Symmetry Analysis of Barotropic Potential Vorticity Equation

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 $\beta \ne 0$.