Nonlinear evolution of the modulational instability under weak forcing and damping

The evolution of modulational instability, or Benjamin-Feir instability is investigated within the framework of the two-dimensional fully nonlinear 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. Evolution of this instability in the presence of damping was considered by Segur et al. (2005a) and Wu et al. (2006). Their results were extended theoretically by Kharif et al. (2010) who considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrodinger equation. 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, 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.