Simulations of crescent water wave patterns on finite depth

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.