Recurrent solutions of the Alber equation initialized by Joint North Sea Wave Project spectra

Abstract

AbstractLinear instability of two-dimensional wave fields and its concurrent evolution in time is here investigated by means of the Alber equation for narrow-banded random surface waves in deep water subject to inhomogeneous disturbances. The probability of freak waves in the context of these simulations is also discussed. The instability is first studied for the symmetric Lorentz spectrum, and continued for the realistic asymmetric Joint North Sea Wave Project (JONSWAP) spectrum of ocean waves with variable directional spreading and steepness. It is found that instability depends on the directional spreading and parameters $\alpha $ and $\gamma $ of the JONSWAP spectrum, where $\alpha $ and $\gamma $ are the energy scale and the peak enhancement factor, respectively. Both influence the mean steepness of waves with such a spectrum, although in different ways. Specifically, if the instability stops as a result of the directional spreading, increase of the steepness by increasing $\alpha $ or $\gamma $ can reactivate it. A criterion for the instability is suggested as a dimensionless ‘width parameter’, $\Pi $. For the unstable conditions, long-time evolution is simulated by integrating the Alber equation numerically. Recurrent evolution is obtained, which is a stochastic counterpart of the Fermi–Pasta–Ulam recurrence obtained for the cubic Schrödinger equation. This recurrence enables us to study the probability of freak waves, and the results are compared to the values given by the Rayleigh distribution. Moreover, it is found that stability–instability transition, the most unstable mode, recurrence duration and freak wave probability depend solely on the dimensionless ‘width parameter’, $\Pi $.