Modutional instability in the presence of forcing and dissipation

The modulational instability of gravity wavetrains on the surface of water in the presence of wind and dissipation is considered. The wind regime is that of validity of Miles' theory and the viscosity is small. A perturbed nonlinear SchrÄodinger equation describing the evolution of a narrow-banded wavepacket is derived. It is found in the presence of wind forcing and dissipation that trains of long gravity waves are less unstable to modulational perturbations than trains of short gravity waves