Rayleigh-B\'enard convection with stochastic forcing localised near the bottom

TL;DR

This paper proves that stochastic forcing near the bottom of a 3D Rayleigh-Bénard convection system stabilizes the flow to a unique stationary state, with stability enhanced by larger noise amplitudes.

Contribution

It establishes stochastic stability for 3D Rayleigh-Bénard convection with localized bottom forcing in the infinite Prandtl number regime, a novel result in the field.

Findings

Flow stabilizes to a unique stationary measure under stochastic forcing.

Stability is guaranteed if the noise amplitude is sufficiently large.

The system's stability holds for any pair of top and bottom temperatures.

Abstract

We prove stochastic stability of the three-dimensional Rayleigh-B\'enard convection in the infinite Prandtl number regime for any pair of temperatures maintained on the top and the bottom. Assuming that the non-degenerate random perturbation acts in a thin layer adjacent to the bottom of the domain, we prove that the random flow periodic in the two infinite directions stabilises to a unique stationary measure, provided that there is at least one point accessible from any initial state. We also prove that the latter property is satisfied if the amplitude of the noise is sufficiently large.