On the statistical properties of surface elevation, velocities and accelerations in multi-directional irregular water waves

This paper presents a detailed investigation of the role played by the wave steepness in connection with the statistical properties of the surface elevation and fluid kinematics in irregular, directionally spread, deep-water wave fields initially given by a JONSWAP spectrum. Using ensembles of large wave fields obtained from fully nonlinear simulations, we first consider the statistical properties of the surface elevation. In that connection we determine the probability density functions (PDFs) of the surface and crest elevations for wave fields of relatively small to unprecedentedly large steepness, and compare them with theoretical results from the literature in order to establish the latter's accuracy. We then consider certain statistical aspects of the fluid kinematics found at the surface and of the fluid kinematics accompanying large crests, which to our knowledge marks the first investigation of these properties in the literature. We first determine the PDFs of the horizontal fluid velocities and accelerations as well as the vertical fluid acceleration at the surface. Next, we investigate the joint PDF of the surface elevation and each of the velocities and accelerations at the surface, and use it to determine the surface elevations for which the velocities and accelerations at the surface are large. We then present an analysis of the largest fluid velocities and accelerations found in the vicinity of large crests, and compute the PDFs of these quantities. Finally, we consider the PDFs of the location at which the largest velocities and accelerations occur relative to the crest.