Derivation of the Equations of Atmospheric Motion in Oblate Spheroidal Coordinates

Abstract

Since Earth is more nearly an oblate spheroid than a sphere, it is of at least theoretical interest to develop the atmospheric equations of motion in spheroidal coordinates. In this system the horizontal unit vectors are oriented eastward and northward along the surfaces of ellipsoids, while the orthogonal unit vector is oriented vertically along the surfaces of intersecting confocal hyperboloids. Using the theory of orthogonal curvilinear coordinates, the spheroidal equations of relative atmospheric motion are derived from the vector equation of absolute motion. With the exception of two terms in the meridional and vertical equations of motion that are unique to the spheroidal system, all of the metric and rotational terms in the spheroidal system correspond to those found in the familiar spherical formulation, but now have coefficients that are functions of both the spheroidal latitude and elevation. The unique spheroidal terms arise from the resolution of the difference between the directions of apparent gravity and Newtonian gravitation, which is neglected in the spherical formulation. The complete spheroidal equations conserve both absolute angular momentum and total kinetic energy, and in the limit as Earth’s focal distance or eccentricity approaches zero, reduce to the familiar spherical equations in both the general and hydrostatic cases. The differences between solutions of the spheroidal and spherical equations are not expected to be significant in most applications, although there is the possibility of the accumulation of systematic differences in long-term integrations.