Localised rotating convection induced by topography

The onset of instability of a rapidly rotating, self-gravitating, Boussinesq fluid in a spherically symmetric cavity containing a uniform distribution of heat sources in the small Ekman number limit (E much less than 1) is characterised by the longitudinal propagation of thermal Rossby waves on a short azimuthal phi-length scale O(E(1/3)). Here we investigate the onset of instability via a steady geostrophic mode of convection which may occur when the outer spherical boundary is deformed. Attention is restricted to topographic features with simple longitudinal dependence proportional to cos m phi and small height of order epsilon/m. Motion is composed of two parts: the larger is geostrophic and follows the geostrophic contours; the smaller is convective and locked to the topography. Analytic solutions are obtained for the case of rigid boundaries when E(1/2) much less than epsilon(2) much less than 1 and E(-1/8) much less than m much less than E(-1/3); onset of instability is characterised by these modes (with geostrophic motion localised radially on a length scale O(E(1/8))) when epsilon(2) much greater than E(2/3)m(3/2). Solutions are obtained for the case of slippery boundaries in different parameter ranges and, in contrast, these are not localised but fill the sphere. In both cases the critical Rayleigh number grows with decreasing E, localising the convection of heat in a neighbourhood close to the surface.