The normal modes of the spherical non-hydrostatic equations with applications to the filtering of acoustic modes

Abstract

Simple calculations suggest that hydrostatic (primitive) equation modelsare adequate for global numerical weather prediction or climate simulation for two-dimensional wavenumbers less than 400. This limit is approximately four times the resolution of present state of the art global models. For resolutions greater than 400, non-hydrostatic models will have to be constructed. Such models have extremely restrictive CFL limits, because of the highfrequencies of the acoustic modes.

The present work suggests a simple procedure for filtering the acoustic modes from such models. The proposed method should be effective and efficient because of the very simple structures of the acoustic modes. For an isothermal basic state, the acoustic modes have universal vertical structures and the horizontal structures are essentially single spherical harmonics. A more realistic basic state introduces a minor complication, namely, that the vertical structures become horizontally scale dependent in the non-hydrostatic regime.

For a model discretization based on a spherical harmonic expansion, a simple acoustic mode filter can be constructed. This procedure simply requires the projection of the state variables onto the acoustic modes, excision of the acoustic modes and re-projection into real space. This can be done every time-step with little overhead because of the simple structures of the acoustic modes. Thus, in principle, a global non-hydrostatic model could beintegrated with long time-steps and virtually no computational penalty.