The influence of topography upon rotating magnetoconvection
Aspects of thermal convection in the Earth's fluid core in the presence of a strong azimuthal magnetic field may be understood by considering a horizontal plane layer, rotating about the vertical z axis, with gravity acting downwards and containing an applied magnetic field aligned in the y (azimuthal) direction. Since the OMB is not smooth, the effects of adding bumps (with axes perpendicular to the applied magnetic field) to the top boundary of the layer are investigated in the magnetogeostrophic limit. The arbitrary geostrophic flow that arises under this limit is evaluated using a modified Taylor constraint. The bumps distort the isotherms so that they are not aligned with equipotential surfaces, leading to an imperfect configuration. This means that a hydrostatic balance is not possible, and motion ensues. This motion takes the form of a steady transverse convection roll, with axis parallel to the bumps. The roll exists for all values of the Rayleigh number, except that value for which the corresponding homogeneous problem in the standard plane layer has a solution. The roll obeys Taylor's constraint, and has no associated geostrophic flow. The stability of this roll to perturbation by oblique rolls (which are preferred for 0(1) values of the Elsasser number) is considered. It is found that the most unstable linear mode consists of a pair of these oblique rolls, aligned so that no geostrophic flow is accelerated by their interaction with the basic state. Hence, the stability results obtained here are identical to those found by perturbing the hydrostatic conduction solution with oblique rolls in the standard layer. Finally, the nonlinear evolution through the Ekman regime of these linear instabilities is considered. It is found that the nonlinear convection behaves similarly to mean field dynamo models which incorporate a geostrophic nonlinearity. Various types of Ekman solution are found, and evolution to Taylor states is observed.