Symmetries and solutions for the inviscid oceanic Rossby wave equation

Abstract

The $(1+1)$-and $(1+2)$-dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the $1D$ 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 $2D$ 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.

Keywords:

• Lie symmetries
• reduction of order
• singularity analysis
• Rossby wave
• propagation properties

1991 AMS SUBJECT CLASSIFICATIONS:

• 34A05
• 34A34
• 34C14
• 22E60
• 35B06
• 35C05
• 35C07

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 ${\mathrm{\Gamma }}_{72}$ to ${\mathrm{\Gamma }}_{74}$ are discussed in cases 3.3 to 3.5.

2 Ablowitz, Ramani and Segur algorithm

Additional information

Funding

AKH is grateful to NBHM Post-Doctoral Fellowship, Department of Atomic Energy (DAE), Government of India, Award No: 0204/3/2021/R&D-II/7242 for financial support and the late Prof. K.M. Tamizhmani for fruitful discussions. PGLL acknowledges the support of the University of KwaZulu-Natal and the Durban University of Technology.