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Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Columnof Stratified Fluid

Published online by Cambridge University Press:  28 January 2013

N. H. Ibragimov  and
R. N. Ibragimov
N. H. Ibragimov
Affiliation:
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
R. N. Ibragimov*
Affiliation:
Department of Mathematics University of Texas at Brownsville, TX 78520, USA
*
∗ Corresponding author. E-mail: Ranis.Ibragimov@utb.edu
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Abstract

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

Keywords

stratified rotating fluidsnonlinear Boussinesq modelexact solutions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 122 - 131
Copyright
© EDP Sciences, 2013

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References

Références

Ali, A., Kalisch, H.. Mechanical balance laws for Boussinesq models of surface water waves. J. Nonlinear Sci. (2012) 22, 371-498. CrossRefGoogle Scholar
Balasuriya, S.. Vanishing viscosity in the barotropic β-plane. J. Math. Anal. Appl. (1997) 214, 128-150. CrossRefGoogle Scholar
Dewan, E., Picard, R., O’Neil, R., Gardiner, H., Gibson, J.. MSX satellite observations of thunderstorm-generated gravity waves in mid-wave infrared images of the upper stratosphere. Geophys. Res. Lett. (1998) 25, 939-942. CrossRefGoogle Scholar
A. Gill. Atmosphere-Ocean Dynamics. New York, etc., Academic Press. (1983)
G.J Haltiner, R.T. Williams. Numerical prediction and dynamic meteorology (1980).
Hsieh, P.A.. Application of modflow for oil reservoir simulation during the Deepwater Horizon crisis. Ground Water. (2011) 49 (3), 319-323. CrossRefGoogle Scholar
Ibragimov, R.N., Ibragimov, N.H.. Effects of rotation on self-resonant internal gravity waves in the ocean. Ocean Modelling, (2010) 31, 80-87. CrossRefGoogle Scholar
N.H. Ibragimov, R.N. Ibragimov. Applications of Lie group analysis in Geophysical Fluid Dynamics. Series on Complexity and Chaos, V2, World Scientific Publishers (2011) .
N.H. Ibragimov, R.N. Ibragimov. Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell. Phys. Lett. A, (2011) 3858-3865.
N.H. Ibragimov, R.N. Ibragimov. Rotationally symmetric internal gravity waves. Int. J. Non-Linear Mech., (2012) 46-52.
Maloney, E.D., Hartmann, D. L.. The Madden–Julian Oscillation, Barotropic Dynamics, and North Pacific Tropical Cyclone Formation. Part I : Observations. J. Atmos. Sci. (2001) 58 (17), 2545–2558. Google Scholar
McCreary, J.P.. Eastern tropical ocean response to changing wind systems with applications to El Niño. J. Phys. Oceanogr. (1976) 6, 632-645. 2.0.CO;2>CrossRefGoogle Scholar
McCreary, J.P.. A linear stratified ocean model of the equatorial undercurrent. Phil. Trans. Roy. Soc. London. (1981) 302, 385-413. CrossRefGoogle Scholar
McCreary, J.P.. Equatorial beams. J. Mar. Res. (1984) 42, 395-430. CrossRefGoogle Scholar
Moore, D.W., Kloosterzeil, R.C., Kessler, W.S.. Evolution of mixed Rossby gravity waves. J. Geophys. Res. (1998) 103 (C3), 5331-5346. CrossRefGoogle Scholar
Nethery, D., Shankar, D.. Vertical propagation of baroclinic Kelvin waves along the west coast of India. J. Earth. Syst. Sci. (2007) 116 (4), 331-339. CrossRefGoogle Scholar
L.V. Ovsyannikov. Lectures on the theory of group properties of differential equations. Novosibirsk University press, Novosibirsk, 1966. English transl., ed. Ibragimov, N., ALGA Publications, Karlskrona, 2009.
Romea, R.D., Allen, J.S.. On vertically propagating coastal Kelvin waves at low latitudes. J. Phys. Oceanogr. (1983) 13 (1), 241-1,254. 2.0.CO;2>CrossRefGoogle Scholar
Shindell, D.T., Schmidt, G.A.. Southern Hemisphere climate response to ozone changes and greenhouse gas increases. Res. Lett., (2004) 31, L18209. CrossRefGoogle Scholar
Staquet, C., Sommeria, J.. Internal Gravity Waves : From instabilities to turbulence. Annu. Rev. Fluid Mech. (2002) 34, 559-593. CrossRefGoogle Scholar
Szoeke, R., Samelson, R.M.. The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. (2002) 32, 2194-2203. 2.0.CO;2>CrossRefGoogle Scholar
J.M. Wallace, P.V. Hobbs.Atmospheric Science : An Introductory Survey. Academic Press, (1977) Inc. 76–77.