Nonlinear statistical characteristics of the multi-directional waves with equivalent energy

Directional distribution is believed to have a significant impact on the statistical characteristics in multi-directional sea states. In real sea states, short-crested waves are discrete not only in frequency but also in direction. For the former one, they are well explained in unidirectional mode, but for the latter one, they are not. In this paper, the kurtosis of short-crested waves with equivalent energy is first discussed. Unimodal-spectrum-multi-direction sea states and bimodal-spectrum-multi-direction sea states are simulated for a long time in a numerical wave basin based on the high-order spectral method. In the equivalent sea-swell sea state, the spatial evolution of kurtosis becomes more inhomogeneous, along with the maximum value of kurtosis being larger and the area where the maximum value occurs wider in the configuration with a crossing angle beta = 40 degrees than that with beta = 0 degrees, while little variations in swell-dominated and wind-sea-dominated states. A positive linear correlation between wavelet group steepness and kurtosis is obtained in a unimodal sea state, but not applied to a crossing sea state characterized by a bimodal spectrum. The exceedance probability of wave height and wave crest distribution at maximum kurtosis is also given. These findings can help predict the probability of extreme waves occurring, guiding the selection of ocean engineering sites to avoid extreme configurations.