Observations and Modeling of Linear and Nonlinear Spatio-Temporal Wave Statistics

We present an analysis of airborne wave observations collected in the Gulf of Tehuantepec. The data includes LIDAR measurements of the surface displacement as a function of two horizontal dimensions and time in fetch-limited conditions, with fetches between 50 and 300 km and winds between 10 and 20 m/s. The spatio-temporal data have an advantage over the commonly used single point time-series measurements allowing direct estimates of the wavelength and wave slope, including spatial information such as the lengths of crests exceeding threshold wave heights and slopes. The statistics of these wave parameters are particularly important for risk assessment of off-shore structures and in other ocean engineering applications. We present an analysis of several statistical wind-wave parameters, including the joint probability distribution function (pdf) of wave amplitudes and wavelengths, the pdf of wave heights, wavenumber vectors, and wave slopes, including the statistics of crests lengths exceeding threshold wave heights or slopes. The empirical findings from the LIDAR data are related to the analytical work by Longuet-Higgins (1957) [1] for a linear spectrum, including the average length of contours surrounding large wave heights. The effect of second-order nonlinearities on the distribution of crest lengths is investigated with numerical stochastic simulations from computed directional wavenumber spectra. The results show that second-order nonlinearities can increase the crest length density of large waves by about a factor of two or more. The results are discussed in the context of predicting wave statistics for ocean engineering applications.