Thermal and magnetically driven convection in a rapidly rotating fluid layer

Abstract

The stability of an electrically conducting Boussinesq fluid which is confined between two horizontal planes a distance d apart is investigated. The fluid is heated from below, cooled from above and the whole system rotates rapidly with angular velocity Ωc about a vertical axis. A weak non-uniform horizontal magnetic field, whose strength is measured by the Alfvén angular velocity ΩM [[Lt ] Ωc, see (1.2)] permeates the fluid and corresponds to the flow of a uniform electric current parallel to the rotation axis. When the modified Rayleigh number R [see (2.1)] is greater than zero and q = κ/λ < 1, where κ and λ are the thermal and magnetic diffusivities respectively, instability sets in as a westward-propagating wave with a low frequency of order κ/d2.When R = 0 and ΩM > 2(ν/λ)½ Ωc, where ν is the viscosity, Roberts & Loper (1979) have isolated an exceptional class of unstable fast inertial waves which grow on the magnetic diffusion time scale τλ = d2/λ. When R < 0 and Γ = τλΩ2M/τc, exceeds some value dependent upon q, a class of unstable slow waves also exists for a range of negative values of R. These waves propagate eastwards (westwards) when q is less (greater) than unity. In this case the fluid is stably stratified and the energy for the disturbance is taken from the magnetic field. The resulting description of the stability boundary for R < 0 in the Γ, R plane extends and clarifies the results of Roberts & Loper (1979), which are valid when both Γ and q are large.