Freak waves as nonlinear stage of Stokes wave modulation instability

European Journal of Mechanics, B/Fluids

Volumn 25, Issue 5, 2006, Pages 677-692

  Zakharov, V E ^a,b   Dyachenko, A I ^b   Prokofiev, A O ^b  

^a University of Arizona   (United States)

^b LANDAU INSTITUTE FOR THEORETICAL PHYSICS   (Russian Federation)

Author keywords

Freak wave; Free surface; Modulation instability

Indexed keywords

COMPUTER SIMULATION; GRAVITY WAVES; NUMERICAL METHODS; PROBLEM SOLVING; SOLITONS; TWO DIMENSIONAL;

FREAK WAVE; FREE SURFACE; MODULATION INSTABILITY; STOKES WAVE;

WAVE PROPAGATION;

EID: 33747140354     PISSN: 09977546     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.euromechflu.2006.03.004     Document Type: Article

References (60)

1. Smith R. Giant waves. Fluid Mech. 77 1976 (2006) 417-431
2. Dean R.G. Freak waves: a possible explanation. In: Torum A., and Gudmestad O.T. (Eds). Water Wave Kinetics (1990), Kluwer 609-612
3. http://bell.mma.edu/~achase/NS-221-Big-Wave.html
4. Earle M.D. Extreme wave conditions during Hurricane Camille. J. Geophys. Res. 80 (1975) 377-379
5. Mori N., Liu P.C., and Yasuda T. Analysis of freak wave measurements in the sea of Japan. Ocean Engrg. 29 (2002) 1399-1414
6. Divinsky V.D., Levin B.V., Lopatikhin L.I., Pelinovsky E.N., and Slyunyaev A.V. A freak wave in the Black sea: Observations and simulation. Dokl. Earth Sci. 395 (2004) 438-443
7. Gerber M. Giant waves and the Agulhas current. Deep Sea Res. (1996)
8. Gutshabash Y.S., and Lavrenov I.V. Swell transformation in the Cape Agulhas current. Izv. Atmos. Ocean Phys. 22 (1986) 494-497
9. Irvine D.E., and Tilley D.G. Ocean wave directional spectra and wave-current interaction in the Agulhas from the shuttle imaging radar-B synthetic aperture radar. J. Geophys. Res. 93 (1988) 15389-15401
10. Lavrenov I.V. The wave energy concentration at the Agulhas current of South Africa. Natural Hazards 17 (1998) 117-127
11. Mallory J.K. Abnormal waves on the south-east of South Africa. Inst. Hydrog. Rev. 51 (1974) 89-129
12. Peregrine D.H. Interaction of water waves and currents. Adv. Appl. Mech. 16 (1976) 9-117
13. Sand S.E., Hansen N.E., Klinting P., Gudmestad O.T., and Sterndorff M.J. Freak wave kinematics. In: Torum A., and Gudmestad O.T. (Eds). Water Wave Kinematics (1990), Kluwer Academic, Dordrecht 535-549
14. Gerber M. The Benjamin-Feir instability of a deep water Stokes wave packet in the presence of a non-uniform medium. J. Fluid Mech. 176 (1987) 311-332
15. Gerber M. The interaction of deep water gravity waves and an annular current: linear theory. J. Fluid Mech. 248 (1993) 153-172
16. Kharif C., and Pelinovsky E. Physical mechanisms of the rogue wave phenomena. Eur. J. Mech. B Fluids 22 (2003) 603-634
17. White B.S., and Fornberg B. On the chance of freak waves at sea. J. Fluid Mech. 355 (1998) 113-138
18. Lighthill M.J. Contribution to the theory of waves in nonlinear dispersive systems. J. Inst. Math. Appl. 1 (1965) 269-306
19. Benjamin T.B., and Feir J.E. The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27 (1967) 417-430
20. Zakharov V.E. Stability of nonlinear waves in dispersive media. J. Teor. Prikl. Fiz. 51 (1966) 668-671 (in Russian)
21. Zakharov V.E. Sov. Phys. JETP 24 (1967) 455-459
22. Feir J.E. Discussion: Some results from wave pulse experiments. Proc. Roy. Soc. London Ser. A 299 (1967) 54-58
23. Zakharov V.E. Stability of periodic waves of finite amplitude on a surface of deep fluid. J. Appl. Mech. Tech. Phys. 9 (1968) 190-194
24. Zakharov V.E., and Shabat A.B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34 (1972) 62-69
25. Dysthe K.B. Note on a modification to the nonlinear Schrodinger equation for application to deep water waves. Proc. Roy. Soc. London Ser. A 369 (1979) 105-114
26. Trulsen K., and Dysthe K.B. A modified nonlinear Schrodinger equation for broader bandwidth gravity waves on deep water. Waves Motion 24 (1996) 281-289
27. Ablovitz M.I., Hammack D., Henderson J., and Scholder C.M. Modulated periodic Stokes waves in deep water. Phys. Rev. Lett. 84 (2000) 887-890
28. Ablovitz M.I., Hammack D., Henderson J., and Scholder C.M. Long-time dynamics of the modulational instability of deep water. Physica D 152-153 (2001) 416-433
29. Peregrine D.H. Water waves, nonlinear Schrodinger equations and their solutions. J. Austral. Math. Soc. B 25 (1983) 16-43
30. D.H. Peregrine, D. Skyner, M. Stiassnie, N. Dold, Nonlinear effects on focused water waves, in: Proc. 21th Intl Conf. on Coastal Engrg., vol. 1, 1988, pp. 732-742 (Chapter 54)
31. Tanaka M. Maximum amplitude of modulated wavetrain. Wave Motion 12 (1990) 559-568
32. K. Trulsen, K.B. Dysthe, Freak waves - a three-dimensional wave simulation, in: Proc. 21st Symposium on Naval Hydrodynamics, 1997, pp. 550-558; http://www.nap.edu/books/0309058791/html/550.html
33. K. Trulsen Simulating the spatial evolution of a measured time series of a freak wave, in: M. Olagnon, G.A. Athanassoulis (Eds.), Rogue Waves 2000: Brest, France, November 2000, Ifremer, 2001, pp. 265-274
34. Trulsen K., Kliakhandler I., Dysthe K.B., and Velarde M.G. On weakly nonlinear modulation of waves on deep water. Phys. Fluids 12 (2000) 2432-2437
35. M. Onorato, A.R. Osborne, M. Serio, T. Damiani, Occurrence of freak waves from envelope equation in random ocean wave simulations, in: M. Olagnon, G.A. Athanassoulis (Eds.), Rogue Waves 2000: Brest, France, November 2000, Ifremer, 2001, pp. 181-192
36. Onorato M., Osborne A.R., and Serio M. The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A 275 (2000) 386-393
37. Onorato M., Osborne A.R., Serio M., and Bertone S. Freak waves in random oceanic sea states. Phys. Rev. Lett. 86 (2001) 5831-5834
38. Onorato M., Osborne A.R., and Serio M. Extreme wave events in directional, random oceanic sea states. Phys. Fluids 14 (2002) L25-L28
39. Zakharov V.E., and Kuznetsov E.A. Optical solitons and quasisolitons. JETP 86 (1998) 1035-1046
40. Zakharov V.E., Dias F., and Pushkarev A.N. One-dimensional wave turbulence. Phys. Rep. 398 (2004) 1-65
41. Majda A., McLaughlin D., and Tabak E. A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci. 7 (1997) 9-44
42. Dommermuth D.G., and Yue D.K.P. A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184 (1987) 267-288
43. West B., Brueckner K., Janda R., Milder D., and Milton R. A new numerical method for surface hydrodynamics. J. Geophys. Res. C 92 11 (1987) 11803-11824
44. Clamond D., and Grue J. A fast method for fully nonlinear-wave computations. J. Fluid Mech. 447 (2001) 337-355
45. L.V. Ovsyannikov, Dinamika Sploshnoi Sredy, M.A. Lavrent'ev Institute of Hydrodynamics Sib. Branch USSR Acad. Sci. 15 (1973) 104-125 (in Russian)
46. Meison D., Orzag S., and Izraely M. Applications of numerical conformal mapping. J. Comput. Phys. 40 (1981) 345-360
47. D. Chalikov, D. Sheinin, Direct modeling of 1-D nonlinear potential waves, in: Ocean Waves, in: Advances in Fluid Mechanics, 1996, pp. 207-258 (Chapter 7)
48. Dyachenko A.I., Kuznetsov E.A., Spector M.D., and Zakharov V.E. Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1996) 73-79
49. Zakharov V.E., Dyachenko A.I., and Vasilyev O.A. New method for numerical simulation of nonstationary potential flow of incompressible fluid with a free surface. Eur. J. Mech. B Fluids 21 (2002) 283-291
50. Dyachenko A.I. On the dynamics of an ideal fluid with free surface. Dokl. Math. 63 (2001) 115-118
51. A.I. Dyachenko, Instability in the Numeric Models of Free Surface Hydrodynamics, International Conference "Frontiers of Nonlinear Physics", St. Petersburg-Nizhny Novgorod, Russia, August 2-9, 2005
52. Shamin R.V. On the existence of smooth solutions to the Dyachenko equations governing free-surface unsteady ideal fluid flows. Dokl. Math. 73 1 (2006) 112-113
53. J.W. Dold, D.H. Peregrine, Water-wave modulation, in: Proc. 20th Internat. Conf. on Coastal Engrg., vol. 1, 1986, pp. 163-175 (Chapter 13)
54. Banner M., and Tian X. On the determination of the onset of wave breaking for modulating surface gravity waves. J. Fluid Mech. 367 (1998) 107-137
55. Henderson K.L.D., Peregrine D.H., and Dold J.W. Unsteady water wave modulations: fully nonlinear solutions and comparison with nonlinear Schrodinger equation. Wave Motion 29 (1999) 341-361
56. Clamond D., and Grue J. Interaction between envelope solitons as a model for freak wave formation. Part I: Long time interaction. C. R. Mecanique 330 (2002) 575-580
57. Balk A. A Lagrangian for water waves. Phys. Fluids 8 (1996) 416-420
58. Song J., and Banner M.L. On determining the onset and strength of breaking for deep water waves. Part 1: Unforced irrotational wave groups. J. Phys. Oceanogr. 32 (2002) 2541-2558
59. Dold J.W. An efficient surface integral algorithm applied to unsteady gravity waves. J. Comput. Phys. 103 (1992) 90-115
60. Dyachenko A.I., and Zakharov V.E. Modulation instability of Stokes wave → Freak wave. JETP Lett. 81 6 (2005) 255-259