Effects of additive noise at the onset of Rayleigh-Bénard convection.

Abstract

The effects of additive Gaussian white noise on the onset of Rayleigh-B\'enard convection are studied by means of a phenomenological model, the stochastic Swift-Hohenberg equation. The strength of the noise term arising from thermal fluctuations is given for both free-slip and rigid horizontal boundaries. As was already pointed out by previous authors this term contains the small parameter ${\mathit{k}}_{\mathit{B}}$T/\ensuremath{\rho}d${\ensuremath{\nu}}^{2}$, where \ensuremath{\rho} is the mass density, d the plate separation, and \ensuremath{\nu} the kinematic viscosity. For typical liquids this parameter is of order ${10}^{\mathrm{\ensuremath{-}}9}$. Experiments involving fluctuation effects may be interpreted in terms of this model if the noise strength is treated as an adjustable parameter, which turns out to be larger than the typical thermal value by four orders of magnitude. The effects of fluctuations on the bifurcation of an infinite system are studied, and the earlier arguments of the present authors leading to a first-order transition are reviewed [Swift and Hohenberg, Phys. Rev. A 15, 319 (1977)]. The conditions under which the multimode model can be approximated by a single-mode stochastic amplitude equation are investigated, and an earlier analytic approximation scheme for calculating the response to a time-dependent Rayleigh number is applied to the multimode model. A comparison with available experimental and numerical simulation data is presented.