Evaluation of Various Approximations in Atmosphere and Ocean Modeling Based on an Exact Treatment of Gravity Wave Dispersion

Various approximations of the governing equations of compressible fluid dynamics are commonly used in both atmospheric and ocean modeling. Their main purpose is to eliminate the acoustic waves that are potentially responsible for inefficiency in the numerical solution, leaving behind gravity waves. The author carries out a detailed study of gravity wave dispersion for seven such approximations, individually and in combination, to exactly evaluate some of the often subtle errors. The atmospheric and oceanic cases are qualitatively and quantitatively different because, although they solve the same equations, their boundary conditions are entirely different and they operate in distinctly different parameter regimes. The atmospheric case is much more sensitive to approximation. The recent “unified” approximation of Arakawa and Konor is one of the most accurate. Remarkably, a simpler approximation, the combined Boussinesq–dynamically rigid approximation turns out to be exactly equivalent to the unified approximation with respect to gravity waves. The oceanic case is insensitive to the effects of any of the approximations, except for the hydrostatic approximation. The hydrostatic approximation is inaccurate at large wavenumbers in both the atmospheric and oceanic cases because it eliminates the entire buoyancy oscillation flow regime and is therefore to be restricted to low aspect ratio flows. For oceanic applications, certain approximations, such as the unified, dynamically rigid, and dynamically stiff approximations, are particularly interesting because they are accurate and approximately conserve mass, which is important for the treatment of sea level rise.