On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis

TL;DR

This paper analyzes the symmetry properties of the primitive equations used in weather modeling, showing that constant rotation can be effectively transformed away, and provides exact solutions and symmetry classifications.

Contribution

It introduces a symmetry-based approach to transform rotating primitive equations into non-rotating ones, enabling the derivation of exact solutions and symmetry group classification.

Findings

The maximal Lie invariance algebra is infinite-dimensional.

Constant rotation can be mapped to a non-rotating frame.

The complete point symmetry group of the primitive equations is computed.

Abstract

Modern weather and climate prediction models are based on a system of nonlinear partial differential equations called the primitive equations. Lie symmetries of the primitive equations with zero external heating rate are computed and the structure of its maximal Lie invariance algebra, which is infinite-dimensional, is studied. The maximal Lie invariance algebra for the case of a nonzero constant Coriolis parameter is mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions for the nonrotating equations. Another important result of the paper is the computation of the complete point symmetry group of the primitive equations using the algebraic method.