The extension of the Miles-Howard theorem to compressible fluids

A statically stable, gravitationally stratified compressible fluid containing a parallel shear flow is examined for stability against infinitesimal adiabatic perturbations. It is found that the Miles–Howard theorem of incompressible fluids may be generalized to this system, so that n 2 ≥ ¼ U ′ 2 throughout the flow is a sufficient condition for stability. Here n 2 is the Brunt–Väissälä frequency and U ’ is the vertical gradient of the flow speed. Howard's upper bound on the growth rate of an unstable mode also generalizes to this compressible system.