Nonlinear Bifurcations to Time-dependent Rayleigh?Benard Convection

Abstract

The single-mode equations of Boussinesq thermal convection have been extended to include a toroidal component of velocity and hence the associated vertical component of vorticity. This formulation allows, under certain determined conditions, the purely poloidal solutions to become unstable to toroidal perturbations via symmetry breaking bifurcations. The bifurcation sequences are governed by a three parameter family: the aspect ratio of the convection cell, the Prandtl number of the fluid and the Rayleigh number of the flow. The initial growth of the vertical vorticity has been found always to be steady. However, in certain parameter ranges there are transitions leading to time-dependent behaviour via a Hopf bifurcation which may be in the form of symmetrical oscillations, asymmetrical oscillations, doubly-periodic behaviour or, possibly, chaos, depending on the form of the transient poloidal phase of the evolution.