Differential rotation and thermal convection in a rapidly rotating hydromagnetic system

Abstract

Abstract As a model for hydromagnetic waves in the Earth's core, we study the linear stability of an electrically conducting fluid confined in a cylindrical annulus. The system is rotating rapidly about the axis of the cylinder with angular velocity ω0 equals; ω012. and the fluid is differentially rotating with velocity U0 equals;U0(s*)lø relative to the rotating frame of reference. [Here, (s*, ø, z*) are cylindrical polar coordinates and 1 x is the unit vector in the direction of increasing x.] A magnetic field B0 equals; B 0(s*)lø and temperature distribution T0(s*) are imposed on the fluid. In an earlier series of papers (Fearn, 1983b, 1984, 1985, 1988a,b) we focused attention on instabilities driven by the field B0. Here we study two facets of buoyancy driven waves. The first is the role of differential rotation. It can act to inhibit convection but may also itself act as a source of energy to drive instability. For values of the Roberts number qequals;k/η ≧ O(1), (k and η are the thermal and magnetic diffusivities), as the strength of U0 is increased, a smooth transition from a buoyancy driven mode of instability to a mode driven by U0 is observed. The second facet is the relationship between the diffusive mode of instability, which is always the preferred mode, and the ideal MAC wave mode. Their relationship is investigated analytically using a narrow gap limit, and numerically. As the unstable temperature gradient is increased, the diffusive mode evolves smoothly into the MAC wave mode.