Nonlinear high-wavenumber Bénard convection

Weakly nonlinear two-dimensional roll cells in Bénard convection are examined in the limit as the wavenumber a of the roll cells becomes large. In this limit the second harmonic contributions to the solution become negligible, and a flow develops where the fundamental vortex terms and the correction to the mean are determined simultaneously, rather than sequentially as in the weakly nonlinear case. Extension of this structure to Rayleigh numbers O(a³) above the neutral curve is shown to be possible, with the resulting flow field having a form very similar to that for strongly nonlinear vortices in a centripetally unstable flow. The flow in this strongly nonlinear regime consists of a core region, and boundary layers of thickness O(a⁻¹) at the walls. The core region occupies most of the thickness of the fluid layer and only mean terms and cos az terms play a role in determining the flow; in the boundary layer all harmonics of the vortex motion are present. Numerical solutions of the wall layer equations are presented and it is also shown that the heat transfer across the layer is significantly greater than in the conduction state.