Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-20T12:42:02.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves

Published online by Cambridge University Press:  26 November 2018

Peter A. E. M. Janssen Open the ORCID record for Peter A. E. M. Janssen [Opens in a new window]  and
Augustus J. E. M. Janssen
Peter A. E. M. Janssen*
Affiliation:
E.C.M.W.F., Shinfield Park, ReadingRG2 9AX, UK
Augustus J. E. M. Janssen
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
*
†Email address for correspondence: p.janssen@ecmwf.int
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863–884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25–36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 790 - 818
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agafontsev, D. S. & Zakharov, V. E. 2015 Integrable turbulence and formation of rogue waves. Nonlinearity 28 (8), 2791.Google Scholar
Akhmediev, N., Eleonskii, V. M. & Kulagin, N. E. 1987 Exact solutions of the first order of nonlinear Schrödinger equation. Theor. Math. Phys. (USSR) 72 (2), 809–818.Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2001 On the predictability of evolution of surface gravity and gravity-capillary waves. Physica D 152–153, 665–675.Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2006 Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech. 561, 181–207.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The desintegration of wavetrains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Cartwright, D. E. & Longuet-Higgins, M. S. 1956 The statistical distribution of the maxima of a random function. Proc. R. Soc. Lond. A 237, 212–232.Google Scholar
Cavaleri, L., Barbariol, F., Benetazzo, A., Bertotti, L., Bidlot, J. R., Janssen, P. & Wedi, N. 2016 The Draupner wave: a fresh look and the emerging view. J. Geophys. Res. C 121 (8), 6061–6075.Google Scholar
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.Google Scholar
Davidson, R. C. 1972 Methods in Nonlinear Plasma Theory, p. 356. Academic Press.Google Scholar
Dean, R. G. 1990 Freak waves: a possible explanation. In Water Wave Kinematics (ed. Torum, A. & Gudmestad, O. T.), pp. 609–612. Kluwer.Google Scholar
Draper, L. 1965 ‘Freak’ ocean waves. Marine Observer 35, 193–195.Google Scholar
Dyachenko, A. I. & Zakharov, V. E. 1994 Is free surface hydrodynamics an integrable system? Phys. Lett. A 190, 144.Google Scholar
El Koussaifi, R., Tikan, A., Toffoli, A., Randoux, S., Suret, P. & Onorato, M. 2018 Spontaneous emergence of rogue waves in partially coherent waves: a quantitative experimental comparison between hydrodynamics and optics. Phys. Rev. E 97, 012208.Google Scholar
Fedele, F. 2015 On the kurtosis of deep-water gravity waves. J. Fluid Mech. 782, 25–36.Google Scholar
Fedele, F., Cherneva, Z., Tayfun, M. A. & Guedes Soares, C. 2010 Nonlinear Schrödinger invariants and wave statistics. Phys. Fluids 22 (3), 036601.Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481.Google Scholar
Janssen, P. 2004 The interaction of Ocean Waves and Wind, Cambridge University Press.Google Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863–884.Google Scholar
Janssen, P. A. E. M.2017 Shallow-water version of the fresh wave warning system. ECMWF Tech. Mem 813.Google Scholar
Janssen, P. A. E. M. & Bidlot, J.-R.2009 On the extension of the freak wave warning system and its verification. ECMWF Tech. Mem. 588.Google Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.Google Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. Jr 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Lighthill, M. J. 1965 Contributions to the theory of waves in non-linear dispersive systems. J. Inst. Maths Applics. 1, 269–306.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep-water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
Mori, N. & Janssen, P. A. E. M. 2006 On kurtosis and occurrence probability of fresh waves. J. Phys. Oceanogr. 36, 1471–1483.Google Scholar
Mori, N., Onorato, M. & Janssen, P. A. E. M. 2011 On the estimation of the kurtosis in directional sea states for freak wave forecasting. J. Phys. Oceanogr. 41, 1484–1497.Google Scholar
Mori, N., Onorato, M., Janssen, P. A. E. M., Osborne, A. R. & Serio, M. 2007 On the extreme statistics of long-crested deep water waves: theory and experiments. J. Geophys. Res. 112, C09011.Google Scholar
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. Phys. Rev. E 70, 067302.Google Scholar
Onorato, M., Residori, S. & Baronio, F. 2016 Rogue and Shock Waves in Nonlinear Dispersive Media, Lecture Notes in Physics, vol. 926. Springer.Google Scholar
Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P. A. E. M., Kinoshita, T., Monbaliu, J., Mori, N., Osborne, A. R., Serio, M., Stansberg, C. T., 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 (1–4), 114502.Google Scholar
Osborne, A. R., Onorato, M. & Serio, M. 2000 The nonlinear dynamics of rogue waves and holes in deep water gravity wave trains. Phys. Lett. A 275, 386–393.Google Scholar
Peregrine, D. H. 1983 Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16–43.Google Scholar
Randoux, S., Walczak, P., Onorato, M. & Suret, P. 2014 Intermittency in integrable turbulence. Phys. Rev. Lett. 113, 113902.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Suret, P., El Koussaifi, R., Tikan, A., Evain, C., Randoux, S., Szwaj, C. & Bielawski, S. 2016 Single-shot observation of optical rogue waves in integrable turbulence using time microscopy. Nature Commun. 7, 13136.Google Scholar
Suret, P., Picozzi, A. & Randoux, S. 2011 Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers. Opt. Express 19 (18), 17852–17863.Google Scholar
Walczak, P., Randoux, S. & Suret, P. 2015 Optical Rogue Waves in integrable turbulence. Phys. Rev. Lett. 114, 143903.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves, Wiley.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.Google Scholar
Zakharov, V. E. & Rubenchik, A. M. 1974 Instability of wave guides and solitons in nonlinear media. Sov. Phys. JETP 38, 494–500; (Engl. Transl.).Google Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focussing and one-dimensional self-modulating waves in nonlinear media. Sov. Phys. JETP 34, 62–69; (Engl. Transl.).Google Scholar