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© World Scientific Publishing Company.We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable - the potential temperature Θ. We consider the case of well-prepared initial data on  flat  torus and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier-Stokes system and the transport equation for the second-order variation of Θ

On singular limits arising in the scale analysis of stratified fluid flows