PATTERN SELECTION IN LONG-WAVELENGTH CONVECTION

The long wavelength of the first instability in Rayleigh-Bénard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems. The evolution of such an instability is described by the equation αt=αα-μ{down triangle, open}2α-{down triangle, open}4α+κ{down triangle, open}·{norm of matrix}{down triangle, open}α{norm of matrix}2{down triangle, open}α+β{down triangle, open}·{down triangle, open}2α{down triangle, open}α-λ{down triangle, open}·α{down triangle, open}α+σ{down triangle, open}2{norm of matrix}{down triangle, open} α{norm of matrix}2, where α is the planform function, μ is the scaled Rayleigh number and κ = ±1. The quantities α,β,λ represent the effects of finite Biot number, asymmetry in the boundary conditions at top and bottom, and departures from the Boussinesq approximation, respectively. The quantity σ=β when the original problem is self-adjoint, but σ ≠ β otherwise. Planform selection is studied for α < 0, κ = +1 using equivariant bifurcation theory. On the square lattice both rolls and squares can be stable, depending on the parameters β, λ and σ. The possible secondary bifurcations located near various codimension-two singularities are analyzed. On the hexagonal lattice the primary bifurcation is always degenerate when β=λ=σ=0. Of the six primary solution branches possible in this case the hexagon branch is stable. When β-σ=λ=0, both rolls and hexagons bifurcate supercritically, but rolls are stable. Finally, when β ≠ σ and/or ψ ≠ 0 a hysteretic transition to H+ or H- occurs depending on sgn(β + λ k2c-σ); if {norm of matrix}β{norm of matrix}, {norm of matrix}λ{norm of matrix}, {norm of matrix}σ{norm of matrix} ≪ stable H+, H- coexist at larger amplitudes, but if {norm of matrix}β + λ k2c-σ{norm of matrix} ≪ {norm of matrix}β{norm of matrix}, {norm of matrix}λ{norm of matrix}, {norm of matrix}σ{norm of matrix} = 0(1), a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls. © 1990.