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.
Author contributions
All the authors contributed equally in the development of this work. Starting from envisaging the work till the preparation of the final draft of the manuscript, the authors worked in tandem.
Disclosure statement
The authors declare that they have no conflict of interests in the submitted work.
Data availability
The authors declare that no datasets were used in the study.
Notes
1 The reductions with respect to to
are discussed in cases 3.3 to 3.5.
2 Ablowitz, Ramani and Segur algorithm