*Math. Model. Nat. Phenom.*Vol. 8, No. 1, 2013, pp. 122-131

## Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid

N. H. Ibragimov^{1} and R. N. Ibragimov^{2}^{∗}

^{1} Department of Mathematics and
Science, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden and Research
Laboratory “Group analysis of mathematical models in natural sciences, technics and
technology” Ufa State Aviation Technical University, 450000
Ufa,
Russia
^{2}
Department of Mathematics University of Texas at
Brownsville, TX
78520,
USA

^{∗} Corresponding author. E-mail: Ranis.Ibragimov@utb.edu

Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformly stratified fluid under the Boussinesq approximation is investigated. The model is constrained by the Kelvin’s conjecture saying that the velocity component normal to the interface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zero everywhere in the domain of fluid motion, not only at the boundary. Three classes of distinctly different exact solutions for the nonlinear model are obtained. The obtained solutions are associated with symmetries of the Boussinesq model. It is shown that one class of the obtained solutions can be visualized as rotating whirlpools along which the pressure deviation from the mean state is zero, is positive inside and negative outside of the whirlpools. The angular velocity is zero at the center of the whirlpools and it is monotonically increasing function of radius of the whirlpools.

Mathematics Subject Classification: 35A20

Key words: stratified rotating fluids / nonlinear Boussinesq model / exact solutions

*© EDP Sciences, 2013*