# Modelling of wave run-up on a macrotidal beach for assessing coastal flooding

Carlos López Solano, Imen Turki, Ángel David Gutiérrez Barceló, Harish Selvam, Ernesto Tonatiuh Mendoza, Stéphane Costa, Edward Salameh, Antonin Migaud, Robert Lafite

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To cite this version:

Carlos López Solano, Imen Turki, Ángel David Gutiérrez Barceló, Harish Selvam, Ernesto Tonatiuh Mendoza, et al.. Modelling of wave run-up on a macrotidal beach for assessing coastal flooding. Results in engineering, 2025, 25, pp.104379. 10.1016/j.rineng.2025.104379. irsn-04963241

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Submitted on 24 Feb 2025

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# Modelling of wave run-up on a macrotidal beach for assessing coastal flooding

Carlos López Solano<sup>a, *, †</sup>, Emma Imen Turki<sup>a</sup>, Ángel David Gutiérrez Barceló<sup>b</sup>, Harish Selvam<sup>c</sup>,
Ernesto Tonatiuh Mendoza<sup>a</sup>, Stephan Costa<sup>a</sup>, Edward Salameh<sup>a</sup>, Antonin Migaud<sup>d</sup>,
Robert Lafite<sup>a</sup>

## ARTICLE INFO

### Keywords:

- Coastal storms

- Flooding

- Wave run-up

- Numerical modelling

- Video monitoring systems

## ABSTRACT

This work examines the impact of coastal storms on a macrotidal beach at Normandy (France), the beach of Villers-sur-Mer, with the aim of assessing coastal flooding. Extreme wave heights and water levels of four typical storms are simulated using the non-hydrostatic model SWASH to estimate the wave run-up. The numerical simulations have been compared to empirical formulations and validated with processed timesteps generated by video monitoring systems (VMS). The validation shows good agreements between observations and simulations with a mean RMSE of 16 cm, which emphasizes the accuracy of this methodological approach. In the light of the overall results, the water level has a predominant effect on the run-up compared to the wave height. However, the combination of high water levels and waves induces the biggest wave run-up, responsible for the most significant flooding on the coast. Additionally, limited changes in water levels during storm events increase the wave height effect on the run-up, meanwhile small waves with short periods cannot uprush the beach. Furthermore, the transfer of storms, integrating the regional scale of the English Channel and the local scale of the beach, is an important step for assessing coastal flooding in semi-open environments and basins, where the wave direction is the conditioning parameter of storm evolution. Storms from WSW within the English Channel generate the highest wave run-up. This research provides an excellent methodological approach and accurate estimations of coastal flooding in Normandy, which is useful for further works on coastal hazards in response to storms.

## 1. Introduction

Coastal areas are very vulnerable zones to extreme wave- and tide-induced inundation during extreme events causing severe damage in such areas, with around 40% of the world's population living within 100 km of the coast [20]. Extreme events are generally caused by a series of complex atmospheric mechanisms interacting with oceanographic processes as well as the coastal topography to increase coastal flood risk [17]. This risk is being amplified in the global context of climate change, rising sea levels and boosting storms in their intensity and the frequency of their occurrence [12].

The severity of coastal flooding close to beaches is increasing when changes in the basin geometry are substantially important with high

nonlinear interactions between tidal and wave drivers in shallow water areas [21,36]. In these environments, the key components describing the coastal inundation induced in shore controlled by wave and sea level variations are related to the wave run-up, a time-varying process which is defined as the vertical distance of the water's edge on the foreshore of the beach [20]. Coastal flooding can be understood as the addition of the high tide to the mean sea level, the surge and the wave run-up (Fig. 1), including the nonlinear interactions between them.

Previous engineering studies have been developed to investigate coastal flooding through the use of different approaches of in-situ measurements or physical and numerical modelling with the aim to estimate the wave run-up, which is crucial for assessing water extension close to the shoreline. Although numerical wave models effectively

<footnote><sup>a</sup> Univ Rouen Normandie, Université Caen Normandie, CNRS, Normandie Univ, M2C UMR 6143, Rouen F-76000, France</footnote>

<footnote><sup>b</sup> SandS, Bajada de San Juan, 77, Santander 39012, Spain</footnote>

<footnote><sup>c</sup> Institute of Hydraulic Engineering and Water Resources Management, RWTH Aachen University, Aachen 52074, Germany</footnote>

<footnote><sup>d</sup> Institute for Radiological Protection and Nuclear Safety, Fontenay aux Roses 92262, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: carlos.lopez-solano@univ-rouen.fr (C. López Solano).</footnote>
simulate nearshore hydrodynamics in beaches, their computational cost
limits widespread use. Coupling approaches based on numerical
modelling, field observations and empirical relationships would offer a
more resource-efficient alternative to reproduce the hydrodynamics of
coastal systems from sandy to gravel and composite beaches [24,34].

Regarding the numerical approach, the use of models for simulating
coastal flooding, including run-up, in the aforementioned applications
requires a detailed approach based on the phase- and depth- resolving
modelling, as provided by 2D and 3D models. However, the applicable
hydrodynamic models are somewhat limited due to the high computa-
tional cost for the 3D problems, which is the case of the well-known
OpenFoam and SPH-based models, considered as high-end models. As
a compromise between resolving depth and reducing the computational
cost, the use of the non-hydrostatic model SWASH [41] with a
multi-vertical layer set-up could be an option. Indeed, SWASH has been
proven to be a reliable model for reproducing the complex interaction
between waves and coastal topography, showing its accuracy for
breaking waves and run-up on beaches [32], extensively validated with
laboratory experiments [6] and with field measurements [26].

Additionally, it is possible to generate wave run-up measurements
from images provided by Video Monitoring Systems (VMS), which
would be challenging due to the need of manual supervision, leading to
increased processing times. This approach has proven to be an effective
solution in monitoring temporal beach changes during extreme storm
events [2,25]. Accurate prediction of wave run-up height is essential for
designing coastal protection works, beach nourishment projects, and
planning coastal management schemes. Coastal engineers typically es-
timate run-up using empirical formulations based on offshore wave
parameters and beach slope, but recently, coastal VMS have facilitated
long-term measurements of wave run-up, with studies like Stockdon
et al. [33] providing extensive analyses. The study by Vousdoukas et al.
[39] presents a semi-automatic procedure for generating wave run-up
data from image timesteps, applied to images from a reflective beach
on the Portuguese coast.

However, the use of the suitable approaches is not always available
due to the complexity of the coastal systems, including their geometrical
morphological characteristics, where the effects of the changing geo-
morphology on the water levels are enhanced, exhibiting a modulation
of the energy from nearshore to coastal areas, as it happens in the En-
glish Channel.

Indeed, the English Channel is considered as a semi-open system where the complex interaction between the different physical processes is exemplified, especially when waves approach from the Atlantic to the basin and then to the Normandy coasts [18,19]. Previous works related to the dynamics of extreme events within the English Channel have been carried out using different approaches including stochastic [37,38] and numerical modelling approaches [3,18,19,30,31]. Bennis et al. [3] investigated the hydrodynamic interactions in Alderney Race (France), considered the most energetic tidal site in Western Europe, using a 3D modeling approach. Soloy et al. [30] analyzed storm clustering along Normandy coasts, using numerical models and remote sensing to understand storm events comprehensively. Some studies also included the coastal effects of these storms, such as beach erosion [5], or their

morphodynamic impacts on changes in the coastline [1,28,29].
Recently, López Solano et al. [19] have investigated the dynamics of
storms within the channel and close to the Normandy coasts, demon-
strating that the extreme waves are modulated between the UK and the
French coasts, exhibiting an amplification of the energy at the shallow
water areas.

This study focuses on assessing coastal flooding at the coast of Normandy in response to various energetic storms, produced between 1980 and 2022 and modelled at the regional scale of the English Channel [19]. The evolution of the different extreme scenarios has been investigated at the local scale of a macrotidal beach by coupling the non-hydrostatic model SWASH, developed for the interaction of nearshore hydrodynamics with coastal morphologies and remote sensing approaches of VMS to determine the wave run-up and the time-evolution of the waterline during storms.

This research was carried out in the framework of the Surface Water and Ocean Topography (SWOT) program of the French National Center of Space Studies (CNES) and focused on gaining insights into coastal flood risks in Normandy by integrating modeling approaches and remote sensing techniques. This work is structured as follows: Section 2 describes the hydro-morphodynamics dataset used in this work and the methodological approach developed for investigating coastal flooding. The results have been deeply explored in Section 3 and discussed in Section 4. Concluding remarks have been outlined in Section 5.

## 2. Materials and methods

This section describes the dataset used and the methodological
approach developed in this work for assessing coastal flooding. The
recent works on the modelling storm evolution within the English
Channel, carried out by López Solano et al. [19], are explored to inte-
grate the extreme hydrodynamics from regional to local scales on the
coast where the wave-run up induced by different energy scenarios has
been investigated.

2.1. *Datasets and study sites*

2.1.1. *Study site: the English Channel and Villers-sur-Mer*

The English Channel is a basin between north France and south UK,
connecting the Atlantic Ocean and the North Sea, a shallow area located
on the continental shelf. The Channel extends over about 560 km with a
varying width between the French and the British shores, from around
240 km in the open boundary of the Atlantic Ocean to <35 km in the
Strait of Dover (Fig. 2).

The entrance of the Channel is of 100 m depth which decreases
progressively close to the Normandy and UK coasts with mean ranges of
15–25 m and 10–20 m, respectively. Globally, the English Channel has
an averaged 60 m depth in its central part.

The irregular variation of the bathymetry of the Channel and the
composite sediments of the seabed with high changes in the bottom
roughness [9] exhibit a complex evolution of the hydrodynamics and its
modulation close to the coastal areas depending on the intensity of the
marine energy and its direction [18].

<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic presentation of wave run-up on the foreshore.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Bathymetry map of the English Channel with the location of the study site, Villers-sur-Mer, and the location of the SCENES buoy used for calibration nearshore.</figcaption></figure>

The impacts of a series of stormy events with different energy scenarios have been investigated at a Normandy beach, Villers-sur-Mer, by combining the effects of extreme waves and storm surges. The water level ranges from around 3 m at the neap tides up to 10 m at the spring tides, being considered a macro- to mega-tidal system [16].

Villers-sur-Mer is a macrotidal sandy beach with a dissipative morphology, with a slope of the intertidal zone of around 1% [29]. This site is monitored by a VMS, deployed since 2019, providing different products with a variable temporal and spatial resolution. One of the hydrodynamic products built from video recordings, the timestacks, will be used further in the present study, which gives useful information about the motion of the shoreline over the slope of the beach.

### 2.1.2. Hydrodynamic and bathymetric data

A large set of hydrodynamic and bathymetric data, collected from various sources have been used to build the numerical set-ups within the English Channel and close to Villers-sur-Mer coast.

At the regional scale of the English Channel, at first, a coarse 2D computational grid in the horizontal plane (2HD) has been developed [19]. This numerical approach is summarized in Fig. 5. Then, a 2D computational grid in the vertical plane (2VD) is deployed to model the beach profiles at the local scale of Villers-sur-Mer coast. Using the non-hydrostatic model SWASH with a high number of vertical layers enhances its accuracy to solve the depth variation in comparison to previous studies [10,26].

The hydrodynamic conditions of high energy storms have been extracted from the recent works, carried out by López Solano et al. [19], at the regional scale of the Channel. Wave, water levels and wind observations from various sources have been explored for the set-up of the numerical modelling. In their work, they have forced the model with waves from the Atlantic-European North-West Shelf - Wave Physics Reanalysis [13,22,23], astronomical tide from FES2014 [4], wind and atmospheric pressure data from the ERA5 on a single level database [11] and bathymetry information extracted from the European Marine Observation and Data Network (EMODnet) Bathymetry portal [7].

Extreme wave outputs of this database, generated with the numerical suite Delft3D, are taken as inputs for using them as forcings of the simulations nearshore with the numerical model SWASH. This simulation aims not just to simulate run-up, but also to integrate a detailed analysis of the effect of specific storms, highlighting how their inner characteristics affect the run-up in the Normandy region.

The bathymetry data used in the numerical model of the 2VD computational grid is extracted from a high-resolution LiDAR campaign carried out in October 2019 with an extension of 9000 × 7000 m over

the beach of Villers-sur-Mer. This adds a level of detail to model the topography not presented in other studies [27,32].

The buoy SCENES located in the Seine estuary (49° 28.844' N–0° 1.932' E) has also been used, with measurements from October 2017 to October 2020.

## 2.2. Methodology

### 2.2.1. Description of the numerical modelling

The simulations of stormy waves impacting on the coast have been carried out using the numerical model SWASH (an acronym of Simulating WAves till SHore), developed at Delft University of Technology. SWASH is a non-hydrostatic wave-flow model intended to be used for predicting the transformation of dispersive surface waves from offshore to the beach. SWASH allows studying the surf zone and swash zone dynamics, wave propagation and agitation in ports and harbors [41]. The governing equations are the Non-Linear Shallow Water (NLSW) equations, including the non-hydrostatic pressure correction term. SWASH can be run in depth-averaging mode (assimilated as a 2HD model) or multi-layered mode (either 3D or 2VD), in which the computational domain is divided into a fixed number of vertical terrain-following layers, as the one used in the study. SWASH improves its frequency dispersion by increasing this number of layers rather than increasing the order of derivatives of the dependent variables like Boussinesq-type wave models do.

### 2.2.2. Forcings and set-ups used for Swash model

The boundary conditions in the profiles of the numerical modeling are defined by the wave spectrum aggregated parameters and the water level. Firstly, four storms were selected from the time series of significant wave height time series at the study site, following different criteria. Storm Ciara has been taken as the strongest event that took place over the past few years. The three other storms have been selected as examples because of their inner characteristics presented in López Solano et al. [19]: Joachim, Andrea and the so-called Storm 19. The most relevant variable has been their direction in origin in the Atlantic Ocean before starting propagating throughout the English Channel. Each one of these three storms belongs to one-directional sector with a Southern, Northern or central component at the entrance of the basin.

The Digital Elevation Model (DEM) in the working domain is obtained by interpolation of the bathymetric data (Fig. 3). The DEM allows the definition of the spatial distribution of multiple profiles along the beach, an original contribution in comparison to previous studies to accurately reproduce the physical conditions of the study site, giving a
<figure><img src="image_5.png" /><figcaption>Fig. 3. a) Bathymetry from the LiDAR campaign used for the set-up of the numerical model and profiles selected for the simulations in Villers-sur-Mer, and geometry of 4 profiles used for the simulations in Villers-sur-Mer relative to the MSL: b) P7, c) P11, d) P15 and e) P19.</figcaption></figure>

unique geographical context. The spatial reference is the projection system EPSG:32,631 WGS84/UTM zone 31N.

The profiles are selected equally spaced each 400 m, avoiding being placed over the groins located in some parts of the beach of Villers-sur-Mer. The 4 chosen profiles are named P7, P11, P15 and P19 from East to West.

The total length of the profiles extracted from the DEM is 1000 m, artificially extended 200 m in the deeper boundary to produce a "conditioning area" with a flat bottom. This facilitates the simulation of waves at the beginning of the computational domain to stabilize the propagation before the interaction with the real profile.

The computational grid is defined over this length with 1200 cells of 1 m width, and for each node between cells, the bathymetry is interpolated (the depth profiles are shown in Fig. 3). Villers-sur-Mer is a sandy beach, so the bathymetry, defined in the model as the bottom layer, is considered as a full impermeable layer. The vertical domain is divided into 10 vertical sigma layers, equally distributed with 10% of the depth of water following the bottom layer. After several tests, this number is appropriate for computing stable simulations, a compromise between obtaining accurate results and a high computational cost of setting up many layers [41].

The simulations of SWASH have been divided into 3-hour sea states throughout the storms, considering the time step limitation in the outputs of the numerical modeling in the English Channel. The database can only provide reliable information on waves every 3 h due to the limitation, at the same time, in the forcing of the NWSHELF reanalysis database used. With the aim of keeping a comparable time between storms, every event is simulated for 3 days, which produces 25 simulations per storm and profile. Wave parameters used to force the model are significant wave height ($H_s$) and peak period ($T_p$), alongside the water level. Since SWASH can only be forced with either waves or sea level, and resolution in time of wave parameters of every sea state cannot be reduced, a constant water level had to be set for each 3-hour simulation. As the main objective of this study is to compare the

maximum run-up between the different storms, the water level chosen for each sea state simulated has been then maximum registered over every 3-hour period in order to obtain the worst scenario associated with every sea state.

Before simulating all 4 storms in Villers-sur-Mer, a correction to the output of water level is applied. Because of the resolution of 1 to 2 km of the computational grid of the English Channel, it is expected that, when the cells are located nearshore, the results do not have the total amplitude in the tidal waves, as reflected in López Solano et al. [19]. The records of sea level of the buoy SCENES are used, representative of the tidal conditions of the study site because of the close location. These corrections based on tidal records increase the accuracy of the results, something that is not addressed in many of the available references.

The buoy measurements only include storm Ciara out of the 4 storms simulated, so the full time series from October 2017 to October 2020 has been used to correct the equivalent time series. The calibration method consists of applying an amplification factor to the tidal ranges of each tidal cycle of the time series for all the values different from 0. The amplification factor is applied according to the similarity of the tidal ranges output of the model to the tidal ranges of the measurements of the buoy, despite considering other hydrodynamic characteristics as the velocity fields. Then, these factors obtained for every value of every tidal cycle during the 3 days of simulation are extrapolated for each of the 4 storms, obtaining the corresponding water levels. As an example, in Fig. 4, the correction applied to Ciara storm is shown, with the original output of the coarse model, the observations of the water level measured by the SCENES buoy and the time series of water level from which the maximum every 3 h is obtained to force the simulation with SWASH.

The simulations are repeated for the 4 profiles, with the 25 simulations corresponding to each 3-hour sea state, defined by $H_s$, $T_p$ and the water level (WL). In Table 1, the variables and dates (UTC) of the 4 storms used for the simulations are shown. As output, the surface elevation of water is obtained to calculate the run-up height.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Time series of water level with the output of the English Channel model in Villers-sur-Mer during the 3 days of simulation of Ciara storm, time series of water level corrected according to the corrections in amplitude obtained from SCENES buoy, and time series of the water level buoy observations.</figcaption></figure>

<table><caption>**Table 1**

Forcings of significant wave height (Hs), peak period (Tp) and water level (WL) for the set-up of the numerical model of every 3-hour sea state of the 4 storms in Villers-sur-Mer.</caption><thead><tr><th rowspan="2">Time</th><th colspan="3">Storm Ciara (09/02/2020 12 h - 12/02/2020 12 h)</th><th colspan="3">Storm Joachim (12/12/2011 21 h - 15/12/2011 21 h)</th><th colspan="3">Storm Andrea (03/01/2012 06 h - 06/01/2012 06 h)</th><th colspan="3">"Storm 19" (28/01/2013 18 h - 31/01/2013 18 h)</th></tr><tr><th>H<sub>s</sub> (m)</th><th>T<sub>p</sub> (s)</th><th>WL (m)</th><th>H<sub>s</sub> (m)</th><th>T<sub>p</sub> (s)</th><th>WL (m)</th><th>H<sub>s</sub> (m)</th><th>T<sub>p</sub> (s)</th><th>WL (m)</th><th>H<sub>s</sub> (m)</th><th>T<sub>p</sub> (s)</th><th>WL (m)</th></tr></thead><tbody><tr><td>00h</td><td>0.91</td><td>3.48</td><td>2.999</td><td>0.64</td><td>2.71</td><td>2.490</td><td>0.73</td><td>3.08</td><td>1.678</td><td>0.64</td><td>2.71</td><td>-0.562</td></tr><tr><td>03h</td><td>0.94</td><td>3.48</td><td>-0.273</td><td>0.67</td><td>3.08</td><td>2.641</td><td>0.98</td><td>3.48</td><td>0.142</td><td>0.77</td><td>3.48</td><td>0.358</td></tr><tr><td>06h</td><td>1.17</td><td>3.95</td><td>0.221</td><td>0.84</td><td>3.48</td><td>0.581</td><td>1.16</td><td>3.95</td><td>-0.077</td><td>0.77</td><td>3.95</td><td>2.938</td></tr><tr><td>09h</td><td>1.54</td><td>5.07</td><td>3.497</td><td>0.84</td><td>3.48</td><td>-0.708</td><td>1.25</td><td>4.47</td><td>1.788</td><td>0.70</td><td>3.48</td><td>2.692</td></tr><tr><td>12h</td><td>1.57</td><td>5.74</td><td>3.416</td><td>1.04</td><td>5.74</td><td>2.932</td><td>1.07</td><td>5.74</td><td>1.920</td><td>0.69</td><td>3.48</td><td>-0.24</td></tr><tr><td>15h</td><td>1.93</td><td>6.50</td><td>0.000</td><td>1.15</td><td>5.74</td><td>3.257</td><td>1.00</td><td>5.74</td><td>0.635</td><td>0.69</td><td>4.47</td><td>0.311</td></tr><tr><td>18h</td><td>1.75</td><td>6.50</td><td>0.074</td><td>1.09</td><td>5.74</td><td>1.275</td><td>1.11</td><td>6.50</td><td>-0.692</td><td>0.67</td><td>4.47</td><td>3.230</td></tr><tr><td>21h</td><td>1.50</td><td>6.50</td><td>3.88</td><td>1.01</td><td>6.50</td><td>-1.927</td><td>1.15</td><td>6.50</td><td>1.378</td><td>0.67</td><td>4.47</td><td>3.097</td></tr><tr><td>24h</td><td>1.26</td><td>6.50</td><td>3.858</td><td>0.94</td><td>5.74</td><td>2.174</td><td>1.02</td><td>5.74</td><td>1.835</td><td>0.69</td><td>3.08</td><td>-0.048</td></tr><tr><td>27h</td><td>1.45</td><td>5.74</td><td>0.165</td><td>0.79</td><td>5.74</td><td>2.848</td><td>0.88</td><td>5.74</td><td>1.207</td><td>0.74</td><td>3.95</td><td>-0.045</td></tr><tr><td>30h</td><td>1.51</td><td>6.50</td><td>-0.825</td><td>0.66</td><td>5.74</td><td>1.349</td><td>0.82</td><td>5.74</td><td>-0.907</td><td>0.81</td><td>3.48</td><td>3.013</td></tr><tr><td>33h</td><td>1.55</td><td>6.50</td><td>3.402</td><td>0.62</td><td>5.74</td><td>-1.705</td><td>0.81</td><td>5.74</td><td>0.855</td><td>0.85</td><td>3.48</td><td>2.985</td></tr><tr><td>36h</td><td>1.51</td><td>6.50</td><td>3.589</td><td>0.72</td><td>5.07</td><td>2.068</td><td>0.90</td><td>5.07</td><td>1.782</td><td>0.80</td><td>3.48</td><td>0.003</td></tr><tr><td>39h</td><td>1.45</td><td>6.50</td><td>0.688</td><td>0.71</td><td>5.07</td><td>3.014</td><td>1.36</td><td>5.74</td><td>1.392</td><td>0.68</td><td>5.07</td><td>-0.107</td></tr><tr><td>42h</td><td>1.34</td><td>6.50</td><td>-1.308</td><td>0.71</td><td>5.07</td><td>1.957</td><td>1.55</td><td>6.50</td><td>-0.527</td><td>0.67</td><td>4.47</td><td>3.208</td></tr><tr><td>45h</td><td>1.29</td><td>6.50</td><td>3.383</td><td>0.67</td><td>5.07</td><td>-1.356</td><td>1.69</td><td>6.50</td><td>0.759</td><td>0.78</td><td>4.47</td><td>3.245</td></tr><tr><td>48h</td><td>1.22</td><td>5.74</td><td>3.767</td><td>0.69</td><td>5.07</td><td>1.170</td><td>1.81</td><td>6.50</td><td>2.298</td><td>0.79</td><td>5.07</td><td>0.194</td></tr><tr><td>51h</td><td>1.16</td><td>5.74</td><td>1.450</td><td>0.68</td><td>4.47</td><td>2.686</td><td>2.08</td><td>6.50</td><td>2.152</td><td>0.76</td><td>5.07</td><td>-1.222</td></tr><tr><td>54h</td><td>1.01</td><td>6.50</td><td>-2.418</td><td>0.75</td><td>4.47</td><td>1.949</td><td>2.13</td><td>7.37</td><td>-0.036</td><td>0.67</td><td>5.07</td><td>2.749</td></tr><tr><td>57h</td><td>0.98</td><td>5.74</td><td>2.492</td><td>1.11</td><td>5.07</td><td>-0.816</td><td>1.93</td><td>7.37</td><td>0.108</td><td>0.60</td><td>5.07</td><td>2.946</td></tr><tr><td>60h</td><td>0.90</td><td>5.74</td><td>3.382</td><td>1.14</td><td>5.74</td><td>1.224</td><td>1.50</td><td>7.37</td><td>2.117</td><td>0.62</td><td>5.07</td><td>0.448</td></tr><tr><td>63h</td><td>0.84</td><td>5.74</td><td>1.825</td><td>1.10</td><td>5.74</td><td>3.070</td><td>1.21</td><td>6.50</td><td>2.062</td><td>0.69</td><td>3.95</td><td>-1.460</td></tr><tr><td>66h</td><td>0.79</td><td>5.74</td><td>-1.897</td><td>1.02</td><td>6.50</td><td>2.658</td><td>1.05</td><td>5.74</td><td>-0.002</td><td>0.76</td><td>3.95</td><td>2.766</td></tr><tr><td>69h</td><td>0.78</td><td>5.74</td><td>2.301</td><td>0.86</td><td>6.50</td><td>-0.406</td><td>0.96</td><td>5.74</td><td>-0.068</td><td>0.73</td><td>4.47</td><td>3.133</td></tr><tr><td>72h</td><td>0.71</td><td>5.74</td><td>3.555</td><td>0.76</td><td>5.74</td><td>0.127</td><td>0.88</td><td>5.74</td><td>2.250</td><td>0.69</td><td>5.07</td><td>1.089</td></tr></tbody></table>

### 2.2.3. Definition and formulations of run-up

According to the EurOtop manual [8], the wave run-up height is represented by $R_{u2\%}$, which is the wave run-up level exceeded by 2% of the individual incident waves, measured vertically from the still water line. From this calculation, a run-up time series associated with the number of incoming waves is obtained for all the waves within a 3-hour sea state, giving a value of $R_{u2\%}$ for every sea state and profile of all the storms.

Two formulations are used in this study. The first formulation is taken from the EurOtop manual [8], where they express a mean value approach for the wave run-up as in Eq. (1).

$$ \frac{R_{u2\%}}{H_{m0}} = 1.65 \cdot \gamma_b \cdot \gamma_f \cdot \gamma_\beta \cdot \xi_{m-1.0} \quad (1) $$

where $R_{u2\%}$ is the wave run-up height exceeded by 2% of the incoming waves, $\gamma_b$ is the influence factor for a berm, $\gamma_f$ is the influence factor for roughness elements on a slope, $\gamma_\beta$ is the influence factor for oblique wave attack and $\xi_{m-1.0}$ is the breaker parameter, surf similarity parameter or Iribarren number is defined by Eq. (2).

$$ \xi_{m-1.0} = \frac{\tan \alpha}{\sqrt{H_{m0}/L_{m-1.0}}} \quad (2) $$

where $\tan \alpha$ is the slope in the front face of the structure, and $H_{m0}$ and $L_{m-1.0}$ are the significant wave height and wavelength in deep waters, in this study, the forcings of the simulations on the boundary.

$L_{m-1.0}$ is calculated as $gT_{m-1.0}^2/(2\pi)$, being $T_{m-1.0}$ the spectral wave period. According to the manual, there is a clear relationship between this spectral wave period $T_{m-1.0}$ and the peak period $T_p$ for conventional single peak wave spectra and can be calculated as $T_p = 1.1 T_{m-1.0}$.

Since there are no berms or bars on the beach of Villers-sur-Mer, the berm factor $\gamma_b$ is set as 1. The roughness factor $\gamma_f$ is also set as 1 to approach the formulation to the smoothness of the bottom level of the numerical model. Finally, the wave attack factor $\gamma_\beta$ is also set as 1 because the 4 storms simulated arrive perpendicularly to the shoreline in this study. Considering these values, the wavelength in deep waters $L_p$ calculated as a function of the peak period $T_p$ as $gT_p^2/(2\pi)$, and making the substitutions, Eq. (1) can be transformed into Eq. (3).

$$ R_{u2\%} = 1.5 \cdot \tan \alpha \cdot \sqrt{H_{m0}L_p} \quad (3) $$

Only dependent on the slope of the beach $\tan \alpha$, and the forcings $H_s$ and $T_p$.

The second formulation compared is the one proposed by Stockdon et al. [33], where the Run-Up 2% is parametrized as in Eq. (4).
$$R_{u2\%} = 1.1 \cdot \left[ \langle \eta \rangle + \frac{S}{2} \right] \quad (4)$$

where $\langle \eta \rangle$ is the setup of the surface of water due to the presence of waves and $S$ is the swash, calculated as in Eq. (5) and Eq. (6), respectively.

$$\langle \eta \rangle \geq 0.35 \cdot \tan \alpha \cdot \sqrt{H_{m0}L_p} \quad (5)$$

$$S = \sqrt{(S_{inc})^2 + (S_{IG})^2} \quad (6)$$

where $S_{inc}$ is the swash uprush related to incident waves and $S_{IG}$ is the additional uprush related to infragravity waves obtained from Eq. (7) and Eq. (8).

$$S_{inc} = 0.75 \cdot \tan \alpha \cdot \sqrt{H_{m0}L_p} \quad (7)$$

$$S_{IG} = 0.06 \cdot \sqrt{H_{m0}L_p} \quad (8)$$

Then, this second formulation of $R_{u2\%}$ of Stockdon et al. [33] would be as in Eq. (9).

$$R_{u2\%} = 1.1 \cdot \left( 0.35 \cdot \tan \alpha \cdot \sqrt{H_{m0}L_p} + \frac{\sqrt{H_{m0}L_p}(0.563 \tan^2 \alpha + 0.004)}{2} \right) \quad (9)$$

Also, it is only dependent on the slope of the beach $\tan \alpha$, and the forcings $H_s$ and $T_p$.

Out of the 4 profiles simulated, 2 are selected as representative: P11 and P15. Both of them have a similar slope between the shoreline and the bathymetric contour line of 10 m. So, an approximative value of tan ($\alpha$) can be set as 0.0264 to obtain the value of $R_{u2\%}$ with the formulations of the EurOtop manual and of Stockdon et al. [33]. The results obtained from the formulations will be compared to the results of the simulations of the 4 storms with the numerical model SWASH.

### 2.2.4. Video monitoring systems and timestacks

The VMS installed at the location provide different products that are useful for the validation of the results. The validation is performed using the timestacks, images formed by stacking side-by-side transsects obtained from snap images at different instants. Each column of a timestack corresponds to a transect at a given instant, so they record the run-up of each wave on one specific profile of the beach during the 10 mins of exposure of each image. By considering the angle of the camera with respect to the transect that composes the timestacks, and the tidal range of the different water levels during a storm, the real-world coordinates of each pixel of the timestack can be calculated, which allows to obtain the horizontal run-up for each instant of the timestack.

From the horizontal run-ups of the timestacks and including the elevation of the surface on the vertical profile, the $R_{u2\%}$ can be extracted every 10 mins during daylight of the 3 days simulated for the 4 storms. The timestacks can be performed when there are no other issues affecting image capture, such as raindrops, insufficient sunlight, foggy weather, or camera malfunctions.

Considering the central camera of the study site as the most useful for validation, the closest profile to the hypothetical perpendicular profile of the camera among the simulated is the one that is used for this purpose. In this case, the closest profile is P11. The morphological conditions of Villers-sur-Mer, with a smooth slope typical of a sandy beach and relatively small waves arriving at this location, facilitate the validation with video.

The VMS of Villers-sur-Mer started the record in March 2019, so, among the four extreme events simulated, only Ciara has available images to perform the validation. And, from the 3 days simulated, only 2 days have processed images, the so-called timestacks, for 10th and 11th February 2020. At this point there are some limitations to consider associated with the simulations and with the VMS. The timestacks are

only available during daylight, approximately from 7:30 in the morning to 18:20 in Villers-sur-Mer at this time of the year and when the water level is not too far from the point of record of the VMS. Regarding the simulations, the main limitation is the time. Each simulation covers 3 h, with each sea state defined by the wave height, the period and the water level. This latter was taken as the maximum water level within the 3 h. This fact provides the ideal instant for performing the comparison between the timestacks and the outputs of the simulation. To be able to compare, the timestack that should be taken is the 10-minute timestack, within the 3 h corresponding to each simulation, that has the highest water level. By doing this, it is sure that the waves are varying over the same part of the profile, and then, the slope is the same. This way, and since the $R_{u2\%}$ is just relative to the number of incoming waves and not the time during which it is calculated, the values can be compared.

After considering all these limitations and looking for the maximum number of available comparisons, 6 sea states among the 25 simulations are the ones that can be validated with timestacks: for 10th and 11th of February, the sea state of 9 h, 12 h and 15 h, each of them with the timestack that has the highest water level. The timestacks are treated using the software developed by Vousdoukas et al. [39], who provide an open-access GUI to extract and process the time series of the cross-shore position of the swash extrema. The cross-shore positions are transformed into elevations using the topographic information provided, which, in this case, corresponds to the profile P11 on which the simulations are performed, located only a few meters away from the profile on which the timestacks are obtained. The swash time series are manually corrected using the GUI interface from the automatically identified time series when necessary. Finally, the two percent exceedance values, the $R_{u2\%}$, are estimated from the swash time series after extracting the local maxima.

A flowchart with the complete approach proposed in the methodology of this study is presented in Fig. 5. It includes the inputs and outputs of each of the two numerical simulations with their respective forcings and the correction and validations of the second one. This offers a robust and adaptable methodology for extreme wave and correlated run-up analysis, making it unique compared to generic studies.

## 3. Results

### 3.1. Validation of the model

A comprehensive validation is performed by evaluating three different aspects of the simulations: the inputs used for the simulations, the outputs of run-up 2% in comparison to theoretical formulations, and the outputs of run-up 2% in comparison to the results extracted from VMS.

#### 3.1.1. Validation of the forcing inputs of Swash

The first validation is applied to the inputs used for the simulations with SWASH of significant wave height ($H_s$) and peak period ($T_p$) with the existing buoys located on the study area. The buoy used is SCENES buoy mentioned before near Villers-sur-Mer, with measurements available for one of the extreme events simulated, the storm Ciara.

The comparison is performed between the records of the buoy and the outputs of the numerical simulation performed in the English Channel obtained on the same exact locations of this buoy (Fig. 6). Both variables show good agreement, despite the differences on the time step on the two sources, to consider accurately enough the inputs of the simulations with SWASH. The other variable the model is forced with is the water level, already corrected as previously explained.

#### 3.1.2. Validation with theoretical formulations

The second validation is applied to the outputs of run-up 2% obtained on the simulations of the 4 storms in Villers-sur-Mer, comparing the outputs of SWASH to the results that can be calculated using theoretical formulations.
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Flowchart of the study, including the numerical simulation of López Solano et al. [19] and the SWASH simulation with their respective forcings, and the validations and corrections nearshore.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Time series comparing the outputs of the numerical simulations in the English Channel with the measurements of the buoy SCENES of (a) significant wave height (Hs) and (b) peak period (Tp), during the 3 days simulated of storm Ciara.</figcaption></figure>

In Table 2, results of the simulations for 2 profiles and the theoretical formulations are presented. To compare the relative changes, for each storm, the cells are colored independently for the numerical results and for the theoretical results conditioning the ranges from green to red. The results are also visually presented in Fig. 7 for each storm in one graphic, showing the values of $R_{u2\%}$ every 3 h obtained from the numerical simulations and the analytical formulations.

From these values, the RMSE is obtained for all the possible combinations of simulations against formulations, having 4 values of RMSE for each storm. In Table 3 all the results and means are presented. This gives a mean RMSE of 0.213 m and 0.277 m for the EurOtop and Stockdon formulations, respectively, for Storm Ciara, a mean RMSE of 0.22 m and 0.314 m for Storm Joachim, a mean RMSE of 0.196 m and 0.322 m for Storm Andrea and a mean RMSE of 0.16 m and 0.226 m for Storm 19. A total averaged RMSE of 0.197 m for the EurOtop [8] for all the storms and an averaged RMSE of 0.285 m for the Stockdon et al. [33] formulations in comparison to the results of $R_{u2\%}$ for the simulations on these profiles is obtained.

The maximum differences happen when the water level set for each sea state is too low or too high with respect to the Mean Sea Level (MSL), producing that the variations of the free surface of water vary over some parts of the profiles where the slope changes rapidly and does not keep a

constant value as it must be defined on the theoretical formulations. On the other hand, when the SWL is set on the part of the profile with an almost constant slope close to the given value of 0.0264, the results for both formulations show a high accuracy with just some millimeters of difference, even with high values of $H_s$ for some sea states. Then, the results of the numerical model can be considered accurate enough in contrast with theoretical formulations.

#### 3.1.2.1. Validation with timesteps from VMS.
Finally, the third and most important validation is done by comparing some specific variables of the simulations to the records of video cameras located at the study site. This comparison is performed to understand the limitations of both datasets used. The outputs of the numerical simulations are obtained with a constant water level set for each 3-hour sea state but chosen as the maximum within the 3 h to be able to compare to the worst scenario recorded by the video cameras.

In Table 4, there is a summary of the characteristics with the $H_s$, the $T_p$, the water level set for the simulation and the water level corresponding to the 10-minute timestamp. The $R_{u2\%}$ calculated from the output of water level of the simulations with SWASH on P11 and the $R_{u2\%}$ calculated from the processed timesteps are also presented.

The results of the validation demonstrate improved performance for
<caption>**Table 2**</caption>

Results of Run-Up 2% (m) from the numerical simulations in P11 and P15 in Villers-sur-Mer and results obtained from the theoretical formulations of the EurOtop manual [8] and Stockdon et al. [33]. Scale of colors are conditioned by the results of the simulations of both profiles and for both formulations independently for each storm.

<table><thead><tr><th rowspan="2">Time</th><th colspan="3">Storm Ciara<br>(09/02/2020 09h –<br>12/02/2020 09h)</th><th colspan="3">Storm Joachim<br>(12/12/2011 21h –<br>15/12/2011 21h)</th><th colspan="3">Storm Andrea<br>(03/01/2012 06h –<br>06/01/2012 06h)</th><th colspan="3">“Storm 19”<br>(28/01/2013 18h –<br>31/01/2013 18h)</th></tr><tr><th>P11</th><th>P15</th><th>EurOtop</th><th>Stockdon</th><th>P11</th><th>P15</th><th>EurOtop</th><th>Stockdon</th><th>P11</th><th>P15</th><th>EurOtop</th><th>Stockdon</th><th>P11</th><th>P15</th><th>EurOtop</th><th>Stockdon</th></tr></thead><tbody><tr><td>00h</td><td>0.34</td><td>0.38</td><td>0.52</td><td>0.59</td><td>0.08</td><td>0.15</td><td>0.34</td><td>0.38</td><td>0.14</td><td>0.18</td><td>0.41</td><td>0.46</td><td>0.11</td><td>0.10</td><td>0.34</td><td>0.38</td></tr><tr><td>03h</td><td>0.28</td><td>0.27</td><td>0.53</td><td>0.60</td><td>0.15</td><td>0.17</td><td>0.39</td><td>0.45</td><td>0.31</td><td>0.32</td><td>0.54</td><td>0.61</td><td>0.24</td><td>0.29</td><td>0.48</td><td>0.54</td></tr><tr><td>06h</td><td>0.49</td><td>0.50</td><td>0.66</td><td>0.75</td><td>0.21</td><td>0.24</td><td>0.50</td><td>0.56</td><td>0.52</td><td>0.49</td><td>0.66</td><td>0.75</td><td>0.45</td><td>0.44</td><td>0.54</td><td>0.61</td></tr><tr><td>09h</td><td>0.99</td><td>1.00</td><td>0.98</td><td>1.11</td><td>0.15</td><td>0.19</td><td>0.50</td><td>0.56</td><td>0.58</td><td>0.63</td><td>0.78</td><td>0.88</td><td>0.27</td><td>0.29</td><td>0.45</td><td>0.51</td></tr><tr><td>12h</td><td>1.23</td><td>1.27</td><td>1.12</td><td>1.27</td><td>0.68</td><td>0.89</td><td>0.91</td><td>1.03</td><td>0.70</td><td>0.76</td><td>0.93</td><td>1.05</td><td>0.23</td><td>0.24</td><td>0.45</td><td>0.51</td></tr><tr><td>15h</td><td>1.56</td><td>1.38</td><td>1.41</td><td>1.59</td><td>0.82</td><td>0.66</td><td>0.96</td><td>1.08</td><td>0.66</td><td>0.74</td><td>0.89</td><td>1.01</td><td>0.38</td><td>0.41</td><td>0.58</td><td>0.65</td></tr><tr><td>18h</td><td>1.22</td><td>1.3</td><td>1.34</td><td>1.52</td><td>0.70</td><td>0.73</td><td>0.93</td><td>1.06</td><td>0.89</td><td>0.80</td><td>1.06</td><td>1.21</td><td>0.48</td><td>0.57</td><td>0.57</td><td>0.65</td></tr><tr><td>21h</td><td>1.66</td><td>1.76</td><td>1.24</td><td>1.40</td><td>0.74</td><td>0.67</td><td>1.01</td><td>1.15</td><td>0.90</td><td>0.83</td><td>1.08</td><td>1.23</td><td>0.47</td><td>0.45</td><td>0.57</td><td>0.65</td></tr><tr><td>24h</td><td>1.56</td><td>1.61</td><td>1.14</td><td>1.29</td><td>0.68</td><td>0.71</td><td>0.87</td><td>0.98</td><td>0.73</td><td>0.70</td><td>0.90</td><td>1.02</td><td>0.16</td><td>0.16</td><td>0.40</td><td>0.45</td></tr><tr><td>27h</td><td>0.88</td><td>0.89</td><td>1.08</td><td>1.22</td><td>0.70</td><td>0.62</td><td>0.80</td><td>0.90</td><td>0.59</td><td>0.65</td><td>0.84</td><td>0.95</td><td>0.32</td><td>0.34</td><td>0.53</td><td>0.60</td></tr><tr><td>30h</td><td>0.88</td><td>0.97</td><td>1.24</td><td>1.41</td><td>0.46</td><td>0.51</td><td>0.73</td><td>0.82</td><td>0.53</td><td>0.58</td><td>0.81</td><td>0.91</td><td>0.32</td><td>0.36</td><td>0.49</td><td>0.55</td></tr><tr><td>33h</td><td>1.29</td><td>1.43</td><td>1.26</td><td>1.43</td><td>0.44</td><td>0.44</td><td>0.71</td><td>0.8</td><td>0.55</td><td>0.53</td><td>0.80</td><td>0.91</td><td>0.35</td><td>0.39</td><td>0.50</td><td>0.57</td></tr><tr><td>36h</td><td>1.24</td><td>1.24</td><td>1.24</td><td>1.41</td><td>0.49</td><td>0.46</td><td>0.67</td><td>0.76</td><td>0.54</td><td>0.52</td><td>0.75</td><td>0.85</td><td>0.27</td><td>0.23</td><td>0.48</td><td>0.55</td></tr><tr><td>39h</td><td>0.97</td><td>1.05</td><td>1.22</td><td>1.38</td><td>0.55</td><td>0.57</td><td>0.66</td><td>0.75</td><td>0.81</td><td>0.91</td><td>1.04</td><td>1.18</td><td>0.46</td><td>0.49</td><td>0.65</td><td>0.74</td></tr><tr><td>42h</td><td>1.05</td><td>0.82</td><td>1.17</td><td>1.33</td><td>0.49</td><td>0.46</td><td>0.66</td><td>0.75</td><td>1.02</td><td>1.07</td><td>1.26</td><td>1.43</td><td>0.44</td><td>0.56</td><td>0.57</td><td>0.64</td></tr><tr><td>45h</td><td>1.10</td><td>1.19</td><td>1.15</td><td>1.31</td><td>0.55</td><td>0.47</td><td>0.65</td><td>0.73</td><td>1.11</td><td>1.22</td><td>1.32</td><td>1.49</td><td>0.55</td><td>0.65</td><td>0.61</td><td>0.70</td></tr><tr><td>48h</td><td>0.88</td><td>1.06</td><td>0.99</td><td>1.12</td><td>0.42</td><td>0.46</td><td>0.65</td><td>0.74</td><td>1.25</td><td>1.39</td><td>1.36</td><td>1.54</td><td>0.52</td><td>0.55</td><td>0.70</td><td>0.79</td></tr><tr><td>51h</td><td>0.74</td><td>0.75</td><td>0.96</td><td>1.09</td><td>0.46</td><td>0.57</td><td>0.57</td><td>0.65</td><td>1.40</td><td>1.62</td><td>1.46</td><td>1.66</td><td>0.60</td><td>0.64</td><td>0.69</td><td>0.78</td></tr><tr><td>54h</td><td>0.77</td><td>0.89</td><td>1.02</td><td>1.15</td><td>0.45</td><td>0.47</td><td>0.60</td><td>0.68</td><td>1.52</td><td>1.70</td><td>1.67</td><td>1.90</td><td>0.54</td><td>0.63</td><td>0.64</td><td>0.73</td></tr><tr><td>57h</td><td>0.79</td><td>0.76</td><td>0.88</td><td>1.00</td><td>0.77</td><td>0.67</td><td>0.83</td><td>0.94</td><td>1.39</td><td>1.41</td><td>1.59</td><td>1.81</td><td>0.51</td><td>0.51</td><td>0.61</td><td>0.69</td></tr><tr><td>60h</td><td>0.86</td><td>0.69</td><td>0.85</td><td>0.96</td><td>0.57</td><td>0.63</td><td>0.95</td><td>1.08</td><td>1.23</td><td>1.26</td><td>1.4</td><td>1.59</td><td>0.40</td><td>0.44</td><td>0.62</td><td>0.70</td></tr><tr><td>63h</td><td>0.63</td><td>0.59</td><td>0.82</td><td>0.93</td><td>0.89</td><td>0.89</td><td>0.94</td><td>1.06</td><td>0.85</td><td>0.94</td><td>1.11</td><td>1.26</td><td>0.41</td><td>0.39</td><td>0.51</td><td>0.58</td></tr><tr><td>66h</td><td>0.59</td><td>0.48</td><td>0.8</td><td>0.9</td><td>0.69</td><td>0.72</td><td>1.02</td><td>1.16</td><td>0.77</td><td>0.72</td><td>0.92</td><td>1.04</td><td>0.42</td><td>0.40</td><td>0.53</td><td>0.61</td></tr><tr><td>69h</td><td>0.59</td><td>0.59</td><td>0.79</td><td>0.89</td><td>0.72</td><td>0.67</td><td>0.94</td><td>1.06</td><td>0.68</td><td>0.69</td><td>0.87</td><td>0.99</td><td>0.48</td><td>0.56</td><td>0.60</td><td>0.68</td></tr><tr><td>72h</td><td>0.78</td><td>0.92</td><td>0.75</td><td>0.85</td><td>0.56</td><td>0.61</td><td>0.78</td><td>0.88</td><td>0.65</td><td>0.64</td><td>0.84</td><td>0.95</td><td>0.43</td><td>0.45</td><td>0.66</td><td>0.74</td></tr></tbody></table>

the highest values of $H_s$, below 5 cm, while exhibiting a more pronounced discrepancy at the lower values. The most significant difference is observed for the sea state with $H_s = 1.16$ m, $T_p = 5.74$ s and $WL = 1.45$ m.

Even if the results are apparently satisfactory, the diverse limitations throughout the process add uncertainties that have to be addressed in future developments. Nevertheless, the overall methodology and results accomplished the aim of the validation.

### 3.1.3. Impacts of physical drivers on the beach of Villers-sur-Mer

For the 4 profiles simulated, P7, P11, P15 and P19, the results of $R_{u2%}$ are obtained for each storm independently in order to compare and analyze the transformation from waves on the boundary of the numerical model to the run-up over the profile of the beach. The results are presented in Table 5 with the scale of colors conditioned to the variation of the $R_{u2%}$ independently for each storm and in Fig. 8 the results of $R_{u2%}$ for each sea state of the four storms.

At first sight, it is remarkable that the highest value of $H_s$ does not produce the maximum value of $R_{u2%}$, for storm Ciara in Villers-sur-Mer, it is the highest value of water level, but in combination with a high $H_s$ gives the greatest $R_{u2%}$, surpassing the 1.65 m for every profile. This is

probably because of the transversal shape of the beach, facilitating the uprush of the water on the upper part and generating the opposite on the down part of the profiles. After the two sea states with the highest water level, the peak of the storm, defined by the maximum $H_s$, produces a run-up of >1.3 m on every profile, reaching a value of 1.55 m on P11. The first two instants, with low values of $H_s$ and $T_p$ barely produce any $R_{u2%}$. In general, the water level apparently presents a bigger influence on the values of $R_{u2%}$ obtained than $H_s$. It can be noticed, too, that all the profiles vary approximately in the same way, growing and decreasing harmonically all along the beach.

In the case of storm Joachim, after the fourth sea state, the $R_{u2%}$ remains in a range between 0.3 m and 0.9 m because the $H_s$ of storm Joachim takes values of ~0.7 m for most of the sea states of the storm, producing a $R_{u2%}$ of 0.4–0.5 m, whatever the water level set is for each simulation. The maximum $R_{u2%}$ is reached from $t = 12$ h to $t = 24$ h, and in the last sea states, because the time series of $H_s$ takes its peak and grows again with a second peak, respectively, but still no values higher than 0.9 m since neither the $H_s$ takes values higher than 1.15 m. For the four first instants, the $T_p$ is too low to produce barely any run-up, in combination with an $H_s$ lower than 0.83 m.
<figure><img src="image_9.png" /><figcaption>Fig. 7. Results of Run-Up 2% from the numerical simulations and for the formulations of EurOtop and Stockdon for the four simulated storms in Villers-sur-Mer: a) Ciara, b) Joachim, c) Andrea and d) Storm 19.</figcaption></figure>

**Table 3**

Results of RMSE (m) calculated with the Run-Up 2% from the numerical simulations in P11 and P15 in Villers-sur-Mer and results obtained from the theoretical formulations of the EurOtop manual [8] and Stockdon et al. [33].

<table><thead><tr><th rowspan="2"></th><th colspan="3">Ciara</th><th colspan="3">Joachim</th><th colspan="3">Andrea</th><th colspan="3">Storm 19</th><th rowspan="2">Mean</th></tr><tr><th>P11</th><th>P15</th><th>Mean</th><th>P11</th><th>P15</th><th>Mean</th><th>P11</th><th>P15</th><th>Mean</th><th>P11</th><th>P15</th><th>Mean</th></tr></thead><tbody><tr><td>EurOtop</td><td>0.20</td><td>0.22</td><td>0.21</td><td>0.22</td><td>0.22</td><td>0.22</td><td>0.21</td><td>0.19</td><td>0.20</td><td>0.17</td><td>0.15</td><td>0.16</td><td>0.197</td></tr><tr><td>Stockdon</td><td>0.27</td><td>0.28</td><td>0.28</td><td>0.32</td><td>0.31</td><td>0.31</td><td>0.34</td><td>0.31</td><td>0.32</td><td>0.24</td><td>0.22</td><td>0.23</td><td>0.285</td></tr></tbody></table>

**Table 4**

Summary of characteristics of the 6 sea states of Villers-sur-Mer for which the validation of run-up 2% with timesteps obtained from VMS has been performed.

<table><thead><tr><th rowspan="2">Date</th><th colspan="5">Simulation</th><th colspan="3">Timestamp</th></tr><tr><th>Hour</th><th>H<sub>s</sub> (m)</th><th>T<sub>p</sub> (s)</th><th>WL (m)</th><th>R<sub>u2%</sub> (m)</th><th>Hour</th><th>WL (m)</th><th>R<sub>u2%</sub> (m)</th></tr></thead><tbody><tr><td>10/02/2020</td><td>09h</td><td>1.49</td><td>6.50</td><td>3.88</td><td>1.659</td><td>10 :10</td><td>3.8</td><td>1.6613</td></tr><tr><td>10/02/2020</td><td>12h</td><td>1.26</td><td>6.50</td><td>3.86</td><td>1.562</td><td>10 :40</td><td>3.88</td><td>1.5881</td></tr><tr><td>10/02/2020</td><td>15h</td><td>1.45</td><td>5.74</td><td>0.16</td><td>0.879</td><td>13 :40</td><td>0.67</td><td>0.9097</td></tr><tr><td>11/02/2020</td><td>09h</td><td>1.29</td><td>6.50</td><td>3.38</td><td>1.098</td><td>10 :20</td><td>3.1</td><td>1.0178</td></tr><tr><td>11/02/2020</td><td>12h</td><td>1.22</td><td>5.74</td><td>3.77</td><td>0.881</td><td>12 :40</td><td>3.83</td><td>0.7807</td></tr><tr><td>11/02/2020</td><td>15h</td><td>1.16</td><td>5.74</td><td>1.45</td><td>0.736</td><td>14 :00</td><td>1.45</td><td>0.3584</td></tr></tbody></table>

Storm Andrea arrives to the nearshore with the highest $H_s$ among the four storms. The water level varies within a range of -0.5 m to 2 m during the 3 days of this storm, so, by staying around this part of the profile, the results of $R_{u2%}$ seem to follow much more the trends of the

time series of $H_s$ than in the previous storms analyzed. After the sea state in which the storm starts to increase stronger, $t = 36$ h, the $R_{u2%}$ takes values greater than 1 m in every profile and during the next 7 sea states, reaching values of 1.6 – 1.7 m. The maximum water level is also reached
<caption>**Table 5**</caption>

Results of Run-Up 2% (m) from the numerical simulations for storms Ciara, Joachim, Andrea and 19 for the 4 selected profiles in Villers-sur-Mer. Scale of colors are conditioned by the results of the simulations of each storm.

<table><thead><tr><th rowspan="2">Time</th><th colspan="4">Storm 19</th><th colspan="4">Storm Andrea</th><th colspan="4">Storm Joachim</th><th colspan="4">Storm Ciara</th></tr><tr><th>P7</th><th>P11</th><th>P15</th><th>P19</th><th>P7</th><th>P11</th><th>P15</th><th>P19</th><th>P7</th><th>P11</th><th>P15</th><th>P19</th><th>P7</th><th>P11</th><th>P15</th><th>P19</th></tr></thead><tbody><tr><td>00h</td><td>0.08</td><td>0.11</td><td>0.10</td><td>0.06</td><td>0.17</td><td>0.14</td><td>0.18</td><td>0.16</td><td>0.11</td><td>0.08</td><td>0.15</td><td>0.13</td><td>0.35</td><td>0.34</td><td>0.38</td><td>0.45</td></tr><tr><td>03h</td><td>0.18</td><td>0.24</td><td>0.2</td><td>0.30</td><td>0.20</td><td>0.31</td><td>0.32</td><td>0.26</td><td>0.16</td><td>0.15</td><td>0.17</td><td>0.19</td><td>0.21</td><td>0.28</td><td>0.27</td><td>0.18</td></tr><tr><td>06h</td><td>0.33</td><td>0.45</td><td>0.44</td><td>0.51</td><td>0.32</td><td>0.52</td><td>0.49</td><td>0.46</td><td>0.23</td><td>0.21</td><td>0.24</td><td>0.26</td><td>0.34</td><td>0.49</td><td>0.50</td><td>0.44</td></tr><tr><td>09h</td><td>0.29</td><td>0.27</td><td>0.29</td><td>0.29</td><td>0.66</td><td>0.58</td><td>0.63</td><td>0.72</td><td>0.15</td><td>0.15</td><td>0.19</td><td>0.13</td><td>0.90</td><td>0.99</td><td>1.00</td><td>0.89</td></tr><tr><td>12h</td><td>0.19</td><td>0.23</td><td>0.24</td><td>0.24</td><td>0.78</td><td>0.70</td><td>0.76</td><td>0.82</td><td>0.81</td><td>0.68</td><td>0.89</td><td>0.83</td><td>1.08</td><td>1.23</td><td>1.27</td><td>1.14</td></tr><tr><td>15h</td><td>0.33</td><td>0.38</td><td>0.41</td><td>0.38</td><td>0.59</td><td>0.66</td><td>0.74</td><td>0.68</td><td>0.66</td><td>0.82</td><td>0.66</td><td>0.84</td><td>1.37</td><td>1.56</td><td>1.38</td><td>1.32</td></tr><tr><td>18h</td><td>0.5</td><td>0.48</td><td>0.57</td><td>0.53</td><td>0.76</td><td>0.89</td><td>0.80</td><td>0.75</td><td>0.67</td><td>0.70</td><td>0.73</td><td>0.72</td><td>1.23</td><td>1.22</td><td>1.30</td><td>1.17</td></tr><tr><td>21h</td><td>0.44</td><td>0.47</td><td>0.45</td><td>0.49</td><td>0.82</td><td>0.90</td><td>0.83</td><td>0.81</td><td>0.71</td><td>0.74</td><td>0.67</td><td>0.67</td><td>1.67</td><td>1.66</td><td>1.76</td><td>1.72</td></tr><tr><td>24h</td><td>0.08</td><td>0.16</td><td>0.16</td><td>0.19</td><td>0.69</td><td>0.73</td><td>0.70</td><td>0.72</td><td>0.72</td><td>0.68</td><td>0.71</td><td>0.73</td><td>1.57</td><td>1.56</td><td>1.61</td><td>1.57</td></tr><tr><td>27h</td><td>0.25</td><td>0.32</td><td>0.34</td><td>0.29</td><td>0.56</td><td>0.59</td><td>0.65</td><td>0.58</td><td>0.62</td><td>0.70</td><td>0.62</td><td>0.73</td><td>0.85</td><td>0.88</td><td>0.89</td><td>0.91</td></tr><tr><td>30h</td><td>0.34</td><td>0.32</td><td>0.36</td><td>0.43</td><td>0.62</td><td>0.53</td><td>0.58</td><td>0.53</td><td>0.50</td><td>0.46</td><td>0.51</td><td>0.59</td><td>1.05</td><td>0.88</td><td>0.97</td><td>1.07</td></tr><tr><td>33h</td><td>0.37</td><td>0.35</td><td>0.39</td><td>0.46</td><td>0.52</td><td>0.55</td><td>0.53</td><td>0.62</td><td>0.53</td><td>0.44</td><td>0.44</td><td>0.54</td><td>1.41</td><td>1.29</td><td>1.43</td><td>1.27</td></tr><tr><td>36h</td><td>0.17</td><td>0.27</td><td>0.23</td><td>0.24</td><td>0.66</td><td>0.54</td><td>0.52</td><td>0.64</td><td>0.54</td><td>0.49</td><td>0.46</td><td>0.49</td><td>1.24</td><td>1.24</td><td>1.24</td><td>1.22</td></tr><tr><td>39h</td><td>0.41</td><td>0.46</td><td>0.49</td><td>0.40</td><td>0.81</td><td>0.81</td><td>0.91</td><td>0.96</td><td>0.53</td><td>0.55</td><td>0.57</td><td>0.58</td><td>1.01</td><td>0.97</td><td>1.05</td><td>0.96</td></tr><tr><td>42h</td><td>0.43</td><td>0.44</td><td>0.56</td><td>0.56</td><td>1.02</td><td>1.02</td><td>1.07</td><td>0.99</td><td>0.56</td><td>0.49</td><td>0.46</td><td>0.47</td><td>1.06</td><td>1.05</td><td>0.82</td><td>0.76</td></tr><tr><td>45h</td><td>0.49</td><td>0.55</td><td>0.65</td><td>0.60</td><td>1.12</td><td>1.11</td><td>1.11</td><td>1.19</td><td>1.19</td><td>0.57</td><td>0.55</td><td>0.47</td><td>0.98</td><td>1.10</td><td>1.19</td><td>1.09</td></tr><tr><td>48h</td><td>0.45</td><td>0.52</td><td>0.55</td><td>0.47</td><td>1.25</td><td>1.25</td><td>1.25</td><td>1.39</td><td>1.38</td><td>0.42</td><td>0.42</td><td>0.46</td><td>0.5</td><td>1.05</td><td>0.88</td><td>0.88</td></tr><tr><td>51h</td><td>0.64</td><td>0.60</td><td>0.64</td><td>0.62</td><td>1.4</td><td>1.40</td><td>1.62</td><td>1.61</td><td>0.40</td><td>0.46</td><td>0.57</td><td>0.47</td><td>0.75</td><td>0.74</td><td>0.75</td><td>0.73</td></tr><tr><td>54h</td><td>0.51</td><td>0.54</td><td>0.63</td><td>0.48</td><td>1.53</td><td>1.52</td><td>1.7</td><td>1.69</td><td>0.49</td><td>0.45</td><td>0.47</td><td>0.43</td><td>0.74</td><td>0.77</td><td>0.89</td><td>0.8</td></tr><tr><td>57h</td><td>0.49</td><td>0.51</td><td>0.51</td><td>0.51</td><td>1.39</td><td>1.39</td><td>1.41</td><td>1.50</td><td>0.72</td><td>0.77</td><td>0.67</td><td>0.6</td><td>0.77</td><td>0.79</td><td>0.76</td><td>0.73</td></tr><tr><td>60h</td><td>0.36</td><td>0.40</td><td>0.44</td><td>0.44</td><td>1.24</td><td>1.23</td><td>1.26</td><td>1.34</td><td>0.56</td><td>0.57</td><td>0.63</td><td>0.61</td><td>0.69</td><td>0.86</td><td>0.69</td><td>0.86</td></tr><tr><td>63h</td><td>0.40</td><td>0.41</td><td>0.39</td><td>0.38</td><td>0.93</td><td>0.85</td><td>0.94</td><td>1.03</td><td>0.84</td><td>0.89</td><td>0.89</td><td>0.85</td><td>0.62</td><td>0.63</td><td>0.59</td><td>0.68</td></tr><tr><td>66h</td><td>0.40</td><td>0.42</td><td>0.40</td><td>0.39</td><td>0.68</td><td>0.77</td><td>0.72</td><td>0.66</td><td>0.70</td><td>0.69</td><td>0.72</td><td>0.64</td><td>0.59</td><td>0.59</td><td>0.48</td><td>0.61</td></tr><tr><td>69h</td><td>0.50</td><td>0.48</td><td>0.56</td><td>0.49</td><td>0.63</td><td>0.68</td><td>0.69</td><td>0.65</td><td>0.61</td><td>0.72</td><td>0.67</td><td>0.61</td><td>0.59</td><td>0.59</td><td>0.59</td><td>0.60</td></tr><tr><td>72h</td><td>0.42</td><td>0.43</td><td>0.45</td><td>0.52</td><td>0.65</td><td>0.65</td><td>0.64</td><td>0.71</td><td>0.54</td><td>0.56</td><td>0.61</td><td>0.49</td><td>0.81</td><td>0.78</td><td>0.92</td><td>0.83</td></tr></tbody></table>

during the instants around the peak of the storm, generating worst scenarios in terms of combination of a high water level with a high $R_{u2%}$. The profiles seem to differ more in their behavior during the 3 days of simulation of Andrea, probably due to the smaller range in the variation of the water level.

For the last simulation, the storm 19, the behavior of $R_{u2%}$ presents an almost flat evolution in time, with barely any variation around ~0.4 or ~0.5 m, since Storm 19 never reaches 1 m of height.

Analyzing the four events together, some straight conclusions are obtained.

The worst scenario is produced when the time series of $H_s$ grows and the water level reaches the high tide, even if this sea state is not the peak of the storm. The $R_{u2%}$ is more influenced by the effects of water level of each state rather than those of $H_s$. Indeed, the combination of medium to

high values of both variables could produce the highest value of $R_{u2%}$. For Villers-sur-Mer, as a sandy beach, the $R_{u2%}$ seems to grow in concordance with the increase of water level and decreases also accordingly, but not with a strong connection.

Nonetheless, Storm Andrea gives an exception compared to the aforementioned results. The results of $R_{u2%}$ during the 3 days simulated on the model follow the patterns of the forcing of $H_s$ much more than for the other events. Even if it is still influenced by the forcing of water level, the trends match with the wave height. This is justified by the smaller range of water level in comparison to the other storms. For the rest, the water level set for the 25 simulations varies >4 m, and even almost 6 m during storm Ciara, but for storm Andrea the range stays under 3 m, producing that the waves propagate only around a limited part of the profile. Considering that the beach profiles do not have a constant slope,
<figure><img src="image_10.png" /><figcaption>Fig. 8. Results of Run-Up 2% from the numerical simulations for a) storm Ciara, b) storm Joachim, c) storm Andrea and d) storm 19 for the 4 profiles.</figcaption></figure>

keeping the interaction between the surface of the water and the topography on a restricted part of the profile produces that the slope remains similar. Then, the water level has less effect on the results, and the $H_s$ has a stronger influence on the $R_{u2%}$.

A few sea states among all the storms, mainly the initial sea states of most of the simulated events, barely produce any $R_{u2%}$, being even 0 m for some of them. This circumstance has a clear explanation because all these sea states have a common characteristic: the forcings of $H_s$ and $T_p$ have very low values, particularly, the peak period. When the $T_p$ is around ~3 s, the waves cannot reach the shore with a wavelength long enough to uprush the beach, especially in combination with a low value of $H_s$.

Finally, comparing the different selected profiles, no difference can be appreciated along the beach, all the profiles seem to vary harmonically independently on their location in front of Villers-sur-Mer.

## 3.2. Evolution of the different storms from the English Channel to the beach

Additionally, a specific regional application is proposed, considering the influence of the four storms affecting the English Channel in correlation to the flooding generated when they arrive at the Normandy coast. Following what is presented in López Solano et al. [19], storm Joachim has a WNW component, with a direction in the Atlantic Ocean of 271° at its peak, that remains as the averaged direction during the

extreme event. Storm Andrea comes from the WSW sector, with a 246° direction at the peak in origin, but with an average direction of 260°, and Storm 19 presents a Southern component, with a direction of 239° at the peak and averaging <245° during the event. As demonstrated, these directions condition their propagation through the English Channel, with Andrea presenting the highest $H_s$ when it reaches the coasts of Normandy, although Joachim is more important in the Atlantic Ocean in terms of energy. This is explained because this extreme event follows the orientation of the basin, around 70° to the North, suffering from a lower refraction and, therefore, arriving with less modifications.

Ciara is the most powerful storm registered over the last few years. In the Atlantic Ocean, like Andrea, it has a direction within the WSW sector, with a direction of 266° at its peak, that stays at 260° during the event. Also, it has the highest $H_s$ in the open boundary of the English Channel (9.44 m) in comparison to the others (8.26 m, 8.42 m and 6.83 m for Joachim, Andrea and Storm 19, respectively).

To track the storms through the English Channel and till the beach of Villers-sur-Mer, the contour lines of $H_s$ of 3 and 6 m associated with these four storms are shown in Fig. 9 on the left column alongside the contour lines of $R_{u2%}$ on the beach of Villers-sur-Mer in the same corresponding instants for the four storms on the right column.

The maximum values of $H_s$ are reached in the English Channel some instants before the maximum $R_{u2%}$ calculated on the beach of Villers-sur-Mer. As can be seen in the left column of Fig. 9, between 9 and 3 h before the maximum $R_{u2%}$, the values of 3 m of $H_s$ pass along the Channel
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Significant wave height contour lines of $H_s = 3$ m and $H_s = 6$ m of the four storms (Ciara, Joachim, Andrea and "Storm 19") in consecutive instants propagating in the English Channel, and run-up 2% contour lines on the beach of Villers-sur-Mer in consecutive and correlated instants. The location of Villers-sur-Mer is marked with a red point.</figcaption></figure>

reaching even the North Sea. In the case of storm Ciara, the highest among the last storms in the area, the contour line of 6 m almost arrives to the coasts of Normandy. After that, the values of $H_s$ start to decrease, with the contour lines of 3 m moving back towards the Atlantic Ocean, and it is in that instant when the highest $R_{U2\%}$ is reached, which is

explained by a higher water level associated with those instants.

Therefore, it can be appreciated how the $R_{U2\%}$ is more affected by the water level of each instant of simulation rather than the $H_s$, and the combination of both gives the highest values of $R_{U2\%}$, happening for the four storms. This confirms the conclusions obtained from the analysis
nearshore, providing added value with the integration of scales.

Nevertheless, the highest values of $R_{u2\%}$ are reached with the most powerful storm, storm Ciara, which is the one with the highest values of $H_s$ in the English Channel, too. With Andrea, they have the optimal direction to propagate through the basin, suffering from fewer modifications and arriving at the coasts of Normandy with the highest $H_s$ and, in consequence, generating the highest $R_{u2\%}$. This establishes a direct influence of the direction at the peak and averaged during a stormy event in the Atlantic Ocean before entering the English Channel, on the flooding measured by the run-up height on the beach of Villers-sur-Mer, representative of the coasts of Normandy.

# 4. Discussion

## 4.1. The use of Swash model

In this study, the numerical model SWASH has been chosen for simulating the wave storm events producing flooding on the coast. SWASH is a preferable option when a high number of simulations have to be carried out. According to the manual of the model [35], 10 or more vertical layers should be included to accurately compute the phase velocity and the breaking wavefront. For the macrotidal sandy beach selected for this study, setting up more than the 10 vertical layers overcomplicates the calculations and produces some of the simulations to finish unexpectedly, not obtaining appropriate results. The set-up adopted with 10 layers is accurate. Furthermore, the "conditioning area" added at the beginning of the profile of >2 wavelengths for the highest periods (in this case, ~200 m) allows the waves to propagate on the real profile with the desired shape. Also, being SWASH a phase-resolving model, the cross-shore grid spacing $\Delta x$ of 1 m on the cells is also appropriate, typically keeping the relation $L_0 / \Delta x$ between 50 and 100 for this kind of models. Increasing the resolution by reducing $\Delta x$ led to a higher number of instabilities after some tests already carried out for this study site.

XBeach was also considered for this study. The model presents two main modes: the XBeach Nonhydrostatic (XB-NH), which also resolves the NLSW equations with a non-hydrostatic pressure correction term, and the XBeach Surfbeat (XB-SB), which solves the wave-action equation, being a phase-averaged model, the reason why this mode of XBeach was straightly discarded. In comparison to XB-NH, SWASH allows the inclusion of a varying number of vertical layers, not averaging the results in depth, while the maximum number of vertical layers in XB-NH is two. Using XB-NH with no definition of the vertical distribution is equivalent to one-layer modelling with SWASH. The use of depth-averaging models such as XB-NH could considerably decrease the computational time, but previous studies [14,40] found that depth-averaged models were unable to accurately estimate the evolution of the surface of water due to their lack of vertical resolution. On the opposite side, there are the Boussinesq-type models and the depth-resolving models that solve the Navier-Stokes equations, but which require a much higher computational demand, not always leading to more accurate results (a full comparison of the use of the different numerical models and their application on the simulation of waves and their interaction with coastal structures can be found in Lashley [15], which was used as a base for the election of the most appropriate model in this study).

## 4.2. Accuracy of the run-up numerical simulations

The validation of the results of run-up obtained from the numerical simulations also remarks the difficulty of finding the most satisfactory formulation that accurately parametrizes the run-up. The two formulations applied in this study [8,33] show similar trends to the results obtained from the numerical simulations in Villers-sur-Mer, the macrotidal sandy beach where the waves propagate over an impermeable layer, particularly the formulation of Stockdon et al. [33]. The main problem

with the parametrization of wave run-up comes from the way the energy is calculated, relying entirely on the offshore wave spectrum using wave statistical and spectral parameters, as stated by Vousdoukas et al. [39], which is a simplification and possible source of errors. Additionally, the present study highlights the limitation in the formulations produced by the introduction of a constant value of the Iribarren number, with just one value of the slope, when, on real profiles, this slope is variable.

For these reasons, the validation of the results of the numerical simulations can be performed using images obtained from VMS. Even considering the diverse limitations of both the numerical modelling and the records from videocameras, the GUI interface developed by Vousdoukas et al. [39] gives similar results to those obtained from SWASH, being the validation restricted to the daylight and the availability of records during the days of the storms selected. In Villers-sur-Mer, the images allow measuring the $R_{u2\%}$ because of the smoother slope of the beach. For this study, the images used for the validation have been the timesteps, presenting wave by wave the run-up on the beach for every interval of 10 mins. The timestamp of 10 mins with the highest associated water level within the 3 h covered by every sea state has been taken for the validation, given that the highest water level of the same period was set for each simulation on the model. The 2% exceedance of the time series of run-up can be compared with every pair of results of outputs of the model and timesteps, because this variable just depends on the number of incoming waves and not on the time during which it is calculated. Taking this into consideration, the results of the comparison are satisfactory, with differences lower than 10 cm for most of the sea states.

## 4.3. Limitations and applicability to other coastal contexts

The simulation of flooding during extreme events and its validation using VMS in a coastal system of Normandy has been investigated for the first time in this area by coupling numerical and analytical methods. The use of a high-resolution bathymetry extracted from a LiDAR campaign shows a methodological advance compared to more conventional studies. Nevertheless, the intertidal bathymetry changes during the storm and should be considered in the numerical model, which could be obtained from enhanced VMS, providing the water extension, including the run-up and the infragravity waves. The role of VMS is very important in coastal systems with complex internal parameters such as morphology and the presence of anthropogenic activities, as in Normandy. Yet, the quality of snap images and the timesteps, worsened during winter periods, is the other limitation of the use of VMS.

Other sites located on the Normandy coasts were considered for the study, more specifically, Etretat and Hautot-sur-Mer. The morphology of these beaches' changes with respect to Villers-sur-Mer, since they are considered as pure gravel and mixed beaches, respectively. Nevertheless, a different set of difficulties were found for them. The validation should be performed considering the presence of a vertical wall at the end of the two beaches, reached by the run-up of waves during the storms and occasionally overtopped. The complication of numerically reproducing the presence of pebbles in gravel and mixed beaches is of great importance. Soloy et al. [31] proved that parametrizing this kind of sediment using the numerical model XBeach leads to more difficulties, highlighting that morphodynamics also play an important role.

# 5. Conclusions

This work investigates coastal flooding assessment through the determination of wave run-up by using the non-hydrostatic numerical model SWASH with VMS observations. This combined approach has been applied to a macrotidal beach, Villers-sur-Mer, at Normandy coast (NW France), to simulate flood extension in response to various energetic scenarios of storms.

The wave run-up has been estimated for four typical storms with different characteristics, Ciara, Joachim, Andrea and the so-called Storm
19. Results have shown that the maximum run-up is produced during storms combining high water levels with energetic wave height. The elevation of wave run-up height increases in accordance with the variation of the water level, not exhibiting a linear correlation between both variations, meanwhile it is strongly modulated by wave height when the water level changes are lower, as observed for the storm Andrea. However, small waves with low heights and periods are limited to reach the coast since their wavelengths are not enough to uprush the beach, and, therefore, no run-up can be produced.

This study also presents an integration of regional and local scales, connecting the propagation of extreme waves during storms in the English Channel to their induced run-up on the beach of Villers-sur-Mer. The highest run-up is produced by storms with a WSW direction in the Atlantic Ocean when they approach the Channel. This range of directions produces a propagation of waves following the orientation of the basin (of around 70°N) and suffers from a weaker refraction compared to those coming from the Southern sector. In spite of their limited energy in the open boundary of the English Channel, these storms reach the coast with a higher wave height, and, consequently, generate more important flooding. The integration of scales also shows that the dynamics of wave run-up are more sensitive to the variation of water levels, as already demonstrated by the overall results.

As further perspectives, and in the framework of the new altimetry mission SWOT, a new approach based on satellite-derived topo-bathymetry and coastal hydrodynamic monitoring will be developed to enhance the physical set-up of the numerical modelling and extend the validation of results.

Furthermore, this work could be supported by the use of hybrid numerical models. Such models couple the far-shore inviscid solver models based on either fully non-linear potential flow theory (FNPT), non-linear shallow water equations (SWE) or the Boussinesq equations with the viscous solver models based on Navier-Stokes equations. This approach could be useful to optimize the computational time and obtain more accurate representation of the near-shore flow features.

## CRediT authorship contribution statement

**Carlos López Solano:** Writing – original draft, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Emma Imen Turki:** Writing – review & editing, Writing – original draft, Visualization, Supervision, Project administration, Methodology, Conceptualization. **Ángel David Gutiérrez Barceló:** Writing – review & editing, Validation, Software, Methodology, Formal analysis, Conceptualization. **Harish Selvam:** Writing – review & editing, Methodology. **Ernesto Tonatiuh Mendoza:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis. **Stephan Costa:** Writing – review & editing, Supervision. **Edward Salameh:** Writing – review & editing, Visualization, Methodology, Formal analysis. **Antonin Migaud:** Writing – review & editing, Supervision, Project administration. **Robert Lafite:** Writing – review & editing, Writing – original draft, Visualization, Supervision, Methodology, Formal analysis, Conceptualization.

## Declaration of competing interest

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

## Acknowledgements

C.L.S. and E.I.T. acknowledge financial support from the Normandy region (France) through the program RIN. This research was also funded by the Institute for Radiological Protection and Nuclear Safety (IRSN) and the Water Authority of Seine Normandy (AESN). We are grateful to the international project SWOT 3MC funded by CNES-TOSCA and

related to the mission of Surface Water and Ocean Topography (SWOT).

## Data availability

Data will be made available on request.

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