Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-07T13:30:19.007Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  11 May 2010

SUMA DEBSARMA ,
K. P. DAS  and
JAMES T. KIRBY
SUMA DEBSARMA
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India
K. P. DAS
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India
JAMES T. KIRBY*
Affiliation:
Center for Applied Coastal Research, University of Delaware, Newark, DE 19716, USA
*
†Email address for correspondence: kirby@udel.edu
Get access Rights & Permissions [Opens in a new window]

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 O(μ2) 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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 281 - 303
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Bingham, H. B. & Agnon, Y. 2005 A Fourier–Boussinesq method for nonlinear water waves. Eur. J. Mech. B 24, 255–274.CrossRefGoogle Scholar
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, 1–23.Google Scholar
Choi, W. & Camassa, R. 1996 a Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83–113.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1996 b Long internal waves of finite amplitude. Phys. Rev. Lett. 77, 1759–1762.CrossRefGoogle ScholarPubMed
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.Google Scholar
Clamond, D. & Grue, J. 2001 A fast method for fully nonlinear water-wave computations. J. Fluid Mech. 447, 337–355.CrossRefGoogle Scholar
Fructus, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323–347.Google Scholar
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, 181–210.Google Scholar
Goullet, A. & Choi, W. 2008 Large amplitude internal solitary waves in a two-layer system of piecewise linear stratification. Phys. Fluids 20, 096601.CrossRefGoogle Scholar
Grue, J. 2002 On four highly nonlinear phenomena in wave theory and marine hydrodynamics. Appl. Ocean Res. 24, 261–274.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusas, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.Google Scholar
Grue, J., Jensen, A, Rusas, P.-O. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181–217.CrossRefGoogle Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
Jackson, C. R. 2004 An atlas of internal solitary-like waves and their properties. http://www.internalwaveatlas.com/Atlas2.index.html.Google Scholar
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225–251.Google Scholar
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, 7995–8008.CrossRefGoogle Scholar
Lynett, P. J. & Liu, P. L. F. 2002 A two-dimensional depth-integrated model for internal wave propagation over variable bathymetry. Wave Motion 36, 221–240.Google Scholar
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, 1–30.CrossRefGoogle Scholar
Nguyen, H. Y. & Dias, F. 2008 A Boussinesq system for two-way propagation of interfacial waves. Physica D 237, 2365–2389.Google Scholar
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, 618–638.Google Scholar
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.Google Scholar
Ostrovsky, L. A. & Grue, J. 2003 Evolution equations for strongly nonlinear internal waves. Phys. Fluids 15 (10), 2934–2948.CrossRefGoogle Scholar
Phillips, O. M. 1977 Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
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, 275–294.CrossRefGoogle Scholar
Segur, H. & Hammack, J. L. 1982 Soliton models of long internal waves. J. Fluid Mech. 118, 285–304.CrossRefGoogle Scholar
Stanton, T. P. & Ostrovsky, L. A. 1998 Observation of highly nonlinear internal solitons over the continental shelf. Geophys. Res. Lett. 25, 2695–2698.CrossRefGoogle Scholar
Voronovich, A. G. 2003 Strong solitary internal waves in a 2.5 layer model. J. Fluid Mech. 474, 85–94.Google Scholar
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, 71–92.Google Scholar
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, 1277–1281.Google Scholar