Lie group method for solving viscous barotropic vorticity equation in ocean climate models

In studying the problem of the nonlinear viscous barotropic non-divergent vorticity equation on $f$- and $\beta$- planes, the method of Lie group has been applied. The method reduces the number of independent variables by one, and consequently, for the case of three independent variables we applied the method successively twice and the nonlinear partial differential equation reduces to ordinary differential equation. Investigation of exact solutions of the viscous barotropic non-divergent vorticity equation on $f$- and $\beta$- planes, via the application of Lie group, provides large classes of new exact solutions which include both Rossby and Rossby–Haurwitz waves as special cases. Also, The Lie symmetries of the viscous barotropic non-divergent vorticity equation with two parameters $F$ and $\beta$, are determined. The possible reductions of the viscous barotropic vorticity equation with two parameters $F$ and $\beta$ have been investigated by means of one- dimensional Lie subalgebras.