Lie reduction and exact solutions of vorticity equation on rotating sphere

TL;DR

This paper performs a systematic Lie symmetry analysis of the barotropic vorticity equation on a rotating sphere, classifies its symmetry subalgebras, and derives exact solutions relevant to atmospheric physics.

Contribution

It extends previous work by classifying all finite-dimensional subalgebras and exhaustively deriving Lie reductions and exact solutions for the vorticity equation on a sphere.

Findings

Classified all finite-dimensional subalgebras of the symmetry algebra.

Derived new exact solutions relevant to atmospheric processes.

Connected results to recent studies on Euler equations in spherical geometry.

Abstract

Following our paper [J. Math. Phys. 50 (2009) 123102], we systematically carry out Lie symmetry analysis for the barotropic vorticity equation on the rotating sphere. All finite-dimensional subalgebras of the corresponding maximal Lie invariance algebra, which is infinite-dimensional, are classified. Appropriate subalgebras are then used to exhaustively determine Lie reductions of the equation under consideration. The relevance of the constructed exact solutions for the description of real-world physical processes is discussed. It is shown that the results of the above paper are directly related to the results of the recent letter by N. H. Ibragimov and R. N. Ibragimov [Phys. Lett. A 375 (2011) 3858] in which Lie symmetries and some exact solutions of the nonlinear Euler equations for an atmospheric layer in spherical geometry were determined.