The asymptotic stability of a bounded rotating fluid heated from below: conductive basic state

Abstract

The asymptotic stability of a rapidly rotating, horizontally bounded fluid, heated from below, is treated using boundary-layer methods. It is shown that the rotational constraint is so strong as to preclude instabilities, if the interior regions of the fluid are considered to be inviscid. The correct formulation allows this constraint to be broken by introducing horizontal diffusive effects into the interior, while vertical diffusion is confined to Ekman layers on the horizontal surfaces; no vertical layers exist. Moreover, the mechanism of instability is (to the lowest order) associated with energy conversions entirely within the interior region. The present formulation elucidates the role of the Ekman layers in producing high-order corrections to the limiting critical Rayleigh number, and the asymptotic results are extended to include higher-order terms. The effect of rigid side walls on the critical Rayleigh number, and on the azimuthal wavenumber, is considered. Except for very tall cylinders, the critical Rayleigh number is unaffected by the presence of side walls; the results for different azimuthal modes of convection are inconclusive, but indicate that no great error occurs if disturbances are assumed axisymmetric.