Statistics of long-crested extreme waves in single and mixed sea states

Abstract

Most of the studies on extreme waves are focused on the systems with single-peak wave spectra. However, according to the statistics of occurrence, the bimodal spectral system is also frequent in real sea conditions. In order to summarize the statistics of extreme waves, irregular wave trains under single-peak and bimodal spectra for long durations are simulated in this paper, based on a two-dimensional High Order Spectral (HOS) numerical wave tank. A large number of configurations have been tested under unimodal and bimodal spectra. The investigation on the wave trains under single-peak spectrum indicates that although in conditions often referred as deep water (k p h > π), the relative water depth has a significant influence on the probabilities of occurrence of extreme waves. A detailed analysis of the combined effect of Benjamin-Feir Index (BFI) and relative water depth is provided. However, the situation is more complex in real sea conditions, which may exhibit multimodal spectra. We focus in this study on long-crested bimodal spectra characterized by the same significant wave height H s and mean zero-crossing period T z of the sea states as the single-peak spectrum. The wave conditions under bimodal spectrum present milder extreme wave statistics than those under single-peak spectrum. In addition, mixed ocean systems with equivalent energy distribution (i.e., Sea-Swell Energy Ratio (SSER) is close to 1.0) and larger separation between partitions (i.e., Intermodal Distance (ID) > 0.10) are the less prominent to extreme waves appearance. The comparison of the mixed sea states and the corresponding single independent systems demonstrates that the complexity of the underlying physics of a given sea state (for instance the presence of modulational instability or other nonlinear process) cannot be deduced by an analysis limited to the statistical content of the combined sea state. The wave energy being distributed among frequencies plays a major role. Additionally, Gram-Charlier distribution can accurately predict the probability of large waves (1.5 < H/H s < 2.0) compared to the MER distribution, but it underestimates the statistics of the wave height distribution when H/H s is larger than 2.0 for both single-peak and bimodal states.

Real-world rogue wave probabilities

“…Despite having been in research focus for almost 25 years, they are still being studied extensively 2 – 7 . By now, we know several ways to produce truly exceptional waves in wave tanks and simulations 8 – 10 .…”

“… Parameter Physical meaning References Crest-trough correlation Correlation coefficient between wave crest heights and trough depths 18 , 21 , 22 Spectral bandwidth Spectral peak width, controls wave group dynamics 11 Mean period Mean wave period 2 Rel. low-frequency energy Relative low-frequency (swell) energy content 2 , 4 , 23 , 24 Directional spread Short-crestedness of waves 6 Ursell number ( ) Non-linear shallow water effects 25 Benjamin–Feir index Degree of non-linearity, modulational instability 26 – 28 Excess kurtosis Proneness to outliers of sea surface elevation 13 , 26 , 29 Steepness Weakly nonlinear corrections, wave breaking 15 , 17 Significant wave height Reference wave height, total energy 14 Skewness Shape asymmetry between wave crests and troughs 13 , 30 Relative depth ( ...…”

Relation between occurrence probability of freak waves and kurtosis/skewness in unidirectional wave trains under single-peak spectra

“…The relative spectral bandwidth f/f p refers to the definition introduced in Ref. [22]. The values are listed in Table 1.…”

“…With respect to the discretization in space, the number of points per peak wavelength is 30, the time step is 1/100 of the wave period, and the order of HOS method is 5, referring to the convergence analysis in Ref. [22].…”

“…For more details of the present implementation and the choice of parameters, readers can refer to Ref. [22].…”