On the kurtosis of deep-water gravity waves

In this paper, we revisit Janssen’s (2003) formulation for the dynamic excess kurtosis of weakly nonlinear gravity waves at deep water. For narrowband directional spectra, the formulation is given by a sixfold integral that depends upon the Benjamin-Feir index and the parameter $R={\sigma }_{\theta }^2\mathrm{/}2{\nu }^2$, a measure of short-crestedness for the dominant waves with $\nu$ and ${\sigma }_{\theta }$} denoting spectral bandwidth and angular spreading. Our refinement leads to a new analytical solution for the dynamic kurtosis of narrowband directional waves described with a Gaussian type spectrum. For multidirectional or short-crested seas initially homogenous and Gaussian, in a focusing (defocusing) regime dynamic kurtosis grows initially, attaining a positive maximum (negative minimum) at the intrinsic time scale $${\tau }_c={\nu }^2{\omega }_0{t}_c=1\mathrm{/}\sqrt{3R},\mathrm{o}\mathrm{r}{t}_c\mathrm{/}{T}_0\approx 0.13\mathrm{/}\nu {\sigma }_{\theta },$$ where ${\omega }_0=2\pi \mathrm{/}{T}_0$ denotes the dominant angular frequency. Eventually the dynamic excess kurtosis tends monotonically to zero as the wave field reaches a quasi-equilibrium state characterized with nonlinearities mainly due to bound harmonics. Quasi-resonant interactions are dominant only in unidirectional or long-crested seas where the longer-time dynamic kurtosis can be larger than that induced by bound harmonics, especially as the Benjamin-Feir index increases. Finally, we discuss the implication of these results on the prediction of rogue waves.