Averaging principle for the stochastic primitive equations in the large rotation limit
Quyuan Lin, Rongchang Liu, Vincent R. Martinez
TL;DR
This work analyzes the stochastic primitive equations on the torus with additive white noise under fast rotation. By applying a stochastic averaging principle and a coupling argument, the authors show that, after rescaling by the rotation, the distribution of strong solutions converges to the unique invariant measure of a limit resonant system, comprising a 2D barotropic Navier–Stokes component with noise and a baroclinic linear heat component driven by the barotropic flow. Convergence to the limit system is established on finite time horizons, and the limit resonant system is proven to exhibit exponential mixing in $H^1$. Consequently, in the large-rotation limit, the dissipative stochastic PE dynamics exhibit nearly unique ergodicity, with the long-time statistics governed by the limit resonant system’s invariant measure $\mu$. The results extend stochastic averaging techniques to the rotating PE, elucidating how fast rotation enforces an effective ergodic behavior despite open questions about ergodicity for fixed finite rotation rates.
Abstract
It is known that the unique ergodicity of the viscous primitive equations with additive white-in-time noise remains an open problem. In this work, we demonstrate that, as the rotational intensity approaches infinity, the distribution of any given strong solution, rescaled by the rotation, is attracted to the unique invariant measure of the stochastic limit resonant system. This suggests that the system exhibits nearly unique ergodicity in the large rotation limit. The proof is based on a stochastic averaging principle combined with a coupling argument.