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Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

Published online by Cambridge University Press:  29 February 2012

N. H. Ibragimov  and
R. N. Ibragimov
N. H. Ibragimov
Affiliation:
Department of Mathematics and ScienceBlekinge 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: [email protected]
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Abstract

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

Keywords

internal gravity wavesinvariant solutions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 52 - 65
Copyright
© EDP Sciences, 2012

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References

Cho, H., Shepherd, T., Vladimirov, V.. Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. J. Atmosph. Sci., (1993), 50 (6), 822-836. 2.0.CO;2>CrossRefGoogle Scholar
Craig, W., Guyenne, P., Kalisch, H.. Hamiltonian long-wave expansions for the free surfaces and interfaces. Comm. Pure Appl. Math., (2005), 58, 1587-1641. 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
Dalziel, S., Hughes, G., Sutherland, B.. Whole field density measurements by synthetic schlieren. Experiments in Fluids, (2000), 28, 322-337. CrossRefGoogle Scholar
Dauxois, T., Young, W.. Near-critical reflection of internal waves. J. Fluid Mech., (1999), 390, 271-295. CrossRefGoogle Scholar
Fjortoft. R, R. , Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geophys. Publ., (1950), 17(6), 1-52. Google Scholar
Flynn, M., Onu, K., Sutherland, B.. Internal wave excitation by a vertically oscillating sphere. J. Fluid Mech., (2003), 494, 65-93. CrossRefGoogle Scholar
C. Garrett. Internal tides and ocean mixing. Science, (2003), 301 (5641), 1858-1859, doi :10.1126/science.1090002.
Garrett, C., Munk, W.. Space time scale of internal waves. A progress report. J. Geophys. Res., (1975), 80, 291-297. CrossRefGoogle Scholar
A. Gill. Atmosphere-Ocean Dynamics. New York, etc., Academic Press, (1983).
J. Hadamard. Lectures on Cauchy’s problem in linear partial differential equations. Yale University Press, New Haven, (1983).
J. Hadamard. The problem of diffusion of waves. Annals of Mathematics, Ser. 2 43 : 510-522, (1942).
N. Ibragimov. Elementary Lie group analysis of ordinary differential equations. John Wiley & Sons, Chichester, (1999).
N. Ibragimov. Transformation Groups Applied to Mathematical Physics. Nauka, Moscow (1983), English. transl., Reidel, Dordrecht.
N. Ibragimov Ed. CRC Handbook of Lie group analysis of differential equations (CRC Press, Boca Raton) ; Vol. 1 (1994, 429 p), Vol. 2 (1995, 546 p.), Vol. 3 (1996, 536 p.).
Ibragimov, N.. A new conservation theorem J. Math. Anal. Appl., (2007), 333 : 311-328. CrossRefGoogle Scholar
Ibragimov, N.. Group analysis - a microscope of mathematical modelling. I : Galilean relativity in diffusion models. Selected works (ALGA Publications, Karlskrona), (2006), 2 : 225-243. Google Scholar
Ibragimov, N.. Conformal invariance and Huygens’ principle. Soviet Mathematics Doklady, (1970), 11(5) : 1153-1157. Google Scholar
Ibragimov, N.. Application of group analysis to liquid metal systems. Archives of ALGA, (2010), 6 : 91-101. Google Scholar
Ibragimov, N.. Lie group analysis of Moffatt’s model in metallurgical industry. Nonlinear Math. Phys., (2011), 18, 143-162. CrossRefGoogle Scholar
Ibragimov, N., Ibragimov, R., Kovalev, V.. Group analysis of nonlinear internal waves in oceans. Archives of ALGA, (2009), 6, 45-54. Google Scholar
N. Ibragimov, R. Ibragimov. Applications of Lie Group Analysis in Geophysical Fluid Dynamics. Series on Complexity, Nonlinearity and Chaos, (2011), Vol 2, World Scientific Publishers, ISBN : 978-981-4340-46-5.
Ibragimov, N., Ibragimov, R.. Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics. Comm. Nonlinear Sci. Num. Simulat., (2010), 15, 1989-2002. CrossRefGoogle Scholar
Ibragimov, R.. Oscillatory nature and dissipation of the internal wave energy spectrum in the deep ocean. Eur. Phys. J. Appl. Phys., (2007), 40, 315-334. CrossRefGoogle Scholar
Ibragimov, R.. Generation of internal tides by an oscillating background flow along a corrugated slope. Phys. Scr., (2008), 78, 065801. CrossRefGoogle Scholar
Ibragimov, R., Pelinovsky, D.. Incompressible viscous fluid flows in a thin spherical shell. J. Math. Fluid. Mech., (2009), 11, 60-90. CrossRefGoogle Scholar
Ibragimov, R., Ibragimov, N.. Effects of rotation on self-resonant internal gravity waves. Ocean Modelling, (2010), 31, 80-87. CrossRefGoogle Scholar
Ibragimov, R., Dameron, M.. Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids. Phys. Scr., (2011), 84, 015402. CrossRefGoogle Scholar
Javam, A., Imberger, J., Armfield, S.. Numerical study of internal wave-wave interactions in a stratified fluid. J. Fluid Mech., (2000), 415, 65-87. CrossRefGoogle Scholar
Kalisch, H., Nguyen, N.. On the stability of internal waves. J. Phys. A, (2010), 43, 495205. CrossRefGoogle Scholar
Kalisch, H., Bona, J.. Modes for internal waves in deep water. Disc. Cont. Dyn. Sys. (2000), 6, 1-19. CrossRefGoogle Scholar
Kistovich, A., Chashechkin, Y.. Nonlinear interactions of two dimensional packets of monochromatic internal waves. Izv. Atmos. Ocean. Phys., (1991), 27 (12) 946-951. Google Scholar
Lam, F., Mass, L., Gerkema, T.. Spatial structure of tidal and residual currents as observed over the shelf break in the Bay of Biscay. Deep-See Res., (2004), I 51, 10751096. Google Scholar
Lombard, P., Riley, J.. On the breakdown into turbulence of propagating internal waves. Dyn. Atmos. Oceans, (1996), 23, 345-355. CrossRefGoogle Scholar
H. Moffatt. High frequency excitation of liquid metal systems. Metallurgical Applications of Magnetohydrodynamics, (1984), (Metals Society, London) 180-189.
Müller, P., Holloway, G., Henyey, F., Pomphrey, N.. Nonlinear interactions among internal gravity waves. Rev. Geophys., (1986), 24, 3, 493-536. CrossRefGoogle Scholar
Nash, J., Kunze, E., Lee, C., Sanford, T.. Structure of the baroclinic tide generated at Keana Ridge, Hawaii. J. Phys. Oceanogr., (2006), 36, 1123-1135. CrossRefGoogle Scholar
P. Olver. Applications of Lie groups to differential equations. Springer-Verlag, New York, 2nd ed. 1993.
L. Ovsyannikov. Group Analysis of Differential Equations. Nauka, Moscow, (1978), English transl., ed. W.F. Ames, Academic Press, New York (1982).
Ramsden, D., Holloway, G.. Energy transfers across an internal wave-vortical mode spectrum. J. Geophys. Res., (1992), 97, 3659-3668. CrossRefGoogle Scholar
J. Riley, R. Metcalfe, M. Weissman. Direct numerical simulations of homogeneous turbulence in density-stratified fluids. Nonlinear properties of internal waves, (1981), 76, edited by B.J. West, pp. 79-112, Americal Institute of Physics, New York.
Shepherd, T.. Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Advances in Geophysics, (1990), 32, 287-338 CrossRefGoogle Scholar
Staquet, C., Sommeria, J.. Internal Gravity Waves : From instabilities to turbulence. Annu. Rev. Fluid Mech., (2002), 34, 559-593. CrossRefGoogle Scholar
Tabaei, A., Akylas, T., Lamb, K.. Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech., (2005), 526, 217-243. CrossRefGoogle Scholar
Tabaei, A., Akylas, T.. Nonlinear internal gravity wave beams. J. Fluid Mech., (2003), 482, 141-161. CrossRefGoogle Scholar
Teoh, S., Imberger, J., Ivey, G.. Laboratory study of the interactions between two internal wave rays. J. Fluid Mech., (1997), 336:91. CrossRefGoogle Scholar
Winters, K., D’Asaro, E.. Direct simulation of internal wave energy transfer. J. Phys. Oceanogr., (1997), 9, 235-243. Google Scholar
Wunsch, C., Ferrari, R.. Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., (2004), 36, 281-314. CrossRefGoogle Scholar
Zhang, H., King, B., Swinney, H.. Resonant generation of internal waves on a model continental slope. Phys. Rev. Let., (2008), 100, 244504.CrossRefGoogle Scholar