On the extension of the Boussinesq approximation for inertia dominated flows

Abstract

We examine the validity of the Boussinesq approximation for shallow flows with stratification where the atmosphere can be incompressible or compressible, with possible existence of strong base flow motion and externally imposed small initial density perturbations. It is known that the “incompressible fluid” Boussinesq approximation when applied to a compressible atmosphere predicts an incorrect Brunt-Väisälä frequency. We deduce a nondimensional parameter for selecting the appropriate equations depending on the compressibility of the medium. An asymptotic reduction of the compressible Navier-Stokes equations is used to elucidate various physical aspects such as small entropy perturbations in the initial conditions, stratification, and the existence of a time-varying, three-dimensional base flow for a general (compressible or incompressible) fluid. The final equations are uniformly valid for combinations of these effects. An extension to the Boussinesq approximation is proposed to take into account the interaction of density perturbations and flow inertia. It is shown that only a minimal modification to the traditional Boussinesq algorithm is required for efficient and accurate numerical solution of the extended equations for many practical flows.