Mechanism of mode selection in Rayleigh—Bénard convection with free—rigid boundaries

Higher harmonic resonances with wavenumber ratio of 1:2, 1:2:3 and so on are shown to take place in Rayleigh—Bénard convection under free—rigid boundary condition. Bifurcation diagrams for two-dimensional motion are obtained for the Prandtl number P = 7. The subharmonic instability is explained by a couple of amplitude equations obtained from weakly nonlinear stability theory. A straightforward extension of the coupled amplitude equations leads to a model which consists of n amplitude equations. The mechanism of mode selection is illustrated by numerical simulations of the model equations.