As a model for convection in the Earth's core we study the linear stability of a rapidly rotating electrically conducting spherical fluid shell, permeated by a toroidal magnetic field B. We look at the effect of introducing a stably stratified layer into the fluid adjacent to the core-mantle boundary (CMB). For values of the Elsasser number (a non-dimensional measure of the magnetic field strength), #x39B; #x226A; 0(1), convection can penetrate significantly into the stable layer. In this weak field case, the constraints of the Taylor-Proudman theorem cause convection to become columnar in structure. As #x39B; is decreased the azimuthal wave number m, corresponding to the most unstable mode, increases. For #x39B; = O(1), convection is still unaffected by the introduction of a stable layer, but is no longer columnar. For #x39B; #x226B; O(1), we find that convection becomes concentrated in the unstably stratified region. Except in the low magnetic field strength regime, our findings agree with previous work of Boda (1988) and #x160;ev#x10D;#xED;k (1989), who studied a similar problem with plane-layer geometry, and with Fearn and Richardson (1991) who considered the cylindrical case.
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