On diffusive instabilities of a rapidly rotating electrically conducting layer of compressible fluid of varying depth

Abstract

Compressibility is added to Busse's (1976) study of convection in a rotating, electrically conducting layer of fluid, of varying depth, which is permeated by an azimuthal magnetic field that is orthogonal to both the rotation vector Ω and the gravitational acceleration g. On the basis of a linear theory, we investigate the kinds of infinitesimal wave motion which the fluid can support when the dominant balance of forces is between pressure gradient and Coriolis force, so that the Proudman–Taylor theorem holds. Unlike Busse, however, we assume that the fluid is statically stable in the sense that the temperature gradient is sub-adiabatic.In the absence of diffusion, these waves are dynamically neutral and take the form of Rossby waves modified by compressibility and magnetic field. The waves are examined in two limits, the adiabatic and the isothermal, and we define two distinct frequencies at which a pure Rossby wave can oscillate. When diffusion is restored, the disparity between these frequencies makes the fluid susceptible to overstability. We prove that all such amplifying waves must propagate eastward, i.e. in the direction of g ∧ Ω, irrespective of the sign of the depth gradient or the magnitudes of the diffusivities. The theorem does not apply if the fluid is incompressible or convectively unstable at the outset, and therefore does not contradict Busse's result that the direction of azimuthal propagation can be altered by the diffusivities. Nevertheless, we suggest reasons why Busse's method of regarding the imaginary part of the marginal stability equation as a dispersion relation is not in general a reliable one.We examine the instabilities by subjecting the neutral waves to a weakly diffusive perturbation. We discover, in particular, a new kind of magnetic instability which is crucially dependent upon both compressibility and the depth gradient. In agreement with the general result described above, the instability takes the form of a slow, eastward propagating, amplifying wave.The principal source for all instabilities is elastic energy, which cannot be tapped in a Boussinesq fluid, since the work done by compression is neglected under that approximation.