On the kurtosis of deep-water gravity waves

In this paper, we revisit Janssen's (J. Phys. Oceanogr., vol. 33 (4), 2003, pp. 863-884) formulation for the dynamic excess kurtosis of weakly nonlinear gravity waves in 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(2)(theta)/2v(2), a measure of short-crestedness for the dominant waves, with v 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 homogeneous and Gaussian, in a focusing (defocusing) regime dynamic kurtosis grows initially, attaining a positive maximum (negative minimum) at the intrinsic time scale tau(c) = v(2) omega(0)t(c) = 1/root 3R, or t(c)/T-0 approximate to 0.13/v sigma(theta), where omega(0) = 2 pi/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 by 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 for the prediction of rogue waves.