Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T06:13:51.996Z Has data issue: false hasContentIssue false
  • English
  • Français

Solitary internal waves with oscillatory tails

Published online by Cambridge University Press:  26 April 2006

T. R. Akylas  and
R. H. J. Grimshaw
T. R. Akylas
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
R. H. J. Grimshaw
Affiliation:
School of Mathematics, University of New South Wales, Kensington, NSW 2033, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 279 - 298
Copyright
© 1992 Cambridge University Press

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

Akylas, T. R. 1991 On the radiation damping of a solitary wave in a rotating channel. In Mathematical Approaches in Hydrodynamics (ed. T. Miloh), pp. 175–181. SIAM.
Amick, C. J. & Toland, J. F. 1981 On solitary water waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9–95.Google Scholar
Amick, C. J. & Toland, J. F. 1992 Solitary waves with surface tension I: trajectories homoclinic to periodic orbits in four dimensions. Preprint.
Beale, J. T. 1991 Exact solitary water waves with capillary ripples at infinity. Commun. Pure Appl. Maths 44, 211–257.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Benney, D. J. 1966 Long nonlinear waves in fluid flows. J. Maths & Phys. 45, 52–63.Google Scholar
Boyd, J. P. 1989 Weakly non-local solitary waves. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), pp. 103–112. Elsevier.
Boyd, J. P. 1991 Weakly non-local solitons for capillary—gravity waves: fifth-degree Korteweg—de Vries equation.. Physica D 48, 129–146.CrossRefGoogle Scholar
Byatt-Smith, J. G. B. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves.. Proc. R. Soc. Lond. A 350, 175–189.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.Google Scholar
Farmer, D. M. & Smith, J. D. 1980 Tidal interaction of stratified flow with a sill in Knight Inlet. Deep-Sea Res. 27 A, 239–254.Google Scholar
Grimshaw, R. & Hooper, A. P. 1991 The non-existence of a certain class of travelling wave solutions of the Kuramoto—Sivashinsky equation.. Physica D 50, 231–238.Google Scholar
Hunter, J. K. & Scheurle, J. 1988 Existence of perturbed solitary wave solutions to a model equation for water waves.. Physica D 32, 253–268.Google Scholar
Hunter, J. K. & Vanden-Broeck, J.-M. 1983 Solitary and periodic gravity—capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219.Google Scholar
Kruskal, M. D. & Segur, H. 1991 Asymptotics beyond all orders in a model of crystal growth. Stud. Appl. Maths 85, 129–182.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave. II.. Proc. R. Soc. Lond. A 340, 471–493.Google Scholar
Pomeau, Y., Ramani, A. & Grammaticos, B. 1988 Structural stability of the Korteweg—de Vries solitons under a singular perturbation.. Physica D 31, 127–143.Google Scholar
Segur, H. & Kruskal, M. D. 1987 Nonexistence of small-amplitude breather solutions of 4 theory. Phys. Rev. Lett. 58, 747–750.Google Scholar
Sun, S. M. 1991 Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3. J. Math. Anal. Appl. 156, 471–504.Google Scholar
Tung, K.-K., Chan, T. F. & Kubota, T. 1982 Large amplitude internal waves of permanent form. Stud. Appl. Maths 66, 1–44.Google Scholar
Turkington, B., Eydeland, A. & Wang, S. 1991 A computational method for solitary internal waves in a continuously stratified fluid. Stud. Appl. Maths 85, 93–127.Google Scholar
Vanden-Broeck, J.-M. 1991 Elevation solitary waves with surface tension.. Phys. Fluids A 3, 2659–2663.Google Scholar
Vanden-Broeck, J.-M. 1992 Long periodic internal waves. Phys. Fluids A (sub judic).Google Scholar
Williams, G. P. & Wilson, R. J. 1988 The stability and genesis of Rossby vortices. J. Atmos. Sci. 45, 207–241.Google Scholar
Yih, C.-S. 1979 Fluid Mechanics. West River Press.