Modified Rayleigh-Benard convection driven by long-wavelength heating from above and below

Classical Rayleigh-Benard convection occurs when a horizontal fluid layer is heated sufficiently strongly from below. In more recent times, there has been increased interest in structured convection, when the heating that is applied is no longer uniform but rather spatially varying in some way. Here, we examine the effect of structured convection in a fluid layer when both the lower and upper boundaries are heated so that their temperatures fluctuate sinusoidally over a common long length scale offset by some phase . While this non-uniform heating can be shown to induce small fluid motions by way of a primary form of convection, some previous computations have shown that the layer is also susceptible to a stronger, secondary form of convection. When the heating is just sufficient to induce this secondary motion, the cells tend to conglomerate near the local hot spots where the underlying heating is at its most intense. The strength of the secondary convection falls off away from the hot spot on a length scale which is appreciably longer than the wavelength of the individual cells but also much shorter than the wavelength of the underlying applied heating. In this work, we derive a second-order amplitude equation that describes the secondary convection and discuss the important features of its solution. These are compared with some direct numerical simulations and are likely to apply for other situations in which long-scale heating gives rise to structured convection patterns.