Rayleigh-Bénard problem with imposed weak through-flow: Two coupled Ginzburg-Landau equations.

Abstract

The Rayleigh-B\'enard instability of a horizontal shear flow in a narrow channel is governed by two competing unstable modes: traveling transverse and stationary longitudinal convection rolls. The dynamics of the mode amplitudes obeys two coupled Ginzburg-Landau equations, which we derive in the present article. For mathematical simplicity we assume free-slip boundaries at the channel sidewalls. The analytical discussion and the numerical simulation of these amplitude equations show transitions between the two convective structures. Propagating fronts, which separate areas of transverse rolls and longitudinal ones, occur in the parameter range where both patterns are stable. Coexisting uniform states with nonzero amplitudes of both modes are unstable solutions of the equations. Results are in qualitative agreement with recent experiments.