The Derivation of a Terrain-Following Coordinate System for Use in a Hydrostatic Model

Abstract

This article uses tensor transformation procedures in order to derive a terrain-following coordinate system that is frequently used in a number of regional and mesoscale hydrostatic models. Tensor transformation procedures are used so as to ensure physical invariance of the primitive equations between the Cartesian and terrain-following systems. Among the major conclusions are as follows: Applying the chain rule to the hydrostatic equation, before transforming from a Cartesian to a terrain-following coordinate system, yields a different set of equations than if the hydrostatic assumption is applied after the tensor transformation is made. The hydrostatic equations in the two terrain-following representations are the same only when the slope of the terrain in the model is much less than 45°. Variations of the metric tensor across a grid volume appear in the set of conservation equations as a result of averaging the equations over a grid volume. Such deviations have always been ignored in existing non-hydrostatic and hydrostatic meteorological models. Care must be taken to assure that parameterizations which are a function of distance above the ground be defined in terms of the original Cartesian system, and not the new generalized vertical coordinate σ. The profile exchange coefficient K(z), for example, cannot be defined simply by replacing z by σ.