The nonlinear dynamics of three-dimensional instabilities of uniform gravity-wave trains evolving to crescent wave patterns is investigated numerically. A new mechanism of generation of oscillating horseshoe patterns is proposed and a detailed discussion on their occurrence in a water wave tank is given. It is suggested that these patterns are more likely to be observed naturally in water of finite depth. A critical wave steepness for the onset of three-dimensional wave breaking due to the nonlinear evolution of quintet resonant interactions corresponding to the phase-locked crescent-shaped structures (class II instability) is provided when the quartet resonant interaction (class I instability) is absent. The nonlinear coupling between quartet resonant interactions (class I instability) and quintet resonant interactions (class II instability) leading to three-dimensional breaking waves, as shown experimentally by Su & Green (1984, 1985), is numerically investigated
A numerical study of the instabilities of Stokes waves on finite depth has been carried out using an efficient fully nonlinear method [D. Clamond and J. Grue, “A fast method for fully nonlinear water-wave computations,” J. Fluid Mech. 447, 337 (2001)]. First, attention is given to five-wave instabilities with k0h=O(1), k0 being the wavenumber and h the depth. Both instabilities leading to breaking and instabilities leading to recurrence are studied, yielding considerably different patterns than on infinite depth. Higher-order instabilities are exemplified, for the first time, by simulations of six- and seven-wave instabilities. Simulations of interactions between four- and five-wave instabilities show that a classical modulational instability can destabilize a three-dimensional perturbation causing crescent waves to appear, in accordance with the hypothesis of [M.-Y. Su and A. W. Green, “Coupled two- and three-dimensional instabilities of surface gravity waves,” Phys. Fluids 27, 2595 (1984)]. Also, a recurrent five-wave instability can boost the energy in a four-wave instability.
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