Quasi-Three-Dimensional Simulation of Crescent-Shaped Waves

Properties and limitations of a quasi-three-dimensional water-wave model put forward by V.P. Ruban and based on the assumption of narrow directional distribution of the wave spectrum are studied. Within the approximate equations of motion, a stability problem for a finite amplitude Stokes wave to three-dimensional perturbations is considered. A zone of instability corresponding to five-wave interactions is examined. It is shown that, despite the fact that the corresponding modes consist of harmonics that are relatively far from the main direction, the increment values are close to those given by the exact equations of motion. The subsequent stages of the three-dimensional instability growth exhibit a plausible dynamics, leading to formation of crescent-shaped waves. A modification to cubic components of the Hamiltonian functional of the model is suggested that eliminates a spurious zone of instability for perturbations propagating almost perpendicularly to the main direction.