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Register a new userRossby waves in shallow water magnetohydrodynamics
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The influence of a toroidal magnetic field on the dynamics of Rossby waves in a thin layer of ideal conductive fluid on a rotating sphere is studied in the shallow water magnetohydrodynamic approximation for the first time. Dispersion relations for magnetic Rossby waves are derived analytically in Cartesian and spherical coordinates. It is shown that the magnetic field causes the splitting of low order (long wavelength) Rossby waves into two different modes, here denoted fast and slow {em magnetic Rossby waves}. The high frequency mode (the fast magnetic Rossby mode) corresponds to an ordinary hydrodynamic Rossby wave slightly modified by the magnetic field, while the low frequency mode (the slow magnetic Rossby mode) has new and interesting properties since its frequency is significantly smaller than that of the same harmonics of pure Rossby and Alfv{e}n waves.
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Westward winds have now been inferred for two hot Jupiters (HJs): 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 HJs 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 $B_{text{dip},text{HAT-P-7b}} gtrsim 6 , mathrm{G} $, a result that is consistent with three-dimensional magnetohydrodynamic simulations and lies below typical surface dipole estimates of inflated HJs. The same analysis suggests the minimum dipole field strength required to reverse the winds of CoRoT-2b is $B_{text{dip},text{CoRoT-2b}} gtrsim 3 , mathrm{kG}$, which considerably exceeds estimates of the maximum surface dipole strength for HJs. 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.
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 $4mbox{ G}$ and $19mbox{ G}$; for WASP-12b and WASP-33b westward hotspots can be explained by $1mbox{ G}$ and $2mbox{ G}$ 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.
We derive analytical solutions and dispersion relations of global magnetic Poincare (magneto-gravity) and magnetic Rossby waves in the approximation of shallow water magnetohydrodynamics. The solutions are obtained in a rotating spherical coordinate system for strongly and weakly stable stratification separately in the presence of toroidal magnetic field. In both cases magnetic Rossby waves split into fast and slow magnetic Rossby modes. In the case of strongly stable stratification (valid in the radiative part of the tachocline) all waves are slightly affected by the layer thickness and the toroidal magnetic field, while in the case of weakly stable stratification (valid in the upper overshoot layer of the tachocline) magnetic Poincare and fast magnetic Rossby waves are found to be concentrated near the solar equator, leading to equatorially trapped waves. However, slow magnetic Rossby waves tend to concentrate near the poles, leading to polar trapped waves. The frequencies of all waves are smaller in the upper weakly stable stratification region than in the lower strongly stable stratification one.
The influence of a toroidal magnetic field on the dynamics of shallow water waves in the solar tachocline is studied. A sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to trapping of the waves near the equator and to an increase of the Rossby wave period up to the timescale of solar cycles. Dispersion relations of all equatorial magnetohydrodynamic (MHD) shallow water waves are obtained in the upper tachocline conditions and solved analytically and numerically. It is found that the toroidal magnetic field splits equatorial Rossby and Rossby-gravity waves into fast and slow modes. For a reasonable value of reduced gravity, global equatorial fast magneto-Rossby waves (with the spatial scale of equatorial extent) have a periodicity of 11 years, matching the timescale of activity cycles. The solutions are confined around the equator between latitudes 20-40, coinciding with sunspot activity belts. Equatorial slow magneto-Rossby waves have a periodicity of 90-100 yr, resembling the observed long-term modulation of cycle strength, i.e., the Gleissberg cycle. Equatorial magneto-Kelvin and slow magneto-Rossby-gravity waves have the periodicity of 1-2 years and may correspond to observed annual and quasi-biennial oscillations. Equatorial fast magneto-Rossby-gravity and magneto-inertia-gravity waves have periods of hundreds of days and might be responsible for observed Rieger-type periodicity. Consequently, the equatorial MHD shallow water waves in the upper overshoot tachocline may capture all timescales of observed variations in solar activity, but detailed analytical and numerical studies are necessary to make a firm conclusion toward the connection of the waves to the solar dynamo.
68 -
Enrique cerda
, Fernando Lund (Departamento de Fisica
, Facultad den Ciencias Fisicas y Matematicas
1993
Vortical flows in shallow water interact with long surface waves by virtue of the nonlinear terms of the fluid equations. Analytical formulae are derived that quantify the spontaneous generation of such waves by unsteady vorticity as well as the scattering of surface waves by vorticity. In a first Born approximation the radiated surface elevation is linearly related to the Fourier transform of the vorticity. The ``dislocated wavefronts that are analogous to the Aharonov-Bohm effect are obtained as a special case.
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