Date/Time
Date(s) - 06/03/2023
11:30 am - 12:30 pm

Speaker: Zhou Zhang (University of Michigan)

Title: Understanding of surface gravity wave turbulence

Abstract: Wave turbulence theory (WTT) provides a statistical description of dispersive waves subject to nonlinear interactions. In this theory, a kinetic equation (KE) describes the wave system’s evolution under assumptions of infinite domain, weak nonlinearity, and phase stochasticity. The KE’s stationary solution yields the Kolmogorov Zakharov (KZ) wave spectra, which have a power-law form in the inertial range, with S(k)~P^beta k^alpha , where P is the constant energy flux from large to small scales and k is the wavenumber. WTT has been applied to various wave systems, including surface gravity waves, capillary waves, internal gravity waves, plasma waves, and gravitational waves. However, experimental studies often reveal deviations from the KZ spectra due to violations of its assumptions. Understanding the spectral behavior of wave systems beyond the WTT framework is therefore crucial.

This work starts by revisiting the derivation of KE from the governing dynamical equations (Euler equations in Zakharov form) of the surface gravity wave system. We focus on clarifying the assumptions in WTT. Next, we conduct a numerical study of surface gravity waves under different forcing conditions (in terms of bandwidth of amplitude) and free-decay conditions to elucidate the mechanisms underlying the variation of wave spectra. We find that the WTT prediction is approached at high nonlinearity levels in all conditions, while spectra deviate from WTT as the nonlinearity level decreases, with the largest deviation rate observed in the narrow-band forcing case. Through a quantitative study of bound waves (waves not satisfying the dispersion relation) and a tricoherence analysis, we show that the general deviation from WTT can be explained by the finite size effect, and that bound waves are responsible for the rapid deviation in the narrow-band forcing case.

We then investigate the energy transfer in gravity waves at the limiting low nonlinearity state, corresponding to the discrete turbulence regime. Based on a kinematic model simulating the generation of active wave modes in a finite discrete wavenumber space, we examine the possibility of energy cascades by exact resonances. We find that with an increase in the initial excited region, the system sharply transitions from a frozen turbulence state to an unlimited energy cascading state. We elucidate the mechanism associated with the sharp transition and the role of angular energy transfer in the cascades by studying the structure of resonant quartets. 

Finally, we study the effects of three-wave interactions, which are considered insignificant in WTT. Based on a numerical technique separating the energy fluxes due to nonlinear interactions at different orders, we verify that nontrivial three-wave interactions are possible in surface gravity waves. We plan to complete a quantitative investigation of their contribution to the overall energy transfer.

Location: Virtual 

Zoom Link: https://mcmaster.zoom.us/j/94822925263?pwd=SitnUVpvQldERlpsMWdHZzZRTjlyZz09