Symmetries and solutions for the inviscid oceanic Rossby wave equation

Abstract

The -and -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method.