Observational Consequences of Shallow-water Magnetohydrodynamics on Hot Jupiters

We use results of shallow-water magnetohydrodynamics (SWMHD) to place estimates on the minimum magnetic field strengths required to cause atmospheric wind variations (and therefore westward venturing hotspots) for a dataset of hot Jupiters (HJs), including HAT-P-7b, CoRoT-2b, Kepler-76, WASP-12b, and WASP-33b, on which westward hotspots have been observationally inferred. For HAT-P-7b and CoRoT-2b our estimates agree with past results; for Kepler-76b we find that the critical dipolar magnetic field strength, over which the observed wind variations can be explained by magnetism, lies between $4\mbox{ G}$ and $19\mbox{ G}$; for WASP-12b and WASP-33b westward hotspots can be explained by $1\mbox{ G}$ and $2\mbox{ G}$ dipolar fields respectively. Additionally, to guide future observational missions, we identify $61$ further HJs that are likely to exhibit magnetically-driven atmospheric wind variations and predict these variations are highly-likely in $\sim 40$ of the hottest HJs.

However, using three-dimensional (3D) magnetohydrodynamic (MHD) simulations, Rogers & Komacek (2014) showed that HJs can exhibit winds that oscillate from east to west, causing east-west hotspot variations. Using continuous Kepler data, westward venturing brightness offsets have since been identified in the atmospheres of the ultra-hot Jupiters (UHJs) HAT-P-7b (Armstrong et al. 2016) and Kepler-76b (Jackson et al. 2019). Furthermore, thermal phase curve measurements from Spitzer have found westward hotspots on the UHJ WASP-12b (Bell et al. 2019) and the cooler CoRoT-2b (Dang et al. 2018); and optical phase curve measurements from TESS found westward brightspot offsets on the UHJ WASP-33b (von Essen et al. 2020). Three explanations for these observations have been proposed: cloud asymmetries confounding optical measurements (Demory et al. 2013;Lee et al. 2016;Parmentier et al. 2016); non-synchronous rotation (Rauscher & Kempton 2014); and magnetism (Rogers 2017). In Hindle et al. (2019), we found that CoRoT-2b would need an implausibly large planetary magnetic field to explain its westward atmospheric winds; concluding that a non-magnetic explanation is more likely. Rogers (2017) and Hindle et al. (2019) respectively used 3D MHD and shallow-water MHD (SWMHD) simulations to show that magnetism resulting from a B dip 6 G dipolar field strength can explain westward hotspots on HAT-P-7b, which is expected to be tidally-locked. Moreover, dayside cloud variability has recently been ruled-out as an explanation of the westward brightness offsets on HAT-P-7b (Helling et al. 2019) and, since all these testcases have near-zero eccentricities, they are expected to be synchronously rotating.
In this work we apply results from Hindle et al. (2021) on a dataset of HJs to calculate estimates of the minimum magnetic field strengths required to drive reversals. These conditions can be used to constrain the magnetic field strengths of UHJs.

REVERSAL CONDITION FROM SHALLOW-WATER MHD
The hottest HJs have weakly-ionised atmospheres, strong zonal winds, and are expected to host dynamodriven deep-seated planetary magnetic fields. If a HJ's atmosphere is sufficiently ionised, winds become strongly coupled to the planet's deep-seated magnetic field, inducing a strong equatorially-antisymmetric toroidal field that dominates the atmosphere's magnetic field geometry (Menou 2012;Rogers & Komacek 2014).
In hydrodynamic (and weakly-magnetic) systems, mid-to-high latitude geostrophic circulations cause a net west-to-east equatorial thermal energy transfer, yielding eastward hotspots, and net west-to-east angular momentum transport into the equator from higher latitudes, driving superrotating equatorial jets (Showman & Polvani 2011). In Hindle et al. (2021), we showed that the presence of a strong equatorially-antisymmetric toroidal field obstructs these energy transporting circulations and results in reversed flows with westward hotspots. The threshold for such reversals can be estimated using (Hindle et al. 2021): where V A,crit is the reversal threshold of the toroidal field's Alfvén speed, with V A,0 and V A,f respectively denoting the thresholds in the zero-forcing-amplitude limit and for a moderate-to-strong pseudo-thermal forcing. Here R is the planetary radius, c g is the shallow-water gravity wave speed, β = 2Ω/R is the latitudinal variation of the Coriolis parameter at the equator (for the planetary rotation frequency Ω), L eq ≡ (c g /β) 1/2 is the equatorial Rossby deformation radius, α = 2πR/L eq is a longitude-latitude lengthscale ratio, τ wave ≡ L eq /c g is the system's characteristic wave time scale (as in Showman & Polvani 2011), and ∆h eq /H determines the magnitude of the shallow-water system's pseudo-thermal forcing profile, for a Newtonian cooling treatment with a radiative timescale, τ rad .

METHOD FOR PLACING MAGNETIC REVERSAL CRITERIA ON HOT JUPITERS
Equation (1) shows that the parameters R, c g , Ω, τ rad , and ∆h eq /H can be used to estimate the minimum magnetic field strengths required for reversals. We apply this simple relation to a dataset of HJs taken from ex- oplanet.eu 1 , using planets with 0.1 M J < M < 10 M J and a < 0.1 AU, where M and M J denote the planetary mass and Jupiter's mass respectively, and a is the semimajor axis. The criteria are calculated using the equilibrium temperature (assuming zero albedos; e.g., Laughlin et al. 2011): for stellar radius, R * , orbital eccentricity, e, and stellar effective temperature, T * . The validity of the shallow-water approximation can be assessed by comparing L eq to the pressure scale height, H ∼ RT eq R 2 /GM , where G is Newton's gravitational constant and R, the specific gas constant, is calculated using the solar system abundances in Lodders (2010). For the sampled HJs, mean(H/L eq ) = 7.5 × 10 −3 , so shallow-water theory is generally expected to capture their leading order atmospheric dynamics well. The shallow-water gravity wave speed is calculated by equating thermal and geopotential energies, yielding c g ≡ √ gH ∼ (RT eq ) 1/2 . Doing so implies ∆h/H ∼ ∆T /T eq , where ∆h are deviations in shallow-water layer thickness from the reference H and ∆T ≡ T day − T eq for the dayside temperature, T day .
An interesting feature of HJs is that the dynamical parameters c g , Ω and R of a HJ are all related to its host star proximity and the mass/radius/luminosity of its host star (i.e., they are all related to T eq ). The consequence of this interdependence is that, for the hottest HJs, L eq /R and τ wave approximately converge to L eq /R ≈ 0.7 and τ wave ≈ 2 × 10 4 s (see Figure 1; top panels). In Figure 1 (bottom panel) we use Equation (1) to plot V A,crit /c g for ∆T /T eq = 0, 0.1, 0.2, 0.3. Taking ∆T ≈ (T day − T night )/2, ∆T /T eq = 0.1, 0.2, 0.3 cover the expected range of relative dayside-nightside variations (e.g., Komacek et al. 2017); whereas ∆T /T eq = 0 shows the zero-amplitude limit. V A,crit /c g varies linearly with ∆T /T eq above ∆T /T eq = 0.1, but approaches the zeroamplitude limit for ∆T /T eq 0.1. A remarkable feature of the HJ dataset is that, due to the aforementioned interdependences, the ratio V A,crit /c g also converges in the large T eq limit for a given ∆T /T eq . Equation (1), the Alfvén speed definition, and the ideal gas law yield where B φ,crit is the critical threshold of the toroidal field magnitude B φ , µ 0 is the permeability of free space, and T and P are the temperature and pressure at which the reversal occurs.
If the electric currents that generate the planet's assumed deep-seated dipolar field are located far below the atmosphere, Menou (2012) showed that B φ can be related to the dipolar field strength, B dip , by the scaling law where R m = U φ H/η is the magnetic Reynolds number for a given magnetic diffusivity, η, zonal wind speed, U φ , and pressure scale height, H. R m estimates the relative importance of the atmospheric toroidal field's induction and diffusion; while U φ /c g scales linearly with ∆h/H ∼ ∆T /T eq in geostrophically or drag dominated flows (Perez-Becker & Showman 2013). Taking a geostrophically-dominated flow yields We fix the constant of proportionality in this scaling by setting U φ ∼ 1.5 × 10 2 m s −1 for the conditions corresponding to the simulations of Rogers (2017). We calculate η following the method of Rauscher & Menou (2013) and Rogers & Komacek (2014), taking where χ e is the ionisation fraction, which is calculated using a form of the Saha equation that takes into account all elements from hydrogen to nickel. It is given by In this sum the number density for each element, n i , and the ionisation fraction of each element, χ e,i , are calculated using χ 2 for density ρ, total number density n, molecular mass µ m , relative elemental abundance (normalised to the hydrogen abundance) a i /a H , the electron mass m e , Plank's constant h, the Boltzmann constant k, and the elemental ionisation potential i . To calculate η, we use the solar system abundances in Lodders (2010) and take T = T eq + ∆T / √ 2, the root-mean-squared temperature for a sinusoidal longitudinal temperature profile. Estimates of R m and B φ,crit are calculated at depths corresponding to P = 10 mbar, at which Rogers & Komacek (2014) found magnetically-driven wind variations. In Figure 2 we plot R m (lefthand panel) and B φ,crit (righthand panel) vs. T eq , for HJs in the dataset (with T eq > 1000 K), taking ∆T /T eq = 0.1, 0.2, 0.3.
Induction of the atmospheric toroidal field is expected to become significant when R m exceeds unity. At P = 10 mbar, R m exceeds unity for T 1500 K, depending on ∆T /T eq . However, due to the highly temperature dependent nature of Equation (8), R m varies significantly when one compares ∆T /T eq = 0.1, 0.3 for a given HJ.
As we see in Section 4.2, B φ is only likely to exceed B φ,crit if the HJ in question is hot enough to maintain a significant atmospheric toroidal field (R m 1). We therefore concentrate our discussion on these hotter HJs; however, we place hypothetical estimates on B φ,crit for all planets in the dataset with T eq > 1000 K (Figure 2, righthand panel). Since, for a given ∆T /T eq , V A,crit /c g is virtually independent of T eq in the hottest HJs, so is B φ,crit , with 100 G B φ,crit 450 G for 0.1 < ∆T /T eq < 0.3; whereas larger L eq /R values can cause B φ,crit to decrease in the cooler HJs (compare with Figure 1). We comment that B φ,crit is generally least severe in the uppermost regions of the atmosphere, where the atmosphere is least dense, explaining why Rogers & Komacek (2014) found the east-west wind variations at these depths.
In Hindle et al. (2021), we highlighted that magneticallydriven wind variations can be viewed as a saturation mechanism for the atmospheric toroidal field, with the reversal mechanism preventing B φ from greatly exceeding B φ,crit . This suggests that B φ should peak in the deepest regions satisfying B φ ∼ B φ,crit , where B φ,crit can be large, then decrease towards the surface, where B φ,crit is smaller. This is consistent with Rogers & Komacek (2014), who found B φ peaks in the mid-atmosphere (and declined to 300 G B φ 450 G at P = 10 mbar in their M7b simulations).

Dipolar magnetic field strengths
In Figure 3 we use Equation (4) to plot T eq vs. B dip,crit , the critical dipolar field (at P = 10 mbar) for  ∆T /T eq = 0.1, 0.2, 0.3. Since the translation of planetary dynamo theory into the HJ parameter regime is not well-understood, we include a physically motivated reference line at B dip,crit = 14 G (the magnitude of Jupiter's magnetic field at its polar surface) and a second reference line at 28 G (twice this). Due to the highly temperature dependent nature of R m , these estimates of B dip,crit carry a high degree of uncertainty (e.g., compare B dip,crit of a given HJ for the different ∆T /T eq choices). Therefore, for useful estimates of B dip,crit , accurate temperature estimates/measurements (at the depth being probed) are required. Generally, T day is not directly calculable from standard planetary/stellar parameters, so measured values should be used where possible. For the five HJs with westward hotspot observations, we use dayside temperatures based on phase curve measurements to estimate B φ,crit and B dip,crit . We present these estimates in Table 1 and add labelled error bars to Figure 3. The UHJs are found to have low-to-moderate B dip,crit requirements. For HAT-P-7b we estimate 3 G < B dip,crit < 4 G at P = 10 mbar 2 , recovering the previously-known result that westward hotspots on HAT-P-7b can be wellexplained by magnetism (Rogers 2017;Hindle et al. 2019). On the UHJs WASP-12b and WASP-33b dipole fields respectively exceeding 1 G and 2 G at P = 10 mbar would explain westward hotspots. Likewise, at P = 10 mbar, a dipole field exceeding B dip,crit for 4 G < B dip,crit < 19 G is required to explain westward hotspots on Kepler-76b. Given the comparison with Jupiter and that Cauley et al. (2019) predicted surface magnetic fields on HJs could range from 20 G to 120 G, these es-timates support the idea that wind reversals on these UHJs have a magnetic origin. If non-magnetic explanations can be ruled out, such estimates of B dip,crit can be used as lower bounds for B dip on UHJs. In contrast, unless CoRoT-2b hosts an unfeasibly large 3 kG dipolar field, its westward hotspots are not explained by magnetism (recovering the result of Hindle et al. 2019). To check our method's fidelity, we also compare predictions to the simulations in Rogers & Komacek (2014), finding good agreement (for both B dip,crit and B φ,crit ).
Using the range ∆T /T eq = (0.1, 0.3) to estimate B φ,crit generally has uncertainties between one-half and one order of magnitude. However, Figure 3 shows that HJs divide into three clear categories: (i) those likely to have magnetically-driven atmospheric wind variations for any choice of ∆T /T eq (T eq 1950 K); (ii) those unlikely to have sufficiently strong toroidal fields to explain atmospheric wind variations, for any choice of ∆T /T eq (T eq 1600 K); and (iii) marginal cases that depend on the magnitude of day-night temperature differences (1600 K T eq 1950 K).
Using the conditions B dip,crit < 28 G, P = 10 mbar, and ∆T /T eq = 0.1, we identify 61 further HJs that are likely to exhibit magnetically-driven wind variations. We present these in Table 2 (Appendix A), which is ordered by ascending B dip,crit (i.e., from most-likely to least-likely to exhibit reversals), to help guide future observational missions. Of these 61 reversal candidates, 37 HJs have weaker reversal requirements than Kepler-76b. Hence, using these fairly conservative criteria, we predict that magnetic wind variations could be present in ∼ 60 and argue that they are highly-likely in ∼ 40 of the hottest HJs. . Critical dipole magnetic field strengths, B dip,crit , at P = 10 mbar. We plot B dip,crit using T = Teq + ∆T , with ∆T /Teq = 0.1, 0.2, 0.3 (blue, orange, red). For a given HJ, these are connected by translucent lines. We include error bars and labels for the planets discussed in this letter (see Table 1) along with reference lines at 14 G (dashed; Jupiter's polar surface magnetic field strength) and 28 G (dotted; twice this). Table 1. Estimates of B φ,crit and B dip,crit at P = 10 mbar, using the tabulated T day , for HAT-P-7b, CoRoT-2b, Kepler-76b, WASP-12b, and WASP-33b.

Planet
T For HJs with intermediate temperatures (1600 K T eq 1950 K), the magnitude of ∆T /T eq (and our simplifying assumptions) plays a significant role in determining whether magnetic wind variations are plausible, so specific dayside temperature measurements should be used for estimates. These intermediate temperatures HJs offer excellent opportunities to fine-tune magnetohydrodynamic theory, via cross-comparisons between observations and bespoke models.

DISCUSSION
We have applied the theory developed in Hindle et al. (2021) to a dataset of HJs to estimate the critical magnetic field strengths B dip,crit and B φ,crit (at P = 10 mbar), beyond which strong toroidal fields cause westward hotspots. The new criterion differs both mathematically and in physical interpretation from the criterion of Rogers & Komacek (2014) and Rogers (2017), which identifies when Lorentz forces from the deepseated dipolar field become strong enough to significantly reduce zonal winds, but doesn't theoretically explain wind variations. However, the estimates made in this work match well with typical magnetic fields in the 3D simulations of Rogers & Komacek (2014) and Rogers (2017), which exhibit wind variations, and also match values resulting from their criterion in these regions of parameter space. This is because, while describing different magnetic effects, both criteria predict the critical magnetic field strengths at which magnetism becomes dynamically-important in HJ atmospheres. Applying the new criterion to the HJ dataset, we found that the brightspot variations on Kepler-76b can be explained by plausible planetary dipole strengths (B dip 4 G using T day = 2850; B dip 19 G using T day = 2300), and that westward hotspots can be explained for B dip 1 G on WASP-12b and B dip 2 G on WASP-33b. The esti-mates of B φ,crit and B dip,crit for HAT-P-7b and CoRoT-2b are consistent with the estimates of Rogers (2017) and Hindle et al. (2019). We then used an observationally motivated set of criteria (B dip,crit < 28 G, ∆T /T eq = 0.1, and P = 10 mbar) to tabulate 65 HJs that are likely to exhibit magnetically-driven wind variations (see Table 2, Appendix A) and predict such effects are highly-likely in ∼ 40 of the hottest HJs.
With exoplanet meteorology becoming increasingly developed, the results of this study suggests that further observations of hotspot variations in UHJs should be expected. A combination of archival data and future dedicated observational missions from Kepler, Spitzer, Hubble, TESS, CHEOPS, and JWST can be used to identify magnetically-driven wind variations and other interesting features at different atmospheric depths. In particular, long time-span studies observing multiple transits of UHJs are likely to be essential in understanding hotspot/brightspot oscillations. Of the studies that have measured westward hotspot/brightspot offets, only the long time-span studies of Armstrong et al. (2016) (HAT-P-7b; 4 years) and Jackson et al. (2019) (Kepler-76b; 1000 days) identify hotspot/brightspot oscillations. In both cases, such oscillations are observed on timescales of ∼10-100 Earth days, which Rogers (2017) noted is consistent with timescales of wind variability in 3D MHD simulations (and the deep-seated magnetic field's Alfvén timescale). Such timescales are of-order or longer than the total time-spans of the other UHJ studies with westward hotspot/brightspot measurements (Bell et al. 2019;von Essen et al. 2020), so it is impossible to tell whether these measurements are part of an oscillatory evolution.
If non-magnetic explanations can be ruled-out for past and future identifications of westward hotspot offsets on UHJs, the coolest planets with wind variations can indicate typical B dip magnitudes on HJs. This has the potential to drive new understanding of the atmospheric dynamics of UHJs and provide important observational constraints for dynamo models of HJs. Parallel to this, future theoretical work can refine estimates of B dip,crit . In many cases combining observational measurements with bespoke 3D MHD simulations offer the best prospect for providing accurate constraints on the magnetic field strengths of UHJs, yet the simple concepts and results of this work can provide useful starting points for such studies and can highlight trends from an ensemble viewpoint. The largest limiting factor in our estimates of B dip,crit is the highly temperature dependent nature of R m . Furthermore, the magnetic scaling law does not account for longitudinal asymmetries in the magnetic diffusivity or the dipolar field strength within the atmospheric region. In future work we shall investigate how these inhomogeneities effect the atmospheric dynamics more closely, using a 3D model containing variable magnetic diffusivity, consistent poloidaltoroidal field coupling, stratification, and thermodynamics. To date, MHD models of HJs have strictly considered dipolar magnetic field geometries for the planetary magnetic field. Dynamo simulations would offer insight into the nature of magnetic fields in the deep interiors of HJs, which, at present, is not well-understood.