A Three-Dimensional Anelastic Model Based on the Vorticity Equation

Abstract

A three-dimensional anelastic model has been developed using the vorticity equation, in which the pressure gradient force is eliminated. The prognostic variables of the model dynamics are the horizontal components of vorticity at all heights and the vertical component of vorticity and the horizontally uniform part of the horizontal velocity at a selected height. To implement the anelastic approximation, vertical velocity is diagnostically determined from the predicted horizontal components of vorticity by solving an elliptic equation. This procedure replaces solving the elliptic equation for pressure in anelastic models based on the momentum equation. Discretization of the advection terms uses an upstream-weighted partially third-order scheme. When time is continuous, the solution of this scheme is quadratically bounded. As an application of the model, interactions between convection and its environment with vertical shear are studied without and with model physics from the viewpoint of vorticity dynamics, that is, the deceleration/acceleration process of the basic flow in particular. The authors point out that the process is purely three-dimensional, especially when the convection is relatively localized, involving the twisting terms and the horizontal as well as vertical transports of vorticity. Finally, it is emphasized that parameterization of cumulus friction is a resolution-dependent problem of vorticity dynamics associated with cumulus convection.