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Waves in Geophysical Fluids pp 49–106Cite as

Weakly nonlinear and stochastic properties of ocean wave fields. Application to an extreme wave event

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Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 489))

Abstract

There has been much interest in freak or rogue waves in recent years, especially after the Draupner “New Year Wave” that occurred in the central North Sea on January 1st 1995. From the beginning there have been two main research directions, deterministic and statistical. The deterministic approach has concentrated on focusing mechanisms and modulational instabilities, these are explained in Chap. 3 and some examples are also given in Chap. 4. A problem with many of these deterministic theories is that they require initial conditions that are just as unlikely as the freak wave itself, or they require idealized instabilities such as Benjamin-Feir instability to act over unrealistically long distances or times. For this reason the deterministic theories alone are not very useful for understanding how exceptional the freak waves are. On the other hand, a purely statistical approach based on data analysis is difficult due to the unusual character of these waves. Recently a third research direction has proved promising, stochastic analysis based on Monte-Carlo simulations with phase-resolving deterministic models. This approach accounts for all possible mechanisms for generating freak waves, within a sea state that is hopefully as realistic as possible. This chapter presents several different modified nonlinear Schrödinger (MNLS) equations as candidates for simplified phase-resolving models, followed by an introduction to some essential elements of stochastic analysis. The material is aimed at readers with some background in nonlinear wave modeling, but little background in stochastic modeling. Despite their simplicity, the MNLS equations capture remarkably non-trivial physics of the sea surface such as the establishment of a quasi-stationary spectrum with ω−4 power law for the high-frequency tail, and nonlinear probability distributions for extreme waves. In the end we will suggest how often one should expect a “New Year Wave” within the sea state in which it occurred.

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Bibliography

  • I. E. Alber (1978) The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A, Vol. 363, 525–546.

    MATH  MathSciNet  Google Scholar 

  • U. Brinch-Nielsen and I. G. Jonsson (1986) Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth. Wave Motion, Vol. 8, 455–472.

    Article  MATH  MathSciNet  Google Scholar 

  • D. E. Cartwright and M. S. Longuet-Higgins (1956) The statistical distribution of the maxima of a random function. Proc. R. Soc. Lond. A, Vol. 237, 212–232.

    MATH  MathSciNet  Google Scholar 

  • D. R. Crawford, P. G. Saffman and H. C. Yuen (1980) Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion, Vol. 2, 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  • K. B. Dysthe (1979) Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A, Vol. 369, 105–114.

    Article  MATH  Google Scholar 

  • K. B. Dysthe and K. Trulsen (1999) Note on breather type solutions of the NLS as models for freak-waves. Physica Scripta, Vol. T82, 48–52.

    Article  Google Scholar 

  • K. B. Dysthe, K. Trulsen, H. E. Krogstad and H. Socquet-Juglard (2003) Evolution of a narrow band spectrum of random surface gravity waves. J. Fluid Mech., Vol. 478, 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  • Y. Goda (2000) Random seas and design of maritime structures. World Scientific.

    Google Scholar 

  • J. Grue, D. Clamond, M. Huseby and A. Jensen (2003) Kinematics of extreme waves in deep water. Appl. Ocean Res., Vol. 25, 355–366.

    Article  Google Scholar 

  • S. Haver (2004) A possible freak wave event measured at the Draupner jacket Januar 1 1995. http://www.ifremer.fr/web-com/stw2004/rw/fullpapers/walk_on_haver.pdf. In Rogue Waves 2004, pages 1–8.

    Google Scholar 

  • P. A. E. M. Janssen (2003) Nonlinear four-wave interactions and freak waves. J. Phys. Ocean., Vol. 33, 863–884.

    Article  Google Scholar 

  • D. Karunakaran, M. Baerheim, and B. J. Leira (1997) Measured and simulated dynamic response of a jacket platform. In C. Guedes-Soares, M. Arai, A. Naess, and N. Shetty, editors, Proceedings of the 16th International Conference on Offshore Mechanics and Arctic Engineering, Vol. II, 157–164. ASME.

    Google Scholar 

  • E. Kit and L. Shemer (2002) Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves. J. Fluid Mech., Vol. 450, 201–205.

    Article  MATH  MathSciNet  Google Scholar 

  • V. P. Krasitskii (1994) On reduced equations in the Hamiltonian theory of weakly nonlinear surface-waves. J. Fluid Mech., Vol. 272, 1–20.

    Article  MATH  MathSciNet  Google Scholar 

  • E. Lo and C. C. Mei (1985) A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech., Vol. 150, 395–416.

    Article  MATH  Google Scholar 

  • E. Y. Lo and C. C. Mei (1987) Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study. Wave Motion, Vol. 9, 245–259.

    Article  MATH  Google Scholar 

  • D. U. Martin and H. C. Yuen (1980) Quasi-recurring energy leakage in the two-spacedimensional nonlinear Schrödinger equation. Phys. Fluids, Vol. 23, 881–883.

    Article  Google Scholar 

  • M. K. Ochi (1998) Ocean waves. Cambridge.

    Google Scholar 

  • M. Onorato, A. R. Osborne and M. Serio (2002) Extreme wave events in directional, random oceanic sea states. Phys. Fluids, Vol. 14, L25–L28.

    Article  MathSciNet  Google Scholar 

  • M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini and C. T. Stansberg (2004) Observation of strongly non-gaussian statistics for random sea surface gravity waves in wave flume experiments. Phys. Rev. E, Vol. 70, 1–4.

    Article  Google Scholar 

  • M. Onorato, A. R. Osborne, M. Serio, D. Resio, A. Pushkarev, V. E. Zakharov and C. Brandini (2002) Freely decaying weak turbulence for sea surface gravity waves. Phys. Rev. Lett., Vol. 89, 144501-1–144501-4.

    Article  Google Scholar 

  • A. R. Osborne, M. Onorato and M. Serio (2000) The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Physics Letters A, Vol. 275, 386–393.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Papoulis (1991) Probability, random variables, and stochastic processes. McGraw-Hill, 3rd edition.

    Google Scholar 

  • O. M. Phillips (1958) The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech., Vol. 4, 426–434.

    Article  MATH  MathSciNet  Google Scholar 

  • Y. V. Sedletsky (2003) The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of a finite-depth fluid. J. Exp. Theor. Phys., Vol. 97, 180–193.

    Article  Google Scholar 

  • H. Socquet-Juglard, K. Dysthe, K. Trulsen, H. E. Krogstad and J. Liu (2005) Probability distributions of surface gravity waves during spectral changes. J. Fluid Mech., Vol. 542, 195–216.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Stiassnie (1984) Note on the modified nonlinear Schrödinger equation for deep water waves. Wave Motion, Vol. 6, 431–433.

    Article  MATH  MathSciNet  Google Scholar 

  • M. A. Tayfun (1980) Narrow-band nonlinear sea waves. J. Geophys. Res., Vol. 85, 1548–1552.

    Google Scholar 

  • K. Trulsen (2001) Simulating the spatial evolution of a measured time series of a freak wave. In M. Olagnon and G. Athanassoulis, editors. Rogue Waves 2000, 265–273. Ifremer.

    Google Scholar 

  • K. Trulsen (2005) Spatial evolution of water surface waves. In Proc. Ocean Wave Measurement and Analysis, Fifth International Symposium WAVES 2005, number 127, 1–10.

    Google Scholar 

  • K. Trulsen and K. B. Dysthe (1996) A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion, Vol. 24, 281–289.

    Article  MATH  MathSciNet  Google Scholar 

  • K. Trulsen and K. B. Dysthe (1997) Freak waves — a three-dimensional wave simulation. In Proceedings of the 21st Symposium on Naval Hydrodynamics, 550–560. National Academy Press.

    Google Scholar 

  • K. Trulsen and K. B. Dysthe (1997) Frequency downshift in three-dimensional wave trains in a deep basin. J. Fluid Mech., Vol. 352, 359–373.

    Article  MATH  MathSciNet  Google Scholar 

  • K. Trulsen, I. Kliakhandler, K. B. Dysthe and M. G. Velarde (2000) On weakly nonlinear modulation of waves on deep water. Phys. Fluids, Vol. 12, 2432–2437.

    Article  MathSciNet  Google Scholar 

  • K. Trulsen and C. T. Stansberg (2001) Spatial evolution of water surface waves: Numerical simulation and experiment of bichromatic waves. In Proc. 11th International Offshore and Polar Engineering Conference, Vol. 3, 71–77.

    Google Scholar 

  • C. C. Tung and N. E. Huang (1985) Peak and trough distributions of nonlinear waves. Ocean Engineering, Vol. 12, 201–209.

    Article  Google Scholar 

  • V. E. Zakharov (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., Vol. 9, 190–194.

    Article  Google Scholar 

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Authors and Affiliations

  1. Mechanics Division, Department of Mathematics, University of Oslo, Norway

    Karsten Trulsen

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  1. Karsten Trulsen
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  1. University of Oslo, Norway

    John Grue  & Karsten Trulsen  & 

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Trulsen, K. (2006). Weakly nonlinear and stochastic properties of ocean wave fields. Application to an extreme wave event. In: Grue, J., Trulsen, K. (eds) Waves in Geophysical Fluids. CISM International Centre for Mechanical Sciences, vol 489. Springer, Vienna. https://doi.org/10.1007/978-3-211-69356-8_2

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  • DOI: https://doi.org/10.1007/978-3-211-69356-8_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-37460-3

  • Online ISBN: 978-3-211-69356-8

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