Hydromagnetic convection in a rapidly rotating fluid layer

Abstract

The linear stability of a rotating, electrically conducting viscous layer, heated from below and cooled from above, and lying in a uniform magnetic field is examined, using the Boussinesq approximation. Several orientations of the magnetic field and rotation axes are considered under a variety of different surface conditions. The analysis is, however, limited to large Taylor numbers, T, and large Hartmann numbers, M. (These are non-dimensional measures of the rotation rate and magnetic field strength, respectively). Except when field and rotation are both vertical, the most unstable mode at marginal stability has the form of a horizontal roll whose orientation depends in a complex way on the directions and strengths of the field and angular velocity. For example, when the field is horizontal and the rotation is vertical, the roll is directed parallel to the field, provided that the field is sufficiently weak. In this case, the Rayleight number, R (the non-dimensional measure of the applied temperature contrast) must reach a critical value, $R_{\text{c}}$, which is $O(T^{\frac{2}{3}})$ before convection will occur. If, however, the field is sufficiently strong $[T=O(M^{4})]$, the roll makes an acute angle with the direction of the field, and $R_{\text{c}}=O(T^{\frac{1}{2}})$, i.e. the critical Rayleigh number is much smaller than when the magnetic field is absent. Also, in this case the mean applied temperature gradient and the wavelength of the tesselated convection pattern are both independent of viscosity when the layer is marginally stable. Furthermore, the Taylor-Proudman theorem and its extension to the hydromagnetic case are no longer applicable even qualitatively. Over the interior of the layer, however, the Coriolis forces to which the convective motions are subjected are, to leading order, balanced by the Lorentz forces. The results obtained in this paper have a bearing on the possibility of a thermally driven steady hydromagnetic dynamo.