Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries

The convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer. It is shown that when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth. Following Childress & Spiegel (1980) the disparity of scales is exploited in an expansion scheme that results in a nonlinear evolution equation for the leading-order temperature perturbation. This equation which does not depend on the vertical co-ordinate, is solved analytically where possible and numerically where necessary; most attention is concentrated on solutions representing two-dimensional rolls. It is found that for any given heat flux a continuum of steady solutions is possible for all wave numbers smaller than a given cut off. Stability analysis reveals, however, that each mode is unstable to one of longer wavelength than itself, so that any long box will eventually contain a single roll, even though the most rapidly growing mode on linear theory has much shorter wavelength.