Effects of the wind on the breaking of modulated wave trains

The effect of wind on the wave breaking induced by the modulational instability is investigated numerically using the open source software Gerris. The two-phase flow is modelled by the two-dimensional Navier-Stokes equations for a single incompressible fluid and a Volume of Fluid technique is employed to capture the air-water interface. The flow is initialized as a fundamental wave component with two side-band perturbations so that the breaking is induced via the Benjamin-Feir instability mechanism. In order to investigate its effect on the wave evolution and on the breaking process, a uniform wind, twice the phase speed, is assigned in the air phase. The simulation covers the initial development of the wind profile, the growth of the modulational instability, the breaking and post breaking phases. Results show the occurrence of air flow separation from the wave crests shortly after the initial start. Pressure and tangential stress acting on the free surface are computed. It is shown that due to the flow separation there is a favourable pressure gradient about the wave crests whereas the tangential stresses are generally in favour of the wave propagation on the back of the wave but are opposed to the propagation along the forward face and in the wave trough. An initial growth of the energy content in water is observed, followed by a dissipation stage which is not related to the breaking process. In agreement with the experiments, the growth rate of the side-bands is reduced when compared to the corresponding no-wind solution. Because of the slower growth, the limiting condition for the onset of the breaking is reached with some delay. At the end of the breaking process, when the downshift is completed, the amplitude of the left side-band in the wind case is somewhat lower than that for the no-wind case. No substantial differences have been found in terms of the total energy dissipated by the whole breaking process although the dissipation rate for the wind case is noticeably higher. The higher dissipation rate observed in the wind case is found to be related to the larger amount of air entrained by the breaking process.