Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T06:04:55.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation of Nonlinear Conservation Laws from the Classical Energy Conservation Law for Internal Gravity Waves in Cylindrical Wave Field

Published online by Cambridge University Press:  17 September 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
R.N. Ibragimov*
Affiliation:
Department of Mathematics University of Texas at Brownsville, TX 78520, USA
*
⋆ Corresponding author. E-mail: Ranis.Ibragimov@utb.edu
Get access

Abstract

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

Keywords

Kelvin internal wavesconservation laws
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 119 - 130
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ali, A., Kalisch, H.. Mechanical balance laws for Boussinesq models of surface water waves J. Nonlinear Sci. 22 (2012), 371-398. CrossRefGoogle Scholar
V. Andreev. O. Kaptsov. V. Pukhnachev, A. Rodionov. Applications of group theoretic methods in hydrodynamics. Novosibirsk, Nauka. (Russian). English translation by Kluwer Academic Publishers (1994).
Balasuriya, S.. Vanishing viscosity in the barotropic β−plane J. Math.Anal. Appl. 214, 128-150. CrossRef
A. Buchnev. Lie group admitted by the equations of motion of an ideal incompressible fluid. Continuum Dynamics. 7 (1971) pp. 212-214. Institute of Hydrodynamics, USSR Acad. Sci., Siberian Branch, Novosibirsk. (Russian).
Cho, H., Shepherd, T., Vladimirov, V.. Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. J. Atmosph. Sci. 50 6 (1993), 822-836. 2.0.CO;2>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. 25 (1998), 939-942. CrossRefGoogle Scholar
Fjortoft, R.. Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geophys. Publ., 17 (6) 1950, 1-52. Google Scholar
A. Gill. Atmosphere-Ocean Dynamics. New York, etc., Academic Press. 1983
G.H. 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. 49 (3), (2011), 319-323. CrossRefGoogle Scholar
Ibragimov, N.H.. Nonlinear self-adjointness in constructing conservation laws. Archives of ALGA 7 (8) (2010-2011), 1-99. Google Scholar
N.H. Ibragimov. Nonlinear self-adjointness in constructing conservation laws. arXiv: 1109.1728v1[math-ph], (2011), 1-104.
N.H. Ibragimov, R.N. Ibragimov. Rotationally symmetric internal gravity waves. J. Non-Linear Mech. (2011), doi:10.1017/j.ijnonlinmec.2011.08.011.
Ibragimov, R.N., Yilmaz, N., Bakhtiyarov, A.S.. Experimental mixing parameterization due to multiphase fluid-structure interaction. Mechanics Research Communications, 38 (2011), 261-266. CrossRefGoogle Scholar
Nethery, D., Shankar, D.. Vertical propagation of baroclinic Kelvin waves along the west coast of India. J. Earth. Syst. Sci. 116 4 (2007), 331-339. CrossRefGoogle Scholar
Kalisch, H., Nguyen, N.T.. On the stability of internal waves. J. Phys. A. 43 (2010), 495205. CrossRefGoogle Scholar
Romea, R.D., Allen, J.S.. On vertically propagating coastal Kelvin waves at low latitudes. J. Phys. Oceanogr. 13 (1983) 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. 31 (2004), L18209. CrossRefGoogle Scholar
Staquet, C., Sommeria, J.. Internal Gravity Waves: From instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (2002), 559-593. CrossRefGoogle Scholar
Szoeke, R., Samelson, R.M.. The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. 32 (2002), 2194–2203.2.0.CO;2>CrossRefGoogle Scholar