The Derivation Analysis and Application of Terrian-Following Coordinate Representation of Atmospheric Motion

Abstract

This article uses tensor transformation procedures in order to derive a terrianfollowingcoordinate system that is frequently used in a number of regional and mesoscalehydrostatic models. Tensor transformation procedures are used so a- +o ensure physicalinvariance of the primitive equations between the Cartesian and terrian following systems.Among the major conclusions are as follows:

1. Applying the chain rule the hydrostatic equation, before transformation from aCartesian to a terrian-following coordinate system, yields a different set of equations than ifthe hydrostatic assumption is applied after the tensor transformation is made. The hydrostaticequations in the two terrian-following representations are the same only when the slope of theterrian in the model is much less than 45O.

2. Variations of the metric tensor across a grid volume appear in the set ofconservation equations as a result of averaging the equations over a grid voume. Suchderivations have always been ignored in existing non-hydrostatic and hydrostatic meteorologicalmodels.