Vincent Gruwez

The European coasts are among the most densely populated of the world, with natural sand dune barriers urbanised and replaced by traditional hard coastal protection structures. Without the needed measures to adapt, the number of people exposed to floods is anticipated to increase 187 million worldwide by the end of the 21st century. In Europe, the coasts of The North Sea, the Baltic Sea, and the Atlantic are anticipated to experience substantial flood risks from sea-level rise, and climate extremes are also expected to impact southern Europe Mediterranean coasts (Vousdoukas et al., 2017). Future coastal management surpasses the current fixed and non-adaptive flood coastal protection setup: Hybrid Nature-based-Solutions (abbreviated as NbS) that can efficiently integrate static hard infrastructure with dynamic aeolian, and vegetated sediments are currently developed along urbanized areas of most of the European sandy coasts, yet still at small scales.

Low-lying countries typically have mildly-sloping beaches as part of their coastal defense system. Many countries in north-western Europe have coastal urban areas that rely on this type of defense system, which consists of a low-crested impermeable sea dike with a relatively short promenade, and a long (nourished) beach in front that acts as a very/extremely shallow foreshore as defined by Hofland et al. (2017). Along the cross-section of this hybrid beach-dike coastal defense system, storm waves are forced to undergo many transformation processes before they finally overtop the dike. These hydrodynamic processes include shoaling, sea-swell (SS) wave energy transfer to sub- (also infragravity or IG waves) and superharmonics via nonlinear wave-wave interactions, wave dissipation by breaking and bottom friction, reflection against the dike, wave run-up and overtopping on the dike, bore impact on a wall or building, and finally reflection back towards the sea interacting with incoming bores on the promenade. Due to breaking of the SS waves and growth of the IG waves on the shallow foreshore, the IG waves can become as important or even dominant at the toe of the dike (Hofland et al., 2017; Lashley et al., 2020), which influences the overtopping process (van Gent, 1999).

Infragravity waves are surface waves with relatively longer periods in comparison to periods of the spectrum-dominant gravity waves. They are characterized by oscillations between 20 and 300 seconds (0.0033 Hz < f < 0.05 Hz), amplitudes that range from a few millimeters to tens of centimeters, and wavelengths of kilometers (Munk, 1950; Holman and Bowen, 1982; Ardhuin et al., 2014). Their forcing is linked to, amongst others, nonlinear interaction between sea swell waves, varying wave heights causing the breaking point of the waves to vary with height, and height variation of incoming waves (Bertin et al., 2018). Infragravity waves play an important role in coastal dynamics (Svendsen, 2005) and have been reported to trigger nearshore hazards such as beach and dune erosion (de Vries et al. 2008; Roelvink et al., 2009), development of seiches in harbors (Melito et al., 2006; Cuomo and Guza, 2017), wave-driven coastal inundation (Gent, 2001; Stockdon et al., 2006), and ice shelves collapsing (Bromirski et al., 2010). Therefore, revealing infragravity wave characteristics is of utmost importance to understand their potential to generate hazards in a certain region, especially at sites strongly influenced by human occupation and activities. Their consideration in coastal safety planning can avoid damages, as several locations have already experienced in the past (Yamanaka et al., 2019). Implementing optimal sampling strategies for observing and characterizing infragravity waves might be challenging. By nature, these waves are hard to measure accurately due to their low amplitude. Their evolving characteristics in an environment marked by pronounced bathymetric features, such as the sand bank systems off the Belgian coast, add a degree of complexity that requires testing of different approaches, and at different sites. Within this context, this work first revisits observational approaches, instrumentation, logistics, and sampling techniques that have been used to study this phenomenon on the Belgian Coast. The advantages, challenges and limitations of different approaches are discussed, and best practices for collecting high-quality data in the field are addressed. To do so, this study explores multi-sensor in situ deployments conducted at four selected sites off the Belgian coast (Figure 1) (Nieuwpoort, Raversijde (inshore and offshore Stroombank), and Trapegeer) within the context of the “Influence of infragravity sea waves during storms on the hydro- and morphodynamic processes along hybrid soft-hard coastal defence structures with a shallow foreshore” project, an FWO-funded initiative being conducted in collaboration between UGent, VLIZ, and KULeuven and with support of Agency for Coastal and Maritime Services (AMDK). More specifically, field observations were conducted using multipurpose mooring frames equipped both with (i) Acoustic Doppler Current Profiles (ADCPs) to sample pressure (0.1% FS), current, and sea surface elevation through acoustic surface tracking and (ii) high-accuracy quartz pressure sensors (accuracy 0.01 % FS). Both ADCPs and pressure sensors were set to measure continuously at 4 Hz being, therefore, able to capture both infra- and gravity waves. Furthermore, the moorings were collocated with standard wave buoys from AMDK. Data was collected continuously for about 3 months, covering storm and calm wave conditions. Finally, the measurements from ADCPs (pressure and acoustic) and pressure sensors were compared and used to derive the infragravity wave characteristics, as well as cross-validated against wave buoy data.

Infragravity (IG) waves are low-frequency waves (0.0033 Hz < f < 0.05 Hz) that exist: (i) being bound to the wave groups of high-frequency sea-swell (SS) waves (0.05 Hz < f < 1 Hz), (ii) generated by the moving breakpoint due to the groupiness of the SS waves (Bertin et al., 2018), and (iii) as free IG waves released from the SS wave groups and reflected from neighbouring coasts (Rijnsdorp et al., 2021). Offshore in deeper water (h/Hm0,o > 4, with h the water depth and Hm0,o the offshore significant wave height), IG waves become most apparent during storm conditions, but are still rather limited in wave height (order of a couple of cm). However, at the toe of coastal structures with a shallow foreshore depth (1 < ht/Hm0,o < 4, with ht the water depth at the toe), the energy of IG waves becomes more significant relative to the SS wave frequency band of the wave spectrum (Hofland et al., 2017), becomes equally important for very shallow foreshores (0.3 < ht/Hm0,o < 1), and even dominates SS wave energy for extremely shallow foreshore depths (ht/Hm0,o < 0.3). IG wave growth on mildly sloped shallow foreshores is a result of IG wave shoaling and SS wave energy transfer to IG waves via nonlinear wave-wave interactions (Bertin et al., 2018), and the relative growth of IG to SS wave energy is further aided by SS wave breaking. Coastal urban areas along the Belgian coast (and in other low-lying countries worldwide) rely on a hybrid beach-dike coastal defense system for protection against flooding. During design storm conditions, the nourished beach erodes to a very to extremely shallow foreshore in front of a low-crested impermeable sea dike. Wave overtopping at this type of structure has been shown to be affected by IG waves, based on physical (Altomare et al., 2016) and numerical modelling (Lashley et al., 2021) for shallow to extremely shallow foreshores. However, field observations of the influence of IG waves on wave overtopping have been scarce (van Gent and Giarusso, 2003) and the influence of free IG waves has not been investigated yet. Indeed, the amount of IG wave energy that is expected to arrive at the coastline, not only depends on IG waves naturally bound to the SS wave groups, but also on free IG waves generated offshore or locally reflected from the coast (edge waves). The Belgian continental shelf is characterized by the prevalence of offshore sandbanks (see Figure 1a), which might be an additional source of free IG waves, due to ss wave breaking on the shallow crest of those banks. These free IG waves are location specific and their existence can only be determined by field observations. Knowledge of the offshore boundary conditions with detailed IG wave information is indispensable for the correct prediction and modelling of surf zone hydrodynamics and beach morphodynamics (Fiedler et al., 2019), and the consequent wave overtopping on the dike. Simultaneous measurements of IG waves, offshore and on the beach-dike system during storm conditions together with wave overtopping has not been done before, as it is complex and needs a specific setup with high precision measurement equipment. In-situ measurements provide an invaluable dataset (i.e., no model nor scale effects) and are essential for validating safety assessment and design methodologies of coastal defense systems.

Low-lying countries typically have mildly-sloping beaches as part of their coastal defence system. Many countries in north-western Europe have coastal urban areas that rely on this type of defence system, which consists of a lowcrested impermeable sea dike with a relatively short promenade, and a long (nourished) beach in front that acts as a very/extremely shallow foreshore as defined by Hofland (2017). Along the cross-section of this hybrid beach-dike coastal defence system, storm waves are forced to undergo many transformation processes before they finally overtop the dike. These hydrodynamic processes include shoaling, sea and swell wave energy transfer to sub- and superharmonics via nonlinear wavewave interactions, wave dissipation by breaking and bottom friction, reflection against the dike, wave run-up and overtopping on the dike, bore impact on a wall or building, and finally reflection back towards the sea interacting with incoming bores on the promenade. Field measurements of all these processes at the same time are very challenging but necessary since these suffer from neither scale nor model effects. Field data are therefore crucial to evaluate design methodologies, which rely on physical and numerical modelling. This paper presents the field setup and the design features of the innovative artificial dike, unique in the world.

Wave overtopping is a critical parameter in the design of coastal defense structures. Nowadays, several empirical formulations based on small-scale experiments are available in the literature to predict the mean overtopping discharge at dikes on shallow foreshores. Although the accuracy of the predictions has improved due to each approach’s contributions, the formulations’ performance depends on their range of applicability. In engineering applications, it is important to know the performance and limitations of the different formulas. This work presents a new experimental dataset focused on very shallow and extremely shallow foreshore conditions for a range of foreshore slopes (i.e., 1/20, 1/35, 1/50, and 1/80) and relative water depths. The recent developments in wave overtopping research on very shallow and extremely shallow foreshore conditions have been reviewed using this dataset to reflect the existing uncertainties and challenges in the wave-overtopping literature. We find that predicting wave overtopping for extremely shallow foreshore conditions still requires improvement. Additional research is needed to understand the (residual) influence on the wave overtopping of the foreshore slope and relative magnitude of the infragravity wave height to the sea-swell wave height at the dike toe, especially for extremely shallow foreshore conditions. The variation in performance of the formulas for different foreshore slopes is demonstrated. Finally, some of the remaining uncertainties that need further exploration are discussed.

Three open source wave models are applied in 2DV to reproduce a large-scale wave flume experiment of bichromatic wave transformations over a steep-sloped dike with a mildly-sloped and very shallow foreshore: (i) the Reynolds-averaged Navier–Stokes equations solver interFoam of OpenFOAM^® (OF), (ii) the weakly compressible smoothed particle hydrodynamics model DualSPHysics (DSPH) and (iii) the non-hydrostatic nonlinear shallow water equations model SWASH. An inter-model comparison is performed to determine the (standalone) applicability of the three models for this specific case, which requires the simulation of many processes simultaneously, including wave transformations over the foreshore and wave-structure interactions with the dike, promenade and vertical wall. A qualitative comparison is done based on the time series of the measured quantities along the wave flume, and snapshots of bore interactions on the promenade and impacts on the vertical wall. In addition, model performance and pattern statistics are employed to quantify the model differences. The results show that overall, OF provides the highest model skill, but has the highest computational cost. DSPH is shown to have a reduced model performance, but still comparable to OF and for a lower computational cost. Even though SWASH is a much more simplified model than both OF and DSPH, it is shown to provide very similar results: SWASH exhibits an equal capability to estimate the maximum quasi-static horizontal impact force with the highest computational efficiency, but does have an important model performance decrease compared to OF and DSPH for the force impulse.

In this paper, a Reynolds-averaged Navier–Stokes (RANS) equations solver, interFoam of OpenFOAM^®, is validated for wave interactions with a dike, including a promenade and vertical wall, on a shallow foreshore. Such a coastal defence system is comprised of both an impermeable dike and a beach in front of it, forming the shallow foreshore depth at the dike toe. This case necessitates the simulation of several processes simultaneously: wave propagation, wave breaking over the beach slope, and wave interactions with the sea dike, consisting of wave overtopping, bore interactions on the promenade, and bore impacts on the dike-mounted vertical wall at the end of the promenade (storm wall or building). The validation is done using rare large-scale experimental data. Model performance and pattern statistics are employed to quantify the ability of the numerical model to reproduce the experimental data. In the evaluation method, a repeated test is used to estimate the experimental uncertainty. The solver interFoam is shown to generally have a very good model performance rating. A detailed analysis of the complex processes preceding the impacts on the vertical wall proves that a correct reproduction of the horizontal impact force and pressures is highly dependent on the accuracy of reproducing the bore interactions.

This study comprises a detailed description of the individual overtopped bore impact processes against a vertical wall, situated on top of a dike. A twin peak force impact signal shape was observed with two distinct peaks during every impact. The two peaks were assigned consecutively to the dynamic components (thickness and velocity) or hydrostatic components (run-up of water at the wall) of the impacting bore. The two peaks were termed dynamic F1 and quasi-static F2 impact respectively. Based on available literature semi-empirical equations to describe either the dynamic F1 or quasi-static F2 impact force were investigated and the prediction accuracy evaluated using impact force data from large-scale experiments. The prediction accuracy of the dynamic F1 impacts was very low. The prediction accuracy of the quasi-static impact F2 was increased based on fitting the hydrostatic theory to the maximum run-up measurement at the wall. Based on these findings 80% of the maximum run-up height was effectively contributing to the maximum quasi-static force F2 on the wall. The results coincided well with previous small-scale studies (Chen et al. 2012). After deconstructing the process chain preceding an impact, using the physically most meaningful parameters to predict the impact force, evaluating on a range of existing approaches, and observing the scattered prediction results, it was concluded that the impact behavior is highly stochastic and statistical analysis would be more beneficial.

This paper introduces the 2D experiments conducted for the CREST project in the wave flume of Ghent University. The experiments focus on wave interactions with low-crested sea dikes fronted by a shallow foreshore and mildly to steeply sloping beaches, which is a very typical situation along the Belgian coast. Foreshore slopes of 1/20, 1/35, 1/50 and 1/80 were tested for a range of low to high energy wave conditions, a variation in wave steepness and two water levels. The main goal was to obtain a dataset in which the effects of the infragravity waves on the wave-structure interactions (i.e. wave overtopping and impact forces) can be studied. The tests included high spatial resolution surface elevation measurement tests, which is new for beaches including a dike in the inner surf zone. From the first results it became clear that the foreshore slope influences the wave transformation up to the dike toe. The influence is apparent comparing to existing (semi-) empirical models for prediction of the spectral wave period at the dike toe and wave overtopping at the dike crest. The high spatial resolution data show a steep increase in infragravity significant wave height in the very shallow area in front of the dike.