Modulational instability in wind-forced waves

• We start from the nonlinear Schrödinger equation (NLSE) for fast-growing waves.
• The wind-forcing terms give rise to the enhancement of the modulational instability.
• In this regime the ratio between wave momentum and norm is not conserved in time.

We consider the wind-forced nonlinear Schrödinger (NLS) equation obtained in the potential flow framework when the Miles growth rate is of the order of the wave steepness. In this case, the form of the wind-forcing terms gives rise to the enhancement of the modulational instability and to a band of positive gain with infinite width. This regime is characterised by the fact that the ratio between wave momentum and norm is not a constant of motion, in contrast to what happens in the standard case where the Miles growth rate is of the order of the steepness squared.