Short-scale convection and long-scale deformationally unstable Rossby wave in a rotating fluid layer heated from below

Abstract

A rotating fluid layer, heated from below, with a deformable upper and nondeformable lower stress free surfaces is considered in the Boussinesq approximation. The system of the differential equations that governs the long-scale Rossby waves and short-scale convection is obtained in the rapid-rotation approximation. Long-scale flows are unstable due to heating and deformation of the upper surface. The neutral stability curves for Rossby waves and convection are obtained for linearized version of the equations. In a slightly supercritical regime the amplitude equations for convection and Rossby waves are derived by the use of the method of multiscale expansions. The properties of the amplitude equations are discussed. The existence of the two weakly supercritical stationary convection regimes is shown by numerical integration of the equations in the rapid-rotation approximation. In one of them, the amplitude of short-scale convection is modulated due to long-scale deformation of the upper surface associated with the excitation of the Rossby wave. In the other regime, the presence of deformation gives rise to alternating regions with and without convection.