Skip to main content
SpringerLink
Log in

Twenty years of progresses in oceanic rogue waves: the role played by weakly nonlinear models

  • Original Paper
  • Published:
Natural Hazards Aims and scope Submit manuscript

Abstract

Here we discuss some of the progresses that have been made in the last 20 years in the field of oceanic rogue waves, focusing on the role played by leading order equations such as the nonlinear Schrödinger and the Korteweg-De Vries equations. For such equations, it is possible, as shown in Onorato et al. (Origin of heavy tail statistics in equations of the nonlinear Schrödinger type: an exact result, 2016. arXiv:1601.04317), to derive a very simple relation in which the variation of the third (for the KdV) and fourth (for the NLS) moment of the probability density function of the wave field can be related to the variation of the spectral bandwidth. These relations give some new perspectives on the formation of rogue waves in a random sea state.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akhmediev N, Korneev V (1986) Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor Math Phys 69(2):1089

    Article  Google Scholar 

  • Akhmediev N, Dudley JM, Solli D, Turitsyn S (2013) Recent progress in investigating optical rogue waves. J Opt 15(6):060201

    Article  Google Scholar 

  • Alber IE, Saffman P (1978) Stability of random nonlinear deep water waves with finite bandwidth spectra. Techincal report—TRW Defense and Space System Group 31326-6035-RU-

  • Annenkov S, Shrira V (2009) Evolution of kurtosis for wind waves. Geophys Res Lett 36(L13603):L13603

    Article  Google Scholar 

  • Annenkov S, Shrira V (2015) Modelling the impact of squall on wind waves with the generalized kinetic equation. J Phys Oceanogr 45(3):807

    Article  Google Scholar 

  • Benjamin TB, Feir JE (1967) The disintegration of wave trains on deep water. Part I. Theory. J Fluid Mech 27:417

    Article  Google Scholar 

  • Chabchoub A, Hoffmann N, Akhmediev N (2011) Rogue wave observation in a water wave tank. Phys Rev Lett 106(20):204502

    Article  Google Scholar 

  • Chabchoub A, Kibler B, Dudley JM, Akhmediev N (2014) Hydrodynamics of periodic breathers. Philos Trans R Soc Lond A 372(2027):20140005

    Article  Google Scholar 

  • Clauss G, Klein M, Onorato M (2011) Formation of extraordinary high waves in space and time. In: Proceedings of the ASME 2011 30th international conference on Ocean, Offshore and Arctic Engineering, June 19-24, 2011, Rotterdam, The Netherlands (2011), pp. 1–13

  • Dysthe KB, Trulsen K (1999) Note on breather type solutions of the NLS as models for freak-waves. Phys Scr T82:48

    Article  Google Scholar 

  • Fedele F (2015) On the kurtosis of deep-water gravity waves. J Fluid Mech 782:25. doi:10.1017/jfm.2015.538

    Article  Google Scholar 

  • Gramstad O, Trulsen K (2007) Influence of crest and group length on the occurrence of freak waves. J Fluid Mech 582:463

    Article  Google Scholar 

  • Gramstad O, Zeng H, Trulsen K, Pedersen G (2013) Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water. Phys Fluids 25(12):122103

    Article  Google Scholar 

  • Henderson K, Peregrine D, Dold J (1999) Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29(4):341

    Article  Google Scholar 

  • Iafrati A, Babanin A, Onorato M (2013) Modulational instability, wave breaking, and formation of large-scale dipoles in the atmosphere. Phys Rev Lett 110(18):184504

    Article  Google Scholar 

  • Islas A, Schober C (2005) Predicting rogue waves in random oceanic sea states. Phys Fluids 17(3):031701

    Article  Google Scholar 

  • Janssen PAEM (2003) Nonlinear four-wave interaction and freak waves. J Phys Ocean 33(4):863

    Article  Google Scholar 

  • Janssen P (2009) On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J Fluid Mech 637(1):1

    Article  Google Scholar 

  • Kharif C, Pelinovsky E, Talipova T, Slunyaev A (2001) Focusing of nonlinear wave groups in deep water. J Exp Theor Phys Lett 73(4):170

    Article  Google Scholar 

  • Kharif C, Giovanangeli JP, Touboul J, Grare L, Pelinovsky E (2008) Influence of wind on extreme wave events: experimental and numerical approaches. J Fluid Mech 594:209

    Article  Google Scholar 

  • Kharif C, Pelinovsky E, Slunyaev A (2009) Rogue waves in the ocean. Springer, Berlin

    Google Scholar 

  • Kibler B, Chabchoub A, Gelash A, Akhmediev N, Zakharov V (2015) Superregular breathers in optics and hydrodynamics: omnipresent modulation instability beyond simple periodicity. Phys Rev X 5(4):041026

    Google Scholar 

  • Kuznetsov E (1977) Solitons in a parametrically unstable plasma. Akademiia Nauk SSSR Doklady 236:575–577

    Google Scholar 

  • Mori N, Onorato M, Janssen PA (2011) On the estimation of the kurtosis in directional sea states for freak wave forecasting. J Phys Oceanogr 41(8):1484

    Article  Google Scholar 

  • Moslem W, Sabry R, El-Labany S, Shukla P (2011) Dust-acoustic rogue waves in a nonextensive plasma. Phys Rev E 84(6):066402

    Article  Google Scholar 

  • Onorato M, Osborne AR, Serio M, Damiani T (2000) Occurrence of freak waves from envelope equations in random ocean wave simulations. Rogue Wave 2000:181

    Google Scholar 

  • Onorato M, Osborne AR, Serio M, Bertone S (2001) Freak waves in random oceanic sea states. Phys Rev Lett 86:5831

    Article  Google Scholar 

  • Onorato M, Osborne AR, Serio M (2002) Extreme wave events in directional random oceanic sea states. Phys Fluids 14(4):25

    Article  Google Scholar 

  • Onorato M, Osborne A, Fedele R, Serio M (2003) Landau damping and coherent structures in narrow-banded 1 + 1 deep water gravity waves. Phys Rev E 67(4):046305

    Article  Google Scholar 

  • Onorato M, Osborne A, Serio M, Cavaleri L, Brandini C, Stansberg C (2006) Extreme waves, modulational instability and second order theory: wave flume experiments on irregular waves. Eur J Mech B Fluids 25:586

    Article  Google Scholar 

  • Onorato M, Waseda T, Toffoli A, Cavaleri L, Gramdstad O, Janssen PAEM, Kinoshita T, Monbaliu J, Mori N, Osborne AR, Serio M, Stansberg C, Tamura H, Trulsen K (2009) Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Phys Rev Lett 102:114502 (4)

    Article  Google Scholar 

  • Onorato M, Proment D, Clauss G, Klein M (2013a) Rogue waves: from nonlinear Schrödinger breather solutions to sea-keeping test. PloS one 8(2):e54629

    Article  Google Scholar 

  • Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F (2013b) Rogue waves and their generating mechanisms in different physical contexts. Phys Rep 528(2):47 Rogue waves and their generating mechanisms in different physical contexts

    Article  Google Scholar 

  • Onorato M, Proment D, El G, Randoux S, Suret P (2016) Origin of heavy tail statistics in equations of the nonlinear Schrödinger type: an exact result. arXiv:1601.04317

  • Osborne A, Onorato M, Serio M (2000) The nonlinear dynamics of rogue waves and holes in deep-water gravity wave train. Phys Lett A 275:386

    Article  Google Scholar 

  • Osborne A, Onorato M, Serio M (2005) Nonlinear Fourier analysis of deep-water, random surface waves: theoretical formulation and experimental observations of rogue waves. In: Proceedings of 14th Aha Huliko Oa winter workshop. Honolulu, Hawaii

  • Pelinovsky E et al (2006) Numerical modeling of the \(K\)d\(V\) random wave field. Eur J Mech B Fluids 25(4):425

    Article  Google Scholar 

  • Peregrine D (1983) Water waves, nonlinear Schrödinger equations and their solutions. J Austral Math Soc Ser B 25(1):16

    Article  Google Scholar 

  • Perić R, Hoffmann N, Chabchoub A (2015) Initial wave breaking dynamics of peregrine-type rogue waves: a numerical and experimental study. Eur J Mech B Fluids 49:71

    Article  Google Scholar 

  • Randoux S, Suret P, El G (2015) Identification of rogue waves from scattering transform analysis of periodized waveforms. arXiv:1512.04707

  • Sergeeva A, Pelinovsky E, Talipova T (2011) Nonlinear random wave field in shallow water: variable Korteweg-de Vries framework. Nat Hazards Earth Syst Sci 11(2):323

    Article  Google Scholar 

  • Shemer L, Alperovich L (2013) Peregrine breather revisited. Phys Fluids 25(5):051701

    Article  Google Scholar 

  • Shemer L, Sergeeva A (2009) An experimental study of spatial evolution of statistical parameters in a unidirectional narrow‐banded random wavefield. J Geophys Res Oceans 114(C1). doi:10.1029/2008JC005077

    Article  Google Scholar 

  • Shrira V, Slunyaev A (2014) Nonlinear dynamics of trapped waves on jet currents and rogue waves. Phys Rev E 89(4):041002

    Article  Google Scholar 

  • Slunyaev A (2006) Nonlinear analysis and simulations of measured freak wave time series. Eur J Mech B Fluids 25(5):621

    Article  Google Scholar 

  • Slunyaev A, Didenkulova I, Pelinovsky E (2011) Rogue waters. Contemp Phys 52(6):571

    Article  Google Scholar 

  • Smith R (1976) Giant waves. J Fluid Mech 77(03):417

    Article  Google Scholar 

  • Solli D, Ropers C, Koonath P, Jalali B (2007) Optical rogue waves. Nature 450(7172):1054

    Article  Google Scholar 

  • Stansberg CT ( 1994) Effects from directionality and spectral Bandwidth on non-linear spatial modulations of deep-water surface gravity waves. In: Edge B (ed) Proceedings of the ASCE in 1995 24th International Conference on Coastal Engineering, October 1994, vol 1, Kobe, Japan, pp 579–593

  • Trulsen K, Dysthe KB (1997) In: Proceedings of the 21st symposium on naval hydrodynamics. National Academy Press, Washington, pp 550–560

  • Trulsen K, Zeng H, Gramstad O (2012) Laboratory evidence of freak waves provoked by non-uniform bathymetry. Phys Fluids 24(9):097101

    Article  Google Scholar 

  • Viotti C, Dias F (2014) Extreme waves induced by strong depth transitions: fully nonlinear results. Phys Fluids 26(5):051705

    Article  Google Scholar 

  • Viotti C, Dutykh D, Dudley JM, Dias F (2013) Emergence of coherent wave groups in deep-water random sea. Phys Rev E 87(6):063001

    Article  Google Scholar 

  • Walczak P, Randoux S, Suret P (2015) Optical rogue waves in integrable turbulence. Phys Rev Lett 114(14):143903

    Article  Google Scholar 

  • Wen L, Li L, Li ZD, Song SW, Zhang XF, Liu W (2011) Matter rogue wave in Bose–Einstein condensates with attractive atomic interaction. Eur Phys J D 64(2–3):473

    Article  Google Scholar 

  • Xiao W, Liu Y, Wu G, Yue DK (2013) Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J Fluid Mech 720:357

    Article  Google Scholar 

  • Yan Z (2011) Vector financial rogue waves. Phys Lett A 375(48):4274

    Article  Google Scholar 

  • Zakharov V (1968) Stability of period waves of finite amplitude on surface of a deep fluid. J Appl Mech Tech Phys 9:190

    Article  Google Scholar 

  • Zakharov VE, Gelash AA (2013) Nonlinear stage of modulation instability. Phys Rev Lett 111:054101

    Article  Google Scholar 

  • Zeng H, Trulsen K (2012) Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom. Nat Hazards Earth Syst Sci 12(3):631

    Article  Google Scholar 

Download references

Acknowledgments

M.O. was supported by MIUR Grant PRIN 2012BFNWZ2. Dr. B. Giulinico, D. Proment, S. Randoux and G. El are acknowledged for discussions. With their collaboration, it was noticed that the Hamiltonian of the NLS equation contains information about the statistical properties of the wave field.

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria, 1, 10125, Torino, Italy

    Miguel Onorato

  2. Laboratoire de Physique des Lasers, Atomes et Molecules, Université de Lille, UMR-CNRS 8523, Lille, France

    Pierre Suret

Authors
  1. Miguel Onorato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Suret
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miguel Onorato.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Onorato, M., Suret, P. Twenty years of progresses in oceanic rogue waves: the role played by weakly nonlinear models. Nat Hazards 84 (Suppl 2), 541–548 (2016). https://doi.org/10.1007/s11069-016-2449-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11069-016-2449-z

Keywords

Search

Navigation