Lie symmetries and exact solutions of barotropic vorticity equation

The Lie group method is used in order to investigate various issues related to symmetries and exact solutions of the barotropic vorticity equation. This is done in a similar but more systematic and complete way as given by Huang and Lou [Phys. Lett. A. 320 (2004) 428–437]. The Lie symmetries of the barotropic vorticity equations on the f and β-planes as well as on the sphere in rotating and resting reference frames are determined. A symmetry background for reduction of the rotating reference frame to the resting one is presented. Oneand two-dimensional inequivalent subalgebras of the Lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations. This gives large classes of exact solutions, which include both Rossby and Rossby–Haurwitz waves as special cases. We also discuss the possibility of partial invariance for the β-plane equation, to further extend the family of its exact solutions.