Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-08T04:32:17.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduced-order precursors of rare events in unidirectional nonlinear water waves

Published online by Cambridge University Press:  11 February 2016

Will Cousins  and
Themistoklis P. Sapsis
Will Cousins
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Themistoklis P. Sapsis*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of short-term prediction of rare, extreme water waves in irregular unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the linear dispersive mixing of different harmonics, can grow significantly due to modulation instability. Here we show how the interplay between (i) modulation instability properties of localized wave groups and (ii) statistical properties of wave groups that follow a given spectrum defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the ‘trigger’ of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves, circumventing the need for solving the governing equations. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the modified nonlinear Schrödinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Instability: Nonlinear instability Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 790 , 10 March 2016 , pp. 368 - 388
Check for updates
Copyright
© 2016 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

Adcock, T. A. A., Gibbs, R. H. & Taylor, P. H. 2012 The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water. Proc. R. Soc. Lond. A 468, 2704–2721.Google Scholar
Adcock, T. A. A. & Taylor, P. H. 2009 Focusing of unidirectional wave groups on deep water: an approximate nonlinear Schrödinger equation-based model. Proc. R. Soc. Lond. A 465, 3083–3102.Google Scholar
Akhmediev, N. & Pelinovsky, E. 2010 Editorial – Introductory remarks on ‘Discussion and debate: rogue waves – towards a unifying concept?’. Eur. Phys. J. Special Top. 185, 1–4.CrossRefGoogle Scholar
Alam, M.-R. 2014 Predictability horizon of oceanic rogue waves. Geophys. Res. Lett. 41 (23), 8477–8485.CrossRefGoogle Scholar
Alber, I. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A 363 (1715), 525–546.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
Berland, H., Skaflestad, B. & Wright, W. M. 2007 EXPINT – a MATLAB package for exponential integrators. ACM Trans. Math. Softw. 33 (1), 4.CrossRefGoogle Scholar
Boccotti, P. 1983 Some new results on statistical properties of wind waves. Appl. Ocean Res. 5 (3), 134–140.CrossRefGoogle Scholar
Boccotti, P. 2008 Quasideterminism theory of sea waves. J. Offshore Mech. Arctic Engng 130 (4), 41102.CrossRefGoogle Scholar
Chabchoub, A., Hoffmann, N., Onorato, M. & Akhmediev, N. 2012 Super rogue waves: observation of a higher-order breather in water waves. Phys. Rev. X 2 (1), 11015.Google Scholar
Chabchoub, A., Hoffmann, N., Onorato, M., Genty, G., Dudley, J. M. & Akhmediev, N. 2013 Hydrodynamic supercontinuum. Phys. Rev. Lett. 111 (5), 054104.CrossRefGoogle ScholarPubMed
Chabchoub, A., Hoffmann, N. P. & Akhmediev, N. 2011 Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106 (20), 204502.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. ASCE 125 (7), 756–760.CrossRefGoogle Scholar
Clauss, G. F., Klein, M., Dudek, M. & Onorato, M. 2014 Application of higher order spectral method for deterministic wave forecast. In Ocean Engineering, vol. 8B, p. V08BT06A038. ASME.Google Scholar
Cousins, W. & Sapsis, T. P. 2014 Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model. Physica D 280, 48–58.CrossRefGoogle Scholar
Cousins, W. & Sapsis, T. P. 2015 The unsteady evolution of localized unidirectional deep water wave groups. Phys. Rev. E 91, 063204.CrossRefGoogle ScholarPubMed
Cox, S. M. & Matthews, P. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2), 430–455.CrossRefGoogle Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108 (1), 73–83.CrossRefGoogle Scholar
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), 1–16.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
Dyachenko, A. & Zakharov, V. 2011 Compact equation for gravity waves on deep water. JETP Lett. 93 (12), 701–705.CrossRefGoogle Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1), 73–79.CrossRefGoogle Scholar
Dysthe, K., Krogstad, H. E. & Müller, P. 2008 Oceanic rogue waves. Annu. Rev. Fluid Mech. 40 (1), 287–310.CrossRefGoogle Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369 (1736), 105–114.Google Scholar
Dysthe, K. B. & Trulsen, K. 1999 Note on breather type solutions of the NLS as models for freak-waves. Phys. Scr. T 1999 (82), 48.CrossRefGoogle Scholar
Dysthe, K. B., Trulsen, K., Krogstad, H. E. & Socquet-Juglard, H. 2003 Evolution of a narrow-band spectrum of random surface gravity waves. J. Fluid Mech 478, 1–10.CrossRefGoogle Scholar
Fedele, F. 2008 Rogue waves in oceanic turbulence. Physica D 237 (14), 2127–2131.CrossRefGoogle Scholar
Fedele, F. 2014 On certain properties of the compact Zakharov equation. J. Fluid Mech. 748, 692–711.CrossRefGoogle Scholar
Fedele, F. & Tayfun, M. A. 2009 On nonlinear wave groups and crest statistics. J. Fluid Mech. 620, 221–239.CrossRefGoogle Scholar
Goullet, A. & Choi, W. 2011 A numerical and experimental study on the nonlinear evolution of long-crested irregular waves. Phys. Fluids 23 (1), 16601.CrossRefGoogle Scholar
Grooms, I. & Majda, A. J. 2014 Stochastic superparameterization in a one-dimensional model for wave turbulence. Commun. Math. Sci. 12 (3), 509–525.CrossRefGoogle Scholar
Haver, S. 2004 A possible freak wave event measured at the Draupner jacket January 1 1995. In Rogue Waves 2004, pp. 1–8. Ifremer.Google Scholar
Henderson, K. L., Peregrine, D. H. & Dold, J. W. 1999 Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29, 341–361.CrossRefGoogle Scholar
Islas, A. L. & Schober, C. M. 2005 Predicting rogue waves in random oceanic sea states. Phys. Fluids 17, 031701.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33 (4), 863–884.2.0.CO;2>CrossRefGoogle Scholar
Kassam, A.-K. & Trefethen, L. N. 2005 Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 (4), 1214–1233.CrossRefGoogle Scholar
Koenderink, J. J. 1984 The structure of images. Biol. Cybern. 50 (5), 363–370.CrossRefGoogle ScholarPubMed
Lindeberg, T. 1998 Feature detection with automatic scale selection. Intl J. Comput. Vis. 30 (2), 79–116.CrossRefGoogle Scholar
Lindgren, G. 1970 Some properties of a normal process near a local maximum. Ann. Math. Stat. 41, 1870–1883.CrossRefGoogle Scholar
Liu, P. C. 2007 A chronology of freaque wave encounters. Geofizika 24 (1), 57–70.Google Scholar
Lo, E. & Mei, C. C. 1985 A numerical study of water wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–416.CrossRefGoogle Scholar
Lucarini, V., Faranda, D. & Wouters, J. 2012 Universal behaviour of extreme value statistics for selected observables of dynamical systems. J. Stat. Phys. 147 (1), 63–73.CrossRefGoogle Scholar
Lucarini, V., Faranda, D., Wouters, J. & Kuna, T. 2014 Towards a general theory of extremes for observables of chaotic dynamical systems. J. Stat. Phys. 154 (3), 723–750.CrossRefGoogle ScholarPubMed
Müller, P., Garrett, C. & Osborne, A. 2005 Meeting Report – Rogue Waves. The Fourteenth ’Aha Huliko’a Hawaiian Winter Workshop. Oceanography 18 (3), 66–75.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R. & Serio, M. 2002a Extreme wave events in directional, random oceanic states. Phys. Fluids 14 (4), L25.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. & Cavaleri, L. 2005 Modulational instability and non-Gaussian statistics in experimental random water-wave trains. Phys. Fluids 17, 078101.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M., Resio, D., Pushkarev, A., Zakharov, V. E. & Brandini, C. 2002b Freely decaying weak turbulence for sea surface gravity waves. Phys. Rev. Lett. 89 (14), 144501.CrossRefGoogle ScholarPubMed
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 (5), 386–393.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24 (3), 281–289.CrossRefGoogle Scholar
Witkin, A. P.1984 Scale-space filtering: a new approach to multi-scale description. In Acoustics, Speech, and Signal Processing, IEEE Intl Conf. on ICASSP ’84, pp. 150–153.Google Scholar
Wu, G. X., Ma, Q. W. & Eatock Taylor, R. 1998 Numerical simulation of sloshing waves in a 3D tank based on a finite element method. Appl. Ocean Res. 20 (6), 337–355.CrossRefGoogle Scholar
Xiao, W., Liu, Y., Wu, G. & Yue, D. K. P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357–392.CrossRefGoogle Scholar
Yuen, H. C. & Fergusen, W. E. 1978 Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21 (8), 1275.CrossRefGoogle 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 (2), 190–194.CrossRefGoogle Scholar