Hostname: page-component-54dcc4c588-tfzs5 Total loading time: 0 Render date: 2025-09-27T19:16:12.868Z Has data issue: false hasContentIssue false
  • English
  • Français

Transverse instability of interfacial solitary waves

Published online by Cambridge University Press:  25 September 2008

TAKESHI KATAOKA
TAKESHI KATAOKA*
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan kataoka@mech.kobe-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of finite-amplitude interfacial solitary waves in a two-layer fluid of finite depth is examined analytically on the basis of the Euler equations. An asymptotic analysis is performed, which provides an explicit criterion of instability in the case of long-wavelength transverse disturbances. This result leads to the general statement that, when the amplitude of the solitary wave is increased, the solution becomes transversely unstable before an exchange of longitudinal stability occurs.

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 611 , 25 September 2008 , pp. 255 - 282
Copyright
Copyright © Cambridge University Press 2008

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Amick, C. J. & Turner, R. E. L. 1986 A global theory of internal solitary waves in two-fluid systems. Trans. Am. Math. Soc. 298, 431–484.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.CrossRefGoogle Scholar
Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153–183.Google Scholar
Calvo, D. C. & Akylas, T. R. 2003 On interfacial gravity–capillary solitary waves of the Benjamin type and their stability. Phys. Fluids 15, 1261–1270.CrossRefGoogle Scholar
Craig, W., Guyenne, P. & Kalisch, H. 2005 Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Maths 68, 1587–1641.CrossRefGoogle Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 2003 On internal fronts. J. Fluid Mech. 479, 145–154.CrossRefGoogle Scholar
Evans, W. A. B. & Ford, M. J. 1996 An integral equation approach to internal (2-layer) solitary waves. Phys. Fluids 8, 2032–2047.CrossRefGoogle Scholar
Funakoshi, M. & Oikawa, M. 1986 Long internal waves of large-amplitude in a two-layer fluid. J. Phys. Soc. Japan 55, 128–144.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusås, P. O. & Sveen, J. K. 1999 Properties of large amplitude internal waves. J. Fluid Mech. 380, 257–278.CrossRefGoogle Scholar
Jeffery, A. & Kakutani, T. 1970 Stability of the Burgers shock wave and the Korteweg–de Vries soliton. Indiana Univ. Math. J. 20, 463–468.CrossRefGoogle Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in a weakly dispersing medium. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kakutani, T. & Yamasaki, N. 1978 Solitary waves on a two-layer fluid. J. Phys. Soc. Japan 45, 674–679.CrossRefGoogle Scholar
Kataoka, T. 2006 The stability of finite-amplitude interfacial solitary waves. Fluid Dyn. Res. 38, 831–867.CrossRefGoogle Scholar
Kataoka, T. & Tsutahara, M. 2004 a Transverse instability of surface solitary waves. J. Fluid Mech. 512, 211–221.CrossRefGoogle Scholar
Kataoka, T. & Tsutahara, M. 2004 b Instability of solitary wave solutions to long-wavelength transverse perturbations in the generalized Kadomtsev–Petviashvili equation with negative dispersion. Phys. Rev. E 70, 016604.Google ScholarPubMed
Kivshar, Y. S. & Pelinovsky, D. E. 2000 Self-focusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117–195.CrossRefGoogle Scholar
Laget, O. & Dias, F. 1997 Numerical computation of capillary–gravity interfacial solitary waves. J. Fluid Mech. 349, 221–251.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Tanaka, M. 1997 On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68.CrossRefGoogle Scholar
Meiron, D. I. & Saffman, P. G. 1983 Overhanging interfacial gravity waves of large amplitude. J. Fluid Mech. 129, 213–218.CrossRefGoogle Scholar
Michallet, H. & Barthélemy, E. 1998 Experimental study of interfacial solitary waves. J. Fluid Mech. 366, 159–177.CrossRefGoogle Scholar
Ostrovsky, L. A. & Shrira, V. I. 1976 Instability and self-refraction of solitons. Sov. Phys. JETP 44, 738–743.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1988 Finite-amplitude solitary waves at the interface between two homogeneous fluids. Phys. Fluids 31, 3550–3559.CrossRefGoogle Scholar
Rusås, P. O. & Grue, J. 2002 Solitary waves and conjugate flows in a three-layer fluid. Eur. J. Mech. B/Fluids 21, 185–206.CrossRefGoogle Scholar
Sha, H. & Vanden-Broeck, J.-M. 1993 Two-layer flows past a semicircular obstruction. Phys. Fluids A 5, 2661–2668.CrossRefGoogle Scholar
Shrira, V. I. 1980 Nonlinear refraction of solitons. Sov. Phys. JETP 52, 44–49.Google Scholar
Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29, 650–655.CrossRefGoogle Scholar
Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987 Instability and breaking of a solitary wave. J. Fluid Mech. 185, 235–248.CrossRefGoogle Scholar
Turner, R. E. L. & Vanden-Broeck, J.-M. 1988 Broadening of interfacial solitary waves. Phys. Fluids 31, 2486–2490.CrossRefGoogle Scholar
Zakharov, V. E. 1975 Instability and nonlinear oscillations of solitons. JETP Lett. 22, 172–173.Google Scholar