Wave turbulence: a solvable problem applied to the Navier–Stokes equations

Sébastien Galtier

doi:10.5802/crphys.221

Article de synthèse

Wave turbulence: a solvable problem applied to the Navier–Stokes equations
[Turbulence d’ondes : un problème résoluble appliqué aux équations de Navier–Stokes]

Sébastien Galtier ¹

¹ Laboratoire de Physique des Plasmas, Université Paris-Saclay, École polytechnique, 91128 Palaiseau, France

Comptes Rendus. Physique, Volume 25 (2024), pp. 433-455.

Résumé
Abstract

La turbulence d’ondes et la turbulence de tourbillons sont les deux régimes que l’on peut rencontrer dans la nature. L’attention des mécaniciens des fluides étant principalement portée sur l’hydrodynamique incompressible, c’est généralement le second régime qui est traité dans les livres, alors que les ondes sont souvent présentes en géophysique et en astrophysique. Dans cette revue, je présente la théorie de la turbulence d’ondes qui est exempte du problème de fermeture rencontré dans le cas classique. Fondamentalement, l’amplitude de l’onde est introduite dans une méthode à échelles de temps multiples comme un petit paramètre pour trouver les équations dites cinétiques à partir desquelles des résultats exacts peuvent être obtenus (spectres en loi de puissance, direction de la cascade, constante de Kolmogorov) et comparés avec les données. Deux applications hydrodynamiques sont considérées avec des ondes capillaires et des ondes inertielles, la première conduisant à une turbulence isotrope et la seconde à une turbulence anisotrope.

Wave turbulence and eddy turbulence are the two regimes that we may encounter in nature. The attention of fluid mechanics being mainly focused on incompressible hydrodynamics, it is usually the second regime that is treated in books, whereas waves are often present in geophysics and astrophysics. In this review, I present the theory of wave turbulence which is free from the closure problem encountered in eddy turbulence. Basically, the wave amplitude is introduced in a multiple time scale method as a small parameter to derive the so-called kinetic equations from which exact results can be obtained (power-law spectra, direction of the cascade, Kolmogorov’s constant) and compared with the data. Two hydrodynamic applications are considered with capillary waves and inertial waves, the first leading to isotropic turbulence and the second to anisotropic turbulence.

Reçu le : 2024-11-04
Accepté le : 2024-11-14
Publié le : 2024-12-12

DOI : 10.5802/crphys.221

Keywords: Hydrodynamics, Turbulence, Wave
Mots-clés : Hydrodynamique, Onde, Turbulence

Affiliations des auteurs :

Sébastien Galtier ¹

¹ Laboratoire de Physique des Plasmas, Université Paris-Saclay, École polytechnique, 91128 Palaiseau, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Wave turbulence: a solvable problem applied to the {Navier{\textendash}Stokes} equations},
     journal = {Comptes Rendus. Physique},
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Sébastien Galtier. Wave turbulence: a solvable problem applied to the Navier–Stokes equations. Comptes Rendus. Physique, Volume 25 (2024), pp. 433-455. doi : 10.5802/crphys.221. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.221/

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