Local analysis of thermal and magnetic instabilities in a rapidly rotating fluid

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Abstract
We examine by means of a local analysis the effects of an azimuthal magnetic field B(r) on the stability of a rapidly-rotating fluid subject to a radial temperature gradient, taking the ratio κ/η of thermal to magnetic diffusivities to be small, as in the Earth's core. According to this theory, previous results for the case B ∝ r are typical of a certain range of magnetic field profiles, but if B decreases with r faster than r −½ we find instead that (i) the critical Rayleigh number increases sharply as the magnetic field strength increases beyond “magnetostrophic” values and (ii) the recently-discovered magnetic instabilities triggered by bottom-heavy density gradients do not occur. If B increases with r faster than r 3/2, on the other hand, the major change to the B∝r picture of events is that the system becomes unstable to comparatively fast magnetic instabilities as soon as the field passes magnetostrophic values.

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