The 1: 2 Mode Interaction in Rayleigh-bÉnard convection with and without Boussinesq Symmetry

Nonlinear two-dimensional Rayleigh–Bénard convection with periodic boundary conditions in the horizontal is studied for spatial periods near the 1:2 steady state mode interaction. The boundary conditions at the bottom are no-slip, and convection is driven by a fixed imposed temperature difference across the layer. Homotopic continuation is used to continue the boundary conditions at the top from no-slip (β = 0) to stress-free (β = 1). When β = 0 and nonBoussinesq effects are absent the system has midplane reflection symmetry and the 1:2 resonance is weak. When β = 1 this symmetry is strongly broken and the resonance is strong. The transition between these two cases is explored for two Prandtl numbers, σ = 10 and σ = 0.1, representing behavior typical of large and low Prandtl numbers, respectively.