NOISE AND PATTERN-FORMATION IN PERIODICALLY DRIVEN RAYLEIGH-BENARD CONVECTION

We present a new model for periodically driven Rayleigh-Benard convect ion with thermal noise, derived as a truncated vertical mode expansion of a mean field approximation to the Oberbeck-Boussinesq equations. T he resulting model includes the continuous dependence on the horizonta l wave number, and preserves the full symmetries of the hydrodynamic e quations as well as their inertial character. The model is shown to re duce to a Swift-Hohenberg-like equation in the same limiting cases in which the Lorenz model reduces to an amplitude equation. The order-dis order transition experimentally observed in the recurrent pattern form ation near the convective onset is studied by using both the present m odel and its above-mentioned limiting form, as well as a generalizatio n of the amplitude equation for modulated driving introduced by Schmit t and Lucke [Phys. Rev. A 44, 4986 (1991)] and the generalized Lorenz model previously introduced by the authors [Phys. Rev. E 55, R3824 (19 97)]. We show that all these models agree with the experimental data m uch closer than previous models like the Swift-Hohenberg equation or t he amplitude equation, though thermal noise alone still seems insuffic ient to lead to a precise fit. The relationship between these models i s discussed, and it is shown that the inclusion of the continuous wave -number dependence, a consistent treatment of the driving time depende nce, and the inclusion of inertial effects are all relevant to the for mulation of a model describing equally well both the time-periodic and static driving cases.