Statistical properties of group height and group length in combined sea states

https://doi.org/10.1016/j.coastaleng.2021.103897Get rights and content

Highlights

  • The envelope spectra in combined sea states are investigated.

  • The distributions of group heights and lengths in three types of mixed sea states are analysed.

  • Secondary waves generated by experiments and simulated data are examined.

  • A mixture Weibull model is applied to fit the marginal distributions of group parameters.

  • A mixture copula model is proposed to describe the joint distribution.

Abstract

The statistical properties of ocean wave groups are extremely important for the analysis of resonant effects on ships and marine structures. The patterns in combined sea states have attracted increasing attention recently. This study focuses on the probability distributions of the group height and group length as defined by an envelope approach in sea states with bimodal spectra. Wave data from laboratory experiments and numerical simulations are used. The envelope spectra in combined sea states might exhibit a double-peaked structure. Although the bimodal envelope spectrum has slight effect on the group height distribution, the group length might have a multimodal and broader distribution. As a result, the joint distribution might be characterised by a multimodal shape. Further, a mixed Weibull distribution of the marginal distributions of the group parameters and a mixed copula model of their bivariate distributions are constructed. For comparison, a conventional Weibull distribution and a single copula model are also applied to fit the distributions. The results suggest that the single models can be used only to fit the group height distribution. By contrast, the mixed models can provide satisfactory descriptions of both the marginal distributions and joint distributions.

Introduction

In the oceans, large waves usually occur as wave groups instead of separate waves, and it is therefore reasonable to study the group structure of sea waves (Kuznetsov and Khaskhatchikh, 1978; Ma et al., 2011; Wang et al., 2018). The importance of the groupiness of sea waves to coastal and ocean engineering is well-recognised. Previous research has shown that when two wave trains have similar spectra and wave statistics, wave groups can cause more damage to marine structures than individual waves dispersed throughout a wave train (Johnson et al., 1978). Medina et al. (1990) reported that some wave grouping characteristics significantly affect the stability of the armour layer of rubble mound breakwaters, which may be described by some groupiness parameters. Balaji et al. (2008) analysed the motion of a buoy and found that it was strongly affected by wave groups. The relevance of group structure to the ship response has also attracted increasing attention recently (Anastopoulos et al., 2016). Moreover, the groupiness of ocean waves is linked to extreme waves (Fedele, 2007), wave breaking (Banner et al., 2000; Svendsen and Veeramony, 2001; Banner and Peirson, 2007), surf beat (Baldock et al., 2000), wave overtopping (Yoshimichi et al., 2000), sediment transport (Baldock et al., 2010), the motion responses of floating bodies (Balaji et al., 2008), and other subjects relevant to marine engineering.

Ocean waves are random and irregular in real sea states. Therefore, it is reasonable to apply the stochastic process approach to explore the statistical properties of variable waves for coastal design. Three methods are commonly used for the statistical description of wave groups. In the first approach, which was proposed by Kimura (1980), the wave height train is treated as a Markov chain, which implies that the correlation coefficient of successive wave heights can represent the wave grouping to some extent. Battjes and Van Vledder (1984) reported that this model adequately predicts the group statistics of observed wave records, including those obtained during storms. The relationships between the groupiness calculated using the Markov chain model and four wave parameters indicating the spectral width and shape were investigated by van Vledder (1992) in an effort to determine which spectral parameters are suitable measures of the wave grouping. In addition, Masson and Chandler (1993) examined the limitations of Kimura's theory by discussing the relationships between the lengths of wave groups and some characteristics of the wave spectrum. They demonstrated that this method should be used with caution. Some possible causes of deviation from the Markov chain model were identified and discussed by Lamberti and Rossi (1993). In the second approach, wave groups are described by applying envelope theory to wave sequences as proposed by Rice (1945, 1958). Here, a wave group is defined in terms of the exceedance of the wave envelope above a certain threshold level. Longuet-Higgins (1984) simplified the Markov chain model without using matrices and reported that these two methods can give almost identical results for the wave group characteristics. Later studies focusing on the nonlinear properties of ocean wave groups were conducted by Tayfun and Lo (1989) and Tayfun (2008). Moreover, some researchers emphasised the statistics of wave groups containing more than one high wave (above the specified threshold height) owing to their significant effect on marine systems (Ochi and Sahinoglou 1989a, b; Dawson et al., 1991). In the third approach, which was introduced by Funke and Mansard (1979), wave groups are analysed using the smoothed instantaneous wave energy history (SIWEH), where the squared wave surface elevation is smoothed over the peak period. Those authors proposed a SIWEH-based groupiness factor, GFs, that describes the wave group activity. The Local Variance Time Series, a similar approach focusing on the time variation of the local variance, was proposed by Thompson and Seelig (1984), who also introduced a parameter comparable to GFs that indicates the extent of wave grouping. Owing to the drawback of GFs resulting from the standard use of the peak period, List (1991) recommended a groupiness factor GFe based on the wave envelope. This parameter has well-defined numerical limits and can describe the degree of wave grouping. Two new parameters, the temporal group steepness and spatial group steepness, which were found to be more responsive than GFs, were introduced by Haller and Dalrymple (1995) to quantify wave grouping. Because ocean waves are non-stationary, Dong et al. (2008) defined the instantaneous wave energy signal based on a continuous wavelet transform and proposed a new groupiness factor, which is less affected by the operational definition than GFs or GFe. More recently, Liu and Li (2018) adopted a wave grouping factor based on the wave height history to investigate the variation of wave groupiness across a fringing reef.

It is well recognised that adequate measures of wave grouping should be included in the engineering design and standard analysis of coastal and ocean activities. Therefore, scientists and engineers have investigated the probability structure of the wave group parameters. Assuming a Markov chain model, Kimura (1980) derived the probability distribution of the run of high waves, which is defined as the number of waves in a group. Note that the transition probabilities were derived in that study under the assumption that the consecutive wave height distribution is a two-dimensional Rayleigh model. Note also that this distribution model can hold only if the spectral width of the wave records is narrow. Stansell et al. (2002) assumed a bivariate Rayleigh distribution and used four methods of estimating the correlation coefficient of adjacent wave heights. They compared the results with measured data collected during storms and noted that the parameter calculation for this model is still a problem for broader-banded seas. Ochi and Sahinoglou (1989a) derived the probability density function of the time duration of a wave group, and thus the number of wave crests in the group. Dawson et al. (1991) demonstrated that the probability distributions of the envelope durations and intervals are both better fitted by the gamma distribution model than the Poisson or exponential distribution applied by Longuet-Higgins (1984). Mase and Iwagaki (1984) demonstrated that at least two parameters, which reflect the group length and group height, respectively, are required to describe the magnitude of wave groups. Consequently, Dong et al. (2015) applied the copula method to establish the joint distribution of the group height and group length.

Waves observed in real sea states can contain various wave systems, such as a mix of swell and wind-generated waves (Aranuvachapun, 1987), the wave spectrum of which can exhibit a double peak. There are a few studies on the statistical properties of wave groups in such wave fields. Rodríguez et al. (2000) applied Kimura's model to obtain the probabilities of the run length and total length for mixed sea states using numerically simulated data. They observed that this model could characterise the frequency distribution of the run length as a single-peak Joint North Sea Wave Project (JONSWAP) spectrum adequately, whereas the statistics of wave groups in mixed sea states could not be described well. They also suggested that the spectral bandwidth parameters can be used to measure the degree of groupiness. Kumar et al. (2003) studied the wave group statistics of sea states with double-peaked spectra using wave data collected along the Indian coast. Neves et al. (2004) reported that the statistics derived using Kimura's theory, the technique of Longuet-Higgins (1984), and the wave spectrum are not good indicators of the wave grouping of multipeaked spectra. Maris et al. (2012) applied the Hilbert–Huang transform to explore the characteristics of waves with a bimodal spectrum. The outcomes were compared to those obtained using the Fourier transform, wavelet transform, and Stockwell transform. In addition, the intrinsic model functions derived using different decomposition methods were examined.

The wave group statistics of combined sea states deserve further attention. In this study, the bivariate distributions of the group height and group length under different degrees of wave grouping are investigated using wave data with double-peaked spectra. The rest of the paper is organized as follows. In Section 2, two group parameters defined by an envelope method are introduced. Section 3 briefly describes the data, which consist of experimental data and simulated data. The patterns of the probability distributions of the group height and group length are also examined. Section 4 applies mixture models to build the marginal distributions of these two parameters, and Section 5 establishes their bivariate distribution using a Gaussian mixture copula model. Finally, the main findings are presented in Section 6.

Section snippets

Wave group statistics

According to the wave envelope approach, a wave group can be defined as the waves between two successive up-crossings of the wave envelope A(t) that exceeds a given threshold level à (Fig. 1). For marine disasters associated with resonance phenomena, the resonant stress on structures can be affected by both the duration and intensity of the driving force. Therefore, as mentioned above, the magnitude of wave groups should be characterised by at least two parameters, one that represents the wave

Laboratory experiments

Data on the higher harmonics can be obtained when waves pass over a bar or other obstacle used for coastal protection. Such waves on the sea side of marine structures, which are known as secondary waves, involve the decomposition of the primary wave and the redistribution of the wave energy among the harmonics (Beji and Battjes, 1993; Kuznetsov and Saprykina, 2012). The nonlinear wave interactions are well known to be responsible for the transfer of wave energy from the initial frequency

Mixture models of marginal distributions of group parameters

The probability distributions of random wave parameters are often assumed to follow a Weibull distribution model. The Weibull probability density function of a variable X is given byf(x)={βα(xα)β1exp[(xα)β]x>00x0,where α and β are the scale and shape parameters, respectively. The Weibull cumulative distribution function can be written asF(x)={1exp[(xα)β]x>00x0.

The marginal distributions of group statistics in combined sea states can have distinctive features owing to the mix of wind–sea

Copula-based model of group height and group length

A multivariate description of the relevant ocean wave parameters is extremely important for engineering design and probabilistic risk analysis. From a stochastic viewpoint, the group height and group length should be considered as dependent variables. There are currently a few types of statistical parametric models of the bivariate distribution. These approaches can be broadly divided into three classes: 1) well-known bivariate distribution functions (Kimura, 1981; Myrhaug et al., 1995); 2) the

Conclusions

The probability distributions of the group height and group length defined by the envelope method in combined sea states were investigated. Data from laboratory experiments and numerical simulations were used. Wind–sea-dominated, swell-dominated, and sea-swell energy equivalent sea states were analysed. It was shown that the double-peaked power spectrum of ocean waves can produce an envelope spectrum with a bimodal structure and thus significantly affect the probabilistic structure of the

CRediT authorship contribution statement

Weinan Huang: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing – original draft. Sheng Dong: Conceptualization, Methodology, Formal analysis, Resources, Writing – original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The study was supported by the NSFC–Shandong Joint Fund (U1706226) and the National Natural Science Foundation of China (51779236).

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