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.

Breather Solutions to a Two-dimensional Nonlinear Schrödinger Equation with Non-local Derivatives

“…First, we note that the breather also emerges for the focusing equation ( 1) with λ = −1 under the same conditions. Second, the breather can also be observed under a forced/dissipated system [28], but with the Rayleigh-Jeans spectrum replaced by the wave turbulence spectrum at high nonlinearity levels. Last but not least, we have performed extensive numerical analysis to verify that the breather we observe is not a numerical artifact.…”

Energy cascade in the Garrett–Munk spectrum of internal gravity waves

“…Deng & Hani 2023) and numerical studies (e.g. Hrabski & Pan 2020, 2022), the WKE should be considered as a result of maintaining sufficient quasi-resonances from the dynamical equation, so using broadening in the delta function is somewhat redundant. Such broadened delta functions, on the other hand, might be physically meaningful if finite size effect is important, as in situations described in Pan & Yue (2017).…”

The structure of energy fluxes in wave turbulence

“…Non-zero inter-scale energy fluxes are a fundamental feature of wave turbulence that is still far from being fully understood – see e.g. the recent works Hrabski & Pan (2022) and Dematteis & Lvov (2021). In geophysical applications, the study of wave turbulence fluxes dates back to the early 1980s for internal waves (McComas & Müller 1981; Holloway, Henyey & Pomphrey 1986) and surface gravity waves (Hasselmann & Hasselmann 1981).…”

“…Subsequent improvements of the approximations to flux computations led to important theoretical and numerical tools that are used up to this day. For the surface gravity wave problem, the numerical schemes currently employed in the WAM global model of wave forecasting (Hasselmann & Hasselmann 1985;Resio & Perrie 1991;Komen et al 1996;Janssen 2004) use approximations of the main resonant wave quartets that are responsible for the direct and inverse cascade of energy and wave action through the wave spectrum. This allows for accurate predictions of the global sea states, explaining, for instance, the formation of the large oceanic swells from an inverse cascade process towards the long waves.…”