Impact statement
The fundamental understanding of wave–seabed interactions is crucial for the establishment of accurate extreme wave statistics and deterministic wave prediction in such water-depth-varying zones. With the increase in wind intensities resulting from global warming and respective change in climate dynamics, it is anticipated that the frequency of rogue wave events, occurring in particular offshore and coastal areas, will increase in the future. It is therefore essential to fully understand the formation and complex dynamics of large-amplitude waves in varying water depth conditions, for instance, when deep-water wave groups are transitioning to shallow-water areas. Moreover, quantifying the role of nonlinearity in such wave shoaling or focusing processes is, among other things, decisive for the estimation of associated wave loads on coastal structures and their impact on the shoreline components.
Background
Ocean water depth is a key parameter in the modeling of waves. In fact, it affects the dispersion relation and characteristic shape features. The change of water depth can be either localized in the form of seamounts and submerged volcanic islands or continuously varying such as continental shelves, encasing nearshore sandbars (Dingemans, Reference Dingemans1997; Svendsen, Reference Svendsen2005), as exemplified in Figure 1.
Figure 1. A schematic exemplifying typical depth variations from a ridge and continental shelf.
Extreme wave formation, being the specific subject of interest of this review article, is not restricted to a particular water depth. In fact, it is known that such large-amplitude waves have been widely reported and recorded not only offshore, but also in coastal zones (Kharif et al., Reference Kharif, Pelinovsky and Slunyaev2008; Dudley et al., Reference Dudley, Genty, Mussot, Chabchoub and Dias2019; Didenkulova, Reference Didenkulova2020; Gemmrich and Cicon, Reference Gemmrich and Cicon2022). Recent studies suggest that extreme wave conditions are likely to increase as a consequence of climate change (Meucci et al., Reference Meucci, Young, Hemer, Kirezci and Ranasinghe2020), even though there are some uncertainties in the modeling and hindcast projections which must be considered (Morim et al., Reference Morim, Wahl, Vitousek, Santamaria-Aguilar, Young and Hemer2023). Regardless of either the superposition principle or wave instability as the underlying focusing mechanism at play, the role of nonlinearity in the wave shoaling transformations is indisputable. The impact of such nonlinear effects is enhanced with the decrease of water depth as a result of Stokes bound harmonics accentuation and contribution to the change of wave shape profile and celerity, see Mei et al. (Reference Mei, Stiassnie and Yue1989), Osborne (Reference Osborne2010), Babanin (Reference Babanin2011).
While the role of modulation instability (MI), which is triggered as a result of four-wave quasi-resonant interaction and third-order nonlinear effects, has been intensively studied in deep water for decades (Waseda, Reference Waseda, Young and Babanin2020), it is only recently that systematic experimental progress has been accomplished in characterizing key statistical features of waves propagation and the role of high-order nonlinear effects when either isolated wave groups or irregular waves propagate atop depth transitions. The presence of MI (Zakharov, Reference Zakharov1968) in an irregular wave field can be quantified by computing the deviation of the surface elevation probability distribution’s fourth spectral cumulant, i.e. the kurtosis, from the value of three, which is typical for a Gaussian process (P. A. E. M. Janssen, Reference Janssen2003; Mori et al., Reference Mori, Onorato and Janssen2011).
Such fundamental understanding is not only crucial to improve wave modeling and prediction, but also to better assess wave loads on coastal installations (Li et al., (Reference Li, Tang, Li, Draycott, van den Bremer and Adcock2023) and provide an accurate nonlinear depth-inversion framework (Martins et al., Reference Martins, Bonneton, De Viron, Turner and Harley2013).
This review paper puts an emphasis on essential recent progress in nonlinear wave modeling and the occurrence of extreme conditions when waves propagate over different types of variable bottom topographies. These advances can be categorized either by the dominance of second-order effects of wave approximation or by the inclusion of third-order contributions to accurately describe the extreme wave dynamics. We will also discuss the essential and complementing laboratory experiments, comprising simplified and complex bathymetries, which have been conducted either for model validation purposes or to drive respective theoretical and numerical progress.
Physical mechanisms and modeling
Extremely large wave events, also known as rogue, freak or monster waves, are characterized by two main features: their sudden appearance out of nowhere and strikingly amplified steepness compared to their surroundings, thus posing a great risk to the safety and reliability of offshore structures as well as coastal management and protection in nearshore waters (Bitner-Gregersen and Gramstad, Reference Bitner-Gregersen and Gramstad2015). Indeed, there have been documented accidents caused by extreme waves in both intermediate and shallow water (Chien et al., Reference Chien, Kao and Chuang2002; Didenkulova and Anderson, Reference Didenkulova and Anderson2010; Gramstad et al., Reference Gramstad, Zeng, Trulsen and Pedersen2013). In order to characterize rogue wave events, a few useful proxies are commonly used, including skewness and kurtosis which correspond to the third and fourth moment of surface elevation, respectively (Janssen, Reference Janssen2003; Dysthe et al., Reference Dysthe, Krogstad and Müller2008; Mori et al., Reference Mori, Onorato and Janssen2011). These proxies are used to measure the degree of deviation from the Gaussian random process, thus indicating the occurrence probability of rogue waves.
The properties of surface gravity waves are affected by a seabed in intermediate or shallow water, leading to a complex interplay with wave nonlinearity, compared with deep-water waves. Noticeably, the MI appearing at the third-order in wave steepness approximation has been recognized as a possible mechanism for the formation of rogue waves in deep water (Benjamin and Feir, Reference Benjamin and Feir1967; Zakharov, Reference Zakharov1968). It can be stabilized for small-amplitude and long-crested waves in water regions for
$ kh\hskip0.35em \lesssim \hskip0.35em 1.36 $
, where
$ k $
and
$ h $
denote the characteristic wavenumber and local water depth, respectively (Johnson, Reference Johnson1977). The threshold value of
$ kh\approx 1.36 $
is essential to the understanding of rogue waves and the evolution of nonlinear energy transfers in finite and shallow water (Janssen and Onorato, Reference Janssen and Onorato2007). That said, for finite-amplitude waves, Benney and Roskes (Reference Benney and Roskes1969), McLean (Reference McLean1982), Toffoli et al. (Reference Toffoli, Fernandez, Monbaliu, Benoit, Gagnaire-Renou, Lefevre, Cavaleri, Davide Proment, Stansberg, Waseda and Onorato2013) showed that the MI, as a combination of quartet resonant wave interaction and wave nonlinearity, can occur even in water regions where
$ kh\lesssim 1.36 $
, when the waves are subject to oblique perturbations. The directional spreading and wave dissipation in random sea states in uniform and finite water depth have been found to lead to considerable deviations from normal statistics (see, e.g., Fernandez et al., Reference Fernandez, Onorato, Monbaliu, Toffoli, Pelinovsky and Kharif2015; Karmpadakis et al., Reference Karmpadakis, Swan and Christou2019, among others), while numerical simulations based on the high-order spectral method (HOSM) have indicated the emergence of significant deviations from normal statistics in random directional sea states in the absence of breaking dissipation and independently of the significance of directional spreading of wave spectra in the same uniform depth conditions (Toffoli et al., Reference Toffoli, Benoit, Onorato and Bitner-Gregersen2009).
Compared with an intermediate uniform depth, the underlying fundamental physics of surface waves experiencing an additional depth decrease becomes more complex, attributing to the linear refraction and diffraction and the interaction between a varying seabed and wave nonlinearity (Kirby and Dalrymple, Reference Kirby and Dalrymple1983; Tsay, Reference Liu and Tsay1984; Dingemans, Reference Dingemans1997). The coupled effects of wave nonlinearity and a varying bathymetry are the focus of this section.
As noted, wave transformation in variable water regions has been intensively studied in the last decade; these studies are mainly arising from new findings associated with the increased likelihood of extremely large wave events in such alternating depth regions. In their experimental observations, Trulsen et al. (Reference Trulsen, Zeng and Gramstad2012) report both a non-homogeneous distribution of skewness and kurtosis of surface displacement and their anomalous behavior in the neighborhood of the water region atop a depth decrease. This suggests an enhanced occurrence probability of rogue waves in this particular region. Similar findings have been reported in other theoretical studies (Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc), numerical simulations (Sergeeva et al., Reference Sergeeva, Pelinovsky and Talipova2011; Zeng and Trulsen, Reference Zeng and Trulsen2012; Gramstad et al., Reference Gramstad, Zeng, Trulsen and Pedersen2013; Viotti and Dias, Reference Viotti and Dias2014; Ducrozet and Gouin, Reference Ducrozet and Gouin2017; Lawrence et al., Reference Lawrence, Trulsen and Gramstad2021; Lyu et al., Reference Lyu, Mori and Kashima2021) and later-on experimental observations (Ma et al., Reference Ma, Dong and Ma2014; Bolles et al., Reference Bolles, Speer and Moore2019; Kashima and Mori, Reference Kashima and Mori2019; Zhang et al., Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019; Trulsen et al., Reference Trulsen, Raustøl, Jorde and Rye2020; Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) in a large range of water depth, as will be discussed in the next section. The numerical simulations based on the standard one-dimensional Boussinesq equations, carried out by Gramstad et al. (Reference Gramstad, Zeng, Trulsen and Pedersen2013), suggest that the non-homogeneous wave statistics can only be observed for waves propagating from a deeper to shallower water region, but not vice versa. This finding is similar to Armaroli et al. (Reference Armaroli, Gomel, Chabchoub, Brunetti and Kasparian2020), which concludes that the MI for long-crested waves propagating atop a slowly increased water depth can be stabilized subject to nonlinear evolution, suggesting a possible increase in the lifetime of unstable wave groups, when the water level experiences a depth increase. The features of extreme waves in a varying water region are in principle complex as they are altered by a number of physical parameters such as the non-dimensional wave depth
$ kh $
; the “mildness” of the depth variation relative to the change of wavelength; Ursell number, which measures the degree of the wave nonlinearity relative to a local water depth; directional spreading; the profile shape of a varying bathymetry; and the difference and ratio of water depths (Sergeeva et al., Reference Sergeeva, Pelinovsky and Talipova2011; Zeng and Trulsen, Reference Zeng and Trulsen2012; Viotti and Dias, Reference Viotti and Dias2014; Ducrozet and Gouin, Reference Ducrozet and Gouin2017; Kashima and Mori, Reference Kashima and Mori2019; Zheng et al., Reference Zheng, Lin, Li, Adcock, Li and van den Bremer2020; Kimmoun et al., Reference Kimmoun, Hsu, Hoffmann and Chabchoub2021; Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc; Lawrence et al., Reference Lawrence, Trulsen and Gramstad2022). The location where the largest probability of extreme waves atop a varying bathymetry may occur has also been found to coincide with the one where the monochromatic surface waves start to break as the waves steepen (Draycott et al., Reference Draycott, Li, Stansby, Adcock and van den Bremer2022). Different from the aforementioned findings, the local peak of kurtosis and skewness near the top of a mildly shoaling slope was not reported in Zeng and Trulsen (Reference Zeng and Trulsen2012) using numerical simulations, confirmed by Lawrence et al. (Reference Lawrence, Trulsen and Gramstad2021). This suggests an enhanced number of extreme waves in a varying water region requires the bathymetry to not vary in an extremely mild manner.
A few fundamental physical mechanisms for the formation of extremely large wave events over depth transitions have been proposed in the last decade (Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc). The second-order nonlinearity dominant mechanisms are first highlighted. In agreement with the second-order dominant physics, as has been pointed out by Gramstad et al. (Reference Gramstad, Zeng, Trulsen and Pedersen2013), these are referred to as the processes in which the underlying physics is considered and approximated up to the second-order in wave steepness. A physics-based statistical model is derived by Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) based on a deterministic wavepacket model (Foda and Mei, Reference Foda and Mei1981; Massel, Reference Massel1983; Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc). As weakly nonlinear waves propagate over an intermediate uniform depth, it has been well known that the waves forced by the second-order nonlinearity are bound (or locked) as they do not obey the linear dispersion relation, see for instance Phillips (Reference Phillips1960), Dalzell (Reference Dalzell1999) and Li and Li (Reference Li and Li2021) among others. Indeed, a second-order, three-dimensional, finite-depth wave theory can well interpret in-situ measurements of short-crested wind waves, which are observed to cause a setup instead of setdown below large wave groups (Toffoli et al., Reference Toffoli, Monbaliu, Onorato, Osborne, Babanin and Bitner-Gregersen2007). In contrast, the statistical model proposed by Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) accounts for the complementary physics of the nonlinear forcing of free waves, attributing to the complex interaction between the second-order bound waves and a varying seabed. Both the additional physics and the statistical model by Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) have been validated by rigorous theoretical derivations as well as numerical and experimental observations. We refer to Foda and Mei (Reference Foda and Mei1981), Massel (Reference Massel1983), Ohyama and Nadaoka (Reference Ohyama and Nadaoka1994), Monsalve Gutiérrez (Reference Monsalve Gutiérrez2017), and Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) for more details. The additionally released free waves carry energy and propagate at a different speed from the bound waves responsible for their generation. The differences in the propagation speed of waves lead to their separation at a distance sufficiently far from the top of depth transitions, thus leading to non-homogeneous wave features (Massel, Reference Massel1983; Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc; Draycott et al., Reference Draycott, Li, Stansby, Adcock and van den Bremer2022). We would also like to stress that another physics-based model has been derived by Majda et al. (Reference Majda, Moore and Qi2019) for shallow-water extreme waves experiencing depth transitions. It is based on truncated Korteweg–de Vries equations and statistical matching conditions of wave fields before and after the depth transition. Both Majda et al. (Reference Majda, Moore and Qi2019) and Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) assume quasi-Gaussian statistics processes for waves on the deeper (constant-water) side of the depth transition.
A second-order statistical non-Gaussian model has been recently derived by Mendes et al. (Reference Mendes, Scotti, Brunetti and Kasparian2022) and Tayfun and Alkhalidi (Reference Tayfun and Alkhalidi2020) with respect to the wave heights and free surface elevation, respectively, for waves atop a local intermediate depth transition. Mendes et al. (Reference Mendes, Scotti, Brunetti and Kasparian2022) neglect the second-order subharmonic bound waves, and this finding has been extended in the following work by Mendes and Kasparian (Reference Mendes and Kasparian2022) to allow for the effects of a varying seabed slope. In contrast to Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc), the statistical models of non-Gaussianity neglect the second-order subharmonic (bound and free) waves, the complex interaction between second-order superharmonic bound waves, a varying seabed and the effect of wave reflection. The non-homogeneity of the wave statistical features predicted by the non-Gaussian models originates from a non-constant depth, meaning that the predicted statistical wave features remain invariant with the space if the water is uniform in a local region. This suggests that the model is expected to fail when accounting for the local peaks of skewness and kurtosis near the flat top region of depth transitions. The non-homogeneity of skewness and kurtosis of the surface elevation has particularly been investigated in a number of papers, for example, Trulsen et al. (Reference Trulsen, Zeng and Gramstad2012), Zeng and Trulsen (Reference Zeng and Trulsen2012), Ducrozet and Gouin (Reference Ducrozet and Gouin2017) and Zheng et al. (Reference Zheng, Lin, Li, Adcock, Li and van den Bremer2020).
It shall be noted that the second-order-based mechanisms are in general insufficient for the predictions of kurtosis evolution as the combined effect of the linear waves and third-order nonlinearity cannot be considered. These higher-order effects play a considerable role in the deviation from Gaussian statistics (Janssen, Reference Janssen2014) and are discussed next.
The mechanism of out-of-equilibrium dynamics of wave fields has been initially discussed by Viotti and Dias (Reference Viotti and Dias2014), and thereafter by a number of works, for example, the review by Onorato and Suret (Reference Onorato and Suret2016) and Trulsen (Reference Trulsen, Velarde, Tarakanov and Marchenko2018). It is referred to as the process of re-adjusting wave fields from one equilibrium to a new one due to local changes in the environmental conditions, for example, varying bathymetries (Viotti and Dias, Reference Viotti and Dias2014; Zhang et al., Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019; Lawrence et al. Reference Lawrence, Trulsen and Gramstad2021, Reference Lawrence, Trulsen and Gramstad2022; Zhang and Benoit, Reference Zhang and Benoit2021; Zhang et al., Reference Zhang, Ma, Tan, Dong and Benoit2023), non-uniform currents (Hjelmervik and Trulsen, Reference Hjelmervik and Trulsen2009; Onorato and Suret, Reference Onorato and Suret2016; Zheng et al., Reference Zheng, Li and Ellingsen2023) or the sudden appearance of a ship (Molin et al., Reference Molin, Kimmoun, Remy and Chatjigeorgiou2014). The out-of-equilibrium dynamics of wave fields mainly arise from the quasi-resonant wave interaction at third-order in nonlinearity, leading to the deviation of statistical properties of surface elevation from Gaussian statistics (Janssen, Reference Janssen2003; Onorato and Suret, Reference Onorato and Suret2016; Tang et al., Reference Tang, Xu, Barratt, Bingham, Li, Taylor, van den Bremer and Adcock2021). This can lead to a change in the spectral bandwidth (Beji and Battjes, Reference Beji and Battjes1993), accompanied by a variation of the skewness (Onorato and Suret, Reference Onorato and Suret2016), and consequently the occurrence of extreme waves (Viotti and Dias, Reference Viotti and Dias2014).
Here, we find the experimental observations by Trulsen et al. (Reference Trulsen, Raustøl, Jorde and Rye2020), which report different non-homogeneous features of the kurtosis of surface elevation and wave kinematics due to long-crested waves atop a submerged bar allowing for transitions between deep-water and intermediate depths, very instructive. These findings have a significant impact on the hydrodynamic loads, and therefore also the design of structures in coastal waters (Bitner-Gregersen and Gramstad, Reference Bitner-Gregersen and Gramstad2015; Trulsen et al., Reference Trulsen, Raustøl, Jorde and Rye2020; Ghadirian et al., Reference Ghadirian, Pierella and Bredmose2023). The experimental results have been confirmed by means of the HOSM (Lawrence et al., Reference Lawrence, Trulsen and Gramstad2021). Later, these findings have been extended to account for two-dimensional bathymetry using HOSM-based numerical simulations by Lawrence et al. (Reference Lawrence, Trulsen and Gramstad2022). So far, no satisfying theoretical explanations have been proposed for the phenomenon reported by Trulsen et al. (Reference Trulsen, Raustøl, Jorde and Rye2020), although the statistical model of surface elevation by Mendes et al. (Reference Mendes, Scotti, Brunetti and Kasparian2022) and Mendes and Kasparian (Reference Mendes and Kasparian2022) can predict non-homogenous statistical features of surface elevation, but are limited to a local water region in which the depth is assumed to vary in space. Especially, whether or not differences between the statistical features of wave kinematics and surface elevation appear in other general contexts is an open question and subject to future studies.
Experimental investigation
Isolated extreme wave creation in a group atop a changing bathymetry in water wave facilities has attracted the attention of experimentalists in wave hydrodynamics since the 1990s (Baldock and Swan, Reference Baldock and Swan1996; Whittaker et al., Reference Whittaker, Fitzgerald, Raby, Taylor, Orszaghova and Borthwick2017). The wave group focus has been modeled based on the wave superposition principle while accounting for higher harmonics corrections in the boundary conditions adopted to initiate the experiments (Ma et al., Reference Ma, Zhang, Chen, Tai, Dong, Xie and Niu2022). Such considerations are crucial for the precise wave generation as well as accurate assessments of flow kinematics (Faltinsen et al., Reference Faltinsen, Newman and Vinje1995; Borthwick et al., Reference Borthwick, Hunt, Feng, Taylor and Stansby2006), swash oscillations on the beach (Baldock and Holmes, Reference Baldock and Holmes1999), sediment transport estimates and scour around a pile (Sumer and Fredsøe, Reference Sumer and Fredsøe2002; Aagaard et al., Reference Aagaard, Hughes, Baldock, Greenwood, Kroon and Power2012), and wave loads on structures (Zang et al., Reference Zang, Taylor, Morgan, Stringer, Orszaghova, Grice and Tello2010; Ghadirian and Bredmose, Reference Ghadirian and Bredmose2019; Li et al., Reference Li, Tang, Li, Draycott, van den Bremer and Adcock2023).
More recently, experimental studies investigating the effect of bathymetry slope change on either quasi-steady (Li et al., Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) or modulationally unstable wave groups (Kimmoun et al., Reference Kimmoun, Hsu, Hoffmann and Chabchoub2021) confirmed that the role of second-order effects is dominant during the extreme wave group transformation on a slope bathymetry. Having said that, the unstable wave groups did not swiftly demodulate over steep slopes when reaching depth regions
$ kh<1.363 $
, known to be the water regime for the MI to be inactive for unidirectional wave propagation (Johnson, Reference Johnson1977; Mei et al., Reference Mei, Stiassnie and Yue1989).
When analyzing more realistic conditions, that is, broadband wave signal of JONSWAP-type representative sea state initialization, as parametrized in Hasselmann et al. (Reference Hasselmann, Barnett, Bouws, Carlson, Cartwright, Enke, Ewing, Gienapp, Hasselmann, Kruseman, Meerburg, Müller, Olbers, Richter, Sell and Walden1973), as well as considering the propagating of the respective irregular waves in variable depth conditions, groundbreaking key findings from laboratory wave data have been reported since the first pioneering study of its kind by Trulsen et al. (Reference Trulsen, Zeng and Gramstad2012). In the latter and as mentioned earlier, it has been shown that a local maximum in skewness and kurtosis occurs on the shallower side of a linear slope, suggesting the increase of extreme wave probability in the neighborhood of the top of the depth change. Follow-up studies continued the investigation of the role of nonlinearity in the extreme wave emergence over a variable floor depth while considering a similar unidirectional experimental setup, that is, as utilized and described by Trulsen et al. (Reference Trulsen, Zeng and Gramstad2012) atop either a submerged bar or different linear slope inclinations (Kashima et al., Reference Kashima, Hirayama and Mori2014; Ma et al., Reference Ma, Dong and Ma2014; Kashima and Mori, Reference Kashima and Mori2019; Zhang et al., Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019). Schematics of a state-of-the-art apparatus are shown in Figure 2 (a) while (b) shows the corresponding evolution of surface elevation kurtosis as measured from the wave gauges. An excellent progress timeline has been provided by Trulsen et al. (Reference Trulsen, Raustøl, Jorde and Rye2020) in their Figure 1. It is worth highlighting the work of Kashima and Mori (Reference Kashima and Mori2019), which suggests that for steep bathymetry slopes, third-order nonlinear effects are still active, even though the dimensionless depth regime
$ kh<1.363 $
is not supposed to allow the quasi-four waves resonant interactions to unfold – a fact, also confirmed in experiments and numerical simulations by Kimmoun et al. (Reference Kimmoun, Hsu, Hoffmann and Chabchoub2021). Moreover, the study by Zhang et al. (Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019) emphasized that advanced numerical simulations, such as the Boussinesq-type model, can excellently reproduce the key statistical features of the experiments and confirmed the simulation results of Gramstad et al. (Reference Gramstad, Zeng, Trulsen and Pedersen2013).
Figure 2. (a) Schematic representation of a state-of-the-art experimental setup to study nonlinear wave shoaling and de-shoaling dynamics. The arrangement includes a computer-controlled wave generator, wave gauges, a wave absorber and a submerged bar. (b) Example of the non-homogeneous distribution of statistics (e.g., kurtosis) of surface elevation for random waves atop a submerged bar, as determined from the measurements.
Follow-up breakthrough contributions discussing the role of the shoal depth and the mismatch of the location of kurtosis peak of the surface elevation and horizontal fluid velocity on the lee side of the shoal have been reported by Trulsen et al. (Reference Trulsen, Raustøl, Jorde and Rye2020), as already elaborated upon in the previous section, while the results by Li et al. (Reference Li, Draycott, Adcock and van den Bremer2021a,Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremerb,Reference Li, Zheng, Lin, Adcock and van den Bremerc) underpinned the generation of new second-order free waves responsible for the wave focusing.
It is also pertinent to note that an abrupt bathymetry change from finite to deep-water conditions can freeze modulationally unstable wave groups to steady packets (Gomel et al., Reference Gomel, Chabchoub, Brunetti, Trillo, Kasparian and Armaroli2021).
An experimental campaign comprising a more sophisticated experimental setup consisting of a submerged bar and an accelerating uniform current revealed that up to a certain shoal depth threshold, the presence of such a flow forcing can enhance the non-Gaussianity of a sea state, thus increasing the frequency of extreme event formation (Zhang et al., Reference Zhang, Ma, Tan, Dong and Benoit2023).
There are also excellent experimental contributions discussing long-wave focusing and tsunami-type wave shoaling behavior (Goseberg et al., Reference Goseberg, Wurpts and Schlurmann2013; Pujara et al., Reference Pujara, Liu and Yeh2015), and the occurrence of rogue waves in opposing currents (Toffoli et al., Reference Toffoli, Waseda, Houtani, Cavaleri, Greaves and Onorato2015). However, these will not be discussed as being beyond the scope of this review.
Summary and outlook
Our brief review article comprises an overview of the latest physical modeling and experimental validation studies addressing the formation of isolated extreme wave events when transitioning from a deep to a shallow environment, and in some cases the other way around through a specific change in the bathymetry. The progress has been particularly significant and impactful over the last decade, underlining the need of studying such flow dynamics and statistics to confront the global warming–related increase of wind speeds and associated wave heights in the future.
Even though the theoretical, numerical and experimental advances have been “overwhelming”, as reported, there are still crucial improvements that have to be made in the modeling. This is to address realistic conditions for different and varying coastal morphologies as well as converging towards common outcomes and conclusions when including directional sea states propagating over a shoal (Bitner, Reference Bitner1980; Cherneva et al., Reference Cherneva, Petrova, Andreeva and Soares2005; Ducrozet and Gouin, Reference Ducrozet and Gouin2017; Lawrence et al., Reference Lawrence, Trulsen and Gramstad2022; Lyu et al., Reference Lyu, Mori and Kashima2023). Whether or not differences between the statistical features of wave kinematics and surface elevation appear in other general contexts and how the differences affect the design standards of coastal structures are open questions for future studies. Moreover, the fast developments of computational capacities will allow the study of this physical problem within the framework of a more advanced numerical framework, such as two-phase flows solving the Navier–Stokes equations or smoothed particle hydrodynamics, applied to realistic domain configurations. Last, but certainly not least, we anticipate that newly developed machine learning algorithms, if fed with high-fidelity data, will play a major role in the operational detection of nearshore extreme waves in the near future.
Open peer review
To view the open peer review materials for this article, please visit http://doi.org/10.1017/cft.2023.21.
Acknowledgments
The authors thank Yuchen He for assistance in the figure preparation.
Financial support
Y.L. acknowledges support from the Research Council of Norway through the POS-ERC project 342480. A.C. is supported by the Hakubi Center for Advanced Research at Kyoto University.
Competing interest
The authors declare no competing interests exist.
Open access
Comments
Dear Editors,
Submitting our review paper “On the formation of coastal extreme waves in water of variable depth” to “Cambridge Prisms: Coastal Futures”, we feel that it is the most suitable journal for our results.
The modelling and experimental investigations related to the formation of extreme waves in transitional water depth zone have particularly thrived over the last decade.
Our brief review recaps the related most significant theoretical developments and groundbreaking experimental advances, which have particularly thrived over the last few years.
As such, our work will immediately attract the attention of the wide scientific community as well as experts in the area.
With best wishes on behalf of the authors,
Dr. Amin Chabchoub