Asymmetric squares as standing waves in Rayleigh-Bénard convection

Possibility of thermal convection in the form of asymmetric squares in a thin layer of Boussinesq fluids of large lateral extension confined between stress-free and conducting flat boundaries is investigated numerically using a seven mode Lorenz-like model. For fluids with moderate and high Prandtl numbers (4<sigma<20) and in a narrow window of high Rayleigh numbers (12<R/R(c)<15), which depend on Prandtl number, the stationary rolls become unstable and asymmetric squares appear as standing waves at the onset of secondary instability. Asymmetric squares, two-dimensional rolls, and again asymmetric squares with their corners shifted by half a wavelength form a stable limit cycle. The oscillatory bifurcation is supercritical.