Ergodicity in Randomly Forced Rayleigh-B\'enard Convection

TL;DR

This paper proves the existence and uniqueness of ergodic invariant measures for stochastic Rayleigh-Bénard convection systems in two and three dimensions, using advanced probabilistic and analytical techniques.

Contribution

It establishes unique ergodicity for stochastic Rayleigh-Bénard convection models, including the infinite Prandtl system, with streamlined proofs and bounds on heat transport.

Findings

Existence of a unique ergodic invariant measure in 2D with sufficient stochastic forcing.

Existence of a statistically invariant state in 3D and unique ergodicity for the infinite Prandtl system.

Application of the background method to derive bounds on the Nusselt number in the stochastic setting.

Abstract

We consider the Boussinesq approximation for Rayleigh-B\'{e}nard convection perturbed by an additive noise and with boundary conditions corresponding to heating from below. In two space dimensions, with sufficient stochastic forcing in the temperature component and large Prandtl number, we establish the existence of a unique ergodic invariant measure. In three space dimensions, we prove the existence of a statistically invariant state, and establish unique ergodicity for the infinite Prandtl Boussinesq system. Throughout this work we provide streamlined proofs of unique ergodicity which invoke an asymptotic coupling argument, a delicate usage of the maximum principle, and exponential martingale inequalities. Lastly, we show that the background method of Constantin-Doering [CD96] can be applied in our stochastic setting, and prove bounds on the Nusselt number relative to the unique invariant measure.