## Abstract

Buoyancy-induced convection is investigated in a rapidly rotating, self-gravitating fluid sphere internally heated by a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance $\overline{\omega}$* from the rotation axis. Attention is restricted to relatively small magnetic field strengths (as measured by the parameter $\Lambda$) such that the dominant force balance remains geostrophic. Convection is then confined to a thin cylindrical annulus, radius $\overline{\omega}$*$_0$, about the rotation axis. A linear analysis is used to find the state of marginal stability together with the corresponding minimum critical value of the modified Rayleigh number, R$_c$ (a measure of the buoyancy force required to maintain convective motions). Two distinct modes of instability are found to operate: `Rossby' and `magnetic'. When no magnetic field is applied (\Lambda = 0), and when the Prandtl number $\sigma$ \simeq 1, the instability takes the form of a thermally driven Rossby wave propagating eastward but with group velocity westward. Modifications to this mode due to a non-zero magnetic field result in R$_c$ being a complicated function R$_c$ ($\Lambda$, q, $\sigma$), where $q = \kappa/\eta$ and $\sigma$ = $\nu/\kappa$ with $\nu$, $\kappa$ and $\eta$ denoting the viscous, thermal and magnetic diffusivities. When it has a significant effect \[$\sigma\leqslantO$(1)\], the magnetic field inhibits the Rossby mode and convection moves towards the axis where the field is weaker ($\overline{\omega}$*$_0$ \rightarrow 0). The presence of the magnetic field also permits new modes of instability which are facilitated by increasing field strength, with R$_c$ \propto \Lambda$^{-1}$. These magnetic modes propagate eastwards (westwards) when q < q$_m$(q > q$_m$) with q$_m$ \approx 2.8. Of particular interest to the Earth is the limit q \simeq 1 where the marginally stable magnetic mode takes the form of an undamped buoyancy wave in an unstably stratified fluid. No diffusive processes are operating at leading order and the only effect opposing convection is the geostrophic constraint.