Point symmetry group of the barotropic vorticity equation

TL;DR

This paper determines the complete point symmetry group of the barotropic vorticity equation on the β-plane using advanced algebraic methods and normalization techniques.

Contribution

It introduces two novel techniques for computing symmetry groups, enhancing the analysis of differential equations' invariance properties.

Findings

The full symmetry group of the equation is explicitly characterized.

Two complementary methods for symmetry analysis are demonstrated.

The techniques can be applied to other differential equations with similar structures.

Abstract

The complete point symmetry group of the barotropic vorticity equation on the $\beta$-plane is computed using the direct method supplemented with two different techniques. The first technique is based on the preservation of any megaideal of the maximal Lie invariance algebra of a differential equation by the push-forwards of point symmetries of the same equation. The second technique involves a priori knowledge on normalization properties of a class of differential equations containing the equation under consideration. Both of these techniques are briefly outlined.