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In this paper, we investigate the low Mach and low Froude numbers limit forthe compressible Navier-Stokes equations with degenerate, density-dependent,viscosity coefficient, in the strong stratification regime. We consider thecase of a general pressure law with singular component close to vacuum, andgeneral ill-prepared initial data. We perform our study in thethree-dimensional periodic domain. We rigorously justify the convergence to thegeneralised anelastic approximation, which is used extensively to modelatmospheric flows.

Low Mach number limit for degenerate Navier-Stokes equations in presence of strong stratification