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 spherica l fluid shell, rotating uniformly in a co-rotating zonal magnetic fiel d, are studied. Various profils of magnetic field and temperature grad ient 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 LAMBDA << 1. (LAMBDA 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 rotatio n rate and magnetic field amplitude (Eltayeb and Kumar, 1977). It is s hown that an infinity of modes exists and both westward and eastward p hase propagation is possible. The sloping boundary of the spherical sh ell tends to oppose westward propagation by imparting an eastward phas e speed, of the same magnitude for all modes of the same wavenumber, w hose magnitude is proportional to the colatitude of the annular cell. The infinity of modes can be divided into two distinct categories depe nding 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 equ atorial plane. Both categories of solutions are present outside the an nular cylindrical surface, C(c), touching the inner core at its equato r but only the analytic continuation of the odd mode is present inside C(c). This result is used to clarify the relationship between wave mo tions in thin and thick spherical shells. The stability of the waves i s investigated and the effects of the profiles of the ambient magnetic field and basic temperature gradient as well as the presence of the i nner core on the preferred mode of convection, are studied. The influe nce of the magnetic number q(= kappa/eta, where kappa and eta 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 q(o) of q is examined in detail. It is found that q(o), which signifies a change from buoyan cy Rossby waves (q < q(o) to slow magnetohydrodynamic waves (q > q(o)) , depends on the profiles of magnetic field and temperature gradient. The changeover can be accompanied by a discontinuity in the colatitude theta(c) or occur smoothly in theta(c) depending on the basic tempera ture gradient.