## Wave TurbulenceWave Turbulence refers to the statistical theory of weakly nonlinear dispersive waves. There is a wide and growing spectrum of physical applications, ranging from sea waves, to plasma waves, to superfluid turbulence, to nonlinear optics and Bose-Einstein condensates. Beyond the fundamentals the book thus also covers new developments such as the interaction of random waves with coherent structures (vortices, solitons, wave breaks), inverse cascades leading to condensation and the transitions between weak and strong turbulence, turbulence intermittency as well as finite system size effects, such as “frozen” turbulence, discrete wave resonances and avalanche-type energy cascades. This book is an outgrow of several lectures courses held by the author and, as a result, written and structured rather as a graduate text than a monograph, with many exercises and solutions offered along the way. The present compact description primarily addresses students and non-specialist researchers wishing to enter and work in this field. |

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### Contents

1 Introduction | 1 |

Part I Primer on Wave Turbulence | 15 |

Part II Wave Turbulence Closures | 54 |

Part III Wave Turbulence Predicitions | 117 |

Part IV Selected Applications | 189 |

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### Common terms and phrases

2D turbulence Alfvén waves anisotropic Bose-Einstein condensation capillary waves Chap component condition const corresponds curve cutoff density derivative described dimensional discrete dispersion relation dissipation scales dynamics Eð1DÞk energy cascade energy spectrum enstrophy evolution example Exercise ﬁnd finite Fjørtoft argument fluid four-wave systems Fourier frequency function Gaussian gravity waves Hamiltonian instability integral interaction coefficient invariant inverse cascade k-space Kartashova Kelvin waves kinetic equation Kolmogorov KZ solutions KZ spectrum Lett linear Lvov magnetohydrodynamic Mech mesoscopic MHD turbulence modes Nazarenko NLS model nonlinear waves nonlocal numerical simulations obtained one-mode particles PDF’s phase Phys physical plasma power-law random Rayleigh distribution regime resonant respect Rossby waves Rossby/drift Sect solitons spectra statistical superfluid thermodynamic variables vortex vortices water waves wave amplitudes wave spectrum wave systems wave turbulence waveaction wavenumber wavevectors weak turbulence weak WT weakly nonlinear WT theory Zakharov zero zonal flows