Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interaction

Didier Clamond; John Grue

doi:10.1016/S1631-0721(02)01496-1

Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interaction
[Interaction de solitons enveloppes comme modèle pour la formation de « freak waves». Partie I : Interaction à long terme]

Didier Clamond ¹ ; John Grue ¹

¹ Department of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway

Comptes Rendus. Mécanique, Volume 330 (2002) no. 8, pp. 575-580.

Résumé
Abstract

On s'interresse à un mécanisme particulier pouvant expliquer l'apparition de vagues de grandes amplitudes (freak waves). On étudie numériquement l'évolution à long terme de paquets d'onde de gravité surfaciques. On compare un modèle complètement non linéaire avec des équations simplifiées de type Schrödinger. On observe que l'interaction d'ondes solitaires enveloppes génèrent des vagues de grandes amplitudes. Cela est prédit par tous les modèles. Toutefois, le modèle complètement non linéaire exhibe un comportement à long terme très different des modèles simplifiés. L'apparition de freak waves y est beaucoup plus fréquente. C'est une conséquence de l'interaction à long terme de solitons enveloppes, qui est totalement différente de celle prédite par les scénario dérivés des équations de type Schrödinger. Les différences fondamentales apparaissent pour des temps supérieurs aux domaı̂nes de validité des équations simplifiées. D'envisageable modèles statistiques, basés sur ces dernières, devraient donc sous-estimer la probabilité d'apparition de freak waves.

We are concerned by a special mechanism that can explain the formation of freak waves. We study numerically the long time evolution of a surface gravity wave packet, comparing a fully nonlinear model with Schrödinger-like simplified equations. We observe that the interaction between envelope solitons generates large waves. This is predicted by both models. The fully nonlinear simulations show a qualitative behaviour that differs significantly from the ones preticted by Schrödinger models, however. Indeed, the occurence of freak waves is much more frequent with the fully nonlinear model. This is a consequence of the long-time interaction between envelope solitons, which, in the fully nonlinear model, is totally different from the Schrödinger scenario. The fundamental differences appear for times when the simplified equations cease to be valid. Possible statistical models, based on the latter, should hence under-estimate the probability of freak wave formation.

Reçu le : 2001-12-26
Accepté le : 2002-05-27
Publié le : 2002-01-01

DOI : 10.1016/S1631-0721(02)01496-1

Keywords: waves, freak wave, soliton enveloppe, interaction
Mot clés : ondes, freak wave, envelope soliton, interaction

Affiliations des auteurs :

Didier Clamond ¹ ; John Grue ¹

¹ Department of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway

@article{CRMECA_2002__330_8_575_0,
     author = {Didier Clamond and John Grue},
     title = {Interaction between envelope solitons as a model for freak wave formations. {Part~I:} {Long} time interaction},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {575--580},
     publisher = {Elsevier},
     volume = {330},
     number = {8},
     year = {2002},
     doi = {10.1016/S1631-0721(02)01496-1},
     language = {en},
}

TY  - JOUR
AU  - Didier Clamond
AU  - John Grue
TI  - Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interaction
JO  - Comptes Rendus. Mécanique
PY  - 2002
SP  - 575
EP  - 580
VL  - 330
IS  - 8
PB  - Elsevier
DO  - 10.1016/S1631-0721(02)01496-1
LA  - en
ID  - CRMECA_2002__330_8_575_0
ER  -

%0 Journal Article
%A Didier Clamond
%A John Grue
%T Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interaction
%J Comptes Rendus. Mécanique
%D 2002
%P 575-580
%V 330
%N 8
%I Elsevier
%R 10.1016/S1631-0721(02)01496-1
%G en
%F CRMECA_2002__330_8_575_0

Didier Clamond; John Grue. Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interaction. Comptes Rendus. Mécanique, Volume 330 (2002) no. 8, pp. 575-580. doi : 10.1016/S1631-0721(02)01496-1. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01496-1/

Bibliographie
Cité par

[1] K.B. Dysthe Note on a modification to the nonlinear Schrödinger equation for application to deep water, Proc. Roy Soc. London A, Volume 369 (1979), pp. 105-114

[2] K. Trulsen; I. Kliakhadler; K.B. Dysthe; M.G. Velarde On weakly nonlinear modulation of waves on deep water, Phys. Fluids, Volume 12 (2000) no. 10, pp. 2432-2437

[3] A.R. Osborne; M. Onorato; M. Serio The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains, Phys. Lett. A, Volume 275 (2000), pp. 386-393

[4] C. Kharif; E. Pelinovsky; T. Talipova; A. Slunyaev Focusing of nonlinear wave groups in deep water, JETP Lett., Volume 73 (2001) no. 4, pp. 170-175

[5] M. Onorato; A.R. Osborne; M. Serio Extreme wave events in directional, random oceanic sea states, Phys. Fluids, Volume 14 (2002) no. 4, p. L25-L28

[6] D. Clamond; J. Grue A fast method for fully nonlinear water-wave computations, J. Fluid Mech., Volume 447 (2001), pp. 337-355

[7] M.L. Banner; X. Tian On the determination of the onset of breaking for modulating surface water waves, J. Fluid Mech., Volume 367 (1998), pp. 107-137

[8] J.D. Fenton The numerical solution of steady water wave problems, Computers & Geosciences, Volume 14 (1988) no. 3, pp. 357-368

[9] J. Satsuma; N. Yajima Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Supp. Prog. Theor. Phys., Volume 55 (1974), pp. 284-306

[10] E. Lo; C.C. Mei A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation, J. Fluid Mech., Volume 150 (1985), pp. 395-416

[11] H.C. Yuen; B.M. Lake Nonlinear dynamics of deep-water gravity waves, Adv. Appl. Mech., Volume 22 (1982), pp. 67-229

Cité par Sources :

Commentaires - Politique