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

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.