Numerical simulation of lowest-order short-crested wave instabilities

A numerical study of doubly periodic deep-water short-crested wave instabilities, arising from various quartet resonant interactions, is conducted using a high-order Boussinesq-type model. The model is first verified through a series of simulations involving classical class I plane wave instabilities. These correctly lead to well-known (nearly symmetric) recurrence cycles below a previously established breaking threshold steepness, and to an asymmetric evolution (characterized by a permanent transfer of energy to the lower side-band) above this threshold, with dissipation from a smoothing filter promoting this behaviour in these cases. A series of class la short-crested wave instabilities, near the plane wave limit, are then considered, covering a wide range of incident wave steepness. A close match with theoretical growth rates is demonstrated near the inception. It is shown that the unstable evolution of these initially three-dimensional waves leads to an asymmetric evolution, even for weakly nonlinear cases presumably well below breaking. This is characterized by an energy transfer to the lower side-band, which is also accompanied by a similar transfer to more distant upper side-bands. At larger steepness, the evolution leads to a permanent downshift of both the mean and peak frequencies, driven in part by dissipation, effectively breaking the quasi-recurrence cycle. A single case involving a class Ib short-crested wave instability at relatively large steepness is also considered, which demonstrates a reasonably similar evolution. These simulations consider the simplest physical situations involving three-dimensional instabilities of genuinely three-dimensional progressive waves, revealing qualitative differences from classical two-dimensional descriptions. This study is therefore of fundamental importance in understanding the development of three-dimensional wave spectra.