Differential rotation and thermal convection in a rapidly rotating hydromagnetic system

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*)l_ø 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*)l_ø 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.

Key Words:

• Earth's core
• hydromagnetic waves
• MAC waves
• thermal convection
• shear instability