Linear stability analysis of the convective flow in a spherical gap withη=0.714

Abstract

A linear stability analysis has been performed of the buoyancy-driven flow of a Boussinesq fluid in a spherical gap where the inner shell is warmer than the outer one (T1>T2) for the radii ratio η=R1R2=0.714. We show that the two-dimensional axisymmetric basic flow becomes unstable with respect to non-axisymmetric perturbations. The stability diagrams, critical Grashof number Grc and wave number mc versus the Prandtl number Pr are presented for both the steady and the oscillatory instabilities. In this way, the results bridge the gap to recent three-dimensional simulations performed in Scurtu et al. (2010), Feldman and Colonius (2013) for air in this η. Furthermore we investigate the energy exchange between the basic flow and perturbations in terms of a Reynolds–Orr-equation.