Numerical simulations of water waves' modulational instability under the action of wind and dissipation
Résumé
Since the work of Benjamin & Feir (1967), water waves propagating in infinite depth are known to be unstable to modulational instability. The evolution of such wave trains is well described through fully nonlinear simulations, but also by means of simplified models, such as the nonlinear Schrödinger equation. Segur et al. (2005) and Wu et al. (2006) studied theoretically and numerically the evolution of this instability, and both concluded that a long term restabilization occurs in these conditions. More recently, Kharif et al. (2010) considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation, and discussed the range of parameters for which this behavior is still valid. This work aims to demonstrate how numerical simulations are useful to analyze their theoretical predictions. Since we are dealing with long term stability, results are especially complicated to obtain experimentally. Thus, numerical simulations of the fully nonlinear equations turn out to be a very useful tool to provide a validation for the model. Here, the evolution of the modulational instability is investigated within the framework of the two-dimensional fully non linear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles theory. The introduction of dissipation in the equations is briefly discussed. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, the long term evolution of the wave trains can be obtained through the numerical simulations, and it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon.