HYDROMAGNETIC-WAVES IN RAPIDLY ROTATING SPHERICAL-SHELLS GENERATED BYMAGNETIC TOROIDAL DECAY MODES

Instabilities in the form of slow azimuthally travelling hydrodynamic waves in a rapidly rotating, stress-free, electrically conducting sphe rical fluid shell are investigated. The instabilities are generated by the toroidal decay mode of the lowest order or a combination of toroi dal decay modes. It is found that the Elsasser number Lambda(c) at the onset of instability is always Lambda(c)=O(10) for various profiles o f the basic magnetic field. It is also found that the hydromagnetic wa ves of the preferred instability propagate eastward [i.e. for solution s proportional to exp i(m phi + omega t), (omega> < 0] and are charact erized by nearly two-dimensional columnar fluid motions attempting to satisfy the Proudman-Taylor theorem, indicating that the most rapidly growing magnetic disturbance arranges itself in such a way that the co rresponding magnetic forces balance only the ageostrophic component of the Coriolis force. Except for the Stewartson-type velocity boundary layer at the equator of the inner core, the velocity and magnetic fiel d of the most unstable mode are always smooth, with length scale compa rable with the shell width. By studying the same system with and witho ut fluid inertia, we show that fluid inertia cannot introduce any new instability of physical relevance or significantly change the main Fea tures of the instability when T-m > 10(5), T-m being the magnetic Tayl or number. By studying the detailed dependence of the instabilities in the whole range 0 < T-m less than or equal to infinity, we are able t o demonstrate that the existence of the Stewartson-type layer is unlik ely to affect the primary properties of the instabilities. In the case of the quadruple symmetry mode, there is no indication of a Stewartso n-type boundary layer; a result of the symmetry properties. For suffic iently large T-m, the most rapidly growing instability always has dipo le symmetry (which allows the formation of two-dimensional fluid motio ns). We have compared our resuls (in the limit T-m --> infinity) with those found using the magnetostrophic approximation. Calculations usin g the latter do not impose stress-free boundary conditions and slightl y lower values of the critical Elsasser number result. Otherwise, the solutions found are broadly similar.