Xavier B Bertin
email: xavierbertin@univ-lr.fr

Kévin Martins

Anouk De Bakker

Teddy Chataigner

Thomas Guérin

Thibault Coulombier

Olivier Viron

Olivier De Viron

Energy Transfers and Reflection of Infragravity Waves at a Dissipative Beach Under Storm Waves

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Introduction

Infragravity (hereafter IG) waves are surface gravity waves with frequencies typically ranging from 0.004 to 0.04 Hz, related to the presence of groups in incident short waves [see Bertin et al., 2018, for a recent review]. IG waves are only a few mm to cm-high in the deep ocean [e.g. [START_REF] Rawat | Infragravity waves across the oceans[END_REF][START_REF] Crawford | The sources of deep ocean infragravity waves observed in the North Atlantic Ocean[END_REF][START_REF] Smit | Infragravity wave radiation across the shelf break[END_REF] but can exceed 1.0 m close to shore under storm conditions [START_REF] Ruessink | Observations of turbulence within a natural surf zone[END_REF][START_REF] Fiedler | Observations of runup and energy flux on a low-slope beach with high-energy, long-period ocean swell[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF].

Although not validated against field observations, the application of XBeach [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF] to the SW of Oléron Island (France) by [START_REF] Baumann | Importance of infragravity waves for the generation of washover deposits[END_REF] even suggests that IG waves could exceed 2.0 m close to shore under very energetic, narrow-banded incident swells.

Therefore, IG waves have an essential contribution to sandy beach [e.g. [START_REF] Guza | Swash oscillations on a natural beach[END_REF][START_REF] Elgar | Observations of infragravity waves[END_REF][START_REF] Reniers | Linear modelling of infragravity waves during Delilah[END_REF]Ruessink et al., 1998a;[START_REF] Guedes | Observations of wave energy fluxes and swash motions on a low-sloping, dissipative beach[END_REF] and tidal inlet hydrodynamics [START_REF] Bertin | Relevance of infragravity waves at a wave-dominated inlet[END_REF][START_REF] Bertin | The closure of a shallow tidal inlet promoted by infragravity waves[END_REF][START_REF] Mendes | Importance of infragravity waves in a wave-dominated inlet under storm conditions[END_REF], sediment transport [e.g. [START_REF] Russell | Mechanisms for beach erosion during storm[END_REF][START_REF] Aagaard | Infragravity wave contribution to surf zone sediment transport -the role of advection[END_REF]De Bakker et al., 2016a] and dune and barrier breaching [e.g. [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF], particularly under storm waves.

At dissipative beaches, IG waves usually dominate the spectrum of the free surface elevation close to shore [START_REF] Guza | Swash oscillations on a natural beach[END_REF][START_REF] Russell | Mechanisms for beach erosion during storm[END_REF]Ruessink et al., 1998b] and therefore control the development of large runup [START_REF] Raubenheimer | Observations and predictions of run-up[END_REF]Ruessink et al., 1998b;[START_REF] Ruggiero | Wave runup, extreme water levels and the erosion of properties backing beaches[END_REF] and overwash along the coast and subsequent washovers in low-lying zones [e.g. [START_REF] Baumann | Importance of infragravity waves for the generation of washover deposits[END_REF]. At such gently-sloping beaches, IG waves are mostly generated through the bound wave mechanism, which corresponds to secondorder nonlinear wave-wave interactions that result in the development of long waves, out of phase with the energy envelope of the short waves. This mechanism was first demonstrated analytically in 1D by [START_REF] Biesel | Équations générales au second ordre de la houle irrégulière[END_REF] and [START_REF] Longuet-Higgins | Radiation stress and mass transport in gravity waves, with application to surf beats[END_REF] and was then extended to 2D random waves by [START_REF] Hasselmann | On the non-linear energy transfer in a gravity-wave spectrum part 1. General theory[END_REF]. These analytical solutions imply that the frequency of the resulting IG waves should decrease under narrow-banded and long-period incident short waves. Yet, [START_REF] Bertin | Relevance of infragravity waves at a wave-dominated inlet[END_REF] observed relatively short-period IG waves (e.g. T < 60 s) under such incident short waves while other authors reported IG wave periods in the range 120 -300 s under shorter period incident waves [e.g.

De [START_REF] De Bakker | Shoreline dissipation of infragravity waves[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF]. This apparent inconsistency suggests that the relationship between the shape of the incident short wave spectra and the frequency of IG waves is not straightforward. Also, these analytical solutions were derived considering a flat bottom while for sloping bottoms, the bound waves increasingly lag behind the wave groups as and when the water depth decreases. This phase lag was observed in both field [e.g. [START_REF] List | A model for the generation of two-dimensional surf beat[END_REF][START_REF] Masselink | Group bound long waves as a source of infragravity energy in the surf zone[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF] and laboratory experiments [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF][START_REF] De Bakker | Infragravity wave propagation and dissipation on a low-sloping laboratory beach[END_REF][START_REF] Padilla | Transfer and dissipation of energy during wave group propagation on a gentle beach slope[END_REF] and was demonstrated semi-analytically [START_REF] Janssen | Long waves induced by short-wave groups over a sloping bottom[END_REF][START_REF] Guérin | On the bound wave phase lag[END_REF]. [START_REF] Van Dongeren | Shoaling and shoreline dissipation of low-frequency waves[END_REF] showed that this phase lag enhances energy transfer from short waves to IG waves. This process is often referred to as "bound wave shoaling", which is somehow misleading because the resulting growth of IG waves is higher than the growth associated with conservative shoaling (i.e. Green's Law).

In the surf zone, depth-induced breaking of short waves results in a shoreward reduction of short wave groupiness, which allows for the release of IG waves that then propagate as free waves [e.g. [START_REF] Janssen | Long waves induced by short-wave groups over a sloping bottom[END_REF][START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF]. [START_REF] Baldock | Dissipation of incident forced long waves in the surf zone-implications for the concept of "bound" wave release at short wave breaking[END_REF] proposed that bound wave release can only occur if short waves are in the shallow-water regime when breaking, which condition is not necessarily met under short period waves and/or storm conditions. Throughout their propagation in the inner surf zone of gently sloping beaches, IG waves transfer energy to higher frequencies and steepen, which promotes their dissipation through depth-limited breaking [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF][START_REF] Van Dongeren | Shoaling and shoreline dissipation of low-frequency waves[END_REF][START_REF] Lin | Infra-gravity wave generation by the shoaling wave groups over beaches[END_REF][START_REF] De Bakker | Shoreline dissipation of infragravity waves[END_REF]. Several authors reported the saturation of wave runup in the IG band under storm waves [Ruessink et al., 1998b;[START_REF] Ruggiero | Wave run-up on a high-energy dissipative beach[END_REF][START_REF] Sénéchal | Wave runup during extreme storm conditions[END_REF], possibly owing to IG wave dissipation close to shore. However, Fiedler et al.

[2015] did not observe any runup saturation under comparable incident conditions and beach slope, suggesting that further research on IG wave dissipation is needed. IG wave energy that is not dissipated in the surf zone can be reflected along the shore, a process that was observed and quantified both in the field [START_REF] Huntley | Field observations of surf beat: 1. Progressive edge waves[END_REF][START_REF] Guza | Observations of surf beat[END_REF][START_REF] Elgar | Reflection of ocean surface gravity waves from a natural beach[END_REF][START_REF] De Bakker | Shoreline dissipation of infragravity waves[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF] and in laboratory experiments [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF][START_REF] Van Dongeren | Shoaling and shoreline dissipation of low-frequency waves[END_REF]. The reflection coefficient R 2 , which corresponds to the ratio between the outgoing and the incoming IG wave energy, was shown to increase with the beach slope and with the period of incident IG waves [START_REF] Van Dongeren | Shoaling and shoreline dissipation of low-frequency waves[END_REF]. As sandy beaches often exhibit steeper slopes in their upper part, a tidal modulation of IG wave reflection is expected and was already reported [START_REF] Okihiro | Infragravity energy modulation by tides[END_REF]. Recent observations of free IG waves across the shelf of Oregon even revealed that such a tidal modulation can be strong in the deep ocean [START_REF] Smit | Infragravity wave radiation across the shelf break[END_REF]. The frequencydependence of IG wave dissipation and reflection processes produces large differences in the cross-shore structure of surface elevation spectra. Indeed, higher frequency IG waves tend to be progressive across the surf zone while lower frequencies tend to develop quasi-standing patterns due to a stronger reflection [START_REF] De Bakker | Shoreline dissipation of infragravity waves[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF].

In a recent review paper, [START_REF] Bertin | Infragravity waves: From driving mechanisms to impacts[END_REF] pointed out that further research on IG waves was needed to better understand their generation mechanisms, their transformations across the surf zone and their reflection along the coast. This study aims to contribute to this effort and to answer more specifically the following questions: how large can IG waves be in the nearshore? What is the relationship between the shape of the incident short-wave spectra and the IG waves spectra along the coast? Do IG waves necessarily suffer strong dissipation and subsequent low reflection at a dissipative beach? To address these questions, we present the analysis of an unpublished dataset collected under storm-wave conditions at a dissipative beach located to the SW of Oléron island (France), complemented with numerical modelling using XBeach [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF].

Study area

This study takes place at Saint-Trojan Beach, a dissipative beach located in the central part of the French Atlantic coast (Figure 1), along the south-western coast of Oléron Island. This stretch of coast corresponds to a 8 km-long sandspit, bounded to the South by the Maumusson Inlet [START_REF] Bertin | Tidal inlet response to sediment infilling of the associated bay and possible implications of human activities: The Marennes-Oléron Bay and the Maumusson Inlet, France[END_REF]. The continental shelf in front of the study area is about 150 km-wide, with a very gently sloping shoreface, the isobath 20 m being found about 10 km away from the shoreline. The tidal regime in this region is semi-diurnal and macrotidal, with a tidal range varying between about 1.5 m during neap tides and 5.5 m during spring tides. Tidal currents remain weak at the studied beach and the impact of tides on short waves remains mostly restricted to water level variations. According to [START_REF] Bertin | Longshore transport estimation and inter-annual variability at a high-energy dissipative beach: Saint Trojan Beach, SW Oléron Island, France[END_REF], yearly-mean deep water wave conditions are characterized by a significant height (H s ) of 2 m, a peak period (T p ) of 10 s and a mean direction of 285 • N . During storms, H s can reach episodically 10 m in deep water and T p can exceed 20 s with a westerly direction [START_REF] Bertin | The contribution of short-waves in storm surges: two case studies in the Bay of Biscay[END_REF]. Due to the very gently sloping shoreface and the wide continental shelf, the most energetic waves suffer strong dissipation and numerical wave models suggest that their H s hardly exceeds 5 m at the breaking point [START_REF] Bertin | Longshore transport estimation and inter-annual variability at a high-energy dissipative beach: Saint Trojan Beach, SW Oléron Island, France[END_REF].

The studied beach is mainly composed of fine and well sorted sands (d 50 = 0.18 -0.22 mm), which together with the energetic wave climate and the macrotidal range cause its morphology to be non-barred and dissipative (Figure 1-B). Small-amplitude intertidal bars can only develop after the persistence of fair weather conditions. Its slope typically ranges from about 0.0015 at the shoreface to 0.015 in the intertidal area [START_REF] Bertin | Longshore transport estimation and inter-annual variability at a high-energy dissipative beach: Saint Trojan Beach, SW Oléron Island, France[END_REF], although a berm usually develops during the course of the summer period, with a slope reaching 0.04. Due to the persistence of low to moderate-energy wave conditions in autumn 2016 and early winter 2017, such a berm was still present during our field campaign. This gently sloping morphology and the presence of a shallow shoreface induce a strong wave refraction, so that the wave angle at breaking is usually small, with typical values smaller than 10 • [START_REF] Bertin | Longshore transport estimation and inter-annual variability at a high-energy dissipative beach: Saint Trojan Beach, SW Oléron Island, France[END_REF].

Field campaign and data processing

The field campaign presented in this study was motivated by the development of the storm Kurt, which started to deepen to the SE of the Newfoundland on the 31/01/2017 and crossed the North Atlantic Ocean following an Eastward track, centered between 48°and 50°. Based on the CFSR reanalysis [START_REF] Saha | The NCEP climate forecast system reanalysis[END_REF], Kurt reached a minimum sea-level pressure around 955 mbar on the 02/02/2017 about 1000 km to the West of the Bay of Biscay, accompanied with a ∼ 500 km-wide band of western winds in the range 25 -35 m.s -1 . This atmospheric setting drove a very large and long period swell, that reached the French coasts in the night of the 02/02/2017 to the 03/02/2017. At the deep water buoy of Biscay (Figure 1-A), the mean wave period increased from 8.0 s to 13.0 s and H s rapidly increased from 3.0 m to almost 10.0 m, which value corresponds to a return period on the order of one year [START_REF] Nicolae-Lerma | Spatial variability of extreme wave height along the atlantic and channel french coast[END_REF]. The wave hindcast described in [START_REF] Guérin | Impacts of waveinduced circulation in the surf zone on wave setup[END_REF] suggests that the wave peak period T p exceeded 20 s. The measurement period encompassed four tidal cycles, from the 01/02/2017 to the 03/02/2017, characterized by a tidal range of 3.5 to 4 m. An Acoustic Doppler Current Profiler (ADCP1) with a frequency of 600 kHz and equipped with a pressure sensor was deployed about 3 km offshore (Figure 1-B). In the intertidal zone, 9 pressure transducers (PT) sampling at 4 Hz were deployed (Figure 1-C) as well as a second ADCP (ADCP2), with a head frequency of 2 MHz mounted with a pressure sensor at the location of PT3. The PTs were buried between 0.05 to 0.10 m of sand, in order to avoid dynamic pressure errors. The position of each sensor was measured with a differential GNSS, using a post-processing technique with a base station settled on the dune crest in front of the instrumented profile [START_REF] Guérin | Impacts of waveinduced circulation in the surf zone on wave setup[END_REF]. This GNSS was also used to survey the intertidal beach topography at every low tide. The analysis of this data revealed surprisingly small morphological changes for storm wave conditions, typically lower than 0.1 m along the cross-shore profile of the instruments.

For each sensor, bottom pressure measurements were first corrected for sea level atmospheric pressure measured at the nearby meteorological station of Chassiron (Figure 1-A). The entire record was split into consecutive bursts of 30 min and only the bursts in which the sensor was continuously submerged were considered. PT9 was never continuously submerged during more than 30 minutes and was therefore discarded for the present study. Bottom pressure power density spectra E p (f ) were computed using Fast Fourier Transforms, with 10 Hanning-windowed, 50% overlapping segments (20 degrees of freedom). These pressure spectra were then converted into elevation spectra E(f ) considering linear wave theory [see for instance [START_REF] Bishop | Measuring waves with pressure transducers[END_REF]. The spectral significant wave height (H m0 ) was computed as:

H m0 = 4 √ m 0 (1)
with

m 0 = fmax fmin E(f )df (2)
where f max was set to 0.4 Hz, a value for which the pressure correction reaches about 13 by 3 m water depth, well below the threshold of 100 to 1000 recommended by [START_REF] Bishop | Measuring waves with pressure transducers[END_REF]. f min is time-varying and defined following [START_REF] Roelvink | Bar-generating cross-shore flow mechanisms on a beach[END_REF] or [START_REF] Hamm | Wave parameters in the nearshore: A clarification[END_REF] as half the continuous peak frequency f p , computed from the offshore ADCP data as:

f p = m 2 0 m -2 m 1 (3)
where

m k = fmax fmin f k E(f )df (4)
The continuous peak frequency was preferred to the discrete one because it is a more stable parameter, particularly under multimodal sea states, as it was the case at the beginning of the field campaign. As the incident peak period doubled during the field campaign, a fixed frequency cutoff would not have allowed for a proper separation between the gravity and the IG bands. In order to characterize the spatio-temporal variations of incoming IG wave frequencies, the mean IG wave period T m02,IG + was also computed based on the incoming IG wave signal, separated from the outgoing one at the ADCP2 and the ADV using the method of [START_REF] Guza | Swash on steep and shallow beaches[END_REF]:

T m02,IG + = m 0 m 2 (5)
4 Numerical modelling

Model description

XBeach is a two-dimensional modelling system that couples a Saint-Venant solver with a simplified wave-action model and a sediment transport model to simulate the generation and propagation of IG waves and the associated dynamics [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF]

Model implementation

A rectilinear grid centered on the instrumented cross-shore profile and extending 4000 m in the alonghore direction was implemented. In the cross-shore direction, this grid starts from the dune crest and extends 11000 m offshore, corresponding to a mean water depth of about 23 m (Figure 1 with time series of water levels recorded at ADCP1 (Figure 1-B). The wave model was forced with time-series of directional wave spectra, originating from a regional application of the WaveWatchIII model [START_REF] Tolman | User manual and system documentation of WAVEWATCH III version 3[END_REF] forced by wind fields from the CFSR reanalysis [START_REF] Saha | The NCEP climate forecast system reanalysis[END_REF]. In deep water, [START_REF] Guérin | Impacts of waveinduced circulation in the surf zone on wave setup[END_REF] showed that this wave hindcast resulted in a normalized root mean squared discrepancy (hereafter NRMSD) of 10% for H m0 and 6% for the mean wave period. Bottom friction was represented by a quadratic bottom shear stress with a constant Chezy coefficient set to 80 m 0.5 .s -1 . The horizontal eddy viscosity was assumed constant (0.5 m 2 .s -1 ) and the minimum water depth was set to 0.01 m. Depth-limited breaking in the wave model was represented using the model of Roelvink [1993a] with a breaking index γ adjusted to 0.38 after calibration. The model was run for the duration of the field experiment (3 days), and time series of short wave height, surface elevation and current velocities were archived at 2 Hz. In order to get insights into IG wave generation mechanisms, additional numerical experiments were performed, where wave forces were switched off either inside or outside the surf zone. The breaking formulation of [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF] was used to determine whether a computational node was located inside or outside the surf zone, i.e. if the local wave height was not exceeding the threshold H max as defined in [START_REF] Roelvink | Surf beat and its effect on cross-shore profiles[END_REF]. In the bound wave case, wave forces were turned off inside the surf zone while in the breakpoint case, wave forces were turned off outside the surf zone and the incoming boundwave was not included along the open boundary, following [START_REF] Pomeroy | The dynamics of infragravity wave transformation over a fringing reef[END_REF] and [START_REF] Bertin | Relevance of infragravity waves at a wave-dominated inlet[END_REF]. Lastly, additional 24 h-long simulations were run with constant water levels and wave forcing at specific times of the measurement period to provide sufficient resolution and number of degrees of freedom for spectral and bispectral analysis used to investigate IG wave energy transfers through nonlinear triad coupling. Spectral estimates were computed following the same methodology as for the field observations (section 3). Due to the random phase summation technique, a substantial scatter in terms of IG waves can be observed from one simulation to another. For model/data comparisons, bulk parameters computed from field observations were compared against the mean of a model ensemble of 10 realizations.

Model post-processing

Incoming and outgoing IG waves along the studied cross-shore profile were separated using a Radon transform [START_REF] Radon | Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten[END_REF]. This technique projects the two-dimensional wave field (here space corresponds to the studied cross-shore profile and time to 1800 s-long samples) into polar space [START_REF] Yoo | Depth inversion in the surf zone with inclusion of wave nonlinearity using video-derived celerity[END_REF]. The Radon transform is a powerful tool to study nearshore waves as it allows the separation of incoming and outgoing wave fields on highresolution data sets without any hypotheses on the hydrodynamics [e.g. [START_REF] Almar | On the use of the Radon Transform in studying nearshore wave dynamics[END_REF][START_REF] Martins | The influence of swash-based reflection on surf zone hydrodynamics: a wave-by-wave approach[END_REF]. The energy conservation properties of the method were assessed by examining the ratio of energy computed using the sum of the separated signals and that of the original total signal. The relative difference never exceeded 4% and was 1.4% on average over the simulated period. Reflection coefficients R 2 were computed as the ratio between the outgoing and the incoming IG wave energy.

In order to analyze IG-wave generation mechanisms, the phase lag between the short wave energy envelope A and incoming IG waves η + was estimated using cross-spectral analysis:

φ f,x = arctan I{Cr A,η + (f, x)} R{Cr A,η + (f, x)} (6)
where Cr is the cross-spectrum of the short wave envelope A and the incoming IG waves η + , separated along the studied cross-shore transect using the Radon transform described above and I and R are the imaginary and the real parts of the cross-spectrum. As this method provides a phase lag per frequency, the averaged values over the IG band (0.005 -0.04 Hz)

were considered, which eliminates the problem of selecting arbitrarily a single frequency while providing more stable phase lag estimates.

Possible energy transfers in the IG band were investigated through bispectral analysis, a technique originally introduced by Hasselmann et al. [1963] and then used in numerous IG

wave studies [e.g. [START_REF] Elgar | Observations of bispectra of shoaling surface gravity waves[END_REF][START_REF] De Bakker | Nonlinear infragravity-wave interactions on a gently sloping laboratory beach[END_REF]. Source terms for nonlinear energy transfers between frequencies S nl were computed using the stochastic formulation of the second order nonlinear wave interaction theory of [START_REF] Herbers | Spectral energy balance of breaking waves within the surf zone[END_REF]:

S nl,f = 3πf h I      f f =0 ∆f B f ,f -f -2 f N -f f =0 ∆f B f ,f      . ( 7
)
where the term

f f =0
B f ,f -f represents the sum interactions in the imaginary part of the bispectrum B, and the term -2

∞ f =0
B f ,f represents the difference interactions, as a particular frequency can contribute simultaneously to both difference and sum interactions.

The 24-h analyzed model samples allow computing these terms with 602 degrees of freedom and a spectral resolution of 0.0019 Hz. Further details on these analysis techniques can be found in [START_REF] Bertin | Infragravity waves: From driving mechanisms to impacts[END_REF].

Results

Characterization of short waves and IG wave heights

As explained above, the field campaign presented in this study was motivated by the development of the storm Kurt, which generated offshore H s of the order of 10.0 m with T p over 20 s. At the ADCP1 (Figure 1-A), the spectral significant wave height in the gravity band (hereafter H m0,G ) increased from 2.0 m at the beginning of the campaign to a maximum of 6.0 m on the 03/02/2017 (Figure 2-B). Over the same period, T p increased from 10 -15 s to more than 20 s. The increase in H m0,G is well captured by XBeach, with a RMSD of 0.3 m and a NRMSD below 10%. In more detail, H m0,G is tidally modulated during the last tidal cycle, which suggests that the surf zone at low tide extended offshore of the ADCP1, located more than 3000 m from the shoreline. At the sensors deployed in the intertidal zone (PT1 to PT9), wave heights were depth-limited and therefore strongly tidally-modulated, with H m0,G roughly equal to half of the water depth (Figure 3). Wave heights are well reproduced by XBeach, with a mean RMSD of 0.12 m over the 8 intertidal sensors, which results in a NRMSD of 15% (Figure 5 individual IG waves exceeding 2.5 m (not shown). Figure 4 shows an example of time series of measured water depth at the ADCP2 (Figure 4-A), where the low-pass filtered signal reveals that individual IG waves exceeded 2.0 m and induced cross-shore velocities ranging from -2.0 m.s -1 (offshore) to +2.0 m.s -1 (onshore)(Figure 4-B). This figure also shows that alongshore velocities were much weaker (Figure 4-C), with the ratio of the alongshore over the cross-shore IG velocity variance being lower than 10%. The extension of this analysis to the whole ADCP2 dataset reveals this ratio is about 19 ± 8% (where ± corresponds to one standard deviation), which means that the IG wave dynamics was dominated by cross-shore motions. H m0,IG are well reproduced by XBeach, except during the second tidal cycle where a 0.2 m negative bias can be observed (Figure 3), a problem that is also found at the ADCP1 (Figure 2-C). Considering the whole set of intertidal sensors, H m0,IG are reproduced by XBeach with a RMSD of 0.14 m, which corresponds to a NRMSD of 20% (Figure 5-B). To be more specific, H m0,IG are particularly underestimated when the incident short waves have a large directional spreading (i.e. > 35°, Figure 5-B), which occurred mainly during the second tidal cycle (Figure 3). While discarding model results for such conditions, the RMSD drops to 0.08 m and the NRMSD to 11%, which corresponds to high predictive skills compared to previously published studies that provide a detailed validation of XBeach [e.g. [START_REF] Van Dongeren | Numerical modeling of low-frequency wave dynamics over a fringing coral reef[END_REF][START_REF] Bertin | Relevance of infragravity waves at a wave-dominated inlet[END_REF][START_REF] Lashley | Nonhydrostatic and surfbeat model predictions of extreme wave run-up in fringing reef environments[END_REF]. The underestimation of IG waves for broad short wave spectra is related to the original surfbeat approach of Xbeach, where the wave energy from different directional bins is simply added up, without considering the interference of the different wave components [START_REF] Roelvink | Improving predictions of swash dynamics in XBeach: The role of groupiness and incident-band runup[END_REF]. To overcome this problem, [START_REF] Roelvink | Improving predictions of swash dynamics in XBeach: The role of groupiness and incident-band runup[END_REF] recently proposed an alternative approach, where the mean wave directions are calculated first and the short wave energy is then propagated along these directions. However evaluating the recent improvements of XBeach is outside the scope of this study, focused on the physical processes associated with IG waves.

Frequency repartition of IG wave energy

In order to investigate the evolution of the IG wave energy frequency repartition with a sufficient spectral resolution, we analyzed 3 h-long time-series of modeled and observed water depth, centered on the high tide of the last tidal cycle (storm Kurt). Incoming and outgoing modeled IG waves were separated using a Radon transform (section 4.3) but the number of intertidal sensors was too limited to apply this technique to field measurements. Figure 6-A and 6-B show similarities between the modeled and the observed spatiofrequential patterns of IG wave energy in the intertidal zone, with a main peak between 0.01 and 0.015 Hz, a super-harmonic peak around 0.02 and 0.025 Hz and a subharmonic peak between 0.001 and 0.005 Hz. From the beach lower part to the beach upper part, these three peaks slide towards higher frequencies and the lower frequency one totally dominates the power density spectra along the shoreline. Such intriguing patterns were already observed in laboratory experiments of IG waves [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF][START_REF] Buckley | Mechanisms of wave-driven water level variability on reef-fringed coastlines[END_REF]] and short waves [START_REF] Martins | The influence of swash-based reflection on surf zone hydrodynamics: a wave-by-wave approach[END_REF] and are explained by interference between incoming and outgoing IG waves, creating nodes and antinodes, whose position with respect to the shoreline varies with the IG wave frequency and the bottom slope. Figure 6-D and 6-E corroborate this hypothesis and show that this pattern disappears in the modeled spectra corresponding to the incoming and outgoing signals. The analysis of the modeled spectra for the incoming signal (Figure 6-D) reveals that, at the shoreface, IG wave energy is mostly located around a first peak at 0.01 Hz and a second broader peak centred around 0.005 Hz. A superharmonic secondary peak is also present around 0.02 Hz. Approaching the shoreline, IG wave energy at higher frequencies decreases while most of the energy is located between 0.002 and 0.006 Hz. The spectra of the outgoing signal (Figure 6-E) suffer less transformation and, compared to the incoming signal, display much less energy at frequencies above 0.015 to 0.02 Hz. This analysis was extended to the whole studied period based on the mean incoming IG wave period T m02,IG + (Figure 7-A), computed from the incoming IG wave signal. For moderate-energy wave conditions, this figure shows that T m02,IG + first decreases from about In water depth less than 1 m, T m02,IG + increases back to 60 -70 and even over 80 s at low tide (Figure 7). Under storm wave conditions (3/02/2017), a different behaviour is observed, with a continuous increase in T m02,IG + from about 70 s in 10 m water depth to 90 -100 s in 2 m water depth. In water depth less than 1 m, T m02,IG + further increases up to 120 s at low tide. The modeled temporal evolution of T m02,IG + fairly matches the observations at the location of the ADCP2 and ADV, with a RMSD of 8 s, corresponding to a 12% error.

Reflection

The comparison between observed and modeled reflection coefficient R 2 shows a good agreement in the intertidal zone, with a RMSD of 0.12, corresponding to a normalized error of about 15% (Figure 8-B and-C). Under moderate-energy incident short waves, both modeled and observed R 2 show a strong tidal modulation, with values below 0.2 at low tide and close to 1.0 at high tide, suggesting full reflection along the waterline. This tidal modulation substantially weakens under storm wave conditions, with R 2 above 0.5 at low tide along the waterline. [START_REF] Radon | Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten[END_REF] and modeled (blue) against observed (circles) timeseries of R 2 at the ADCP2 (B) and the ADV (C).

Such a tidal modulation of R 2 was already reported by [START_REF] Okihiro | Infragravity energy modulation by tides[END_REF], who explained the higher reflection at high tide by the steeper slope of the beach upper part.

In the present study, the beach slope increases from about 1:100 in the lower part of the intertidal zone to 1:25, which could directly explain the observed tidal modulation of IG wave reflection. [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF] proposed that IG wave reflection is related to a normalized beach slope parameter β H , defined as:

β H = β ω g H , ( 8
)
where β is the beach slope, ω is the IG wave angular frequency, g is the gravitational acceleration and H corresponds to the height of the incoming IG wave. According to these authors, a large reflection would occur for β H substantially above unity. Considering incoming IG waves of height 0.5 m and frequency 0.01 Hz, β H would increase from about 0.7 at low tide to 3.0 at high tide, which explains the observed tidal modulation of R 2 .

Outside the surf zone, model results suggest R 2 well above unity at high tide, which was already observed in the field by [START_REF] Elgar | Reflection of ocean surface gravity waves from a natural beach[END_REF] and [START_REF] Sheremet | Observations of nearshore infragravity waves: Seaward and shoreward propagating components[END_REF]. These authors interpreted this behaviour by the fact that IG waves were gaining energy between their sensors and the shoreline, before being reflected from a steep beach face.

Discussion

Generation mechanisms

The IG waves observed during our field campaign correspond to the largest values reported in the literature [START_REF] Ruessink | Observations of turbulence within a natural surf zone[END_REF][START_REF] Fiedler | Observations of runup and energy flux on a low-slope beach with high-energy, long-period ocean swell[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF], while our adaptive frequency cutoff to separate gravity and IG bands is more conservative than the fixed frequency cutoff used in these studies. At gently sloping beaches, it is usually admitted that IG waves are generated through the bound wave mechanism [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF][START_REF] Bertin | Infragravity waves: From driving mechanisms to impacts[END_REF]. Considering the analytical solution proposed by [START_REF] Hasselmann | On the non-linear energy transfer in a gravity-wave spectrum part 1. General theory[END_REF] to compute the bound wave based on directional wave spectra, energy transfers from the gravity band to the IG band increase when the wave energy is large and distributed over narrow spectra.

Therefore, the very energetic and narrow-banded swell associated with the storm Kurt would directly explain the development of such large IG waves. In order to verify this hypothesis, we compared the incoming IG wave heights H m0,IG + computed from our baseline simulation with those computed from simulations where only either the bound wave or the breakpoint mechanisms were represented (see section 4.2). This comparison reveals only modest differences between the baseline simulation (Figure 9-A) and the simulation where only the bound wave is considered (Figure 9-B). Conversely, the simulation where only the breakpoint mechanism is considered (Figure 9-C) shows much smaller IG waves, with H m0,IG + ranging from 0.1 to 0.2 m during first three tidal cycles and 0.4 m around the peak of the storm. This analysis confirms that IG waves were mostly generated through the bound wave mechanism, independently from the tidal phase or the incident short-wave energy.

In accordance with this first finding, the modeled incoming IG waves are negatively correlated with the short wave energy envelope (hereafter WEE, Figure 10-A). Strongest correlations are found from the breaking point up to water depths of about 12 m, with coefficients ranging from -0.4 to -0.9 over the studied period (always significant at 95%, mean -0.54 ± 0.16). In more details, the strongest negative correlations are not found at zero lag and incoming IG waves are lagging behind the WEE by a few seconds. Figure 10-B shows the modeled spatio-temporal evolution of the normalized phase lag between incoming IG waves and the WEE, shifted by π for the sake of readibility. For moderate-energy incident conditions, this phase lag continuously increases from about π/20 by 10 m water depth to about π/5 by 4 m water depth, which corresponds to the beginning of the surf zone. In the inner surf zone, the correlation cancels out and then switches to positive (Figure 10-A), due to the modulation of short wave heights by IG waves, a process that is well documented [START_REF] Guza | Swash on steep and shallow beaches[END_REF][START_REF] Tissier | Infragravity-wave modulation of short-wave celerity in the surf zone[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF]. For storm wave conditions, a phase lag of about π/10 is already present by 10 m water depth (i.e. 3000 m from the shoreline, Figure 10) and increases up to π/5 at the beginning of the inner surf zone. According to [START_REF] Van Dongeren | Shoaling and shoreline dissipation of low-frequency waves[END_REF], this phase lag enhances energy transfers from the short waves to the IG waves. Due to the gentle slope of the studied beach, this process is active over several kilometers under storm waves, which would explain the development of such large IG waves. However, our field dataset entails several limitations that prevent from verifying these values. First, the ADCP1 was located more than 3000 m from the intertidal profile, which implies travel times of IG waves ranging from 400 to 500 s, depending on the tidal phase. Over such a long distance/duration, IG waves suffer multiple transformations (see next sections) so that we were unable to correlate the offshore WEE with the IG waves at the intertidal sensors. Second, the width of the surf zone ranged from 200 m to more than 3000 m, so that the intertidal sensors were always located inside the surf zone, where the correlation between IG waves and the WEE is biased positively as explained above.

Energy transfers

In order to better explain the spatio-temporal repartition of IG wave energy, source terms for nonlinear energy transfers between IG wave frequencies S nl were computed using bispectra following [START_REF] Herbers | Spectral energy balance of breaking waves within the surf zone[END_REF] based on 24 h-long runs corresponding to moderateenergy and storm conditions, both at low and high tide (Figure 11). This figure reveals that, at high tide, substantial energy transfers from frequencies in the range 0.01-0.03 Hz towards higher frequencies start to occur about 200 m from the shoreline (Figure 11-A). At low tide, similar energy transfers occur but starting about 500 m from the waterline (Figure 11-B). Surprisingly, the overall patterns of energy transfers are not very different for storm conditions, except that they affect lower IG wave frequencies (0.002 Hz to 0.02 Hz) (Figure 11-C and -D). Note that for these high-energy cases, the overall patterns are shifted onshore because the mean water level is higher due to a larger wave setup. For moderate-energy wave conditions, the cross-shore evolution of Snl (Figure 11-A and -B) suggests that the decrease in mean incoming IG wave period T m02,IG + with the water depth (Figure 7) and its subsequent tidal modulation is explained by energy transfers from principal components to higher harmonics via non-linear coupling between triads. In shallow depth (i.e. less than 1 m), the increase in T m02,IG + is explained by the breaking of the higher-frequency IG waves, a process well supported at gently-sloping beaches by field observations [START_REF] De Bakker | Shoreline dissipation of infragravity waves[END_REF][START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF], laboratory experiments [START_REF] Battjes | Shoaling of subharmonic gravity waves[END_REF][START_REF] Van Dongeren | Shoaling and shoreline dissipation of low-frequency waves[END_REF][START_REF] Padilla | Transfer and dissipation of energy during wave group propagation on a gentle beach slope[END_REF] and numerical modelling [START_REF] Ruju | Radiation stress and low-frequency energy balance within the surf zone: A numerical approach[END_REF]De Bakker et al., 2016b;[START_REF] Mendes | Infragravity wave energy changes on a dissipative barred beach: A numerical study[END_REF]. For storm wave conditions, a different situation occurs, where T m02,IG + continuously increases across the surf zone, so that T m02,IG + almost doubles from 10 m water depth to the shoreline (Figure 7). While the increase in T m02,IG + close to the shoreline can also be attributed to the breaking of the highest frequency IG waves, an additional mechanism should be active in intermediate water depths where IG waves cannot break. Indeed, classical interactions between pairs of frequencies could hardly explain this behaviour as the computation of S nl reveals virtually no energy transfers towards very low frequencies, as for moderate-energy conditions (Figure 11). Analyzing field observations collected under storm waves at a steep rocky shore in western Brittany, [START_REF] Sheremet | Observations of large infragravity runup at Banneg Island, France[END_REF] reported the development of 300 s period IG waves along the shore. These authors applied a phase-resolving triad numerical model, which failed to reproduce such low frequency IG waves and concluded that, as in the present study, it does not result from classical non-linear triad interactions. While the development of IG wave super-harmonics at gently-sloping beaches is reasonably well documented in the literature and explained through IG-IG nonlinear interactions [START_REF] Elgar | Observations of bispectra of shoaling surface gravity waves[END_REF][START_REF] De Bakker | Nonlinear infragravity-wave interactions on a gently sloping laboratory beach[END_REF], 2016b;[START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF][START_REF] Padilla | Transfer and dissipation of energy during wave group propagation on a gentle beach slope[END_REF], the transfer of energy towards subharmonic frequencies received much less attention [De Bakker et al., 2016b]. To further look into this phenomenon, we analyzed timestack samples of modeled incoming free surface elevation, close to high tide under moderate energy (first tidal cycle) and storm waves (fourth tidal cycle, storm Kurt) (Figure 12-A and-B). On this figure, the troughs of the WEE were automatically detected at x = 3000 m and their location was then computed across the surf zone using the wave group velocity C g , given by the linear wave theory and the representative frequency used in XBeach, which corresponds to 1/T m-1,0 . For moderate energy, Figure 12-A shows that IG wave crests 1 to 12 follow the WEE troughs, which means that they propagate at C g and shows that they are bound to the wave groups. This behaviour illustrates the dominance of the bound wave mechanism in IG wave generation, as explained in section 6.1. This situation strongly contrasts with storm wave conditions, where IG wave crests depart from the WEE troughs at different locations across the surf zone, meaning that they no longer travel at C g but behave as free waves (Figure 12-B). For instance, focusing on the IG wave crests number 1 to 4, crest 1 seems bound to the wave group until x = 500 m, crest 2 and 4 until x = 2300 m and crest 3 until x = 1000 m. Although less impressive, this process can also be observed for IG wave crests 6 and 10. Considering a mean water depth of 12 m at x = 3000 m and a representative frequency of 0.064 Hz at the time of Figure 12-B, bound IG waves propagate at C g = 9.8 m.s -1 while free IG waves propagate at 10.84 m.s -1 . Whereas such differences only represent 11 % and further less when approaching the coastline, they result in traveltime differences ranging from 25 to 30 s once integrated over such a large surf zone. This allows the merging of individual crests into longer-period IG waves close to shore, as for instance crest 2 with crest 1 and crests 4 with crest 3 (Figure 12-B). A closer look at the model results however suggests that the behaviour of the near-shore IG wave-field is more complex. For instance, new IG wave crests appear inside the surf zone, such as between crests number 2 and 3 at x = 1500 m or between crests 11 and 12 at x = 2300 m. As these IG wave crests coincide with the crests of the WEE, they may correspond to free IG waves generated through the breakpoint mechanism [START_REF] Symonds | Two-dimensional surf beat: Long wave generation by a time-varying breakpoint[END_REF]. This hypothesis is supported by Figure 9-C, which shows that the breakpoint mechanism is active 2000 m from the shoreline under storm waves. Future studies based on extensive field measurements and/or phase resolving models will have to investigate the detailed processes responsible for IG wave merging and the possible contribution of breakpoint-generated free IG waves.

Small beach slopes and resulting wide surf zones exacerbate this process as it provides individual IG waves with more time to travel at slightly different speed and catch up with one another. Although it could not be evidenced here, a modulation of individual IG wave celerity by longer IG waves might also be possible in this region of the shoreface [START_REF] Tissier | Infragravity-wave modulation of short-wave celerity in the surf zone[END_REF], which would also promote the merging of individual waves. Ultimately, we propose that the interactions of incident bound and free IG waves along the gently sloping shoreface of Oléron Island explain the increase of the incoming IG wave period by a factor of two across the surf zone under storm waves (Figure 7). This mechanism also suggests that the frequency of IG waves close to shore is not only controlled by the shape of the incident short wave spectra [i.e. [START_REF] Hasselmann | On the non-linear energy transfer in a gravity-wave spectrum part 1. General theory[END_REF] but also by the width of the surf zone and hence the slope of the beach. This mechanism will have to be verified in the field, although instrumenting a surf zone several kilometre-wide is very challenging.

Implications for other studies

In this study, we observed reflexion coefficient R 2 to be very high for a dissipative beach. For instance, De Bakker et al. [2014] and [START_REF] Inch | Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions[END_REF] reported values for R 2 of the order of 0.5 or below for beaches of comparable mean bottom slope. The presence of a steeper upper beach in Saint Trojan can explain the higher reflexion observed at high tide, but not at low tide under storm wave conditions, where R 2 hardly drops below 0.5. We propose that the high reflection that we observe under storm waves is related to energy transfers towards subharmonic frequencies (section 6.2), so that the resulting 200 to 300 s period IG waves are too long to suffer a substantial dissipation, even when propagating over a mild slope. Albeit not addressing the underlying mechanisms, a few other studies reported the development of IG waves with frequencies around 0.005 Hz under storm waves and suggested that they had a key role in runup [START_REF] Sheremet | Observations of large infragravity runup at Banneg Island, France[END_REF][START_REF] Ruggiero | Wave run-up on a high-energy dissipative beach[END_REF]. Indeed, such low frequency IG waves suffer less dissipation when propagating in shallow depth and may therefore propagate farther inland and induce larger damages in low-lying coastal zones compared to higher-frequency IG waves. The generation of low-frequency IG waves through merging of bound and free waves across wide surf zones appears therefore as a process of key importance and will have to be investigated at other sites, such as the very gently sloping coastlines of Bangladesh, which are extremely vulnerable to flooding under tropical hurricanes [START_REF] Krien | Towards improved storm surge models in the northern bay of bengal[END_REF]. This process should also be investigated at large estuaries and tidal inlets, where the ebb deltas can drive the development of kilometer-wide surf zones.

Besides, it is nowadays well admitted that free IG waves in the ocean are the main source of background-free oscillations of the solid earth, also referred to as "the hum of the Earth" [e.g. [START_REF] Rhie | A study of the relation between ocean storms and the Earth-s hum[END_REF][START_REF] Webb | The earth's 'hum' is driven by ocean waves over the continental shelves[END_REF]. As better understanding the generation of the hum could help refining the use of microseism background signals to investigate the structure of the solid Earth, understanding IG wave reflection along the coast is of key importance. Correlating deep ocean pressure measurements with short-wave bulk parameters over the North Atlantic Ocean, [START_REF] Crawford | The sources of deep ocean infragravity waves observed in the North Atlantic Ocean[END_REF] proposed that the main source for free IG wave generation was located between the Western part of the Iberic Peninsula and NW Africa, namely because they display steep beaches. Our study suggests that strong reflection can also occur along dissipative beaches under storm waves, which will have to be considered in future studies on deep ocean IG waves.

Conclusions

This study aimed at improving our understanding of the generation and transformation of IG waves, combining an unpublished data set with numerical modeling. The development of exceptionally large IG waves was primarily explained by very energetic and narrow-banded incident short waves associated with the storm Kurt. The analysis of model results revealed that the bound wave mechanism was clearly dominant over the breakpoint mechanisms in the generation of IG waves. Model results also showed that the gently sloping shoreface of the study area leads to the presence of a phase lag between IG waves and WEE over several kilometers under storm waves, which certainly enhanced IG wave generation through a dynamic shoaling, although our dataset did not allow to further analyze this process. The XBeach modelling system was found capable of reproducing the generation and transformation of such IG waves, which is an important result as XBeach is becoming more and more popular while detailed hydrodynamic validations are lacking under storm conditions. Model results and field observations revealed that large energy transfers occur in the IG band across the surf zone, with the development of superharmonics but also the transfers of energy towards subharmonic frequencies. While the former behaviour is reasonably well documented in the literature and explained by non-linear coupling between triads, the latter was reported in a few studies only and we propose that it is (at least partly) related to the merging of IG waves across the surf zone. According to our modelling results, IG wave merging seems to be related to the combination of free and bound IG waves in the surf zone, traveling at different celerities over a long distance. This mechanism will have to be verified on the field and is potentially of key importance under extreme storms, because low-frequency IG waves suffer little dissipation and may therefore propagate farther inland and induce larger damages compared to higher frequency IG waves. This mechanism also results in IG-wave reflection surprisingly high for a dissipative beach, which has implications for future studies on deep ocean IG waves.

Figure 1 .

 1 Figure 1. (A) Location of the study area in the Bay of Biscay, with the Biscay buoy () and Chassiron Meteorological station (). (B) Detailed bathymetry of the study area (m relative to mean sea-level), showing the location of the offshore ADCP1 () and the instrumented cross-shore profile (dashed line). The coordinates are in meter (Lambert-93 projection) (C) Zoom-in on the intertidal part of the instrumented cross-shore profile, showing the location of the instruments used in this study (+) with respect to the beach topography (blue line).

  . Based on directional wave spectra provided along the open boundary, XBeach applies random phase summation to reconstruct time series of the free surface elevation. A Hilbert transform is applied to derive time-series of short-wave energy varying at the scale of wave groups and imposed as boundary conditions in the wave action model. IG waves are represented explicitly in the Saint-Venant model and are forced by the gradient of wave radiation stress computed from the wave model. The incoming bound-wave is computed following Herbers et al. [1994] and is imposed along the open boundary of the Saint-Venant model.

  -B). The spatial resolution ranges from 20 m along the open boundary to 1 m in the beach upper part (Figure 1-B). A sensitivity analysis revealed that such a fine resolution was required to adequately represent IG waves in shallow water and their reflection along the shoreline. Along the offshore open boundary, XBeach was forced

Figure 2 .Figure 3 .

 23 Figure 2. Time series of observed (circles) and modeled (solid lines) bulk parameters and spectra at the location of the ADCP1 (Figure 1-B): (A) water depth; (B) Hm0 in the gravity band; (C) Hm0 in the IG band; (D) continuous peak (red) and mean (blue) wave periods and (E) power density spectra of observed water levels computed, where the white dashed line corresponds to the adaptive frequency cutoff between gravity and IG bands as described in section 3.

Figure 4 .

 4 Figure 4. Time series of water depth (A), near bottom cross-shore (B) and longshore velocity (C) at 2 Hz (blue) and low-pass filtered (black), measured at the ADCP2/PT3 (Figure 1-C) on the 03/02/2017 at 5h00.

Figure 5 .

 5 Figure 5. Modelled against observed Hm0 in the gravity (A) and in the IG (B) bands, where symbol color corresponds to the directional spreading of the incident short waves at the open boundary of the modeled domain (Figure 1-C).

FrequencyFigure 6 .Figure 7 .

 67 Figure 6. Spatio-frequential repartition of IG wave power spectral density modeled (A) and observed (B) for the total signals in the intertidal zone and modeled along the studied cross-shore transect for the total (C), the incoming (D) and the outgoing (E) signals.

  s in 10 m water depth to about 50 s in 2 m water depth. For a given location along the studied cross-shore profile, T m02,IG + exhibits therefore a substantial tidal modulation.

Figure 8 .

 8 Figure8. (A) Spatio-temporal variations of reflection coefficients R 2 computed over the studied period using a Radon transform[START_REF] Radon | Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten[END_REF] and modeled (blue) against observed (circles) time-

Figure 9 .

 9 Figure 9. Modeled incoming IG wave height H m0,IG + in the baseline simulation (A), the bound wave case (B) and the breakpoint case (C). The black dotted line corresponds to the offshore limit of the surf zone, as defined in section 4.2

Figure 10 .

 10 Figure 10. (A) Modeled correlation coefficient between the WEE and incoming IG waves and (B) modeled normalized phase lag, shifted by π (i.e. phase lag in radians shifted by π and then normalized by π) between incoming IG waves and the WEE. The black solid lines correspond to isolines of the mean water depth.

FrequencyFigure 11 .

 11 Figure11. Source terms for nonlinear energy transfers between IG wave frequencies along the studied cross-shore profile computed for each frequency following[START_REF] Herbers | Spectral energy balance of breaking waves within the surf zone[END_REF] over 24 h stationary runs for moderate and high-energy consitions, both at low and high tide.

Figure 12 .

 12 Figure 12. Timestack sample of modeled incident surface elevation (η IG + ) along the studied cross-shore transect around high tide for (A) moderate incident energy and (B) high-energy (storm Kurt). Numbers correspond to IG wave crests and the dashed black lines correspond to the troughs of the WEE, automatically detected at x = 3000 m and then computed across the surf zone.

Acknowledgments

This research was carried out in the scope of the Regional Chair Program EVEX, funded by the Region Poitou-Charentes (France). KM greatly acknowledges the financial support from the University of Bordeaux, through an International Postdoctoral Grant (Idex, nb. 1024R-5030). AdB was partly funded by the program Prestige of Campus France (grant agreement n • PCOFUND-GA-2013-609102). We warmly thank the developping team of XBeach for making the code available. The processed field data and model input files required to run the XBeach simulations presented in this paper are available through a Zenodo repository (DOI:10.5281/zenodo.3753018).


Shenyu Liu

Aneel Tanwani

Daniel Liberzon

Average Dwell-Time Bounds for ISS and Integral ISS of Switched Systems using Lyapunov Functions

The problem of input-to-state stability (ISS), and its integral version (iISS), is considered for switched systems with inputs and resets. The individual subsystems are assumed to be ISS (resp. iISS) with nonlinear decay rates in dissipation inequalities associated with the Lyapunov function of each subsystem. The change in the value of Lyapunov functions at switching instants is described by a nonlinear growth function. A generalized lower bound is computed for average dwell-time (ADT) to guarantee ISS/iISS of the switched system. In particular, an explicit formula of ADT lower bound is given for switched bilinear systems with zero-input-stable subsystems.

I. Introduction

Among the tools that have been developed for stability analysis of switched systems, the notion of dwell-time or average dwell-time (ADT) has played a significant role in characterization of stability conditions [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF], [START_REF] Liberzon | Switching in Systems and Control[END_REF]. In autonomous switched systems, a common recipe for computing such bounds is to associate a Lyapunov function with individual subsystems. Assuming exponential decay in the value of Lyapunov function during flows, and exponential growth at switching instants, lower bounds on ADT can be computed as a function of these decay and growth rates which guarantee asymptotic stability for that particular class of slowly switching signals.

For dynamical systems with inputs, the notion of input-to-state stability (ISS), or its close variant integral ISS (iISS), provides an elegant method to quantify the performance in control related applications. In the pioneering works on ISS [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] and iISS [START_REF] Sontag | Comments on integral variants of ISS[END_REF], these stability notions have also been characterized using Lyapunov functions. For switched systems, converse results regarding the existence of ISS and iISS systems first appeared in [START_REF] Mancilla-Aguilar | On converse Lyapunov theorems for ISS and iISS switched nonlinear systems[END_REF], and some implications relating ISS and iISS for switched systems have been developed in [START_REF] Haimovich | ISS implies iISS even for switched and time-varying systems (if you are careful enough)[END_REF].

Combining the aforementioned two directions of research, it is rather natural to ask whether for slowly switching systems, these Lyapunov characterizations can be combined with ADT notions to obtain sufficient conditions for ISS/iISS of switched systems. This question has indeed been addressed in the literature in different settings, and it is also the central topic of this paper. Early work dealing with ADT bounds for switched systems using Lyapunov functions for individual systems appears in [START_REF] Vu | Input-to-state stability of switched systems and switching adaptive control[END_REF]. Just like for autonomous systems, a major restriction of this work is that both the dissipation rates in the inequalities associated with the derivative of Lyapunov functions and the changes in the value of the Lyapunov functions at switching instants are assumed to be linear. While the dissipation rates are relaxed to be nonlinear in the study of ISS/iISS of impulsive systems, which are similar to switched systems, growth in the value of Lyapunov functions at impulses are not allowed in [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF]. Switching dynamics with reset maps can also be captured by the framework of hybrid systems [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] and ISS characterizations via Lyapunov functions for hybrid systems have been developed in [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF], [START_REF]Robust input-to-state stability for hybrid systems[END_REF]. Relaxing certain assumptions in these works, along with some developments based on converse results, iISS characterizations via Lyapunov functions for hybrid systems appear in [START_REF] Noroozi | A characterization of integral input-to-state stability for hybrid systems[END_REF].

Continuing with the research on computing ADT lower bounds which render the switched system ISS or iISS, the most relevant results for this paper have recently appeared in [START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF]. Compared to the earlier works, [START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF] considers a family of subsystems where the corresponding Lyapunov functions do not necessarily admit an exponential decay during flows. Similarly, a nonlinear estimate is assumed for growth in the value of Lyapunov functions of the individual subsystems at switching times. In this article, the formula of ADT reported in [START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF] for ISS switched systems is extended to iISS switched systems under some mild assumptions and with some appropriate modifications.

We also examine an application of our results in the context of bilinear switched systems, where the flow dynamics are bilinear with respect to the disturbances. A logarithmic Lyapunov function for (nonswitching) bilinear systems has been proposed in [START_REF] Chaillet | Combining iISS and ISS with respect to small inputs: The strong iISS property[END_REF]. The same Lyapunov function can be used to show that individual systems are iISS if the unforced systems are asymptotically stable. The growth in the value of Lyapunov function at switching instants is also described by a sublinear function and hence the inequalities do not satisfy the assumptions of exponential growth/decay which are often found in the literature for computing ADT bounds. However, using the results derived in this paper, we obtain lower bounds on ADT with logarithmic Lyapunov functions which result in the bilinear switched system being iISS.

II. Preliminaries

In this section, we introduce the basic notation required for introducing the class of switched systems studied in this paper, and the definitions of ISS, iISS of switched systems under slow switching.

A function α : R 0 -→ R 0 is said to be positive definite if it is continuous, α(0) = 0 and α(s) > 0 for all s > 0. If α is also strictly increasing, then it is said to be of class K. In addition if α is also unbounded, then it is said to be of class β(s,•) is decreasing and β(s, t) -→ 0 as t -→ ∞ for each fixed s ∈ R 0 ; see [START_REF] Khalil | Nonlinear Systems[END_REF]Chapter 4] for their use in formulation of common stability notions. In addition, we require class KLL functions: A function

K ∞ . A function β : R 0 × R 0 -→ R 0 is said to be of class KL if β(•, t) is of class K for each t ∈ R 0 ,
β : R 0 × R 0 × R 0 -→ R 0 is a class KLL function if β(•, •, j) is a class KL function for each j 0 and β(•, t, •) is a class KL function for each t 0.
Let P ⊂ N be a set of either finite or infinite cardinality. For each p ∈ P, there is a locally Lipschitz vector field f p : R n × R m -→ R n . Moreover, these vector fields have the common equilibrium property, that is, f p (0, 0) = 0, for each p ∈ P. The differential equations ẋ = f p (x, ω), p ∈ P

are the dynamics of the subsystems or modes of the switched system. For each pair (p, q) ∈ P × P, there is also a jump map g p,q : R n × R m -→ R n with the common equilibrium property that g p,q (0, 0) = 0. Let Σ be the set of all right-continuous mappings from R 0 to P with a locally finite number of discontinuities, called switching signals. For each switching signal σ ∈ Σ, define

T (σ) := {t > 0 : σ(t) = σ(t -)}; in other words, T (σ)
is the collection of times when switches occur. With this data, we consider switched dynamical systems with inputs, described by,

ẋ(t) = f σ(t) (x(t -), ω(t)) if t ∈ T (σ), (2a)
x(t) = g σ(t),σ(t -) (x(t -), ω(t)) if t ∈ T (σ), (2b)
where ω : R 0 -→ R m is locally essentially bounded on R 0 and bounded on T (σ). We denote the solution of (2) with initial state x 0 , input ω and switching signal σ by x(•; x 0 , ω, σ). When x 0 , ω, σ are clear from the context, the solution is also abbreviated by x(•). Note by definition of (2), x(•; x 0 , ω, σ) obeys some differential equation when there is no switch, and it jumps when there is a switch.

For each σ ∈ Σ, N σ (s, t) denotes the number of switches of σ over the time interval [s, t). A switching signal σ admits an average dwell time (ADT) of τ a > 0 if σ ∈ Σ τa , where

Σ τa := σ ∈ Σ : ∃N 0 ∈ N s.t N σ (s, t) N 0 + t -s τ a ∀t > s 0 . (3) Definition 1. Switched system (2) is uniformly input- to-state stable (ISS) over Σ τa if there exist β ∈ KL, γ ∈ K such that |x(t; x 0 , ω, σ)| β(|x 0 |, t) + γ(ess sup s∈[0,t] |ω(s)|) (4) for all t 0, x 0 ∈ R n and σ ∈ Σ τa . Definition 2. Switched system (2) is uniformly integral input-to-state stable (iISS) over Σ τa if there exist α 0 ∈ K ∞ , β ∈ KL, γ ∈ K such that α 0 (|x(t; x 0 , ω, σ)|) β(|x 0 |, t) + t 0 γ(|ω(s)|)ds (5)
for all t 0, x 0 ∈ R n , and σ ∈ Σ τa .

Remark II.1. In [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF], iISS for impulsive systems is defined via the following inequality

α 0 (|x(t)|) β(|x 0 |, t) + t 0 γ(|ω(s)|)ds + t k ∈T (σ) γ(|ω(t k )|) (6)
where the effects of impulses are captured by the last summation term. The estimate in ( 5) is indeed a stronger version of the more general iISS definition in [START_REF] Haimovich | ISS implies iISS even for switched and time-varying systems (if you are careful enough)[END_REF], and in this article we only develop conditions which lead to iISS in the sense of (5). Our proposed conditions for establishing iISS, as in [START_REF] Mancilla-Aguilar | On converse Lyapunov theorems for ISS and iISS switched nonlinear systems[END_REF], are therefore stronger than one would expect in general for the estimate [START_REF] Haimovich | ISS implies iISS even for switched and time-varying systems (if you are careful enough)[END_REF]. This can indeed be seen from the upcoming assumption (L3) that the jump dynamics is independent of input disturbance.

III. Assumptions and Main Result

We now state some assumptions on the data of system (2) required for the statement of the main result. Assumption 1. There exist C 1 Lyapunov functions V p : R n -→ R 0 , satisfying the conditions:

(L1) There exist α, α ∈ K ∞ such that α(|x|) V p (x) α(|x|), ∀x ∈ R n , p ∈ P. (7) (L2) There exist a positive definite function α and γ ∈ K such that ∂ ∂x V p (x), f p (x, ω) -α(V p (x)) + γ(|ω|). (8) for all x ∈ R n , ω ∈ R m .
(L3) There exists χ ∈ K ∞ , such that for each (p, q) ∈ P × P,

V q (g p,q (x, ω)) χ(V p (x)), ∀x ∈ R n , ω ∈ R m . ( 9
)
Remark III.1. For each subsystem in (1), (L1) and (L2) provide a necessary and sufficient condition for the subsystem to be iISS [START_REF] Angeli | A characterization of integral input-to-state stability[END_REF]. Uniform bounds α, α in (L1) exist when P has a finite cardinality or when V p are uniformly bounded with respect to p ∈ P. In addition, if it is assumed that α in (L2) is K ∞ , then each subsystem described by ( 1) is ISS.

Based on Assumption 1, we introduce the function ψ : R 0 -→ R 0 as ψ(r) := min{α(r), cr} [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF] where c > 0 is some constant. The next assumption that we need relates to the regularity of the function ψ.

Assumption 2. The function ψ is globally one-sided Lipschitz with constant c 0 0, in the sense that, for any

w v 0, ψ(w) -ψ(v) c 0 (w -v).
The last assumption is adopted from [START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF]. It provides a lower bound on ADT such that the switched system is ISS/iISS. Assumption 3. The supremum

ζ * := sup s>0 χ(s) s 1 ψ(r) dr ( 11
)
is finite.

It is noticed that for linear systems and bilinear systems, proper choices of the Lyapunov functions do give finite value of ζ * . However, it is also noticed that the nonlinear switched system in [17, Section 3] does not have a finite ζ * . Indeed, that system is not iISS no matter how slowly it switches; it is only quasi-iISS under slow switching in the sense defined in [START_REF] Russo | Quasi-integral input-to-state stability for switched nonlinear systems[END_REF].

Now we are ready to present our main result:

Theorem III.2. Consider the switched system (2) and suppose that Assumptions 1-3 hold. Then the system (2) is uniformly iISS over Σ τa for any τ a > ζ * . In addition, if in (L2) α ∈ K ∞ , then the system (2) is uniformly ISS over Σ τa for any τ a > ζ * .

IV. Integral ISS of Bilinear Systems

In this section, we demonstrate how Theorem III.2 can be used to find a lower bound on the ADT for the switched system with bilinear subsystems

ẋ = f p (x, ω) = A p x + m j=1 B p,j xω j + C p ω, x + = x (12)
to be iISS. For this system [START_REF] Noroozi | A characterization of integral input-to-state stability for hybrid systems[END_REF] we assume

p ∈ P = {1, 2, • • • P }, x ∈ R n , ω = ω 1 • • • ω m ∈ R m and A p ∈ R n×n
are all Hurwitz, B p,j ∈ R n×n and C p ∈ R n×m . It is known from [START_REF] Sontag | Comments on integral variants of ISS[END_REF] that each bilinear subsystem, with A p being Hurwitz, is iISS. In particular, this is enough to guarantee that Assumption 1 holds.

A. Assumption 1 holds if each A p is Hurwitz

Indeed, for each p ∈ P, we can consider the iISS-Lyapunov function V p (x) as (see, e.g., [START_REF] Chaillet | Combining iISS and ISS with respect to small inputs: The strong iISS property[END_REF])

V p (x) := ln(1 + x M p x) ( 13
)
where M p is a symmetric positive definite matrix that satisfies

M p A p + A p M p + Q p = 0 ( 14
)
for some symmetric positive definite matrices

Q p ∈ R n×n . Item (L1) in Assumption 1 is satisfied with α(s) = ln(1 + as 2 ), α(s) = ln(1 + as 2 )
where

a = max p∈P |λ(M p )|, a = min p∈P |λ(M p )| ( 15
)
and λ(•) and λ(•) denote the maximum and minimum eigenvalues respectively. Next, to show that (L2) hold, it can be algebraically verified that √ a s, we conclude [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF]. In addition, by definition we see that α is positive definite and γ ∈ K so that item (L2) in Assumption 1 is verified: each subsystem of ( 12) is iISS.

d dx V p (x), f p (x, ω) - b|x| 2 1 + x M p x + c 1 |x| 2 + c 2 |x| 1 + x M p x |ω| - b(e Vp(x) -1) ae Vp(x) + c 1 a + c 2 2
Finally, we check (L3) and compute the map χ in (9). To do so, let us introduce a constant µ 1 such that

x M q x µ x M p x ( 17
)
for every p, q ∈ P and x ∈ R n . Consequently, we get

V q (x) = ln(1 + x M q x) ln(1 + µx M p x) = ln(1 + µ(e Vp(x) -1)).
We conclude that (9) holds with

χ(s) = ln µ e s -1 + 1 . ( 18
)

B. Assumption 2, Assumption 3 and ADT bound

Next, we construct the map ψ as defined in [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF], and compute ζ * in [START_REF]Robust input-to-state stability for hybrid systems[END_REF] From Theorem III.2, it holds that the switched bilinear system ( 12) is iISS for each σ ∈ Σ τa , when

τ a > a b ln µ. ( 19
)
An immediate observation resulting from the foregoing expression is that, when there exists a common Lyapunov function, all M p 's are the same and we can choose µ = 1 which results in the lower bound on ADT in [START_REF] Liu | Average dwell-time bounds for ISS and integral ISS of switched systems using Lyapunov functions[END_REF] being 0. This is indeed the case because the switched system is iISS with arbitrary switching [START_REF] Mancilla-Aguilar | On converse Lyapunov theorems for ISS and iISS switched nonlinear systems[END_REF].

V. Technical Tools and Proofs

As mentioned in the introduction, our results of ISS/iISS switched system rely on rewriting the switched system into a hybrid system in the framework of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] and then showing ISS/iISS of that hybrid system. In this section we provide a brief description of hybrid systems, followed with some supporting lemmas used in the proof of Theorem III.2.

A. Modeling as hybrid systems

Consider the hybrid dynamical system with inputs, which is described as:

ξ ∈ F(ξ, d) (ξ, d) ∈ C, ξ + ∈ G(ξ, d) (ξ, d) ∈ D, (20)
where the state trajectory ξ evolves in X , and the disturbance d takes values in R m . The sets C, D are subsets of X ×R m and are called flow, and jump sets, respectively. The evolution of the state is thus described by F (during flows) and by G (at jump instants), which are set-valued mappings from X × R m to X . It is assumed that the system data (F, G, C, D) satisfies certain basic assumptions [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], [START_REF]Robust input-to-state stability for hybrid systems[END_REF] such that, for each input d : dom d -→ R m , the system (20) admits a local solution, called hybrid arc, ξ : dom ξ -→ X , where dom ξ ⊂ R × N. For an initial state ξ 0 and an input d, a hybrid arc of ( 20) is denoted by ξ(

:= max{i ∈ N | (s, i) ∈ dom d}.
We can also characterize ISS and iISS using Lyapunov functions, as stated in the following two lemmas [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF], [START_REF] Noroozi | A characterization of integral input-to-state stability for hybrid systems[END_REF]:

Lemma V.1. A hybrid system (20) is ISS with respect to A if there exists a smooth function V : X -→ R 0 and functions α 1 , α 2 , α c , α d ∈ K ∞ and γ c ∈ K such that α 1 (|ξ| A ) V (ξ) α 2 (|ξ| A ) (23) for all ξ ∈ X , ∂ ∂ξ V (ξ), f -α c (|ξ| A ) + γ c (|d|) ( 24
)
for all (ξ, d) ∈ C, f ∈ F(ξ, d), and

V (g) -V (ξ) -α d (|ξ| A ) ( 25
)
for all (ξ, d) ∈ D, g ∈ G(ξ, d).

Lemma V.2. A hybrid system is iISS with respect to

A if there exists a smooth function V : X -→ R 0 and functions α 1 , α 2 ∈ K ∞ , positive definite functions α c , α d and γ c ∈ K such that (23) holds for all ξ ∈ X , (24) holds for all (ξ, d) ∈ C, f ∈ F(ξ, d) and (25) holds for all

(ξ, d) ∈ D, g ∈ G(ξ, d).
We call a V satisfying the inequalities in Lemma V.1 (resp. Lemma V.2) a hybrid ISS (resp. iISS) Lyapunov function. Remark V.3. Note that Lemma V.2 differs from Lemma V.1 in the sense that α c , α d are only assumed to be positive definite instead of class K ∞ , which also implies that iISS is strictly weaker than ISS.

B. ADT switching and hybrid systems

One can write the switched system (2), along with the ADT constraint on the switching signal, more naturally in the form of [START_REF] Arcak | A unifying integral ISS framework for stability of nonlinear cascades[END_REF]. To do so, we let

ξ :=   x p τ   ∈ R n × P × [0, N 0 ] =: X . ( 26
)
The hybrid model capturing the dynamics of the switched system, driven by an R m -valued disturbance d, and a switching signal σ ∈ Σ τa , is compactly written as ξ ∈ F(ξ, d) :

   ẋ = f p (x, d) ṗ = 0 τ ∈ [0, 1 τa ] if (ξ, d) ∈ C (27a) ξ + ∈ G(ξ, d) :    x + ∈ G p (x, d) p + ∈ P τ + = τ -1 if (ξ, d) ∈ D (27b)
where

C = X × R m , D = R n × P × [1, N 0 ] × R m , and G p (x, d) = q∈P g p,q (x, d).
The attractor of the system (27) is the compact set

A := {0} n × P × [0, N 0 ]. (28)
A similar modeling of a switched system with ADT constraint in hybrid system framework can also be found in [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF]. Comparing ( 21) and ( 22) with ( 4) and ( 5) respectively, we have the following straightforward result:

Corollary V.4. The switched system (2) with its switching signal admitting ADT of τ a is ISS (resp. iISS) if the hybrid system (27), with augmented state variable defined in (26), is ISS (resp. iISS) with respect to A.

C. Supporting Lemmas

Define the function ϕ : R 0 -→ R 0 by

ϕ(s) := exp s 1 2c ψ(r) dr s > 0, 0 s = 0. (29)
where we recall that c and ψ are introduced in (10). We state two properties associated with ψ and ϕ. Due to space constraints, the proofs are omitted and can be found in the online technical report [START_REF] Liu | Average dwell-time bounds for ISS and integral ISS of switched systems using Lyapunov functions[END_REF].

Lemma V.5. It holds that ϕ ∈ K ∞ .
Proof. It suffices to show that ϕ is continuous at 0 and lim s-→∞ ϕ(s) = ∞. Note that 2c ψ(r) 2 r and hence when s > 1,

Define

ψ v : [v, ∞) -→ R + : ψ v (r) = ψ(v) + c 0 (r -v).
Recall that c 0 is the global one-sided Lipschitz constant of ψ, so by Assumption 2, we have ψ v (r) ψ(r) for all r v. Hence, continuing from (30), we have ln(c 1 )

w v 2c ψ(r) dr w v 2c ψ v (r) dr = ψv(w) ψv(v) 2c s d ψ -1 v (s) = 2c c 0 ψv(w) ψv(v) 1 s ds = 2c c 0 ln ψ v (w) ψ v (v) 2c c 0 ln ψ(w) ψ(v) , which results in ψ(w) ψ(v) c c 0 2c 1 .
Theorem III.2 is a direct consequence of Lemma V.1, Lemma V.2, Corollary V.4, and the following Lemma, which we prove in the remainder of this section. Lemma V.7. Consider the hybrid system (27). Suppose Assumptions 1-3 hold and

τ a > ζ * . For any ζ > 0 such that τ a > ζ > ζ * , V (ξ) := ϕ -1 (e 2cζτ ϕ(V p (x))) ( 31
)
is a hybrid iISS Lyapunov function with respect to the compact set A. In addition if we further assume that in (L2) α ∈ K ∞ , then (31) is an ISS Lyapunov function with respect to A.

Proof. Before we start the proof, we note that ϕ -1 in (31) is well-defined because ϕ ∈ K ∞ , as stated in Lemma V.5.

To check the "sandwich" condition (23), we simply define α 1 (s) := ϕ -1 (ϕ(α(s))), α 2 (s) := ϕ -1 (e 2cζN0 ϕ(α(s))).

We now want to show (24) and we only consider the non-trivial case when x = 0. Define

W (ξ) := e 2cζτ ϕ(V p (x)). ( 32
) Then V (ξ) = ϕ -1 (W (ξ)). Pick (ξ, d) ∈ C and f ∈ F(ξ, d), we have ∂ ∂ξ W (ξ), f = 2cζe 2cζτ ϕ(V p (x)) τ + e 2cζτ ϕ (V p (x)) ∂ ∂x V p (x), f p 2cζW (ξ) τ a + 2cW (ξ) ψ(V p (x)) -α(V p (x)) + γ(|d|) 2c ζ τ a -1 W (ξ) + 2cW (ξ) ψ(V p (x)) γ(|d|)
where we have used the fact that ϕ (s) = 2cϕ(s) ψ(s) in the first inequality, and the property ψ(s) α(s) for the second inequality above. Note that since we have picked ζ < τ a , ζ τa < 1, it holds that a := 1 -ζ τa > 0. Next, to compute the derivative of V along the differential inclusion, notice that (ϕ -1 ) (s) =

1 ϕ (ϕ -1 (s)) = ψ(ϕ -1 (s)) 2cs
, which results in

∂ ∂ξ V (ξ), f = (ϕ -1 ) (W (ξ)) ∂ ∂ξ W (ξ), f ψ(ϕ -1 (W (ξ))) 2cW (ξ) -2caW (ξ) + 2cW (ξ) ψ(V p (x)) γ(|d|) = -aψ(ϕ -1 (W (ξ))) + ψ(ϕ -1 (W (ξ))) ψ(V p (x)) γ(|d|) = -aψ(V (ξ)) + ψ(V (ξ)) ψ(V p (x)) γ(|d|).
To further simplify the right-hand side, let v := V p (x), w = V (ξ); then clearly ϕ(w) = W (ξ) = e 2cζτ ϕ(V p (x)) e 2cζN0 ϕ(v), and from Lemma V.6,

∂ ∂ξ V (ξ), f -aψ(V (ξ)) + ψ(V (ξ)) ψ(V p (x)) γ(|d|) -aψ(V (ξ)) + e c0ζN0 γ(|d|) -aψ • α 1 (|ξ| A ) + e c0ζN0 γ(|d|).
Hence, (24) holds with α c (s) = aψ • α 1 (s), γ c (s) = e c0ζN0 γ(s). It follows from the definition of ψ in (10) that, if α is positive definite (resp. α ∈ K ∞ ), then α c is also positive definite (resp. α c ∈ K ∞ ). In addition, γ c ∈ K.

To show (25), pick (ξ, d) ∈ D and ḡ ∈ G(ξ, d). By the definition of ϕ in (29) and ζ * in [START_REF]Robust input-to-state stability for hybrid systems[END_REF] we have

ϕ χ(V p (x)) = exp χ(Vp(x)) 1 2c ψ(r) dr = exp χ(Vp(x)) Vp(x) 2c ψ(r) dr exp Vp(x) 1 2c ψ(r) dr e 2cζ * ϕ(V p (x)).
Therefore, When α ∈ K ∞ , 1 ψ(r) decreases to 0 as r increases to infinity; therefore, (34) holds for all v > 0 only when the interval of integration grows to infinity with respect to v. In other words, we conclude that κ ∈ K ∞ .

V (g) = ϕ -1 e 2cζτ + ϕ(V p + (x + )) ϕ -1 e 2cζ(τ -1) ϕ χ(V p (x)) ϕ -1 e 2cζ(τ -1)+2cζ * ϕ V p (x) ϕ -1 e 2c(ζ * -ζ) ϕ(V (ξ)) . Define κ(v) := v -ϕ -1 e 2c(ζ * -ζ) ϕ(v) , ( 33
As a final remark, we conclude from Lemma V.2 (resp. Lemma V.1) that V is an iISS (resp. ISS) Lyapunov function.

VI. Conclusions

Within the context of stability of switched systems under slow switching, this paper provides lower bounds on ADT which guarantee ISS or iISS properties. In particular, we considered the case where each subsystem is ISS (resp. iISS). Using the nonlinear supply functions associated with these subsystems, and the growth functions describing the change in the value of Lyapunov functions at switching instants, a lower bound on ADT is computed which guarantees ISS (resp. iISS) for a class of switching signals satisfying such a bound. It is seen that switched bilinear systems indeed fall within the framework studied in this paper, and we have provided conditions under which slowly switching bilinear systems are seen to be iISS. Among several possible extensions, it is desirable to adopt this framework under relaxed hypotheses which allow for destabilizing effect of disturbances in jump dynamics, and a wider class of supply functions associated with individual subsystems. Since iISS property has found utility in analyzing interconnections of dynamical systems [START_REF] Arcak | A unifying integral ISS framework for stability of nonlinear cascades[END_REF], one can also use the results of this paper to analyse stability of interconnections of switched systems under slow switching.

  √ a |ω| for all p ∈ P, where b is defined by b := min p∈P |λ(Q p )|, (16) and c 1 , c 2 are given by c 1 : = √ m max j∈{1,...,m},p∈P |λ(B p,j M p + M p B p,j )|, c 2 : = 2 max p∈P |λ(M p C p )|. Thus by setting α(s) := b(e s -1)

  . Notice that α (s) = b ae s b a for all s 0, we can pick c = b a such that ψ(s) = α(s) = b(e s -1) ae s . Using (11), the lower bound on ADT is ζ

1 2c 1 2c 1 2cψ(r) dr c 1 exp v 1 2cψ

 1111 cr dr = 2 ln s so lim s-→∞ ϕ(s) lim s-→∞ s 2 = ∞. On the other hand when s < 1, cr dr = 2 ln s so lim s-→0 + ϕ(s) lim s-→0 + s 2 = 0 and thus ϕ is continuous at 0. Lemma V.6. Let c 1 > 0 and w v > 0 such that ϕ(w) c 1 ϕ(v). Then under Assumption 2, ψ(w) ψ(v) c c 0 2c 1 . Proof. Fix w v > 0. Then, ϕ(w) c 1 ϕ(v) means exp w (r) dr . Take logarithm on both sides and subtract the integral on the right from the left, we get w v 2c ψ(r) dr ln(c 1 ).

  ) the inequality (25) is seen to hold with α d (s) = κ • α 1 (s). Recall that ζ > ζ * and ϕ ∈ K ∞ ; thus by construction κ is positive definite, so is α d . Lastly we show that α d ∈ K ∞ when α ∈ K ∞ and it suffices to show that κ ∈ K ∞ . It follows from (33) that ϕ (v -κ(v) = e 2c(ζ * -ζ) ϕ(v) and, after plugging (29) in and taking logarithm, we get v-κ(v) dr = ζ -ζ * > 0.(34)

S. Liu and D. Liberzon were supported by the NSF grant CMMI-1662708 and the AFOSR grant FA9550-17-1-0236. A. Tanwani was supported by ANR project ConVan with grant number ANR-17-CE40-0019-01.