<figure><img src="image_1.png" /></figure>

# Observations and simulations of the meteotsunami generated by the Tonga eruption on 15 January 2022 in the Mediterranean Sea

P. Heinrich,<fnref n="1" /> A. Gailler,<fnref n="1" /> A. Dupont,<fnref n="1" /> V. Rey,<fnref n="2" /> H. Hébert<fnref n="1" /> and C. Listowski<fnref n="1" />

Accepted 2023 February 23. Received 2023 February 23; in original form 2022 September 29

## SUMMARY

The 15 January 2022 eruption of the Hunga-Tunga volcano generated a Lamb pressure wave propagating all over the globe and triggering a tsunami throughout the planet. A first atmospheric wave arrived 16 hr after the eruption on the French Mediterranean coasts and propagated southward. A second one originating from Africa was observed 4 hr later with an attenuated amplitude. The first wave can be roughly defined either by a *N* wave or a sinusoid signal with a period close to 50 min and an amplitude of 130 Pa. In the Mediterranean Sea, the tsunami was recorded by almost all standard coastal tide gauges or pressure gauges. The French tide gauge stations recorded water waves with amplitudes ranging from a few centimetres to 10 cm and with periods ranging from 10 min to 1 hr. Numerical simulation of the tsunami is performed by the operational code Taitoko for different atmospheric sources. Non-linear shallow water equations are solved by a finite-difference method, using the nested multigrid approach. The tsunami is generated by calculating analytically the atmospheric pressure gradient in the momentum equations. Comparisons of time-series between numerical solutions and records are satisfactory for most tide-gauges along the French Mediterranean coast. Sensitivity analysis on the atmospheric source and on the resolution is performed. For most tide-gauge stations, numerical results show that the wave forms depend first on local resonance phenomena.

**Key words:** Numerical simulation; Tonga volcano; Tsunami propagation.

## 1 INTRODUCTION

On 15 January 2022, at 4hr14 UTC, the Hunga-Tonga volcano (175.38°W, 20.57°S) located in the Islands of Tonga in the south-west Pacific Ocean, explosively erupted. This eruption triggered a tsunami that was recorded unexpectedly by most tide-gauge stations around the world. In the Pacific Ocean, Tsunami Warning Centers were not able to deal with such event efficiently, since the existing tsunami warning systems are designed for earthquake-generated tsunamis. Nevertheless, tsunami warning messages were issued based on tsunami signal records (IOC 2022a).

The Tonga tsunami caused damage locally in the Tonga islands but also across the Pacific Ocean. Damaging waves struck harbours and coasts in New Zealand, Rarotonga, Hawaii, the United States West Coast, and as far away as Peru, Chile and Japan (e.g. Lynett 2022; Imamura et al. 2022; Terry et al. 2022). The French Mediterranean coasts were reached 16 hr after the explosion by water waves with maximum amplitudes of up to 10 cm. In western Mediterranean Sea, the tsunami was recorded by most stations of the French Institute (Service Hydrographique et Océanographique de la Marine) as well as by HTM-NET stations (Rey et al. 2020). As in the Pacific Ocean, no warning centre from the NEAM (Northeast

Atlantic and Mediterranean) region could issue any warning message, at least at the information level, since every TSP (Tsunami Service Provider) of the area, including the French Cenalt, monitors earthquake sources only (Gailler et al. 2018).

As reported by Carvajal et al. (2022), Kubota et al. (2022) or Sekizawa & Kohyama (2022), the leading tsunami waves are associated to an atmospheric source propagating with the sound speed ($c \sim 310 \text{ m s}^{-1}$). As a result, the leading waves were detected much earlier than expected (Kubota et al. 2022) considering the tsunami celerity calculated as $\sqrt{gd}$ (where $g = 9.81 \text{ m s}^{-2}$ and *d* is the water depth).

In the atmosphere, the explosion of the Tonga volcano generated various types of atmospheric waves including infrasound (Matoza et al. 2022; Vergoz et al. 2022), Rossby waves and gravity waves (Amores et al. 2022; Kubota et al. 2022). Among them, the Lamb wave (Lamb 1932; Bretherton 1969) propagates at the sound speed over very long distances with minor losses of energy. This wave is non-dispersive and is characterized by a purely horizontal propagation in the troposphere with a maximum pressure amplitude at the earth surface. As a consequence, this wave dominates far away from the source and may circle the planet multiple times (Lynett 2022; Terry et al. 2022). Barograph observations show that

<footnote n="1">¹Commissariat à l'Energie Atomique et aux Energies Alternatives, Centre DAM-Ile-de-France, 91297 Arpajon, France. E-mail: philippe.heinrich@cea.fr</footnote>

<footnote n="2">²Toulon, Aix-Marseille University, CNRS/INSU, IRD, Mediterranean Institute of Oceanography, 83041 Toulon, Cedex 09, France</footnote>
its amplitude ranged from 50 to 300 Pa worldwide (Denamiel et al. 2022).

Similarly to meteorological fronts, the propagation of this atmospheric wave generated ‘forced’ water waves (Monserrat et al. 2006). Due to the celerity $c$ of the Lamb wave, the Froude number $\frac{c}{\sqrt{gd}}$ is much larger than one and the Proudman resonance (Proudman 1929) is not activated for such tsunamis. In other terms, amplitudes of offshore water waves depend only on the intensity of atmospheric forcing and are generally uniform across the oceans in the world (Carvajal et al. 2022; Denamiel et al. 2022). Along the coasts, shoaling, reflection and refraction of water waves are responsible for resonances and amplification of water waves. Kubota et al. (2022) also investigates the effects of atmospheric gravity waves propagating at velocities (around $200 \text{ m s}^{-1}$) close to the mean tsunami celerity in the Pacific Ocean, which could result in resonances between atmospheric and water waves and thereby in amplification of water heights with travel distance.

In the past, the Lamb wave and the associated tsunami were first observed following the Krakatoa volcanic eruption in 1883 (Symons 1888; Press & Harkrider 1966). The explosion-induced atmospheric waves after Krakatoa eruption circled the Earth three times (Murty 1977) and generated tsunami waves recorded by 35 tide gauges distributed throughout the world (Monserrat et al. 2006). As for the 2022 event, arrival times of the tsunami at these sites were in agreement with arrival times of atmospheric sound waves.

Considering the similarities between atmospheric Lamb waves and oceanic shallow water waves, the atmospheric wave propagation is simulated by Amores et al. (2022) and Kubota et al. (2022) by solving vertically-integrated shallow water equations. By contrast, Omira et al. (2022) or Lynett (2022) performed tsunami simulations using an empirical atmospheric pressure model calibrated by weather stations in the Pacific. Following this latter approach, the paper describes tsunami simulation using the Lamb wave model as the generation mechanism. The associated atmospheric pressure wave is defined analytically from pressure records of HTM-NET stations along the French Mediterranean coast. The objective is to analyse and interpret the water heights recorded on the French Mediterranean coasts over a period of 10 hr after the Lamb wave arrival.

In Section 2, records in Europe from three technologies are briefly analysed. In Section 3, spectra of sea level time-series of a few HTM-NET and SHOM stations in the Mediterranean Sea are investigated. Section 4 is devoted to numerical simulations of the tsunami generation and propagation in the Mediterranean Sea and results are compared to tide gauge records. Sensitivity studies on the tsunami source and the grid resolution are analysed.

## 2 ANALYSIS OF INFRASOUND, SEISMIC AND TIDE-GAUGE RECORDS

Europe was crossed on the 15 and 16 January 2022 by two pressure waves recorded by most infrasound, seismic and tide-gauge stations (Figs 1 and 2). These three technologies are merged together to track the Lamb wave propagation across Europe, using signals from 15 stations distributed over Europe and selected approximately along the great circle passing through the source and its antipodal point (Fig. 1). The objective is to calculate the celerity of these waves and to check their azimuths.

Fig. 2 shows the time-series of the 15 stations as a function of the distance from the volcano. The first wave travels from the volcano towards the antipodal point in Northern Africa, the second wave

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Location of infrasound (red points), seismic (green triangles) and tide-gauge (blue squares) stations in western and northern Europe selected to track the Lamb wave propagation. The great circle (dashed red line) is orientated south-north and passed through the Honga-Tonga source and its antipodal point in southern Algeria (not represented).</figcaption></figure>

travels northwards from this antipodal point. Slopes defined by the first significant crest or trough are drawn and labelled in Fig. 2 by the letters *R* and *R'*. For both waves, the celerity of the Lamb wave is inferred from these slopes and is estimated at $310 \text{ m s}^{-1}$ (with an uncertainty of $\pm 5 \text{ m s}^{-1}$). Finally, the azimuth of seismic and infrasound signals is calculated from west-east and south-north components. Taking into account both technologies, the mean azimuth is $6.8^\circ$ which corresponds approximately to the longitude of the antipodal point (4.6$^\circ$E).

## 3 ANALYSIS OF HTM-NET AND SHOM RECORDED WATER WAVES

Sea level time-series are provided either by the SHOM institute or by Toulon University for HTM-NET data (Rey et al. 2020). The network of HTM-NET stations includes 18 gauges, located on the French coasts and recording sea temperatures, atmospheric pressures as well as water surface elevations (https://htmnet.mio.osu.pytheas.fr/). The sampling period of HTM-NET and SHOM stations are 2 and 1 min, respectively. Only stations with an acceptable signal-to-noise ratio and moderate data gaps are retained. Out of 30 stations, signals of 17 stations are analysed and compared to simulations.

All data (pressure or water waves) are bandpass filtered with cut-off periods between 2 min and 2 hr. As usual with tsunamis,
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Waveforms recorded by infrasound (red), seismic (black) and tide-gauge (blue) stations over a period of about 45 000 s (or ~14 hr). Stations are sorted according to their latitude from south (top panel) to north (bottom panel). A bandpass filter with a passband of 25–60 min is applied. Signals are then resampled at 100 s. The slopes of the two black dotted lines labelled by the letters R and R' define the atmospheric wave celerity measured at around 310–315 m s⁻¹. The origin time is 16h40 UTC (15 January).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Location of 30 tide-gauge stations and bathymetry with a resolution of 250 m. Red and green points are active HTM-NET and SHOM stations, respectively. Orange points are non-active HTM-NET stations on the 15 January 2022.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4. Location of 16 tide gauge stations (mentioned in the article) and bathymetry with a resolution of 250 m. The 17th selected tide-gauge (not shown) is located at Ile Rousse in northern Corsica.</figcaption></figure>

the period of the Tonga tsunami in the Mediterranean Sea varies from place to place and with time at each place. Fig. 5 shows the time-series and spectra of water waves for eight tide-gauge stations distributed from Sète to Monaco and in Corsica, over a period of 13 hr. The recorded water waves within the first hours (after 20 hr UTC on the 15th January) are characterized by very small amplitudes (around 1 cm). Later, amplitudes grow progressively and reach up to 10 cm a few hours after the tsunami arrival which is attributed to resonances in coastal areas and/or interferences of water waves following different paths. For most gauges, the fundamental periods range from 20 to 50 min (Fig. 5). It has to be noted that recorded amplitudes are larger in the Gulf of Lion (e.g. at Sète or La Ciotat) than those recorded in the east part of the basin (at Monaco and Ile Rousse in Fig. 5) characterized by steep submarine slopes.

## 4 SIMULATION OF THE METEOTSUNAMI

Simulations are performed by the code Taitoko developed at CEA (Jamelot et al. 2019; Heinrich et al. 2021). This nested multigrid code solves either the non-linear shallow water or the weakly nonlinear Boussinesq equations in a spherical or Cartesian coordinates system. Considering the periods of recorded water waves (Section 3), frequency dispersion may be neglected for this tsunami.

### 4.1 Short description of the code Taitoko

We choose to solve the non-dispersive non-linear long waves equations or shallow water (SW) equations (1–1), (1–2) and (1–3) in a system of spherical coordinates (φ, θ), where φ and θ are the longitude and the latitude, respectively. Continuity and momentum equations read (Murty 1984):

$$ \frac{\partial \eta}{\partial t} = - \frac{1}{R \cos \theta} \left( \frac{\partial u h}{\partial \varphi} + \frac{\partial v h \cos \theta}{\partial \theta} \right) \quad (1) $$

$$ \frac{\partial u}{\partial t} = - \frac{1}{R \cos \theta} \left( u \frac{\partial u}{\partial \varphi} + \cos \theta v \frac{\partial u}{\partial \theta} - \sin \theta u v \right) - \frac{g}{R \cos \theta} \frac{\partial \eta}{\partial \varphi} - F_{\varphi} \quad (2) $$

$$ \frac{\partial v}{\partial t} = - \frac{1}{R \cos \theta} \left( u \frac{\partial v}{\partial \varphi} + \cos \theta v \frac{\partial v}{\partial \theta} + \sin \theta u^2 \right) - \frac{g}{R} \frac{\partial \eta}{\partial \theta} - F_{\theta} \quad (3) $$

$$ F_{\varphi} = - \frac{1}{\rho_w R} \frac{\partial p_s}{\partial \varphi}, \quad F_{\theta} = \frac{1}{\rho_w} \frac{g}{R \cos \theta} \frac{\partial p_s}{\partial \theta}, \quad (4) $$

where R is the earth's radius, η is the surface elevation, d is the water depth, h = η + d is the flow depth, (u, v) are the depth-averaged velocities along (φ, θ), (F_φ, F_θ) are the pressure forces, p_s(φ, θ, t) is
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Spectrogram, bandpass filtered (3600–60 s) and non-filtered water surface elevation records at the tide gauge stations of Sète (3.70°E, 43.40°N), La Ciotat (5.61°E, 43.17°N), Ifremer (5.88°E, 43.10°N) in the Little Bay of Toulon, Giens (6.11°E, 43.04°N), Les Oursinières (6.02°E, 43.08°N), Porquerolles (6.20°E, 43.00°N), Monaco (7.42°E, 43.73°N), Ile Rousse (8.93°E, 42.64°N, North Corsica). Location of gauges is indicated in Figs 3 and 4.</figcaption></figure>

the atmospheric surface pressure. The bottom friction force (based on the Manning's formulation) is neglected in this study.

The non-linear long wave equations are solved by means of a staggered-grid finite difference method in which *u* and *v* are calculated at the cell interface and the water depth *h* at the cell centre. The finite-difference scheme is implicit and uses an iterative procedure

based on a second-order Strong Stability-Preserving Runge-Kutta method (Heinrich *et al.* 2021).

The system of multiple nested grids allows to model multiscale processes with increasing resolution: coastal areas are covered by high-resolution grids that are included in low-resolution grids covering deep water regions. In this code, only one-way communications
between grids have been implemented: data (velocities and water surface elevation) at the boundaries of the child nested grid are interpolated from data of the parent grid.

## 4.2 Modelling of atmospheric pressure waves

The propagation of atmospheric pressure waves is not simulated by Taitoko. The effects of these waves are introduced as external forces in the momentum equations (1.2 and 1.3). Two types of waves are implemented in the code, sinusoidal waves and N waves.

The first one is written as:

$$p_s(\varphi, \theta, t) = -A \sin \alpha \quad (5)$$

$$\alpha = \frac{2\pi}{T} \left( \frac{(r - r_s)}{c} \left( \text{idir} - (t - t_s) \right) \right), \quad (6)$$

where A is the amplitude (in Pa), T is the sinus period, r is the radial distance between the antipodal point in southern Algeria and the grid point defined by (φ, θ), c is the sound speed (set at 310 m s⁻¹), idir (idir = +1 or −1, for waves propagating from or towards the antipodal point, respectively) and t is the time. The wave propagation starts at t = t_s for a distance r_s from the antipodal point. These formulas (5) and (6) are similar to those of Sekizawa & Kohyama (2022) in their 1-D shallow water simulation.

Following the approach of Lynett (2022), the second type of waves is a N wave that is calculated as:

$$p_s(\varphi, \theta, t) = -A (1 + e^{-q \alpha}) \tan^{-1} s \quad (7)$$

$$s = \sin \alpha / (1 + e^{-q\alpha} - \cos \alpha), \quad (8)$$

where q is the slope of the N wave (q = 0.1), A is the amplitude and α is defined by eq. (6). In the next section, the wave amplitude is multiplied by a factor of 1.25 for the negative part of the N wave in order to fit recorded data.

## 4.3 Sensitivity analysis on the source

Tsunami generation is exclusively based on the introduction of external forces due to atmospheric pressure waves. The source sensitivity is analysed by comparing simulated and recorded tsunami waves.

### 4.3.1 Tsunami propagation for atmospheric sinusoidal waves or N waves

For each type of atmospheric waves, tsunami simulation is performed by introducing two pressure waves as analysed in Section 2 and defined in Section 4.2. The first Lamb wave originates from the volcano and propagates towards the antipodal point (4.62°E, 20.57°N). The propagation of this wave, defined in eq. (6), starts in this study from the point S = (6°E, 44.5°N) close to Toulon on the French coast and located at a distance of 17370 km from the volcano. Assuming a sound velocity around 310 m s⁻¹, the associated starting time is t_s = 19hr45 UTC. The second Lamb wave propagates northwards from the antipodal point, starting at t_s = 22hr10 UTC and reaching Toulon about 2 hr later.

The amplitude and the period of the first and second sinusoidal waves are (A₁ = 130 Pa, T₁ = 3000 s) and (A₂ = 60 Pa, T₂ = 2800 s), respectively. The equivalent static sea level changes in response to the steady pressure anomalies are −Δp/ρg ~ −1.3 and −0.6 cm, respectively. The amplitude and the period of the first and

second N waves are, respectively (A₁ = 100 Pa, T₁ = 3000 s) and (A₂ = 50 Pa, T₂ = 2800 s). Fig. 6 shows the first calculated pressure wave crossing the Mediterranean Sea at t = 30 min and t = 60 min after the starting time.

Considering the sound velocity, the Mediterranean Sea is crossed by atmospheric pressure waves within approximately 1 hr. Fig. 7 shows the pressure records for six stations, selected among the 18 HTM-NET stations, starting at 19hr45 UTC and over a period of 7 hr. These time-series are compared with the two types of analytical signals defined in Section 4.2. As shown in Fig. 7, the second wave starting from the antipodal point arrives on the French coasts about 4hr30min after the first wave.

Pressure records are very similar at all HTM-NET stations. The time lag between recorded signals is lower than a few minutes at all stations, which is in agreement with the sound velocity and the maximum south-north distance between HTM-NET stations, lower than 100 km in latitude. The maximum amplitudes of the first and second recorded arches are about 120 and 60 Pa, respectively. These two arches are modelled either by two sinusoidal signals (with periods of 3000 and 2800 s, respectively) or by two N waves with the same periods (Fig. 7). Qualitatively, recorded pressure waves are reproduced well by both models. Oscillations recorded between the two Lamb waves and after the 2nd Lamb wave passing could be due to gravity waves propagating near the surface, triggered or not by the volcano (Vergoz et al. 2022). Their effect is expected to be negligible for the meteotsunami simulation as their amplitude is similar to the second Lamb wave whose impact is negligible (see Section 4.3.2).

The associated tsunami is simulated for these two types of pressure waves. The shallow water simulation covers the western Mediterranean Sea using a resolution of 250 m × 250 m and a time step of 1 s. The mesh is composed of 4800 × 3400 cells with depths ranging from a minimum of 10 m at the coast (to cope with the insufficient resolution along the coast) to 3530 m offshore. The CPU time for a 10-hr simulation is 95 min, 47 min or 27 min according to the number of cores on a HPC machine, 140, 280 or 560. Results are presented in Fig. 8 for six tide-gauge stations distributed on the French coast (from west to east) and one station in northern Corsica. In this figure, the associated pressure waves are referred to as ‘double Nwave’ and ‘double sinus’ waves. Comparisons between recorded and simulated time-series are satisfactory for almost all tide-gauges stations. For all stations along the northern Mediterranean coast, the first computed wave is a trough of a few millimetres, which is in agreement with the solution of the Cauchy–Poisson problem (Proudman 1929). This solution writes as the addition of a ‘forced’ water wave propagating at the sound speed c and two ‘free’ water waves propagating at ±√gd :

$$\eta(x, t) = \frac{\eta_0(x - ct)}{1 - Fr^2} - \frac{\eta_0(x - \sqrt{gd}t)}{2(1 - Fr)} - \frac{\eta_0(x + \sqrt{gd}t)}{2(1 + Fr)}, \quad (9)$$

where η₀(x, t) = −p_s(x,t)/ρg is the hydrostatic response to the pressure change p_s(x, t) and Fr is the Froude number Fr = c/√gd. Along the French Mediterranean coasts and for small water depths (d < 1000 m), the Froude number is larger than 3 and the predominant term is the second one in eq. (9). This term is associated with a ‘free’ water wave of opposite sign to the atmospheric signal η₀(x − √gd t) / (2(Fr − 1)) = −p_s/ρg propagating at +√gd. As a result, the amplitude of the first water wave is generally negative at stations located in the Gulf of Lion and in Côte d’Azur. Offshore, the Froude number is smaller and the first term also contributes to the generation of the first water
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Pressure wave (in Pa) calculated at *t* = 30 and 60 min after 19hr45 UTC for a sinusoidal signal with a period of 3000 s. This wave propagates towards the antipodal point in southern Algeria. Isobaths in solid lines (0 and 2500 m) or dashed lines (1000 m).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Comparison between recorded atmospheric pressure time-series (dashed black line) and those calculated analytically for two *N* waves (red) or for two sinusoidal signals (green) at Redonne (5.22°E, 43.33°N), Le Brusc (5.80°E, 43.08°N), Les Oursinières (6.02°E, 43.08°N), La Capte (6.15°E, 43.06°N), Port Cros (6.38°E, 43.01°N) and Menton (7.52°E, 43.78°N). The origin time is 19hr45 TU on the 15th January (or 15hr30 after the eruption). Recorded time-series are bandpass filtered. Locations of HTM-NET gauges are indicated in Figs 3 and 4.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Comparison between recorded time-series of surface elevation (dashed black line) and those simulated using the double *N* wave (blue), the ‘double sinus’ (green) and the ‘single sinus’ sources (red) at tide gauges of Sète (3.67°E, 43.40°N), La Ciotat (5.61°E, 43.17°N), Les Oursinières, Giens (6.11°E, 43.04°N), Porquerolles (6.20°E, 43.00°N), Monaco (7.42°E, 43.73°N) and Ile Rousse (8.93°E, 42.64°) (in Corsica). Recorded time-series are bandpass filtered. Locations of gauges are indicated in Figs 3 and 4. The arrival time of pressure waves at the station of Sète is indicated by two red arrows. The other stations are impacted almost simultaneously within a short time interval of about 7 min.</figcaption></figure>

wave. As a result, the amplitude of the first water wave propagating in the southern direction may become positive. This combination may account for the first positive water wave calculated at Ile Rousse located in northern Corsica (Fig. 8). The half-period of this first wave is about 20 min.

For stations located in the Gulf of Lion, west of Levant islands (Porquerolles and Port-Cros islands), amplitudes of water waves are lower than a few centimetres during the first 2 hr. Later, maximum amplitudes reach up to 10 cm between 3 and 6 hr following the first atmospheric wave (Fig. 8), which is attributed to local resonances along the coast. Conversely, the bathymetry east of the Levant islands is deeper and stations located east of the Saint-Tropez gulf are less subject to resonances and wave amplifications. Amplitudes at Monaco and Ile Rousse do not exceed 3 cm.

As expected, both simulations produce similar results for almost all stations in terms of amplitudes or periods (Fig. 8). A closer examination of computed signals indicates that shorter oscillations (with periods around 10 min) and larger amplitudes are produced at some stations by the ‘*N*-wave’ simulation, which is attributed to the

shorter periods associated with the *N*-wave source. These results are confirmed by the sensitivity study to the atmospheric wave period in Section 4.3.2.

### 4.3.2 Sensitivity of the sinusoidal source

The sensitivity analysis in Section 4.3.2 consists first in removing the second sinusoidal wave (the associated simulation is then called ‘single sinus’ simulation) and second in varying periods of the first sinusoidal wave.

Fig. 8 shows the computed time-series at the seven selected tide-gauge stations for the ‘double sinus’ and ‘single sinus’ simulations. As expected, the computed signals are identical during the first 5 hr. Later, for almost all stations, computed results are close to each other with amplitudes slightly attenuated in the ‘single sinus’ simulation. The second pressure wave with an amplitude of 60 Pa is much less energetic with a ratio of about 5 (the wave energy is proportional to the square of its amplitude and to the wavelength).
<figure><img src="image_10.png" /><figcaption>**Figure 9.** (a) Comparison between recorded time-series of surface elevation (dashed black line) with those calculated by the model with high (red) and low (green) resolution at Sête, Fos-sur-Mer, Redonne, Cassis, La Ciotat and Le Brusc (from west to east). The resolution of the child grid covering Redonne is 25 m. Locations of gages are indicated in Fig. 4. (b) Same as in Fig. 9(a) at the tide gauges of Ifremer (Petite Rade or Little Bay), Lazaret (Petite Rade or Little Bay), Saint Louis (Bay of Toulon), Les Oursinières, Giens, La Capte and Porquerolles (from west to east). The resolution of the child grid covering Les Oursinières is 25 m. (c) Same as Fig. 9(a) at the tide gauges of Port Cros, Monaco, Menton and Ile Rousse.</figcaption></figure>

As a result, its effects on the water waves on the French coasts are weak. In other terms, the Mediterranean coasts are first sensitive to the first and largest pressure wave which excites local resonance in bays and harbours, at least over a period of 8 hr (Fig. 8).

The sensitivity of the sinusoidal source is further studied by vary-
ing periods of the first atmospheric wave from 1800 to 4200 s every
600 s and results (not shown) are compared to those obtained in the
3000 s ‘single sinus’ simulation. For most stations, the sinusoidal
sources with periods larger than 3600 s cannot reproduce local res-
onances and amplitudes of water waves may be strongly attenuated.
This phenomenon can be observed in particular for water waves
with periods of 20 min at La Ciotat, Giens or Les Oursinières and
for water waves with periods of 30 min at Monaco. Conversely, si-
nusoidal simulations with periods lower than 2400 s reproduce local
resonances. It is observed that the 2400 s sinusoidal and the *N*-wave
sources are close to each other and in agreement with records.

4.4 High resolution simulations for *N*-wave atmospheric
waves

In the Mediterranean Sea, high-resolution is generally required for
standard meteotsunamis (Denamiel *et al.* 2019) since amplification
of waves generally occurs in narrow channels and small harbours.

The nested multigrid approach is used here to simulate both
the offshore propagation and the coastal impact of water waves.
Three grid levels are used. The parent grid is the one used in pre-
vious sections and covers the western Mediterranean Sea with a
resolution of 250 m. Seven coastal regions, from Sete to Men-
ton and in Corsica, are covered by an intermediate grid with a
resolution of 25 m. In each of these regions, a child grid with a
resolution of 5 or 10 m is built using lidar data at 5 m resolution
(litto3D®) and covers the harbour where the tide-gauge is located.
The two exceptions are the small havens of Redonne (10 km west
of Marseille) and Les Oursinières (in the eastern Large Bay of
Toulon) which are covered by the intermediate grid with a resolu-
tion of 25 m only. Considering their dimensions (less than 100 m),
these havens are modelled neither in the 250-m nor in the 25-m
grids.

For most stations, high-resolution simulations do not improve
significantly the results (Fig. 9a, b, c). The largest differences be-
tween simulations are observed at Fos-sur-Mer, at Le Brusc, in the
Little Bay (or ‘Rade’) of Toulon, to a lesser extent at St Louis and
in Menton. Computed water surfaces every 2 min (not shown) may
highlight amplifications of water waves for some stations. However,
this amplification is generally complex and due to a combination of
shoaling, reflection and refraction.
<figure><img src="image_11.png" /><figcaption>**Figure 9 – continued**</figcaption></figure>

In the Gulf of Lion, the computed water surfaces show the west-east propagation along the coast of edge waves. At Sète, these edge waves with periods of about 1 hr reach amplitudes larger than 5 cm over several hours (Fig. 9a). At Fos-sur-Mer, the computed signal in the coarse grid resolution differs significantly from the record. The mismatch in time as well as the wave amplification are due to the deformation of the bay geometry in the 250-m resolution grid. In the reality, the harbour of Fos-sur-Mer is protected by a dune ridge of several km and includes several deep and rectangular basins in one of which the tide-gauge station is located. Numerical results show that the presence of the dune ridge in the child grid attenuates wave heights and delays wave arrivals in agreement with records.

At Redonne, results are similar in the coarse (250 m) and intermediate (25 m) grids. Time-series records are poorly reproduced within the first hours (Fig. 9a), which could be due to the absence of this haven in both grids.

Stations of Cassis, La Ciotat and Le Brusc exhibit similar water waves with periods of about 30 min. Computed water surfaces show that their origin is a combination of offshore waves, edge waves and reflected waves. Amplifications are more or less pronounced according to the local bathymetry. At Le Brusc, larger amplitudes are calculated in the child grid (with a resolution of 5 m) in agreement

with records (Fig. 9a). A closer examination shows that bathymetric data are characterized by smooth slopes in the child grid, which favours amplification.

According to Rey *et al.* (2020), the Little Bay of Toulon may be considered as a closed basin and is submitted to a resonance (or a ‘seiching’) with a period of 20 min. This phenomenon is well reproduced (Fig. 9b) in the 5-m and 250-m grids at both HTM-NET stations (Ifremer and Lazaret). Larger amplitudes in the coarse grid are attributed to the absence of irregularities along the coast line.

The small harbour of Saint-Louis is located outside of the Little Bay, only 1 km northeast of this bay (Fig. 4). The tide-gauge is located at the entrance of the harbour closed by a 200-m long dike. The records of this station (every 2 min) are instantaneous and characterized by high noise levels. The recorded wave periods shorter than 10 min (Fig. 9b) are not reproduced by the model.

Regions located west and east of the Giens peninsula are shallow regions with water depths smaller than 20 m. Both regions form semi-closed basins subjected to resonances which are reproduced satisfactorily in all grids (Fig. 9b). West of the peninsula, the period of water waves is about 25 min and computed amplitudes are higher than those recorded. The basin located east of the peninsula is much larger and periods at Porquerolles and La Capte are about 50 min.
<figure><img src="image_12.png" /><figcaption>Figure 9 – continued</figcaption></figure>

Farther east, the bathymetry in front of Monaco is deeper with strong submarine slopes. Recorded and computed time-series are in agreement and do not exhibit amplifications (Fig. 9c).

The tide gauge station at Menton is located at the entrance of the harbour protected by a 600-m-long dyke which is modelled only in the 5-m child grid. Amplitudes in the coarse and intermediate grids are below 2 cm. In the 5-m child grid, amplitudes are two times larger and shorter periods (about 10 min) are calculated (Fig. 9c), which could be attributed to the harbour resonance. It has to be noted that recorded data are characterized by a high noise level and comparisons with simulations may be irrelevant.

## 5 CONCLUSION

Numerical simulations of the ‘Tonga’ tsunami are performed to analyse and interpret the sea level time-series recorded along the French Mediterranean coasts, following the explosion of the volcano. Different atmospheric sources and different grid resolutions have been tested. Results are compared with water waves recorded by HTM-NET and SHOM tide-gauges within the first 10 hr following the arrival of the atmospheric pressure wave.

The tested atmospheric sources are single sinusoidal waves, double sinusoidal or double *N* waves. These experiments indicate that the second source originating from the antipodal point may be neglected. The sensitivity analysis on the sinusoidal sources shows that periods between 2400 and 3000 s produce the best results, qualitatively. For these periods, time-series are close to those simulated using the *N*-wave source. Unlike meteotsunamis, the associated periods of this event are long and high resolution grids are not required. The numerical simulations show that the 250-m resolution of the parent grid covering the western Mediterranean Sea seems to be sufficient for reproducing recorded signals at most stations. The 5-m or 10-m resolutions are probably required in order to reproduce local amplifications or local resonances in areas that are not modelled by coarse resolution such as small harbours or havens protected by dikes.

Tsunamis of volcanic origin as well as meteotsunamis are forced by atmospheric pressure waves. Meteotsunamis, associated to an atmospheric storm front of a few hectopascals, have the potential to produce substantial damages to houses and infrastructures (Rabinovich et al. 2009; Denamiel et al. 2019; Romero et al. 2019). These events can be forecasted in the Mediterranean Sea since meteorological winds and surface pressure may be calculated accurately over a period of a few days. In our case, the main Lamb pressure wave was recorded by hundreds of barometers of weather stations and reached the Mediterranean coasts 16 hr after the explosion. As a consequence, it would have been technically possible to issue tsunami warning information in the Mediterranean Sea.

Tsunamis originating from volcanic eruption should now be accounted for in the future procedures of the tsunami warning systems. In particular, procedures have been suggested by IOC/Unesco in the aftermath of the 2022 explosion (IOC 2022b) to implement detection processing in the Pacific Ocean regardless of an earthquake characterization. This detection would be based on observed tsunami wave amplitudes and on tsunami simulations such as those exposed in this paper. Such events could be then monitored, not only in the Pacific Ocean, but also in any oceanic basin subject to this particular Lamb wave forcing.

## ACKNOWLEDGMENTS

The coarse bathymetry was built from GEBCO gridded data (https://www.gebco.net) and finer topo-bathymetric grids were built from IGN and SHOM (HOMONIM and litto3d®) data available at https://geoservices.ign.fr/rgealti and https://diffusion.shom.fr. We are thankful to SHOM institute for providing us tide-gauge signals along the Mediterranean coasts. HTM-NET observational network has received recurrent cofinancing since 2013 from CNRS/INSU for the national program SOERE, from ILICO-DYNALIT and also from the Toulon Provence Méditerranée (TPM) urban community. T. Missamou, responsible technician and D. Mallarino, webmaster, are acknowledged for the maintenance of the stations, the web site
and data post-processing. We also thank the reviewers for their comments.

## DATA AVAILABILITY

HTM-NET data are available on direct request to the project manager on its own website (https://htmnet.mio.osupytheas.fr). French water level data are also available on the SHOM data web site (data.shom.fr). The tide-gauge records, 'PT17' and 'OSTE', are provided by the Flanders Marine Institute (VLIZ; Intergovernmental Oceanographic Commission). Seismic data for France come from the CEA seismic network and all the complementary seismic stations may be downloaded through the IRIS Web Services (https://service.iris.edu). Finally, IMS infrasound data (except the AD1F station provided by CEA) are available from the CTBTO for scientific purposes through the virtual Data Exploitation Centre (https://www.ctbto.org/specials/vdec/). Numerical results from this paper are available on request to P. Heinrich (philippe.heinrich@cea.fr). The source of the code Taitoko is not publicly available for now but its detailed description is available on request to P. Heinrich or A. Jamelot (jamelot@labogeo.pf).

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