Geometric approximations for global atmospheres on moderately oblate planets

Certain geometric approximations, such as the widely used traditional shallow-atmosphere, spherical-geoid (TSA-SG) and deep-atmosphere, spherical-geoid (DA-SG) approximations, boil down to the specification of a spatial metric tensor. In order to eliminate the leading-order errors due to the SG and TSA approximations, a sequence of four geometric approximations of decreasing accuracy at high altitudes is obtained. Their Lagrangians possess a simple, closed-form analytical expression. The approximations capture to leading order the oblateness of the planet, the widening of atmospheric columns with height, the horizontal and vertical variations of gravity, and the non-traditional part of the Coriolis force. Furthermore, for the last two approximations, the horizontal metric is conformal (proportional) to the spherical metric, which may simplify analytical and numerical formulations of the equations of motion.