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

  • Will Cousins (a1) and Themistoklis P. Sapsis (a1)
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.

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Corresponding author
Email address for correspondence: sapsis@mit.edu
References
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AdcockT. A. A., GibbsR. H. & TaylorP. H. 2012 The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water. Proc. R. Soc. Lond. A 468, 27042721.
AdcockT. A. A. & TaylorP. H. 2009 Focusing of unidirectional wave groups on deep water: an approximate nonlinear Schrödinger equation-based model. Proc. R. Soc. Lond. A 465, 30833102.
AkhmedievN. & PelinovskyE. 2010 Editorial – Introductory remarks on ‘Discussion and debate: rogue waves – towards a unifying concept?’. Eur. Phys. J. Special Top. 185, 14.
AlamM.-R. 2014 Predictability horizon of oceanic rogue waves. Geophys. Res. Lett. 41 (23), 84778485.
AlberI. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A 363 (1715), 525546.
BenjaminT. B. & FeirJ. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417430.
BerlandH., SkaflestadB. & WrightW. M. 2007 EXPINT – a MATLAB package for exponential integrators. ACM Trans. Math. Softw. 33 (1), 4.
BoccottiP. 1983 Some new results on statistical properties of wind waves. Appl. Ocean Res. 5 (3), 134140.
BoccottiP. 2008 Quasideterminism theory of sea waves. J. Offshore Mech. Arctic Engng 130 (4), 41102.
ChabchoubA., HoffmannN., OnoratoM. & AkhmedievN. 2012 Super rogue waves: observation of a higher-order breather in water waves. Phys. Rev. X 2 (1), 11015.
ChabchoubA., HoffmannN., OnoratoM., GentyG., DudleyJ. M. & AkhmedievN. 2013 Hydrodynamic supercontinuum. Phys. Rev. Lett. 111 (5), 054104.
ChabchoubA., HoffmannN. P. & AkhmedievN. 2011 Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106 (20), 204502.
ChoiW. & CamassaR. 1999 Exact evolution equations for surface waves. J. Engng Mech. ASCE 125 (7), 756760.
ClaussG. F., KleinM., DudekM. & OnoratoM. 2014 Application of higher order spectral method for deterministic wave forecast. In Ocean Engineering, vol. 8B, p. V08BT06A038. ASME.
CousinsW. & SapsisT. P. 2014 Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model. Physica D 280, 4858.
CousinsW. & SapsisT. P. 2015 The unsteady evolution of localized unidirectional deep water wave groups. Phys. Rev. E 91, 063204.
CoxS. M. & MatthewsP. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2), 430455.
CraigW. & SulemC. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108 (1), 7383.
CrawfordD. R., SaffmanP. G. & YuenH. C. 1980 Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion 2 (1), 116.
DommermuthD. G. & YueD. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267288.
DyachenkoA. & ZakharovV. 2011 Compact equation for gravity waves on deep water. JETP Lett. 93 (12), 701705.
DyachenkoA. I., KuznetsovE. A., SpectorM. D. & ZakharovV. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1), 7379.
DystheK., KrogstadH. E. & MüllerP. 2008 Oceanic rogue waves. Annu. Rev. Fluid Mech. 40 (1), 287310.
DystheK. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369 (1736), 105114.
DystheK. B. & TrulsenK. 1999 Note on breather type solutions of the NLS as models for freak-waves. Phys. Scr. T 1999 (82), 48.
DystheK. B., TrulsenK., KrogstadH. E. & Socquet-JuglardH. 2003 Evolution of a narrow-band spectrum of random surface gravity waves. J. Fluid Mech 478, 110.
FedeleF. 2008 Rogue waves in oceanic turbulence. Physica D 237 (14), 21272131.
FedeleF. 2014 On certain properties of the compact Zakharov equation. J. Fluid Mech. 748, 692711.
FedeleF. & TayfunM. A. 2009 On nonlinear wave groups and crest statistics. J. Fluid Mech. 620, 221239.
GoulletA. & ChoiW. 2011 A numerical and experimental study on the nonlinear evolution of long-crested irregular waves. Phys. Fluids 23 (1), 16601.
GroomsI. & MajdaA. J. 2014 Stochastic superparameterization in a one-dimensional model for wave turbulence. Commun. Math. Sci. 12 (3), 509525.
HaverS. 2004 A possible freak wave event measured at the Draupner jacket January 1 1995. In Rogue Waves 2004, pp. 18. Ifremer.
HendersonK. L., PeregrineD. H. & DoldJ. W. 1999 Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29, 341361.
IslasA. L. & SchoberC. M. 2005 Predicting rogue waves in random oceanic sea states. Phys. Fluids 17, 031701.
JanssenP. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33 (4), 863884.
KassamA.-K. & TrefethenL. N. 2005 Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 (4), 12141233.
KoenderinkJ. J. 1984 The structure of images. Biol. Cybern. 50 (5), 363370.
LindebergT. 1998 Feature detection with automatic scale selection. Intl J. Comput. Vis. 30 (2), 79116.
LindgrenG. 1970 Some properties of a normal process near a local maximum. Ann. Math. Stat. 41, 18701883.
LiuP. C. 2007 A chronology of freaque wave encounters. Geofizika 24 (1), 5770.
LoE. & MeiC. C. 1985 A numerical study of water wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395416.
LucariniV., FarandaD. & WoutersJ. 2012 Universal behaviour of extreme value statistics for selected observables of dynamical systems. J. Stat. Phys. 147 (1), 6373.
LucariniV., FarandaD., WoutersJ. & KunaT. 2014 Towards a general theory of extremes for observables of chaotic dynamical systems. J. Stat. Phys. 154 (3), 723750.
MüllerP., GarrettC. & OsborneA. 2005 Meeting Report – Rogue Waves. The Fourteenth ’Aha Huliko’a Hawaiian Winter Workshop. Oceanography 18 (3), 6675.
OnoratoM., OsborneA. R. & SerioM. 2002a Extreme wave events in directional, random oceanic states. Phys. Fluids 14 (4), L25.
OnoratoM., OsborneA. R., SerioM. & CavaleriL. 2005 Modulational instability and non-Gaussian statistics in experimental random water-wave trains. Phys. Fluids 17, 078101.
OnoratoM., OsborneA. R., SerioM., ResioD., PushkarevA., ZakharovV. E. & BrandiniC. 2002b Freely decaying weak turbulence for sea surface gravity waves. Phys. Rev. Lett. 89 (14), 144501.
OsborneA. R., OnoratoM. & SerioM. 2000 The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A 275 (5), 386393.
TrulsenK. & DystheK. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24 (3), 281289.
WitkinA. 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.
WuG. X., MaQ. W. & Eatock TaylorR. 1998 Numerical simulation of sloshing waves in a 3D tank based on a finite element method. Appl. Ocean Res. 20 (6), 337355.
XiaoW., LiuY., WuG. & YueD. K. P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357392.
YuenH. C. & FergusenW. E. 1978 Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21 (8), 1275.
ZakharovV. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190194.
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Journal of Fluid Mechanics
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