Hexagons and rolls in periodically modulated Rayleigh-Bénard convection

A laterally infinite liquid layer is heated from below in a time-periodic fashion with dimensionless frequency ω and amplitude δ. A Lorenz-like truncation of the hydrodynamic equations is derived and used to study the pattern competition between hexagons and rolls near threshold. This approximation yields a realistic estimate, valid for arbitrary ω and δ, of the jump in convection amplitude at the subcritical bifurcation from conduction to hexagonal convection, and of the range of stability of hexagons near threshold. The jump is predicted to be unobservably small for typical parameter values, and the stability range of hexagons turns out to be small but potentially observable for suitable choices of parameters. An earlier calculation by Roppo, Davis, and Rosenblat [Phys. Fluids 27, 796 (1984)], which is limited to the range δ${\pi }^2$) but predicts observable hexagon effects in that range, is shown to overestimate those effects considerably. The present model is suitable for studying the dynamics of pattern competition by straightforward numerical techniques.