The effects of slow spatial variations on Benard convection

The onset of thermal convection in a fluid layer is considered when the temperature difference between the boundaries is modulated on a scale much larger than the separation of the horizontal boundaries. Using a two-scaling method, it is shown that the disturbance is largely confined to the neighbourhood of 'hot-spots' where the temperature difference between the boundaries is greatest. The increase in the critical Rayleigh number can be predicted by means of a local analysis. In an exceptional case in which the amplitude of the variation is small the disturbance is wider-ranging, and a local analysis is not sufficient to predict the increase in the critical Rayleigh number.