This paper investigates the average shape of the largest waves arising in finite water depths. Specifically, the largest waves recorded in time histories of the water surface elevation at a single point have been examined. These are compared to commonly applied theories in engineering and oceanographic practice. To achieve this both field observations and a new set of laboratory measurements are considered. The latter concern long random simulations of directionally spread sea states generated using realistic Joint North Sea Wave Project (JONSWAP) frequency spectra. It is shown that approximations related to the linear theory of quasi-determinism (QD) cannot describe some key characteristics of the largest waves. While second-order corrections to the QD predictions provide an improvement, key effects arising in very steep or shallow water sea states are not captured. While studies involving idealized wave groups have demonstrated significant changes arising as a result of higher-order nonlinear wave–wave interactions, these have not been observed in random sea states. The present paper addresses this discrepancy by decomposing random wave measurements into separate populations of breaking and nonbreaking waves. The characteristics of average wave shapes in the two populations are examined and their key differences discussed. These explain the mismatch between findings in earlier random and deterministic wave studies.
The largest waves in the ocean have long attracted oceanographic and engineering interest. Regarding offshore and coastal structures, both the size and the shape of these waves represent key design parameters. For example, jacket-type structures require a deck elevation that is high enough to avoid potentially catastrophic wave-in-deck loading (Ma and Swan 2020). In principle, it is the largest waves, or waves with a very low probability of exceedance in a severe sea state, that will give rise to this type of loading. These are typically defined by the integration of the short-term crest height distribution over the long term distribution of sea state parameters (DNV 2010), the latter typically based on hindcast models.
Following this approach, a “design” wave is fitted to the required crest elevation and the wave kinematics calculated. These are then used to perform wave loading calculations—an essential part for any structural design. Traditionally, these design waves were defined using regular wave theories. More correctly, they should represent asymptotic approximations to random wave theories. In adopting this approach, very long random wave testing can be substituted by the investigation of selected wave events, the latter commonly defined by the theory of quasi-determinism (QD) following (Lindgren 1972; Boccotti 1983; Tromans et al. 1991).
The justification for using representative wave events has been extensively examined in the literature. Typically, the assessment involves comparisons between the average shape of (random) measured surface elevation time histories and available analytical theories; broad agreement being generally reported (Phillips et al. 1993b; Jonathan and Taylor 1997; Tayfun and Fedele 2007). However, when one of these analytical representations is used as the input to a study of idealized wave groups in a fully nonlinear (numerical or experimental) simulation, different results arise. This refers to significant changes in the magnitude and symmetry (both vertical and horizontal) of the fully nonlinear waves (Johannessen and Swan 2001, 2003; Gibbs and Taylor 2005; Gibson et al. 2007; Adcock et al. 2015). This apparent discrepancy raises a number of important questions: Are these nonlinear effects important in random seas? If they are, why are they not observed in the average shape of the largest waves recorded therein?
Considering the statistical distribution of crest heights, several studies have illustrated that higher-order nonlinearities can play an important role (Onorato et al. 2009; Shemer et al. 2010). More importantly, recent results by Latheef and Swan (2013) and Karmpadakis et al. (2019) have shown that the competing mechanisms of nonlinear amplifications and wave breaking have a profound effect on crest height statistics. To illustrate this effect, the distribution of crest heights ηc normalized by their significant wave height Hs is shown on Fig. 1. These results relate to a very steep, laboratory-generated, short-crested sea state with Hs = 15.3 m and effective water depth kpd = 1.22 reported by Karmpadakis et al. (2019). The measured data are compared to the predictions of the commonly applied Forristall (2000) distribution. The latter is defined by
where αF and βF are the scale and shape coefficients defined in terms of the sea state steepness and Ursell parameter and Q is the probability of exceedance. The Forristall (2000) distribution was derived as a fit to second-order numerical simulations. In comparing the measured data with the second-order distribution, two important observations can be made. First, nonlinear amplifications (beyond second order) are present, appearing as increases above the Forristall distribution in the range: 10−3 < Q < 10−1. Second, the largest crest heights in the tail of the distribution fall below the Forristall distribution. This demonstrates the dissipative effects of wave breaking for Q < 10−3. Taken together, these two competing effects have a profound influence on the crest height distribution and have important design implications.
Building upon these observations of the crest heights, the present paper addresses the questions raised above concerning the significance of nonlinear amplifications and wave breaking on the average shape of the largest waves. To achieve this, experimental measurements are supplemented by field and numerical data, the intention being to examine the characteristics of nonbreaking and breaking waves separately. The contents of this paper are arranged as follows. First, a brief overview of relevant work in the field is provided in section 2. The adopted methodology and details of the datasets are presented in section 3. The findings arising from this study are discussed in section 4, and the main conclusions summarized in section 5.
Adopting linear random wave theory (LRWT), the water surface elevation correct to first order, η(1)(x, t), can be expressed as
where x = (x, y) is the horizontal coordinate vector, i denotes an individual wave harmonic of amplitude ai and initial phase ψi, and k = (kx, ky) = (k cosθ, k sinθ) is the wavenumber vector associated with the cyclic frequency ω and direction θ via the linear dispersion relation. Adopting this approach the water surface elevation represents a zero-mean, random Gaussian process (Ochi 1998). As such, Lindgren (1972), Boccotti (1983), and Tromans et al. (1991) used the asymptotic properties of Gaussian theory to derive the most probable shape of the largest waves. This approach is commonly referred to as the theory of quasi-determinism or QD wave (Boccotti 2000). Alternatively, Tromans et al. (1991) relabeled these events as “NewWaves,” and this notation has been adopted in some design codes (ISO:19901-1 and API 2MET). Irrespective of the name adopted, the average shape of the largest waves arising in a (stationary) sea state was shown to be proportional to its normalized autocorrelation function r(τ). Removing the spatial dependence from Eq. (2), the temporal QD wave profile at a single location is given by
where Sηη(ω) is the energy density function, τ is the time lag [measured from the maximum of r(τ)], and A is a scaling factor that can be adjusted to approximate the maximum crest elevation ηmax. Theoretically, this approximation is valid for ηc/ση → ∞, where ηc is the crest height and ση the standard deviation of the surface elevation time series. However, as discussed in section 4, the practical application of this model requires the definition of a large but finite value for this ratio.
While the QD wave profile provides a good approximation for linear sea states (Boccotti et al. 1993; Phillips et al. 1993a,b), real seas are nonlinear, particularly those of interest in design. As a result, the largest waves arising in these seas will inevitably exhibit some level of nonlinear behavior (Guedes Soares and Pascoal 2005). At a second-order of wave steepness, the free surface elevation is given by the sum of the linear part [Eq. (2)] and the second-order bound contributions. The latter are further divided into the frequency-difference terms η(2−) and the frequency-sum terms η(2+), as described by Longuet-Higgins and Stewart (1960) and Sharma and Dean (1981). These are given by
where the interaction kernels Mij− and Mij+ are given in the appendix and Ψ = (kx − ωt + ψ) for each (i, j) wave harmonic. Considering these contributions, the frequency-difference terms represent slowly varying terms (or group terms), while the frequency-sum terms are high-frequency oscillations. Taken together, the total surface elevation according to second-order random wave theory (SORWT) is η(2) = η(1) + η(2−) + η(2+). To capture the effects arising at a second order of wave steepness, Jensen (1996, 2005), Fedele and Arena (2005), Tayfun (2006a), and Tayfun and Fedele (2007) have derived analytical corrections to the linear QD wave profile. These second-order corrections have been shown to provide a better approximation to the average profile of the largest waves recorded in field data, with evidence provided by Tayfun and Fedele (2007). For this reason, both the linear and second-order QD wave profiles are examined in the present study, the latter being obtained explicitly from SORWT (Arena 2005).
An alternative method to account for nonlinearities has been applied by Johannessen and Swan (2003), Walker et al. (2004), Taylor and Williams (2004), Santo et al. (2013), and Whittaker et al. (2016), among others. In this case, the average profiles of waves with the largest crest heights and deepest toughs are employed to decompose the nonlinear contributions. A Stokes-type expansion is then used to obtain the nonlinear wave profile. This method has been shown to be quite versatile in terms of the order of nonlinearity that can be included—Walker et al. (2004) incorporated effects up to a fifth order of wave steepness.
One particular category of large ocean waves relates to so-called rogue waves. These represent wave events that are significantly larger than the surrounding wave field in a given sea state. The most common definition is that proposed by Haver and Andersen (2000) in which ηmax > 1.25Hs or Hmax > 2Hs, where Hs is the significant wave height and the ratios are based upon a 20-min record. Such events are commonly said to be responsible for a number of marine accidents (Kharif and Pelinovsky 2003). Increasing evidence in the literature suggest that a physical mechanism that leads to the formation of these wave events arises through the spatiotemporal focusing of individual wave harmonics (Christou and Ewans 2014; Cavaleri et al. 2016; Benetazzo et al. 2017), with the linear focusing underpinning the QD wave being enhanced by higher-order nonlinearities. While some studies suggest that effects higher than second-order are insignificant (Fedele et al. 2016), others have shown evidence of their importance in experimental and field measurements (Latheef and Swan 2013; Gibson et al. 2014; Karmpadakis et al. 2019). It is clear that in a linear sense a QD wave profile and a focused wave are identical—the input spectrum for the latter being the Fourier transform of Eq. (3). Nonlinear effects can therefore be investigated by generating focused waves either experimentally (Baldock and Swan 1996; Johannessen and Swan 2001) or numerically (Johannessen and Swan 2003; Bateman et al. 2012; Adcock and Taylor 2016). The aforementioned studies have provided significant insights into the nonlinear physics that drive the formation of large wave events. Two characteristic changes relate to increased crest height elevations above second-order theory and front-back asymmetry of the largest wave event at the time of focusing, the latter being induced by its movement toward the front of the wave group. Taking into account that neither of these nonlinear changes is captured by the analytical QD theories, it is worth considering whether they are relevant to the definition of the average shape of the largest waves in random seas.
While many of the aforementioned studies investigate the shape of the largest waves in deep-water conditions, fewer studies have considered shallower water conditions (Whittaker et al. 2016). Considering the significance of large waves in finite water depths (Nikolkina and Didenkulova 2011; Karmpadakis 2019), the potential mechanisms of nonlinear amplifications (Slunyaev et al. 2002; Katsardi et al. 2013; Fernandez et al. 2014), and the effects of wave breaking (Katsardi and Swan 2011; Karmpadakis et al. 2020), this study concentrates on finite water depth conditions.
3. Data sources and methods of analysis
Three complementary sources of data have been used in the present paper. These include the analysis of surface elevation measurements recorded at the field, laboratory observations, and numerical simulations. In each case the average shapes of the largest waves are compared to theory, and the effects of nonlinearity and wave breaking investigated.
a. Field data
The field data used within this study were recorded using wave radars mounted on the side of fixed offshore platforms. These were part of an extensive field data analysis project (the LoWiSh Joint Industry Project) including measurements from 10 different locations in the central and southern North Sea (Karmpadakis et al. 2020). In the present study, data from the shallowest and deepest locations are considered, the two platforms being located in water depths of 7.7 m (close to the Dutch coast) and 45 m (in the Danish sector), respectively. In both cases the free surface was recorded using Saab wave radars with high sampling rates (4 Hz ≤ fs ≤ 5.12 Hz), with the accuracy estimated to be ±6 mm. Indeed, these measurements are in accordance with the highest standards in platform-based observations, with a recent review of the operational characteristics of the instrument type being provided by Ewans et al. (2014). More importantly, the use of recordings from fixed instruments avoids potential issues that have been observed in the analysis of large waves using wave buoys. These include the linearization of the measured waves and the movement around the largest, three-dimensional, wave crests (James 1986; Magnusson et al. 1999; Dysthe et al. 2008). Moreover, the adopted sampling rate guarantees that the nonlinear characteristics of the largest waves can be captured with sufficient accuracy. In this respect, low sampling rates have been shown to underestimate the largest crest elevations (Tayfun 1993; Stansell et al. 2002).
In seeking to obtain a high-quality database, the raw surface elevation records were processed according to the strict quality control (QC) procedures outlined by Christou and Ewans (2014). In effect, this involves the application of a series of flags to identify potential sources of error, the latter including instrument lock-ins, sensor drifts, and unrealistic spikes in the surface elevation records. When erroneous measurements were identified, the full 20-min record was abandoned. All remaining (unflagged) records were then processed using standard spectral and zero-crossing analysis methodologies. It is also important to note that any tidal fluctuations or storm surges were removed prior to the commencement of the analysis. Having completed the analysis of each 20-min record, appropriate met-ocean parameters, such as the significant wave height Hs and peak period Tp, were used to bin the resulting sea states into small groups with similar characteristics. The largest waves recorded in the sea states within each of these data bins were then extracted and are presented in the analysis that follows.
b. Experimental data
In generating the laboratory wave data, a large number of random sea states were simulated in the directional wave basin at Imperial College London. This wave basin has plan dimensions of 10 m × 20 m and a movable horizontal bed, with the water depth in the present tests being set to d = 0.5 m. The basin is equipped with 56 individually controlled, bottom-hinged wave paddles positioned along the 20-m length. The wavemakers operate on the basis of a theoretical transfer function with active, force-feedback, wave absorption (Spinneken and Swan 2012). The combination of the active absorption system and a perforated parabolic beach on the opposite side of the wavemakers ensures that the maximum reflection coefficients were less than 5%. Moreover, there is no build-up of reflected wave energy within the wave basin during a long random generation (Masterton and Swan 2008).
The time histories of water surface elevation η(t) were recorded using 32 resistance-type wave gauges. These gauges are composed of two thin steel wires (of diameter 1.5 mm) spaced 10 mm apart and were calibrated daily to maintain an accuracy of ±0.5 mm. The sampling rate was sufficiently high (fs = 128 Hz) to ensure that η(t) was measured accurately and no postprocessing or filtering was required. Importantly, Haley (2016) has demonstrated that this configuration can accurately record the surface elevation of both very steep and breaking waves. This was confirmed by comparing the output of these wave gauges with high-speed video imaging. The layout of wave gauges for the experiments presented herein consists of a 5 × 5 array in the center of the wave basin, with seven additional wave gauges placed along the centreline of the wave basin. Full details of this layout are given in Karmpadakis et al. (2019). Importantly, the minimum distance between the upstream wave gauge and the wavemakers (l = 2.3 m) was larger than 3d. This ensures that there were no evanescent wave modes present in the measured data. The operational characteristics of this facility yield results that are spatially homogeneous in the working area of basin (Latheef and Swan 2013). As such, unless otherwise stated, the results presented correspond to measurements at the central wave gauge; the latter being representative of the full wave gauge array.
All the experiments involve random, directionally spread, sea states. To ensure that the sea states under consideration correspond to realistic conditions in the field, they were defined on the basis of the Joint North Sea Wave Project (JONSWAP) spectrum (Hasselmann et al. 1973); the spectral density function Sηη for each case is defined by
where ω is the circular wave frequency (ω = 2π/T), T the corresponding wave period, ωp circular wave frequency at the spectral peak, β = 1.25, σ = 0.07 for ω ≤ ωp, and σ = 0.09 for ω > ωp. To simulate sea states with finite frequency bandwidth, the peak enhancement factor, γ, was set to 2.5 for all test cases. The Phillips parameter α was adjusted in each test case so that the target Hs could be obtained for a given spectral peak period Tp. Although the JONSWAP spectrum does not represent the present state-of-the-art when modeling real seas (see, e.g., Lenain and Melville 2017), it is widely applied in engineering practice and has been adopted as the basis for many earlier studies. Moreover, it provides a reasonable description of the field data employed in the present study. To generate directionally spread sea states, a wrapped-normal directional spreading function (DSF) was applied to the unidirectional spectra defined in Eq. (6). The functional form of the DSF is given by
where θ is the angle of propagation, measured relative to the x axis, σθ is the standard deviation of the frequency-independent directional spreading, and A is a normalizing factor such that . The directional spectrum is thus given by F(ω, θ) = Sηη(ω)D(ω, θ). Further details concerning the effective generation of directionally spread seas are given in Latheef et al. (2017). Given the nature of the input conditions, the target sea states can be uniquely defined by 3 parameters: (Hs, Tp, σθ). In relating the experimental test cases to field measurements, scaling based on Froude number similarity has been applied. Specifically, a length scale of ls = 1:100 and a corresponding time scale of have been adopted throughout these tests.
The analysis in the present paper focuses on test cases with kpd = 1.22 (Tp = 1.4 s) and σθ = 10°, with a variety of sea state steepnesses being examined. These cases correspond to a wide variety of realistic sea state conditions in which a detailed investigation of the effects of nonlinearity and wave breaking is conducted. In validating the results obtained in these conditions, additional sea states were also considered. In total, these involve 3 different effective water depths, kpd = 1.53, 1.22, and 1.02, each with directional spreads of σθ = 0°, 10°, and 20° and a range of sea state steepnesses Sp. A summary of the relevant experimental test cases is provided in Table 1.
The selection of the test cases presented in Table 1 was primarily driven by the need to investigate the changes induced by increasing sea state steepness in finite water depths. In this respect, the sea state steepness Sp is defined as
where g = 9.81 m s−2 is the gravitational acceleration. The significant wave height Hs in each test case was selected to provide an incremental increase in Sp, such that ΔSp = 0.01. Taken together, the sea states presented herein vary from near linear (Sp = 0.01) to extremely steep (Sp = 0.06), the latter being characterized by extensive wave breaking.
For each of these test cases, 20 random simulations or seeds were undertaken, with the duration of a single simulation being 1024 s. Given the adopted scaling, each simulation (approximately) corresponds to a 3-h sea state at field scale. Furthermore, the target spectrum used as input to the wavemakers composed of frequency components lying in the range 0.4 Hz < f < 2.5 Hz and had a resolution of Δf = 1/1024 Hz. This yields a set of 2145 individual wave components that define each random seed. The amplitude of each wave component was defined by the target JONSWAP spectrum without being further randomized. This means that the (one-dimensional) energy spectra for each seed within the same test case are identical. The initial phase ψ of each individual wave component was chosen randomly from a uniform distribution lying in the range [0, 2π). Additionally, each individual wave component was assigned a direction of propagation θ. These varied between −45° ≤ θ ≤ 45° and were randomly sampled from the target DSF defined in Eq. (7). It should be noted that the adoption of this method leads to the generation of individual frequency components propagating in a single direction and is effectively a modification of the single summation method (Miles and Funke 1989), often called the random directional method (RDM). Latheef et al. (2017) have shown that the RDM method has significant advantages when it comes to generating ergodic, directionally spread sea states.
A subtle but very important point regarding this experimental investigation concerns the treatment of the initial phases and directions of propagation between the seeds of different test cases. Specifically, a set of random phases and directions was defined for each seed in the lowest steepness case (e.g., case B1–10 with Sp = 0.01 and σθ = 10°; see Table 1). For the cases with larger Sp (but the same Tp and σθ), the sets of phases and directions were kept unchanged. As a result, the amplitude of the individual wave components is the only variable that changes between the same seeds in sea states of different steepnesses. This methodology leads to a collection of 20 random simulations (or seeds) for each sea state. However, each single seed has the same “random” characteristics in all test cases with the same effective water depth and directional spreading. In other words, the inverse Fourier transform of the input directional spectra of the same seed in different sea states provides time histories that are exactly aligned, with the only difference between them being the scale of η(t). This alignment is clearly shown in Fig. 2, which presents the same 5-s segment of the surface elevation η(t) time histories from seed 11 in cases A1–10, A2–10, and A3–10, their steepnesses being Sp = 0.01, 0.02, and 0.03, respectively. The main advantage of this method lies in the ability to perform direct comparisons between individual wave events within random, directionally spread sea states with increasing steepness. As such, its application is critical in identifying the effects driven by increases in nonlinearity, the latter arising at second-order of wave steepness and above.
c. Numerical simulations
To take full advantage of the experimental method described above, numerical simulations were also performed using second-order random wave theory (SORWT) based upon Sharma and Dean (1981). These were generated using the same input conditions and output locations as in the experiments. This allows the direct superposition of experimental and numerical results. Figure 3 presents examples of time histories of the water surface elevation η(t) recorded on the centreline of the wave basin with direct comparisons to the predictions of SORWT. Figure 3a concerns a near-linear sea state (Sp = 0.01, kpd = 1.02, σθ = 10°) and shows very good agreement between the experimental and numerical results. This agreement indicates that second-order random wave theory is sufficient to describe the wave field and acts to validate the accuracy of the adopted wave generation. More importantly, the agreement in both space and time indicates that the wave field is not contaminated by any spurious waves or significant reflections, with further validation being provided in Karmpadakis et al. (2019). Figure 3b concerns the same time segment, but corresponds to a more nonlinear sea state (Sp = 0.02). Overall, the experimental and numerical results show good agreement, but some differences arise during the formation of the largest wave event. For example, the wave recorded at the last wave gauge, around t ≈ 964 s, is larger than its second-order counterpart. Karmpadakis et al. (2019) have attributed this increase to the effects of higher-order nonlinear interactions and discussed its implications for the short-term distribution of crest heights (see Fig. 1). In Fig. 3c, the sea state steepness has been further increased to Sp = 0.03. In this case, the wave event identified above is shown to be smaller than the SORWT prediction at all locations and arrives at the last wave gauge earlier. Considering that the surrounding wave field is described reasonably well by the numerical simulations, this decrease in the surface elevation indicates that the wave recorded in the experiment has broken. Clearly, this is not something that can be captured using second-order random wave theory.
Following a similar approach, Fig. 4 presents results arising in sea states with increasing steepness recorded at the central wave gauge. Figure 4a concerns time segments for sea states with Sp = 0.01, Sp = 0.02, and Sp = 0.03, with the experimental data again being compared to SORWT. In the less steep cases, the numerical results provide an accurate description of the wave field. However, discrepancies are apparent in the steepest case (Sp = 0.03). These are demonstrated in two ways. First, the crest height of the largest wave event in the experiment is larger than its SORWT counterpart. Second, the relative elevation of the wave troughs adjacent to the largest experimental wave crest is reduced compared to the second-order simulation. In explaining these changes, the effects of higher-order nonlinear interactions, arising at third order and above, need to be considered. At a third order of approximation, these interactions consist of both bound and resonant (or near-resonant) terms. The former contribute to the total surface elevation, but their magnitude becomes progressively smaller at higher orders of nonlinearity. In contrast, resonant interactions act to modify the free wave spectrum and induce changes in the amplitude and phasing of the individual wave harmonics. As such, their contribution is to increase the total crest height elevation and change the shape of (at least) the largest wave event presented in this figure. In Fig. 4b, a near-linear sea state (Sp = 0.01) is compared with a nonlinear (Sp = 0.03) and a highly nonlinear sea state (Sp = 0.06). While similar conclusions can be drawn for the first two steepnesses, it is clear that the surface elevations for Sp = 0.06 show marked differences. The height of the largest wave crest in the experiment is smaller than the corresponding second-order crest height. However, the same wave exhibits a larger (than SORWT) crest when a sea state with Sp = 0.03 is considered. The reduced crest heights recorded in the steepest sea states provide direct evidence of the dissipative effect of wave breaking.
Taken together, the methodology described above is used to provide temporal wave profiles from random experimental simulations that are influenced solely by the sea state steepness, with the alignment of the corresponding waves being clearly defined. By comparing these results with the predictions of second-order random wave theory, the effects of nonlinearity and wave breaking can be explicitly identified.
4. Discussion of results
While the study of individual wave events, such as those presented in section 3, is very informative, the focus of the present study lies in the characteristics of the largest waves. This is addressed by investigating their average shape (in time) in both field and laboratory measurements. To obtain these average wave shapes, a common approach is to extract a short time segment of the surface elevation η(t) around the largest crest heights. The (shortened) time histories are then shifted in time such that the maximum crest elevation occurs at t = 0 s and averaged. While this approach has been widely applied in the literature (Phillips et al. 1993b; Jonathan and Taylor 1997; Guedes Soares and Pascoal 2005; Whittaker et al. 2016), the exact number of waves being averaged varies between studies. In many field related studies, the 20 largest waves arising in sea states of 30-min duration are used, while in others some fraction of ηc/ση is preferred. As indicated in section 2, the theoretical limit of ηc/ση → ∞ cannot be applied in practice.
In an effort to select an appropriate (and consistent) number of individual waves to include in the averaging process, a large number of numerical simulations were performed using linear random wave theory (adopting the methodology outlined in section 3). To this end, 50 random seeds, each of 3-h duration (at field scale), were used to extract the largest waves corresponding to different percentiles, with the latter including the largest (0.1%, 0.2%, 0.5%, 1%, 2%, and 5%) waves in each seed. The inclusion of a larger number of waves inevitably leads to a downscaling of the resulting profile , since smaller waves are included in the averaging process. In contrast, the inclusion of only a small number of waves within the averaging process increases the statistical variability and (consequently) introduces deviations from the symmetric profile of a QD solution.
In seeking to define the optimal number of waves to be included within the averaging process, two metrics are defined. First, the ratio between the maximum of the average wave profile at each percentile and the maximum corresponding to the smallest percentile is considered. The second metric is the root-mean-square (RMS) error ϵrms between the average wave profiles and the scaled QD wave profile ηQD, with the scaling factor based upon the maximum crest height ηmax in each case. This second metric is defined by
where i corresponds to each observation and N is the total number of time steps in the time histories, t ∈ [−2Tp, 2Tp]. Figure 5 presents the values of these metrics for each percentile under consideration, with the two vertical axes having different scales. These show that the average maximum crest height decreases monotonically, with the inclusion of more (smaller) waves, while the minimum rms error ϵrms is observed for the 1% percentile. Specifically, ϵrms reduces toward the 1% percentile, as more waves are included, but then increases for larger percentiles. This indicates that the smaller waves added for increased percentiles violate the asymptotic assumptions of the QD wave profile. In such cases an alternative representation should be sought (Lindgren 1970, 1972). Using this guidance, the 1% of the largest waves is used when calculating the average wave shapes for the remainder of the present paper. This percentile corresponds to a ratio ηc/ση ≈ 3, which has been proposed by some earlier studies (Phillips et al. 1993a) and is justified theoretically by Cartwright and Longuet-Higgins (1956). It is also important to note that similar results regarding the statistical variability arise if the Lindgren variance is used as an alternative (Lindgren 1972), the latter quantifying the statistical variability when moving away from the largest crest height. However, the approach adopted herein relates directly to the experimental method and provides a simpler alternative.
Having selected the optimal number of waves to consider, the average profile of the largest waves can be readily calculated. While field observations generally provide the most accurate representation of realistic conditions, it is seldom possible to record sufficiently long time series in severe sea states. In this respect, laboratory experiments can be used to address the lack of data in the steepest sea states. However, the experimental generation of large waves in shallow water is extremely difficult due to unrepresentative energy losses by bed friction and the inherent limitations of wavemaking theory. Specifically, the latter limitations are related to the increased importance of the second-order difference terms as d reduces. To address these issues, the experimental and field datasets are used in a complementary manner to provide comparisons to the QD predictions.
These comparisons are shown in Fig. 6. In all cases the average wave profiles have been normalized by their maximum elevation ηmax. As such, the (vertical) deviations between the theoretical and measured wave profiles appear as differences in the depth of the adjacent wave troughs. In two examples (Figs. 6a,c), the light gray lines correspond to the individual measured wave profiles used within the averaging process—the first relating to laboratory data and the second to field data. Comparisons between these profiles show that the observed variability is similar in the two datasets. Figures 6a and 6b present comparisons between experimental results and the QD wave profiles for kpd = 1.22 and σθ = 10°. In the moderate sea state (Sp = 0.01) presented in Fig. 6a, the linear and second-order corrected QD wave profiles are closely aligned, given the limited nonlinearity, and agree very well with the measured data. In contrast, Fig. 6b considers the nonlinear sea state (Sp = 0.04) and shows that the measured profile deviates markedly from the theoretical predictions. These deviations are apparent both in the adjacent wave troughs and the front slope (∂η/∂t) of the largest wave, with the measured data exhibiting both a steeper gradient and a narrower crest. These observations indicate nonlinear contributions that are not captured by the QD models. However, it is important to note that the second-order correction to the QD wave profile provides notably better predictions than its linear counterpart.
Following a similar approach, the average wave profiles recorded in the field are also compared to theory. In defining the sea states to consider at each of the two locations, the data-binning methodology (section 3) was adopted. To achieve the largest possible homogeneity, the selection was based upon a maximum variation of ±5% in the sea state parameters (Hs, Tp, and T1), where T1 defines the mean wave period. Figures 6c and 6d relate to measurements from the deepest location (d = 45 m), while Fig. 6e relates to the shallowest location (d = 7.7 m). Interestingly, the sea states in Figs. 6c and 6d are characterized by a similar effective water depth (kpd ≈ 1.4) but different steepnesses Sp = [0.018, 0.027]. In the former case, it can be seen that the linear QD wave profile does not agree well with the measurements. In contrast, the second-order corrected QD wave profile closely follows the measured average profile. Similar conclusions arise when the steeper case is considered (Fig. 6d). However, the improvement provided by the second-order correction is not sufficient to approximate the measured profile. Considering that these correspond to steeper sea states the explanation lies in the effects of nonlinearity arising above second order. In addition, wave breaking will also be present; the extent to which it influences the results is examined in what follows. While these results are in agreement with the findings of Guedes Soares and Pascoal (2005), most studies in the literature (section 2) do not identify such deviations, with the absence of reliable data in sufficiently steep sea states being the most probable explanation. More importantly, the results presented in Fig. 6e exhibit clear nonlinear behavior with steep front and back wave slopes (∂η/∂t), a sharp wave crest and flat wave troughs—all of them being indicative of the small effective water depth (kpd ≈ 0.75). As such, the linear QD wave profile presents widely different predictions, the second-order correction being outside its range of validity (Tayfun 2006b). This last example is indicative of the vast majority of the results obtained in this water depth, irrespective of sea state steepness. This raises significant concerns regarding the applicability of the QD wave approach in very shallow effective water depths.
With the laboratory data used to extrapolate findings into steeper sea states that have either not been encountered in the field or for which insufficient data is available, it is crucial to verify that the two independent data sources provide the same results in those cases where there is an overlap between the two. Figure 7 presents an example of a direct comparison between the average profiles of the largest waves recorded in the laboratory and the field for similar sea states. To achieve this, a nearest neighbor algorithm was employed to identify the sea states from the deepest field location (d = 45 m) that matched experimental cases in terms of the nondimensional parameters (Sp, kpd). A requirement for T1/Tp ≈ 0.82 (DNV 2010) was also enforced to obtain sea states with peak enhancement factors similar to the experiments (γ = 2.5). The observed agreement indicates that the experimental measurements can accurately describe the conditions encountered in the field. More importantly, the fact that this agreement is observed in a sea state with Sp = 0.03 provides additional confidence because noteworthy nonlinear effects have been observed in the crest height statistics for similar sea state conditions (Karmpadakis et al. 2019).
To elaborate on this, the average wave shapes arising in experimentally generated sea states with increasing steepness are further investigated on Fig. 8. The conditions correspond to kpd = 1.22 and σθ = 10° and have been normalized with respect to the standard deviation of the free surface ση and the mean period T1. Figure 8a presents results corresponding to Sp = 0.01, 0.02, and 0.03, while Fig. 8b relates to sea states with Sp = 0.04, 0.05, and 0.06. It is clear that in the former case, an increase in sea state steepness leads to more nonlinear average wave profiles, with the nonlinearities being manifested as increases in the crest elevation, steepening of the wave slopes (∂η/∂t), and flattening of the wave troughs. In contrast, in the steepest sea states (Fig. 8b) the reverse trend is observed, particularly considering the maximum crest height. Indeed, increases in the sea state steepness lead to a reduction in the maxima of the average profiles. Importantly, the average profiles in Fig. 8a show little or no evidence of horizontal asymmetry, while some minor asymmetries are observed in Fig. 8b, with the wave troughs preceding the largest crest being marginally shallower than those that follow.
If these results are examined in isolation, the aforementioned observations could largely be attributed to bound nonlinear interactions and the limiting influence of wave breaking, with the former used to justify the nonlinear changes observed in Fig. 8a and the latter the energy dissipation in Fig. 8b. As a consequence, an observed agreement with a weakly nonlinear QD profile would not seem unreasonable, as suggested in the literature (section 2). However, any interpretation that nonlinear resonant (or near-resonant) effects are not significant is misleading. This is because the population of the largest measured waves will likely include both breaking and nonbreaking waves, with their characteristics potentially averaging out important nonlinear changes. To illustrate this, the distributions of crest heights ηc arising in all sea states (Sp = 0.01–0.06) are considered in Fig. 9. For each sea state, these are based upon a zero-crossing analysis of the time histories of each individual seed. Given that the crest heights in each seed (of the same sea state) represent random samples of the same population, they can be combined into a single larger sample, ranked in descending order and plotted against their probability of exceedance Q. In this way, results with much lower probabilities of exceedance, lying at the tail of the distribution, can be examined. Subsequently, the five largest crest heights arising in the second-order simulation for Sp = 0.01 are identified and correlated to their corresponding wave events in the laboratory measurements. As the steepness of the sea states is scaled-up these wave events are tracked taking advantage of the (time) alignment of the coupled numerical and experimental datasets (section 3). These are then superimposed on Fig. 9, with their corresponding probability of exceedance Q being calculated on the basis of their rank at each sea state.
In examining these results, it is clear that the waves that exhibit the largest crest height in the near-linear case Sp = 0.01 do not maintain their rank (as the largest) in the steeper cases. In fact, they are redistributed toward larger probabilities of exceedance from Sp = 0.02 onward. The range of probabilities which they occupy is also clearly broadening as the sea state steepness is increased. In the steepest case their probabilities range from 10−1 to 3 × 10−3 and only one is still ranked in the largest five waves in the three sea states with Sp > 0.03. It is worth keeping in mind that if the corresponding statistics were generated on the basis of SORWT results, these waves would maintain their rank, since no energy transfers or wave breaking are incorporated. For the experimental results, the migration toward larger probabilities of exceedance is justified by the occurrence of wave breaking and the associated wave energy dissipation. This implies that eventually the largest waves in a linear (or second-order) simulation are more susceptible to wave breaking and will not remain the largest as the steepness of the sea state is increased. As a result, their place in the ordered set of crest heights will be occupied by a different wave which will correspond to a smaller linear (or second order) equivalent. This result has far-reaching implications with respect to the interpretation of crest height (or wave height) distributions. Generally, it is well established that the occurrence of wave breaking leads to crest height reductions and, consequently, waves moving toward larger probabilities of exceedance. However, thus far this movement was considered to be relatively small; the largest waves were considered to remain in the tail of the distribution despite the reductions, as discussed by Battjes and Groenendijk (2000). In view of the results presented herein it is clear that this is not the case, with the breaking waves being characterized by probabilities that are typically associated to small nonbreaking waves. In this respect, the novelty of the adopted approach of coupling numerics and experiments, as well as sea states with different steepness, becomes apparent and is shown to be very insightful.
With the aim to further clarify these points, this approach is extended to define separate populations of breaking and nonbreaking waves. More specifically, the total population of the normalized crest heights in the SORWT simulations is partitioned into bins of width for . The corresponding normalized crest heights from the laboratory simulations are identified in the same manner as above and the ratio between the two is calculated on a wave by wave basis. As such, r > 1 means that the measured crest height is larger than SORWT, while the opposite is true for r < 1. This ratio is then used to detect whether an individual (zero crossing) wave is being amplified (by nonlinearity) or dissipated (by breaking). To avoid the influence of small fluctuations in the measured water surface a “buffer” of 5% in the calculated values for r is imposed. Therefore, a wave is labeled as breaking if r < 0.95, and amplified if r > 1.05—a sensitivity analysis on the width of the buffer zone showing that no qualitative changes arise when different bands are considered. In this context, the term “breaking” refers to waves that are exhibiting some level of dissipation with respect to the predictions of SORWT. In this sense, these include waves that have already broken when they arrive at the measuring location. Therefore, this criterion is different to the classic geometric, kinematic, and dynamic criteria (Babanin 2011; Perlin et al. 2013) that identify incipient breaking and should not be interpreted as such.
The aforementioned definitions are applied to the partitioned data to derive the conditional probabilities of amplification Pa and breaking Pb as the ratio between the number of waves in each population over the total number of waves contained in each bin. These probabilities are shown in Fig. 10 for all the sea states with kpd = 1.22 and σθ = 10°. Considering the probability of amplification in Fig. 10a, it can be seen that as increases the probability of a (second-order) wave being amplified reduces across all sea state steepnesses. Moreover, as the sea state steepness increases, the probability of amplification for the smallest (second-order) waves is also increased, while it is rapidly reduced for the largest waves. Considering the probability of breaking in Fig. 10b, the opposite trends are observed; the largest (second-order) waves are progressively more likely to break as and Sp increase. Wave breaking is observed even in the most moderate sea states with small Sp. In undertaking this analysis, it is worth noting that data bins with fewer than 5 points have been excluded in the calculation of both probabilities. When the two plots are examined together, it becomes clear that the reason why the largest (second-order) waves are not further amplified is because they are breaking, for example, for Sp = 0.06. These results justify the observations discussed earlier with respect to tracking the relative rank of crest heights in the total population of waves (Fig. 9). This has clear implications when it comes to selecting individual wave events for a wide range of design applications or the calculation of extremal statistics. The main conclusion is that the largest wave in a fully nonlinear sense will not necessarily stem from a wave that is found in the tail of a linear or second-order crest height distribution. In contrast, the largest waves in the steepest sea states may well correspond to much smaller linear or second-order waves.
In effect, the results presented in Fig. 10 clearly show that across a wide range of breaking and nonbreaking waves will be present in sea states of varying steepness. The question that immediately arises is whether the average shapes of the largest waves in these two populations have the same characteristics. To address this, the same approach of wave classification is employed to investigate the largest 1% of waves, with the latter referring to the experimentally measured data instead of SORWT simulations. After classifying each wave as breaking and nonbreaking the wave profiles of each population are extracted from the experimental and numerical time histories, time-shifted and averaged. Given that the waves included in the averaging process correspond to the same events in the numerical and experimental datasets, the comparisons of their average profiles are deterministic. As such there is significantly more information carried than simply comparing averages from uncorrelated samples. Such comparisons have not previously been presented in the literature.
First, the average profiles of the largest nonbreaking waves are considered. Figure 11 presents comparisons between the corresponding linear, second-order and experimental wave profiles. Considering data with kpd = 1.22 and σθ = 10°, Figs. 11a and 11b show results for a very moderate (Sp = 0.01) and a steep (Sp = 0.04) sea state, respectively. In the former, the largest nonbreaking waves agree well with the second-order results and are only marginally larger than the linear predictions. This confirms that the waves are weakly nonlinear, with the dominant nonlinear effects arising at a second-order of wave steepness. In contrast, the comparison in the steeper sea state shows two important effects. First, the maximum crest heights observed experimentally are larger than the second-order predictions. The magnitude of this difference is at least comparable to the difference between LRWT and SORWT, suggesting that it cannot be justified solely by higher-order bound contributions, with the magnitude of the latter being one order of magnitude smaller (Fedele et al. 2016). Considering the deterministic nature of these comparisons, this observation supports the importance of resonant and near-resonant interactions (Slunyaev et al. 2002; Fernandez et al. 2014) in these random records. This is further supported by the amplifications in the crest height statistics above the (second-order) Forristall (2000) model for the same data discussed by Karmpadakis et al. (2019). Second, when considering the depth of the adjacent wave troughs another important difference is observed. The following wave trough in the experimental measurements appears to be shallower than the corresponding SORWT and LRWT prediction. This indicates the aforementioned higher-order interactions act to change the shape of the waves in a way that bound interactions cannot. In drawing an analogy with the study of focused wave groups, similar increases in the following wave troughs have been reported by Johannessen and Swan (2003), among others. In that respect the nonlinearities are manifested as a movement of the largest wave event toward the front of the group leading to a trough asymmetry. Conversely, if the measured data are considered in isolation, it is obvious that their profile is not symmetric but has a front-back asymmetry, with the following trough being deeper than the preceding. This trend is observed in all the sea states (Sp > 0.02) in this water depth (kpd = 1.22). In addition, two representative examples are included for a deeper (kpd = 1.53) and shallower (kpd = 1.02) sea state, both with Sp = 0.03 in Figs. 11c and 11d. In examining these examples the same conclusions are reached regarding nonlinear changes in the wave crest and wave shape thereby extending these findings to a wider range of effective water depths.
The results presented in Fig. 11 have addressed the average shape of the largest nonbreaking waves. In performing the same analysis on the population of breaking waves, their average shapes can again be extracted. These are now compared to the average wave profiles from the nonbreaking population to illustrate the differences between them. Figures 12a–d present these comparisons for sea state steepnesses between Sp = 0.03 and Sp = 0.06 for kpd = 1.22. In comparing the profiles of the breaking and nonbreaking waves, two important observations arise. First, the broken waves are characterized by smaller maximum crest heights, a clear manifestation of energy dissipation. Second, the breaking wave profiles exhibit a horizontal asymmetry that is opposite to the asymmetry of the nonbreaking waves, the observation being consistent across all steepnesses. This means that the wave troughs preceding the largest crest elevations are deeper than the following wave troughs for the largest broken waves. Noting that these results represent a breakdown of the average wave profiles shown on Fig. 8, the lack of significant asymmetries observed in the latter can be justified. In effect, the two different types of asymmetries for breaking and nonbreaking wave populations largely cancel out, leading to a more symmetric wave profile when all waves are considered (for the largest 1%). This does not, however, imply that a weakly nonlinear QD wave profile is appropriate, rather that important effects have been cancelled out by addressing two very different wave populations.
To verify that the interpretation of asymmetry presented so far is indeed important, we examine the statistics of the geometry of the largest waves. In this way, the results arising from the analysis of average wave profiles can be generalized. To achieve this, an asymmetry parameter β is defined as
where ηpt and ηft are the preceding and following trough depths, respectively. As such, if β < 1 a profile is exhibiting the characteristic nonbreaking asymmetry, while the opposite is true for β > 1. This metric is used to examine the individual wave profiles corresponding to the largest 1% of waves arising in each individual sea state. More specifically, the asymmetry parameter is calculated separately for the total, breaking and nonbreaking populations of waves. Figure 13 presents the values of β for all sea state steepnesses in kpd = 1.22, with Fig. 13a relating to σθ = 10° and Fig. 13b relating to σθ = 20°. In this respect, the findings of this analysis are extended toward sea states with different directional spreading. Additionally, the 95% confidence intervals have been added on the results of the total wave population as an indication of the variability in the estimates of β. In interpreting the results on Fig. 13 it is clear that in both cases the nonbreaking wave population has β < 1, the breaking population has β > 1 and that the total population lies between the two fluctuating around β = 1. This is exactly the same behavior as discussed with respect to the average wave shapes and provides a significant validation of the results presented earlier. Moreover, a secondary trend in the values of the asymmetry parameter β for the total and nonbreaking populations can be observed. This refers to a decreasing trend in β for increasing steepness toward a local minimum (e.g., Sp = 0.04 in Fig. 13a) followed by an increasing trend for larger steepnesses. This is implies that the nonlinear effects have a larger influence for increasing steepness until a critical steepness is reached. Beyond this point the effects of wave breaking become progressively more important, with the latter leading to some degree of saturation. While this secondary trend is less clear for σθ = 20° it certainly coincides with the observed nonlinear effects in the crest height statistics which obtain a maximum for Sp = 0.04 (Karmpadakis et al. 2019).
Interestingly, comparisons between Figs. 14a and 14b also allow some comments to be drawn concerning the role of directionality. For the nonbreaking wave population (β < 1) it is clear that the higher-order amplifications remain significant, but are perhaps rather smaller with increases in the directional spread. Furthermore, in the breaking wave population (β > 1) an increase in the directional spread leads to smaller β values suggesting that breaking is rather less important, particularly for lower Sp values. These effects are consistent with the crest height distributions reported in Latheef and Swan (2013), Latheef et al. (2017), and Karmpadakis et al. (2019), following the expected reduction in the individual wave steepness with increasing directionality.
Finally, using these results an answer is provided as to why some studies of field data report symmetric average wave profiles even in relatively severe sea state conditions (Christou and Ewans 2014; Gemmrich and Thomson 2017), while others have recorded asymmetries (Myrhaug and Kjeldsen 1986; Guedes Soares et al. 2004). The extent to which either type of asymmetry can be identified critically depends on the competing effects of nonlinear amplifications and the dissipative effects of wave breaking. Clearly, the method adopted in the experimental part of this study cannot be applied to field measurements where coupled simulations cannot be generated. However, the asymmetry parameter β can be used to assess whether any statistically significant trends are apparent in the present field data. To achieve this, the normalized zero up-crossing and down-crossing wave heights arising in all available sea states are sorted and their probability of exceedance Q calculated. Using the ratio of the (ordered) wave heights the asymmetry parameter is calculated and plotted against Q for the two available water depths on Fig 14. Considering d = 45 m, it can be seen that β fluctuates consistently around 1 for the vast majority of the data with some increases observed for the largest wave heights located in the tail of the distribution. In contrast, for d = 7.7 m the asymmetry parameter is consistently larger than 1 indicating a strong presence of wave breaking. In interpreting these findings, the behavior of β for the deeper location indicates the presence of both breaking and nonbreaking waves which effectively cancel out their effects. This is consistent with the expected behavior in intermediate and deep water locations and explained above (Fig. 10). In contrast, for the shallowest location the observed asymmetry implies that wave breaking is the dominating process. Indeed, a recent analysis of wave height statistics at this location (Karmpadakis et al. 2020) has shown that there is a very strong presence of breaking waves, primarily driven by depth limitations. Since the majority of the largest waves are breaking, the observed values of β > 1 are consistent with the analysis presented herein. Clearly, this example relates to the general population trends at each field location and not to the characteristics of individual sea states, with the latter being a natural extension of the present work.
5. Concluding remarks
The present paper has investigated the characteristics of the largest waves arising in random, directionally spread sea states in finite water depths. This has been achieved using field, experimental and numerical data. The average profiles of the largest waves for a wide range of sea states have been compared to the theory of quasi-determinism (QD). While this undoubtedly provides a marked improvement over the “equivalent” regular waves commonly adopted in engineering design, it is not without its limitations. Specifically, comparisons to linear QD wave profiles show good agreement for near-linear sea states. With an increase in the sea state steepness, the second-order corrected QD wave profile incorporates some of the nonlinearity of the wave profile and provides a better approximation. However, very steep sea states or sea states in shallow water show significant departures from the theoretical predictions.
When considering the total population of the largest 1% of waves in sea states with varying steepness, it was found that their average profile was either (horizontally) symmetric or characterized by very small asymmetries between the wave troughs adjacent to the largest crest. This is inconsistent with the observations of fully nonlinear focused wave groups that develop strong asymmetries due to the nonlinear physics arising at third-order and beyond. To address this discrepancy, a novel coupling approach was employed to generate random time-histories of directionally spread seas that are phase-aligned for increasing sea state steepnesses. These data were generated both experimentally and numerically; the latter used linear and second-order random wave theory. Taking advantage of this coupling, the total population of the largest (1%) waves was subdivided into two smaller populations of nonbreaking and breaking waves. When the average profiles of the breaking and nonbreaking waves are examined separately it was shown that they develop opposite asymmetries. In many sea states these two asymmetries effectively cancel out. When the total population of large waves is considered, this produces a symmetric wave profile and the inappropriate conclusion that a weakly nonlinear QD wave profile is relevant. Interestingly, the asymmetric profile observed for the largest nonbreaking waves has the same characteristics as that of nonlinear focused waves. Importantly, the higher-order nonlinear wave-wave interactions that have been shown to produce significant amplifications in crest height statistics (Karmpadakis et al. 2019) have been shown to induce characteristic changes in the shape of the largest nonbreaking waves, a result that has not previously been established from random wave records.
The coupling of the phase-aligned data has also allowed the tracking of (the same) individual waves in sea states of different steepnesses. This has shown that the largest waves arising in a linear (or second-order) simulation do not maintain their rank (as largest) in a fully nonlinear (experimental) simulation. While some mobility in the probability domain was expected due to energy dissipation by wave breaking, these waves were expected to remain at the tail of the crest height (or wave height) distributions. However, the present results show that this is not the case, emphasizing the importance of both nonlinear evolution (above second-order) and, particularly, wave breaking. This has clear implications in the consideration of extremal statistics of crest heights, wave heights and the associated wave shapes. Building upon these data, a first attempt is made to quantify the conditional probabilities of amplification and wave breaking based upon the magnitude of the underlying linear (or second-order) waves. Again, this emphasizes the importance of wave breaking when seeking to describe the individual waves defining the tail of the crest height distribution in steep sea states. Further work to quantify the variation of this effect with effective water depth, directional spread, and spectral bandwidth is presently ongoing.
The authors are grateful to the following sponsors of the LoWiSh (Limits on Waves in Shallow Water) JIP for funding part of this research: Shell, Mærsk, BP, Total, Exxon, Woodside, and Equinor. The authors would also like to thank Shell and Mærsk for providing access to the field data, whilst confirming that the views expressed herein are entirely their own.
Second-Order Interaction Kernels