The Propagation and Stability of Linear Wave Motions in Rapidly Rotating Spherical Shells: Weak Magnetic Fields

The propagation properties and stability of wave motions in a spherical fluid shell, rotating uniformly in a co-rotating zonal magnetic field, are studied. Various profils of magnetic field and temperature gradient are examined. The fluid is viscous and electrically and thermally conducting. The analysis is applicable when the rotation rate is very high and the Elsasser number A- 1. (A measures the ratio of Lorentz to Coriolis forces). The wave motions occur in the form of annular cylindrical cells whose thickness is determined by the rotation rate and magnetic field amplitude (Eltayeb and Kumar, 1977). It is shown that an infinity of modes exists and both westward and eastward phase propagation is possible. The sloping boundary of the spherical shell tends to oppose westward propagation by imparting an eastward phase speed, of the same magnitude for all modes of the same wavenumber, whose magnitude is proportional to the colatitude of the annular cell. The infinity of modes can be divided into two distinct categories depending on their parity with respect to the equatorial plane. The even (odd) solution is associated with a temperature perturbation and axial vorticity which are symmetric (anti-symmetric) with respect to the equatorial plane. Both categories of solutions are present outside the annular cylindrical surface, C touching the inner core at its equator but only the analytic continuation of the odd mode is present inside Cc. This result is used to clarify the relationship between wave motions in thin and thick spherical shells. The stability of the waves is investigated and the effects of the profiles of the ambient magnetic field and basic temperature gradient as well as the presence of the inner core on the preferred mode of convection, are studied. The influence of the magnetic number q(= k/n, where k and n are the thermal and magnetic diffusivities, respectively) on the location and direction of phase propagation of the unstable wave is also examined. In particular the changeover of the critical mode from an eastward to a westward propagating magnetic mode at a value qo of q is examined in detail. It is found that qc, which signifies a change from buoyancy Rossby waves (q<q0) to slow magnetohydrodynamic waves (q=q0), depends on the profiles of magnetic field and temperature gradient. The changeover can be accompanied by a discontinuity in the colatitude θC or occur smoothly in C depending on the basic temperature gradient.