Wave Crest Distributions Research Articles

Nonlinear random wave fields within a Boussinesq system

The dynamics of nonlinear random fields is important for understanding wave turbulence. In this work, we use a Boussinesq system to examine the distinctions between unidirectional and bidirectional waves. Our study demonstrates that in both scenarios, the wave spectra reach a stationary state. Moreover, we show that the occurrence of rogue waves is more probable in the unidirectional case. In the unidirectional case, the probability distribution of wave crests exceeds the one predicted by the Rayleigh distribution once the spectra reach the stationary state. Conversely, in the bidirectional case, the opposite trend is observed. The discovery of various types of rogue waves, including massive wave trains commonly known in the literature as “two sisters” and “three sisters” are found.

Nonlinear statistical characteristics of the multi-directional waves with equivalent energy

Directional distribution is believed to have a significant impact on the statistical characteristics in multi-directional sea states. In real sea states, short-crested waves are discrete not only in frequency but also in direction. For the former one, they are well explained in unidirectional mode, but for the latter one, they are not. In this paper, the kurtosis of short-crested waves with equivalent energy is first discussed. Unimodal-spectrum-multi-direction sea states and bimodal-spectrum-multi-direction sea states are simulated for a long time in a numerical wave basin based on the high-order spectral method. In the equivalent sea-swell sea state, the spatial evolution of kurtosis becomes more inhomogeneous, along with the maximum value of kurtosis being larger and the area where the maximum value occurs wider in the configuration with a crossing angle β = 40° than that with β = 0°, while little variations in swell-dominated and wind-sea-dominated states. A positive linear correlation between wavelet group steepness and kurtosis is obtained in a unimodal sea state, but not applied to a crossing sea state characterized by a bimodal spectrum. The exceedance probability of wave height and wave crest distribution at maximum kurtosis is also given. These findings can help predict the probability of extreme waves occurring, guiding the selection of ocean engineering sites to avoid extreme configurations.

A Study on the High-Order Spectral Model Capability to Simulate a Fully Developed Nonlinear Sea States

Modeling a nonlinear ocean wave is one of the primary concerns in ocean engineering and naval architecture to perform an accurate numerical study of wave-structure interactions. The high-order spectral (HOS) method, which can simulate nonlinear waves accurately and efficiently, was investigated to see its capability for nonlinear wave generation. An open-source (distributed under the terms of GPLv3) project named "HOS-ocean" was used in the present study. A parametric study on the "HOS-ocean" was performed with three-hour simulations of long-crested ocean waves. The considered sea conditions ranged from sea state 3 to sea state 7. One hundred simulations with fixed computational parameters but different random seeds were conducted to obtain representative results. The influences of HOS computational parameters were investigated using spectral analysis and the distribution of wave crests. The probability distributions of the wave crest were compared with the Rayleigh (first-order), Forristall (second-order), and Huang (empirical formula) distributions. The results verified that the HOS method could simulate the nonlinearity of ocean waves. A set of HOS computational parameters was suggested for the long-crested irregular wave simulation in sea states 3 to 7.

Second-order Stokes wave-induced dynamic response and instantaneous liquefaction in a transversely isotropic and multilayered poroelastic seabed

The ocean waves exhibit obvious non-linearity with asymmetric distribution of wave crests and troughs, which could induce significantly different effect on the seabed compared to the commonly used linear wave theory. In this paper, a semi-analytical solution for a transversely isotropic and multilayered poroelastic seabed under non-linear ocean wave is proposed by virtue of the dual variable and position (DVP) method. The ocean wave and seabed are, respectively, modelled using second-order Stokes theory and Biot’s complete poroelastodynamic theory. Then the established governing equations are decoupled and solved via the powerful scalar potential functions. Making use of the DVP scheme, the layered solutions are finally gained by combining the boundary conditions of the seabed. The developed solutions are verified by comparing with existing solutions. The selected numerical examples are presented to investigate the effect of main parameters on the dynamic response of the seabed and evaluate the corresponding liquefaction potential. The results show that the anisotropic stiffness and permeability, degree of saturation and stratification have remarkable influence on the dynamic response and liquefaction behavior of the seabed. The present solution is a useful tool to estimate the stability of transversely isotropic and layered seabed sediment in the range of non-linear ocean wave.

Numerical and experimental study of a FORM-based design wave applying the HOS-NWT nonlinear wave solver

This paper presents a comparative study of long-time irregular waves and equivalent design waves (EDW) in terms of geometric similarity and probability of exceedance (POE) distribution of the wave crest. For a proper comparison between the two wave types, the same nonlinear model was applied in the wave generation by application of the Higher-Order Spectral-Numerical Wave Tank (HOS-NWT), a fully nonlinear propagation solver.Numerical and experimental Monte-Carlo results were obtained through a number of realizations of a given sea state, and results were used as a reference for the EDW results. The experimental measurements and numerical simulations for a given sea state were analyzed and compared in terms of the wave spectrum estimation and the POE distribution of wave crests. The agreement between experimental and numerical results in wave quality seemed sufficient for it to be used as a reference for EDW cases.The First-Order Reliability Method (FORM) approach was applied in the calculation of EDW. The geometrical similarity between the measured EDW wave signal and the corresponding irregular wave signals measured in a given sea state was reviewed. It confirmed that the FORM-based EDW generates a comparable wave profile. In the statistical analysis, however, the results showed that for some EDW cases in relatively severe sea states, the POE estimates by the FORM method appeared to have conservative values compared to the Monte-Carlo reference POE distribution, whereas for the EDW cases in moderate sea states, the estimated POEs were in very good agreement with the empirical wave crest POE distribution of a given sea state.

Influence of Rainfall Infiltration on Stability of Granite Residual Soil High Slope

The seepage evolution characteristics and stability change rule of granite residual soil inside the high cutting slope under the action of rainfall infiltration were studied. Based on the saturated-unsaturated seepage theory, the numerical simulation of the rainwater infiltration path inside the granite residual soil slope was carried out under the condition of rainwater infiltration. The evolution characteristics of the seepage field of the granite residual soil cutting high slope under the condition of rainfall infiltration were revealed, and the influence of the rainwater infiltration on the stability of the granite residual soil slope was mainly studied. The research results show that with the accumulation of rainfall infiltration, the pore water pressure of the granite residual soil slope increases continuously, and the pore water pressure of the soil below the platform increases the fastest. With the increase in elevation, the water content of the slope presents a wave crest distribution. In the early stage of rainfall, the seepage velocity vector distribution on the slope surface is relatively uniform. In the middle stage of rainfall, the seepage velocity vector is the most dense at the platform, and the rainwater migration rate is accelerated. In the later stage of rainfall, the infiltrated rainwater at the slope angle of the first-level slope spreads to the surrounding area. With the continuous rainfall, the slope stability safety factor gradually decreased unevenly, and after the rainfall stopped, it gradually increased with the rainwater infiltration.

Varying ocean wave statistics emerging from a single energy spectrum in an experimental wave tank

As it strongly impacts the design of offshore structures, the realistic reproduction of sea states in experimental and numerical wave tanks is of great interest to the ocean engineering community. The vast majority of wave qualification procedures rely on the accurate control of i) the wave energy spectrum and ii) the wave crest statistics at a target location in the domain. However the control of the wave field is strongly challenged by nonlinear phenomena such as breaking and high-order nonlinearities, which are at the origin of significant variations of the wave properties along the tank. Considering this issue, the most common industry methodologies focus on reproducing the wave energy spectrum at the target position. The wave crest statistics are compared to reference distributions. The present study aims to explore the limitations of such a practice, investigating in detail the wave field properties for various target locations over a long domain. We address the problem within the framework of deep water long-crested irregular waves. In this respect, using the Ecole Centrale de Nantes (ECN) experimental facility, a specific sea state is consecutively generated at three positions of the wave tank using a dedicated procedure based on wave maker motion iterative corrections. For such nonlinear wave conditions the wave crest statistics are known to be enhanced along the tank by high-order nonlinearities. As a result, configurations characterized by identical wave spectra lead to the generation of different wave crest distributions, revealing an increasing number of extreme events as the spectrum is generated farther from the wave maker. The data yielded by the study provide convincing evidence that controlling the wave field at a target location by correcting the sole energy spectrum is insufficient. Particular attention must be paid to the nonlinear spatial dynamics of the wave field in order to control the wave crest statistics.

A fully nonlinear potential flow wave modelling procedure for simulations of offshore sea states with various wave breaking scenarios

An accurate representation of a given sea state is crucial for the study of hydrodynamic loads on offshore structures. It is straightforward to check the quality of the reproduced regular waves in a numerical wave tank (NWT). However, many more parameters need to be considered to ensure the quality of irregular waves. In this paper, a fully non-linear potential flow (FNPF) wave model is used to reproduce irregular sea states with different severity of wave breaking. The numerical model solves the velocity potential from the Laplace equation and the free surface boundary conditions using a finite difference method on a σ-coordinate grid. A comprehensive procedure is introduced to ensure the quality of the reproduced full-scale sea states. The effect of wave spectrum discretisation techniques and breaking wave algorithms are compared for an optimal performance. The evaluation of the simulation results takes into account the kurtosis and the wave crest distribution in addition to the wave spectra. The vertical arrangement of the σ-coordinate grid plays an important role in representing the dispersion relation and a grid optimisation method is explained. The current study provides a working procedure that reproduces high-fidelity irregular sea states with breaking waves in an efficient FNPF NWT.

Data driven analysis on the extreme wave statistics over an area

In this paper we analyse ocean wave crest statistics over different sized areas using data driven methods. We use second order numerical simulations to generate extreme crest data. We consider a simplistic Gumbel distribution fit as well as using a Random Forest Model to map the sea-state parameters to extreme crest values. Our simulations are compared with the existing distributions in the literature. We find that existing distributions perform well for more straightforward cases but that as more parameters are introduced the data science approach can capture features other methods cannot. Our approach also highlights the importance of different parameters such as steepness or length in the mean wave direction. We conclude that machine learning model is promising approach to predicting wave crest distributions in complex scenarios.

A new wave crest distribution based on modified Box-Cox transformation and Rayleigh distribution

A modified Box-Cox transformed Rayleigh (MBR) distribution for the wave crests is presented in this paper. The basic idea contains two steps: firstly, non-Gaussian elevations are transformed into Gaussian elevations by a modified Box-Cox transformation. Afterwards the crests of the transformed elevations are described by the Rayleigh distribution. Comparison between the MBR and Tayfun distributions shows that these two models are the same if terms smaller than O(ε2) are ignored, where ε is the elevations’ skewness. Finally, verifications and comparisons are presented for two sets of field measurement. Based on the results, the MBR distribution gives better results than the Forristall model for very large wave crests. Moreover, the MBR model matches the observations better than the Tayfun distribution since it obtains larger probabilities for larger crest in addition to smaller probabilities for small crests.

Joint Distribution of the Wave Crest and Its Associated Period for Nonlinear Random Waves of Finite Bandwidth

The theoretical treatment of statistical properties relevant to nonlinear random waves of finite bandwidth, such as the joint distribution of wave crest and its associated wave period, is an overdue task hampered by the complicated form of the analytical model for sea surface elevation. In this study, we first derive the wave crest distribution based on the simplified version of the Longuet-Higgins’ wave model and proceed to derive the joint distribution of the wave crest and its associated period, and the conditional wave period distribution with a given wave crest, which are of great engineering value. It is shown that the bandwidth of the wave spectrum has a significant influence on the crest distribution, and the significant wave crest is getting larger in an increasing manner as nonlinearity is increased as expected. It also turns out that the positive correlation of wave crest height with its associated period is extended to more massive waves as nonlinearity is enhanced contrary to the general perception in the coastal engineering community that the wave crest is a statistically independent random process with wave period over large waves. The peak period decreases due to the destructive interference of second-order free harmonics.

On the Interaction between Wind Stress and Waves: Wave Growth and Statistical Properties of Large Waves

AbstractStatistical properties and development of wave fields with different wind forcings are investigated through parametric laboratory experiments. Thirty different, random sea states simulated using a JONSWAP spectrum are mechanically generated in deep-water conditions. Each of the random simulated sea states is exactly repeated but subjected to a range of different wind speeds to study the interaction between wind stress and the existing random sea state waves, especially the isolated effect of the wind stress on the largest waves. Wave crest distributions are sensitive to the wind at the extreme end such that there is an observed deviation from second-order theory for the largest (lowest probability) waves at high wind speed. Because the local wave steepness increases with wind speed, eventually reaching a breaking point, the growth of extreme waves (relative to the significant wave height) due to wind stress is shown to be limited by wave breaking. Even when large waves are breaking, the data reveal that amplitude modulation of wave groups is enhanced substantially as the wind speed increases due to the difference in growth rates between the highest and the lowest wave crests in a wave group. However, there is no evidence of an increase in modulation instability with the wind speed, suggesting that the wind–wave interaction under strong wind forcing dominates the wave growth mechanism over nonlinear wave interactions in a broadband wave field.

On kurtosis and extreme waves in crossing directional seas: a laboratory experiment

We examine the statistical properties of extreme and rogue wave activity in crossing directional seas, to constrain the probabilistic distributions of wave heights and wave crests in complex sea states; such crossing seas alter the statistical structure of surface waves and are known to have been involved in several marine accidents. Further, we examine the relationship between the kurtosis as an indicator of nonlinearity in the spectrum and the directionality and crossing angles of the sea-state components. Experimental tests of two-component directionally spread irregular waves with varying frequency, directional spreading and component crossing angles were carried out at the Ocean Basin Laboratory in Trondheim, Norway. The results from the experiments show that wave heights are well described by a first-order (linear) statistical distribution, while for the wave crest heights several cases exceed a second-order distribution. The number of rogue waves is relatively low overall, which agrees with previous findings in directionally spread seas. The kurtosis and wave and crest height exceedance probabilities were more affected by varying the directional spreading of the components than by varying the crossing angles between components; reducing the component directional spreading increases the kurtosis and increases the exceedance probabilities. The kurtosis can be estimated quite well for two-component seas from the directional spreading using an empirical relationship based on the two-dimensional Benjamin–Feir index when the effects of bound modes are included. This result may allow forecasting of the probability of extreme waves from the directional spreading in complex sea states.

Extreme wave crest distribution by direct numerical simulations of long-crested nonlinear wave fields

The wave crest height qualification checks are required during the wave calibration before the model test in wave basin. However, the reliable criteria of nonlinear wave crest probability distribution in 3-h duration (full-scale) has not been well established yet. We investigate wave crest-height statistics of long-crested nonlinear wave fields using high-order spectral (HOS) method, which can take the effects of both second-order bound waves and third-order free waves into account. The energy dissipation effects due to wave breaking were included by employing an eddy viscosity model. Sensitivity analyses to the wave breaking onset criterion have been performed. Validation is provided by comparing the obtained numerical results with the available calibration test data. Based on extensive and direct numerical simulations, semi-empirical single realization distributions for wave calibration have been developed through 3-parameter Weibull fitting and systematic regression analyses. Particular attention has been paid to the tail of upper bound of wave crest distributions. The effects of wave steepness and water depth on the maximum wave crest height in 3-h duration have been examined. It is found that with the increase of wave steepness, the extreme wave crest height increases until it reaches a critical value. In addition, for the scale water depth kph < 1.36, the maximum crest height decreases as the water depth increases, while in the opposite case the maximum crest height increases as the water depth increases. Moreover, it is confirmed that that the fourth-order nonlinearity does not have significant effects on the distribution of the wave crest height.

Individual wave height and wave crest distributions based on field measurements from the northern North Sea

This paper concerns full-scale wave measurements of individual wave heights and crest heights at one location with water depth 190 m in the northern North Sea from 2004 up to date. The surface elevation was recorded by a Saab WaveRadar REX from one location in the northern North Sea. The measured individual wave heights and crest heights have been compared to well-known theoretical distributions. For wave heights, the Rayleigh distribution is found to form a conservative upper bound, whilst the Forristall distribution is considered to show a good accuracy, though somewhat conservative in the most severe sea states. For crest heights, the Rayleigh distribution is found to be nonconservative, whilst the Forristall distributions for long- and short-crested sea both provide reasonable descriptions. Based on the present study, the Forristall distributions for individual wave height and crest heights in long-crested seas are considered to provide the most appropriate descriptions of these measured wave data.

Nonlinear wave crest distribution on a vertical breakwater

The probability distribution of the nonlinear, up to the second order, crest height on a vertical wall is determined under the assumption of finite spectral bandwidth, finite water depth and long-crested waves. The distribution is derived by relying on the Quasi-Deterministic representation of the free surface elevation on the vertical wall. The theoretical results are compared against experimental data obtained by utilizing a compressive sensing algorithm for reconstructing the free surface elevation on the wall. The reconstruction is pursued by starting from recorded wave pressure time histories obtained by utilizing a sequence of pressure transducers located at various levels. The comparison demonstrates an excellent agreement between the proposed distribution and the experimental data, while, notably, the deviation of the crest height distribution from the Rayleigh one is considerable.

LH-moment estimation for statistical analysis on the wave crest distributions of a deepwater spar platform model test

The design of fixed and compliant offshore platforms requires the reliable estimation of extreme values with small probabilities of exceedance based on an appropriate probability distribution. The Weibull distribution is commonly utilised for the statistical analysis of wave crests, including near-field wave run-ups. The parameters are estimated empirically from experimental or onsite measurements. In this paper, the data set of wave crests from a Spar model test was statistically analysed by using the method of LH-moments for parameter estimation of the Weibull distribution. The root-mean-square errors (RMSEs) and the error of LH-kurtosis were used to examine the goodness-of-fit. The results for the first four LH-moments, the estimated parameters, and the probability distributions showed that the level of the LH-moments has a significant influence. At higher levels, the estimation results gave a more focused representation of the upper part of the wave crest distributions, which indicates consistency with the intention of the method of LH-moments. The low tail RMSE values of less than 2.5% demonstrated that a Weibull distribution model estimated by using high-level LH-moments can accurately represent the probability distribution of large extreme wave crests for incident waves, wave run-ups, and moon pool waves. Goodness-of-fit test on the basis of comparison of sampling LH-kurtosis and theoretical LH-kurtosis was recommended as a procedure for selecting an optimum level.

Efficient Computation of Nonlinear Crest Distributions for Irregular Stokes Waves

AbstractThis paper concerns the computation of nonlinear crest distributions for irregular Stokes waves, and a numerical algorithm based on the Fast Fourier Transform (FFT) technique has been developed for carrying out the nonlinear computations. In order to further improve the computational efficiency, a new Transformed Rayleigh procedure is first proposed as another alternative for computing the nonlinear wave crest height distributions, and the corresponding computer code has also been developed. In the proposed Transformed Rayleigh procedure, the transformation model is chosen to be a monotonic exponential function, calibrated such that the first three moments of the transformed model match the moments of the true process. The numerical algorithm based on the FFT technique and the proposed Transformed Rayleigh procedure have been applied for calculating the wave crest distributions of a sea state with a Bretschneider spectrum and a sea statewith the surface elevation datameasured at the Poseidon platform. It is demonstrated in these two cases that the numerical algorithm based on the FFT technique and the proposed Transformed Rayleigh procedure can offer better predictions than those from using the empirical wave crest distribution models. Meanwhile, it is found that our proposed Transformed Rayleigh procedure can compute nonlinear crest distributions more than 25 times faster than the numerical algorithm based on the FFT technique.

Nonlinear crest distribution for shallow water Stokes waves

This article concerns the calculation of nonlinear crest distribution for shallow water Stokes waves. The calculations have been carried out by incorporating a second order nonlinear wave model into an asymptotic analysis method. This is a new approach to the calculation of wave crest distribution, and, as all of the calculations are performed in the probability domain, avoids the need for long time-domain simulations. The accuracy and efficiency of this new approach for calculating the wave crest distribution are validated by comparing the results predicted using it with those predicted by using the Monte Carlo simulation (MCS) method, by using a previous Transformed Rayleigh method, by using some existing wave crest distribution formulas, and by using the measured surface elevation data at the Poseidon platform in the Japan Sea.

Space–time long-term statistics of ocean storms

This paper presents an approach for estimating the long-term statistics of random wave crests occurring over a certain space–time domain. Such a problem is relevant in a number of marine engineering applications, as classical analyses based, exclusively, on time domain approaches underestimate wave crest amplitudes associated with a given return period.The return period of a certain wave crest is derived by combining the Equivalent Power Storm model, used in long-term statistical analyses, with the Euler Characteristic (EC) of an excursion set concept, recently applied to the study of sea wave statistics. In this regard, the paper shows that by computing the average EC, the probability distribution of the wave crests can be derived. Specifically, an explicit solution valid for finite crest thresholds can be derived by an approximation of the EC. Thus, removing the limitation of actual solutions, which are valid only for extremely large wave crests. In addition, return period of nonlinear wave crests are derived. In this context, Forristall distribution is adopted and introduced on a heuristic basis in the EC framework.The EC approximation and the reliability of the nonlinear crest distribution is assessed against Monte Carlo data by comparing distributions of maximum wave crests both in a sea state and in a sea storm. Then, return value estimations are discussed for a number of cases.