Finite-amplitude thermal convection and geostrophic flow in a rotating magnetic system

Abstract

An electrically conducting Boussinesq fluid is confined between two rigid horizontal planes. The fluid is permeated by a strong uniform horizontal magnetic field and the entire system rotates rapidly about a vertical axis. By heating the fluid from below and cooling it from above the system becomes unstable to small perturbations when the adverse temperature gradient becomes sufficiently large. Attention is restricted to small values of the Ekman number E and the ratio q of the thermal and magnetic diffusivities (see (1.2) and (1.3) below). In this parameter range marginal convection is steady and its character depends on the relative sizes of the Coriolis and Lorentz forces as measured by the parameter λ (see (1.1) below). When λ [ges ] 2/3½, motion consists of a single roll, whose axis is perpendicular to the applied magnetic field. On the other hand, when λ < 2/3½, two distinct rolls are possible: the axis of each roll lies oblique but with equal angle to the applied magnetic field. Only the latter case is discussed here.Once the Rayleigh number R exceeds its critical value Rc only one of the two sets of single rolls remains stable, while its squared amplitude increases linearly with R – Rc. For certain values of the parameters λ and τ (see (1.6) below) a second critical value may be isolated at which the system becomes unstable to unidirectional geostrophic flow perturbations aligned with the applied magnetic field. The instability sets in as either a steady or oscillatory shear flow dependent on the values taken by λ and τ. In both cases, after the secondary instability sets in, the roll amplitude remains largely insensitive to further increase in the Rayleigh number with the consequence that the geostrophic flow is stabilized. The amplitude of the shear, on the other hand, increases with R, adjusting its magnitude to ensure stability of the convection rolls.