LOCALIZED ROTATING CONVECTION INDUCED BY TOPOGRAPHY

The onset of instability of a rapidly rotating, self-gravitating, Bous sinesq fluid in a spherically symmetric cavity containing a uniform di stribution of heat sources in the small Ekman number limit (E much les s 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 deform ed. Attention is restricted to topographic features with simple longit udinal 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 lo cked to the topography. Analytic solutions are obtained for the case o f rigid boundaries when E(1/2) much less than epsilon(2) much less tha n 1 and E(-1/8) much less than m much less than E(-1/3); onset of inst ability is characterised by these modes (with geostrophic motion local ised radially on a length scale O(E(1/8))) when epsilon(2) much greate r than E(2/3)m(3/2). Solutions are obtained for the case of slippery b oundaries in different parameter ranges and, in contrast, these are no t localised but fill the sphere. In both cases the critical Rayleigh n umber grows with decreasing E, localising the convection of heat in a neighbourhood close to the surface.