Complete point symmetry group of the barotropic vorticity equation on a rotating sphere

TL;DR

This paper determines the complete point symmetry group of the barotropic vorticity equation on a rotating sphere using Lie algebra invariance methods, identifying all continuous and discrete symmetries.

Contribution

It introduces a novel approach to find the symmetry group of the spherical vorticity equation by analyzing megaideals of its Lie algebra.

Findings

Only two independent discrete symmetries identified.

Complete symmetry group characterized for the spherical vorticity equation.

Method applicable to other differential equations with similar symmetry structures.

Abstract

The complete point symmetry group of the barotropic vorticity equation on the sphere is determined. The method we use relies on the invariance of megaideals of the maximal Lie invariance algebra of a system of differential equations under automorphisms generated by the associated group. A convenient set of megaideals is found for the maximal Lie invariance algebra of the spherical vorticity equation. We prove that there are only two independent (up to composition with continuous point symmetry transformations) discrete symmetries for this equation.