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 $P={\sum }_{\Omega }{P}_{\Omega }$, 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 study the wave-turbulence closure model. Our results show that time series of $P$ closely follows a Gaussian distribution, with its standard deviation several times its mean value $\overline{P}$. This large standard deviation is shown to mainly result from the fluctuation (in time) of the quasi-resonances, i.e., $P_{\Omega \ne 0}$. The scaling of spectral level with $\overline{P}$ exhibits ${\overline{P}}^{1/3}$ and ${\overline{P}}^{1/2}$ 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 ($P_{\Omega \ne 0}$) 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 $f(\Omega )$, sometimes argued to be a $sinc$ function in the theory. Our numerical data show that consistent behavior of $f(\Omega )$ can only be observed upon averaging over a large number of quartets, but with $f(\Omega )$ showing $f\sim 1/{\Omega }^{\beta }$ dependence with $\beta$ taking values between $1.3$ and $1.6$.

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Cited by in corpus (3)

• Numerical investigation of turbulence of surface gravity waves
• Feynman rules for forced wave turbulence
• The Structure of Energy Fluxes in Wave Turbulence