M. Aziz Tayfun

On nonlinear wave groups and crest statistics

We present a second-order stochastic model of weakly nonlinear waves and develop theoretical expressions for the expected shape of large surface displacements. The model also leads to an exact theoretical expression for the statistical distribution of large wave crests in a form that generalizes the Tayfun distribution (Tayfun, J. Geophys. Res ., vol. 85, 1980, p. 1548). The generalized distribution depends on a steepness parameter given by μ = λ 3 /3, where λ 3 represents the skewness coefficient of surface displacements. It converges to the Tayfun distribution in narrowband waves, where both distributions describe the crests of all waves well. In broadband waves, the generalized distribution represents the crests of large waves just as well whereas the Tayfun distribution appears as an upper bound and tends to overestimate them. However, the theoretical nature of the generalized distribution presents practical difficulties in oceanic applications. We circumvent these by adopting an appropriate approximation for the steepness parameter. Comparisons with wind-wave measurements from the North Sea suggest that this approximation allows both distributions to assume an identical form with which we can describe the distribution of large wave crests fairly accurately. The same comparisons also show that third-order nonlinear effects do not appear to have any discernable effect on the statistics of large surface displacements or wave crests.

Effects of spectrum band width on the distribution of wave heights and periods

Abstract A theoretical expression for the joint distribution of crest-to-trough wave heights and zero up-crossing periods is developed from a modified extension of presently available results relevant to wave envelopes and periods under narrowband conditions. The resulting probability structure, in particular, the marginal density for crest-to-trough heights is explored explicitly with respect to effects associated with the spectrum band width. Results are given to show that the Rayleigh distribution, which is often used in predicting crest-to-trough heights, can not be regarded as accurate. The analysis is extended further to restate and establish the validity of an approximate expression for the distribution of crest-to-trough heights, which was suggested in a previous study (Tayfun, 1981).

Nonlinear effects on the distribution of crest-to-trough wave heights

The statistical distribution of the crest-to-trough heights of narrowband nonlinear sea waves is derived in a semi-closed form. A quantitative comparison of the resulting density and exceedance probability distributions with those of the linear theory is given. It is shown that the nonlinearity of waves, even with steepnesses typical of extreme sea states, has an insignificant influence on the distribution of crest-to-trough heights.

Expected Shape of Extreme Waves in Storm Seas

The theoretical expected structure of large nonlinear waves can be described, using the Gram-Charlier approximations of Jensen et al. (1995) and Jensen (1996, 2005), and the quasi-deterministic model of Fedele & Arena (2005). The second-order narrow-band approximation offers a simpler alternative to these models, as recently suggested in Tayfun & Fedele (2006). Herein, this latter alternative is elaborated further, deriving theoretical expressions for predicting the expected shape of large waves, conditional on somewhat more general constraints than those previously considered. The theoretical results are verified favorably with oceanic measurements gathered at deep and transitional water depths in the North Sea. Some comparisons of the present model with those of Jensen et al. (1995) and Fedele & Arena (2005) are also given, showing that all three models do in fact reasonably well in representing the expected profile of large waves in storm seas.Copyright

The Effect of Third-Order Nonlinearities on the Statistical Distributions of Wave Heights, Crests and Troughs in Bimodal Crossing Seas

This paper investigates the effect of third-order nonlinearities on the statistical distributions of wave heights, crests, and troughs of waves mechanically generated in a deep-water basin and simulating two crossing systems characterized by bimodal spectra. The observed statistics exhibits various effects of third-order nonlinearities, in a manner dependent on both the distance from the wave-maker and the angle between the mean directions of the component wave systems. In order to isolate and demonstrate the effects of third-order nonlinearities by themselves, the vertically asymmetric distortions induced by second-order bound waves are removed from the observed time series. It appears then that the distributions of wave crests, troughs and heights extracted from the nonskewed records clearly deviate from the Rayleigh distribution, suggesting that the waves are characterized by nonlinear corrections of higher-order than the typical of second-order waves. Nonetheless, some models developed for weakly nonlinear second-order waves can still be used in describing wave heights, crests and troughs in mixed seas, provided that the relevant distribution parameters are modified, so as to reflect the effects of third-order corrections and some basic characteristics of the mixed seas.

Nonlinear effects of the distribution of amplitudes of sea waves

The effect of nonlinearities, such as wave-breaking and vertical asymmetry associated with sea waves, on the distribution of wave amplitudes is explored. Semiclosed theoretical expressions are derived to describe the distributions of breaking-limited crest and trough amplitudes for Stokes-type nonlinear sea waves. These are compared with the conventional Rayleigh distribution appropriate to linear wave amplitudes. The construction of nonlinear wave envelopes with the fast Fourier transform technique is described. The technique can be utilized to enlarge the data base in empirical analyses of field records which typically contain limited information on amplitude characteristics. The theoretical distributions and the proposed data enlargement technique are demonstrated with the analysis of a nonlinear wave record.

Envelope and Phase Statistics of Large Waves

A theoretical expression derived previously for describing the joint distribution of the envelope and phase of nonlinear waves is verified with wind-wave measurements collected in the North Sea. The same distribution is explored further to derive the marginal and conditional distributions of wave envelopes and phases. The nature and implications of these are examined with emphasis on the occurrence of large waves and associated phases. It is shown that the wave-phase distribution assumes two distinct forms depending on whether if envelope elevations exceed the significant envelope height or not. For envelope elevations sufficiently larger than this threshold, the wave-phase distribution approaches a simple limit form, indicating that large surface displacements can occur only above the mean sea level. Comparisons with the North Sea data confirm these theoretical results and also suggest that large surface displacements and thus large wave heights arise from the constructive interference of spectral components with different amplitudes and phases. Further, large waves with high and sharp crests do not display any secondary maxima and minima. They appear more regular or narrow-banded than relatively low waves, and their heights and crests do not violate the Miche-Stokes type upper limits.Copyright

Space-Time Extremes in Sea Storms

We present a stochastic model of sea storms to predict the maximum height of the wave surface over a given area during storms. To do so, we exploit the theory of Euler Characteristics of random excursion sets combined with a generalization of Boccotti’s equivalent triangular storm model (Boccotti, 2000) that describes an actual storm history in the form of a generic power law (Fedele and Arena, 2010). An analytical solution for the return period of extreme wave events over a given area and the associated statistical properties are given. We then assess the relative validity of the new model and its predictions by analyzing wave measurements retrieved from NOAA-NODC buoys moored offshore of the Atlantic and Pacific coasts.Copyright

EXTREME WAVES OF SEA STORMS

We present a stochastic model of sea storms for describing long-term statistics of extreme wave events. The formulation generalizes Boccotti’s equivalent triangular storm model (Boccotti 2000) by describing an actual storm history in the form of a generic power law. The latter permits the derivation of analytical solutions for the return periods of extreme wave events and associated statistical properties. Finally, we assess the relative validity of the new model and its predictions by analyzing wave measurements retrieved from two NOAA-NODC buoys in the Atlantic and Pacific Oceans.Copyright

Wave-Height Distributions and Nonlinear Effects

Theoretical distributions for describing the crest-to-trough heights of linear waves are reviewed briefly. To explore the effects of nonlinearities, these and two approximations that follow from the Tayfun [28] model are generalized to nonlinear waves via the second-order quasi-deterministic model of Fedele and Arena [7]. The potential utility of Gram-Charlier type approximations [17, 18, 29, 31] in representing the statistics of nonlinear wave heights is also explored. All models and a fifth-order Stokes-Rayleigh type model recently proposed by Dawson [5] are compared with linear and nonlinear waves simulated from the JONSWAP spectrum representative of long-crested extreme seas, and also with observational data gathered during two severe storms in the North Sea. The results indicate first that nonlinearities do not appear to affect the crest-to-trough wave heights significantly. Most models and their nonlinear extensions yield similar and reasonable predictions of the data trends observed. The present comparisons do not confirm the efficacy of Gram-Charlier type approximations in modeling the statistics of unusually large wave heights.Copyright