Z. Cherneva

Nonlinear Schrödinger invariants and wave statistics

Third-order quasiresonant interactions among free waves and associated modulational instabilities can significantly affect the statistics of various surface features in narrowband waves. In particular, modulational instabilities tend to induce intermittent amplifications on the surface displacements, causing their statistics to deviate from the linear Gaussian and second-order models. Herein, we investigate the nature of such instabilities on the statistical and spectral characteristics of deep-water waves generated in a large wave basin. We analyze the spectral changes that occur as waves propagate along the basin, develop bounds on the spectrum bandwidth, and interpret various statistics based on third-order Gram–Charlier distributions.

Distribution of Wave Height Maxima in Storm Sea States

The effect of the coefficient of kurtosis, as a measure of third order nonlinearity, on the distribution of wave height maxima has been investigated. Measurements of the surface elevation during a storm at the North Alwyn platform in the North Sea have been used. The mean number of waves in the series is around 100. The maximum wave statistics have been compared with nonlinear theoretical distributions. It was found that the empirical probability densities of the maximum wave heights describe qualitatively the shift of the distribution modes toward higher values. The tendency for the peak of distribution to diminish with an increase in the coefficient of kurtosis up to 0.6 is also clearly seen. However, the empirical peak remains higher than the theoretically predicted one. The exceedance probability of the maximum wave heights was also estimated from the data and was compared with the theory. For the highest coefficients of kurtosis, estimated at nearly 0.6, the theoretical distribution approximates very well the empirical data. For lower coefficients of kurtosis, the theory tends to overestimate the exceedance probability of the maximum wave heights.

Influence of the Third Order Nonlinearity on the Distribution of Wave Height Maxima in an Offshore Basin

The effect of the coefficient of kurtosis, λ40 , on the distribution of wave height maxima has been investigated. The data set used consists of water surface displacements in irregular deep water unidirectional wave fields generated in an offshore basin and defined by the JONSWAP spectrum. The full-scale records are of almost 3h17min duration. The measurements have been performed at ten equidistant gauges along the basin, which permits to follow the changes in wave statistics away from the wave-generator. Subsequently, the records have been split into series of different length, corresponding to N = 100, 200 and 300 waves, and the probability density functions and the exceedance probabilities of the maximum wave heights have been constructed conditional on λ40 . They have been compared with the modified Edgeworth-Rayleigh model of Mori and Janssen [1] applied to the maximum wave heights. The theoretical expressions are formulated as a simple function of the coefficient of kurtosis and the number of waves in the sample. The coefficient of kurtosis, reflecting the third order nonlinearity, is found to increase with the distance from the wave-maker. The considered theoretical density curves describe only qualitatively the shift of the empirical mode towards higher values. The tendency of the peak of the distribution to diminish with increase of λ40 has been observed. However, the most probable wave height remains underestimated by the theory for all classes of λ40 , regardless of the length of the time series. Finally, the probability that a certain normalized height level Hmax /Hs will be exceeded increases with the increase of λ40 , as being theoretically predicted, although it is overestimated by the theory in the lower range of values of λ40 and underestimated over the higher range of values of λ40 .Copyright

Non-Gaussian Wave Groups Generated in an Offshore Wave Basin

The main goal of the present paper is to study the differences of the descriptors of thewave groups in the nonlinear case in comparison with the same parameters for aGaussian process. The data analyzed are from a deep water basin of Marintek. Theyconsist of sequence of five identical independent experimental runs of unidirectionalwaves measured at ten fixed points situated in different distances from the wave maker.Each series contain about 1800 waves. Thus the statistics collected from a given gaugecomprise about 9000 waves combined in a number of wave groups. Because the seriesdescribe a process significantly different from the Gaussian one, an upper and lowerenvelopes are introduced as lines which connect the peaks of the crests and the lowerpoints of the troughs respectively. Spline functions are applied to calculate these enve-lopes. Then, the mean high run and mean group length are estimated for different lev-els, their ensemble average over five experimental runs is found for every gauge and iscompared with the results of the theory of Gaussian process. It is found that the valuesof the mean time intervals of the groups correlate with coefficient of kurtosis of theprocess. It is determined also that mean group length is shorter and the mean high runis larger for the nonlinear wave groups in comparison with the Gaussian wave groups.The modification of wave groups in space and time is investigated in the work as well.Wigner time-frequency spectrum with Choi-Williams kernel is applied to describe theprocess of entire modulation and demodulation of the groups. It is found that beforeformation of the high wave a wave down-shifting takes place. At this stage the localspectrum is relatively narrow and the group shrinks continuously. Close to the focusthe time-frequency spectrum is very wide and the group has a triangle form. Furtherthe high wave breaks and the wave group acquires the form of “three sisters.” Thetransform of the group continues by its disintegration, the local spectrum stands nar-row and an up-shifting is observed. [DOI: 10.1115/1.4006394]Keywords: wave groups, wave envelope, nonlinear waves, mean high wave run, meangroup length

A PHYSICAL MODEL OF WIND WAVES IN THE COASTAL ZONE

The variability of the amplitude-frequency structure of wind waves in space and time during their transformation in the coastal zone are considered. Wave time series, measured synchronously in 15 points along the wave propagation, obtained at field and laboratory experiments, were used for the analysis. Free surface elevation time series were represented as a sum of first and second harmonics with amplitudes slowly varying in time (or envelopes of the waves of corresponding frequency bands). Relative changes of these amplitudes in space and time were studied also. It was revealed, that at the initial stage of the wave transformation, the changes of amplitudes of the first and the second harmonics are similar and amplitudes of the second harmonics are proportional to the squared amplitudes of the first harmonics. At this stage the variability of parameters of individual irregular waves can be explained by Stokes theory. Nearer to the coast the instantaneous values of the amplitudes of the first and the second harmonics varies in time chaotically and is not possible to construct a simple model of the variability of the parameters of individual irregular waves. The main reason for this effect is the backward energy transfer from the second to the first harmonics of the waves during nearly resonant non-linear triad interactions.Copyright

Comparison of Distributions of Wave Heights From Nonlinear Schröedinger Equation Simulations and Laboratory Experiments

Numerical simulations of the nonlinear Schrodinger (NLS) equation are performed by using random initial wave conditions characterized by the JONSWAP spectrum and compared with four different sea states generated in the deep water wave basin of Marintek. The comparisons show that the numerical simulations have a high degree of agreement with the laboratory experiments although a little overestimation can be observed, especially in the severe sea state. Thus the simulations still catch the main characteristics of the extreme waves and provide an important physical insight into their generation. The coefficient of kurtosis λ40 presents a similar spatial evolution trend with the abnormal wave density and the nonlinear Gram-Charlier (GC) model is used to predict the wave height distribution. It is demonstrated again that the theoretical approximation based on the GC expansion can describe the larger wave heights reasonably well in most cases. However, if the sea state is severe, wave breaking can significantly decrease the tail of wave height distribution in reality and the discrepancy occurs comparing with the numerical simulation. Moreover, the number of waves also plays an important role on the prediction of extreme wave height.Copyright

Non-Gaussian Wave Groups Generated in an Offshore Wave Basin and Their Statistics

The main goal of this work is to investigate the wave groups using data from a deep water basin. Available data are for unidirectional waves measured at several fixed points situated in different distances from the wave maker. Previous works of many authors show that such series describe a process which differs significantly from the Gaussian one. Omitting the usual envelope definition by the Hilbert transform an upper and lower envelopes are introduced. Then the mean high run, mean group length and their distributions are found and compared with the theoretical results for Gaussian process.Copyright

Nonlinear Wave Statistics

Third-order quasi-resonant interactions among free waves and associated modulational instabilities can significantly affect the statistics of various surface features in narrowband waves. In particular, modulational instabilities tend to induce intermittent amplifications on the surface displacements, causing their statistics to deviate from the linear Gaussian and second order models. Herein, we investigate the nature of such instabilities on the statistical and spectral characteristics of deep-water waves generated in a large wave basin. We analyze the spectral changes that occur as waves propagate along the basin, develop bounds on the spectrum bandwidth, and interpret various statistics based on third-order Gram-Charlier distributions.Copyright

Statistics of Nonlinear Waves Simulated in a Wave Basin

Modulational instabilities induced by third-order nonlinear interactions among freely propagating waves can cause the statistics of various surface features to deviate significantly from the predictions based on the linear Gaussian and second-order models. This study analyzes deep-water waves simulated in a wave basin and characterized with such instabilities, and compares the statistics of the wave heights, crests and troughs amplitudes observed with a variety of theoretical approximations based on Gram-Charlier expansions. The results indicate that the theoretical approximations describe the empirical distributions observed reasonably well, for the most part. Further comparisons also show that the heights and crests of the largest waves do not exceed Miche-Stokes type upper limits.Copyright