Advertisement

Modulation instability of Stokes wave → freak wave

  • A. I. Dyachenko
  • V. E. Zakharov
Plasma, Gases

Abstract

Formation of waves of large amplitude (freak waves, killer waves) at the surface of the ocean is studied numerically. We have observed that freak waves have the same ratio of the wave height to the wave length as limiting Stokes waves. When a freak wave reaches this limiting state, it breaks. The physical mechanism of freak wave formation is discussed.

PACS numbers

02.60.Cb 47.15.Hg 92.10.−c 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Smith, J. Fluid Mech. 77, 417 (1976).ADSMATHMathSciNetGoogle Scholar
  2. 2.
    R. G. Dean, in Water Wave Kinetics, Ed. by A. Torum and O. T. Gudmestad (Kluwer Academic, Dordrecht, 1990), p. 609.Google Scholar
  3. 3.
    I. V. Lavrenov, Nat. Hazards 17, 117 (1998).CrossRefGoogle Scholar
  4. 4.
  5. 5.
    V. D. Divinsky, B. V. Levin, L. I. Lopatikhin, et al., Dokl. Earth Sci. 395, 438 (2004).Google Scholar
  6. 6.
    M. J. Lighthill, J. Inst. Math. Appl. 1, 269 (1965).MATHMathSciNetGoogle Scholar
  7. 7.
    T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).ADSGoogle Scholar
  8. 8.
    V. E. Zakharov, Zh. Éksp. Teor. Fiz. 51, 668 (1966) [Sov. Phys. JETP 24, 455 (1967)].Google Scholar
  9. 9.
    J. E. Feir, Proc. R. Soc. London, Ser. A 299, 54 (1967).ADSGoogle Scholar
  10. 10.
    V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968).Google Scholar
  11. 11.
    V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).MathSciNetGoogle Scholar
  12. 12.
    K. B. Dysthe, Proc. R. Soc. London, Ser. A 369, 105 (1979).ADSMATHGoogle Scholar
  13. 13.
    K. Trulsen and K. B. Dysthe, Wave Motion 24, 281 (1996).MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. I. Ablovitz, D. Hammack, J. Henderson, and C. M. Scholder, Phys. Rev. Lett. 84, 887 (2000); Physica D (Amsterdam) 152–153, 416 (2001).ADSGoogle Scholar
  15. 15.
    M. Onorato, A. R. Osborne, M. Serio, and T. Damiani, in Rogue Waves 2000: Proceedings of a Workshop, Brest, France, 2000, Ed. by M. Olagnon and G. A. Athanassoulis (Ifremer, Plouzanu, France, 2001), p. 181.Google Scholar
  16. 16.
    M. Onorato, A. R. Osborne, and M. Serio, Phys. Lett. A 275, 386 (2000).ADSMathSciNetGoogle Scholar
  17. 17.
    M. Onorato, A. R. Osborne, M. Serio, and S. Bertone, Phys. Rev. Lett. 86, 5831 (2001).ADSCrossRefGoogle Scholar
  18. 18.
    M. Onorato, A. R. Osborne, and M. Serio, Phys. Fluids 14, L25 (2002).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    D. H. Peregrine, J. Aust. Math. Soc. B, Appl. Math. 25, 16 (1983).MATHMathSciNetGoogle Scholar
  20. 20.
    D. H. Peregrine, D. Skyner, M. Stiassnie, and J. W. Dold, in Proceedings of 21st International Conference on Coastal Engineering (1988), Vol. 1, Chap. 54, p. 732.Google Scholar
  21. 21.
    K. Trulsen and K. B. Dysthe, in Proceedings of 21st Symposium on Naval Hydrodynamics (1997), p. 550; http://www.nap.edu/books/0309058791/html/550.html.Google Scholar
  22. 22.
    K. Trulsen, in Rogue Waves 2000: Proceedings of a Workshop, Brest, France, 2000, Ed. by M. Olagnon and G. A. Athanassoulis (Ifremer, Plouzanu, France, 2001), p. 265.Google Scholar
  23. 23.
    K. Trulsen, I. Kliakhandler, K. B. Dysthe, and M. G. Velarde, Phys. Fluids 24, 32 (2000); D. Clamond and J. Grue, C. R. Acad. Sci., Ser. Mec. 330, 575 (2002).MathSciNetGoogle Scholar
  24. 24.
    C. Kharif and E. Pelinovsky, Eur. J. Mech. B/Fluids 22, 603 (2003).MathSciNetCrossRefGoogle Scholar
  25. 25.
    A. A. Kurkin and E. N. Pelinovsky, Freak Waves: Facts, Theory and Modeling (Nizhni Novgorod Univ., 2004).Google Scholar
  26. 26.
    V. E. Zakharov and E. A. Kuznetsov, JETP 86, 1035 (1998).ADSCrossRefGoogle Scholar
  27. 27.
    V. E. Zakharov, F. Dias, and A. N. Pushkarev, Phys. Rep. 398, 1 (2004).ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    A. Majda, D. McLaughlin, and E. Tabak, J. Nonlinear Sci. 7, 9 (1997).MathSciNetGoogle Scholar
  29. 29.
    A. I. Dyachenko, E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov, Phys. Lett. A 221, 73 (1996).ADSGoogle Scholar
  30. 30.
    V. E. Zakharov, A. I. Dyachenko, and O. A. Vasilyev, Eur. J. Mech. B/Fluids 21, 283 (2002).MathSciNetCrossRefGoogle Scholar
  31. 31.
    V. E. Zakharov, Am. Math. Soc. Trans., Ser. 2 182, 167 (1998).MATHGoogle Scholar
  32. 32.
    A. I. Dyachenko, in Proceedings of the II International Conference on Frontiers of Nonlinear Physics (Nizhny Novgorod-St. Petersburg, Russia, 2004).Google Scholar
  33. 33.
    A. I. Dyachenko, Dokl. Math. 63, 115 (2001).MATHGoogle Scholar
  34. 34.
    J. W. Dold and D. H. Peregrine, in Proceedings of 20th International Conference on Coastal Engineering (1986), Vol. 1, Chap. 13, p. 163.Google Scholar
  35. 35.
    D. G. Dommermuth and D. K. P. Yue, J. Fluid Mech. 184, 267 (1987).ADSGoogle Scholar
  36. 36.
    J. W. Dold, J. Comput. Phys. 103, 90 (1992).MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    W. Tsai and D. Yue, Annu. Rev. Fluid Mech. 28, 249 (1996).ADSMathSciNetGoogle Scholar
  38. 38.
    J. A. Zufiria and P. G. Saffman, Stud. Appl. Math. 74, 259 (1986).MathSciNetGoogle Scholar
  39. 39.
    D. Meison, S. Orzag, and M. Izraely, J. Comput. Phys. 40, 345 (1981).MathSciNetGoogle Scholar
  40. 40.
    J. Song and M. L. Banner, J. Phys. Oceanogr. 32, 2541 (2002).MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2005

Authors and Affiliations

  • A. I. Dyachenko
    • 1
  • V. E. Zakharov
    • 1
    • 2
  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

Personalised recommendations