On the properties of energy flux in wave turbulence

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 P in the stationary state to elucidate its distribution and scaling with spectral level. More importantly, we perform a quartet-level decomposition of P = Sigma P-Omega(Omega), with each component P-Omega representing the contribution from quartet interactions with frequency mismatch Omega, in order to explain the properties of P as well as to study the wave turbulence closure model. Our results show that the time series of P closely follows a Gaussian distribution, with its standard deviation several times its mean value (P) over bar. This large standard deviation is shown to result mainly from the fluctuation of the quasi-resonances, i.e. P-Omega not equal 0. The scaling of spectral level with (P) over bar exhibits (P) over bar (1/3) and (P) over bar (1/2) 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 (P-Omega not equal 0) and exact-resonances (P-Omega=0) 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 f(Omega). Our numerical data show that consistent behaviour of f(Omega) can be observed only upon averaging over a large number of quartets, but with such f(Omega) showing a somewhat different form from the theory.