Idealised simulations of the deep atmosphere of hot jupiters: Deep, hot, adiabats as a robust solution to the radius inflation problem
F. Sainsbury-Martinez, P. Wang, S. Fromang, P. Tremblin, T. Dubos, Y. Meurdesoif, A. Spiga, J. Leconte, I. Baraffe, G. Chabrier, N. Mayne, B. Drummond, F. Debras
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2019November 18, 2019
Idealised simulations of the deep atmosphere of hot jupiters: deep, hot, adiabats as a robust solution to the radius inflation problem
F. Sainsbury-Martinez (cid:63) , P. Wang , , S. Fromang , P. Tremblin , T. Dubos , Y. Meurdesoif , A. Spiga , , J. Leconte ,I. Bara ff e , , G. Chabrier , , N. Mayne , B. Drummond , and F. Debras Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ, Université Paris-Saclay Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France Ecole Normale Superieure de Lyon, CRAL, UMR CNRS 5574 Laboratoire AIM, CEA / DSM-CNRS-Université Paris 7, Irfu / Departement d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette,France Laboratoire de Météorologie Dynamique (LMD / IPSL), Sorbonne Université, Centre National de la Recherche Scientifique, ÉcolePolytechnique, École Normale Supérieure, Paris Laboratoire des Sciences du Climat et de l’Environnement / Institut Pierre-Simon Laplace, Université Paris-Saclay, CEA Paris-Saclay Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geo ff roy Saint-Hilaire, 33615 Pessac, France. Astrophysics Group, University of Exeter, Exeter, Devon IRAP, Université de Toulouse, CNRS, UPS, Toulouse, FranceReceived 02 August 2019; accepted 14 Nov 2019
ABSTRACT
Context.
The anomalously large radii of hot Jupiters has long been a mystery. However, by combining both theoretical arguments and2D models, a recent study has suggested that the vertical advection of potential temperature leads to an adiabatic temperature profilein the deep atmosphere hotter than the profile obtained with standard 1D models.
Aims.
In order to confirm the viability of that scenario, we extend this investigation to three dimensional, time-dependent, models.
Methods.
We use a 3D General Circulation Model (GCM), DYNAMICO to perform a series of calculations designed to explore theformation and structure of the driving atmospheric circulations, and detail how it responds to changes in both the upper and deepatmospheric forcing.
Results.
In agreement with the previous, 2D, study, we find that a hot adiabat is the natural outcome of the long-term evolution ofthe deep atmosphere. Integration times of order 1500 years are needed for that adiabat to emerge from an isothermal atmosphere,explaining why it has not been found in previous hot Jupiter studies. Models initialised from a hotter deep atmosphere tend to evolvefaster toward the same final state. We also find that the deep adiabat is stable against low-levels of deep heating and cooling, as longas the Newtonian cooling time-scale is longer than ∼ Conclusions.
We conclude that the steady-state vertical advection of potential temperature by deep atmospheric circulations consti-tutes a robust mechanism to explain hot Jupiter inflated radii. We suggest that future studies of hot Jupiters are evolved for a longertime than currently done, and, when possible, include models initialised with a hot deep adiabat. We stress that this mechanism stemsfrom the advection of entropy by irradiation induced mass flows and does not require (finely tuned) dissipative process, in contrastwith most previously suggested scenarios.
Key words.
Planets and satellites: interiors - Planets and satellites: atmospheres - Planets and satellites: fundamental parameters -Planets: HD209458b - Hydrodynamics
1. Introduction
The anomalously large radii of highly irradiated Jupiter-like ex-oplanets, known as hot Jupiters, remains one of the key unre-solved issues in our understanding of extrasolar planetary atmo-spheres. The observed correlation between the stellar irradiationof a hot Jupiter and its observed inflation (for examples, seeDemory & Seager 2011; Laughlin et al. 2011; Lopez & Fort-ney 2016; Sestovic et al. 2018) suggests that it is linked to theamount of energy deposited in the upper atmosphere. Severalmechanisms have been suggested as possible explanations (seeBara ff e et al. 2009; Bara ff e et al. 2014; Fortney & Nettelmann2010, for a review). These solutions include tidal heating and (cid:63) e-mail: [email protected] physical (i.e. not for stabilisation reasons) dissipation (Leconteet al. 2010; Arras & Socrates 2010; Lee 2019), ohmic dissipa-tion of electrical energy (Batygin & Stevenson 2010; Perna et al.2010; Batygin et al. 2011; Rauscher & Menou 2012), deep de-position of kinetic energy (Guillot & Showman 2002), enhancedopacities which inhibit cooling (Burrows et al. 2007) or ongoinglayered convection that reduces the e ffi ciency of heat transport(Chabrier & Bara ff e 2007). At present time, however, there isno consensus across the community on a given scenario becausethe majority of these solutions require finely tuned physical en-vironments which either deposit additional energy deep withinthe atmosphere or a ff ect the e ffi ciency of vertical heat transport.Recently, Tremblin et al. (2017), hereafter PT17, suggested amechanism that naturally arises from first physical principles. Article number, page 1 of 13 a r X i v : . [ a s t r o - ph . E P ] N ov & A proofs: manuscript no. main
Their argument goes as follows: consider the equation for theevolution of the potential temperature Θ , which is equivalent toentropy in this case: D Θ Dt = Θ HT c p , (1)where D / Dt stands for the Lagrangian derivative in sphericalcoordinates, H is the local heating or cooling rate, c p is the heatcapacity at constant pressure, and Θ is defined as a function ofthe temperature T and pressure, P : Θ = T (cid:18) P P (cid:19) γ − γ , (2)where P is a reference pressure and γ = C p / C v is the adiabaticindex. In a steady state, Equation 1 reduces to u · ∇ Θ = Θ
HT c p , (3)where u is the velocity. In the deep atmosphere, radiative heatingand cooling both tend to zero (i.e. H →
0) because of large at-mospheric opacities. In this case (with H → | u | (cid:44)
0, see section 3.2), the potential tem-perature Θ must remain constant for Equation 3 to be valid. Inother words, the temperature-pressure profile must be adiabaticand satisfy the scaling: P ∝ T γγ − . (4)We emphasise that this adiabatic solution is an equilibrium thatdoes not require any physical dissipation. There is an internal en-ergy transfer to the deep atmosphere, through an enthalpy flux,but there is no dissipation from kinetic, magnetic, or radiative en-ergy reservoirs to the internal energy reservoir. Dissipative pro-cesses D dis would act as a source term with u · ∇ Θ ∝ D dis andwould drive the profile away from the adiabat.Physically, as discussed by PT17, this constant potential tem-perature profile in the deep atmosphere is driven by the verticaladvection of potential temperature from the outer and highly ir-radiated atmosphere to the deep atmosphere by large scale dy-namical motions where it is almost completely homogenisedby the residual global circulations (which themselves can belinked to the conservation of mass and momentum, and the largemass / momentum flux the super-rotating jet drives in the outeratmosphere). The key point is that it causes the temperature-pressure profile to converge to an adiabat at lower pressures thanthose at which the atmosphere becomes unstable to convection.As a result, the outer atmosphere connects to a hotter internaladiabat than would be obtained through a standard, ’radiative-convective’ single column model. This potentially leads to alarger radius compared with the predictions born out of these1D models.Whilst PT17 was able to confirm this hypothesis through the useof a 2D stationary circulation model, there are still a numberof limitations to their work. Maybe most importantly, the mod-els they used only considered the formation of the deep adia-bat within a 2D equatorial slice. The steady-state temperature-pressure profiles at other latitudes remains unknown, as well asthe nature of the global circulations at these high pressures inthe equilibrated state. Strong ansatzes were also made about thenature of the meridional (i.e. vertical and latitudinal) wind at theequator, with their models prescribing the ratio of latitudinal tovertical mass fluxes, that could potentially a ff ect the proposed scenario. The purpose of this paper is to reduce and constrainthese assumptions and limitations and to demonstrate the viabil-ity of a deep adiabat at equilibrium. This is done by means ofa series of idealised 3D GCM calculations designed such as toallow us to fully explore the structure of the deep atmosphericcirculations in equilibrated hot Jupiter atmospheres, as well asinvestigate the time-evolution of the deep adiabat. As we demon-strate in this work, the adiabatic profile predicted by PT17 nat-urally emerges from such calculations and appears to be robustagainst changes in the deep atmosphere radiative properties. Thisis the core result of this work.The structure of the work is as follows. Our simulations proper-ties are described in section 2, where we introduce the GCM DY-NAMICO, used throughout this study. We then demonstrate that,when using DYNAMICO, not only are we are able to recoverstandard features observed in previous short-timescale studiesof hot Jupiter atmospheres (section 3.1), but also that, when thesimulations are extended to long-enough time-scales, an adia-batic profile develops within the deep atmosphere (section 3.2).We then explore the robustness of our results by presenting a se-ries of sensitivity tests, including changes in the outer and deepatmosphere thermal forcing (section 3.3). Finally, in section 4,we provide concluding remarks, including suggestions for futurecomputational studies of hot Jupiter atmospheres and a discus-sion about implications for the evolution of highly irradiated gasgiants.
2. Method
DYNAMICO is a highly computationally e ffi cient GCM thatsolves the primitive equation of meteorology (see Vallis 2006for a review and Dubos & Voitus 2014 for a more detailed dis-cussion of the approach taken in DYNAMICO) on a sphere (Du-bos et al. 2015). It is being developed as the next state–of–theart dynamical core for Earth and planetary climate studies at theLaboratoire de Météorologie Dynamique and is publicly avail-able . It has recently been used to model the atmosphere of Sat-urn at high resolution (Spiga et al. 2020). Here, we present somespecificities of DYNAMICO (section 2.1) as well as the requiredmodifications we implemented to model hot Jupiter atmospheres(section 2.2). Briefly, DYNAMICO takes an energy-conserving Hamiltonianapproach to solving the primitive equations. This has beenshown to be suitable for modelling hot Jupiter atmospheres(Showman et al. 2008; Rauscher & Menou 2012), although thismay not be valid in other planetary atmospheres (Mayne et al.2019). Rather than the traditional latitude-longitude horizon-tal grid (which presents numerical issues near the poles dueto singularities in the coordinate system - see the review ofWilliamson 2007 for more details), DYNAMICO uses a stag-gered horizontal-icosahedral grid (see Thuburn et al. 2014 fora discussion of the relative numerical accuracy for this type ofgrids) for which the number of horizontal cells N is defined bythe number of subdivisions d of each edge of the main spherical DYNAMICO is available at http: // forge.ipsl.jussieu.fr / dynamico / wiki,and our hot Jupiter patch available at https: // gitlab.erc-atmo.eu / erc-atmo / dynamico_hj.Article number, page 2 of 13. Sainsbury-Martinez et al.: Idealised simulations of the deep atmosphere of hot jupiters: Quantity (units) Description Valuedt (seconds) Time-step 120 N z Number of Pressure Levels 33 d Number of Sub-divisions 20 N ( ◦ ) Angular Resolution 3 . P top (bar) Pressure at Top 7 × − P bottom (bar) Pressure at Bottom 200 g (m.s −
2) Gravity 8.0 R HJ (m) HJ Radius 10 Ω (s − ) HJ Angular Rotation Rate 2 . × − c p (J.kg − .K − ) Specific Heat 13226.5 R (J.kg − .K − ) Ideal Gas Constant 3779.0 T init ( K ) Initial Temperature 1800Table 1: Parameters for Low Resolution Simulationsicosahedral : N = d + . (5)As for the vertical grid, DYNAMICO uses a pressure coordi-nate system whose levels can be defined by the user at runtime.Finally, the boundaries of our simulations are closed and stress-free with zero energy transfer (i.e. the only means on energyinjection and removal are the Newtonian cooling profile and thehorizontal, numerical, dissipation). Note that, unlike some otherGCM models of hot Jupiters (e.g. Schneider & Liu 2009; Liu& Showman 2013; Showman et al. 2019), we do not include anadditional frictional (i.e. Rayleigh) drag scheme at the bottomof our simulation domain, instead relying on the hyperviscosityand impermeable bottom boundary to stabilise the system.As a consequence of the finite di ff erence scheme used in DY-NAMICO, artificial numerical dissipation must be introducedin order to stabilise the system against the accumulation ofgrid-scale numerical noise. This numerical dissipation takes theform of a horizontal hyper-di ff usion filter with a fixed hyper-viscosity and a dissipation time-scale at the grid scale, labelled τ dissip , which serves to adjust the strength of the filtering (thelonger the dissipation time, the weaker the dissipation). Tech-nically DYNAMICO includes three dissipation timescales, eachof which either di ff uses scalar, vorticity, or divergence indepen-dently. However, for our models, we set all three timescales tothe same value. It is important to point out that the hypervis-cosity is not a direct equivalent of the physical viscosity of theplanetary atmosphere, but can be viewed as a form of increasedartificial dissipation that both enhances the stability of the code,and accounts for motions, flows, and turbulences which are un-resolved at typical grid scale resolutions. This is known as thelarge eddy approximation and has long been standard practice inthe stellar (e.g. Miesch 2005) and planetary (e.g Cullen & Brown2009) atmospheric modelling communities. Because it acts at the Specifically, to generate the grid we start with a sphere that consistsof 20 spherical triangles (sharing 12 vertex, i.e. grid, points) then, wesubdivide each side of each triangle d times, using the new points togenerate a new grid of spherical triangles with N total vertices. Thesevertices then from the icosahedral grid. grid cell level, the strength of the dissipation is resolution depen-dent at a fixed τ dissip (this can be seen in our results in Figure 7).In a series of benchmark cases, Heng et al. (2011) (hereafterH11) have shown that both spectral and finite-di ff erence baseddynamical cores which implement horizontal hyper-di ff usion fil-ters can produce di ff erences of the order of tens of percent inthe temperature and velocity fields when varying the dissipationstrength. We also found such a similar sensitivity in our mod-els: for example, the maximum super-rotating jet speed variesbetween 3000 ms − and 4500 ms − as the dissipation strengthis varied. The dissipation strength must thus be carefully cali-brated. In the absence of significant constraints on hot Jupiterzonal wind velocities, this was done empirically by minimis-ing unwanted small-scale numerical noise as well as replicatingpublished benchmark results (An alternative, which is especiallyuseful in scenarios where direct or indirect data comparisons areunavailable, is to plot the spectral decomposition of the energyprofile and adjust the di ff usion such that the energy accumula-tion on the smallest scales is insignificant). We found that setting τ dissip = ff usion of the same order ofmagnitude as, for example, H11. Note that, due to di ff erences inthe dynamics between those of Saturn and that observed in hotJupiters, and in particular due to the presence of the strong super-rotating jet, we must use a significantly stronger dissipation tocounter grid-scale noise than that used in previous atmosphericstudies calculated using DYNAMICO (Spiga et al. 2020). In our simulations of hot Jupiter atmospheres using DYNAM-ICO, we do not directly model either the incident thermal radia-tion on the day-side, or the thermal emission on the night-side, ofthe exoplanet. This would be prohibitively computationally ex-pensive for the long simulations we perform in the present work.Instead we use a simple thermal relaxation scheme to modelthose e ff ects, with a spatially varying equilibrium temperatureprofile T eq and a relaxation time-scale τ that increases with pres-sure throughout the outer atmosphere. Specifically, this is doneby adding a source term to the temperature evolution equationthat takes the form: ∂ T ( P , θ, φ ) ∂ t = − T ( P , θ, φ ) − T eq ( P , θ, φ ) τ ( P ) . (6)This method, known as Newtonian Cooling has long been ap-plied within the 3D GCM exoplanetary community (i.e. Show-man & Guillot (2002), Showman et al. (2008), Rauscher &Menou 2010, Showman & Polvani (2011), Mayne et al. 2014a,Guerlet et al. 2014 or Mayne et al. 2014b), although it is gradu-ally being replaced by coupling with simplified, but more com-putationally expensive, radiative transfer schemes (e.g. Show-man et al. 2009, Rauscher & Menou 2012 or Amundsen et al.2016) due to its limitations (e.g. it is di ffi cult to use to probe indi-vidual emission or absorption features, such as non-equilibriumatmospheric chemistry or stellar activity).The forcing temperature and cooling time-scale we use withinour models have their basis in the profiles Iro et al. (2005) calcu-lated via a series of 1D radiative transfer models. These modelswere then parametrised by H11, who created simplified day-sideand night-side profiles. The parametrisation used here is basedupon this work, albeit modified in the deep atmosphere since this Article number, page 3 of 13 & A proofs: manuscript no. main is the focus of our analysis. As a result, it somewhat resembles aparametrised version of the cooling profile considered by Liu &Showman (2013).Specifically, T eq is calculated from the pressure dependent night-side profile ( T night ( P )) according to the following relation: T eq ( P , θ, φ ) = T night ( P ) + ∆ T ( P ) cos ( θ ) max (cid:2) , cos( φ − π ) (cid:3) , (7)where ∆ T is the pressure dependent day-side / night-side temper-ature di ff erence, ∆ T ( P ) = ∆ T if P < P low ∆ T log( P / P low ) if P low < P < P high P > P high , (8)in which we used ∆ T =
600 K, P low = .
01 bar and P high = T night is parametrised as aseries of linear interpolations in log( P ) space between the points (cid:18) T , P (cid:19) = (800 , − ), (1100 ,
1) & (1800 , . (9)For P >
10 bar, we set T eq = T night = T day = τ is linearly interpo-lated, in log( P ) space, between the points (cid:18) log (cid:18) τ (cid:19) , P (cid:19) = (2 . , − ), (5 , . ,
10) & (log( τ ) , . (10)For P >
10 bar, we consider a series of models that lie betweentwo extremes: at one extreme we set log( τ ) (which we defineas the decimal logarithm of the cooling time-scale τ at the bot-tom of our model atmospheres: i.e. at P =
220 bar) to infinity,which implies that the deep atmosphere is radiatively inert, withno heating or cooling. As for the other extreme, this involves set-ting log( τ ) = .
5, which implies that radiative e ff ects do notdiminish below 10 bar. In section 3 we explore results at the firstextreme, with no deep radiative dynamics. Then, in section 3.3.3,we explore the sensitivity of our results to varying this prescrip-tion.
3. Results
The default parameters used with our models are outlined in Ta-ble 1, with the resultant models, as well as the simulation specificparameters, detailed in Table 2.In section 3.2, we use the results of models A and B to demon-strate the validity of the work of PT17 in the time-dependent,three-dimensional, regime. We next explore the robustness andsensitivity of our results to numerical and external e ff ects in sec-tion 3.3. Note that, throughout this paper, all times are eithergiven in seconds or in Earth years - specifically one Earth yearis exactly 365 days. We start by exploring the early evolution of model A , testinghow well it agrees with the benchmark calculations of H11. Themodel is run for an initial period of 30 years in order to reachan evolved state before we take averages over the next five yearsof data. Note that this model was also used to calibrate the hor-izontal dissipation ( τ dissip ). In Figure 1, we show zonally andtemporally-averaged plots of the zonal wind and the temperatureas a function of both latitude and pressure. Model Description A The base low resolution model, in which thedeep atmosphere is isothermally initialised B Like model A , but with the deep atmosphereadiabatically initialised C Mid Resolution version of model A ( d = D High Resolution version of model A ( d = E → I Highly evolved versions of model A , whichhave reached a deep adiabat and then had deepisothermal Newtonian cooling introduced atvarious strengths: For E log( τ ) = . F log( τ ) = G log( τ ) = H log( τ ) =
20, and I log( τ ) = . J & K Highly evolved versions of model A whichhave reached a deep adiabat, and then had theirouter atmospheric Newtonian cooling modi-fied to reflect a di ff erent surface temperature:1200K in model J and 2200K in model K Table 2: Models discussed in this workWe find that the temperature (left panel) is qualitatively simi-lar to that reported by both H11 and Mayne et al. (2014b). Thetemperature range we find ( ∼ →∼ ∼ → ff erences between the various set-ups and numerical implementations of the GCMs, as well as thevariations that occur when adjusting the length of the temporalaveraging window.The zonal wind displays a prominent, eastward, super-rotatingequatorial jet that extends from the top of the atmosphere downto approximately 10 bar (Note that, as we continue to run thismodel for more time, the vertical extent of the jet increases,eventually reaching significantly deeper that 100 bar after 1700years). It exhibits a peak wind velocity of ≈ − , depend-ing upon the averaging window considered, in good agreementwith the work of both H11 and Mayne et al. (2014b) who foundpeak jet speeds on the order of 3500 → − . In the upperatmosphere, it is balanced by counter-rotating (westward) flowsat extratropical and polar latitudes. The zonal wind is also di-rected westwards at all latitudes below ∼
50 bar, with this windalso contributing to the flows balancing the large mass and mo-mentum transport of the super-rotating jet.The di ff erences we find between our models and the referencemodels are not unexpected. As discussed by H11, the jet speedand temperature profile are indeed highly sensitive not only tothe numerical scheme adopted by the GCM (i.e. spectral vs finitedi ff erence - see their Figure 12) but also to the form and magni-tude of horizontal dissipation and Newtonian cooling used. Inour models, unlike H11, we explicitly set our deep ( P > Article number, page 4 of 13. Sainsbury-Martinez et al.: Idealised simulations of the deep atmosphere of hot jupiters:(a) Temperature Contours (b) Zonal Wind Contours
Fig. 1: Two plots designed to aid the comparison of the early evolution of model A with prior studies (e.g. Heng et al. (2011) andMayne et al. 2014b) in-order to benchmark DYNAMICO in the hot Jupiter regime. The figures show the zonally and temporallyaveraged temperature (a) and zonal wind (b) profiles, both of which are commonly used to benchmark hot Jupiter models.long time-scale evolution is explored in detail in the followingsection. As discussed by PT17, and in section 1, an adiabatic profile inthe deep atmosphere (i.e. P > ∼ →∼
10 bar) should be a goodrepresentation of the steady state atmosphere. In order to con-firm that this is the case, we performed a series of calculationswith a radiatively inert deep atmosphere (i.e. no deep heating orcooling, as required by the theory of PT17).We explore this using two models, A and B , which only di ff er inboth their initial condition and their duration. In model A , the at-mosphere, including the deep atmosphere, is initially isothermalwith T = T – P profile (as shown inFigure 2a). As a consequence of the long time-scales requiredfor the model to reach equilibrium, and the computational costof such an endeavour, model A (and B ) is run at a relatively lowresolution . We will investigate the sensitivity of our results tospatial resolution in section 3.3.1. As for model B , it is identi-cal to model A except in the deep atmosphere, where it is ini-tialised with an adiabatic T – P profile for P >
10 bar. As a resultof this model being initialised close to the expected equilibriumsolution, model B was then run for only 100 years in order toconfirm the stability of the steady-state. In both cases, we findthat the simulation time considered is long enough such that thethermodynamic structure of the atmosphere has not changed formultiple advective turnover times t adv ∼ π R HJ / u φ .Figure 2 shows that both models have evolved to the same steadystate: an outer atmosphere whose T − P profile is dictated by theNewtonian cooling profile, and a deep adiabat which is slightlyhotter ( ∼ P =
10 bar (1800K).This is reinforced by the latitudinal and longitudinal temperatureprofile throughout the simulation domain. In Figure 4 we plot However this does not mean that our models have problems con-serving angular momentum, they maintain 97 .
44% of the initial angularmomentum after over 1700 years of simulation time (which compareswell to other GCMs: Polichtchouk et al. 2014). the zonal wind and temperature profile at three di ff erent heights(pressures). Here we can see that, in the outer atmosphere (pan-els a and b ) the profile is dominated by the newtonian cooling,with horizontal advection (and the resulting o ff set hotspot) start-ing to become significant as we move towards middle pressures.As for the deep atmosphere (snapshot in panel c and time aver-age in panel d ), here we start to see evidence of both the heatingand near-homogenisation of the deep atmosphere. Note that werefer to the atmosphere as nearly homogenised because the tem-perature fluctuations at, for example, P =
10 bar are less than1% of the mean temperature.Importantly, this convergence to as deep adiabat not only occursin the absence of vertical convective mixing (an e ff ect whichis absent from our models, which contain no convective driv-ing), but also at a significantly lower pressure ( P =
10 bar) thanthe pressure ( ∼
40 bar for HD209458b - Chabrier et al. 2004) atwhich we would expect the atmosphere to become unstable toconvection (and so, in the traditional sense, prone to an adiabaticprofile).Therefore, the characteristic entropy profile of the planet iswarmer than the entropy profiles calculated from standard 1Dirradiated models. We will discuss the implications of this resultfor the evolution of highly irradiated gas giant in section 4.In model A , the steady state described above is very slow toemerge from an initially isothermal atmosphere. This is illus-trated in Figure 3 which shows the time evolution of the T - P profile. It takes more than 500 years of simulation time to stopexhibiting a temperature inversion in the deep atmosphere, letalone the > B .As will be further discussed in section 4, this slow evolutionof the deep adiabat is probably one of the main reason whythis result has not been reported by prior studies of hot Jupiteratmospheres.The mechanism advocated by PT17 relies on the existenceof vertical and latitudinal motions that e ffi ciently redistributepotential temperature. In order to determine their spatial struc-ture, we plot in Figure 5 the zonally and temporally averaged Article number, page 5 of 13 & A proofs: manuscript no. main − − − P r e ss u r e ( b a r) T model( A ) T forced T init (a) Isothermal Initialisation − − − P r e ss u r e ( b a r) T model( B ) T forced T init (b) Adiabatic Initialisation Fig. 2: Equatorially averaged (i.e. the zonal-mean at the equator) T – P profiles, in orange, for two evolved models that were eitherisothermally (a) or adiabatically (b) initialised. In both cases there is no forcing below 10 bars (i.e. when P >
10 bar), and theforcing above this point is plotted in dark grey. In both cases, the models have been run long enough such that their T – P profileshave fully evolved from their initial states, either isothermal (a) or adiabatic (b) for P >
10 bar, as shown by the light grey dashedline, to the same steady state, a deep adiabat that corresponds to T surface = ∼ ∼ − − − P r e ss u r e ( b a r) T model ( A ) T forced T init Fig. 3: Time evolution of the equatorially averaged T – P profile within model A covering the > P >
10 bar, whilstthe dark grey line shows the forcing profile for P <
10 bar. The time evolution is represented by the intensity of the lines, withthe least evolved (and thus lowest visual intensity) snapshot starting at t ≈
30 years followed by later snapshots at increments ofapproximately 60 yearsmeridional mass-flux stream function and zonal wind velocityfor model A .Starting with the zonal wind profile (grey lines) we can see ev-idence for a super-rotating jet that extends deep into the atmo-sphere, with balancing counter flows at the poles and near thebottom of the simulation domain. In the deep atmosphere, thisjet has evolved with the deep adiabat, extending towards higherpressures as the developing adiabat (almost) homogenises (andhence barotropises) the atmosphere. This barotropisation on longtimescales seems similar to the drag-free simulation started froma barotropic zonal wind in Liu & Showman (2013) The meridional mass-flux stream function is defined accord-ing to Ψ = π R HJ g cos θ (cid:90) PP top u φ dP . (11)We find that the meridional (latitudinal and vertical) circu-lation profile is dominated by four vertically aligned cellsextending from the bottom of our simulation atmospheres towell within the thermally and radiatively active region locatedin the upper atmosphere. These circulation cells lead to the Article number, page 6 of 13. Sainsbury-Martinez et al.: Idealised simulations of the deep atmosphere of hot jupiters:(a) Snapshot of the zonal wind and temperature profile at P = P =
455 mbar(c) Snapshot of the zonal wind and temperature profile at P = P = Fig. 4: Zonal wind (arrows) and temperature profile (map) at three di ff erent pressures within a fully evolved (i.e. steady-state deepadiabat) model A . The first three panels ( a , b , and, c ) show snapshots of the profiles, whilst the last ( d ) shows the time-averagedprofiles at the same pressure as panel ( c ) in order to illustrate the variable nature of the deep atmosphere. Note that the period of thetime average was approximately seven and a half years.formation of a strong, deep, down-flow at the equator (whichcan be linked to the high equatorial temperatures in the upperatmosphere), weaker, upper atmosphere, downflows near thepoles, and a mass conserving pair of upflows at mid latitudes( θ = ◦ → ◦ ). The meridional circulation not only leads tothe vertical transport of potential temperature (as high potentialtemperature fluid parcels from the outer atmosphere are mixedwith their ‘cooler’ deep atmosphere counterparts), but also tothe almost complete latitudinal homogenisation of the deepatmosphere (with only small temperature variations remaining).In a fully radiative model, these circulations would also mixthe outer atmosphere, leading to the equilibrium temperatureprofiles we instead impose via Newtonian cooling (see, forexample, Drummond et al. 2018a,b for more details about the3D mixing in radiative atmospheres). Note that the vertical extent the zonal wind, and the structureof the lowest cells in the mass-flux stream function, appearto be a ff ected by the bottom boundary, suggesting that theyextend deeper into the atmosphere. Whilst this is interesting andimportant, it should not a ff ect the final state our P-T profilesreach, but does suggest that models of hot-Jupiters should berun to higher pressures to fully capture the irradiation drivendeep flow dynamics.The primary driver of the latitudinal homogenisation arefluctuations in the meridional circulation profile, which are vis-ible within individual profile snapshots, but are averaged outwhen we take a temporal average. This includes contributionsfrom spatially small-scale velocity fluctuations at the interface ofthe large-scale meridional cells. Evidence for these e ff ects can be Article number, page 7 of 13 & A proofs: manuscript no. main
Fig. 5: Zonally and temporally-averaged (over a period of ≈ A . Clockwise circulations onthe meridional plane are shown in red and anticlockwise circula-tions are shown in blue. Additionally the zonally and temporallyaveraged zonal wind is plotted in black (solid = eastward, dashed = westward).seen in snapshots of the zonal and meridional flows, in an RMSanalysis of the zonal velocity, and of course in deep temperatureprofile that these advective motions drive. The first reveals com-plex dynamics, such as zonally-asymmetric and temporally vari-able flows, that are hidden when looking at the temporal average,but which mask the net flows when looking at a snapshot of thecirculation. The second reveals spatial and temporal fluctuationson the order of 5 →
10m s − in the deep atmosphere. Finally thethird (as plotted in panels c and d of Figure 4, which show snap-shots or the time average of the zonal wind and temperature pro-file, respectively) reveals small scale temperature and wind fluc-tuations, which are likely associated with the deep atmospheremixing, that are lost when looking at the average, steady, state.However, a more detailed analysis of the dynamics of this ho-mogenisation, as well as the exact nature of the driving flowsand dynamics, is beyond the scope of this paper. Although inter-esting in its own right, the mechanism by which the circulation isset up in the deep atmosphere of our isothermally initialised sim-ulations might not be relevant to the actual physical mechanismhappening in hot Jupiters with hot, deep, atmospheres. As a con-sequence of both the meridional circulations described above,and the zonal flows that form as a response to the strong day-side / night-side temperature di ff erential, the deep atmosphere T – P profile is independent of both longitude (Figure 6a) and lat-itude (Figure 6b). Only in the upper atmosphere ( P <
10 bar)do the temperature profiles start to deviate from one another, re-flecting the zonally and latitudinally varying Newtonian forcing.Taken together, the two panels of Figure 6 confirm that the lat-itudinal and vertical steady-state circulation, the super-rotatingeastward jet, and any zonally-asymmetric flows act to advect po-tential temperature throughout the deep atmosphere, leading atdepth to the formation of a hot adiabat without the need for anyconvective motions.
Having confirmed that a deep adiabatic temperature profile con-necting with the outer atmospheric temperature profile at P =
10 bar is a good representation of the steady state within our hot Jupiter model atmospheres, we now explore the robustness ofthis result.
We start our exploration of the robustness of our results by con-firming that the eventual convergence of the deep atmosphere onto a deep adiabat appears resolution independent.Figure 7 shows the T – P profiles obtained for three models at thesame time ( t ≈ ff erent resolutions (our‘base’ resolution model, A , a ‘mid-res’ model, C , and a ‘high-res’ model, D ). The mid resolution model ( C ) has almost reachedthe exact same equilibrium adiabatic profile as the low resolu-tion case ( A ): comparing this with the time-evolution of model A (Figure 3) confirms that they are both on the path to the sameequilibrium state, and that a significant amount of computationaltime would be required to reach it. This becomes even clearerwhen we look at a high resolution model ( D ). Here we find that,despite the long time-scale of the computation, the deep atmo-sphere still exhibits a temperature inversion, suggesting, in com-parison to Figure 3, that the model has a long way to go until itreaches the same, deep adiabat, equilibrium.In general, we have found the better the resolution the moreslowly the atmosphere temperature profile evolves towards theadiabatic steady state solution. This stems most likely from thefact that horizontal numerical dissipation, on a fixed dissipationtime-scale, decreases with increasing resolution. Note that wekept the horizontal dissipation timescale constant due to both thecomputational expense of the parameter study required to set thecorrect dissipation at each resolution, and the numerical dissipa-tion independence of the steady-state in the deep atmosphere.Evidence for the impact of the small-scale flows on this slowevolution can be seen in the temporal and spatial RMS profiles ofthe zonal flows, which reveal that, as we increase the resolutionby a factor 2, the magnitude of the small-scale velocity fluctua-tions decreases by roughly the same factor. These results are inagreement with the e ff ect of changing the numerical dissipationtimescale ( τ dissip ) at a fixed resolution, where longer timescalesalso slow down the circulation, thereby increasing the time re-quired to reach a steady T - P profile in the deep atmosphere (notshown). Despite these numerical limitations, it remains clear thatthe, the presence, and strength, of any numerical dissipation doesnot a ff ect the steady state solutions of the simulation, which re-mains as an adiabatic P-T profile in the deep atmosphere. function We next explore how the deep adiabat responds to changes in theouter atmosphere irradiation and thermal emission (via the im-posed Newtonian cooling). The aim is not only to test the robust-ness of the deep adiabat, but also to explore the response of theadiabat to changes in the atmospheric state. As part of this study,the two scenarios we consider were initialised using the evolvedadiabatic profile obtained in model A, but with a modified outeratmosphere cooling profile such that T night = J )or T night = K ). Figure 8 shows the equilibrium T – P profiles (solid lines) as well as snapshots of the T – P pro-files after only 200 years of ‘modified’ evolution (dashed lines).It also includes a plot of model A to aid comparison.Model J evolves in less than 200 years towards a new steadystate profile that corresponds to the modified cooling profile.The deep adiabat reconnects with the outer atmospheric profile Article number, page 8 of 13. Sainsbury-Martinez et al.: Idealised simulations of the deep atmosphere of hot jupiters: − − − P r e ss u r e ( b a r) T night T day T init (a) Longitudinal Variations − − − P r e ss u r e ( b a r) T init T forced (b) Latitudinal Variations Fig. 6: Snapshots of the latitudinally (a) or longitudinally (b) averaged T-P profile within model A at various longitudes or latitudesrespectively. In each plot, the solid lines represent the various T-P profiles considered. At low pressures ( P <
10 bar), the dashedlines represent either the days-side, night-side and equilibrium profiles (in the longitudinal plot, a), or the reference Newtoniancooling (in the latitudinal plot, b), whereas at high pressures ( P >
10 bar), the dashed lines (light grey) represent the initial state ofthe atmosphere. − − − P r e ss u r e ( b a r) Low Res ( A )Mid Res ( C )High Res ( D ) T forced T init Fig. 7: Equatorially averaged T-P profile snapshots for three ini-tially isothermal (see grey dashed line in the deep atmosphere)models run with the same dissipation time ( t dissip = ff erent horizontal resolutions (Models A (yellow), C (lightgreen), and D (orange)).at P =
10 bar and ∼ ff -set found in our 1800K models, A and B ). The meridional masscirculation (not shown) displays evidence for the same qualita-tive flows driving the vertical advection of potential temperatureas models A and B . However it also shows signs that it is stillevolving, suggesting that the steady state meridional circulationtakes longer to establish than the vertical temperature profile.In model K , we find that, 200 years after modifying the outeratmospheres cooling profile, the deep atmosphere has not yetreached a steady state. In fact it takes approximately 1000 yearsof evolution for it to reach equilibrium, which we show as a solidline in Figure 8. This confirms that, model K , although slow to − − − P r e ss u r e ( b a r) K )1800K ( A )1200K ( J ) T forced Fig. 8: Equatorially averaged T – P profiles for three models: A (green), J (yellow) and K (orange). The orange ( K ) and yellow( J ) models have had their outer atmosphere cooling modifiedsuch that T eq = T – P profiles whilst the dashed linesrepresent the T – P profiles 200 years after the outer atmospheresforcing was adjusted (shown in dark grey for each model). Notethat, after 200 years of ‘modified’ evolution, only the 2200Kmodel has not reached equilibrium.evolve relative to the cooling case (model J ), does eventuallysettle onto a deep, equilibrium, adiabat. Additionally, this ad-justment occurs significantly faster than the equivalent evolutionof a deep adiabat from an isothermal start.Based on the results of this section, we conclude that it isfaster for the deep atmosphere to cool than to warm when itevolves toward its adiabatic temperature profile. In order to un-derstand this time-scale ordering, we have to note that the onlyway for the simulation to inject or extract energy is through the Article number, page 9 of 13 & A proofs: manuscript no. main fast Newtonian forcing of the upper atmosphere and also thatthe thermal heat content of the deep atmosphere is significantlylarger than that of the outer layers. The deep ( d ) and upper ( u )atmospheres are connected by the advection of potential temper-ature that we will rewrite in a conservative form as an enthalpyflux: ρ c p T u and we simplify the process to two steps betweenthe two reservoirs (assuming they have similar volumes): injec-tion / extraction by enthalpy flux and Newtonian forcing in theupper atmosphere. – In the case of cooling, the deep atmosphere contains toomuch energy and needs to evacuate it. It will setup a circu-lation to evacuate this extra-energy to the upper layers withan enthalpy flux that would lead to an upper energy contentset by ρ u c v T u ∼ ρ u c v T u , init + ρ d c v ( T d , init − T d , eq ) if we ignorefirst Newtonian cooling. T u would then be very large essen-tially because of the density di ff erence between the upperand lower atmosphere. The Newtonian forcing term propor-tional to − ( T u − T u , eq ) /τ is then very large and can e ffi cientlyremove the energy from the system. – In the case of heating, the deep atmosphere does not containenough energy and needs an injection from the upper layers.This injection is coming from the Newtonian forcing and canat first only inject ρ u c v ( T u , eq − T u , init ) in the system. The en-thalpy flux will then lead to an energy content in the deep at-mosphere given by ρ d c v T d ∼ ρ d c v T d , init + ρ u c v ( T u , eq − T u , init )if we assume that all the extra-energy is pumped by the deepatmosphere. Because of the density di ff erence and the lim-ited variations in the temperature caused by the forcing, thetemperature change in the deep atmosphere is small and willrequire more injection from the upper layers to reach equi-librium. However, even in the most favourable scenario inwhich all the extra energy is transferred, the Newtonian forc-ing cannot exceed − ( T u , init − T u , eq ) /τ which explains why itwill take a much longer time to heat the deep atmospherethan to cool it. It is unlikely that the atmosphere will suddenly turn thermally in-ert at pressures greater than 10 bar. Rather, we expect the thermaltime-scale will gradually increase with increasing pressure. Inthis section, we examine the sensitivity of the deep atmosphericflows, circulations, and thermal structure to varying levels ofNewtonian cooling. Additionally we are motivated to quantifythe maximum amount of Newtonian cooling under which thedeep atmosphere is still able to maintain a deep adiabat.To explore this, we consider five models each with di ff erent cool-ing time-scales at the bottom of the atmosphere (i.e. five di ff er-ent values of log( τ )). From this, we can then linearly inter-polate the relaxation time-scale in log( P ) space between 10 and220 bar. The resultant profiles are plotted in Figure 9, and canbe split into three distinct groups: 1) The relaxation profile withlog( τ ) = . E ) represents a case with rapid Newto-nian cooling that does not decrease with increasing pressure; 2)The case log( τ ) =
11 (model F ) is a simple linear continuationof the relaxation profile we use between P = τ ) =
15, 20, 22.5 (models
G, H and I ),represent heating and / or cooling processes that get progressivelyslower in the deep atmosphere, in accordance with expectations Newtonian Relaxation Time (s)10 − − P r e ss u r e ( b a r) log( τ ) = E )log( τ ) =
11 ( F )log( τ ) =
15 ( G )log( τ ) =
20 ( H )log( τ ) = I ) Fig. 9: Newtonian cooling relaxation time-scale profiles used inthe models shown in Figure 10. Note that a smaller value of τ means more rapid forcing towards the imposed cooling profile(which in all cases is isothermal in the deep atmosphere, where P >
10 bar), and that the relaxation profiles are identical for P <
10 bar (grey line). − − − P r e ss u r e ( b a r) log( τ ) = . E )log( τ ) =
11 ( F )log( τ ) =
15 ( G )log( τ ) =
20 ( H )log( τ ) = . I ) T forced T init Fig. 10: Snapshots of the T – P profile for five, initially adiabaticsimulations (coloured lines - based on model B , and with thesame outer atmosphere cooling profile (dark grey)) which arethen forced to a deep isothermal profile (grey dashed line) withvarying log( τ ) (Equation 10).born out from 1D atmospheric models of hot Jupiter atmospheres(see, for example, Iro et al. 2005).The results we obtained are summarised by the T – P profiles weplot in Figure 10. For low levels of heating and cooling in thedeep atmosphere (models G, H and I ), the results are almost in-distinguishable from models A and B , with only a decrease inthe outer atmosphere connection temperature of a few Kelvin inmodel G . We find a more significant reduction in the tempera-ture of the T – P when we investigate model F , in which we setlog( τ ) =
11. In particular, there is a deepening of the connec-tion point between the outer atmosphere and the deep adiabat,which only becomes apparent for P >
20 bar in this model. Thisresult suggest that model F falls near the pivot point between Article number, page 10 of 13. Sainsbury-Martinez et al.: Idealised simulations of the deep atmosphere of hot jupiters:(a) log( τ ) =
15 (b) log( τ ) = Fig. 11: Zonally and temporally-averaged (over a period of ≈
30 years) stream-function for for two models with either a relatively‘strong’ ( G - left - log( τ ) =
15) or very weak ( H - right - log( τ ) =
20) isothermal relaxation (cooling) in the deep atmosphere( P >
10 bar). Clockwise circulations on the meridional plane are shown in red and anticlockwise circulations are shown in blue.Additionally the zonally and temporally averaged zonal wind is plotted in black (solid = eastward, dashed = westward).models in which the deep atmosphere is adiabatic and those thatrelax toward the imposed temperature profile. This is confirmedby model E , in which τ = .
5, where we find that the deepadiabat has been rapidly destroyed (in <
30 years), such thatthe deep T – P profile corresponds to the imposed cooling pro-file throughout the atmosphere. This occurs because the New-tonian time-scale has become smaller than the advective time-scale, which means that the imposed temperature profiles domi-nates over any dynamical e ff ects.Before closing this section, let us briefly comment on the merid-ional circulation profiles obtained in those models that convergeonto a similar deep adiabatic temperature profile (models G, H and I ). For all of them, we recover the same qualitative struc-ture we found for model A , characterised by meridional cells ofalternating direction that extend from the deep atmosphere tothe outer regions. The finer details of the circulations, however,di ff er from the ones seen in model A . This is illustrated in Fig-ure 11 which displays the meridional circulation and zonal flowprofiles for models G (Figure 11a) and H (Figure 11b). As theNewtonian cooling becomes faster in the deep atmosphere, thenumber of meridional cells increases (see also Figure 5), to thepoint that, in model E , no deep meridional circulation cells ex-ists and the deep circulation profile is essentially unstructured.Despite these di ff erences in the shape of the meridional circula-tion, the steady state profiles obtained in these simulations in thedeep atmosphere is again an adiabatic PT profile provided theNewtonian cooling is not (unphysically) strong.
4. Conclusion and discussion
By carrying out a series of 3D GCM simulations of irradiatedatmospheres, we have shown in the present paper that: – If the deep atmosphere is initialised on an adiabatic PT pro-file, it remains, as a steady state, on this profile, – If the deep atmosphere is initialised on a too hot state, itrapidly cools down to the same steady state adiabatic pro-file, − − − − P r e ss u r e ( b a r)
25 years2.5 years0.25 years600K Adiabat700K Adiabat
Fig. 12: Evolution of the sub-stellar point (i.e. day-side)Temperature-Pressure profile in a simulation (detailed inAmundsen et al. 2016) calculated using the Met O ffi ce GCM,the Unified Model, (Mayne et al. 2014a) and including a robusttwo-stream radiation scheme (Amundsen et al. 2014). Here weshow snapshots of the T-P profile at 0.25 (purple), 2.5 (green),and 25 (orange) Earth years, along with two example adiabats(grey dotted and dashed lines) designed to show how the deepatmosphere gets warmer and connects to steadily warmer adia-bats as the simulation progresses. Note that this progression is,at the end of the simulated time, ongoing towards a deep, hot,adiabat, albeit at an increasingly slow rate. – If the deep atmosphere is initialised on a too cold state, itslowly evolves towards the steady state adiabatic profile.Furthermore, in all the above cases, the deep adiabat forms atlower pressures that those at which we would expect, from 1Dmodels, the atmosphere to be convectively unstable. We havealso shown that this steady-state adiabatic profile is stable tochanges in the deep Newtonian cooling and is independent ofthe details of the flow structures, provided that the velocities are
Article number, page 11 of 13 & A proofs: manuscript no. main not completely negligible. The hot adiabatic deep atmosphere isthe natural final outcome of the simulations, for various resolu-tions, even though the time-scale to reach steady-state is longerat higher resolution when starting from a too cold initial state.When the simulations are initialised on a too cold profile,the time-scale to reach the steady state is of the order of t ∼ t < ∼ P < partially evolveddeep atmosphere , the structure of which is directly comparableto the early outputs of our isothermally initialised calculation.Examples of this early evolution of the deep atmosphere to-wards a deep adiabat (as seen in the early outputs plotted in Fig-ure 3) include Figure 6 of Rauscher & Menou (2010) (where thedeep temperature profile shows signs of heating from its initialisothermal state, albeit only on the irradiated side of the planet),Figure 7 of Amundsen et al. (2016) (where we see a clear shiftfrom their initially isothermal deep atmosphere towards a deepadiabat), and Figure 8 of Kataria et al. (2015) (where we againsee a temperature inversion and a push towards a deep adiabatfor Wasp-43b). It is tempting to think that if these simulationswere run longer, they would evolve to a similar, deep adiabaticstructure (with a corresponding increase in the exoplanetary ra-dius). In order to investigate this possibility, we have extendedthe model of Amundsen et al. 2016, run with the Unified Modelof the Met O ffi ce (which includes a robust two-stream radiationscheme that replaces the Newtonian Cooling in our models), fora total of ≈
25 Earth years. The results are shown in Figure 12,where we plot the pressure-temperature profile at three di ff erenttimes, along with examples of the approximate deep adiabat thatbest matches each snapshot. We see that the deep atmosphererapidly converges towards a deep adiabat with further verticaladvection of potential temperature warming up this adiabat asthe simulation goes on. Since this process keeps going on duringthe simulation, the result not only reinforces our conclusions butsuggests that our primary Newtonian cooling profile representsa reasonable approximation of the incident irradiation and radia-tive loss.The results obtained in the present simulations suggest that fu-ture hot Jupiter atmosphere studies should be initialised with ahot, deep, adiabat starting at the bottom of the surface irradia-tion zone ( P ∼
10 bar for HD209458b). Furthermore, in a sit-uation where the equilibrium profile in the deep atmosphere isuncertain, we suggest that this profile should be initialised witha hotter adiabat than expected rather than a cooler one. The sim-ulation should then be run long enough for the deep atmosphereto reach equilibrium. This is in agreement with the results ofAmundsen et al. (2016), who also suggested that future GCMmodels should be initialised with hotter profiles than currentlyconsidered. For instance, recent 3D simulations of HD209458bhave been initialised with a hotter interior T-P profile (for ex-ample, one of their models is initialised with an isotherm thatis 800 K hotter than typically used in GCM studies, thus bring-ing the deep atmosphere closer towards its deep adiabat equilib-rium temperature), and show important di ff erences, on the time-scales considered, between the internal dynamics obtained withthis set-up, and the ones obtained with a cooler, more standard,deep atmospheric profile (see, Lines et al. 2018a,b, 2019). Usingaforementioned more correct atmosphere initial profiles shouldnot only bring these models towards a more physical hot Jupiterparameter regime (with then a correct inflated radius), but also provide a wealth of information on how the deep adiabat re-sponds to changes in parameter and computational regime. The results obtained in the present GCM simulations have strongimplications for our understanding of the evolution of highly ir-radiated gas giants. As just mentioned, we first emphasise thatsimulations initialised from a too cold state are not relevant forthe evolution of inflated hot Jupiters (although it could be ofsome interest for re-inflation, but this is beyond the scope of thispaper). Indeed, inflated hot Jupiters are primarily in a hot initialstate and, as far as the evolution is concerned, only the steadystate of the atmosphere matters. The shorter timescales neededto reach this steady state are irrelevant for the evolution (with atypical Kelvin Helmholtz timescale of ∼ ff erent dissipation mech-anisms, whether they involve kinetic energy, or ohmic and tidaldissipation. This is in stark contrast with the present mechanism,in which entropy (potential temperature) is advected from the topto the bottom of the atmosphere. High entropy fluid parcels aremoved from the upper to the deep atmosphere and toward highlatitude while low entropy fluid parcels come from the deep at-mosphere and are deposited in the upper atmosphere. This grad-ually changes the entropy profile until a steady state situation isobtained. Although an enthalpy (and mass and momentum) fluxis associated with this process, down to the bottom of the atmo-sphere (characterised by some specific heat reservoir), this doesnot require a dissipative process (from kinetic, magnetic or ra-diative energy reservoirs into the internal energy reservoir).In order to characterise this deep heating flux, and confirm thatour hot, deep, adiabat would not be unstable due to high tem-perature radiative losses, we also explored the vertical enthalpyflux in our model and compared it to the radiative flux, as calcu-lated for a deep adiabat using ATMO (PT17). This analysis re-veals that the vertical enthalpy flux dominates the radiative fluxat all P > P =
10 bar, we find a net vertical enthalpy flux ( ρ c p T u z ) of − . × ergs − cm − compared to a outgoing radiative flux of7 . × ergs − cm − , suggesting that any deep radiative lossesare well compensated by energy (enthalpy) transport from thehighly irradiated outer atmosphere. This result is reinforced byUM calculation we show in Figure 12, which intrinsically in-cludes this deep radiative loss and show no evidence of coolingdue to deep radiative e ff ects. Article number, page 12 of 13. Sainsbury-Martinez et al.: Idealised simulations of the deep atmosphere of hot jupiters:
This (lack of a requirement for additional dissipative processes)is of prime importance when trying to understand the evolutionof irradiated planets. Whereas dissipative processes imply an ex-tra energy source in the evolution ( (cid:82) M ˙ (cid:15) dm , where ˙ (cid:15) is the energydissipation rate, to be finely tuned), to slow down the planet’scontraction, there is no need for such a term in the present pro-cess. Indeed, as an isolated substellar object (i.e. without nuclearenergy source) cools down, its gravitational potential energy isconverted into radiation at the surface, with a flux σ T ff . Let usnow suppose that the same object is immersed into an isotropicmedium characterised by a pressure P =
220 bars and a temper-ature T ∼ dR / dt ≈ , σ T ≈ σ T as the radiative and convective cool-ing flux at the interior-atmosphere boundary. Indeed, convectionwill become ine ffi cient in transporting energy and any remainingradiative loss in the optically thick deep core will be compen-sated by downward energy transport from the hot outer atmo-sphere. For a highly irradiated gas giant, the irradiation flux isnot isotropic, but the combination of irradiation and atmosphericcirculation will lead to a similar situation, with a deep atmo-sphere adiabatic profile of ∼ σ T ≈
0. The evolutionof the planet is stopped ( dS / dt ≈ Acknowledgements.
FSM and PT would like to acknowledge and thank the ERCfor funding this work under the Horizon 2020 program project ATMO (ID:757858). NJM is part funded by a Leverhulme Trust Research Project Grant andpartly supported by a Science and Technology Facilities Council ConsolidatedGrant (ST / R000395 / / WHIPLASH). GC was supportedby the Programme National de Planétologie (PNP) of CNRS-INSU cofunded byCNES. IB thanks the European Research Council (ERC) for funding under theH2020 research and innovation programme (grant agreement 787361 COBOM).FD thanks the European Research Council (ERC) for funding under the H2020research and innovation programme (grant agreement 740651 NewWorlds). BDacknowledges support from a Science and Technology Facilities Council Con-solidated Grant (ST / R000395 / ffi / K000373 / / K0003259 /
1. DiRAC is part of the National E-Infrastructure.Finally the authors would like to thank the Adam Showman for their careful andinsightful review of the original manuscript.
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