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
Mots-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

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     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},
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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/

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