Numerical Studies of Wave Turbulence in Finite Domains

Abstract

Nonlinear wave systems are ubiquitous in nature, and when many incoherent dispersive waves interact, there is the potential for wave turbulence (WT). Just as in flow turbulence, systems in WT exhibit inter-scale energy cascades, power-law inertial-range spectra, and even intermittency. Unlike in flow turbulence, however, a natural analytical closure for field statistics has been developed. By closing the hierarchy of moment equations that determine field statistics, spectral evolution can be expressed as a Boltzmann-like Wave Kinetic Equation (WKE). The WKE and its supporting closure make formal predictions for the steady power-law inertial-range spectra (known as the Kolmogorov-Zakharov (KZ) spectra), the energy cascade strength and direction, and much more. In addition to being of great theoretical interest, the WKE has been widely employed as a reduced-order model for spectral evolution in practical applications such as global ocean wave forecasting models.

The WT closure and the WKE are derived in the large-domain and infinitesimal wave amplitude limit (together, the kinetic limit), and they describe the average effect of the wave-wave interactions that drive spectral evolution. When a wave system is realized on a finite domain with finite wave amplitude, this assumption of the kinetic limit does not hold. As a result, WKE predictions such as the KZ spectrum become questionable. Numerical and physical experiments in bounded domains often describe steeper spectra and weaker energy cascades than theory predicts. In extreme cases, coherent structures can form that even lead to the breakdown of the kinetic wave description. While recent theory for predicting finite-size effects is in fairly good agreement with observations, it is a largely qualitative model built on kinematic relationships, considering finite-size effects by comparing Fourier domain discreteness to nonlinear broadening of the dispersion relation. For a given domain size, this theory predicts that finite-size effects will dominate when nonlinear broadening becomes much smaller than characteristic Fourier-space frequency spacing.

In this dissertation, we work towards a more quantitative, dynamics-based understanding of finite-size effects through numerical studies of the Majda-McLaughlin-Tabak (MMT) model. First, we explore a limitation of the aforementioned kinematic model: we show that weakly nonlinear wave dynamics in a finite-domain are shaped by the structure of the Discrete Resonant Manifold of wave-wave interactions, which in some cases can support WKE-like dynamics even when nonlinear broadening goes to zero. Next, we explore the properties of the energy cascade in a bounded domain as nonlinear broadening goes to zero. In addition to showing the importance of quasi-resonant interactions to kinetic behavior, we develop an interaction-based energy flux decomposition that allows for a direct, dynamical measurement of nonlinear broadening and a novel and effective study of the WT closure. This tool is then used to study WT in the kinetic limit for a one-dimensional model, where we show numerically that, as the domain is made larger and nonlinearity is made weaker, the error of the WT closure is reduced for a statistically steady WT field. A final study explores a novel, almost-periodic coherent structure in the two-dimensional MMT model that emerges when nonlinearity is weak, where we draw a possible connection to Kolmogorov-Arnold-Moser Theory. We conclude with discussion.