# Estimation of harbor and bay resonances by MMS-FEM model with application to the Bay of Toulon, France

Kostas Belibassakis, Vincent Rey

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Kostas Belibassakis, Vincent Rey. Estimation of harbor and bay resonances by MMS-FEM model with application to the Bay of Toulon, France. Oceanologia, 2024, 66 (4), 10.5697/LOZC6742. hal-04914661

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# Estimation of harbor and bay resonances by MMS-FEM model with application to the bay of Toulon France

Kostas Belibassakis<sup>(1,*)</sup>, Vincent Rey<sup>(2)</sup>

(1) School of Naval Architecture and Marine Engineering, National Technical University of Athens, Zografos 15780, Athens, Greece. e-mail: kbel@fluid.mech.ntua.gr

(2) Université de Toulon, Aix-Marseille Université, CNRS, IRD, MIO, Toulon, France. e-mail: vincent.rey@univ-tln.fr

(*) Corresponding author

## Abstract

Bay and harbor resonances are investigated in this work, taking into account the variable bathymetry of the semi-enclosed basin. The Modified Mild-Slope (MMS) equation is implemented for the description of combined refraction-diffraction effects, from which the eigenperiods and eigenmodes are calculated by means of a low-order Finite Element Method (FEM scheme). The model is first applied to coastal-port region of Toulon, France, illustrating the versatility of the model to easily include coastal structures such as detached breakwater. Next, the present model is applied to the extended nearshore area of Toulon including the gulf of Giens showing the applicability of the developed MMS-FEM model for the estimation of harbor and bay resonances, as well as more extended nearshore regions where variable bottom topography effects become important. The calculated resonant frequency depends on the domain characteristics and the size of the open sea boundary and accurately reproduces the measurements within Toulon Bay. On the other hand, for open bays such as the gulf of Giens, a discrepancy is observed between calculated and measured eigenperiods which is due to a very wide opening of the sea boundary that cannot accurately describes the seiching. This underlines the difficulty of accurately calculating the resonance frequency for open bays, in contrast to the classic studies carried out for ports, which are considered virtually closed basins, and confirms the complementary nature of long-term water level measurements and numerical calculations, for a better quantification of the risks associated with energetic meteorological and/or oceanographic events.

**Keywords:** harbor resonances; bay resonances; Modified Mild-Slope model; FEM; Toulon bay;

## 1. Introduction

In a wide range of applications taking place in the marine environment, including the analysis of marine structures, the study of coastal processes, as well as coastal and port engineering studies, information concerning the wave climate in nearshore and coastal areas is derived by means of Offshore-to-Nearshore transformation of wave conditions; see, e.g., Athanassoulis et al (2002). This is due to the fact that local buoy measurements are usually not available at the site of interest, and reliable information from satellite measured data and operational wave models with global
coverage and long duration is available at some distance from the coast. The above information, combined with results from directional extreme value analysis of the wave conditions in the local nearshore-coastal area contributes also to climatic change effects (Vanem 2017), as well as the identification of extreme phenomena by the identification of possible matching with the marina resonant frequencies (Karathanasi et al 2020). Such resonant oscillations of water level generate strong currents at the nodes (bay and port entrance or within the port for closed basin conditions) and significant amplitude at the boundaries (antinodes), with possible damage on the anchored boats, and risks of flooding on the shore. Indeed, resonant oscillations, with a period of several minutes, are part of infragravity waves, with significant impacts on the coast (Bertin et al, 2018).

Harbor and bay resonances excited by incident waves is an important topic in coastal and ocean engineering; see, e.g., Rabinovich (2009). In the case of the smallest eigenfrequencies or excitation by shallow water waves it is examined in the context of long waves Mei (1994, Chap.5). For intermediate water depth incident waves, in domains characterized by general bathymetry the eigenvalue problem can be considered in the framework of long-wave equation models (see Yalciner & Pelinovsky 2004 and the references cited therein) or the mild-slope equation: see, e.g., Karathanasi et al (2020). In particular, the generation and propagation of focused transient wave groups and their interactions with the harbor are simulated using fully nonlinear Boussinesq model by Gao et al. (2020), and the hydrodynamic characteristics of transient harbor resonance excited by double solitary waves is studied by Gao et al. (2021). Furthermore, investigation of the effects of periodic undulating topography on harbor oscillations through Bragg resonance processes has been studied for regular or irregular wave groups (Gao et al., 2021; Gao et al., 2023; Gao et al, 2024).

As mentioned above, the effects of general bathymetry variations as well as the existence of port structures constitute additional factors which are important. The estimation of wave oscillations in larger basins and extended geographical regions as bays and gulfs must include properly these parameters. The computational cost of time-dependent numerical solution is then increased, especially in the case of intermediate depth conditions, and the efficient determination of eigenfrequencies and corresponding eigenmodes constitutes a useful tool. In this work, a novel model based on the Finite Element Method (FEM) is presented for the identification of port and bay resonances based on the Modified Mild-Slope Equation MMSE (Massel 1993, Chamberlain & Porter 1995), which is appropriate to account for the effects of bathymetric variations in extended range of non-dimensional wavenumbers, and reduces to the Linearised Shallow Wave Equation (LSWE) for small frequency. Thus, the MMSE analysis covers both the shallow and the intermediate water depth conditions, corresponding to the lowest eigenfrequencies and the low-medium part of the spectrum, respectively. However, the coefficients of the MMSE are dependent on the wave frequency through the phase and group velocity of waves, and thus iterations are required for the numerical solution of the non-linear eigenvalue problem.

As a special case of interest, results are presented from the study of the long waves in the bay of Toulon France. Located in a Mediterranean urban area, the bay of Toulon is separated into two basins by a 1.4-km long seawall, as shown in Fig.1. The Little Bay is semi-enclosed and connected to the Large Bay by a fairway channel. This channel is the site of significant water mass exchange as a result of both wind-driven currents and bathymetry, and it is therefore a focal point for marine contamination; see Dufresne et al (2014). Millot et al (1981) have shown the presence of resonant oscillations within the Little Bay of Toulon, the main oscillations corresponding to periods of 20 and 45 min. In their study, the seiching of period 20min was ascribed to the oscillation within the
<figure><img src="image_2.png" /><figcaption>**Figure 1.** The extended coastal-sea area of Toulon and positions of various stations.</figcaption></figure>

Little Bay considered as a closed basin, while the period of 45min corresponded to the fundamental mode of oscillation of the whole Bay of Toulon.

In the extended coastal area of Toulon a network of hydrographic/oceanographic stations HTM-NET (https://htmnet.mio.osupytheas.fr) is operating (Rey et al 2020), as shown in Fig.1. In particular recent events leading to resonant oscillations within the bays in Toulon area have been recorded (Rey et al, 2022) as follows: (i) atmospheric front on 19 September 2020, (ii) storm with energetic sea state on 5 December 2020, (iii) tsunami wave due to an earthquake in Algeria on 18 March 2021, (iv) atmospheric wave from the Tonga volcanic eruption in the Pacific Ocean, on 15 January 2022; see Heinrich et al (2023). The frequency spectra of time-series of four days duration during these events are presented in Fig.2 for the tide gauge of the Shom and HTM-NET stations of Ifremer, Lazaret, Tamaris in the little bay of Toulon and Saint Elme, Port Saint Louis, Les Ourisnières and Giens around the little bay (see Fig. 1). They give information on the seiching periods. Since intermittent oscillations are observed, time series give a better estimate of their amplitude, as showed in Fig. 3 for the event of the atmospheric wave due to the Tonga volcanic eruption. These data will be used in the present study to show the applicability of the developed MMS-FEM model for the estimation of harbor and bay resonances.

In the next section, the present MMS-FEM model is described in detail. Then, in section 3 results from the application of the model to the calculation of eigenfrequencies and eigenmodes of Toulon Bay are presented and discussed, and the effect of the detached breakwater is discussed in section 4. Subsequently, in section 5 the present method is used to estimate the eigenperiods in the extended nearshore area including the Gulf of Giens. It is shown that the calculated resonant frequency depends on the domain characteristics and the size of the open sea boundary, and accurately reproduces the measurements within Toulon Bay. On the other hand, for open bays such as the Gulf of Giens, a discrepancy is observed between calculated and measured eigenperiods which is due to a very wide opening of the sea boundary that cannot accurately describe the seiching.
<figure><img src="image_3.png" /></figure>

**Figure 2.** Periods of resonant oscillation at stations (a) Tide gauge (Shom), (b) Ifremer, (c) Lazaret, (d) Tamaris in the little bay of Toulon, and around, from west to east, (e) Saint Elme, (f) Les Oursinières, (g) Giens, corresponding to storm (black), tsunami (blue), atmospheric front (red), atmospheric wave (green) events.

This underlines the difficulty of accurately calculating the resonance frequency for open bays, in contrast to the classic studies carried out for ports, which are generally virtually closed basins, and confirms the complementary nature of long-term water level measurements and numerical calculations, for a better quantification of the risks of submersion associated with energetic meteorological and/or oceanographic events.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Water level series during the Meteo-tsunami wave due to Tonga volcanic eruption in the Pacific Ocean, on 15 Jan 2022, at 4.14 am UTC. (a) within the little bay of Toulon, stations Ifremer, Tamaris and Lazaret, (b), zoom of figure (a), (c) stations Oursinières, between the large bay of Toulon and the gulf of Giens, and station Giens, within the gulf of Giens.</figcaption></figure>

## 2. MMS-FEM model for bay-port resonances

A standard model describing propagating-refracted-diffracted wavefield $\varphi(\mathbf{x})$ over a general bathymetry domain, which is characterized by slowly varying depth $h(\mathbf{x})$, is the modified mild-slope equation (MMSE):

$$ \nabla(\overline{c_g \nabla \varphi(\mathbf{x})}) + \overline{c_g k^2 (1 + \psi(\mathbf{x})) \varphi(\mathbf{x})} = 0, \quad \mathbf{x} = (x, y) \in \Omega, \qquad (1) $$

where $\varphi(\mathbf{x})$ is the complex wave potential and $\Omega$ denotes the domain; see Massel (1993), Chamberlain and Porter (1995). In the above equation, $c(\mathbf{x}) = \omega/k(\mathbf{x})$ is the phase velocity and $c_g(\mathbf{x}) = c(1 + 2kh/\sinh(2kh))/2$ is the corresponding group velocity, where the angular frequency of the monochromatic incident wave is denoted by $\omega$, and the local wavenumber by $k = k(\mathbf{x})$. The latter, at a specific frequency $\omega$, is obtained from the water-wave dispersion relation,
$$ \omega^2 = gk(\mathbf{x})\tanh[k(\mathbf{x})h(\mathbf{x})], \qquad (2) $$

formulated at the local depth, with g being the acceleration of gravity. Moreover, the function $\psi(\mathbf{x})$ involves terms associated with bottom slope ($\nabla h$)$^2$ and curvature $\nabla^2 h$, and can be found in Massel (1993, Eq. 34); see also Miles & Chamberlain (1998).

For studying the port resonances in enclosed basins the MMSE Eq.(1) is supplemented by appropriate boundary conditions along the boundary $\partial\Omega = \partial\Omega_N \cup \partial\Omega_D$, which is separated into the port-bay wall parts $\partial\Omega_N$ and the port-bay entrance parts $\partial\Omega_D$, as indicated in Fig.4 by using red solid lines and black dashed lines, respectively.

## 2.1 Formulation as a non-linear eigenvalue problem

Following several authors, the problem of detecting the port-bay resonances can be treated by formulating it as a non-linear eigenvalue problem. In this case, the MMSE is supplemented by a Neumann condition $\partial\varphi/\partial n = 0$, $\mathbf{x} \in \partial\Omega_N$ where $\mathbf{n}$ indicates the normal vector on the solid port walls (directed outwards the domain), and a homogeneous Dirichlet condition $\varphi(\mathbf{x}) = 0$, $\mathbf{x} \in \partial\Omega_D$ along the port-bay entrance, as shown by the dashed segment AB in Figure 4(a). The latter are considered approximately appropriate for the studied problem indicating that the entrance interface AB connecting the finite port domain with the infinite exterior region where the port oscillation vanishes is a nodal line; see, e.g., Rabinovich (2009). A difficulty arises from the fact that the coefficients of the MMS Eq. (1) are dependent on the solution introducing implicit nonlinearity.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Domain $\Omega$ and FEM discretization. (a) Port-bay domain with open-sea boundary at the port-bay entrance (AB) shown by using dashed line. (b) Same as before with the domain extended to the open sea region (lower semi-circular subdomain).</figcaption></figure>
In the present work the above nonlinear eigenvalue problem is numerically treated by using a low-order FEM model, described below. Let $\Omega_h = \bigcup_{i=1,N_e} T_i$ denote the finite-element discretization of the domain by means of the $N_e$ totally triangular elements, and introduce the following approximation:

$$ \varphi(\mathbf{x}) = \sum_{j=1}^{N} \varphi_j N_j(\mathbf{x}), \qquad (3) $$

where $N_j(\mathbf{x})$, $j=1,N$ denote the linear (pyramid-like) shape functions associated with each j-node of the triangulation corresponding to the finite element subdivision. Accordingly the MMSE (1) is put in the following discrete form

$$ \sum_{m=1}^{N} K_{jm} \varphi_m + \omega^2 M_{jm} \varphi_m = 0, \quad j=1, \dots, N, \qquad (4) $$

where the FEM-stiffness and mass matrices are defined by

$$ K_{jm} = (cc_g \nabla N_j, \nabla N_m), \quad M_{jm} = (pN_j, N_m), \quad \text{with} \quad p(\mathbf{x}) = c^{-1}(\mathbf{x})c_g(\mathbf{x})(1+\psi(\mathbf{x})), \qquad (5) $$

and $(f(\mathbf{x}), g(\mathbf{x})) = \int_{\Omega_h} f(\mathbf{x})g(\mathbf{x})d\mathbf{x}$ denotes the discrete inner product of functions $f,g$ in the domain. In the above discrete formulation the Neumann boundary condition for $\mathbf{x} \in \partial\Omega_N$ is satisfied as a natural one. The Dirichlet boundary condition for $\mathbf{x} \in \partial\Omega_N$ is satisfied as an essential one by setting the values of the unknowns $\varphi_j$ for all j-nodes on the port-bay entrance boundary $\mathbf{x} \in \partial\Omega_D$ equal to zero.

In the case of constant depth region $h(x)=h_M$ and assuming shallow water conditions the coefficient $cc_g = gh_M$ is constant and also $p=1$. Thus, the eigenvalue equation (4) is solved by calculating the corresponding eigenvalues $\kappa_n^2, n=1,2,3,\dots$ of the Laplace equation in the horizontal domain $\Omega_h$

$$ \sum_{m=1}^{N} (\nabla N_j, \nabla N_m) \varphi_m + \kappa^2 (N_j, N_m) \varphi_m = 0, \quad j=1, \dots, N, \qquad (6) $$

and obtain the following eigenfrequencies as follows:

$$ \omega_n^2 = \kappa_n^2 gh_M, \quad n=1,2,\dots \qquad (7) $$

In the case of variable bathymetry domain $h(\mathbf{x})$ and general wave conditions, the numerical solution of the nonlinear eigenvalue problem (4) is calculated by iterations. The iterative scheme
starts by an initial ($\ell=0$) prediction of the eigenfrequencies, which is obtained by means of the LSWE using the average depth $h_M$ in the domain

$$ \omega_n^2 \approx \omega_{n,0}^2 = \kappa_n^2 g h_M, \quad n=1,2,\dots. \qquad (8) $$

Next, starting from the lowest eigenfrequency ($n=1$) the coefficients $c(\mathbf{x})c_g(\mathbf{x})$ and $p(\mathbf{x})$ are calculated in the domain $\Omega_h$, the stiffness and mass matrices are updated for each frequency $\omega_n$, $n=1,2,3,\dots$, the eigenvalue problem (4) is solved, and this part is iterated ($\ell=1,2,3,\dots$) until convergence $|\omega_{n,\ell} - \omega_{n,\ell-1}| < \varepsilon$, where $\ell$ denotes the iteration index and $\varepsilon$ is the tolerance. In the examples presented below 3 iterations are found to be enough for numerical convergence. The corresponding eigenperiods are given by

$$ T_n = 2\pi / \omega_n, \quad n=1,2,3,\dots. \qquad (9) $$

## 2.2 Formulation as a forced wave problem

The problem of resonant oscillations in a semi-closed basin is alternatively treated by using a forced wave formulation, as discussed by Mei (1994, Chap.5) in the case of long waves, using the LSWE model as the field equation. In this work, the latter model is extended by using the MMSE (1) as the field equation, which is applicable to intermediate water depth conditions while it is also fully compatible with shallow-water conditions. As before, the numerical treatment will be based on a FEM formulation using an artificial extension of the semi-closed port-bay basin to the open sea as depicted in Fig.4b, where the depth is considered to be constant $h=h_I$, equal to the average depth at the port-entrance line AB.

The MMSE model (1) for constant depth reduces to the standard Helmholtz equation for the incident wave field

$$ \nabla^2 \varphi_I(\mathbf{x}) + k^2 \varphi_I(\mathbf{x}) = 0, \quad \mathbf{x} = (x, y) \in \Omega_I, \qquad (10) $$

where $\Omega_I$ denotes the exterior domain. The local constant wavenumber $k$ is obtained from the dispersion relation at the given frequency $\omega$ for water depth $h=h_I$. Next, a general representation is set-up for the wave field in $\Omega_I$ consisting in the incident wave propagating at an angle $\beta$ in the exterior region, as well as the reflected wave component and the scattered field, as illustrated in Fig.4b, as follows:

$$ \varphi_I(\mathbf{x}) = \exp(ik \cos \beta x + k \sin \beta y) + R \exp(ik \cos \beta x + k \sin \beta y) + \varphi_S(\mathbf{x}), \quad \mathbf{x} = (x, y) \in \Omega_I, \qquad (11) $$

where $R$ denotes the reflection coefficient. The scattered field is given by

$$ \varphi_S(\mathbf{x}) = \sum_{m=0} A_m H_m^{(1)}(kr) \cos(m\theta), \quad \mathbf{x} = (x, y) \in \Omega_I, \qquad (12) $$
with $r = |x|$ and $\theta = \tan^{-1}(y/x)$. We note here that the use of cosine terms in Eq.(12) is selected in order to fulfill the solid Neumann boundary condition in the walls segments $x \in (-\infty, A) \cup (B, \infty)$ of the exterior domain. On the basis of the above, the forced oscillation problem is formulated by the MMSE Eq.(1) in the basin, supplemented by the Neumann boundary condition along the boundary $\partial\Omega_N$,

$$ \frac{\partial \varphi}{\partial n} = 0, \quad \mathbf{x} \in \partial \Omega_N, \qquad (13) $$

In conjunction with matching conditions on the segment AB, modelling the port-bay entrance,

$$ \frac{\partial \varphi}{\partial n} = \frac{\partial \varphi_l}{\partial n}, \quad \varphi = \varphi_l, \quad x \in AB. \qquad (14) $$

The general representation in the exterior subdomain given by Eqs.(11),(12) can be used to set-up a non-local boundary condition at the entrance segment of the form

$$ \frac{\partial \varphi}{\partial n} - \mathcal{D}(\varphi) = b(x), \quad x \in AB, \qquad (15) $$

where $\mathcal{D}(\varphi)$ denotes the Dirichlet-to-Neumann (DtN) map. This operator on the boundary data on AB is constructed from the expansion (11) in the exterior domain. Standard tools for expressing the DtN operator are offered by boundary integral representations of the wave potential in the exterior subdomain.

However, for port-bay regions of relatively large entrance AB, the scattered component is small and can be approximately omitted. In this case, the above equation (15) takes the form

$$ \frac{\partial \varphi}{\partial n} - ik \sin \beta \varphi = -2ik \sin \beta \exp(ik \cos \beta x), \quad x \in AB, \qquad (16) $$

with the normal vector $n$, as before, directed to the exterior of $\Omega$, and $y=0$ at the entrance AB; see Fig.4b. With the present definition of coordinate axes the propagation direction $\beta=90^\circ$ corresponds to normally incident waves on AB.

Using the present FEM model, the problem of forced oscillations in the semi-closed port bay area and a given incident wave frequency and direction $(\omega, \beta)$ consists of the system

$$ \sum_{m=1}^{N} K_{jm} \varphi_m + \omega^2 M_{jm} \varphi_m = f_j, \quad j = 1, \dots, N, \qquad (17) $$

where for the interior nodes of the discretization in the basin $j \in J_\Omega$, the FEM-stiffness and mass matrices and the forcing term are given by

$$ K_{jm} = (cc_g \nabla N_j, \nabla N_m), \quad M_{jm} = (pN_j, N_m), \quad \text{and} \quad f_j = 0, \qquad (18) $$
with $p(\mathbf{x}) = c^{-1}(\mathbf{x})c_g(\mathbf{x})(1+\psi(\mathbf{x}))$, while for the nodes $j \in J_{AB}$, belonging at the segment AB the above terms are modified due to the mixed-type condition Eq.(16) as follows

$$K_{jm} = (cc_g \nabla N_j, \nabla N_m), M_{jm} = (pN_j, N_m) + \int_{AB} p(x) N_j N_m dx, \text{ and } f_m = \int_{AB} b(x) N_m dx. \quad (19)$$

with the additional terms defined by the integrals along the port-bay entrance segment AB. The extension to include the scattering effect by means of the DtN operator $\mathcal{D}(\varphi)$ can be represented by using boundary integral representations based on single-layer distributions on AB

$$\varphi_s(\mathbf{x}) = \int_{AB} \sigma_k H_0^{(1)}(k_l |\mathbf{x}-\mathbf{s}|) ds, \quad (20)$$

from which the relation between $\varphi_s(x)$ and $\partial\varphi_s(x)/\partial n, x \in AB$ can be conveniently expressed, and such an extension is left to be studied in future work.

### 3. Application to Toulon bay-port resonances

An aerial photo of the Toulon bay-port area with local view of the harbor is shown in Fig. 5.

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Aerial photo of the Toulon bay-port region and local port area shown by dashed box.</figcaption></figure>

The detailed bathymetry of Toulon bay-port and coastal area is presented in Figure 6. In order to study the port-bay oscillations by the present method the semi-closed domain is defined using the segment AB shown in Fig. 6 by using a dashed line to represent the port-bay entrance. The part corresponding to the studied area is defined as the region north of the entrance AB of length 5.9km and the mean depth in the semi-closed domain is $h_M=7.72m$. The depth along the entrance AB is variable, however, for a great part it remains relatively constant around $h_I=50m$.
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Bathymetry of Toulon port-bay and nearshore/coastal region. The part corresponding to the studied area is region north of the entrance AB of length 5.9km as shown by the dashed line. The mean bathymetry in the studied area is $h_M=7.72\text{m}$ and is about 50m at the entrance.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Coarse FEM mesh in Toulon port-bay region. The number of elements is $N_e=22343$, and total nodes $N=12523$. the number of nodes located at the entrance AB is 52.</figcaption></figure>

A coarse FEM mesh in the above semi-closed Toulon port-bay region is obtained by triangulation as shown in Fig.7, and is used to obtain and discuss first results. The number of elements of the above mesh is $N_e=22343$, and the total nodes $N=12523$. Moreover, the number of nodes located at the entrance post-bay segment AB located at $y=0$ is $N_{AB}=52$.
The first LSWE eigenperiods in Toulon bay-harbor calculated for the mean constant depth $h_M = 7.72m$ as obtained by the present FEM model are listed in Table 1, where also the results are compared against the ones obtained by the MMSE in the variable bathymetry topography in Fig. 8. Moreover, the first 4 calculated eigenmodes are plotted in Figs.9-12, respectively. The results are normalized as calculated by using the Matlab® function eigs.m concerning the discrete system, Eq.(17). We clearly observe from the above results that for the majority of the first calculated eigenperiods the wave conditions are very shallow and the effects of dispersion on the calculated eigenvalues are negligible. In the specific example, for the part of the spectrum above mode index 1400 (listed in the right-part of table 1), the eigenperiods reduce to a value such that intermediate wave conditions are encountered in the domain and the dispersion effects become distinguishable. Also, it is clearly seen that the depth variation effects are significant for the whole band of the spectrum and the assumption of constant average depth is not appropriate for relatively large geographical domains as the one considered here and bathymetric changes have to be taken into account.

<figure><img src="image_9.png" /><figcaption>**Figure 8.** (a) First 300 eigenfrequencies approximated by LSWE for mean constant depth $h_M = 7.72m$ in the domain $\Omega$ compared with the corresponding MMS eigenfrequencies in Toulon port-bay region. (b) Same as before for the largest eigenfrequencies. The dashed line depicts the diagonal.</figcaption></figure>

<table><caption>**Table 1.** First eigenperiods in minutes as calculated by the non-linear eigenvalue problem in Toulon port-bay region using the present MMS -FEM model, and as estimated by LSWE at the mean constant depth $h = 7.72m$.</caption><thead><tr><th>n</th><th>T<sub>n,0</sub> (min)<br>LSWE<br>(h=const)</th><th>T<sub>n,0</sub> (min)<br>LSWE</th><th>T<sub>n</sub> (min)<br>MMSE</th><th>n</th><th>T<sub>n,0</sub> (min)<br>LSWE<br>(h=const)</th><th>T<sub>n,0</sub> (min)<br>LSWE</th><th>T<sub>n</sub> (min)<br>MMSE</th></tr></thead><tbody><tr><td>1</td><td>55.5661</td><td>34.7802</td><td>34.7804</td><td>1491</td><td>0.3397</td><td>0.3134</td><td>0.3199</td></tr><tr><td>2</td><td>22.8517</td><td>16.4550</td><td>16.4552</td><td>1492</td><td>0.3394</td><td>0.3133</td><td>0.3198</td></tr><tr><td>3</td><td>19.5099</td><td>12.1890</td><td>12.1892</td><td>1493</td><td>0.3393</td><td>0.3132</td><td>0.3197</td></tr><tr><td>4</td><td>15.7163</td><td>11.0603</td><td>11.0608</td><td>1494</td><td>0.3391</td><td>0.3129</td><td>0.3196</td></tr><tr><td>5</td><td>14.4398</td><td>8.8116</td><td>8.8125</td><td>1495</td><td>0.3386</td><td>0.3127</td><td>0.3195</td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>**Figure 9.** Calculated 1st MMSE eigenmode in Toulon bay-port area.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10.** Calculated 2nd MMSE eigenmode in Toulon bay-port area.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 11.** Calculated 3rd MMSE eigenmode in Toulon bay-port area.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 12.** Calculated 4th MMSE eigenmode in Toulon bay-port area.</figcaption></figure>

Since the MMSE reduces to LSWE at low frequencies, the first (lowest) eigenfrequencies calculated by the two models coincide, as presented Table 1 for the first 6 modes in the Toulon bay region, and the corresponding eigenmodes are plotted in Figs.9-12, respectively. The effect of intermediate water depth conditions comes into play for much larger values of frequencies starting at mode index around 1400 concerning the calculated eigenvalues where the corresponding eigenperiods drop down to approx. 0.3min (see Table 1). These modes could be of interest of port resonances excited by wind waves and swell in the considered region (see e.g., Karathanassi et al 2020).
The second eigenperiod in the domain as calculated by the present non-linear eigenvalue problem is $T=16.45\text{min}$ and, as it is observed in Fig.10, it leads to a concentration of energy both in the Lazaret-Tamaris as well as in the Ifremer locations (see Fig.3). The above prediction could be considered as the approximation of the recorded value $T=17.8-20\text{min}$ in the case of Meteo-tsunami wave from Tonga volcanic eruption in the Pacific Ocean on 15 Jan 2022. It is also worth noticing here that due to the size of the semi-closed domain as defined in the present analysis, larger eigenperiods as the values $T= 45.7-48.7\text{min}$, recorded at Ifremer station (see Fig.3) are not obtained. In order to provide predictions for such a period, an extended domain should be used, in order to detect the eigenmodes corresponding to possible energy exchanges between neighbour basins.

Furthermore, the detection of eigenfrequencies using the formulation of wave forced problem with the present MMSE-FEM model is examined, and results concerning the mean energy in the subregion of Lazared-Tamaris regions are presented in Fig.13 for incident waves $\beta=90\deg$ (red lines) and $\beta=30\deg$ (black lines) from the exterior region. Incident long wave frequencies corresponding to all periods from $T=16\text{min}$ to $20\text{min}$ are systematically studied and it is clearly observed a strong peak for incident wave period $T=16.2\text{min}$, which is a much better prediction of the eigenperiod observed in the above local area of the Toulon bay-port region.

<figure><img src="image_14.png" /><figcaption>**Figure 13.** Calculated mean energy in the Lazaret-Tamaris local area of the Toulon semi-enclosed bay-port region in the range of incident wave periods $15\text{min} < T < 19\text{min}$, for incident wave directions $\beta=30\deg$ (black line) and $\beta=90\deg$ (red lines). The detected eigenperiod of $T=16.2\text{min}$ is indicated by using a vertical dashed arrow.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 14.** Modulus of calculated modes corresponding to the eigenperiod 16.2min, as obtained from the present MMSE-FEM model for (a) $\beta$=90deg and (b) $\beta$=30deg incident waves in the semi-closed Toulon bay-port region.</figcaption></figure>

The calculated wave modes corresponding to the detected eigenperiod $T=16.2\min$ are presented in Fig.14 for obliquely incident waves ($\beta$=30deg) normally incident waves ($\beta$=90deg) in the semi-closed Toulon bay-port region, as obtained from the present MMSE-FEM model. In the above plots we observe an increase of energy density moving from east to west in the semi-enclosed region, which is compatible with the same features of the first eigenmode shown in Fig.9 obtained by the approximation using the present MMSE-FEM solution of the considered non-linear eigenvalue problem.

**4. The effect of detached breakwater on the estimation of Toulon bay-port resonances**

Previous results did not take into account the effect of detached breakwater shown in Fig.5, which divides the bay of Toulon into two basins: the Little Bay, which is semi-enclosed and includes the military harbour, and the Large Bay, which opens onto to the Mediterranean Sea. These two basins are linked by a 30m deep, 1km wide channel located south of the seawall; see also Dufresne et al (2014).
In order to show the effect of the seawall on the calculated resonant modes, the present model is updated by the inclusion of additional breakwater boundary and the coarse FEM mesh in the above region is obtained by triangulation as shown in Fig.15, and is used to obtain and discuss first results. The number of elements of the above mesh is $N_e=22485$, and the total nodes $N=12610$, and the number of nodes located at the entrance post-bay segment AB located at $y=0$ is $N_{AB}=48$. We observe in Fig. 15 the mesh refinement at the breakwater tip, in order to capture diffraction phenomena.

<table><caption>**Table 2.** First eigenperiods (in min) as calculated by the non-linear eigenvalue problem in Toulon port-bay region using the present MMS-FEM model, and as estimated by LSWE without the breakwater (left part) and with the effect of the breakwater (right part). Effects on change of water depth on the resonant periods are given in the two right-hand columns.</caption><thead><tr><th colspan="3">without the breakwater</th><th colspan="5">with breakwater</th></tr><tr><th>n</th><th>T<sub>n,0</sub> (min)<br>LSWE</th><th>T<sub>n</sub> (min)<br>MMSE</th><th>n</th><th>T<sub>n,0</sub> (min)<br>LSWE</th><th>T<sub>n</sub> (min)<br>MMSE</th><th>T<sub>n</sub> (min)<br>h-0.5m</th><th>T<sub>n</sub> (min)<br>h+0.5m</th></tr></thead><tbody><tr><td>1</td><td>34.7802</td><td>34.7804</td><td>1</td><td>41.1384</td><td>41.1385</td><td>41.7206</td><td>40.5756</td></tr><tr><td>2</td><td>16.4550</td><td>16.4552</td><td>2</td><td>16.5106</td><td>16.5109</td><td>16.9793</td><td>16.1266</td></tr><tr><td>3</td><td>12.1890</td><td>12.1892</td><td>3</td><td>12.2019</td><td>12.2021</td><td>12.5722</td><td>1.8895</td></tr><tr><td>4</td><td>11.0603</td><td>11.0608</td><td>4</td><td>11.5919</td><td>11.5925</td><td>1.8522</td><td>1.4023</td></tr><tr><td>5</td><td>8.8116</td><td>8.8125</td><td>5</td><td>8.8782</td><td>8.8789</td><td>9.4208</td><td>8.6723</td></tr><tr><td>6</td><td>8.1730</td><td>8.1732</td><td>6</td><td>8.2234</td><td>8.2241</td><td>8.4919</td><td>8.0132</td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>**Figure 15.** Coarse FEM mesh in Toulon port-bay region. The number of elements is $N_e=22485$ and total nodes $N=12610$ and the number of nodes located at the entrance AB is 48.</figcaption></figure>
Once again, the predictions by the present MMSE model match the ones by LSWE for low frequencies. The first (lowest) eigenfrequencies calculated by the two models are listed in Table 2 for the first 4 modes in the Toulon bay region, and the corresponding eigenmodes are plotted in Figs.16-19, respectively. We observe in these figures the fulfilment of Neumann boundary condition on the seawall since the equipotential lines intersect the boundary perpendicularly. As an effect of the detached breakwater a small increase of the calculated eigenperiods is observed. Taking into account that the water level in the region varies of about $\pm0.5m$ from lower tide and high atmospheric pressure conditions to high tide and low atmospheric conditions, the effect of the mean water level on the calculated first eigenperiods is also included in the last two columns of Table 2.

Moreover, results concerning the mean energy in the subregion of Lazaret-Tamaris sites are presented in Fig.20 for incident waves $\beta=90\deg$ (red lines) and $\beta=30\deg$ (black lines) from the exterior region. Incident wave frequencies corresponding to all periods from $T=15\min$ to $19\min$ are systematically studied and, as before, a strong peak is clearly observed for incident wave period $T=16.2\min$, a prediction quite close to the second calculated eigenperiod. The corresponding wave mode is presented in Fig. 21 for normally incident waves ($\beta=90\deg$) in the semi-closed Toulon bay-port region, as obtained from the present MMSE-FEM model. From the above results we conclude that the first eigenperiods calculated by the eigenvalue problem formulation using the Dirichlet condition assumption on the entrance boundary provides a good estimation in the case normally incident waves, however, the detailed distribution of energy on the focal areas are more accurately predicted by the present forced wave problem formulation.

<figure><img src="image_17.png" /><figcaption>**Figure 16.** Calculated 1st MMSE eigenmode in Toulon bay-port area.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 17. Calculated 2nd MMSE eigenmode in Toulon bay-port area.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 18. Calculated 3rd MMSE eigenmode in Toulon bay-port area.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>**Figure 19.** Calculated 4th MMSE eigenmode in Toulon bay-port area.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 20.** Calculated mean energy in the Lazaret-Tamaris local area of the Toulon semi enclosed bay-port region in the range of incident wave periods $15 \text{ min} < T < 19 \text{ min}$, for incident wave directions $\beta=30^\circ$ (black line) and $\beta=90^\circ$ (red lines). The detected eigenperiod of T=16.2min is indicated by using a vertical dashed arrow.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure 21.** Calculated modes corresponding to the eigenperiod 16.2min, as obtained from the present MMSE-FEM model for β=90deg incident waves from South in the semi-closed Toulon bay-port region: (a) real part, (b) modulus.</figcaption></figure>

**5. Estimation of eigenperiods in the extended nearshore area including the gulf of Giens**

In this section, the present method is applied to calculate the eigenperiods in an extended nearshore area around Toulon including the gulf of Giens (see Fig.22). The open sea boundary is indicated by a thick dashed line.

<figure><img src="image_23.png" /><figcaption>**Figure 22.** Bathymetry of Toulon port-bay and nearshore/coastal region including the gulf of Giens in the western part. The part corresponding to the studied area is region north of the entrance AB of length 5.9km, as shown by the dashed line.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**Figure 23.** Coarse FEM mesh in the extended Toulon-Giens nearshore region. The number of elements is N<sub>e</sub>=18884 and total nodes N=10198 and the number of nodes located at the entrance of the extended nearshore region AB is 55.</figcaption></figure>

<table><caption>**Table 3.** First eigenperiods (in min) as calculated by the present model in (a) the Toulon port-bay region and (b) the extended nearshore region including the gulf of Giens, by using the present MMS-FEM model.</caption><thead><tr><th>n</th><th>T<sub>n</sub> (min)<br>(a) Toulon bay region</th><th>T<sub>n</sub> (min)<br>(b) Extended nearshore area</th></tr></thead><tbody><tr><td>1</td><td>41.1385</td><td>43.7148</td></tr><tr><td>2</td><td>16.5109</td><td>22.2752</td></tr><tr><td>3</td><td>12.2021</td><td>18.1535</td></tr><tr><td>4</td><td>11.5925</td><td>14.8549</td></tr><tr><td>5</td><td>8.8789</td><td>13.0911</td></tr><tr><td>6</td><td>8.2241</td><td>12.2263</td></tr></tbody></table>

The FEM mesh defined in the extended Toulon-Giens nearshore region is presented in Fig.23. The first 6 calculated eigenperiods are listed in Table 3, and are compared to the corresponding prediction by the present model (including the breakwater) in the Toulon bay area. The corresponding calculated eigenmodes for the first three eigenperiods in the extended area are illustrated in Figs.24, 25 and 26, respectively. We observe that the first eigensolution is compatible with the corresponding result obtained by analysis of the local Toulon bay (Fig.16) presenting energy concentration in the Ifremer station site. Similarly, the third eigensolution in the extended nearshore region (Fig.26) shows compatibility with the second one obtained by analysis of the
local Toulon bay (Fig.17) presenting energy concentration in the Tamaris-Lazaret station site. Both calculations recover the periods of the observations when considering an extended zone area. On the other hand, the second eigensolution in the extended nearshore region illustrated in Fig.25, with calculated period *T*=22.3min, indicates concentration of energy in the gulf of Giens. The above result presents a discrepancy with the observed value which is around 29min as shown in Fig.2(d). This is attributed on the one hand to the wide opening of the calculation domain towards the open sea (virtuel boundary), and on the other hand to the proximity of the open boundary to the gulf of Giens, for which the resonant oscillation is certainly further offshore than modelled, the measured period being higher.

Finally, the measured maxima of amplitudes (half-ranges) in the stations and for the four events described in the introduction are calculated through either direct analysis of the time series or by calculation of the maximum amplitude of the Fourier coefficient, for *T*=44min and *T*=18min, using a moving window for coexisting seiching periods or for noisy signals. Error estimation is about ±0.5cm. Data are presented in Table 4(a) and (b). The relative amplitude in the stations of Lazaret, Tamaris and Shom with respect to Ifremer station are indicated in parentheses. Corresponding results obtained by the present MMS-FEM calculations concerning the relative amplitude, based on the results presented in Figs. 16 and 17, respectively, are provided in the last column of Table 4. We observe that the model predictions are in good agreement with the measured relative amplitudes, and can then accurately extrapolate amplitudes in the whole Toulon bay. Let us also note here that seiching at *T*≈18 min is the more important in the South part of the little bay with a node in the central part of the bay (see Figs. 17 and 26), while seiching at *T*≈44min, which corresponds to an oscillation in phase in the whole little bay, is more pronounced in the North part, as shown in Figs. 16 and 24.

<table><thead><tr><th>(a) Seiching<br>T≈44 min</th><th>(i) Atmospheric<br>front</th><th>(ii) Storm</th><th>(iii) Tsunami</th><th>(iv) Atmospheric<br>wave (Tonga)</th><th>MMS -FEM<br>model</th></tr></thead><tbody><tr><td>Lazaret</td><td>6 (0.75)</td><td>4 (0.73)</td><td>-</td><td>5 (0.77)</td><td>(0.61)</td></tr><tr><td>Tamaris</td><td>6 (0.75)</td><td>4 (0.73)</td><td>2 (0.67)</td><td>4 (0.62)</td><td>(0.61)</td></tr><tr><td>Ifremer</td><td>8 (1)</td><td>5.5 (1)</td><td>3 (1)</td><td>6.5 (1)</td><td>(1)</td></tr><tr><td>Shom</td><td>-</td><td>4 (0.73)</td><td>3 (1)</td><td>6 (0.92)</td><td>(0.93)</td></tr></tbody></table>

<table><caption>**Table 4.** Measured amplitude (cm), with an error of about ±0.5cm, in the case of four events, and relative amplitude in the stations Lazaret, Tamaris and Shom with respect to Ifremer station indicated in parentheses. The corresponding prediction by the present MMS-FEM model is listed in the last column.</caption><thead><tr><th>(b) Seiching<br>T≈18 min</th><th>(i) Atmospheric<br>front</th><th>(ii) Storm</th><th>(iii) Tsunami</th><th>(iv) Atmospheric<br>wave (Tonga)</th><th>MMS -FEM<br>model</th></tr></thead><tbody><tr><td>Lazaret</td><td>9 (1.50)</td><td>11 (1.69)</td><td>-</td><td>18 (1.50)</td><td>(1.78)</td></tr><tr><td>Tamaris</td><td>7 (1.17)</td><td>9 (1.38)</td><td>6 (1)</td><td>14 (1.17)</td><td>(1.51)</td></tr><tr><td>Ifremer</td><td>6 (1)</td><td>6.5 (1)</td><td>6 (1)</td><td>12 (1)</td><td>(1)</td></tr><tr><td>Shom</td><td>-</td><td>4 (0.62)</td><td>3 (0.50)</td><td>6 (0.50)</td><td>(0.93)</td></tr></tbody></table>
<figure><img src="image_25.png" /><figcaption>**Figure 24.** Calculated 1st eigenmode in the extended Toulon-Giens nearshore area.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 25.** Calculated 2nd eigenmode in the extended Toulon-Giens nearshore area.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Figure 26.** Calculated 3rd eigenmode in the extended Toulon-Giens nearshore area.</figcaption></figure>

**6. Conclusions**

In the present work, bay and harbor resonances are investigated, taking into account the variable bathymetry of the semi-enclosed basin. The effects of general bathymetry variations, as well as the existence of port structures, constitute important factors concerning the estimation of wave oscillations in larger basins and extended geographical regions as bays and gulfs. In particular, the Modified Mild-Slope (MMS) equation is implemented for the description of combined refraction-diffraction effects, from which the eigenperiods are calculated by means of a low-order FEM-scheme, which is proved to be a very efficient computational tool concerning the estimation of eigenfrequencies and corresponding eigenmodes. It is demonstrated that the first part of the spectrum of greater eigenperiods is well approximated by the Shallow Wave Equation which offers a simplification to which the MSE reduces. However, as the eigenperiods decrease to values such that intermediate wave conditions are encountered in the domain and the dispersion effects become important the MMS model is found more appropriate. For the detailed analysis of the wave conditions in various subregions a forced wave formulation is presented based on the present MMS-FEM model that is able to provide better prediction of the eigen-characteristics and energy focusing in local areas. Next, the present model is applied to coastal-port region of Toulon, France and in an extended nearshore region including the gulf of Giens. It is validated through comparisons with water level data from the tidegauge of Toulon (Shom) and from the network HTM-NET. The model can also be used to calculate level variations (and fluid velocities) throughout the domain, and to highlight the areas potentially most affected by these resonant oscillations. Furthermore, the applicability of the developed MMS-FEM model is illustrated for
the estimation of harbor and bay and more extended nearshore region resonances. It is shown that the depth variation effects are significant for the whole band of the spectrum and the assumption of constant average depth is not appropriate for relatively large geographical domains and should be taken into account. This underlines the difficulty of accurately calculating the resonance frequency for open bays, in contrast to the classic studies carried out for ports, which are generally virtually closed, and confirms the complementary nature of long-term water level measurements and numerical calculations, for a better quantification of the risks of submersion associated with energetic meteorological and/or oceanographic events. Future work will include the consideration of larger geographical regions, including the Coriolis effects that have been ignored in the present analysis, with application to the prediction of harmonic tides of the Eastern and Western Mediterranean Sea basins.

## Acknowledgments

Support of K. Belibassakis to visit University of Toulon is acknowledged. T. Missamou, from the Mediterranean Institute of Oceanography (MIO), is acknowledged for the installation and the maintenance of the HTM-NET stations, D. Mallarino from the OSU Pytheas institute for the HTM-NET website maintenance. Shom is acknowledged for the water level data provided from the tidegauge of Toulon.

## References

1. Athanassoulis, G.A.; Belibassakis, K.A.; Gerostathis, Th. 2002. The POSEIDON nearshore wave model and its application to the prediction of the wave conditions in the nearshore/coastal region of the Greek Seas. *The Global Atmosphere and Ocean System* 8(2-3), 101–117.

2. Vanem, E. 2017. A regional extreme value analysis of ocean waves in a changing climate. *Ocean Engineering* 144, 277–295.

3. Rabinovich, A.B. 2009. Seiches and Harbor Oscillations. In *Handbook of Coastal and Ocean Engineering*; Young C.K., World Scientific: Singapore, pp. 193–236.

4. Mei, C.C. *The applied dynamics of ocean surface waves*. World Scientific: Singapore, 1994.

5. Yalciner A. C.; Pelinovsky E. 2006. A short cut numerical method for determination of periods of free oscillations for basins with irregular geometry and bathymetry. *Ocean Engineering* 34, 747–757.

6. Karathanasi F., Karperaki A., Gerostathis Th., Belibassakis K., 2020. Offshore-to-Nearshore Transformation of Wave Conditions and Directional Extremes with Application to Port Resonances in the Bay of Sitia-Crete, *Atmosphere* 11(3), 280.

7. Bertin, X., De Bakker A., Van Dongeren Ap, Coco G., Andre G., Ardhuin F., Bonneton P., Bouchette F., Castelle B., Crawford W. C., Davidson M., Deen M., Dodet G., Guerin T., Inch K., Leckler F., Mccall R., Muller H., Olabarrieta M., Roelvink D., Ruessink G., Sous D., Stutzmann E., Tissier M., 2018. Infragravity waves: from driving mechanisms to impacts, Earth Science Review 177, 774-799, https://doi.org/10.1016/j.earscirev.2018.01.002

8. Gao J., Ma Xiaozhou, Zang J., Dong G., Ma Xiaojian, Zhu Y., Zhou L., 2020. Numerical investigation of harbor oscillations induced by focused transient wave groups. *Coastal Engineering* 158: 103670.
9. Gao J., Ma X., Chen H., Zang J., Dong G., 2021. On hydrodynamic characteristics of transient harbor resonance excited by double solitary waves. *Ocean Engineering* 219: 108345.

10. Gao J., Ma X., Dong G., Chen H., Liu Q., Zang J., 2021. Investigation on the effects of Bragg reflection on harbor oscillations. *Coastal Engineering* 170: 103977.

11. Gao J., Shi H., Zang J., Liu Y., 2023. Mechanism analysis on the mitigation of harbor resonance by periodic undulating topography. *Ocean Engineering* 281: 114923.

12. Gao J., Hou L., Liu Y., Shi H., 2024. Influences of Bragg reflection on harbor resonance triggered by irregular wave groups. *Ocean Engineering* 305: 117941.

13. Massel, S.R. Extended refraction-diffraction equation for surface waves. *Coastal Eng* 1993, 19, 97-127.

14. Chamberlain P.G.; Porter D. The modified mild-slope equation. *J. Fluid Mech.* 1995, 291, 393-407.

15. Dufresne Ch., Duffa C. Rey V., 2014. Wind-forced circulation model and water exchanges through the channel in the Bay of Toulon, *Ocean Dynamics* DOI 10.1007/s10236-013-0676-3.

16. Millot, C., Broyard, R., Metais, O., Tine, J., 1981. Les oscillations propres de la Rade de Toulon, *Oceanologica Acta*, **4**(3), 259-262 (in French).

17. Rey, V., Dufresne C., Fuda, J. L., Mallarino, D., Missamou, T., Paugam, C., Rougier, G. and Taupier-Letage, I., 2020. On the use of long term observation of water level and temperature along the shore for a better understanding of the dynamics: Example of Toulon area, France, *Ocean Dynamics*, 2020, https://doi.org/10.1007/s10236-020-01363-7.

18. Rey, V., Paugam, C., Dufresne, C., Mallarino, D., Missamou, T. et Fuda, J. L., 2022. Seiches à l'échelle de baies : origines et identification des périodes propres d'oscillations à partir des données d'observations sur le long terme en Provence à partir du réseau HTM-NET, *XVIIème journée génie civil - génie côtier*, Chatou, 11-13 (in French).

19. Heinrich, P., Gailler, A., Dupont, A., Rey, V., Hébert, H Listowski, C., 2023. Observation and simulation of the meteotsunami generated in the Mediterranean Sea by the Tonga eruption on 15 Jan 2022, *Geophysical Journal Int.*, 234, 2, 903-914, https://doi.org/10.1093/gji/ggad092.

20. Miles, J.W.; Chamberlain, P.G. 1998. Topographical scattering of gravity waves. *J Fluid Mech* 361, 175-188.

21. Karperaki, A.; Papathanasiou, T.K.; Belibassakis, K.A. 2019. An optimized, parameter-free PML-FEM for wave scattering problems in the ocean and coastal environment. *Ocean Engineering* 179, 307-324.