Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study

Abstract

The two-dimensional evolution of Stokes waves over a long fetch has been studied only by means of the cubic Schrödinger equation which is accurate up to the third order in wave slope (Yuen and Ferguson [1], Martin and Yuen [2] and Litvak et al. [3]). For uniform Stokes waves unstable sidebands are known to induce modulation and demodulation at the beginning but energy leakage to higher frequencies leads to chaos eventually and invalidates the theory. For one-dimensional evolution, the fourth-order extension by Dysthe [4] has been shown to corroborate better with experiments that the third-order (Lo and Mei [5]). In this paper we pursue further the implications of the fourth-order theory on two-dimensional evolutions. For the instability of uniform wave trains due to oblique sidebands, we find that the narrower instability region and the presence of a higher-order dispersion term tend to delay the trend toward chaos. We also study the evolution of a single three-dimensional envelope with various aspect ratios. Comparisons between the third- and fourth-order theories are made. Specifically, if the initial envelope is elongated in the direction of the crests, new groups are first born near the center, with ridges parallel to the crests, then everything flattens out. If the initial envelope is elongated in the direction of wave propagation, then the tendency of group-splitting is reduced. Furthermore, the envelope elongates crest-wise to form a ridge before eventual flattening.