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Fully nonlinear higher-order model equations for long internal waves in a two-fluid system

  • SUMA DEBSARMA (a1), K. P. DAS (a1) and JAMES T. KIRBY (a2)

Abstract

Fully nonlinear model equations, including dispersive effects at one-order higher approximation than in the model of Choi & Camassa (J. Fluid Mech., vol. 396, 1999, pp. 1–36), are derived for long internal waves propagating in two spatial horizontal dimensions in a two-fluid system, where the lower layer is of infinite depth. The model equations consist of two coupled equations for the displacement of the interface and the horizontal velocity of the upper layer at an arbitrary elevation, and they are correct to O2) terms, where μ is the ratio of thickness of the upper-layer fluid to a typical wavelength. For solitary waves propagating in one horizontal direction, the two coupled equations reduce to a single equation for the elevation of the interface. Solitary wave profiles obtained numerically from this equation for different wave speeds are in good agreement with computational results based on Euler's equations. A numerical approach for the propagation of solitary waves is provided in the weakly nonlinear case.

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Corresponding author

Email address for correspondence: kirby@udel.edu

References

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Bingham, H. B. & Agnon, Y. 2005 A Fourier–Boussinesq method for nonlinear water waves. Eur. J. Mech. B 24, 255274.
Camassa, R., Choi, W., Michallet, H., Russas, P.-O. & Sveen, J. K. 2006 On the realm of validity of strongly nonlinear asymptotic approximations for internal wave. J. Fluid Mech. 549, 123.
Choi, W. & Camassa, R. 1996 a Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83113.
Choi, W. & Camassa, R. 1996 b Long internal waves of finite amplitude. Phys. Rev. Lett. 77, 17591762.
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 136.
Clamond, D. & Grue, J. 2001 A fast method for fully nonlinear water-wave computations. J. Fluid Mech. 447, 337355.
Fructus, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323347.
Gobbi, M. F., Kirby, J. T. & Wei, G. E. 2000 A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O((kh)4). J. Fluid Mech. 405, 181210.
Goullet, A. & Choi, W. 2008 Large amplitude internal solitary waves in a two-layer system of piecewise linear stratification. Phys. Fluids 20, 096601.
Grue, J. 2002 On four highly nonlinear phenomena in wave theory and marine hydrodynamics. Appl. Ocean Res. 24, 261274.
Grue, J., Jensen, A., Rusas, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257278.
Grue, J., Jensen, A, Rusas, P.-O. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181217.
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.
Jackson, C. R. 2004 An atlas of internal solitary-like waves and their properties. http://www.internalwaveatlas.com/Atlas2.index.html.
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225251.
Liu, A. K., Chang, Y. S., Hsu, M. K. & Liang, N. K. 1998 Evolution of nonlinear internal waves in the East and South China Seas. J. Geophys. Res. 103, 79958008.
Lynett, P. J. & Liu, P. L. F. 2002 A two-dimensional depth-integrated model for internal wave propagation over variable bathymetry. Wave Motion 36, 221240.
Madsen, P. A., Bingham, H. B. & Liu, H. 2002 A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 130.
Nguyen, H. Y. & Dias, F. 2008 A Boussinesq system for two-way propagation of interfacial waves. Physica D 237, 23652389.
Nwogu, O. 1993 An alternative form of Boussinesq equations for nearshore wave of Boussinesq equations for nearshore wave propagation. J.Waterway Port Coast. Ocean Engng 119, 618638.
Orr, M. H. & Mignerey, P. C. 2003 Nonlinear internal waves in south China sea: Observation of the conservation of the depression internal waves, J. Geophys. Res. 108 (C3), 3064, doi:10.1029/2001JC001163.
Ostrovsky, L. A. & Grue, J. 2003 Evolution equations for strongly nonlinear internal waves. Phys. Fluids 15 (10), 29342948.
Phillips, O. M. 1977 Dynamics of the Upper Ocean. Cambridge University Press.
Ruiz de Zárate, A., Vigo, D., Nachbin, A. & Choi, W. 2009 A higher-order internal wave model accounting for large bathymetric variations. Stud. Appl. Math. 122, 275294.
Segur, H. & Hammack, J. L. 1982 Soliton models of long internal waves. J. Fluid Mech. 118, 285304.
Stanton, T. P. & Ostrovsky, L. A. 1998 Observation of highly nonlinear internal solitons over the continental shelf. Geophys. Res. Lett. 25, 26952698.
Voronovich, A. G. 2003 Strong solitary internal waves in a 2.5 layer model. J. Fluid Mech. 474, 8594.
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 7192.
Zeng, K. & Alpers, W. 2004 Generation of internal solitary waves in Sulu Sea and their refraction by bottom topography studied by ERS SAR imagery and numerical model. Intl J. Remote Sens. 25, 12771281.
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Fully nonlinear higher-order model equations for long internal waves in a two-fluid system

  • SUMA DEBSARMA (a1), K. P. DAS (a1) and JAMES T. KIRBY (a2)

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