A multi-scale limit of a randomly forced rotating $3$-D compressible fluid

TL;DR

This paper analyzes the limit behavior of a scaled 3D compressible Navier-Stokes-Coriolis system with stochastic forcing, showing convergence to a 2D incompressible Navier-Stokes system under specific scaling conditions.

Contribution

It establishes the convergence of weak martingale solutions of the 3D rotating compressible fluid equations to a 2D incompressible model in the singular limit where the Mach and Froude numbers are proportional to a small parameter.

Findings

Weak solutions converge in probability as epsilon approaches zero.

The 3D system reduces to a 2D incompressible system in the limit.

The analysis includes stochastic forcing and centrifugal effects.

Abstract

We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll 1$, whereas the Rossby number scales like $\varepsilon^m$ for $m>1$ large, then we show that any family of weak martingale solution to the $3$-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as $\varepsilon\rightarrow0$, to the $2$-D incompressible Navier--Stokes system with a corresponding random forcing term.