Rayleigh-Benard convection driven by a long wavelength heating

Natural convection in a two-dimensional horizontal layer has been investigated. The layer is confined between two parallel horizontal plates. The upper plate is kept isothermal, while the lower plate has an externally imposed, long wavelength, spatially sinusoidal heating with the amplitude expressed in terms of the Rayleigh number Ra and the wavelength characterized by the wave number alpha. Only steady-state flow structures and their bifurcations have been considered. The detailed analysis has been carried out for two Prandtl numbers, i.e. Pr = 0.7 and Pr = 7, and only small differences in the bifurcation diagrams have been observed. When Ra < Ra (cr) = 427, convection has a simple topology consisting of one pair of counter-rotating rolls per heating period. Secondary motion in the form of rolls aligned in the direction of the primary rolls and concentrated around the hot spots occurs for Ra > 427. When 427 < Ra < similar to 470 and alpha < similar to 0.14, the secondary motion is described by the supercritical pitchfork bifurcation. One of the branches of this bifurcation is associated with an odd number of secondary rolls per half wavelength, with rolls above the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half wavelength, with the rolls above the hot spots co-rotating with the primary rolls. The new rolls are pinched off in pairs when alpha decreases. When Ra > similar to 470 and alpha > similar to 0.14, bifurcation assumes the form of "bifurcation from infinity". The main branch is associated with one pair of rolls per heating period for alpha > 0.25. Decrease in alpha along this branch results in the formation of secondary rolls, with the rolls at the hot spot co-rotating with the primary rolls. The lower part of the other branch is associated with one pair of rolls per heating period in the limit alpha -> 0. Increase in alpha results in pinching off a single roll which counter-rotates with respect to the primary roll at the hot spot.