Introduction

The numerical investigation of specific meteorological problems requires the selection of a suitable coordinate system. In many cases the best choice is quite obvious. Attempts to use the same coordinate system for entirely different geometries usually introduce additional mathematical complexities, which should be avoided. For example, it is immediately apparent that the rectangular Cartesian system is not well suited for the treatment of problems with spherical symmetry. The inspection of the metric fundamental quantities gij or gij and their derivatives helps to decide which coordinate system is best suited for the solution of a particular problem. The study of the motion over irregular terrain may require a terrain-following coordinate system. However, it is not clear from the beginning whether the motion is best described in terms of covariant or contravariant measure numbers. We will discuss this situation later.

From the thermo-hydrodynamic system of equations consisting of the dynamic equations, the continuity equation, the heat equation, and the equation of state we will direct our attention mostly toward the equation of motion using covariant and contravariant measure numbers. We will also briefly derive the continuity equation in general coordinates. In addition we will derive the equation of motion using physical measure numbers and assuming that the curvilinear coordinate lines are orthogonal.

The starting point of the analysis is the equation of motion in the absolute coordinate system. The description of the motion in a rotating and time-dependent coordinate system, in general, requires knowledge of the rotational velocity of a point in the atmosphere as well as the deformation velocity of the material surface on which the point is located.