Covariant equations for geophysical fluids with an asymmetric gravitational potential and their hydrostatic limit

Abstract

Starting from the governing equations in tensor form for non‐hydrostatic fluid dynamics, the hydrostatic limit is obtained by introducing a scaling parameter. This allows the use of the same fundamental tensor equations to describe the dynamics of non‐hydrostatic and hydrostatic geophysical fluids, possibly combined with geometric approximations, while preserving the dynamical consistency of the governing equations in arbitrary coordinates. This provides the tools to reproduce in a systematic manner the quartet of equations—deep/non‐hydrostatic, deep/hydrostatic, shallow/non‐hydrostatic, and shallow/hydrostatic—presented elsewhere. This quartet is generalised to the case where the gravitational potential has no spatial symmetries.