On the physical constraints for the exceeding probability of deep water rogue waves

(manuscript accepted into Applied Ocean Research https://www.researchgate.net/publication/344786014)

Abstract

Nearly four decades have elapsed since the first efforts to obtain a realistic narrow-banded model for extreme wave crests and heights were made, resulting in a couple of dozen different exceeding probability distributions. These models reflect results of numerical simulations and storm records measured from oil platforms, buoys, and more recently, satellite data. Nevertheless, no consensus has been achieved in either deterministic or operational approaches. Typically, distributions found in the literature analyze a very large set of waves with large variations in sea-state parameters while neglecting homogeneous smaller samples, such that we lack a suitable definition for the sample size and homogeneity of sea variables, also known as sampling variability (Bitner-Gregersen et al., 2020). Naturally, a possible consequence of such sample size inconsistency is the apparent disagreement between several studies regarding the prediction of rogue wave occurrence, as some studies can report less rogue wave heights while others report more rogue waves or the same statistics predicted by Longuet-Higgins (1952), sometimes a combination of the three in the very same study (Stansell, 2004; Cherneva et al., 2005). In this direction, we have obtained a dimensionless parameter capable of measuring how large the deviations in sea state variables can be so that accuracy in wave statistics is preserved.  In particular, we have defined which samples are too heterogeneous to create an accurate description of the uneven distribution of rogue wave likelihood among different storms (Stansell, 2004). Though the literature is rich in physical bounds for single waves, here we describe empirical physical limits for the ensemble of waves (such as the significant steepness) devised to bound these variables within established and prospective wave distributions. Furthermore, this work supplies a combination of sea state parameters that provide guidance on the influence of sea states influence on rogue wave occurrence. Based on these empirical limits, we conjecture a mathematical model for the dependence of the expected maximum of normalized wave heights and crests on the sea state parameters, thus explaining the uneven distribution of rogue wave likelihood among different storms collected by infrared laser altimeters of the North Alwyn oil platform discussed in Stansell (2004). Finally, we demonstrate that for heights and crests beyond 90% of their thresholds (H>2H_1/3 for heights), the exceeding probability becomes stratified, i.e. they resemble layers of probability curves according to each sea state, suggesting the existence of a dynamical definition for rogue waves rather than purely statistical.

 

References

Bitner-Gregersen, E. M., Gramstad, O., Magnusson, A., Malila, M., 2020. Challenges in description of nonlinear waves due to sampling variability. J. Mar. Sci. Eng. 8, 279.

Longuet-Higgins, M., 1952. On the statistical distribution of the heights of sea waves. Journal of Marine Research 11, 245–265.

Stansell, P., 2004. Distribution of freak wave heights measured in the north sea. Appl. Ocean Res. 26, 35–48.

Cherneva, Z., Petrova, P., Andreeva, N., Guedes Soares, C., 2005. Probability distributions of peaks, troughs and heights of wind waves measured in the black sea coastal zone. Coastal Engineering 52, 599–615.