Evaluation of a Hydrostatic, Height-Coordinate Formulation of the Primitive Equations for Atmospheric Modeling

Abstract

The hydrostatic form of the primitive equations described by Ooyama is evaluated by comparing nonhydrostatic and hydrostatic integrations of a dry axisymmetric model with a specified entropy (heat) source. In this formulation, pressure is a diagnostic variable, so that the hydrostatic approximation can be included simply by replacing the vertical momentum equation with a diagnostic vertical velocity equation. his diagnostic equation is a one-dimensional (height) second-order elliptic equation that can be solved using a direct method. Results show that hydrostatic solutions are very sensitive to the accuracy of the method used to solve the diagnostic vertical velocity equation. However, this sensitivity can be eliminated by adding an extra term to the diagnostic equation that ensures that the solution does not drift away from hydrostatic balance due to numerical approximation. When the extra term is added, this formulation of the primitive equations allows for the design of a numerical model in height coordinates that can be used in hydrostatic and nonhydrostatic regimes.