Session: 12-01-01 Wave theories I

Paper Number: 124977

124977 - Theoretically-Based Probability Density Functions for the Surface Elevation in Highly-Nonlinear Irregular Seas 

Until very recently, the probability density function (p.d.f.) based on the cumulant generating function for the surface elevation in irregular seas - free of assumptions involving narrow-bandedness and small directionality - was only known exactly to first order, where it is Gaussian. At higher orders, this p.d.f. had only been represented approximately (Longuet-Higgins 1963) e.g. as a Gram-Charlier series. In a recent development, the present authors (Fuhrman, Klahn & Zhai 2023) have shown that to second-order the p.d.f. may, in fact, be represented exactly in terms of the Airy function. Notably, this new theoretical result shows that the p.d.f. has an inherently heavier positive tail (i.e. increased probability density of large surface elevations, typical of e.g. rogue waves) than had been explained previously, and that at second order it is governed solely by the skewness.  In the present work, we extend these investigations further, and develop a novel methodology enabling first-time numerical determination of the theoretical p.d.f. to any desired order in nonlinearity.  For this purpose, a new ordinary differential equation (o.d.e.) is first derived, governing the p.d.f. to any order in nonlinearity. Asymptotic solutions to this o.d.e. are then found analytically in the limit of large surface elevation, newly providing the theoretical form of the positive tail of the p.d.f. beyond second order. These are important in their own right, as they make theoretically clear how higher-order cumulants (involving high-order statistical moments such as the kurtosis, hyperskewness and hyperkurtosis) may affect the positive tail, which is shown to get heavier at each successive order. The asymptotic solutions are finally utilized to provide necessary boundary conditions, such that the governing o.d.e. may be solved numerically, thus enabling novel determination of the full theoretical p.d.f. to effectively any order. Successful comparisons with challenging data sets are made, involving cases where second order theory fails, confirming accuracy of obtained higher-order p.d.f.s.  Results up to fifth order in nonlinearity will be shown, though it is again emphasized that the methodology developed may be extended to even higher orders, should this be of interest.

References

Fuhrman, D.R., Klahn, M. & Zhai, Y. (2023) A new probability density function for the surface elevation in irregular seas. J. Fluid Mech. 970, A38.

Longuet-Higgins, M.S. (1963) The effect of non-linearities on statistical distributions in the theory of sea waves. J. Fluid Mech. 698, 304-334.

Presenting Author: David R. Fuhrman Technical University of Denmark

Presenting Author Biography: David R. Fuhrman is Professor of Coastal Dynamics at the Technical University of Denmark (DTU), Department of Civil and Mechanical Engineering. He additionally serves as Editor in Chief of Applied Ocean Research, as Associate Editor of J. Waterway, Port, Coastal & Ocean Engineering, and on the Editorial Boards of Coastal Engineering and OpenFOAM Journal. His research focuses on many aspects of coastal and ocean engineering, including surf zone dynamics, sediment transport, coastal morphology, scour, turbulence modelling, pollutant (microplastic) transport, wave boundary layers, and nonlinear wave hydrodynamics and statistics.

Authors:

Mathias Klahn Odeon A/S
Yanyan Zhai Technical University of Denmark
David R. Fuhrman Technical University of Denmark