Abstract
We consider the motion of a compressible viscous fluid in the asymptotic regime of low Mach and high Reynolds numbers under strong stratification imposed by a conservative external force. Assuming a bi-dimensional character of the flow, we identify the limit system represented by the so-called lake equation – the Euler system supplemented by an anelastic type constraint imposed by the limit density profile. The key ingredient of the proof are new “frequency localized” estimates of Strichartz type.