Rogue wave statistics in (2+1) Gaussian seas I: Narrow-banded distribution

Abstract

Originally discussed in the context of ocean statistics for oil rig platform decks and satellite data, spatiotemporal exceeding probability distributions have recently been thought to explain the failure of second and third-order nonlinear models of distributions at a fixed point in space. To investigate this possibility, we derive a simplified model for the (2+1) exceeding probability distribution of ocean waves based on the Gaussian random field approach, with a particular interest in rogue waves. Using standard statistical tools, we show that (2+1)) Longuet–Higgins’ distribution does not depend on the spatiotemporal scale of the ocean surface, rather on the directional spectrum and the dimensionless wave height. Moreover, we use extreme value theory Gumbel (1958)[24] to find the expected maximum wave height in (2+1) dimensions, showing that Piterbarg’s framework provides a smaller expected maximum than Adler’s approach. Unfortunately, both are known for assigning maximum heights that do not saturate when the surface area is very large. Taking into account the mean rogue wave profile, we derive a new expression for the maximum wave height that provides realistic estimates. Finally, comparing the theory for the exceeding probability and expected maximum dimensionless height with the North Alwyn data set of Stansell (2004)[47] we conclude that both (2+1) versions of Longuet-Higgins (1952, 1980)[31,34] distributions fail to provide an accurate description of both return period and maximum observed wave heights, thus undermining the argument that the error in ocean wave statistics can be cured by extending point statistics to higher-dimensions.