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On two approaches to the problem of instability of short-crested water waves

Published online by Cambridge University Press:  26 April 2006

Sergei I. Badulin ,
Victor I. Shrira ,
Christian Kharif  and
Mansour Ioualalen
Sergei I. Badulin
Affiliation:
Institut de Recherche sur les Phenomenes Hors Equilibre, Laboratoire Interaction Ocean Atmosphere, 163 Avenue de Luminy — Case 903, 13288 Marseille Cedex 9, France Permanent address: P. P. Shirshov Institute of Oceanology Russian Academy of Sciences, 23 Krasikov str., Moscow 117218, Russia.
Victor I. Shrira
Affiliation:
Institut de Recherche sur les Phenomenes Hors Equilibre, Laboratoire Interaction Ocean Atmosphere, 163 Avenue de Luminy — Case 903, 13288 Marseille Cedex 9, France Permanent address: P. P. Shirshov Institute of Oceanology Russian Academy of Sciences, 23 Krasikov str., Moscow 117218, Russia.
Christian Kharif
Affiliation:
Institut de Recherche sur les Phenomenes Hors Equilibre, Laboratoire Interaction Ocean Atmosphere, 163 Avenue de Luminy — Case 903, 13288 Marseille Cedex 9, France
Mansour Ioualalen
Affiliation:
Institut de Recherche sur les Phenomenes Hors Equilibre, Laboratoire Interaction Ocean Atmosphere, 163 Avenue de Luminy — Case 903, 13288 Marseille Cedex 9, France Permanent address: ORSTOM BP A5, Noumea Cedex, New Caledonia, France.
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Abstract

The work is concerned with the problem of the linear instability of symmetric short-crested water waves, the simplest three-dimensional wave pattern. Two complementary basic approaches were used. The first, previously developed by Ioualalen & Kharif (1993, 1994), is based on the application of the Galerkin method to the set of Euler equations linearized around essentially nonlinear basic states calculated using the Stokes-like series for the short-crested waves with great precision. An alternative analytical approach starts with the so-called Zakharov equation, i.e. an integro-differential equation for potential water waves derived by means of an asymptotic procedure in powers of wave steepness. Both approaches lead to the analysis of an eigenvalue problem of the type {\rm det}|{\boldmath A}-\gamma{\boldmath B}|=0 where A and B are infinite square matrices. The first approach should deal with matrices of quite general form although the problem is tractable numerically. The use of the proper canonical variables in our second approach turns the matrix B into the unit one, while the matrix A gets a very specific ‘nearly diagonal’ structure with some additional (Hamiltonian) properties of symmetry. This enables us to formulate simple necessary and sufficient a priori criteria of instability and to find instability characteristics analytically through an asymptotic procedure avoiding a number of additional assumptions that other authors were forced to accept.

A comparison of the two approaches is carried out. Surprisingly, the analytical results were found to hold their validity for rather steep waves (up to steepness 0.4) for a wide range of wave patterns. We have generalized the classical Phillips concept of weakly nonlinear wave instabilities by describing the interaction between the elementary classes of instabilities and have provided an understanding of when this interaction is essential. The mechanisms of the relatively high stability of short-crested waves are revealed and explained in terms of the interaction between different classes of instabilities. A helpful interpretation of the problem in terms of an infinite chain of interacting linear oscillators was developed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 303 , 25 November 1995 , pp. 297 - 326
Copyright
© 1995 Cambridge University Press

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