Shallowness of circulation in hot Jupiters -- Advancing the Ohmic dissipation model

The inflated radii of giant short-period extrasolar planets collectively indicate that the interiors of hot Jupiters are heated by some anomalous energy dissipation mechanism. Although a variety of physical processes have been proposed to explain this heating, recent statistical evidence points to the confirmation of explicit predictions of the Ohmic dissipation theory, elevating this mechanism as the most promising candidate for resolving the radius inflation problem. In this work, we present an analytic model for the dissipation rate and derive a simple scaling law that links the magnitude of energy dissipation to the thickness of the atmospheric weather layer. From this relation, we find that the penetration depth influences the Ohmic dissipation rate by an order of magnitude. We further investigate the weather layer depth of hot Jupiters from the extent of their inflation and show that, depending on the magnetic field strength, hot Jupiter radii can be maintained even if the circulation layer is relatively shallow. Additionally, we explore the evolution of zonal wind velocities with equilibrium temperature by matching our analytic model to statistically expected dissipation rates. From this analysis, we deduce that the wind speed scales approximately as $1/\sqrt{T_\mathrm{eq}-T_0}$, where $T_0$ is a constant that equals $T_0 \sim 1000~\mathrm{K}-1800~\mathrm{K}$ depending on planet-specific parameters (radius, mass, etc.). This work outlines inter-related constraints on the atmospheric flow and the magnetic field of hot Jupiters and provides a foundation for future work on the Ohmic heating mechanism.


Introduction
Generic models of giant planetary structures hold that the radii of evolved Jovian-class planets cannot significantly exceed the radius of Jupiter (Stevenson 1982). Accordingly, measurement of the substantially enhanced radius of HD 209458 b -clocking in at 1.39 R (0.73 M , 1476.81 K; Stassun et al. 2017) -came as a genuine surprise (Charbonneau et al. 2000;Henry et al. 2000). Subsequent detections demonstrated that HD 209458 b is not anomalous; rather, inflated radii are a common attribute of the hot Jupiter (HJ) class of exoplanets (Bodenheimer et al. 2003;Laughlin et al. 2011). This begs the obvious question of what physical mechanism inflates the radii of HJs.
Although a number of hypotheses have been proposed -for example: tidal heating (e.g., Bodenheimer et al. 2001), thermal tides (e.g., Arras & Socrates 2010), advection of potential temperature (e.g., Tremblin et al. 2017), or the mechanical greenhouse effect (e.g., Youdin & Mitchell 2010) -recent data indicate that explicit predictions of the Ohmic dissipation (OD) mechanism (e.g., Batygin & Stevenson 2010;Perna et al. 2010a,b;Wu & Lithwick 2012;Spiegel & Burrows 2013;Ginzburg & Sari 2016) are consistent with the modern data set. Specifically, the functional form of the inflation profile, and its (Gaussian) dependence on the planetary equilibrium temperature, is reflected in the data (Thorngren & Fortney 2018;Sarkis et al. 2021). 1 In light of this, in this work we aim to advance the theoretical understanding of the OD mechanism and pose the 1 The other inflation models scale like power laws, meaning that their efficiency keeps increasing with equilibrium temperature. simple question of how Ohmic heating depends on the depth to which large-scale atmospheric circulation penetrates. Additionally, we examine if the empirical inflation profile derived from the data themselves can constrain the shallowness of circulation layers on these highly irradiated planets.
General circulation models (GCMs) based on "primitive" equations of hydrodynamics routinely find winds in the km s −1 range that penetrate to depths on the order of 1 to 10 bars (e.g., Showman et al. 2015;Rauscher & Menou 2013). Menou (2020), however, argues that this depth can actually be significantly greater. Analytic models based on force-balance considerations (Menou 2012;Ginzburg & Sari 2016) or models that regard the HJ atmosphere as a heat engine (Koll & Komacek 2018) have predicted similarly fast winds, with magnetic drag dominating at high equilibrium temperatures. On the contrary, anelastic magnetohydrodynamic (MHD) simulations find rather shallow circulation layers with slow wind velocities in general (e.g., Rogers & Komacek 2014;Rogers & Showman 2014;Rogers 2017).
Given that each of these models is subject to its own strengths and limitations, an alternative constraint on HJ circulation patterns is desirable. Accordingly, by extending the Ohmic heating model to include varying zonal wind geometries, we demonstrate the existence of a simple scaling law that allows us to predict the wind depth of HJs from the extent of their inflation under the assumption that Ohmic heating constitutes the dominant interior energy dissipation mechanism.
Another point of considerable interest is the evolution of wind patterns with equilibrium temperature. Force balance arguments suggest strong magnetic damping of equatorial winds Article number, page 1 of 8 arXiv:2201.08209v2 [astro-ph.EP] 10 Feb 2022 A&A proofs: manuscript no. 42588corr at high equilibrium temperatures, T eq (Menou 2012). In contrast, MHD simulations show flow reversal (i.e., eastward jets become westward) with increasing T eq (Batygin et al. 2013;Heng & Workman 2014;Rogers & Komacek 2014). Bypassing the question of wind direction, in this work we explore the mean equatorial wind velocity from energetic grounds using statistical evidence of the heating efficiency (Thorngren & Fortney 2018).
The remainder of the manuscript is organized as follows. In Sect. 2 we outline the HJ and OD model. We present the resulting circulation profiles in Sect. 3. We discuss our results and conclude in Sect. 4.

Conductivity profile
The machinery of the OD mechanism is underlined by the fact that -owing to the thermal ionization of alkali metals that are present in trace abundances -HJ atmospheres are rendered weakly conductive. Accordingly, the first step in our analysis is to delineate the conductivity profile. We divided the HJ into radiative and convective layers, with the radiative-convective boundary (RCB) located at 100 bars. 2 In practice, the functional form of the conductivity profile is only important in the vicinity of the RCB. In the radiative layer, we approximated the temperature as isothermal by averaging over the plane-parallel static gray atmosphere (Guillot 2010): where T is the temperature in the atmosphere, T int the intrinsic temperature, T eq the equilibrium temperature of the HJ, τ the optical depth, ζ := κ v κ th the opacity ratio, and E n (z) := ∞ 1 t −n e −zt dt the exponential integral function. We adopted Guillot (2010) values for the visible and thermal opacity of HD 209458 b: κ v = 0.004 cm 2 g −1 and κ th = 0.01 cm 2 g −1 , respectively.
Below the RCB, we assumed a polytropic relation P ∝ ρ γ , where P is the pressure and γ is the polytropic index. We obtained γ by fitting along an adiabat from the hydrogen-helium equation of state of Chabrier & Debras (2021). Because the conductivity rises rapidly in the interior, the contribution of the deep interior to the heating is small. Thus, our polytropic fit suffices for our purposes. To this end, we note that anomalous heating need not be deposited at the center of the convective envelope to sustain the planetary radius in an inflated state. That is to say, energy deposited sufficiently deep in the envelope, a radius that translates to a pressure level on the order of 10 kbar (Batygin et al. 2011;Spiegel & Burrows 2013;Ginzburg & Sari 2016;Komacek & Youdin 2017, and the references therein), is sufficient. Reinflating a HJ from a cold state, however, requires deeper heat deposition (e.g., Thorngren et al. 2021). Generally, concerns of Ohmic reinflation of close-in giant planets (Hartman et al. 2016;Grunblatt et al. 2017) -while fascinating -are beyond the scope of this work. The ionization fraction of alkali metals is determined by Saha's equation: where the subscript j denotes the respective alkali metal, n + j its positively ionized number density, I j its ionization potential, and n j its total number density and k is the Boltzmann constant, m e the electron mass, n e the electron number density, and the reduced Planck constant. We can solve this equation in the weakly ionization limit (n + j n j ) to obtain the electron number density. Throughout this study, we use the proto-solar abundances of Lodders (2003), Z = 0.0149, to obtain n j .
Assuming a classical free electron gas, the conductivity, σ, in the isothermal layer simplifies to an exponential function (Batygin & Stevenson 2010): where the subscript RCB denotes the respective quantity value at the RCB, r the radial distance, and H the pressure scale height. The adiabatic conductivity expression is more complex. While a semi-analytic profile can in principle be readily derived, for simplicity we approximate this layer with an exponential profile as well: where H fit is the free parameter used to fit the profile to the adiabatic expression.
Recently, Kumar et al. (2021) undertook a sophisticated calculation of σ. We did not reproduce their treatment but compare their results with our simple estimate in Fig. 1. Crucially, all conductivity profiles show a drop close to R ∼ 1.365 R = 0.98 R p , and we modeled this with an insulating boundary condition at r = R.

Velocity profile
Most HJs orbit their host star on circular orbits (Bonomo et al. 2017). Tidal de-spinning will lock those planets, creating a con- stantly irradiated dayside and a nightside that faces away from the host star (Hut 1981). This unbalance in heating leads to a pressure gradient, which balances with the planet's rotation and drag to create strong winds on HJs (for a review, see Showman et al. 2012). For HD 209458 b, for example, Snellen et al. (2010) observed wind velocities of around (2 ± 1) km s −1 in the upper atmosphere.
Various GCMs (e.g., Heng et al. 2011;Parmentier et al. 2013;Showman et al. 2015;Menou 2019), analytical theory (e.g., Showman & Polvani 2010, and observations of hotspot shifts (e.g., Knutson et al. 2007) indicate that these winds manifest as a single dominant eastward jet. In the presence of 10 -100 G magnetic fields (Yadav & Thorngren 2017;Cauley et al. 2019), significant induction of electrical currents is unavoidable. While the conductivity of the atmosphere is insufficient for the resulting Lorentz force to fully suppress the circulation (Batygin et al. 2013), conductivity rises rapidly in the convective interior, ensuring the dominance of the magnetic drag (e.g., Liu et al. 2008;Rogers & Showman 2014). As a consequence, zonal winds will be dissipated completely at or below the RCB (velocity u = 0 for r ≤ r RCB ).
This led us to parametrize the velocity field in the radiative layer by u(r, θ) = v max · e ξr − e ξr RCB e ξR − e ξr RCB sin θ ·φ, where v max is the maximum velocity at the equator, R the OD radius, θ the polar angle, andφ the basis vector in azimuthal direction. The ξ is a free parameter of dimension 1/length that controls the radial form of the zonal jet by damping or enhancing the penetration depth. To simplify the comparison between various velocity profiles 3 and give a physical intuition, we introduced the (dimensionless) relative half-velocity radius, δ half , which is the relative distance between the RCB and the OD radius where v decreased by 1/2 (Fig. 2). If δ half = 0.5, the velocity decreases linearly toward the RCB. If 0.5 < δ half < 1, the velocity drops off more quickly and the zonal jets are confined to a smaller region in the upper atmosphere. Lastly, if 0 < δ half < 0.5, the velocity decreases more slowly and the zonal jet is more extended.
3 In principle, a number of velocity profiles would fit our requirements.
For example, the power law r−r RCB R−r RCB ξ also works. However, the derivative of this profile diverges for 0 < ξ < 1, introducing unnecessary complications into the analytical treatment.  To obtain the OD rate, P, we need to compute

Ohmic dissipation rate
where dV is the volume element and j ind is the induced current density. We can obtain j ind from Ohm's law: where B is the HJ's magnetic field, which we approximate by a dipole, and Φ, the electric potential, is obtained by solving a variation of Poisson's equation that comes from the continuity of the current density: The radius of an inflated gas giant is predominately determined by its interior entropy (Zapolsky & Salpeter 1969). Heat deposited into the radiative layer radiates away before influencing the adiabat noticeably (Spiegel & Burrows 2013). Consequently, Eq. 5 is restricted to evaluation in the convective zone when determining the radius inflation.

Weather layer depth
We computed P in the range of 0 < δ half < 1, which represents both shallow and deep zonal flows (Fig. 2). The zonal wind profile influences the OD rate by an order of magnitude (Fig. 3). This is expected since the volume that contributes to the generation of the induced current (Eq. 6) increases. More precisely: the OD rate scales roughly like the volume integral, udV.
In addition, the anomalous heating rates of Thorngren & Fortney (2018) are easily reproduced for magnetic fields stronger than ∼ 12 G. Moreover, for HD 209458 b, we expect δ half = 0.82 Article number, page 3 of 8 A&A proofs: manuscript no. 42588corr  (Snellen et al. 2010). As P ∝ e −r/H , v ∝ e ξr ∝ P −ξH ≈ P −0.52 , which falls more slowly than the v ∝ P −5/2 found by Ginzburg & Sari (2016) and confines the circulation to the top 3% of the HJ. In other words, our results indicate that a shallow weather layer is sufficient to explain the radius anomaly.

Zonal wind velocities
In the last section, we derived the flow geometry (δ half ) given the equatorial wind velocity. Conversely, we can ask what zonal wind velocity is required, given a flow geometry, to inflate a HJ sufficiently. For this, we probed three HJ masses -0.5 M , 1 M , and 3 M -for three zonal flow configurations: respectively, a linear decrease (δ half = 0.5), a shallow weather layer similar to that of HD 209458 b (δ half = 0.8), and a very shallow weather layer (δ half = 0.9). The intrinsic temperature, HJ radius, and magnetic field strength were taken from the results of Thorngren &Fortney (2018) andYadav &Thorngren (2017).
The resulting velocities decrease with equilibrium temperature like a power law (Fig. 4). This originates from a roughly linear increase in the anomalous heating power and the magnetic field with equilibrium temperature (i.e., P ∝ T eq and B ∝ T eq ). Since P ∝ v 2 B 2 , this leads to v ∝ √ P/B ∝ T −1/2 . The physical interpretation for this effect is the damping of zonal flows due to strong magnetic drag.
Once we reach higher equilibrium temperatures, however, the magnetic field strength stays roughly constant (Yadav & Thorngren 2017). Therefore, the velocity stays roughly constant as well. Furthermore, lower δ half values (i.e., more extended zonal jets) lead to lower velocities since OD is more efficient (see also the discussion below).
On the lower end of the spectrum, the equilibrium temperatures are too low to ionize sufficient amounts of alkali metals and planetary magnetic fields are weak. Hence, the zonal winds flow with little resistance, but also deposit little power into the planet's interior. Ultimately, the velocities diverge, indicating that OD cannot inflate efficiently at low T eq -an arguably trivial result. These extremely high velocities can be interpreted as upper bounds, which lose their physical meaning once the speed of sound is significantly exceeded and shock-induced dissipation sets in.

Discussion
In this work, we have explored the physics of OD in HJs and derived an analytic expression for the power deposited into the convective interior as a function of wind penetration depth. This equation (see Eq. A.3) gives a simple and robust order-ofmagnitude estimate.
We found that the wind profile has an order-of-magnitude influence on the power output. This effect must be taken into account in any detailed model of OD. Additionally, the scaling law allows unobservable planetary parameters to be constrained from direct observations. If, for example, one obtained the equatorial wind velocity of a HJ via spectroscopy, our scaling relation would predict the flow geometry with little effort.
Shallow weather layers appear sufficient to explain the radius anomaly. For HD 209458 b, we predict that the flow velocity drops by a factor of 2 at ∼ 0.4 bar compared to the wind velocity at the photosphere. While this velocity slope is rather steep, the Richardson number still exceeds 0.25, indicating that the Kelvin-Helmholtz instability never sets in. The weather layer size is on the lower end of GCM simulations and on the higher end of MHD simulations (Sect. 1). This model does not apply to low equilibrium temperatures (∼ 1000 K), though, where a radius excess above the Jovian radius can be attributed to an extended radiative layer -without the need for an interior heating mechanism.
Lastly, we found that, for a constant δ half , the equatorial wind speed decreases with equilibrium temperature like a power law v max (T eq ) ∝ (T eq − T 0 ) −α , α ≈ 1/2, where T 0 is a constant that equals T 0 ∼ 1000 K−1800 K depending on planet-specific parameters (radius, mass, etc.) -before saturating at a few hundred meters per second. This profile shows a similar decrease to that in Menou (2012), Rogers & Komacek (2014), and Koll & Komacek (2018) at around T eq ≈ 1500 K (Fig. 5). In contrast, though, Menou (2012) suggests more rapid velocity damping due to magnetic drag toward higher equilibrium temperatures. Moreover, Rogers & Komacek (2014) arrive at considerably lower wind velocities (and hence OD rates) due to their numerical treatment (∼ 2 × 10 18 W for B = 15 G and v = 160 m s −1 at depth). All of these models predict lower velocities than recently observed in ultra-hot Jupiters (e.g., 5.5 km s −1 for WASP-76 b, Seidel et al. 2021). In this vein, our model suggests that only a very shallow circulation layer is required to explain the radius inflation in such a system, although a massive high-metallicity core could undoubtably alter this assertion.
More fundamentally, our model assumptions are rendered invalid in the ultra-hot Jupiter regime ( 2000 K), where magnetic Reynolds numbers greatly exceed unity. Another distinction in atmospheric wind measurements between T eq < 2000 K HJs and ultra-hot Jupiters may lie in the fact that the sampled pressure levels of the atmosphere are vastly different. Ultimately, a more detailed analysis of MHD effects (e.g., Rogers 2017;Hindle et al. 2019;Beltz et al. 2021) is required to understand their extremely fast wind velocities. While recent theoretical evidence points toward enhanced bulk metallicities of HJs (e.g., Thorngren et al. 2016;Müller et al. 2020;Schneider & Bitsch 2021, see also Appendix B), those effects are subdominant to the uncertainty in the magnetic field and the velocity (P ∝ √ Zv 2 B 2 ). Nonetheless, since OD exclusively predicts a correlation between alkali metal abundance and radius inflation, atmospheric enrichment in alkali metals, as observed by Welbanks et al. (2019), might be an important contributing factor for this mechanism. For this study, though, as we do not claim to predict the exact anomalous heating power for a specific planet but rather introduce a general and robust extension to the Ohmic heating paradigm, they are of little concern for the validity of our results.
As the atmospheric composition, the wind velocity, and the dipole field strength of HJs are currently poorly constrained, we emphasize that this study can only yield an estimate of the broad circulation profile, rather than a precise prediction. 4 Likewise, more observational data of wind velocities in HJs, especially in the T eq ∼ 1500 K range, would aid in constraining δ half empirically. Lastly, OD in the ultra-hot Jupiter regime remains poorly understood. Future studies could build upon recent analytical (Hindle et al. 2019(Hindle et al. , 2021 or numerical (Beltz et al. 2021) work to account for those objects self-consistently. Nonetheless, this work shows that OD cannot only explain the radius inflation problem, but also allows us to peak deep into HJ atmospheres and, thereby, gives us new insights into their remarkable structure.