Wavelength selection in Rayleigh-Bénard convection

Rayleigh-Bénard convection is considered in the presence of a weakly nonuniform temperature distribution and weakly nonuniform layer height with their gradients perpendicular to the (parallel) convection rolls. In the infinite Prandtl number limit the phase equation that describes the slow spatial and temporal evolution of the local wave number is derived. Evaluating the equation near threshold and for stress-free boundary conditions we find nonuniversal wavenumber selection, forced phase diffusion with moving patterns, and other measurable effects.