = 250 K reference ( top axis ). !!""
10 0005000200020 0003000150015 0007000 P ! bar " T ! K " !""
100 300 1000150020002500 ! ! ! " ! " ad ! !"" P ! bar " ! ! ad Figure 5.
Profiles of a stirred atmosphere ( blue curves ) with K zz ≈ K zz, crit and no dissipation compared to the RE case with K zz = 0 ( dotted black curves ). Both join an adiabat with T = 250K ( dashed red and gray curves ). Mixing pushes the RCB ( graydots ) to high pressure (which remains finite because K zz is 0 . K zz, crit ). ( Left: ) The temperature profile shows modestchanges near P deep ( blue squares ), as shown in the inset. ( Right: )Lapse rate ∇ , relative to ∇ ad . Turbulent diffusion smoothes thetransition towards the adaibat. The inset plot of 1 − ∇ / ∇ ad showsthe marginal stability of the stirred atmosphere up to high pres-sures. At a critical value of K zz = K zz, crit , P c diverges toinfinity while F c drops to zero. This upper limit to dif-fusion is K zz, crit ≈ / s for the adiabat and T deep chosen in Fig. 3. Of course a real planet cannot extendto infinite pressure, to say nothing of the plane-parallelapproximation. The point is that our steady state modelcannot energetically support K zz > K zz, crit .Stronger turbulence could in principle exist, since weare saying nothing about what forces turbulence. In anon-equilibrium state with K zz > K zz, crit , the down-welling eddy flux would then increase the internal en-tropy, and inflate the planet.Fig. 4 shows that higher entropy planets have a highersteady state K zz, crit . Thus by inflating the planet, strongmixing brings the planet’s energy balance into equilib-rium. An ultimate upper limit to K zz is that the planetnot over-inflate and exceed its observed radius. The K zz values invoked in S09, from 10 to > cm / s wouldimply significant or (on the upper end) excessive infla-tion. For comparison AB06 showed (see their Fig. 11)that entropy changes of ∆ S ≈ k B /m p (the scale on ourFig. 4) can expand a hot Jupiter’s radius by ∼
10 —25%. Youdin & MitchellAccurate determination of the maximum K zz allowed fora given planet requires more detailed modeling (includingglobal structure with realistic opacities and EOS) thanwe perform. However our results strongly suggest that K zz values invoked in the literature have significant, oreven excessive, effects on energetics.Fig. 4 also shows that K zz, crit increases with decreas-ing T deep . Thus our constraints on mixing are much morestringent for hot Jupiters than for more distant plan-ets, including Jupiter itself. Recall that thermal inver-sions lower T deep and that mixing can sustain thermalinversions by keeping opacity sources aloft in the strato-sphere. A planet can accommodate strong mixing withsome combination of thermal inversions to lower T deep and increased internal entropy. It is hard to predict ifthermally inverted planets should be more inflated —due to the presumed presence of turbulence — or lessinflated — because lower T deep promotes cooling and in-hibits our mechanical greenhouse effect. Observations donot indicate an obvious correlation. Planets with signa-tures of inversions exhibit varying degrees of inflation.See Miller et al. (2009) for a comparison of observed tomodel radii of transiting planets.While strong mixing pushes P c → ∞ , the depth ofthe isothermal layer, P deep , is relatively unchanged bymixing. (We explore structure in detail below.) Thusplanets with higher entropy or lower T deep have shallowerisothermal layers. Specifically P deep ∝ ( T deep /T ) / ∇ ad from equations (13) and (16).It is hardly surprising that planets which can ac-commodate more mixing (larger K zz, crit ) have shallowerstratified layers to mix (smaller P deep ). In principlestrong mixing could destroy the deep isotherm altogetherby pushing P deep to optically thin regions. This proba-bly requires unrealistically large core entropies. Alterna-tively as interior temperatures rise, blackbody emissionat depth may find opacity windows at wavelengths longerthan the infrared.4.1.2. Structure and Energetics of Stirred Atmospheres F rad -F eddy F rad ( K zz = 0) F tot !!"" P ! bar " F ! e r g c m s " Figure 6.
Energetic balance for the profiles in Fig. 5. In radiativeequilbrium ( black dotted curve ) the radiative flux is constant. Witheddy diffusion and no dissipation the total flux ( gray dotted curve )is constant. The radiative flux ( blue curve ) leaving the interior islow, but F rad first increases and then decreases outward, peakingnear P deep ( blue square ). The downwelling eddy flux ( dashed redcurve ) offsets changes to the radiative flux. We now consider the structure and energetic balance of“stirred atmospheres” with K zz ≈ K zz, crit . We comparethese solutions to standard RE atmospheres of § K zz = 0. Since our model is self-similar, the behavioris independent of the parameters ( T , T deep ) chosen forillustration.Fig. 5 ( top panel ) shows that the temperature profileof the stirred atmosphere is very similar to the RE case.In the stirred atmosphere, the RCB lies at much higherpressure below an extended “pseudo-adiabat,” whichlies very close to the original adiabat. The inset to Fig. 5( top panel ) focuses on the region near P deep (which dropsslightly to 480 bar from the original 610 bar) where thestirred atmosphere is hotter, by at most 60 K. The stirredatmosphere is slightly colder below 250 bar, though thisdifference of at most 3 K is not visible. The stirred at-mosphere is very modestly thicker, by 0 . H deep , where H deep = RT deep /g ≈ . R J .Fig. 5 ( bottom panel ) plots the lapse rate. Mixingsmoothes the transition towards the adiabat. The in-set shows the smooth decline of 1 − ∇ / ∇ ad along thepseudo-adiabat, which gradually reduces the amplitudeof the downwelling eddy flux, from equation (20).Fig. 6 shows that the energetics of the stirred at-mosphere differs significantly from the RE case. With K zz = 0 there is no eddy flux and the radiative fluxis constant down to the RCB, here at P c ≈ . P c → ∞ and F c → K zz all the way to K zz, crit , but choose not to for visualization.The radiative and eddy fluxes change with height.Their sum — the total flux — remains constant be-cause we ignore dissipation. Fig. 6 shows that thefluxes behave differently above and below P deep , i.e.along the isothermal and pseudo-adiabatic regions, re-spectively. Along the pseudo-adiabat, the radiative fluxdeclines with depth as F rad ∝ P − / , as we derivedfor the core flux in equation (8). The radiative cooling( dF rad /dP <
0) in this region balances heating by eddydiffusion ( dF eddy /dP > T - P profile.Since F eddy scales as 1 − ∇ / ∇ ad , it is very sensitive tosmall changes in ∇ along the pseudo-adiabat (see eq. [20]and the bottom inset of Fig. 5).The energy balance along the isotherm, i.e. above P deep is different. With ∇ ≪
1, the eddy flux F eddy ≈− ρgK zz ∝ − P grows in magnitude with depth (a dif-ferent scaling holds if we vary K zz with height, see § dF eddy /dP <
0) balances radia-tive heating ( dF rad /dP > F rad is sensitiveto small changes in ∇ ≪ K zz, crit using our knowl-edge that F c → P deep is cru-cial. Here the eddy flux reaches its peak negative value F eddy , deep ≈ − ρ deep gK zz /
2. The thermal profile con-strains F rad to be roughly F c ( K zz = 0), the core flux ofthe RE atmosphere. We set F rad , deep ≈ F c ( K zz = 0) This is not to be confused with the pseudo-adiabat that de-scribes moist convection in Earth’s atmosphere. ot Jupiter Atmospheres 9to account for the slightly hotter atmosphere near P deep .Energetic balance, F eddy , deep + F rad , deep = F c →
0, thengives K zz, crit ≈ F c ( K zz = 0) ρ deep g ∝ T / ∇ ad T / ∇ ad − / ∇ ′∞ deep ! α (32)where ∇ ′∞ = (2 + α ) / (5 − β ) and our parameters give K zz, crit ∝ T / /T / . These scalings agree with theresults of Fig. 4.4.2. Spatially Varying K zz The above analysis ( § K zz are set by the balance of eddy and ra-diative fluxes near P deep , which itself scales with internalentropy and T deep . To test the robustness of this find-ing, we include a depth dependence to K zz ∝ P ζ . Forwinds driven near the photosphere, i.e. the top of ouratmospheres, one might expect stronger diffusion in theupper atmosphere, i.e. ζ <
0. On the other hand if tur-bulence is triggered by shear layers with the convectiveinterior, perhaps ζ >
0. As discussed in § [and section5?] detailed dynamical simulations can help determineplausible diffusion profiles.Fig. 7 shows the effect of varying ζ with other param-eters fixed (at our standard values of T deep = 1500 K, T = 250 K, α = 1, β = 0 as in e.g. Fig. 3). These plotsshow the strongest possible mixing, which (as we foundfor constant K zz ) drives the RCB to infinite depths andreduces the core flux to zero.The maximum mixing near P deep is relatively un-changed, except when the mixing at the top of the at-mosphere is quite strong. Quantitatively we comparevalues of K zz, deep , defined as the maximum value of K zz at a reference P = 550 bar, which is P deep of the radia-tive equilibrium atmosphere. For constant K zz we found K zz, crit = K zz, deep = 1665 cm / s. For mixing that in-creases with depth as ζ = 0 .
5, 1 .
0, and 1 . K zz, deep de-clines by a modest 5%, 6% and 5%, respectively. Whenmixing declines with depth as ζ = − . − .
0, and − . K zz, deep increases by 14%, 58% and 300%, respectively.We cannot consider models with ζ . − . ζ = − .
4. We explore this issuefurther in appendix A.The top panel of Fig. 7 shows the flux profiles for sev-eral ζ values. The plot shows both radiative and eddyfluxes, which obey F rad = − F eddy because the net flux, F → horizontal black dotted line ) withoutany mixing. The explanation for these flux profiles mir-rors the discussion of Fig. 6 in § K zz are com-pensated by (1 − ∇ / ∇ ad ) — see equation (20) — whichis small and sensitive to slight changes in ∇ close to theadiabat.The flux in the low pressure, isothermal region scalesas − F eddy ∝ ρK zz ∝ P ζ (see eq. [20]). This explainswhy the flux components, F rad = − F eddy , increase with !! !!!!!! T deep ! T !
250 K1 100 10 P ! bar " F r ad ! " F edd y ! e r g $ c m s % " K zz ! P - . K zz = c o n s t . F rad ( K zz = K zz ! P -1 K zz ! P !! !! !!!!
100 300 100015002000 !! !!!!!! T !
250 K T deep !
10 0005000200020 0003000150015 0007000 P ! bar " T ! K " Figure 7.
Models with a depth-dependent K zz ∝ P − ζ for ζ =1 , , − − . dotted green, dashed red, dot-dashed purple andsolid blue curves, repspectively ) and no dissipation. Strong mixingpushes P c → ∞ while P deep is plotted with colored squares. ( Top :)Radiative flux, which is equal and opposite to the eddy flux. Lower ζ values give strong mixing and larger fluxes at the top of theatmosphere. ( Bottom :) Thermal profiles show that strong upperatmosphere mixing (low ζ ) heats the upper atmosphere and resultsin a more gradual approach to the adiabat. height if ζ < −
1. Driving larger radiative fluxes in theupper atmosphere requires a steeper dT /dP . The tem-perature profiles in Fig. 7 ( bottom panel ) reflect this.The ζ = − ζ = − . P deep . We thus find that mixing at the top ofthe atmosphere is more effective — compared to uniformor bottom-focused mixing — at lifting (i.e. heating) the T − P profile. The additional heat in this case is providedby a downward flux of mechanical energy across the topboundary.4.2.1. Limits on Mixing Near the Photosphere
We now consider what might constrain K zz near thetop of the atmosphere, since we find that internal entropymostly constrains diffusion near P deep . Our ζ ≤ − F eddy to a fraction f ∗ ∼
1% of the insolation F ∗ ∼ σT K zz, top < f ∗ F ∗ ρ top g (33) ≈ cm s (cid:18) P top . (cid:19) − (cid:18) T deep (cid:19) f ∗ , scaled for a downwelling flux that originates at a 0.1 barphotosphere. While the efficiency f ∗ and mechanismsof generating a downward mechanical flux are uncertain,the energetic difficulties of mixing at K zz ≫ cm / sis evident.Alternatively, we could attempt to constraint mixing inthe upper atmosphere by appealing to our ζ = − . top ). Smaller ζ values would be needed for a largerflux, but these do no give consistent solutions. In ap-pendix A we analyze the ζ ≈ − . ζ = − . K zz ≈ · ( P/ bar) − . cm / s. A larger internal en-tropy could support a larger K zz (as we showed for con-stant K zz models in Fig. 4). Therefore this constraintis not inconsistent with equation (33), which is a morerobust constraint. ! = ! o ! = ! o + ! buoy ! ! ! ! Ε o ! erg " g s $% K zz , c r i t ! c m " s % Figure 8.
Increasing dissipation reduces K zz, crit , shown for fixed T deep = 1500 K and T = 250 K. The solid blue curve only includesa constant floor to the dissipation, ǫ o , while the dashed curve alsoincludes our prescription for dissipation in stratified regions. Including Dissipation
We now consider the effect of adding dissipation to ourmodels with eddy diffusion. The total flux F will now in-crease with height due to dissipation. The coupled equa-tions (23) and (24) govern the steady state structure. Tounderstand the effect of dissipation on the turbulent heatflux, we return to the simpler case of spatially uniform K zz .With dissipation we still find an upper limit to diffu-sion, K zz, crit , for a given T and T deep . However K zz, crit declines with increasing dissipation. Fig. 8 shows this forboth a dissipation rate ǫ o that is constant with height( solid blue curve ) and the full dissipation prescription( dashed red curve , see eq. [31]). The full prescriptionincludes ǫ buoy , our estimate of the minimum dissipation !!"" ! P ! bar " F ! e r g c m s " -F eddy F rad F tot Figure 9.
Similar to Fig. 6 except dissipation is included. Thetotal flux is no longer constant and the core flux is larger. Bothof these effects reduce the magnitude of F eddy which can nolonger offset as much of F rad . Consequently K zz, crit is reducedto 900 cm / s. The dissipation profile ǫ = ǫ o + ǫ buoy includes afloor ǫ o = 5 × − erg / (gs), that corresponds to f ǫ = 0 .
01, i.e.weaker dissipation near the RCB than in stratified regions. due to stratified turbulence. This additional dissipationfurther reduces K zz, crit .We confirm that the dissipation-free estimates of K zz, crit in previous sections represent a conservative up-per bound. Lowering K zz, crit means that weaker tur-bulent diffusion will inflate the planet. Admittedly, thecases shown in Fig. 8 do not prove that all dissipationprofiles will lower K zz, crit . However we investigated theeffects of both spatially-varying K zz (as in § ǫ . In all cases adding dissipation re-duced K zz, crit from the dissipation-free value.4.3.1. How Dissipation Lowers K zz, crit We explore the energetics of how dissipation lowers K zz, crit . Fig. 9 shows flux balance with dissipation, andcan be compared to Fig. 6. Notice that the peak valueof − F eddy now falls well short of F rad at the relevantpressure, P deep . This is because the total flux F = F c + Z P c P ǫg dP (34)now includes the integrated dissipation, causing F togreatly exceed F c , the small loss of heat from the core.From the RCB to P deep , F rad also increases with height.This rise is not significantly affected by dissipation, with F rad ∝ P − / along the pseudo-adiabat as before. Themagnitude of − F eddy = F rad − F is smaller at P deep because dissipation increases F without comparably in-creasing F rad .We find that P deep (and also ρ deep ) do not significantlychange when we add dissipation. Indeed the T − P or ∇ profiles for the solution in Fig. 6 are indistinguishablefrom the stirred atmospheres in Fig. 5, except the RCBis not pushed as deep, “only” to P c ≈
11 kbar.The limiting value of K zz = 2 | F eddy | / ( ρ deep g ) dropsbecause the smaller eddy flux is not compensated by alower ρ deep . It seems possible that some dissipation pro-file could heat the atmosphere and lower P deep and ρ deep enough to increase K zz, crit . We did not find this to be thecase. One reason is that too much dissipation can affectthe location of the RCB, a subject we address below.ot Jupiter Atmospheres 11For completeness we explain the flux balance at smallpressure illustrated in Fig. 9. The decline in − F eddy ≈ ρgK zz with height in isothermal regions again resultsfrom declining density. However F rad does not declinetowards low pressure (as was seen in Fig. 6), because thetotal flux F is larger with dissipation. The fact that theescaping flux matches the flux from the radiative equi-librium solution ( dotted black line ) is a coincidence (withsome significance, see below). This coincidence occursbecause the dominant dissipation is ǫ ≈ ǫ buoy . Largeror smaller choices of dissipation would give a larger orsmaller (respectively) net F and escaping F rad .This coincidence has some significance. Downwardeddy fluxes reduce the loss of heat from the core, as wehave discussed extensively. However we now see that thisloss is matched by the flux due to the dissipation of thatturbulence — provided our prescription for the minimum ǫ buoy is correct. This replacement is intriguing, but doesnot alter our discussions of evolutionary consequences:turbulent dissipation in radiative regions is powered notby interior heat, but by external means (such as forcedatmospheric circulation).4.3.2. The Effect of Dissipation on the RCB
Ignoring dissipation, we obtained the limiting K zz, crit as P c → ∞ . With dissipation, K zz, crit occurs at finite P c ,provided there is dissipation at the RCB. As noted above,the RCB occurs at 11 kbar with K zz = K zz, crit in Fig. 9.Appendix B derives the relation between dissipation andthe maximum P c in equation (B3).We consider it physically desirable to restrict P c tofinite pressures. Infinite P c obviously violates some ofour idealizations, notably the plane parallel and ideal gasapproximations. It is encouraging that dissipation alters P c without qualitatively changing the insights (namely K zz, crit ) gleaned from the non-dissipative model.Restricting P c to finite values was not crucial for theenergetic balance arguments above. Though F c is largerfor smaller P c , it is still too small to be the main factorthat limits the eddy flux.To illustrate some of these points consider the case ǫ = ǫ buoy , i.e. with no floor, ǫ o , to the dissipation. Inthis case there is no dissipation at the RCB, and we findthat eddy diffusion can still push P c → ∞ . Nevertheless ǫ buoy by itself does still lower K zz, crit , by ∼ /
3. Thiscan be seen in the ǫ o → Varying the Opacity Law and EOS
All of our plots and numerical estimates have used anopacity law κ ∝ P and an ideal gas EOS with ∇ ad = 2 / § ∇ ∞ > ∇ ad and α > − κ ∝ T ( α = 0 and β = 2), that approxi-mates H − opacities for T > κ ∝ P , this law alsohas ∇ ∞ = (1 + α ) / (4 − β ) = 1 / > ∇ ad .The behavior of K zz, crit — shown in equation (32)— is particularly important for interpreting our results. The scaling with internal entropy is quite steep with K zz, crit ∝ T . and K zz, crit ∝ T for our standard andalternate opacities, respectively. Generally, the entropydependance becomes steeper for larger α (as in the aboveexample) and also for a smaller ∇ ad , but does not dependon β . Recall that the burial of the turbulent heat fluxinto the convective interior can bring an atmosphere with K zz > K zz, crit towards energetic equilibrium. A steeperdependance of K zz, crit on entropy means that less infla-tion is needed to enforce this equilbrium.The scaling of K zz, crit with T deep controls how our in-flation mechanism depends on the level of irradiation.Also the development of thermal inversions will lower T deep for a fixed level of irradiation. We find K zz, crit ∝ T − . or K zz, crit ∝ T − , again for the standard andalternate opacities, respectively. The T deep dependancebecomes more steeply negative for larger α (again thedominant effect in our example) and also for larger β (less important in our example) and smaller ∇ ad .These scalings emphasize that a more detailed treat-ment of turbulent heat fluxes in hot Jupiters should in-clude realistic opacities and equations of state. Non-powerlaw behavior could have significant consequences.For instance, Guillot et al. (1994) demonstrated the im-portance of an opacity window near ∼ COMPARISON WITH PREVIOUS WORK5.1.
Simulations of Atmospheric Circulation
Hydrodynamic simulations of hot Jupiter atmo-spheres have been studied in local (Burkert et al. 2005;Li & Goodman 2010, hereafter LG10) and global (e.g.Cooper & Showman 2005; Rauscher & Menou 2010;Showman et al. 2009; Dobbs-Dixon & Lin 2008) models.A major goal of these studies is to determine the circu-lation induced by stellar irradiation. We first discuss es-timates of K zz from these simulations, and then addressthe question of whether radiatively forced turbulence ex-tends throughout the radiative zone, as we have assumed.Other sources of turbulence — perhaps involving mag-netic fields — may also exist, but we do not analyze themhere.The global circulation models of Showman et al.(2009) estimate K zz ∼ cm / s at mbar pressures.This does not contradict our constraint on upper atmo-spheric mixing from the efficiency of radiative forcing[equation (33)] when extrapolated to such low pressures.Moreover at low pressures radiative losses can lower theeddy flux (see § K zz . We caution that K zz estimates from GCMs arerough, since they only resolve relatively large scale flowpatterns. The Showman et al. (2009) K zz estimate arisesfrom multiplying measured vertical speeds by the scale-height H . While a useful guide to what is possible, thisdoes not constitute a direct measurement of diffusion.Local hydrodynamic simulations can better resolveturbulent flows, though as usual with Reynolds num-bers far lower than reality. The calculations of LG102 Youdin & Mitchellfind an effective turbulent viscosity of ν t ∼ .
001 —0 . c s H ∼ — 10 cm / s. It is unclear if thisviscosity should be interpreted as a mixing coefficient.Their simulation with ν t = 0 . c s H has an RMS ver-tical speed w ∼ . c s . Thus assuming ν t ∼ K zz is notconsistent with the simple estimate K zz ∼ wℓ becauseit gives a length scale ℓ ∼ H/
20, smaller than the gridspacing of H/
10. It should not be surprising that simpleestimates based on 3D isotropic turbulence fail, given theorganized structure in their 2D “turbulent” state.Moreover the forcing in LG10 was not by irradiation,but chosen to be large enough to drive super-sonic flowsdespite artificial viscosities that are large for numericalreasons. Thus attempting to interpret the Carnot effi-ciency of stellar irradiation may overextend their results.Despite these caveats, if the LG10 simulations apply nearthe mbar weather layer, there is again no contradictionwith equation (33).Global simulations indicate that the shear layer maynot extend throughout the radiative zone. For instanceShowman et al. (2009) find that strong ( ∼ km/s) zonalwinds terminate at ∼
10 bar. They argue that circula-tion stops because the planet is horizontally isothermalat these depths, removing the local forcing. However, hotJupiter atmospheres have a large separation between theradiative timescales in the weather layer and the deepradiative layer. It is possible (and arguably likely) thatunresolved or long timescale dynamics could push theshear layer even deeper. This possibility should be ex-plored in future modeling studies.How far turbulence can extend below the shear layeris also uncertain. LG10 force a shear layer of with ∼ H , but find that turbulence (or at least some disorderedmotion) extends throughout their box of size ∼ H (seetheir Fig. 10). Turbulence that extends a full 5 H belowa shear layer could thus extend to P ∼ e ·
10 bar ∼ . Thermal Inversions via TiO Diffusion
We now provide a more detailed interpretation of ourresults in terms of the S09 constraints on the mixing re-quired to loft TiO and create thermal inversions. Weemphasize that the constraints on K zz , and especiallyon the depth dependance of K zz depends sensitively onwhether or not there is a cold trap. Wherever TiO con-denses, one must consider the mixing of dust grains, notjust molecules.First consider the case where there is no cold trap,and TiO is always in the vapor phase. In the S09 analy-sis, this is only possible for the most intensely irradiatedplanet, WASP 12b, in part because thermal inversionslower T deep and favor condensation at depth. The mixingof TiO vapor requires K zz ∼ cm / s at P ∼ K zz ∝ P − inan isothermal atmosphere. Thus the constraint is weakerat depth. In a model where the actual K zz ∝ P − (asin § K zz ∼
10 cm / s at the kbarRCB. In this case our model predicts a minimal effectof the eddy flux on the planet’s evolution, though the dissipation of turbulence above the RCB could still besignificant.We briefly summarize how this K zz ∼ cm / s limitand the depth dependance arises, and refer the readerto S09 for details. With only molecular viscosity, i.e.collisions, the TiO scaleheight would be hydrostatic, andthus ∼
30 times smaller than the dominant H species.S09 conclude that the turbulent diffusion must exceedthe molecular diffusion D TiO by a factor ∼ ∼
14) scaleheights between P ∼ mbar andthe kbar RCB. Estimating D TiO as usual — the productof the mean free path and thermal speed — gives aninverse scaling with density, and thus also with pressurein isothermal regions. The numerical value of D TiO ∼ cm / s at P ∼ mbar, combined with the factor of100 excess needed for efficient turbulent mixing, givesthe K zz ∼ cm / s constraint.Now consider the case where grains do condense some-where in the radiative zone, which is true for most hotJupiters (especially those with inversions). The K zz re-quired to loft TiO increases to 10 — 10 cm / s depend-ing on the size of grains and the depth of the cold trap.A rough estimate of K zz ∼ v term L follows from equat-ing the diffusion timescale, L /K zz , to the dust settlingtimescale L/v term , where L is the depth of the cold trapand v term is the grain’s terminal speed. While the ter-minal speed will increase with particle size, it does notdepend on atmospheric density for the relevant viscous(i.e. Stokes’) drag. Thus unlike the case of molecularviscosity, the K zz required for mixing condensed grainsdoes not decline with depth.We can thus conclude that the K zz required to mixTiO and create thermal inversions in a hot Jupiter arelikely excessive, provided TiO condenses somewhere inthe radiative zone. This follows from Fig. 4 which showsthat K zz ∼ cm / s appear off-scale. The internal(specific) entropy of a planet would have to increase by & k B /m p for a planet with T deep & K zz, crit as shown in Fig. 8.However, we cannot firmly declare that the TiO hy-pothesis fails. This is largely due to the approximatenature of our treatment of opacities and the equation ofstate. A more conclusive analysis of the TiO hypothesiswould require a detailed atmospheric model that includesthe eddy fluxes and turbulent dissipation described inthis work. Such a study would also have to abandon thefixed flux bottom boundary condition used in most at-mospheric models (including S09). Instead the bottomboundary should be a fixed adiabat, chosen to matchthe observed radius. This is subject as usual to assump-tions about composition and presence of a core. Howeverallowing the flux to float is needed for a consistent deter-mination of the RCB. This is crucial for understandingthe coupled relationship between compositional mixingand cooling history. CONCLUSIONS6.1.
Summary of Results ot Jupiter Atmospheres 13We have investigated how forced turbulent mixing af-fects the energetic balance and structure of the stablystratified, radiative layers of hot Jupiters. It is crucialto understand how the radiative layer matches onto theconvective interior at the radiative-convective boundary(RCB). This regulates the rate at which the planet coolsand therefore controls the evolution of the planet’s ra-dius.Previous work has invoked turbulent eddy diffusion ofmolecular species (and dust). This mixing changes theopacity to provide better-fitting model spectra of transit-ing planets, especially those that appear to have thermalinversions (S09). We find that turbulent mixing of thiskind does not just redistribute chemical species but alsosignificantly affects energetics.Forced turbulent mixing in stable, radiative regionsdrives a downward flux of energy that pushes the RCBdeeper in the atmosphere, lowering the planet’s coolingrate. We found an upper limit to the strength of turbu-lent diffusion, K zz, crit , that can be achieved in steady-state. Beyond this limit, the downward flux of energywill heat the convective interior and inflate the planet.We did not directly model this entropy growth becauseour model was steady state and did not include over-shoot across the RCB. The deep, marginally stable layerin our solutions would not strongly inhibit overshoot, soheat burial by this mechanism is a likely outcome. Oursolutions indicate that interior heating brings the planetback towards steady state, because higher entropy plan-ets have larger K zz, crit (see Fig. 4). Our mechanism isthus a “mechanical greenhouse effect” with the role ofsunlight in the traditional greenhouse being played byforced turbulent mixing.For non-uniform turbulent mixing we find that our con-straint on K zz, crit applies near the RCB. More specifi-cally this constraint applies at a pressure P deep , whichlies below the deep isothermal region where the radiativelayer is transitioning towards convective instability. Ourconstraints on turbulence in the upper atmosphere areless stringent. However, the downward flux of mechani-cal energy likely cannot exceed a small fraction (probablyat the percent level) of the stellar irradiation if it is sup-plied by weather-layer winds. If turbulence is too weakat the bottom of the radiative layer, it will not dredge upheavy molecular species — particularly those that con-dense onto dust grains — to serve as opacity sources nearthe observable photosphere.Turbulence also deposits heat in the radiative atmo-sphere when it decays. Non-turbulent sources of energydissipation — including non-linear wave breaking andohmic dissipation — also affect energetic balance. Wefind that including energy dissipation reduces K zz, crit ,and thus makes it easier to inflate a planet for a givenlevel of forced turbulence.We find a characteristic scale of K zz, crit ∼ –10 cm / s for typical hot Jupiter parameters, even ignor-ing dissipation. This is orders of magnitude below valuesquoted in the literature of 10 — 10 cm / s for the mix-ing of chemical species (Spiegel et al. 2009; Zahnle et al.2009). We caution that our quantitative results shouldbe taken as illustrative, due to the approximations de-tailed in § Applications and Extensions
Guillot & Showman (2002) first noted that dissipationin radiative layers can slow planetary contraction bypushing the RCB to higher pressures. They imagineddissipation concentrated in the upper regions of the at-mosphere, sourced by insolation driven winds. Other au-thors (e.g. Bodenheimer et al. 2003) have included theGuillot & Showman (2002) dissipation prescription tomodel exoplanet radii.However we are unaware of previous works that in-clude the mechanical flux of energy due to turbulence orwaves — our F eddy — in radiative layers. Incorporatingthis effect in detailed planetary evolution models wouldrequire a more precise treatment than this exploratorystudy. Notably it would require a global model with arealistic equation of state and opacity, not the power lawsconsidered here.In this paper we treat diffusion and dissipation byforced turbulence as free parameters to allow a generalanalysis. This approach is justified since we currentlylack a detailed understanding of these processes. Thusour model can be used to test how any specific model forturbulence and/or energy dissipation affects the struc-ture and evolution of the planet. For instance, the workof Batygin & Stevenson (2010) consider ohmic dissipa-tion in radiative and convective regions, but do not re-compute the effect of the dissipation on the structure ofthe radiative layer. Our point is not to critique, but toemphasize that a more consistent treatment of energeticsmay make it easier to inflate a planet — by any numberof mechanisms.We also hope to include the effects of our study in de-tailed 3D radiation-hydrodynamical simulations of exo-planet atmospheres. This would involve adding sub-gridphysical prescriptions for eddy fluxes to global circulationmodels (GCM) such as those by Showman et al. (2009);Rauscher & Menou (2010) and/or non-hydrostatic simu-lations (Dobbs-Dixon et al. 2010). Goodman (2009) dis-cusses the need to include explicit sources of dissipation.Li & Goodman (2010) present a first step towards sub-grid modeling of turbulence due to Kelvin-Helmholtz in-stabilities. Breaking of vertically propagating gravitywaves represent another possible source of turbulence(Showman et al. 2009). When all these effects are takeninto account, the inflated radii of transiting planets maynot be surprising after all.We have focused on hot Jupiters, but the physics wedescribe in principle applies to other atmospheres. Ourlimits on eddy diffusion become less severe as objects aremore weakly irradiated (see Fig. 4). Thus our mecha-nism would not inflate longer period gas giants, includingJupiter and Saturn. More intrinsically luminous objectslike brown dwarfs will similarly be less affected. Theexpanding inventory of transiting exoplanets, at widerseparations, is an excellent test of radius evolution mod-els.Finally we comment on a possible connection tothe atmosphere of Venus. Venus has a marginallystable pseudo-adiabat beneath a thick cloud deck(Schubert et al. 1980). The clouds shield sunlight from4 Youdin & Mitchellwarming the surface of Venus. Because of this, Venus’ at-mosphere would be nearly isothermal were it not for somepoorly understood mechanical stirring process. Our well-stirred hot Jupiter atmospheres also have marginally sta-ble pseudo-adiabats at depth. Perhaps Venus is a very-well-stirred analog of a hot Jupiter. Obvious differencesexist, for instance the non-negligible fraction of sunlightthat reaches the Venusian surface versus the radiant fluxescaping the core of hot Jupiters. Pursuing analogiessuch as these should improve our understanding of worldsnear and far.ANY and JLM thank David Spiegel for extensive dis-cussions and Peter Goldreich for wise counsel. ANYthanks William Hubbard and Roger Yelle for insightsinto planetary cooling and eddy diffusion, respectively.JLM thanks Martin Pessah and Shane Davis for help-ful discussions. JLM and ANY thank the Institutefor Advanced Study for hosting them during the earlystages of this work. This research was supported in partby the National Science Foundation under Grant No.PHY05-51164 for ANY to visit the KITP. ANY thanksTravis Barman, Lars Bildsten, Brad Hansen and EmilyRauscher for useful feedback during the KITP program“The Theory and Observation of Exoplanets.” We thankthe anonymous referee, Kristen Menou, David Spiegel,Tristan Guillot and David Stevenson for comments thatimproved the submitted manuscript. REFERENCESArras, P., & Bildsten, L. 2006, ApJ, 650, 394Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt,P. H. 2003, A&A, 402, 701Batygin, K., & Stevenson, D. J. 2010, ApJ, 714, L238Bodenheimer, P., Laughlin, G., & Lin, D. N. C. 2003, ApJ, 592,555 Burkert, A., Lin, D. N. C., Bodenheimer, P. H., Jones, C. A., &Yorke, H. W. 2005, ApJ, 618, 512Burrows, A., Guillot, T., Hubbard, W. B., Marley, M. S.,Saumon, D., Lunine, J. I., & Sudarsky, D. 2000, ApJ, 534, L97Burrows, A., Hubeny, I., Budaj, J., Knutson, H. A., &Charbonneau, D. 2007, ApJ, 668, L171Chabrier, G., Barman, T., Baraffe, I., Allard, F., & Hauschildt,P. H. 2004, ApJ, 603, L53Cooper, C. S., & Showman, A. P. 2005, ApJ, 629, L45Dobbs-Dixon, I., Cumming, A., & Lin, D. N. C. 2010, ApJ, 710,1395Dobbs-Dixon, I., & Lin, D. N. C. 2008, ApJ, 673, 513Fortney, J. J., Baraffe, I., & Militzer, B. 2009, ArXiv e-printsGoodman, J. 2009, ApJ, 693, 1645Griffith, C. A., & Yelle, R. V. 1999, ApJ, 519, L85Guillot, T. 2010, ArXiv e-printsGuillot, T., Gautier, D., Chabrier, G., & Mosser, B. 1994, Icarus,112, 337Guillot, T., & Showman, A. P. 2002, A&A, 385, 156Hansen, B. M. S. 2008, ApJS, 179, 484Hubbard, W. B. 1977, Icarus, 30, 305Hubeny, I., & Burrows, A. 2007, ApJ, 669, 1248Hubeny, I., Burrows, A., & Sudarsky, D. 2003, ApJ, 594, 1011Ingersoll, A. P. 1969, J. Atmos. Sci., 26, 1191Iro, N., B´ezard, B., & Guillot, T. 2005, A&A, 436, 719Knutson, H. A., Howard, A. W., & Isaacson, H. 2010, ArXive-printsKombayashi, M. 1967, J. Meteor. Soc. Japan., 45, 137Li, J., & Goodman, J. 2010, ArXiv e-printsMiller, N., Fortney, J. J., & Jackson, B. 2009, ApJ, 702, 1413Perna, R., Menou, K., & Rauscher, E. 2010, ArXiv e-printsRauscher, E., & Menou, K. 2010, ApJ, 714, 1334Richardson, L. J., Deming, D., Horning, K., Seager, S., &Harrington, J. 2007, Nature, 445, 892Schubert, G., Covey, C., del Genio, A., Elson, L. S., Keating, G.,Seiff, A., Young, R. E., Apt, J., Counselman, C. C., Kliore,A. J., Limaye, S. S., Revercomb, H. E., Sromovsky, L. A.,Suomi, V. E., Taylor, F., Woo, R., & von Zahn, U. 1980,J. Geophys. Res., 85, 8007Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916Showman, A. P., Cho, J., & Menou, K. 2009, ArXiv e-printsShowman, A. P., Fortney, J. J., Lian, Y., Marley, M. S.,Freedman, R. S., Knutson, H. A., & Charbonneau, D. 2009,The Astrophysical Journal, 699, 564Spiegel, D. S., Silverio, K., & Burrows, A. 2009, ApJ, 699, 1487Zahnle, K., Marley, M. S., Freedman, R. S., Lodders, K., &Fortney, J. J. 2009, ApJ, 701, L20
APPENDIX A. ZERO FLUX SOLUTIONS
In addition to the solution technique described in § F = 0. As we show in § F c →
0, and the flux remains zero in the absence of dissipation. Thisalternate technique allows us to check our results. It allows integration from the top down, compared to the bottomup integration from the RCB (which recedes to infinite pressure for the zero flux solution). We do not need to specifythe parameter ψ c (eq. [26]), which diverges for these zero flux solutions. With the reduced model we can analyticallyexplore solution properties. We will do this below to explain why we cannot obtain solutions with ζ . − . § F = 0 we can rearrange equation (23) as dTdP = ∇ ad TP (cid:18) ∇ ad k rad TρgK zz P (cid:19) − . (A1)We can integrate from P = 0 with the boundary condition T (0) = T deep . The solutions approach an adiabatic profile T ∝ P ∇ ad at large pressure (shown below). One could find solutions by integrating with various K zz values, iteratinguntil you land on a desired adiabat. Instead we again take a self-similar approach, non-dimensionalizing the pressureusing K zz and the opacity law. We choose a physical scale for the pressure by matching the self-similar solution ontoa chosen adiabat. This fixes the value of K zz (at a reference pressure if K zz is not uniform). The special value of K zz that gives a zero flux solution for a given adiabat is the limiting K zz, crit discussed in § K zz is constant. Ifthe final term in parenthesis vanishes we have ∇ → ∇ ad . We can show that an adiabatic solution is consistent byassuming T ∝ P ∇ ad and then confirming that the final term ∇ ad k rad TρgK zz P ∝ T − β P α ∝ P (2+ α )( ∇ ad / ∇ ′∞ − ≪ P → ∞ . With ∇ ′∞ ≡ (2 + α ) / (5 − β ) we can show that the exponent in the last proportionality of equation (A2) isindeed negative. This is because ∇ ′∞ > ∇ ∞ (which is generally true, for our opacity ∇ ′∞ = 3 / > /
2) and becausereasonable opacities have α > − ∇ ∞ > ∇ ad (see § K zz .Now consider a spatially varying K zz ∝ P ζ . In this case the assumption of an adiabat at depth gives ∇∇ ad →
11 + c ζ P e ζ (A3)as P → ∞ , where c ζ is constant and e ζ = ∇ ad (5 − β ) − (2 + α + ζ ). Consistency (i.e. ∇ → ∇ ad ) requires that e ζ < ζ > − (2 + α ) + ∇ ad (5 − β ) = − / ≈ − .
57. More negative values of ζ do not approach an adiabat even at infinitepressure.This explains the behavior of ζ . − . § ζ ). For the time being, we thus consider the irradiationefficiency constraint in equation (33) as our best limit on turbulent mixing in the upper atmosphere. B. MAXIMUM DISSIPATION AT RCB
This appendix derives how dissipation at the RCB restricts the RCB depth P c . We use the consistency requirementthat d ∇ /dP ≥ ∇∇ ad = F + F iso ∇ ad k rad T /P + F iso (B1)and at the RCB (denoted by the ‘c’ subscript) we have F = F c = ∇ ad ( k rad T /P ) c . The gradient of ∇ at the RCB is ddP (cid:18) ∇∇ ad (cid:19) c = 1 F c + F iso , c ddP (cid:20) F − ∇ ad k rad TP (cid:21) c (B2)where the derivatives of F iso cancel. Thus the limit on dissipation at the RCB, using equation (18) and requiring( d ∇ /dP ) c ≥
0, is ǫ c ≤ (1 + α ) (cid:18) − ∇ ad ∇ ∞ (cid:19) F c gP c = 6 F c g P c . (B3)Since F c /P c ∝ P − / , the maximum depth of the RCB declines with increasing dissipation at the RCB.This result is very useful in finding K zz, crit values with dissipation. Without dissipation we could easily find K zz, crit by increasing ψ c (eq. [26]) to arbitrarily large values (which results in Fig. 3), or by using the zero flux solutions ofappendix A. Instead we add a twist to the self similar technique of § f ǫ . We have thus coupled dissipation to K zz mathematically. (It doesn’tmatter if they are unrelated physically, since f ǫ can be adjusted to give any desired level of dissipation.) Then we cansolve equations (26), (29), (30) and (B3) for ψ c = 6 T deep f ǫ ∇ ad T c . (B4)Since T c is not known until we obtain a solution, we use an iteration procedure: guess a value for T c (but start withsomething too low), use the estimate of ψ c from equation (B4) to integrate the model equations (23) and (24), use theresulting T c to refine ψ c and repeat. Though a bit convoluted, this procedure converges.Finally note that our constraint on dissipation at the RCB does not ensure stability at all P < P cc