On the properties of energy flux in wave turbulence

Abstract

We study the properties of energy flux in wave turbulence via the Majda-McLaughlin- Tabak (MMT) equation with a quadratic dispersion relation. One of our purposes is to resolve the inter-scale energy flux in the stationary state to elucidate its distribution and scaling with spectral level. More importantly, we perform a quartet-level decomposition of, with each component representing the contribution from quartet interactions with frequency mismatch, in order to explain the properties of as well as to study the wave turbulence closure model. Our results show that the time series of closely follows a Gaussian distribution, with its standard deviation several times its mean value. This large standard deviation is shown to result mainly from the fluctuation of the quasi-resonances, i.e.. The scaling of spectral level with exhibits and at high and low nonlinearity, consistent with the kinetic and dynamic scalings, respectively. The different scaling laws in the two regimes are explained through the dominance of quasi-resonances and exact-resonances in the former and latter regimes. Finally, we investigate the wave turbulence closure model, which connects fourth-order correlators to products of pair correlators through a broadening function. Our numerical data show that consistent behaviour of can be observed only upon averaging over a large number of quartets, but with such showing a somewhat different form from the theory.