Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem

TL;DR

This paper analyzes the symmetries of the two-layer quasi-geostrophic model, classifies its invariant subalgebras, and derives exact solutions including classical Rossby waves, enhancing understanding of geophysical fluid dynamics.

Contribution

It provides a comprehensive symmetry analysis and constructs new exact solutions for the two-layer model, including rediscovering classical Rossby wave solutions.

Findings

Complete set of point symmetries identified

Optimal subalgebras constructed for reduction

Exact solutions including Rossby waves obtained

Abstract

The quasi-geostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Where possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.