Statistics of the spectral fluctuations in wave turbulence
Abstract
We study the k-space fluctuations of the waveaction about its mean spectrum in the turbulence of dispersive waves. We use a minimal model based on the Random Phase Approximation (RPA) and derive evolution equations for the arbitrary-order one-point moments of the wave intensity in the wavenumber space. The first equation in this series is the familiar Kinetic Equation for the mean waveaction spectrum, whereas the second and higher equations describe the fluctuations about this mean spectrum. The fluctuations exhibit a nontrivial dynamics if some long coordinate-space correlations are present in the system, as it is the case in typical numerical and laboratory experiments. Without such long-range correlations, the fluctuations are trivially fixed at their Gaussian values and cannot evolve even if the wavefield itself is non-Gaussian in the coordinate space. Unlike the previous approaches based on smooth initial k-space cumulants, our approach works even for extreme cases where the k-space fluctuations are absent or very large and intermittent. We show, however, that whenever turbulence approaches a stationary state, all the moments approach the Gaussian values.Keywords
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