G Le Cozannet

R Thieblemont

L Bricheno

M. D 'anna

B Castelle

D Idier

J Rohmer

Maurizio D'anna
email: m.danna@brgm.fr

Uncertainties in Shoreline Projections to 2100 at Truc Vert Beach (France): Role of Sea-Level Rise and Equilibrium Model Assumptions

HAL is a

Introduction

Ongoing climate change is one of the largest concerns of our time, and its largest impacts on the world's environment are yet to come. Global mean sea-level rise is accelerating since 1870, and is expected to continue rising over the 21 st century, although acceleration may be avoided if the Paris Agreement 'below 2°C climate warming' target is met [START_REF] Church | Sea Level Change[END_REF]Oppenheimer et al., 2019). In addition, global wave power is adapting to the sea surface temperature since the late 1940's [START_REF] Reguero | A recent increase in global wave power as a consequence of oceanic warming[END_REF], and is expected to change along with storminess by 2100 [START_REF] Morim | A global ensemble of ocean wave climate projections from CMIP5-driven models[END_REF].

Sandy beaches provide precious natural, structural and social-economical resources to coastal communities [START_REF] Ghermandi | A global map of coastal recreation values: Results from a spatially explicit meta-analysis[END_REF][START_REF] Poumadère | Coastal vulnerabilities under the deliberation of stakeholders: The case of two French sandy beaches[END_REF], and constitute about one third of the ice-free coasts worldwide [START_REF] Luijendijk | The State of the World's Beaches[END_REF]. Open sandy beaches constantly evolve in response to multiple environmental drivers occurring on different time scales, making sandy shoreline dynamics strongly sensitive to sea-level rise and wave climate change [START_REF] Ranasinghe | Assessing climate change impacts on open sandy coasts: A review[END_REF][START_REF] Ranasinghe | On the need for a new generation of coastal change models for the 21st century[END_REF]. Meanwhile, the expected growth of population density in low-lying coastal areas during the twenty-first century [START_REF] Merkens | Gridded population projections for the coastal zone under the Shared Socioeconomic Pathways[END_REF]Neumann et al., 2015) increases the need for efficient adaptation plans of coastal communities [START_REF] Oppenheimer | Sea Level Rise and Implications for Low-Lying Islands, Coasts and Communities[END_REF].

The spatial heterogeneity of sea-level rise (SLR), wave-climate change, time scales of adaptation, and vulnerability of coastal communities raises the need for shoreline projections with their related uncertainties that provide full support to risk-informed decision making process [START_REF] Hinkel | Meeting user needs for sea level rise information: A decision analysis perspective[END_REF][START_REF] Losada | A planning strategy for the adaptation of coastal areas to climate change: The Spanish case[END_REF][START_REF] Toimil | Estimating the risk of loss of beach recreation value under climate change[END_REF][START_REF] Toimil | Climate Change Driven coastal erosion modelling in temperate sandy beaches methods and uncertainty treatment[END_REF][START_REF] Wainwright | Moving from deterministic towards probabilistic coastal hazard and risk assessment: development of a modelling framework and application to Narrabeen Beach, New South Wales, Australia[END_REF]. However, limits in our understanding and modelling capacity of the primary processes driving shoreline change, together with the uncertainties associated to the future climate (e.g. carbon emission scenario, SLR, storminess, etc.), undermine the confidence in future shoreline estimates proportionally to the time scale of application [START_REF] Ranasinghe | On the need for a new generation of coastal change models for the 21st century[END_REF][START_REF] Toimil | Climate Change Driven coastal erosion modelling in temperate sandy beaches methods and uncertainty treatment[END_REF]. Many studies focused on the effects of SLR uncertainties [START_REF] Athanasiou | Uncertainties in projections of sandy beach erosion due to sea level rise: an analysis at the European scale[END_REF][START_REF] Cozannet | Uncertainties in sandy shorelines evolution under the Bruun rule assumption[END_REF], 2019;[START_REF] Thiéblemont | Deep uncertainties in shoreline change projections: an extra-probabilistic approach applied to sandy beaches[END_REF][START_REF] Vousdoukas | Sandy coastlines under threat of erosion[END_REF] and changes in storminess based on data extrapolation and/or empirical models [START_REF] Allenbach | Black Sea beaches vulnerability to sea level rise[END_REF][START_REF] Casas-Prat | Future wave-driven coastal along the Catalan coast (NW Mediterranean)[END_REF][START_REF] Toimil | Managing coastal erosion under climate change at regional scale[END_REF][START_REF] Vousdoukas | Sandy coastlines under threat of erosion[END_REF] on future shoreline uncertainties. However, these studies do not explicitly resolve wave-driven shoreline change, and it is advocated that new methods have to be developed to predict the impacts of SLR on the coast [START_REF] Cooper | Sandy beaches can survive sea-level rise[END_REF]. Short-and long-term variability in wave energy, as well as the chronology of storm events, can strongly affect future shoreline manuscript submitted to Journal of Geophysical Research: Earth Surface 4 patterns [START_REF] Besio | Time clustering of wave storms in the Mediterranean Sea[END_REF][START_REF] Cagigal | A multivariate, stochastic, climate-based wave emulator for shoreline change modelling[END_REF][START_REF] Coco | Beach response to a sequence of extreme storms[END_REF][START_REF] Dissanayake | Comparison of storm cluster vs isolated event impacts on beach/dune morphodynamics[END_REF][START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF]. Recently, [START_REF] Cagigal | A multivariate, stochastic, climate-based wave emulator for shoreline change modelling[END_REF] developed and used a stochastic climatebased wave emulator to generate ensembles of wave time series at several beaches, and addressed shoreline response to different wave chronologies. Based on the same emulator, [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF] analytically investigated the uncertainties in shoreline predictions associated to the inherent variability of the wave climate in the context of equilibrium shoreline modelling. [START_REF] Kroon | Ranking uncertainty: Wave climate variability versus model uncertainty in probabilistic assessment of coastline change[END_REF] showed the significant effects of wave-climate variability and model uncertainty on the short-term (1 year) probabilistic assessment of coastline change at the Sand Engine (Netherlands).

The authors used a one-line model, i.e. resolving wave-driven longshore sediment transport gradients and resulting shoreline evolution, as this stretch of coast is longshore transport dominated.

Currently, there are no studies addressing the time evolution of the effects that uncertainties in future SLR and model parameters have on shoreline projections to the end of the 21 st century while explicitly resolving wave-driven shoreline response. The recent development of equilibrium shoreline models opened the way to skilful simulation of wave-driven shoreline response on crossshore transport dominated sites, which are ubiquitous worldwide, on time scales from hours (storm events) to decades, with low computational cost [START_REF] Antolínez | Predicting climate driven coastlines with a simple and efficient multi-scale model[END_REF][START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF][START_REF] Lemos | Equilibrium modelling of the beach profile on a macrotidal embayed low tide terrace beach[END_REF][START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF]Splinter et al., 2014a;[START_REF] Vitousek | A model integrating longshore and cross-shore processes for predicting long-term shoreline response to climate change[END_REF][START_REF] Yates | Equilibrium shoreline response: Observations and modeling[END_REF]. Equilibrium shoreline models are based on the principle that the shoreline dynamically moves towards a time-varying equilibrium condition [START_REF] Wright | Morphodynamic variability of surf zones and beaches: a synthesis[END_REF], which can be expressed as a function of the current shoreline position [START_REF] Yates | Equilibrium shoreline response: Observations and modeling[END_REF] or antecedent wave conditions [START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF]. While the two latter equilibrium formulations show similar skill against shoreline observations on a multi-year timescale [START_REF] Castelle | Equilibrium shoreline modelling of high energy meso-macrotidal multiple barred beach[END_REF][START_REF] Montaño | Blind testing of shoreline evolution models[END_REF], the accuracy of one approach over the other in different wave forcing scenarios is unclear, particularly on long timescales (multi-decadal). In addition, in this type of models, sediment transport processes are described by semi-empirical relationships that require site-specific calibration against observed shoreline data, introducing further uncertainty [START_REF] D'anna | Impact of model free parameters and sea-level rise uncertainties on 20-years shoreline hindcast: the case of Truc Vert beach (SW France)[END_REF][START_REF] Splinter | How much data is enough? The importance of morphological sampling interval and duration for calibration of empirical shoreline models[END_REF]. Implementations of cross-shore equilibrium models into probabilistic frameworks recently showed that uncertainties in the calibration of model free parameters [START_REF] D'anna | Impact of model free parameters and sea-level rise uncertainties on 20-years shoreline hindcast: the case of Truc Vert beach (SW France)[END_REF] and in future wave conditions [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF] have a significant impact on model predictions. In addition, recent studies found an inherent connection between the seasonality of wave climate and shoreline model parameters that defines the frequency of shoreline response, for several beaches along the Australian coast [START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF][START_REF] Splinter | Rapid adjustment of shoreline behaviour to changing seasonality of storms: observations and modelling at an open-coast beach[END_REF]. SLR-driven shoreline retreat is often estimated using the [START_REF] Bruun | Sea-level rise as a cause of shore erosion[END_REF] model. This model relates the rate of shoreline erosion to the SLR rate and the average slope of the active beach profile, defined between the seaward and landward limits of cross-shore sediment exchange. The seaward limit of the active beach profile is commonly identified by the depth of closure (Hallermaier, 1978). As local scale bathymetric surveys are scarce and the estimation of the depth of closure is essentially empirical, the active beach profile slope is typically associated with large uncertainties [START_REF] Nicholls | Assessing erosion of sandy beaches due to sea-level rise[END_REF][START_REF] Ranasinghe | Estimating coastal recession due to sea level rise: beyond the Bruun rule[END_REF].

In this work, we aim at deepening our understanding in the role and impact of different uncertainties in shoreline projections. We perform a Global Sensitivity Analysis (GSA) [START_REF] Saltelli | Global Sensitivity Analysis: The premier[END_REF] to unravel the respective contributions of SLR, depth of closure, and shoreline model free parameters uncertainties. The framework is applied to the cross-shore dominated Truc Vert beach (SW France) using two different wave-driven shoreline models, the Bruun model, and stateof-the-art SLR and wave projections for two future Representative Concentration Pathways (RCP) scenarios. The likely range provided along with median SLR estimates in IPCC reports does not cover the full uncertainty range of mean sea level projections. Hence, there remains a probability of up to 33% that sea-level rise exceeds the likely range. Therefore, we also assess shoreline projections in the deterministic high-end SLR scenario, which remains unlikely but plausible and is associated with large impacts [START_REF] Stammer | Framework for high-end estimates of sea level rise for stakeholder applications[END_REF]. The remainder of this paper includes: an outline of Truc Vert beach, the data, the shoreline models, and the method (Section 2); a description of the GSA input variables' probability distributions and the numerical modelling setup (Section 3); and the presentation of the results (Section 4). Discussion and conclusions are provided in Section 5 and 6, respectively.

2

Study site, data and method

Truc Vert beach

Truc Vert is a meso-macrotidal wave dominated sandy beach located in the south of the Gironde coast, southwest France, which extends roughly 100 km between the Gironde river estuary and the south of the Arcachon basin (Figure 1a,b). Truc Vert is backed by a high (~20 m) and wide (~250 m) coastal dune system [START_REF] Robin | 150 years of foredune initiation and evolution driven by human and natural processes[END_REF]. The wave climate is characterized by strong seasonal energy fluctuations, and strong interannual winter energy variability (Castelle et al., 2018a;[START_REF] Charles | Climate change impact on waves in the Bay of Biscay, France[END_REF][START_REF] Robinet | Statistical modeling of interannual shoreline change driven by North Atlantic climate variability spanning manuscript submitted to Journal of Geophysical Research: Earth Surface 59 2000-2014 in the Bay of Biscay[END_REF], the latter associated to large-scale climate patterns of atmospheric variability in the northeast Atlantic region [START_REF] Castelle | A new climate index controlling winter wave activity along the Atlantic coast of Europe: The West Europe Pressure Anomaly[END_REF]. Monthly-averaged significant wave height ranges from 1.1 m in August with dominant W-NW direction to 2.4 m in January with dominant W direction. Truc Vert beach has been intensively monitored since 2003 with monthly to bi-monthly topographic DGPS surveys, with additional daily topographic surveys and high-resolution bathymetric surveys collected during the ECORS'08 field campaign [START_REF] Parisot | Treatment of topographic and bathymetric data acquired at the Truc-Vert Beach (SW France) during the ECORS field experiment[END_REF], see [START_REF] Castelle | 16 years of topographic surveys of ripchannelled high-energy meso-macrotidal sandy beach[END_REF] for detailed description of the datasets. Since 2017, highresolution digital elevation model covering 4 km of beach-dune are also derived seasonally from photogrammetry of UAV images [START_REF] Laporte-Fauret | Low-Cost UAV for High-Resolution and Large-Scale Coastal Dune Change Monitoring Using Photogrammetry[END_REF].

The beach morphology is highly dynamic and responds primarily to cross-shore processes driven by the temporal variability of the incident wave climate [START_REF] Castelle | Equilibrium shoreline modelling of high energy meso-macrotidal multiple barred beach[END_REF][START_REF] Robinet | Statistical modeling of interannual shoreline change driven by North Atlantic climate variability spanning manuscript submitted to Journal of Geophysical Research: Earth Surface 59 2000-2014 in the Bay of Biscay[END_REF][START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF]. Overall, this segment of coastline has been observed to be reasonably stable over the past decades [START_REF] Castelle | Spatial and temporal patterns of shoreline change of a 280-km high-energy disrupted sandy coast from 1950 to 2014: SW France[END_REF], although the interannual distribution of winter wave energy may result in episodic severe beach and dune erosion [START_REF] Castelle | Impact of the winter 2013-2014 series of severe Western Europe storms on a doublebarred sandy coast: beach and dune erosion and megacusp embayments[END_REF][START_REF] Masselink | Extreme wave activity during 2013/2014 winter and morphological impacts along the Atlantic coast of Europe[END_REF].

Wave data: historical and projections

While a dataset of future waves is required to simulate future shoreline change, hindcast wave data were also needed for the present study in order to: [START_REF] Thiéblemont | Deep uncertainties in shoreline change projections: an extra-probabilistic approach applied to sandy beaches[END_REF] run the shoreline models on the past period and estimate the distribution of the model parameters; and (2) support the correction of the wave projection dataset.

2. 1.1 Hindcast wave data (19941.1 Hindcast wave data ( -2020) ) Historical wave data (Hs, Tp, and Dm), from January 1994 to January 2020, was extracted from the NORGAS-UG regional hindcast model [START_REF] Michaud | Improvements of the new French coastal wave forecasting system and application to a wavecurrent interaction study[END_REF] at the grid point co-located with the in situ CANDHIS wave buoy (44°39'9" N; 1°26'48" W) moored in ~50 m depth offshore of Truc Vert beach (Figure 1a). The NORGAS-UG model covers the French Atlantic coastal area using an unstructured mesh grid with resolution of 10 km offshore, increasing to 200 m nearshore. The wave model was validated against several French and international wave buoy data, and showed 0.96-0.99 correlations coefficients, 0.15-0.21 m RMSE, and -0.02 to 0.04 m bias [START_REF] Michaud | Improvements of the new French coastal wave forecasting system and application to a wavecurrent interaction study[END_REF]. The hindcasted wave time series shows the typical seasonal and interannual modulation of the incident wave climate at Truc Vert beach (Figure 2a).

Future wave climate (2020 -2100)

Wave-driven shoreline change at cross-shore transport dominated sites is controlled by the variability in incident wave energy including temporal clustering and chronology of storm wave events [START_REF] Splinter | A relationship to describe the cumulative impact of storm clusters on beach erosion[END_REF][START_REF] Dissanayake | Comparison of storm cluster vs isolated event impacts on beach/dune morphodynamics[END_REF][START_REF] Angnuureng | Shoreline resilience to individual storms and storm clusters on a meso-macrotidal barred beach[END_REF]. Thus, the assessment of future shoreline evolution at Truc Vert requires a continuous wave time series with high resolution (e.g. few hours). [START_REF] Bricheno | Future Wave Conditions of Europe, in Response to High-End Climate Change Scenarios[END_REF] 1a), in ~50 m depth, for both RCP8.5 and RCP4.5 scenarios. To reduce the bias in modelled future waves, we analysed the seasonal quantiles of the 1994-2004 portion of BW18 historic wave time series (extracted at the same location as the 2020-2100 wave data) and the seasonal quantiles of the NORGAS-UG hindcast, and set-up a seasonal quantile-quantile correction that we applied to the 2020-2100 wave dataset (details in Text S1 of Supporting Information). The corrected BW18 wave time series for RCP8.5 and RCP4.5 scenarios are shown in Figure 2b and Figure 2c, respectively. Hereon, we refer to BW18 as the corrected wave time series. (e) winter mean Hs (dashed lines) of 2020-2100 corrected [START_REF] Bricheno | Future Wave Conditions of Europe, in Response to High-End Climate Change Scenarios[END_REF] wave projections, for RCP4.5 (blue) and RCP8.5 (orange) scenarios. For RCP8.5 (RCP4.5), the trend of summer and winter mean Hs are -2 mm/year (-1 mm/year) and -0.05 mm/year (-0.05 mm/year).

These trends were tested to be statistically significant (more than 99% significance) using

Student's t-tests. (f) Quantile-quantile comparison between RCP4.5 and RCP8.5's 3-month average of Hs 2 Tp projections for the four seasons (black crosses) and for the full datasets (grey circles).

Here, we adopted Hs 2 Tp (m 2 s) as a representative variable for deep-water wave energy. The and the 2030 (2060 and 2068) winter following a negative trend of winter energy. Although in both RCP scenarios the incident wave energy fluctuates with a similar interannual period with nearly the same average Hs 2 Tp (52 and 54 m 2 s for RCP8.5 and RCP4.5, respectively), the RCP4.5 scenario associates slightly higher energy during Autumn, Summer and Spring (Figure 2f). The 2020-2100 summer mean wave height (𝐻 𝑠 ̅̅̅ 𝑠𝑢𝑚𝑚𝑒𝑟 ) fluctuates between 0.9 m and 1.7 m, with a statistically significant decrease of 2 mm/year (1 mm/year) rates for the RCP8.5 (RCP4.5) (Figure 2d). Future winter mean wave height (𝐻 𝑠 ̅̅̅ 𝑤𝑖𝑛𝑡𝑒𝑟 ), which is a key driver of cross-shore wavedominated shoreline evolution [START_REF] Dodet | Beach recovery from extreme storm activity during the 2013-14 winter along the Atlantic coast of Europe[END_REF], varies between 1.5 m and 3 m with a manuscript submitted to Journal of Geophysical Research: Earth Surface 13 statistically significant decreasing trend under 0.05 mm/year in both RCP scenarios (Figure 2e). This is consistent with previous regional projections [START_REF] Charles | Climate change impact on waves in the Bay of Biscay, France[END_REF][START_REF] Perez | Statistical multi-model climate projections of surface ocean waves in Europe[END_REF][START_REF] Morim | Robustness and uncertainties in global multivariate wind-wave climate projections[END_REF].

2.3 Mean sea level and vertical land motion

Past mean sea level reconstruction

As SLR-driven shoreline retreat is explicitly accounted for in the calibration of the shoreline models, past MSL information is required. We reconstructed the geocentric MSL change in the Bay of Biscay over the period 2012-2020 using a Kalman filter approach assimilating available tide gauge records in this region (Rohmer & Le Cozannet, 2019). The resulting SLR rates are roughly constant at 2.1 ±0.1 mm/year (median ± σ). Local relative MSL change at Truc Vert beach was calculated by adding the effect of vertical land motion to the relative regional sea level estimate. Vertical land motion in Truc Vert area was estimated using the near Cap-Ferret permanent GNSS station from the SONEL database (Santamaria-Gomez et al., 2017), which provides data from 2005 to 2012, when the station was decommissioned. The GNSS station measures the effects of Glacial Isostatic Adjustment and current gravitational, rotational and deformation changes associated to ongoing glaciers and ice-sheets melting [START_REF] Frederikse | The causes of sea-level rise since 1900[END_REF]. We subtract their effects from the observed GNSS records over the observation period to assess residual vertical ground motions obtaining a subsidence rate of 1.2 ±0.6 mm/yr. This results in a roughly constant SLR rate of 3.3 ±0.7 mm/yr over the past decade (see Figure S5 of Supporting Information). The observed lowering ground might be due to slow subsidence of the former Leyre riverbed [START_REF] Klingebiel | Carte Géologique et structure du Bassin de la Leyre[END_REF].

The pointwise Cap-Ferret GNSS station information may not be exactly that of the surrounding area. This is part of the residual uncertainties of our study.

Future mean sea level projections

State-of-the-art GMSL projections until 2100 are available from the Special Report of Ocean and Cryosphere in a Changing Climate (SROCC, Oppenheimer et al., 2019). SROCC estimates build on the Fifth Assessment Report (AR5, [START_REF] Wong | Coastal systems and low-lying areas[END_REF] with a revised assessment of the Antarctic dynamics contribution based on new evidence on marine ice-sheets instabilities since the AR5.

SROCC provides median values of each sea level change contribution with associated likely range for several RCP scenarios. Unlike other IPCC reports, the SROCC defines the likely range as the 17 th -83 rd percentiles of the distribution of sea-level rise [START_REF] Oppenheimer | Sea Level Rise and Implications for Low-Lying Islands, Coasts and Communities[END_REF]. We reproduced the SROCC global MSL projections to Truc Vert beach following [START_REF] Thiéblemont | Likely and High-End Impacts of Regional Sea-Level Rise on the Shoreline Change of European Sandy Coasts Under a High Greenhouse Gas Emissions Scenario[END_REF] and considering the regional fingerprints of each mechanism contributing to sea-level changes, including the effect of Glacial Isostatic Adjustment [START_REF] Slangen | Projecting twenty-first century regional sea-level changes[END_REF]. This results in regional relative 2020-2100 SLR estimate (median and likely range) of 0.63 ±0.26 m and 0.37 ±0.16 m for the RCP8.5 and RCP4.5 scenarios, respectively.

Residual vertical land motion, which is assumed to be due to slow-ongoing geological processes (see subsection 2.3. 1 and Klingebiel & Legigan, 1992), is assumed to remain constant (1.2 ±0.6 mm/yr) over the 21 st century. Due to the large uncertainty (0.6 mm/yr) of the subsidence rate, the stability of the area is not excluded, but has a very low probability (2.1%). The inclusion of ground motion results in a local relative MSL rise of 0.73 ±0.27 m and 0.47 ±0.17 m from 2020 to 2100 for RCP8.5 and RCP4.5 scenarios, respectively (see Figure S5 in Supporting Information). Further detail on future SLR is provided in Section 3.1.

Shoreline change models

Here, we use two equilibrium-based models to assess wave-driven shoreline response: the Yates et al. ( 2009) model, and an adaptation of the ShoreFor model [START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF]Splinter et al. 2014a). As the Truc Vert bathymetry iso-contours are essentially shore-parallel, breaking wave conditions were computed directly from offshore wave conditions using the [START_REF] Larson | Direct formula to compute wave height and angle at incipient breaking[END_REF] formula. Chronic shoreline retreat induced by SLR was estimated using the [START_REF] Bruun | Sea-level rise as a cause of shore erosion[END_REF] model.

As shoreline change at Truc Vert is known to be dominated by cross-shore sediment transport with negligible gradients in longshore transport [START_REF] Castelle | Equilibrium shoreline modelling of high energy meso-macrotidal multiple barred beach[END_REF]Splinter et al., 2014a), we did not compute longshore sediment transport. The following subsections describe the two wave-driven shoreline models and the Bruun model.

Wave-driven shoreline models and free parameters

Equilibrium shoreline models are based on the principle that local wave climate drives the shoreline towards a time-varying equilibrium position at a rate that depends on the instantaneous wave thrust (e.g. wave power or energy) available to move the sediment, and the dynamic disequilibrium state of the beach [START_REF] Wright | Morphodynamic variability of surf zones and beaches: a synthesis[END_REF]. The [START_REF] Yates | Equilibrium shoreline response: Observations and modeling[END_REF] model and ShoreFor differ primarily in the formulation of the respective disequilibrium conditions.

ShoreFor model

The ShoreFor model (hereafter SF) adopts a disequilibrium condition based on the wave history, expressed as a disequilibrium of dimensionless fall velocity (ΔΩ) and its standard deviation (σΔΩ).

The governing equation for shoreline change rate reads:

𝑑𝑌 𝑑𝑡 = 𝑘 𝑠 +/-P 0.5 ∆Ω σ ∆Ω + 𝑏 (1)
where, ks +/-(m s -1 W -0.5 ) is a response rate parameter, P(W) is the wave power at breaking, and b(m/s) is a linear term trend. Following [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF], the disequilibrium term ΔΩ at a given time is defined as the difference between the equilibrium dimensionless fall velocity (Ωeq(Φ)) and the offshore dimensionless fall velocity (Ωo), where Ωeq(Φ) is a function of the sediment size, prior wave conditions, and the free parameter Φ. The parameter Φ (days) is a site-specific 'beach memory', and defines the time over which a given wave event has an impact over the equilibrium state of the beach. The ks +/-parameter is the shoreline response rate, and assumes different values for accretion (ks + , ΔΩ >0) and erosion (ks -, ΔΩ <0) events, which are driven by different processes associating different time scales. The values of the ks +/-parameter for accretion and erosion conditions are considered proportional through a coefficient r (ks -=rks + ). The r coefficient is not a model free parameter but is defined by the wave forcing, and is such that no trend in wave forcing results in no trend in the modelled shoreline position over the simulated period:

𝑟 = | ∑ 〈𝐹 + 〉 𝑁 𝑖=1 ∑ 〈𝐹 -〉 𝑁 𝑖=1 | (2) 𝐹 = P 0.5 ∆Ω(Φ) σ ∆Ω (3)
where N is the full length of the simulated period, F + and F -are the forcing during accretion (ΔΩ >0) and erosion (ΔΩ <0) events, respectively, and 〈. 〉 denotes an operation that removes the linear trend. Here, the sign of ΔΩ does not change the absolute value of F +/-. For an extended description of SF the reader is referred to [START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF] and Splinter et al. (2014a). In SF, the model free parameters to be calibrated at a given site are ks + , Φ and b. Physically, the ks +/-(m s -1 W -0.5 ) is a measure of the efficiency of wave forcing to drive shoreline change (as described by Splinter et al., 2014a), which can also be interpreted as a time scale of shoreline response [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF]. Indeed, a low efficiency corresponds to a slow shoreline response and a longer time scale, and vice versa. Φ(days) is a time scale for the duration of the impact that past waves exerted on the beach, and provides the ability for the model equilibrium condition to evolve along with longterm wave energy trends. The parameter b(m/s) is a linear term that encapsulates the effect of slow processes, other than wave-driven equilibrium based, which may drive chronic shoreline change (e.g. wind driven sediment transport) and that are not explicitly resolved in the model. We note here that, while accounting for the effects of slow processes using a constant linear trend (i.e. b) can improve the model skill for simulated periods within the decade, the application of such trend over longer time scales (decades to centuries) becomes increasingly inaccurate [START_REF] D'anna | Impact of model free parameters and sea-level rise uncertainties on 20-years shoreline hindcast: the case of Truc Vert beach (SW France)[END_REF]. Therefore, given the long time scale of our application and the absence of secondary processes (e.g. longshore gradients in sediment transport) that may drive long-term shoreline trends at Truc Vert, we set b=0.

Yates model

In Yates' model (hereafter Y09) the disequilibrium condition is defined as a function of the current shoreline position, and the cross-shore rate of shoreline change is calculated as follows:

𝑑𝑌 𝑑𝑡 = 𝑘 𝑦 +/-𝐸 0.5 (𝐸 𝑒𝑞 (𝑌) -𝐸 ) (4)
where E (m 2 ) is the wave energy, ky +/-(m 2 s -1 /m) is the response rate parameter, Y(m) is the present shoreline position, and Eeq(Y) is the wave energy in equilibrium with the current shoreline position Y through a linear relationship:

𝐸 𝑒𝑞 (𝑌) = 𝑎 1 𝑌 + 𝑎 2 (5)
where a1 (m 2 /m) and a2 (m 2 ) are free model parameters. The ky +/-parameter is analogous to ks +/-of SF in that it represents the efficiency rate of the incident wave forcing to shoreline change, or a time scale parameter (see the analytical derivation of the Y09 time scale of shoreline response in [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF]. In the Y09 model no assumption is made on a possible relationship between the ky + and ky -, which are both considered model free parameters and, as well as a1 and a2, require specific calibration for each field site application. Contrarily to SF, here the equilibrium state formulation (Equation 5) does not depend on recent wave conditions, making this model insensitive to wave-climate variability on timescales longer than the calibration period. Instead, Equation 5 depends on the current shoreline position (Y), introducing the potential for feedbacks between Y09 and shoreline change induced by other cross-shore processes (e.g. SLR). Herein, such processes are resolved independently and linearly superposed, so that no feedback is enabled.

Physically, ky +/-, once again is a measure of the shoreline reactivity to the incident wave forcing, and is expressed in (m s -1 /m). Although the dimensions of a1 and a2 are 'energy/meter' and 'energy', respectively, the role of these parameters in the model is purely empirical. A rearrangement of the terms in Equations 2-3 results in combinations of model parameters that are representative of equilibrium time and spatial scales [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF]. However, here we use Y09 in its original form, where a1 and a2 are treated as empirical parameters.

Sea-level driven shoreline recession

We include SLR-driven shoreline recession using the [START_REF] Bruun | Sea-level rise as a cause of shore erosion[END_REF] model, which is based on the equilibrium beach concept and cross-shore balance of sediment volume. While the reliability of this model is highly debated for its oversimplification of the reality [START_REF] Cooper | Sea-level rise and shoreline retreat: time to abandon the Bruun Rule[END_REF][START_REF] Ranasinghe | Estimating coastal recession due to sea level rise: beyond the Bruun rule[END_REF], its simple linear formulation has been extensively used worldwide. In addition, Truc Vert beach is a relatively undisturbed beach-dune environment with large accommodation space, which makes this sites in line with most of the Bruun Rule underlying assumptions. The Bruun model assumes that under rising sea level, on time scales larger than years, the average beach profile translates upwards and landwards. The resulting shoreline retreat (dYSLR/dt) depends on SLR and the average slope of the active beach profile, here extending from the dune crest down to the depth of closure (DoC), defined as the depth beyond which sediment exchange is considered negligible [START_REF] Bruun | The Bruun Rule of Erosion by Sea-Level Rise: A Discussion on Large-Scale Two-and Three-Dimensional Usages[END_REF][START_REF] Wolinsky | A unifying framework for shoreline migration: 2. Application to wave-dominated coasts[END_REF]:

𝑑𝑌 𝑆𝐿𝑅 𝑑𝑡 = 𝑆𝐿𝑅 𝑟𝑎𝑡𝑒 𝑡𝑎𝑛𝛽 ( 6
)
where SLRrate is the rate of SLR (m/time), and tanβ is the average profile slope defined between the DoC and the dune crest. We estimated the DoC according to [START_REF] Hallermeier | Uses for a calculated limit depth to beach erosion[END_REF], and the corresponding tanβ=0.023 using the beach profile reported in Figure 1c.

Global Sensitivity Analysis

Numerical modelling of shoreline change inherits the uncertainties associated to input variables and their complex interactions, affecting the robustness of the shoreline projections. While numerical modelling provides a 'key-hole' to observe the explicit interactions among defined sets of variables, sensitivity analysis provides a way to understand the role of input variables uncertainties in shoreline predictions. Here, we use the framework proposed by D'Anna et al.

(2020), who used a variance-based Global Sensitivity Analysis (GSA) [START_REF] Saltelli | Global Sensitivity Analysis: The premier[END_REF][START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[END_REF] to investigate the relative contributions of the uncertainties affecting input variables to the uncertainties of modelled shoreline predictions, and their evolution in time. The method consists in propagating the input uncertainties through the model obtaining a probabilistic estimate of the shoreline projections, and performing a GSA which decomposes the variance of model results into several contributions, each one associated with an input variable. Each of these contributions is used to estimate a measure of the model results sensitivity to the input uncertainties with a sensitivity index known as first-order Sobol' index (Si). The Si (0-1) quantifies the ratio of output's manuscript submitted to Journal of Geophysical Research: Earth Surface 20 variance associated with the uncertainties of a given input Xi, i.e. the reduction in the output variance that would occur if the uncertain input Xi was set to its true value, and is defined as:

𝑆 𝑖 = 𝑉𝑎𝑟(𝐸(𝑌|𝑋 𝑖 )) 𝑉𝑎𝑟 (𝑌) ( 7
)
where Var is the variance operator, E is the expectation operator, Y is the modelled shoreline position, and Xi is the i-th uncertain input variable. Further details on GSA and Si are provided in Text S3 of Supporting Information.

Here, we address the relative impact of uncertainties associated to SLR, DoC and of model free Prioritization" (as defined by [START_REF] Saltelli | Global Sensitivity Analysis: The premier[END_REF]. At a given time, the Factors' Prioritization identifies the main driver of model results uncertainty (associating the largest Si), that is, the uncertain input variable that would most reduce the output's variance when fixed to its true value.

The method can be summarized in three steps:

1) Definition of probability distribution associated to each stochastic input variable (SLR, DoC and model free parameters);

2) Generation of ensemble modelled shoreline projections, by means of a Monte-Carlo-based procedure (with accounts for dependence among the input parameters); and

3) Computation of first-order Sobol' index time series for each uncertain input variable.

manuscript submitted to Journal of Geophysical Research: Earth Surface

21

The GSA results are interpreted as the repartition of the variance of shoreline projections into normalized portions (between 0 and 1) imputed to the uncertain input variables. For instance, at a given time, Si,SLR = 0.3 means that uncertainties in future SLR alone are responsible for 30% of the variance in shoreline projections. However, the magnitude of a Si alone is not sufficient to identify the main driver of the shoreline projections' variance, which is defined by comparing the values of Si for all input variables and ranking them in terms of importance.

Figure 3 synthesizes the generalized method and details for the Truc Vert probabilistic applications (excluding the additional high-end SLR deterministic scenario).

Figure 3 Flowchart of the method applied herein, summarized for a general case (black box), and for the Truc Vert application (red box) in the four application scenarios.

3 Input probability distributions for future projections

Probabilistic sea-level rise

Sea-level projections inherit uncertainties associated with physical unknowns and modelling of the contributing processes. While many efforts were dedicated to assess such uncertainties, there is no single approach to define MSL probability distributions yet [START_REF] Jackson | A probabilistic approach to 21st century regional sea-level projections using RCP and High-end scenarios[END_REF][START_REF] Jevrejeva | Probabilistic Sea Level Projections at the Coast by 2100[END_REF][START_REF] Kopp | Probabilistic 21st and 22nd century sea-level projections at a global network of tide-gauge sites[END_REF]. We produced probabilistic relative MSL projections, conditional to the RCP8.5 and 4.5 scenarios, defining time varying normal probability distributions characterized by the yearly median and standard deviations obtained in Section 2.3.2 (Figure 4a,b),

following [START_REF] Hunter | Towards a global regionally varying allowance for sea-level rise[END_REF]. In the high-emission scenario (RCP8.5), the large uncertainty associated with Antarctic ice sheet dynamics generates a skewness of the distributions in the second half of the 21 st century, enhancing the amount of possible extreme SLR [START_REF] Grinsted | Sea level rise projections for northern Europe under RCP8.5[END_REF][START_REF] Jackson | A probabilistic approach to 21st century regional sea-level projections using RCP and High-end scenarios[END_REF][START_REF] Kopp | Probabilistic 21st and 22nd century sea-level projections at a global network of tide-gauge sites[END_REF]. The upper tail of the skewed probability distribution is very much debated [START_REF] Jevrejeva | Probabilistic Sea Level Projections at the Coast by 2100[END_REF] and is not represented by the Gaussian distributions. Therefore, in addition to the Gaussian distribution reflecting the SROCC assessment [START_REF] Oppenheimer | Sea Level Rise and Implications for Low-Lying Islands, Coasts and Communities[END_REF], we consider a high-impact, low probability high-end sea level scenario that might take place for high greenhouse gas emissions (RCP8.5; black line in Figure 4b) The possibility that the subsidence rate revealed by the Cap-Ferret GPS station is not representative of the Truc Vert area (located at 8 km distance) constitutes a residual uncertainty that cannot be quantified, and is not accounted in this study due to the lack of quantitative information supporting an alternative scenario for residual vertical ground motions.

following

Depth of closure

The active beach profile slope is critical to SLR-driven erosion rate (Section 2.4), and strongly depends on the depth of closure (DoC). The DoC was calculated from the wave climate using the [START_REF] Hallermeier | Uses for a calculated limit depth to beach erosion[END_REF] formula. Given that DoC depends on the period of time over which the Hallermeier formula is applied [START_REF] Nicholls | Assessing erosion of sandy beaches due to sea-level rise[END_REF], we iteratively applied the Hallermeier formula over a 1-year moving window of the future wave climate with a 30-days step. For both RCP8.5 and RCP4.5 scenarios, the latter procedure generated an ensemble of possible DoC values well fitted by a Gaussian distribution (Figure 4c). The DoC probability distribution shows higher median and standard deviation values in the RCP4.5 (µ = 17.2 m; σ = 1.75 m) than in the RCP8.5

(µ = 16.3 m; σ = 0.95 m). This results from the more frequent occurrence and larger wave heights associated to isolated extreme events in the RCP4.5 scenario, compared to the RCP8.5 scenario.

Model parameters

Numerical models are associated with uncertainties owing to the choice of modelling approach and to the estimation of model free parameters. We accounted for the uncertainty conditional to the choice of modelling approach assessing the shoreline projections using the Y09 and the SF models described in Section 2.4.1, in two separated scenarios. Both models rely on shoreline observations to calibrate the respective free parameters, and inherit uncertainties due to the quality and amount of available data [START_REF] Splinter | How much data is enough? The importance of morphological sampling interval and duration for calibration of empirical shoreline models[END_REF], to possible non-stationarity of the parameters in respect to the wave climate [START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF], and to the optimization method. Uncertainties affecting model free parameters of the Y09 model (ky +/-, a1, a2) and the SF model (ks + , Φ) are synthetized by their associated joined probability distribution. We follow the approach developed in D'Anna et al. ( 2020), who calibrated the SF model free parameters using the Simulated Annealing algorithm [START_REF] Bertsimas | Simulated annealing[END_REF], and determined their joint probability distribution by fitting an empirical multivariate distribution (multivariate kernel function) on an ensemble of model parameters combinations. The authors built the latter ensemble selecting all parameters combinations that produced a RMSE < 10 m against observed shoreline data during the optimization process. Unlike D'Anna et al. ( 2020), here we calibrated the models between January 2012 and December 2019, where no long-term trend in shoreline position is observed, in line with the assumption of the SF parameter b=0 (see Section 2.4.1). In addition, we used the Nash-Sutcliffe [START_REF] Nash | River flow forecasting through conceptual models part I-a discussion of principles[END_REF] efficiency score (NS) instead of the RMSE to determine the models' performance (as for instance in [START_REF] Kroon | Ranking uncertainty: Wave climate variability versus model uncertainty in probabilistic assessment of coastline change[END_REF]. The NS measures the model skill in comparison to the 'mean model' (defined as the observed mean shoreline position), based on the error's variance, and it is calculated as follows:

𝑁𝑆 = 1 - ∑ (𝑌 𝑚 𝑛 -𝑌 𝑜 𝑛 ) 2 𝑁 𝑛=1 ∑ (𝑌 𝑜 ̅ -𝑌 𝑜 𝑛 ) 2 𝑁 𝑛=1 ( 8
)
where N is the number of observations, Ym n and Yo n are the n-th modelled and observed shoreline positions, respectively, and 𝑌 𝑜 ̅ is the mean of the observed shoreline positions. The NS coefficient can range between -∞ and 1, where NS = 1 corresponds to a model perfectly reproducing the observations, NS = 0 to a model with skill comparable to the 'mean model', and NS < 0 corresponds to models less skilful than the 'mean model'. We obtained the probability distribution using combinations of parameters that resulted in a NS ≥ 0.25 (compared to the maximum NS = 0.63), which corresponds to a max RMSE of ~10 m consistently with D'Anna et al. (2020). We defined the latter threshold with the iterative procedure described in Text S4 of Supporting Information.

This procedure results in the probability distributions of ky +/-, a1, and a2 for Y09, and ks + and Φ for SF shown in Figure 4d,e, with the range of possible parameters values reported in Table 1.

Model setup of shoreline projections

Four ensembles of 3000 possible shoreline trajectories from 2020 to 2100 were generated using the SF and Y09 shoreline change models, and the Bruun Rule, for the two RCP8.5 and RCP4.5 scenarios (Table 2). Wave-driven shoreline response (short-term) and SLR (long-term) were computed individually and then linearly combined, so that no feedback mechanisms occur between the models, in line with previous applications (D 'Anna et al., 2020;[START_REF] Vitousek | A model integrating longshore and cross-shore processes for predicting long-term shoreline response to climate change[END_REF]. August). We analysed the decadal shoreline trends by filtering the modelled shoreline time series with a 5-year running mean. In addition, for RCP8.5 scenario, a deterministic high-end-SLR simulation was run with both shoreline models using the optimized model parameters (Table 1)

and the median DoC. It is to be noted that the GSA results (i.e. Sis) are calculated on the likely range (variance) of the model results, regardless of the envelope of modelled shoreline positions.

Shoreline projections

The four future scenarios in Table 2 resulted each one in 3000 shoreline evolution simulations spanning 2020-2100 (Figure 5 and Figure 6). 3). Indeed, the occurrence of extreme winters can produce significant landward shifts of the envelope of shoreline positions, as observed during the 2084-2085 winter (Figure 5c,d). 3). Similarly to the RCP4.5, here we observe for both models some important shifts in shoreline position distribution owing to extreme winters such as 2086's winter (Figure 6c,d).

In the high-end SLR scenario, both models predict a shoreline position within the envelope of probabilistic projections until 2090, before the shoreline moves further inland during the last decade (Figure 6d,e). The modelled 5-year averaged shoreline position in 2100 is of 88 and 74 m for Y09 and SF, respectively (Table 3). The most landward shoreline position observed throughout the simulation is 107 m with Y09, and 86 m with SF (black dashed line in Figure 6c,d).

The likely (envelope) ranges erosion produced by the combined Y09+Bruun models at the end of the simulated period are comparable (larger) to the standalone application of the Bruun Rule (Table 3, and in Figure 5

Global Sensitivity Analysis

In both RCP8.5 and 4.5 scenarios and for both shoreline model applications, the GSA shows that over the first 30 years of simulation the variance of modelled shoreline projections is driven primarily by the uncertainties in model free parameters, while the effects of SLR uncertainties on shoreline position become increasingly significant after 2050 (Figure 7 and Figure 8). The Sis of the Y09 and SF response rate parameters (ky +/-and ks + , respectively) and the SF beach memory parameter (Φ) show seasonal (6 months) and decadal modulation with a decreasing trend as shoreline projections become more sensitive to SLR (Figure 7c,d and Figure 8c,d). Variations in ky +/-and ks + are the primary source of shoreline projection uncertainties during accretion periods.

However, the response rate parameters' uncertainties have a stronger impact on seasonal scale when using the Y09 model (Figure 7c), and a larger impact on interannual scale when using the SF model (Figure 8c), due to the different response of the models to incident wave energy variability. Seasonal modulation is also observed for the Sis of the Y09 empirical parameters (a1 and a2), although the correlation between the variability in incident wave conditions and the parameters' Sis, (both filtered of their seasonal signal with a 1-year running mean) is negligible (R 2 =~0.06 for a1, and R 2 =~0.03 for a2). However, the estimated a1's and a2's Sis remain below 20% during most of the simulated period with occasional peaks up to 45% (Figure 7e,f). The primary effects of SLR uncertainties emerge at different times, which depend both on the RCP scenario and on the shoreline model. When using Y09, a positive trend in SLR's Si emerges in the 2050-2060 period, with SLR's Si exceeding those of model parameters since approximately 2060-2070, for both RCP scenarios (Figure 7g). Instead, with SF in the RCP8.5 (RCP4.5) scenario, such quasi-monotonic trend appears later, approximately during the 2070s' (2060s') and only exceeds the model parameters' Sis after 2085 (2080) (Figure 8e). For all scenarios, DoC's Si slowly increases, with similar trends as SLR's Si, and reaches approximately 5% and 10%, in the RCP8.5 and 4.5 scenarios, respectively. This difference is probably due to the larger uncertainties of SLR in the RCP8.5 scenario (Figure 4b), and to the larger variance of the DoC probability distribution obtained for the RCP4.5 scenario (Figure 4c).

Discussion

Sea-level rise

While observed shoreline erosion in Aquitaine is not yet attributed to SLR, sooner or later a SLRdriven signal will emerge from the current shoreline change variability, as sea levels are committed to rise by meters over the coming centuries [START_REF] Anderson | Doubling of coastal erosion under rising sea level by mid-century in Hawaii[END_REF][START_REF] Oppenheimer | Sea Level Rise and Implications for Low-Lying Islands, Coasts and Communities[END_REF]. Our results suggest that these times of emergence of a SLR-driven erosive trend could be visible during the 2 nd half of the 21 st century, possibly by 2070. This is consistent with the fact that uncertainty (17 th -83 th percentiles) in future sea level grows from roughly 15 cm by the mid 21 st century to 30 cm (RCP4.5) and 50 cm (RCP8.5) in 2100. Yet, this result relies on our modelling assumptions, including the Bruun Rule and the Yates or ShoreFor models.

The GSA applications to four simulated scenarios indicate that uncertainties in the modelled 2020-2100 shoreline projections at Truc Vert are primarily caused by uncertainties in model free parameters between the present day and 2050. The effects SLR uncertainties always emerge as a significant contribution to the shoreline change uncertainties in the second half of the century. We also observed that the time evolution of Sis and the onset of SLR uncertainties effects are conditional to the RCP scenario (in agreement with Le [START_REF] Cozannet | Quantifying uncertainties of sandy shoreline change projections as sea level rises[END_REF], the choice of shoreline model, and the variability of forcing wave climate.

The shoreline trajectory obtained in the deterministic high-end SLR scenario exceeds the envelope of probabilistic projections in the last simulated decade. Truc Vert beach is remote and backed by a high (~20 m) and wide (~250 m) dune system, so that shoreline retreat is not limited by nonerodible geological outcrops or coastal structure. While such large erosion does not threaten any human assets close to Truc Vert beach, such scenario, though unlikely, questions adaptation planning in other eroding urbanised coastal areas with limited accommodation space in southwest France.

Shoreline models

While the SF and Y09 models are both based on the equilibrium beach concept, the respective model structures and parameters associate different physical interpretations and shoreline behaviours (Section 2.4.1). Therefore, the uncertainty associated with the choice of the equilibrium modelling approach cannot be measured by direct confrontation of the Sis obtained with the two models, but requires consideration of the different model responses to the forcing conditions.

The results obtained for the two disequilibrium approaches (Y09 and SF) show similar seasonal and interannual shoreline cycles, although with notably different amplitudes. Such behaviours are rooted in the different expressions of the equilibrium physics adopted in the two wave-driven models (i.e. the mechanism that would drive the shoreline to an equilibrium position under constant wave conditions). [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF] analytically show that the type of equilibrium condition is critical for the short-and long-term response of the shoreline model. On one hand, Y09's equilibrium condition depends on the current shoreline position, and is not influenced by storm events that occurred prior to a given time scale that is implicitly defined by the model calibration (see 'Appendix A' of [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF]. On the other hand, SF's equilibrium state is determined by the (time varying) past wave conditions with an explicit 'beach memory' function, and evolves in time accordingly. This means that, in absence of other processes, the Y09 modelled shoreline oscillates persistently around the same position regardless of the temporal variability of wave energy. Instead, SF can only achieve such a stable mean shoreline trend when forced with a periodic long-term wave climate [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF]. Hence, in presence of longterm trends of wave energy, Y09 emphasizes the short-term shoreline erosion/accretion in order to re-establish the equilibrium shoreline position, while SF adapts to the wave climate pattern. The latter results in larger amplitudes of seasonal fluctuations and in attenuation of long-term fluctuations, compared to SF.

The combined Y09 and Bruun models simulated shoreline ranges at 2100 are overall comparable to the ranges of the standalone Bruun Rule, indicating that in this scenario the net erosion modelled by 2100 is essentially driven by SLR. In fact, Y09 constrains the shoreline response to long-term wave climate shifts to a limited range (as described above) while the linearly added contribution of the Bruun model determines the shoreline trend. Instead, SF can produce wave-driven longterm shoreline trends that are combined with the Bruun retreat. This effect is observed in both RCP4.5 and 8.5 scenarios, where the decreasing wave energy trend (Figure 2d,e) is translated by SF into shoreline accretion trends, resulting in less erosion than the Buun model alone (Table 3).

Such properties of the two model behaviours highlight the different model sensitivities to longterm variability of the wave climate, which can have implications on the uncertainties in shoreline projections. Including the uncertainty of long-term wave climate variability in the ensemble projections would allow investigating the uncertainties related to the different behaviours of the shoreline models.

Model free parameters

Resolving process-based shoreline response to time-varying incident wave energy revealed that uncertainties in model parameters have the largest impact over the first simulated 30 years, regardless of the cross-shore shoreline model choice. Over this period, Y09 and SF uncertainties in response rate parameters (ky +/-and ks + , respectively) are responsible for most of the results uncertainties, which increases during low energy winters (on seasonal scale), and is particularly emphasized for SF in correspondence of extended low energy periods. This suggests that the assumption of a linear relationship between SF's response rate parameters (ks -= r ks + ) may not hold in the context of long-term simulations, as it might depend on the evolution of waves properties [START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF]. In fact, [START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF] found that, such relation is not The behaviour of the model free parameters' Sis highlights, once again, the importance of wave energy variability in determining the impact of the parameters uncertainties on shoreline projections. This was also observed in previous studies, which showed that changes in wave regime can alter the model parameters and the functional relations between them [START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF][START_REF] Splinter | Rapid adjustment of shoreline behaviour to changing seasonality of storms: observations and modelling at an open-coast beach[END_REF]. As a perspective of future work, one way to reduce the effects of model free parameters' uncertainties on modelled shoreline may be to employ non-stationary parameters that can adjust to changes in wave-climate regimes [START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF]. The use of non-stationary parameters would also imply a dynamic value of the r parameter, reducing uncertainties associated to the assumption of a linear relationship, between SF's response rate parameters. In addition, rearranging the Y09 parameters so that the new parameters have a similar order of magnitudes may increase the efficiency of model calibration, reducing model parameters uncertainties [START_REF] Vitousek | The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions[END_REF].

The role of wave time series

Our results indicate that the shoreline erosion is not only associated with large winter energy, but also depends on the trends of past winter wave energy and the internal variability of high-energy events within the season. For instance, in the RCP4.5 scenario the winters 2084-2085 and 2059-2060 show similar 3-month averaged Hs 2 Tp peak (164 m 2 s and 172 m 2 s, respectively), but they are preceded by several years of negative and positive winter energy trend, respectively (Figure 2b).

This results in the winter 2084-2085 producing a rapid landward shift of shoreline position distribution, and the winter 2059-2060 driving more moderate annual changes while contributing to a long-term erosive trend (Figure 5c,d). We also observe that the interannual patterns of shoreline evolution are clearly correlated to those of winter wave energy. These behaviours underline the critical role of high/low energy winters interannual cycles, as well as storms sequencing, in wave-driven shoreline response, in line with previous studies [START_REF] Dissanayake | Comparison of storm cluster vs isolated event impacts on beach/dune morphodynamics[END_REF][START_REF] Besio | Time clustering of wave storms in the Mediterranean Sea[END_REF][START_REF] Dodet | Beach recovery from extreme storm activity during the 2013-14 winter along the Atlantic coast of Europe[END_REF]. In addition, the temporal variability of wave climate (e.g. seasonal distribution of storm events) has been observed to affect the frequency (or 'mode') of shoreline response [START_REF] Splinter | Rapid adjustment of shoreline behaviour to changing seasonality of storms: observations and modelling at an open-coast beach[END_REF][START_REF] Ibaceta | Enhanced Coastal Shoreline Modeling Using an Ensemble Kalman Filter to include Nonstationarity in Future Wave Climates[END_REF].

Therefore, we further investigated the potential role of future waves uncertainties in shoreline projections performing the GSA on an additional ensemble of 3000 simulations forcing the Y09 and SF models with 100 different wave time series. We generated 100 random synthetic wave series using the method proposed by [START_REF] Davidson | Annual prediction of shoreline erosion and subsequent recovery[END_REF], which consists in building continuous series of wave conditions by sampling 1-month portions from a reference dataset of existing wave data (e.g. historic wave data) at a given location. The method generates synthetic wave time series with random, though realistic, chronology of wave events, while maintaining the seasonal and yearly character of the wave climate. However, this assumes a long-term stationarity of the generated wave time series. We used the BW18 projections for the RCP8.5 scenario as reference wave data. We individually applied the [START_REF] Davidson | Annual prediction of shoreline erosion and subsequent recovery[END_REF] method over 8 windows of 10 years from 2020 to 2100 in order to preserve the long-term (>10 years) characteristics of the reference time series while providing enough sampling reference data (Figure 9a). For instance, all the synthetic events from 2030 to 2040 were generated using monthly samples from the 2030-2040 reference dataset.

When using the latter approach to generate ensemble waves the SF model shows some limitations.

Therefore, here we exploit only the test results obtained with Y09. The results of the SF test application and the aforementioned limitations are illustrated in Text S5 and Figure S4 of Supporting Information. The GSA shows that introducing uncertainties in the temporal distribution of wave events (Figure 9a) has a large impact on the variance of model results (Figure 9b) and, in turn, on the relative contributions of the remaining uncertain input parameters (Figure 9c-h). In fact, accounting for uncertainty in wave events chronology (though in a simplistic way) increases the overall model variance throughout the entire simulated period (Figure 6c and Figure 9b), and associates a dominating Si (up to 0.3) over the first half of the simulated period (Figure 9i). However, SLR's Si still emerges after 2060 and dominate shoreline projections uncertainties over the last two simulated decades. We also observe that the inclusion of wave chronology uncertainty attenuates the interannual variability of all Sis while preserving the seasonal and 10-year signals (Figure 9ch). This is a natural consequence of the method used to generate the wave series ensemble. In fact, the [START_REF] Davidson | Annual prediction of shoreline erosion and subsequent recovery[END_REF] method is designed to preserve the seasonal variability, while its application to fixed time windows of the reference time series constrains the ensemble members to maintain the 10-year variability. The black lines in Figure 9c-i show the time evolution of Sis obtained removing the seasonal signal from the model results with a 1-year running average. When the seasonal variability of the results is removed, the SLR's Si compensates the fluctuations of the model parameter's Sis, resulting in an increased trend.

The test application illustrated above suggests that including uncertainties in short-term wave chronology can significantly impact the uncertainties of shoreline projections and the relative contributions of the remaining uncertain input variables. Further, introducing uncertainties on long-term non-stationarity of wave conditions would overcome the SF limitations occurring in this specific application, and may unveil new implications of the different Y09 and SF equilibrium approaches in the context of probabilistic long-term shoreline projections.

Assumptions and limitations

Wave projections are affected by uncertainties owing to the choice of the Global Climate Model [START_REF] Morim | A global ensemble of ocean wave climate projections from CMIP5-driven models[END_REF] and random variability of wave events. Although our results are based on deterministic BW18's wave projections, in the northeast Atlantic region the estimated future wave statistics have been observed to be mostly sensitive to the RCP scenario [START_REF] Morim | A global ensemble of ocean wave climate projections from CMIP5-driven models[END_REF]). Yet, the use of deterministic wave projections hides a potentially large impact of the uncertain waveclimate variability on both shoreline predictions uncertainty and behaviour of the shoreline models.

In addition, accounting for uncertainties in wave projections may also increase the uncertainties in DoC, which were based on one deterministic wave time series in the present study. However, to the authors' knowledge there is no other dataset of continuous 2020-2100 wave projections, over the north Atlantic area, with a sufficient spatial resolution to resolve the site-specific regional scale processes. This underlines the need of continuous wave time series (obtained with different wave models of fine enough spatial resolution, different climate models, for different RCP scenarios), as well as tools allowing generating continuous realistic future wave time series, such as climate based stochastic wave emulators [START_REF] Anderson | Time-varying emulator for short and long-term analysis of coastal flood hazard potential[END_REF][START_REF] Cagigal | A multivariate, stochastic, climate-based wave emulator for shoreline change modelling[END_REF].

In the current work, we assumed that MSL 2020-2100 projections are normally distributed.

However, the MSL distribution may be skewed towards higher values due to additional uncertainty related to Antarctic ice-sheet melting in the RCP8.5 scenario. We simulated a deterministic RCP8.5 high-end SLR scenario to define a low-probability/high-impact scenario for projected shoreline erosion. Yet, our high-end SLR scenario is based on a particular combination of highend contributions to sea-level rise, which makes no consensus in the scientific community [START_REF] Bamber | Ice sheet contributions to future sea-level rise from structured expert judgment[END_REF][START_REF] Edwards | Projected land ice contributions to 21st century sea level rise[END_REF][START_REF] Stammer | Framework for high-end estimates of sea level rise for stakeholder applications[END_REF]. While this is not included in the GSA, the use of a skewed probability distribution may lead to an earlier onset of SLR uncertainties in shoreline projections.

The Bruun Rule, used in our application to estimate SLR-driven shoreline recession, builds on several strong assumptions that reduce the applicability of this model to a limited range of beaches [START_REF] Cooper | Sandy beaches can survive sea-level rise[END_REF]. As the Truc Vert is an uninterrupted natural cross-shore transport dominated beach, with large sediment availability, most underlying assumption of the Bruun model are satisfied. However, alternative models to address beach response to SLR, such as ShoreTrans [START_REF] Mccarroll | A novel rules-based shoreface translation model for predicting future coastal change: ShoreTrans[END_REF], can be implemented in this framework.

Coupling the Bruun Rule with Y09 and SF allows accounting for long-term effects of SLR while resolving short-term shoreline response to the wave climate. The Y09 and SF models do not explicitly resolve sediment sediment exchange between the different beach compartments (e.g.

upper beach and dune), and may fail reproducing episodic shoreline changes such as short-term accretion following to dune erosion events. However, if such events occur during the model calibration period, as in our applications (i.e. winter 2013-2014), their influence on the bulk shoreline response is partially accounted.

Here, we investigated the main effect of the uncertainties in input variables (Sis). While the estimated Si of the DoC remains relatively low over the simulated period, in all simulated scenarios, the interaction of DoC and SLR uncertainties (i.e. second-order Sobol' index) may have a larger impact. However, estimating robust interaction terms would require a larger ensemble of simulations (several tens of thousands). Furthermore, in order to rigorously conclude on the negligible character of some uncertainties, GSA should be conducted within the factors' fixing setting (i.e., investigating the 'total effect' of uncertain variables, [START_REF] Saltelli | Global Sensitivity Analysis: The premier[END_REF]. In the presence of dependence among the inputs, more advanced GSA indices should be used for this purpose. In particular, a method that employes the so-called Shapley effects has recently been proposed and showed very promising results [START_REF] Iooss | Shapley effects for sensitivity analysis with correlated inputs: comparisons with Sobol' indices, numerical estimation and applications[END_REF]. While the direct application of this method requires computational cost of several order of magnitudes larger than the Sobol' indices (see [START_REF] Iooss | Shapley effects for sensitivity analysis with correlated inputs: comparisons with Sobol' indices, numerical estimation and applications[END_REF], [START_REF] Broto | Variance reduction for estimation of Shapley effects and adaptation to unknown input distribution[END_REF] successfully implemented a more computationally efficient sampling-based method for GSA using Shapley indices. This may be an interesting perceptive for future works.

Conclusions

We performed a Global Sensitivity Analysis on probabilistic 2020-2100 shoreline projections at the cross-shore transport dominated Truc Vert beach in southwest France. Time varying first-order Sobol' indices were calculated for sea-level rise, depth of closure, and model free parameters for two different cross-shore shoreline models (Yates and ShoreFor) and two RCP scenarios (RCP4.5 and RCP8.5). We show that uncertainties in shoreline projections are initially driven by uncertainties in model free parameters, with the effects of SLR uncertainties only emerging in the second half of the 21 st century. However, the relative effects of SLR and model parameters uncertainties on shoreline projections do not only depend on the shoreline modelling approach and RCP scenarios, but their time evolution is also related to the forcing wave-climate variability. We also emphasize the importance of accounting for uncertainties related to the temporal distribution of wave energy, and therefore the need of ensembles of synthetic wave time series that account for the inherent variability of the wave climate.

Figure 1

 1 Figure 1 (a) Location of Truc Vert beach (green), wave hindcast grid point co-located with the CANDHIS in situ wave buoy (red), and wave projections grid point (yellow); (b) Picture of Truc Vert beach and dune landscape (photo by V. Marieu); (c) 4 km alongshore-averaged beach-dune profile from merged 2008 topo-bathymetry (submerged beach) and 2018 UAV-photogrammetry digital elevation model (emerged beach and dune); (d) Mean shoreline (1.5-m beach profile elevation proxy) positions between 2011 and 2020 derived from the bimonthly topographic surveys.

Figure 2

 2 Figure 2 Wave data offshore of Truc Vert, including: time series of Hs (black lines) and 3-month averaged Hs 2 Tp (red lines) for (a) the 1994-2020 wave hindcast from NORGAS-UG model (44°39'9" N; -1°26'48" W), and (b) RCP8.5 and (c) RCP4.5 scenarios corrected 2020-2100 Bricheno and Wolf (2018) wave projections; linear trends (solid lines) of annual (d) summer and

RCP8. 5

 5 and 4.5 2020-2100 wave series show a strong interannual modulation of incident wave energy, which is in line with current wave climate characteristics offshore of Truc Vert. Both scenarios of the BW18 wave projections show several peaks of the 3-month average Hs 2 Tp that are comparable to the 2013-2014 outstanding high-energy winter (Hs 2 Tp = 178 m 2 s) experienced at Truc Vert (Figure2a-c). For the RCP8.5 (RCP4.5) scenario, the projected 3-month average Hs 2 Tp reaches at least 90% of the2013-2014 peak in 2030, 2080, 2086, and 2099 (2060, 2068, 2073 and 2085) (Figure2b,c). While characterized by similar integrated intensity, these winters are preceded by different multi-annual energy trends, with the RCP8.5 (RCP4.5) 2080, 2086 and 2099 (2060 and 2068) winters following a positive trend of wave energy (similarly to the 2013-2014 winter),

  parameters on shoreline projections (Y) and their evolution in time for the two different modelling approaches described in Section 2.4. Identifying the main source of model results uncertainties through time is a fundamental step towards improving the reliability of long-term shoreline projections. Following D'Anna et al. (2020), we computed the Sis using the modularized samplebased approach by[START_REF] Li | An efficient modularized sample-based method to estimate the first-order Sobol' index[END_REF], which allows accounting for the statistical dependence between model free parameters, and we estimate Sis for the purpose of "Factors'

  the same assumptions as Thiéblemont et al. (2019) (see Text S2 in Supporting Information for details).

Figure 4

 4 Figure 4 Probability distributions of: relative mean sea level over the period 2020-2100, including the likely (dark shaded areas) and 5 th to 95 th percentile (light shaded areas) ranges, for (a) RCP4.5

For

  each model and RCP scenario, 3000 simulations were run with different combinations of model free parameters, DoC and SLR time series, sampled from the respective probability distributions. Shoreline change was computed with a 3-hour time step from the 1 st January 2020 to the 31 st December 2099 starting from the same shoreline position (Y0 =0), and model outputs were recorded with a 2-weeks resolution. As the characteristics of the MSL probability distribution are time-dependent, we randomly sampled percentile values and extracted the corresponding MSL at each year. The ensemble projections character was synthetized by the likely range, defined here at each time step as the variance, and the envelope (min and max) of modelled shoreline positions, acknowledging that the latter is dependent on the number of simulations and the tails of the probability distributions. The impact of individual winters on shoreline projections is qualitatively discussed observing the distributions of shoreline positions corresponding to the most seaward and landward median shoreline position within each simulated annual cycle (1 st September to 31 th

  Figure 5c,d and Figure 6c,d represent the distribution of 3000 modelled shoreline positions at each recorded output time. All scenarios show a net erosion by 2100, mostly driven by SLR (Table3). All model ensembles also show large interannual variability that is essentially enforced by the interannual variability in incident winter-mean wave height (Figure5a,b and Figure6a,b). In the RCP8.5 (RCP4.5) scenario we observe a long-term shoreline change pattern responding to alternating sequences of high-and low-energy winters with a period of ~20 years (~10 years) and even longer (Figure5a,e,f and Figure6a,e,f).

Figure

  Figure 5c,d (Figure 6c,d) show several episodes of rapid erosion driven by isolated extreme energy

Figure 5

 5 Figure 5 (a) Time series of winter mean wave height of the BW18 RCP4.5 projections (dashed line) with corresponding 5-year average (solid line); (b) BW18 RCP4.5 wave height time series (black line), and 3-month average Hs 2 Tp time series (red line); RCP4.5 scenario 2020-2100 shoreline projections at 14-days resolution obtained using (c) Y09 and (d) SF; and 5-year running mean shoreline projections modelled with (e) Y09, (f) SF, and the standalone Bruun Rule (green bars). Dark (light) blue shaded areas indicate the likely (envelope) range, i.e. variance (min-max), of shoreline position, and solid light line median shoreline position. The dashed vertical line

  ,6,e,f with Bruun model predictions in green). With the combined SF+Bruun models, the likely (envelope) ranges of shoreline positions obtained show ~10 m (~15 m) less erosion than the Bruun Rule.

Figure 6

 6 Figure 6 (a) Time series of winter mean wave height of the BW18 RCP8.5 projections (dashed line) with corresponding 5-year average (solid line); (b) BW18 RCP8.5 wave height time series (black line), and 3-month average Hs 2 Tp time series (red line); RCP8.5 scenario 2020-2100 shoreline projections at 14-days resolution obtained using (c) Y09 and (d) SF; and 5-year running mean shoreline projections modelled with (e) Y09, (f) SF, and the standalone Bruun Rule (green bars). Dark (light) shaded areas indicate the likely (envelope) range , i.e. variance (min-max), of

Figure 7

 7 Figure 7 Global Sensitivity Analysis results over the period 2020-2100 using the Yates model in the RCP4.5 (blue lines) and RCP8.5 (orange lines) scenarios. (a) RCP4.5 and (b) RCP8.5 Ensemble shoreline projections (shaded areas) over 2020-2100; First-order Sobol' index time

Figure 8

 8 Figure 8 Global Sensitivity Analysis results over the period 2020-2100 using the ShoreFor model in the RCP4.5 (blue lines) and RCP8.5 (orange lines) scenarios. (a) RCP4.5 and (b) RCP8.5 Ensemble shoreline projections (shaded areas) over 2020-2100; First-order Sobol' index time series for (c) ks + , (d) Φ, (e) sea-level rise, and (f) depth of closure, with respective linear fit (solid straight lines).

  manuscript submitted to Journal of Geophysical Research: Earth Surface 40 necessarily linear, indicating that the value of r may vary dynamically with changes in wave regimes. While the Sis of the remaining model parameters (Φ for SF; a1 and a2 for Y09) show a definite seasonality, their variability on longer time scales is unclear. However, Φ's Si, which exhibit relatively high values (up to 90%) at the beginning of the simulation, shows an overall decaying trend for both RCP scenarios applications. The a1 and a2's Sis remain weak, though not negligible, (<20%) over all the simulated period.

Figure 9

 9 Figure 9 Ensemble of 3000 Yates simulations forced using (a) 3-month average energy (Hs 2 Tp) of 100 random wave time series from 2020 to 2100 generated with the Davidson et al. (2017) method based on the BW18 wave projections for the RCP8.5 scenario; (b) Ensemble shoreline projections

Table 1

 1 Optimised combinations of cross-shore model free parameters, and respective range of variation in the probability distributions.

	Model	Model parameter	Optimised value	Distribution range
		ks + [m 1.5 s -1 W 0.5 ]	4.4 x10 -8	[2; 7.4] x10 -8
	ShoreFor	Φ [days]	1193	[400; 1423]
		ky + [m 2 s -1 /m]	0.87	[0.24 ; 2]
		ky -[ m 2 s -1 /m]	0.5	[0.1 ;1.5 ]
	Yates	a1 [m 2 /m]	-0.008	[-0.02 ; -0.004]
		a2 [m 2 ]	0.49	[0.33 ; 1]

Table 2

 2 Probabilistic future scenarios for two Representative Concentration Pathways (RCP) and two different wave-driven modelling approaches, using the Bruun Rule and 3000 different combinations of model parameters, SLR percentile and DoC.

	Future scenario	SLR-driven shoreline change	Wave-driven shoreline change	# Combinations of uncertain variables
	RCP 4.5	Bruun Rule	ShoreFor (SF)

Acknowledgments and Data

This work is co-financed by the BRGM and Make Our Planet Great Again (MOPGA) national program (Grant 927923G) and the BRGM (French Geological Survey). BC funded by Agence Nationale de la Recherche (ANR) grant number ANR-17-CE01-0014. GLC and RT are supported by H2020 ERA4CS ECLISEA (Grant 690462).

This study includes the monitoring study site of Truc Vert labelled by the Service National d'Observation (SNO) Dynalit (https://www.dynalit.fr) with additional support from Observatoire Aquitaine de l'Univers (OASU) and Observatoire de la Côte Aquitaine (OCA). The authors thank the colleagues, including S. Bujan, S. Ferreira and V. Marieu involved in the topographic data; SONEL for past sea levels and vertical land motion data (https://www.sonel.org/-GPS-.html); and the Integrated Climate Data Center at the University of Hamburg (https://icdc.cen.unihamburg.de/en/ar5-slr.html) for SLR projections data.

The authors are grateful to the reviewers (G. Coco, S. Vitousek and Anonymous reviewer), Associated Editor and Editor for their constructive remarks, which greatly improved this article.

Data availability statement

All data used (wave, mean sea level and shoreline data) and produced (shoreline projections) in this work are available at https://data.mendeley.com/datasets/gnvkx44t63/draft?a=a32c56ba-f495-4652-9eda-964dea31cec6.

Conflicts of interest

The authors declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported.