Numerical Investigations of Surface Gravity Wave Turbulence Within and Beyond Kinetic Regime

Abstract

Wave turbulence theory (WTT) provides a statistical description of systems with many dispersive waves subject to nonlinear interactions. In this framework, a kinetic equation (KE) is derived to model the spectral evolution of wave fields. The analytical stationary solution of the KE yields a wave spectrum with a power-law form in the inertial range, which is known as the Kolmogorov-Zakharov (KZ) spectra. In the context of surface gravity waves, the KE was first developed by Hasselmann and then widely used in physical oceanography as the theoretical foundation for wave-forecasting models. Despite its extensive application, the Hasselmann equation is derived in the infinite domain and weak nonlinearity limits which cannot be rigorously achieved in any real system. In practical conditions (say a wave tank), the KE is only applicable when the nonlinearity level is high enough to overcome the wavenumber discreteness caused by the finite domain size. Additionally, the validity of the KE for surface gravity waves is based on the dominance of four-wave resonant interactions as the resonant triad is absent. However, it is found in recent works that bound waves generated from non-resonant interactions can play a role in the formation of the wave spectra.

In this dissertation, we aim to obtain a comprehensive understanding of surface gravity wave turbulence through a set of numerical and theoretical investigations. We start from a review of the theoretical derivation of the KE and the KZ spectra from the primitive Euler equations, with the emphasis on clarifying the assumptions used in the WTT framework. Then a numerical study is conducted simulating Euler equations for surface gravity waves under different forcing/free-decay conditions. We demonstrate that the kinetic regime, where the predictions from KE become valid, is approached at sufficiently high nonlinearity level with sufficient quasi-resonances. With the decrease of nonlinearity, steeper spectra and reduced energy flux are observed suggesting deviations from WTT. Through the spatial-temporal analysis and a tricoherence study, we elucidate the mechanisms of bound waves and finite-size effect leading to the behaviors at low nonlinearity level.

Based on our understanding obtained from the previous study, we further investigate the mechanisms affecting the properties of wave fields beyond the kinetic regime. Due to the finite-size effect, the discrete wave turbulence regime is realized when the nonlinearity level is sufficiently low such that exact resonances dominate the nonlinear dynamics of the wave field. We implemented a kinematic model to examine the characteristics of exact resonances in a large discrete wavenumber domain based on an efficient computational strategy. Besides, we perform a numerical study with the focus on non-resonant triad interactions which are the origins of bound waves. With a decomposition technique, we numerically track the contributions from the triad and quartet interactions to the spectral evolution. We elucidate the role of non-resonant triads in the formation of resonant quartets and the distribution of energy in bound and free modes. Finally, we explore the wave dynamics at the spectral tail considering both finite-size effect and non-resonant interactions. We confirm that the end of the inertial range in wave spectra is determined by the balance between the viscosity and nonlinear interactions. Our results also indicate the significance of non-resonant triad interactions at large wavenumbers.