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 $P$ in the stationary state to elucidate its distribution and scaling with spectral level. More importantly, we perform a quartet-level decomposition of $P=\sum _\varOmega P_\varOmega$ , with each component $P_\varOmega$ representing the contribution from quartet interactions with frequency mismatch $\varOmega$ , 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 $\bar {P}$ . This large standard deviation is shown to result mainly from the fluctuation of the quasi-resonances, i.e. $P_{\varOmega \neq 0}$ . The scaling of spectral level with $\bar {P}$ exhibits $\bar {P}^{1/3}$ and $\bar {P}^{1/2}$ 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 ( $P_{\varOmega \neq 0}$ ) and exact-resonances ( $P_{\varOmega =0}$ ) 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 $f(\varOmega )$ . Our numerical data show that consistent behaviour of $f(\varOmega )$ can be observed only upon averaging over a large number of quartets, but with such $f(\varOmega )$ showing a somewhat different form from the theory.