Reviewing and clarifying the derivation of the hydrostatic primitive equations

Abstract

AbstractHaving adopted a spherical geopotential approximation, textbooks generally derive the hydrostatic primitive equations from the fully compressible Euler equations by making three further approximations—usually termed “hydrostatic”, “shallow”, and “traditional”. However, the derivation given by Phillips in 1966 indicates that the traditional approximation is one of the consequences of the consistently applied shallow approximation. This is demonstrated here without recourse to the “vector‐invariant” form of the momentum equation used by him. This new derivation is not only simpler, but also in accord with the known existence of a quartet of dynamically consistent models, depending on whether approximations of shallow and/or hydrostatic type are or are not made in the fully compressible Euler equations in spherical geometry. A brief survey of other derivation strategies is given. To provide further insight, this includes the approximation of Lagrangian density—a scalar quantity—followed by application of either Hamilton's principle or the Euler–Lagrange equations. Relatedly, the consistent specific forms of the spherical geopotential approximation for the (deep) Euler equations and for the (shallow) hydrostatic primitive equations are obtained and discussed. The analysis herein can be generalised to spheroidal geometry. This is of interest, since the Earth is more realistically represented as an oblate spheroid than as a sphere, and opens the way to the development of more accurate dynamical cores for global atmospheric and oceanic modelling.