Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves

In this paper we highlight that extreme events such as freakwaves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen ( J. Phys. Oceanogr. , vol. 33, 2003, pp. 863–884) obtained an evolution equation for the ensemble averageof the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrodinger (NLS) equation, the excess kurtosisis under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele ( J. Fluid Mech. , vol. 782, 2015, pp. 25–36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosisand the generation of extreme events.