Stochastic resonance in the Bénard system

In this paper the effect of small stochastic perturbations on a dynamical system describing the Bénard thermal convection is studied. In particular, the two-dimensional Oberbeck-Boussinesq equations governing the dynamics of three interacting Rayleigh rolls with increasing horizontal wave numbers (i.e., three horizontal modes in the Fourier transform) are reduced to a system of gradient type. The aim is to study the transition paths between the stable steady states, when a stochastic perturbation is taken into account, and the occurrence of stochastic resonance, when the system is perturbed by white noise and the first (gravest) mode is forced by an external periodic component. Results show that i) random transitions between stable steady states representing a clockwise and a counter-clockwise circulation occur through the two saddle points associated with the second mode and not through the (unstable) conductive state nor the saddle points related to the third mode; ii) the introduction of the third mode, as well as of others of smaller spatial scales, does not affect transitions that remain confined along the trajectories linking stable convective states through the saddle points associated with the second mode; iii) the system exhibits a stochastic resonance behavior leading to large amplification of the small amplitude periodic component compared to the one leading to the classical (one-dimensional) stochastic resonance.