Non-equilibrium evolution of wave fields, as occurring over sudden bathymetry variations, can produce rogue seas with anomalous wave statistics. We handle this process by modifying the Rayleigh distribution through the energetics of second-order theory and a non-homogeneous reformulation of the Khintchine theorem. The resulting probability model reproduces the enhanced tail of the probability distribution of unidirectional wave tank experiments. It also describes why the peak of rogue wave probability appears atop the shoal, and explains the influence of depth on variations in peak intensity. Furthermore, we interpret rogue wave likelihoods in finite depth through the
$H$
–
$\sigma$
diagram, allowing a quick prediction for the effects of rapid depth change apart from the probability distribution.
Shoaling surface gravity waves induce rogue wave formation. Though commonly reduced to water waves passing over a step, nonequilibrium physics allows finite slopes to be considered in this problem. Using nonhomogeneous spectral analysis of a spatially varying energy density ratio, we describe the dependence of the amplification as a function of the slope steepness. Increasing the slope increases the amplification of rogue wave probability until this amplification saturates at steep slopes. In contrast, the increase of the down slope of a subsequent de-shoal zone leads to a monotonic decrease in the rogue wave probability, thus featuring a strong asymmetry between shoaling and de-shoaling zones. Due to the saturation of the rogue wave amplification at steep slopes, our model is applicable beyond its range of validity up to a step, thus elucidating why previous models based on a step could describe the physics of steep finite slopes. We also explain why the rogue wave probability increases over a shoal while it is lower in shallower water.
Bathymetric changes have been experimentally shown to affect the occurrence of rogue waves. We recently derived a non-homogeneous correction to the spectral analysis, allowing us to describe the evolution of the rogue wave probability over a shoal. Here, we extend this work to the evolution of the excess kurtosis of the surface elevation, that plays a central role in estimating rare event probabilities. Furthermore, we provide an upper bound to the excess kurtosis. In intermediate and deep water regimes, a shoal does not affect wave steepness nor bandwidth significantly, so that the vertical asymmetry between crests and troughs, the excess kurtosis and the exceedance probability of wave height stay rather constant. In contrast, in shallower water, a sharp increase in wave steepness increases the vertical asymmetry, resulting in a growth of both the tail of the exceedance probability and the excess kurtosis.
Regarding wave statistics, nearly every known exceeding probability distribution applied to rogue waves has showndisagreement with its peers. More often than not, models and experiments have shown a fair agreement with theRayleigh distribution whereas others show that the latter underpredicts extreme heights by almost one order of magnitude. Virtually all previous results seem to be microcosms, special cases of the underlying essence of this phenomenon.The present work focuses on the apparent contradiction among the majority of previous works. Based on the issueof strong uneven distribution of rogue waves found in Stansell (2004), a new exceeding probability distribution forrogue waves and the analysis of their uneven occurrence is conceived. The proposed distribution is a geometricalcomposition of the most popular models for wave records (Longuet-Higgins, 1952; Haring et al., 1976; Tayfun, 1980)with additional algebraic structures. The suggested distribution also obeys empirical physical bounds obtained fromthe analysis of nearly 350,000 waves from storms recorded in North sea and supports the qualitative likelihood ofappearance interpretation based on the symbiosis among three sea state parameters.
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 dozen different exceeding probability distributions. These models reflect results of numerical simulations and storm records measured off of oil platforms, buoys and more recently satellite data. Nevertheless, no consensus has been achieved in either deterministic or operational approaches. Moreover, a minor issue with the established distributions is that they are not bounded by more than one physical limit while others are not bounded at all. Though the literature is rich in physical bounds for single waves, here we describe physical limits for the ensemble of waves that have not yet been addressed. As previous studies have shown, the exceeding probability distribution does not depend unequivocally on one sea state parameter, thus, this work supplies a combination of sea state parameters that provide guidance on the sea state influence on rogue wave occurrence. Based on specific bounds, we conjecture the dependence of the expected maximum of normalized wave heights (also known as abnormality index) and crests on the aforementioned sea-state parameters instead of the total number of waves in the wave record. Finally, we introduce a new dimensionless parameter that is capable of explaining the uneven distribution of rogue waves in the different storms pointed out by Stansell(2004).
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