Statistical Description of Nonlinear Waves

Traditionally, wave parameters and their statistics has been derived from time series measurements of wave elevation. Recently, due to introduction in oceanography of stereo video camera systems, increasing attention has started to be given to spatial wave data and statistics. The present study is addressing temporal and spatial statistics of nonlinear waves giving focus to individual wave parameters. A directionally spread rogue-prone sea state observed in the North Sea is used as an example in the analysis which is based on numerical HOSM (Higher Order Spectral Method) simulations. The nonlinear order in the HOSM solver is set to M = 3, which includes the leading order nonlinear dynamical effects, including the effect of modulational instability. The following wave parameters are investigated: surface elevation, wave crests and wave troughs. The results demonstrate that the maximum spatial crest in a wave record can be up to 70% higher than the temporal crest. Further, the study indicates that the Gram-Charlier series can be used to fit the probability density function of surface elevation. It discusses applicability of the methodology based on the Gram-Charlier series for approximation of distributions of individual wave parameters of extreme and rogue waves and recommends further exploitation of this methodology. The results are discussed in the context of marine structures’ design.