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.

Automated summary

We study the properties of energy flux in wave turbulence using the Majda-McLaughlin-Tabak equation with a quadratic dispersion relation. Our study aims to understand the distribution and scaling of the inter-scale energy flux in the stationary state. Additionally, we perform a quartet-level decomposition to analyze the contributions from quartet interactions with frequency mismatch. We find that the time series of the energy flux closely follows a Gaussian distribution and that the scaling with spectral level exhibits different behaviors at high and low nonlinearity. Furthermore, our investigation of the wave turbulence closure model reveals that consistent behavior can be observed only when averaging over a large number of quartets.