Planform selection by finite-amplitude thermal convection between poorly conducting slabs

Three-dimensional finite-amplitude thermal convection in a fluid layer is considered in the case where the boundaries of the layer are much poorer conductors than the fluid. It can be shown that if the conductive heat flux through the layer is not too large, the horizontal scale of motion is much greater than the layer depth. Then a ‘shallow water theory’ approximation leads to a nonlinear evolution equation for the leading-order temperature perturbation, which can be analysed in terms of a variational principle. It is proved that the preferred planform of convection is a square cell tesselation, as found in a rather more restricted parameter range by Busse & Riahi (1980), in contrast to the roll solutions that obtain for perfectly conducting boundaries. It is also shown that the preferred wavelength of convection increases slowly with amplitude.