Layered incompressible fluid flow equations in the limit of low Mach number and strong stratification

We investigate the low Mach number regimes of the complete Navier–Stokes–Fourier system which describes an evolution of a viscous, heat conducting gas with a heat source term interpreted as a radiation cooling in models of atmospheric flows. This investigation is performed in the context of weak solutions on an arbitrary large time interval and for the ill-prepared initial data. The physically expected limiting equations as the Mach and Froude numbers tend to zero at the same rate are layered incompressible flow equations with strong stratification. We give a rigorous mathematical proof of this result.