Advertisement

Rogue Waves in the Ocean, the Role of Modulational Instability, and Abrupt Changes of Environmental Conditions that Can Provoke Non Equilibrium Wave Dynamics

  • Karsten Trulsen
Chapter
Part of the Springer Oceanography book series (SPRINGEROCEAN)

Abstract

Modulational instability is an efficient mechanism for the generation of rogue waves in the limit of narrow-banded and long-crested wave fields. While such wave fields are easily achieved in laboratories, there appears to be lacking evidence that known occurrences of rogue waves in the ocean (e.g. Draupner “New Year” wave, Andrea wave) or ship accidents that could have been provoked by rogue waves (e.g. the Prestige accident) actually happened in sea states favorable for the modulational instability to have played an important role. The absence of modulational instability does not mean that nonlinear interactions are unimportant. Here we point out recent results that suggest large deviations from Gaussian statistics can happen due to nonlinearity in the absence of modulational instability, the key ingredient seems to be that the wave field is brought into a state of non-equilibrium.

Notes

Acknowledgements

This work has been supported by the Research Council of Norway through the project “EXtreme wave WArning criteria for MARine structures” (ExWaMar) RCN 256466.

References

  1. 1.
    Alber, I. E. (1978). The effects of randomness on the stability of two-dimensional surface wavetrains. Proceedings of the Royal Society of London A, 363, 525–546.CrossRefGoogle Scholar
  2. 2.
    Alber, I. E, Saffman, P. (1978). Stability of random nonlinear deep water waves with finite bandwidth spectra. Technical Report, 31326–6035–RU–TRW Defense and Space System GroupGoogle Scholar
  3. 3.
    Benjamin, T. B. (1967). Instability of periodic wavetrains in nonlinear dispersive systems. Proceedings of the Royal Society of London A, 299, 59–75.CrossRefGoogle Scholar
  4. 4.
    Benjamin, T. B., & Feir, J. E. (1967). The disintegration of wave trains on deep water. Journal of Fluid Mechanics, 27, 417–430.CrossRefGoogle Scholar
  5. 5.
    Benney, D. J., & Roskes, G. J. (1969). Wave instabilities. Studies in Applied Mathematics, 48, 377–385.CrossRefGoogle Scholar
  6. 6.
    Bitner-Gregersen, E. M, Gramstad, O. (2016). Rogue waves—Impact on ships and offshore structures. Technical Report, 05–2015, DNV-GLGoogle Scholar
  7. 7.
    Bitner-Gregersen, E. M., Fernandez, L., Lefèvre, J. M., Monbaliu, J., & Toffoli, A. (2014). The North Sea Andrea storm and numerical simulations. Natural Hazards and Earth System Science, 14, 1407–1415.CrossRefGoogle Scholar
  8. 8.
    Cavaleri, L., Barbariol, F., Benetazzo, A., Bertotti, L., Bidlot, J. R., Janssen, P., et al. (2016). The Draupner wave: A fresh look and the emerging view. Journal of Geophysical Research: Oceans, 121, 6061–6075.Google Scholar
  9. 9.
    Chabchoub, A., Hoffmann, N. P., & Akhmediev, N. (2011). Rogue wave observation in a water wave tank. Physical Review Letters, 106, 204,502.CrossRefGoogle Scholar
  10. 10.
    Chabchoub, A., Akhmediev, N., & Hoffmann, N. P. (2012). Experimental study of spatiotemporally localized surface gravity water waves. Physical Review Letters, 86, 016,311.Google Scholar
  11. 11.
    Chabchoub, A., Kibler, B., Dudley, J. M., & Akhmediev, N. (2014). Hydrodynamics of periodic breathers. Philosophical Transactions of the Royal Society of London A, 372, 20140,005.CrossRefGoogle Scholar
  12. 12.
    Crawford, D. R., Saffman, P. G., & Yuen, H. C. (1980). Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion, 2, 1–16.CrossRefGoogle Scholar
  13. 13.
    Davey, A. (1972). The propagation of a weak nonlinear wave. Journal of Fluid Mechanics, 53, 769–781.CrossRefGoogle Scholar
  14. 14.
    Dommermuth, D. (2000). The initialization of nonlinear waves using an adjustment scheme. Wave Motion, 32, 307–317.CrossRefGoogle Scholar
  15. 15.
    Donelan, M. A., & Magnusson, A. K. (2017). The making of the Andrea wave and other rogues. Scientific Reports, 7, 44,124.CrossRefGoogle Scholar
  16. 16.
    Draper, L. (1964). ‘Freak’ ocean waves. Oceanus, 10, 13–15.Google Scholar
  17. 17.
    Dysthe, K., Krogstad, H. E., & Müller, P. (2008). Oceanic rogue waves. Annual Review of Fluid Mechanics, 40, 287–310.CrossRefGoogle Scholar
  18. 18.
    Dysthe, K. B. (1979). Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proceedings of the Royal Society of London A, 369, 105–114.CrossRefGoogle Scholar
  19. 19.
    Dysthe, K. B., & Trulsen, K. (1999). Note on breather type solutions of the NLS as models for freak-waves. Physica Scripta, T82, 48–52.CrossRefGoogle Scholar
  20. 20.
    Fedele, F., Brennan, J., Ponce de León, S., Dudley, J., & Dias, F. (2016). Real world ocean rogue waves explained without the modulational instability. Scientific Reports, 6, 27,715.CrossRefGoogle Scholar
  21. 21.
    Gramstad, O., & Trulsen, K. (2007). Influence of crest and group length on the occurrence of freak waves. Journal of Fluid Mechanics, 582, 463–472.CrossRefGoogle Scholar
  22. 22.
    Gramstad, O., Zeng, H., Trulsen, K., & Pedersen, G. K. (2013). Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water. Physics of Fluid, 25, 122103.CrossRefGoogle Scholar
  23. 23.
    Hasimoto, H., & Ono, H. (1972). Nonlinear modulation of gravity waves. Journal of the Physical Society of Japan, 33, 805–811.CrossRefGoogle Scholar
  24. 24.
    Haver, S. (2000). Evidences of the existence of freak waves. In: 2000 Rogue Waves (pp. 129–140). Ifremer.Google Scholar
  25. 25.
    Haver, S. (2004). A possible freak wave event measured at the Draupner jacket Januar 1 1995. Rogue Waves, 2004, 1–8.Google Scholar
  26. 26.
    Janssen, P. A. E. M. (2003). Nonlinear four-wave interactions and freak waves. Journal of Physical Oceanography, 33, 863–884.CrossRefGoogle Scholar
  27. 27.
    Kharif, C., Pelinovsky, E., Slunyaev, A. (2009). Rogue waves in the ocean. SpringerGoogle Scholar
  28. 28.
    Kuznetsov, E. A. (1977). Solitons in a parametrically unstable plasma. Soviet physics, Doklady, 22, 507–508.Google Scholar
  29. 29.
    Longuet-Higgins, M. S. (1963). The effect of non-linearities on statistical distributions in the theory of sea waves. Journal of Fluid Mechanics, 17, 459–480.CrossRefGoogle Scholar
  30. 30.
    Ma, Y. C. (1979). The perturbed plane-wave solutions of the cubic Schrödinger equation. Studies in Applied Mathematics, 60, 43–58.CrossRefGoogle Scholar
  31. 31.
    Magnusson, A. K., & Donelan, M. A. (2013). The Andrea wave characteristics of a measured North Sea rogue wave. Journal of Offshore Mechanics and Arctic Engineering, 135, 031,108.CrossRefGoogle Scholar
  32. 32.
    Mallory, J. K. (1974). Abnormal waves on the south east coast of South Africa. The International Hydrographic Review, 51, 99–129.Google Scholar
  33. 33.
    Masuda, A., Kuo, Y. Y., & Mitsuyasu, H. (1979). On the dispersion relation of random gravity waves. Part 1. Theoretical framework. Journal of Fluid Mechanics, 92, 717–730.CrossRefGoogle Scholar
  34. 34.
    Molin, B., Remy, F., Kimmoun, O., & Jamois, E. (2005). The role of tertiary wave interactions in wave-body problems. Journal of Fluid Mechanics, 528, 323–354.CrossRefGoogle Scholar
  35. 35.
    Molin, B., Kimmoun, O., Remy, F., & Chatjigeorgiou, I. K. (2014). Third-order effects in wave-body interaction. European Journal of Mechanics-B/Fluids, 47, 132–144.CrossRefGoogle Scholar
  36. 36.
    Onorato, M., & Suret, P. (2016). Twenty years of progresses in oceanic rogue waves: the role played by weakly nonlinear models. Natural Hazards, 84, 541–548.CrossRefGoogle Scholar
  37. 37.
    Onorato, M., Osborne, A. R., Serio, M., & Bertone, S. (2001). Freak waves in random oceanic sea states. Physical Review Letters, 86, 5831–5834.CrossRefGoogle Scholar
  38. 38.
    Onorato, M., Osborne, A. R., & Serio, M. (2002a). Extreme wave events in directional, random oceanic sea states. Physics of Fluids, 14, L25–L28.CrossRefGoogle Scholar
  39. 39.
    Onorato, M., Osborne, A. R., Serio, M., Resio, D., Pushkarev, A., Zakharov, V. E., et al. (2002b). Freely decaying weak turbulence for sea surface gravity waves. Physical Review Letters, 89(14), 144,501.CrossRefGoogle Scholar
  40. 40.
    Onorato, M., Osborne, A. R., Serio, M., Cavaleri, L., Brandini, C., & Stansberg, C. T. (2004). Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments. Physical Review Letters, 70, 1–4.Google Scholar
  41. 41.
    Onorato, M., Osborne, A. R., Serio, M., Cavaleri, L., Brandini, C., & Stansberg, C. T. (2006). Extreme waves, modulational instability and second order theory: wave flume experiments on irregular waves. European Journal of Mechanics - B/Fluids, 25, 586–601.CrossRefGoogle Scholar
  42. 42.
    Onorato, M., Cavaleri, L., Fouques, S., Gramstad, O., Janssen, P. A. E. M., Monbaliu, J., et al. (2009a). a) Statistical properties of mechanically generated surface gravity waves: A laboratory experiment in a three dimensional wave basin. Journal of Fluid Mechanics, 627, 235–257.CrossRefGoogle Scholar
  43. 43.
    Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P. A. E. M., et al. (2009b). (b) Statistical properties of directional ocean waves: The role of the modulational instability in the formation of extreme events. Physical Review Letters, 102, 114,502.CrossRefGoogle Scholar
  44. 44.
    Osborne, A. R., Onorato, M., & Serio, M. (2000). The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Physics Letters A, 275, 386–393.CrossRefGoogle Scholar
  45. 45.
    Peregrine, D. H. (1983). Water-waves, non-linear Schrödinger-equations and their solutions. Journal of the Australian Mathematical Society B, 25, 16–43.CrossRefGoogle Scholar
  46. 46.
    Pierson, W. J. (1955). Wind generated gravity waves. Advances in Geophysics, 2, 93–178.CrossRefGoogle Scholar
  47. 47.
    Raustøl, A. (2014). Freake bølger over variabelt dyp. Master’s thesis, University of OsloGoogle Scholar
  48. 48.
    Sand, S. E., Ottesen Hansen, N. E., Klinting, P., Gudmestad, O. T., & Sterndorff, M. J. (1990). Freak wave kinematics. In O. T. Gudmestad, A. Tørum (Ed.), Water Wave Kinematics (pp. 535–549). Kluwer.Google Scholar
  49. 49.
    Sergeeva, A., Pelinovsky, E., & Talipova, T. (2011). Nonlinear random wave field in shallow water: variable Korteweg-de Vries framework. Natural Hazards and Earth System Sciences, 11, 323–330.CrossRefGoogle Scholar
  50. 50.
    Stokes, G. G. (1847). On the theory of oscillatory waves. Transactions of the Cambridge Philosophical Society, 8, 441–455.Google Scholar
  51. 51.
    Tamura, H., Waseda, T., & Miyazawa, Y. (2009). Freakish sea state and swell-windsea coupling: Numerical study of the Suwa-Maru incident. Geophysical Research Letters, 36, L01,607.CrossRefGoogle Scholar
  52. 52.
    Tayfun, M. A. (1980). Narrow-band nonlinear sea waves. Journal of Geophysical Research, 85, 1548–1552.CrossRefGoogle Scholar
  53. 53.
    Tick, L. J. (1959). A non-linear random model of gravity waves I. Journal of Mathematics and Mechanics, 8, 643–651.Google Scholar
  54. 54.
    Toffoli, A., & Bitner-Gregersen, E. M. (2011). Extreme and rogue waves in directional wave fields. Open Ocean Engineering Journal, 4, 24–33.CrossRefGoogle Scholar
  55. 55.
    Trulsen, K., & Dysthe, K. B. (1996). A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion, 24, 281–289.CrossRefGoogle Scholar
  56. 56.
    Trulsen, K., & Dysthe, K. B. (1997). Freak waves—a three-dimensional wave simulation. In: Proceedings of the 21st Symposium on Naval Hydrodynamics, National Academy Press (pp 550–560).Google Scholar
  57. 57.
    Trulsen, K., Kliakhandler, I., Dysthe, K. B., & Velarde, M. G. (2000). On weakly nonlinear modulation of waves on deep water. Physics of Fluids, 12, 2432–2437.CrossRefGoogle Scholar
  58. 58.
    Trulsen, K., Zeng, H., & Gramstad, O. (2012). Laboratory evidence of freak waves provoked by non-uniform bathymetry. Physics of Fluids, 24, 097,101.CrossRefGoogle Scholar
  59. 59.
    Trulsen, K., Nieto Borge, J. C., Gramstad, O., Aouf, L., & Lefèvre, J. M. (2015). Crossing sea state and rogue wave probability during the Prestige accident. Journal of Geophysical Research: Oceans, 120, 7113–7136.Google Scholar
  60. 60.
    Viotti, C., & Dias, F. (2014). Extreme waves induced by strong depth transitions: Fully nonlinear results. Physics of Fluids, 26, 051,705.CrossRefGoogle Scholar
  61. 61.
    Viste-Ollestad I, Andersen TL, Oma N, Zachariassen S (2016) Granskingsrapport etter hendelse med fatalt utfall på COSL Innovator 30. desember 2015. Tech. rep., PetroleumstilsynetGoogle Scholar
  62. 62.
    Waseda, T., Tamura, H., & Kinoshita, T. (2012). Freakish sea index and sea states during ship accidents. Journal of Marine Science and Technology, 17, 305–314.CrossRefGoogle Scholar
  63. 63.
    Waseda, T., In, K., Kiyomatsu, K., Tamura, H., Miyazawa, Y., & Iyama, K. (2014). Predicting freakish sea state with an operational third-generation wave model. Natural Hazards and Earth System Science, 14, 945–957.CrossRefGoogle Scholar
  64. 64.
    Zakharov, V. E. (1968). Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9, 190–194.CrossRefGoogle Scholar
  65. 65.
    Zeng, H., & Trulsen, K. (2012). Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom. Natural Hazards and Earth System Science, 12, 631–638.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway

Personalised recommendations