Shallow Water Magnetohydrodynamics for Westward Hotspots on Hot Jupiters

Draft version February 8, 2019 Typeset using LATEX twocolumn style in AASTeX61

SHALLOW WATER MAGNETOHYDRODYNAMICS FOR WESTWARD HOTSPOTS ON HOT JUPITERS

A. W. Hindle,1 P. J. Bushby,1 and T. M. Rogers1

1School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

ABSTRACT Westward winds have now been inferred for two hot Jupiters: HAT-P-7b and CoRoT-2b. Such observations could be the result of a number of physical phenomena such as cloud asymmetries, asynchronous rotation or magnetic fields. For the hotter hot Jupiters magnetic fields are an obvious candidate, though the actual mechanism remains poorly understood. Here we show that a strong toroidal magnetic field causes the planetary-scale equatorial magneto-Kelvin wave to structurally shear as it travels, resulting in westward tilting eddies, which drive a reversal of the equatorial winds from their eastward hydrodynamic counterparts. Using our simplified model we estimate that the equatorial winds of HAT-P-7b would reverse for a planetary dipole field strength Bdip,HAT-P-7b & 6 G, a result that is consistent with three-dimensional magnetohydrodynamic simulations and lies below typical surface dipole estimates of inflated hot Jupiters. The same analysis suggests the minimum dipole field strength required to reverse the winds of CoRoT- 2b is Bdip,CoRoT-2b & 3 kG, which considerably exceeds estimates of the maximum surface dipole strength for hot Jupiters. We hence conclude that our magnetic wave-driven mechanism provides an explanation for wind reversals on HAT-P-7b, however other physical phenomena provide more plausible explanations for wind reversals on CoRoT-2b.

Keywords: magnetohydrodynamics (MHD) – planets and satellites: atmospheres – planets and satellites: individual (CoRoT-2b, HAT-P-7b, HD 189733b)

Corresponding author: Alexander Hindle [email protected] 2 Hindle et al. night day night 1. INTRODUCTION Observations of hot Jupiters (HJs) generally measure a peak brightness offset eastward of the substellar point (Knutson et al. 2009; Wong et al. 2016). Simi- H + ∆heq h heq H larly, equatorial superrotation is an archetypal feature ρ, u, B of hydrodynamic models of tidally-locked, strongly irra- no B diated, short-period planets (Showman & Guillot 2002; flux Cooper & Showman 2005; Langton & Laughlin 2007; Dobbs-Dixon & Lin 2008). Further, Showman & Polvani (2011)(SP11) showed that such systems will always z produce eastward equatorial jets, which are driven by ρl, ul = 0 interactions between the mean flow and the system’s x . . linear equatorial shallow water hydrodynamic (SWHD) . waves. However, recent continuous Kepler measure- Figure 1. The reduced gravity SWMHD model has two ments of HAT-P-7b and thermal phase observations of layers: an active layer sits upon an infinitely-deep inactive CoRoT-2b made by the Spitzer Space Telescope found fluid layer, where both layers have constant densities (ρ and westward venturing peak brightness and hotspot offsets ρl respectively). No magnetic flux is permitted across the (Armstrong et al. 2016; Dang et al. 2018). These obser- interface. The layer thickness, h, is relaxed towards the im- vations suggest the existence of a mechanism that can posed radiative equilibrium thickness profile, heq, over a radiative timescale, τ . also drive westward equatorial winds. rad Based on their magnetohydrodynamic (MHD) simulations, Rogers & Komacek(2014) predicted that west- The reduced gravity SWMHD model, as illustrated ward wind variations would occur as the result of strong in Figure1, has two constant density layers: an upper, coupling between a planet’s flow and magnetic field. meteorologically active layer and an infinitely-deep, qui- Furthermore Rogers(2017) highlighted that, assuming escent lower layer. In the absence of forcing the active wind reversals are magnetically-driven, observations of layer has a thickness H, which is physically analogous westward hotspot offsets lead to a direct constraint on to the pressure scale height. the magnetic field strengths of a given HJ. While Rogers In the limit H/L 1, where L is some typical hori- (2017) demonstrated that westward flows developed in zontal length scale, vertical acceleration becomes vanish- the strong field case, the actual mechanism for wind re- ingly small and the system lies in magneto-hydrostatic versals remained unknown. balance: gravitational acceleration balances the total Here we demonstrate that a shallow water wave-driven (gas plus magnetic) vertical pressure gradient, and the mechanism can explain the wind reversals. Firstly, horizontal velocity and magnetic fields become indepen- we demonstrate that a shallow water magnetohydro- dent of the vertical coordinate, z. Consequently, the dynamic (SWMHD) model can reproduce both east- MHD equations can be integrated over z (while requir- ward hotspot offsets in hydrodynamic cases and west- ing that interfaces between vertical layers are material ward hotspot in the presence of a strong toroidal mag- surfaces, with no magnetic flux across them) to give the netic field, suggesting magnetically-driven wind reversal reduced gravity SWMHD equations. For a local Carte- is a shallow phenomenon. We then highlight magnetic sian system in the equatorial beta-plane approximation, modifications to equatorial SWMHD waves and present the evolution of the active layer of the reduced gravity a wave-driven reversal mechanism, which is consistent SWMHD model is governed by the equations: with the hydrodynamical theory of SP11. We conclude du + βy(z × u) = −g∇h + (B · ∇)B by discussing the possible consequences of these con- b dt (1) cepts for HAT-P-7b and CoRoT-2b. u + R − + Dν , τdrag 2. REDUCED GRAVITY SHALLOW WATER ∂h h − h + ∇ · (hu) = eq ≡ Q, (2) MAGNETOHYDRODYNAMIC MODEL ∂t τrad We adapt the SWMHD model of Gilman(2000) and dA = Dη, (3) use a reduced gravity SWMHD model. This is the MHD dt analogue of the reduced gravity SWHD models used to where h(x, y, t), u(x, y, t) ≡ (u, v), and B(x, y, t) ≡ study HJs in hydrodynamic systems (e.g. Langton & (Bx,By) denote the active layer thickness, the hori- Laughlin 2007; Showman & Polvani 2010; SP11). zontal active layer velocity field and the horizontal ac- Shallow water magnetohydrodynamics for westward hotspots on hot Jupiters 3 tive layer magnetic field (in units of velocity) respec- which has previously been used in comparable SWHD tively. The horizontal gradient and Lagrangian time models (e.g. Showman & Polvani 2010; SP11; PBS13). derivative operators are defined by ∇ ≡ (∂x, ∂y) and We also include simple Rayleigh drag in Equation (1) d/dt ≡ ∂/∂t + u · ∇ respectively. for direct comparison with these SWHD models. The system is defined in terms of a magnetic flux 3. NUMERICAL TREATMENT AND SOLUTIONS function, A(x, y, t), which satisfies hB = ∇ × Abz thus guaranteeing the SWMHD divergence-free condition, We evolve the system, by solving Equations (1)–(3) on ∇ · (hB) = 0, remains satisfied everywhere for all time. the domain −π < x/R < π, −π/2 < y/R < π/2, from a We take the system’s origin (x, y) = (0, 0) to be the flat rest state (h = H and u = 0 everywhere) for SWHD modelled planet’s substellar point, therefore our system solutions, then impose a background magnetic field (A = is compared to spherical geometries with the approx- A0) once a hydrodynamic steady state is achieved for imate coordinate transforms φ ≈ x/R and θ ≈ y/R SWMHD solutions. (where φ and θ denote the azimuthal and latitudinal We apply periodic boundary conditions on u, h and coordinates, and R denotes the planetary radius). The A in the x direction and require v = 0, ∂u/∂y = 0, reduced gravitational acceleration is denoted by the con- ∂A/∂x = 0 (h is chosen to conserve mass) at y boundary stant g, which is defined as in Perez-Becker & Showman points. The equations are solved on a 256×255 grid in x (2013)(PBS13) rather than Vallis(2006), and the lati- and y, with spatial derivatives taken pseudo-spectrally tudinal variation of the Coriolis parameter at the equa- in x and using fourth-order finite difference ghost point tor is given by β ≡ df/dy|y=0 = 2Ω/R, for planetary schemes in y. We integrate the system forwards in time rotation frequency Ω. using an adaptive third-order Adam-Bashforth scheme In numerical simulations we include explicit viscous (Cattaneo et al. 2003). diffusion (Gilbert et al. 2014): The system is driven by relaxing h towards the prescribed radiative equilibrium layer thickness profile: −1 T Dν = h ∇ · νh ∇u + (∇u) , (4)  x y H + ∆heq cos R cos R dayside where ν is the kinematic viscosity. Furthermore, we heq = (7) treat the induction equation with the explicit magnetic H nightside, diffusion (A. D. Gilbert et al., in preparation): where H is the nightside equilibrium thickness and ∆heq is the difference in h between the nightside and the D = η(∇2A − h−1∇h · ∇A), (5) eq η substellar point. This profile is the Cartesian analogue where η is the magnetic diffusivity. of the spherical forcing prescription used in compa- The prescribed Newtonian cooling term, Q, relaxes rable hydrodynamic models (e.g. Langton & Laughlin the system towards the imposed radiative equilibrium 2007; Showman & Polvani 2010; SP11). profile, heq, over a radiative timescale, τrad, by trans- HJs have weakly ionised photospheres. Consequently, ferring mass upwards from the infinitely-deep inactive strong zonal flows crossing the assumed deep-seated layer to the active layer in “heating” regions and vice planetary dipolar magnetic field are believed to induce versa in “cooling” regions. atmospheric toroidal fields. Menou(2012) showed that The vertical mass transport, R, represents the effect the strengths of the dipolar field, Bdip, and the toroidal of Newtonian cooling on the momentum equations. In field, Bφ, can be approximately related by the scaling cooling regions (Q < 0) mass is transported downwards law Bφ ∼ RmBdip, where the magnetic Reynolds num- into the infinitely-deep inactive layer and the specific ber (Rm) is temperature dependent and exceeds unity momentum of both layers is conserved without any hor- for hotter HJs( Teq & 1300 K). Hence, in such systems izontal acceleration. Conversely, in regions of heating the toroidal field is expected to dominate the dipolar (Q > 0) mass with no horizontal velocity is transported field in equatorial regions. upwards into the active layer, causing the horizontal de- Numerically, we implement an equatorially-antisymmetric celeration of the active layer. In heating regions it is azimuthal background magnetic field though a back- required that Newtonian cooling has no effect on the ground flux function, which we impose initially and temporal evolution of the specific momentum, ∂(hu)/∂t, allow to evolve. The imposed background flux function hence, from Equations (1) and (2), hR + uQ must sum takes the form: to zero, giving 1/2 −y2/2L2 A0(y) = −e HVALme m , (8)  0 for Q < 0 where the background Alfvénspeed, VA, determines the R = (6) uQ background magnetic field strength of the system. We − h for Q ≥ 0, 4 Hindle et al.

Figure 2. Contours of g(h − H) are plotted for (quasi-)steady solutions, with a forcing amplitude of ∆heq/H = 0.001 (linear regime), and the radiative/drag timescales τrad = τdrag = 1 Earth day. Wind velocity vectors are overplotted as black arrows, lines of constant horizontal magnetic flux (A) are overplotted as white lines (with solid/dashed lines representing positive/negative magnetic field values), and hotspots (maxima of h on the equatorial line) are marked by white crosses. The system origin lies at the substellar point and velocity vectors are independently normalised for each subplot. set the latitudinal decay length of the magnetic field to the system’s dynamical timescale (τdyn/τη ∼ 0.08) and √ 1/2 Lm = Leq/2, where Leq ≡ ( gH/β) is the equatorial a dynamically relevant quasi-steady state emerges, be- Rossby deformation radius. fore diffusion causes the magnetic field to decay. We Following (PBS13), we choose parameters to match present numerical SWHD and SWMHD solutions in those typical of HD 189733b where possible. This HJ these steady and quasi-steady states respectively and has a planetary radius R = 8.2 × 107 m, a planetary plot g(h − H), the geopotential above the nightside- −5 −1 rotation rate, Ω = 3.2 × 10 s , and gravity waves equilibrium reference state, in Figure2 for VA = 0 and √ −1 √ with a speed of gH = 2 km s (PBS13). The vis- VA = gH/4 (top/bottom panels respectively). Energy cous and magnetic diffusivities are assigned the con- (heat) redistribution is traced via the geopotential, with stant values of ν = 108 m2 s−1 and η = 3 × 107 m2 s−1 high geopotential regions analogous to high temperature respectively. These are typical values in the radiative regions (PBS13). √ zones of HJs but, in reality, the day-night temperature Strikingly, the quasi-steady solution for VA = gH/4 variations on HJs cause longitudinal variations in dif- (lower panel of Figure2) exhibits a westward hotspot fusion coefficients, which can be orders of magnitude offset (marked by a white cross). This is in stark con- for η. We fix the atmospheric pressure scale height trast to SWHD systems (and SWMHD systems with √ H = 400 km (PBS13) and vary the background mag- VA gH/4), which always have an eastward hotspot netic field strength via the free parameter VA, present- offset. ing solutions in the weakly-forced (∆heq/H = 0.001) Solutions in this “strong field limit” have larger geopo- and therefore approximately linear regime, with radia- tential gradients, caused by the role of magnetic tension tive/drag timescales corresponding to moderately effi- (geopotential gradients increase sharply as V is raised √ A cient energy redistribution (τrad = τdrag = 1 Earth day) beyond VA = gH/4), and the shape of the geopotential (PBS13). contours undergoes a phase transition as the magnetic After an initial transient period, SWHD solutions field is increased: the eastward-pointing chevron-shaped reach steady state. For SWMHD systems, the mag- contours, in the zero or weak field regime, transition netic diffusion timescale is relatively large compared to into the westward-pointing chevron-shaped contours in Shallow water magnetohydrodynamics for westward hotspots on hot Jupiters 5 the strong field limit. Since SP11 showed the eastward- pointing chevron-shaped flow patterns to play a major V 2k2 β2 2k3βV 2 role in the formation of eastward zonal jets, this latter µ = A + + A R2gH gH R2ω(ω2 − gHk2) point is of particular interest concerning wind reversals. (10) k3βV 2 1/2 + A + d2 , R2ω3 4. THE MAGNETIC MODIFICATION OF gHV 2k4 EQUATORIAL SHALLOW WATER WAVES d = A . (11) R2ω2(ω2 − gHk2) In a hydrodynamic study, SP11 showed that SWHD systems will always produce eastward equatorial jets In Equations (9)–(11) the azimuthal wavenumber, k, that are driven by interactions between the mean flow and oscillation frequency, ω, are linked by the disper- and the linear equatorial SWHD waves. The dominant sion relation: standing, planetary-scale equatorial waves induced by ω2 kβ gHV 2k4 day-night thermal forcing are the n = 1 Rossby and − k2 − − A = (2n + 1)µ, (12) gH ω R2ω2(ω2 − gHk2) Kelvin waves. The superposition of these modes causes the emergence of eastward-pointing chevron-shaped for n = 0, 1, 2, 3,... . (geopotential and velocity) phase tilts (e.g. upper panel, We solve Equation (12) using numerical root finding Figure2). These cause eddies to pump eastward mo- techniques and find that, as in hydrodynamic theory mentum from high latitudes to the equator, driving an (e.g. Matsuno 1966), there are two bounded n = 0 solu- eastward equatorial jet. tions and three bounded n ≥ 1 solutions. Completeness The question addressed in this section is how this pro- is obtained by replacing the missing/third n = 0 so- cess is modified in the presence of a magnetic field. We lution with a magneto-Kelvin solution, which has the show that magnetism can cause the superposition of characteristic v = 0 everywhere and is often called the planetary-scale, free equatorial SWMHD waves to ac- n = −1 mode (e.g. Matsuno 1966; Zaqarashvili 2018). quire phase tilts that are opposite in direction to their Setting v = 0 everywhere and seeking non-trivial so- SWHD counterparts, then link this to a reversal mech- lutions to the linearised versions of Equations (1)–(3), 2√ 1/2 anism. one finds (for VA < (βR gH) ) the single bounded We linearise Equations (1)–(3), in the absence of forc- magneto-Kelvin solution: ing, drag, and diffusion, about the background state 2 2 1/2 {u0, v0, h0,Bx,0,By,0} = {0, 0,H,B0(y), 0}, where H is V y ω(y) = k gH + A , (13) constant and B0 = VAy/R. This system has previously R2 been solved by Zaqarashvili(2018), who studied it in V 2 β y2 terms of the solar tachocline, and we repeat this analysis uˆ ∝ exp A − √ , (14) R2gH gH 2 for the HJ parameter space (see Section3 for parameter choices), highlighting important features concerning where the exponential profile’s argument is approxi- HJs. 4 4 4 2 mated with accuracy O(VAy /R (gH) ). Perturbations to the background state are sepa- Due to the forms of heq, the linearised continu- rated using the plane wave ansatz, {u1, h1, B1} = ity equation and the eigenfunctions’ derivatives (e.g. {uˆ(y), hˆ(y), Bˆ (y)}ei(kx−ωt). The resulting ODE, (with Abramowitz & Stegun 1965), the n = ±1 modes are al- terms up to y2 only1) is then solved using the bound- ways expected to play an important role in systems with ary conditions, |v| → 0 as |y| → ∞, yielding bounded Leq ∼ R (as found on HJs). In Figure3 we plot geopo- solutions of the form (Zaqarashvili 2018): tential contours of the n = 1 (fast) magneto-Rossby mode (left panel) and the magneto-Kelvin mode (right 2 √ 2 panel), for k = 1/R (the forcing wavenumber) . vˆ (y) = H ( µy)e−(µ+d)y /2, (9) n n The structure of the (fast) magneto-Rossby modes do not significantly vary from their hydrodynamic vari- where d + µ > 0, Hn(ξ) is the Hermite polynomial of ants (see Matsuno 1966), although their deformation order n, for n = 0, 1, 2, 3,... , and length does increase with increasing VA. The magneto-

1 2 2 2 2 2 2 The implied assumptions, that |VAk y /ω R | 1 and The East/West magneto-inertial gravity waves remain similar 2 2 2 2 2 2 |VAk y /(ω − gHk )R | 1, remain valid for the discussed to their SWHD forms, which are known to provide an insigniﬁcant solutions. contribution to linear solutions (Matsuno 1966). 6 Hindle et al.

Figure 3. Geopotential perturbations contours, gh1, and vectors of velocity perturbations,√ u1, for two different mode types at k = 1/R. The n = 1 (fast) magneto-Rossby mode is plotted (left panel) for VA = gH/4. The magneto-Kelvin mode is plotted √ 2 (right panel) for VA = gH/4 at t = τtransf ≡ VAτadv/gH. Plots are made for the parameter choices discussed in Section3. Kelvin mode is the most significantly magnetically- the magneto-Kelvin wave’s structure, the resultant su- modified free wave solution as it acquires a latitudinally- perposition of magneto-Kelvin and n = 1 magneto- dependent contribution to its dispersion relation, which Rossby standing waves has a structural form resembling causes the wave to structurally shear as it propagates a westward-pointing chevron (such as the one seen in eastwards. We estimate the degree of structural shear Figure2). This change in the wave structure would re- in linear quasi-steady solutions by plotting the free verse the sign of the convergence of the meridional flux 2 magneto-Kelvin wave at τtransf ≡ VAτadv/gH, the of zonal eddy momentum. Hence, the structural phase timescale for the wave to transfer a local thickness tilts caused by the waves would pump eastward momen- perturbation, h1, to surrounding regions in the strong tum from the equator to higher latitudes and, provided field limit3 this pumping remains the dominant zonal acceleration We find that the structural deformation of the process, this would drive a westward equatorial jet. magneto-Kelvin wave becomes qualitatively significant We comment that this assumption cannot be guaran- √ at VA ∼ gH/4. This transition in nature is consis- teed without consideration of the forced linear solutions, tent with the numerical solutions discussed in Section3, which we omit from this Letter, but will investigate in which also transitions in nature, obtaining a westward- a future paper. √ chevron phase shift at VA ∼ gH/4. We hence propose the following adjustment to the 5. DISCUSSION mechanism of SP11 to account for magnetism: the hydrodynamic mechanism remains valid for low to We have demonstrated that magnetically modified moderate toroidal field strengths, however, when the waves lead to westward directed winds in a SWMHD toroidal field strength becomes large enough to deform model. We found that the SWMHD model we pre- sented can capture the physics of magnetically-induced wind reversals, which have only previously been stud- 3 We estimate τtransf by considering approximate scalings of ied via full three-dimensional MHD simulations (Rogers terms in the continuity equation (h1/τtransf ∼ HU/L) and momentum equation for a rotationless non-diffusive SWMHD model. & Komacek 2014; Rogers 2017). We showed that the magnetic modification of the planetary-scale equatorial For SWHD models and moderately magnetic√ SWMHD models, U/τtransf ∼ gh1/L, hence τtransf = Leq/ gH (PBS13). We find waves causes the superposition of the magneto-Kelvin numerically that for strong magnetic fields the pressure gradient 2 and n = 1 magneto-Rossby waves to reverse in struc- and Lorentz force approximately balance, yielding gh1/L ∼ VA/L, 2 ture in the strong field limit. Hence we used arguments hence τtransf = VAτadv/gH, where τadv ≡ Leq/U is the hydrodynamic advection timescale defined in PBS13. of simple linear wave dynamics to explain the magnetic wind reversal mechanism. Shallow water magnetohydrodynamics for westward hotspots on hot Jupiters 7

Understanding the magnetic-reversal mechanism in greatly exceeds 250 G, the maximum surface dipole esti- terms of a shallow MHD phenomenon provides informa- mate for HJs (Yadav & Thorngren 2017). We conclude tion about the magnetic fields on HJs. Repeating the that wind reversals on HAT-P-7b are highly likely to numerical analysis of Section3 in the parameter spaces be magnetically-driven, whereas other explanations such of HAT-P-7b and CoRoT-2b, we find that the mini- as cloud asymmetries (Demory et al. 2013; Parmentier mum toroidal field strengths sufficient to magnetically et al. 2016; Lee et al. 2016; Roman & Rauscher 2017) 1/2 reverse winds are Bφ,HAT-P-7b & 3 (P/ 1 bar) kG and or asynchronous rotation (Rauscher & Kempton 2014) 1/2 Bφ,CoRoT-2b & 1 (P/ 1 bar) kG, where P is the atmo- appear more plausible on CoRoT-2b. spheric pressure/depth of the reversal and the ideal gas There are several interesting questions that we do not law is used to convert from velocity units. These minima address in this Letter. First, it is unclear how a highly can be linked to dipolar field strengths using the scal- temperature dependent (and hence horizontally vary- ing laws of Menou(2012), yielding Bdip,HAT-P-7b & 6 G ing) magnetic Reynolds number will effect the toroidal- and Bdip,CoRoT-2b & 3 kG. We comment that the strik- poloidal scaling relationship, and hence the dynamics of ing difference between the two dipole field minima is the wind reversal process. Furthermore, vertical mag- a consequence of the temperature dependence of the netic fields have also been assumed to be small compared magnetic Reynolds number (Perna et al. 2010; Menou to horizontal fields in our model. Three-dimensional 2012). The minimum dipole strength in the atmosphere simulations are required to avoid this approximation. of HAT-P-7-b agrees with the three-dimensional simulations of Rogers(2017) and lies well below the range We acknowledge support from STFC for A. W. Hin- 50–100 G predicted for most inflated HJs (Yadav & dle’s studentship, the Leverhulme grant RPG-2017-035 Thorngren 2017). The dipole field strength necessary and thank Andrew Gilbert, Andrew Cummings, and Na- to magnetically-reverse the winds on CoRoT-2b (3 kG) talia Gómez-Pérezfor useful conversations leading to the development of this manuscript.

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