A Comparison of the Accuracy of Three Anelastic Systems and the Pseudo-Incompressible System

Abstract

The accuracy of three anelastic systems (Ogura and Phillips; Wilhelmson and Ogura; Lipps and Hemler) and the pseudo-incompressible system is investigated for small-amplitude and finite-amplitude disturbances. Based on analytic solutions to the linearized, hydrostatic mountain wave problem, the accuracy of the Lipps and Hemler and pseudo-incompressible systems is distinctly superior to that of the other two systems. The linear dispersion relations indicate the accuracy of the pseudo-incompressible system should improve and the accuracy of the Lipps and Hemler system should decrease as the waves become more nonhydrostatic. Since analytic solutions are not available for finite-amplitude disturbances, five nonlinear, nonhydrostatic numerical models based on these four systems and the complete compressible equations are constructed to determine the ability of each ?sound proof? system to describe finite-amplitude disturbances. A comparison between the analytic solutions and numerical simulations of the linear mountain wave problem indicate the overall quality of the simulations is good, but the numerical errors are significantly larger than those associated with the pseudo-incompressible and Lipps and Hemler approximations. Numerical simulations of flow past a steady finite-amplitude heat source for an isothermal atmosphere and an atmosphere with an elevated inversion indicate the Lipps and Hemler and pseudo-incompressible systems also produce the most accurate approximations to the compressible solutions for finite-amplitude disturbances.