A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow

An incompressibility approximation is formulated for isentropic motions in a compressible stratified fluid by defining a pseudo-density ρ∗ and enforcing mass conservation with respect to ρ∗ instead of the true density. Using this approach, sound waves will be eliminated from the governing equations provided ρ∗ is an explicit function of the space and time coordinates and of entropy. By construction, isentropic pressure perturbations have no influence on the pseudo-density. A simple expression for ρ∗ is available for perfect gases that allows the approximate mass conservation relation to be combined with the unapproximated momentum and thermodynamic equations to yield a closed system with attractive energy conservation properties. The influence of pressure on the pseudo-density, along with the explicit (x, t) dependence of ρ∗ is determined entirely by the hydrostatically balanced reference state. Scale analysis shows that the pseudo-incompressible approximation is applicable to motions for which M min(1,R), where M is the Mach number and R the Rossby number. This assumption is easy to satisfy for small-scale atmospheric motions in which the Earth’s rotation may be neglected and is also satisfied for quasi-geostrophic synoptic-scale motions, but not planetary-scale waves. This scaling assumption can, however, be relaxed to allow the accurate representation of planetary-scale motions if the pressure in the time-evolving reference state is computed with sufficient accuracy that the large-scale components of the pseudo-incompressible pressure represent small corrections to the total pressure, in which case the full solution to both the pseudoincompressible and reference-state equations has the potential to accurately model all non-acoustic atmospheric motions.