Comparison of methods for reducing the error of the pressure gradient force in sigma coordinate models

Abstract

Four methods for computing the pressure gradient force in the sigma coordinate system are compared for flow over steeply sloping terrain. Three of the four schemes considered were the well known schemes by Corby et al. (1972), Janijic (1977) and Arakawa and Suarez (1983). The fourth scheme, proposed in this paper, is based on the second order accuracy vertical interpolation of geopotential. The scheme is designed in such a way as to achieve the cancellation in the energy equation of the contributions of the pressure gradient force and the “alpha-omega” term of the thermodynamic equation. Initially, the atmosphere was at rest and in hydrostatic equilibrium. The motions generated in such an atmosphere are excited only by the pressure gradient force error. The integrations were performed in a two-dimensional (χ, σ) domain with a triangular mountain defined in the middle of the domain. A strong inversion was located to the left of the mountain. In the rest of the domain, temperature was assumed to vary linearily either with lnp orp κ. The Janjic and Arakawa-Suarez schemes had comparable errors which were significantly smaller than those of the Corby et al. scheme. The error of the proposed scheme was extremely low. However, this is paid too by reduced computational efficiency.