Symmetry Analysis of Barotropic Potential Vorticity Equation

TL;DR

This paper analyzes the symmetry properties of the barotropic potential vorticity equation, classifies its Lie subalgebras, and derives group-invariant solutions, extending results to cases with non-zero beta parameter.

Contribution

It provides a detailed symmetry classification and solution construction for the potential vorticity equation, including the case when the beta parameter is non-zero.

Findings

Existence of a transformation to set β=0 when F≠0.

Classification of Lie subalgebras for the equation.

Derivation of group-invariant solutions extended to β≠0.

Abstract

Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, $F$ and $\beta$, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth' angular rotation, respectively. In the present paper it is shown that in the case $F\ne 0$ there exists a well-defined point transformation to set $\beta = 0$. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination $F\ne 0$ and $\beta = 0$. Based upon this classification, distinct classes of group-invariant solutions is obtained and extended to the case $\beta \ne 0$.