Turbulence d'ondes dans des modèles unidimensionnels

doi:10.1016/S1620-7742(00)01258-7

Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics

Volume 328, Issue 10, October 2000, Pages 757-762

https://doi.org/10.1016/S1620-7742(00)01258-7 Get rights and content

Résumé

Dans cette note, nous évaluons la validité de la théorie de la turbulence faible par l'étude numérique d'une équation non-linéaire d'ondes dispersives à trois paramètres. Les solutions prédites en loi d'échelle sont déterminées explicitement puis comparées aux résultats numériques. Quelque soit le signe de la non linéarité, nous montrons que le régime de turbulence faible est fortement influencé par la présence de structures cohérentes. Ce sont d'une part les collapses et d'autre part les quasi-solitons.

Abstract

In this note, we assess the validity of weak turbulence theory by numerically investigating a three-parameter nonlinear dispersive wave equation. The predicted power-law solutions are explicitly determined and then compared with the numerical results. For both signs of nonlinearity, we show that the weakly turbulent regime is strongly influenced by the presence of coherent structures. These are wave collapses and quasisolitons.

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Cited by (2)

One-dimensional wave turbulence
2004, Physics Reports
Citation Excerpt :
Our parameter β is the opposite of the parameter β in the original paper by MMT. The extension λ=±1 in Eq. (2.1) was first treated independently in [18] and [38]. This extension is nontrivial because the balance between nonlinear and dispersive effects may change according to the sign of λ.
The problem of turbulence is one of the central problems in theoretical physics. While the theory of fully developed turbulence has been widely studied, the theory of wave turbulence has been less studied, partly because it developed later. Wave turbulence takes place in physical systems of nonlinear dispersive waves. In most applications nonlinearity is small and dispersive wave interactions are weak. The weak turbulence theory is a method for a statistical description of weakly nonlinear interacting waves with random phases. It is not surprising that the theory of weak wave turbulence began to develop in connection with some problems of plasma physics as well as of wind waves. The present review is restricted to one-dimensional wave turbulence, essentially because finer computational grids can be used in numerical computations.
Most of the review is devoted to wave turbulence in various wave equations, and in particular in a simple one-dimensional model of wave turbulence introduced by Majda, McLaughlin and Tabak in 1997. All the considered equations are model equations, but consequences on physical systems such as ocean waves are discussed as well. The main conclusion is that the range in which the theory of pure weak turbulence is valid is narrow. In general, wave turbulence is not completely weak. Together with the weak turbulence component, it can include coherent structures, such as solitons, quasisolitons, collapses or broad collapses. As a result, weak and strong turbulence coexist. In situations where coherent structures cannot develop, weak turbulence dominates.
Even though this is primarily a review paper, new results are presented as well, especially on self-organized criticality and on quasisolitonic turbulence.
Kolmogorov spectra of weak turbulence in media with two types of interacting waves
2001, Physics Letters, Section A: General, Atomic and Solid State Physics
A system of one-dimensional equations describing media with two types of interacting waves is considered. This system can be viewed as an alternative to the model introduced by Majda, McLaughlin and Tabak in 1997 for assessing the validity of weak turbulence theory. The predicted Kolmogorov solutions are the same in both models. The main difference between both models is that coherent structures such as wave collapses and quasisolitons cannot develop in the present model. As shown recently these coherents structures can influence the weakly turbulent regime. It is shown here that in the absence of coherent structures weak turbulence spectra can be clearly observed numerically.

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Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics

Turbulence d'ondes dans des modèles unidimensionnels

Résumé

Abstract

One-dimensional wave turbulence

Kolmogorov spectra of weak turbulence in media with two types of interacting waves