Current Effects on Nonlinear Interactions of Shallow-Water Waves

Abstract

This study focuses on the effects of depth-uniform currents on near-resonant triad interactions of gravity waves in shallow water. First, we derive the evolution equations for triad interactions based on Yoon and Liu's equations with a constant water depth and an ambient current. An opposing current increases the magnitude of phase mismatch, resulting in a reduction of the degree of interactions and vice versa for a following current. Second, numerical experiments on harmonic generation and nonlinear shoaling of shallow-water waves with opposing and following currents are given by the use of an enhanced Boussinesq model with higher dispersion accuracy for wave/current interaction. The numerical model predictions that are in good agreement with the physical experiment data in the absence of ambient currents serve as a benchmark to reveal the current effects on nonlinear interactions of shallow-water waves. Although an opposing current increases wave heights, it actually reduces the Ursell number, beat-length, and ratio of the energy in higher harmonics to the energy in the primary wave in comparison with the pure wave motion in shallow water. Conversely, it is found that a following current intensifies the extent of triad interactions. This is in contrast to the current influence on quadruplet interactions of deepwater waves.