Rigorous justification of the hydrostatic approximation limit of viscous compressible flows.

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as "well-prepared" initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to zero. • Research demonstrates the rigorous limit of the compressible Navier-Stokes to the primitive equations under small aspect ratio. • Data for the compressible Navier-Stokes equations can be ill-prepared or well-prepared, allowing for a wide range of input. • Research finds that the asymptotic limit mechanism resembles the acoustic wave in low Mach number studies, a new insight for this problem. [ABSTRACT FROM AUTHOR]