On the Alber equation for random water-waves

We investigated the instability of two-dimensional wave fields and its recurrent evolution in time by means of the Alber equation for narrow-banded random surface waves in deep water, subject to inhomogeneous disturbances. The study is started from symmetric spectra and then continued for realistic asymmetric JONSWAP spectrum of ocean waves. Criteria for the instability, both in one-dimensional and two-dimensional cases, are suggested. For the unstable conditions, long-time evolution is simulated and the recurrent evolution is obtained which enables us to study the probability of freak waves. The results are compared to the values given by the Rayleigh distribution and to the Forristall distribution. In particular, it is found that the transition to instability, the most unstable mode, the recurrence duration and the freak wave probability all depend solely on one parameter which is ?