Bathymetry surveys were conducted using a single-beam echo sounder, BioSonics DT/DE Series Digital Echosounder, in the Cala Millor beach and a multi-beam echo sounder, R2Sonic2020, in Playa de Palma. The final spatial resolution is 1 m cross-shore and 2 m alongshore in Cala Millor and 0.5 m × 0.5 m in Playa de Palma. These measurements were complemented with topographies of the aerial beach obtained using a survey-grade RTK-GPS (Real Time Kinematic Global Position System) mounted in a backpack carried by a human walker. These detailed beach topo-bathymetries were surveyed under calm conditions.

In Cala Millor, nearshore hydrodynamic data were obtained from three directional Acoustic Waves and Currents (AWACs) sensors located at 8, 12 and 25 m water depths; the AWACs were deployed as part of the RISKBEACH field survey, which covered 12 March to 14 April 2014. Offshore hourly hydrodynamic data have been recovered from Capdepera buoy, located 36.45 km northeast of Cala Millor at 48 m depth (see Fig. 1 for location). The buoy has been operative during the period 1989–2014 as part of Puertos del Estado (the Spanish holding of harbours) buoys network. On the other hand, in Playa de Palma, wave data came from a coastal buoy located at 23 m depth and an acoustic Doppler current profiler (ADCP) deployed at 17 m depth, both operating since January 2012 as part of the SOCIB beach monitoring programme.

Five and fourteen video cameras are used to measure the coastline position along Cala Millor and Playa de Palma beaches, respectively. These cameras are part of the video-based coastal zone monitoring system called SIRENA developed by SOCIB and IMEDEA (Mediterranean Institute of Advanced Studies). Departing from images taken at 7.5 Hz the SIRENA system generates statistical products that provide quantitative information of hydrodynamics and morphodynamics after specific post-processing (Nieto et al., 2010). Specifically, the coastline is routinely obtained from the time image consisting of the addition of all images captured during 10 min (a total of 4500 images) and applying a post-processing of cluster classification. After applying different corrections to overcome the coarser resolution of the far-field camera images as well as rectifying the perspective projection, the coastline is georeferenced in a world coordinate system.

The processing of camera images involves two types of errors related to the intrinsic and extrinsic calibrations. After images have been optically corrected, the extrinsic calibration relates pixel position with real-world coordinates, and thus errors are associated with the georeferencing (Simarro et al., 2017). Typically, resolution ranges between 0.5 and 2 pixels for Cala Millor and from 0.5 to 5 pixels for Playa de Palma. Conversely, pixel resolution decreases with distance, but higher resolution (∼ 0.2 m) is obtained at the shore since cameras are oriented to measure this part at the centre of the image. Only pixels where errors are less than 3 m have been considered in this study. Exemplarily, the pixel resolution is added to Fig. 15 in one area with the lowest radial resolution in Playa de Palma.

With the aim of simulating the shoreline changes under given offshore conditions, the SWAN (Booij et al., 1999) and SWASH (Zijlema et al., 2011) models have been combined to resolve the wave processes from deep waters up to the swash zone. SWAN is a third-generation wave model that solves the spectral action balance equation for the propagation of wave spectra (http://swanmodel.sourceforge.net/). This model allows an accurate and computationally feasible simulation of waves in relatively large areas. On the other hand, SWASH is a phase-resolving non-hydrostatic model governed by the nonlinear shallow water equations with the addition of a vertical momentum equation and non-hydrostatic pressure in horizontal momentum equations (http://swash.sourceforge.net/). Due to its computational cost, the application of SWASH is restricted to small areas. The combination of both models allows for high-resolution and accurate results with less computational cost.

For the present study, SWAN simulations have been performed in a stationary mode over two regular nested grids. In Cala Millor, the coarser grid covers a domain of 21 km × 21 km, with its lowest left vertex at 39.53∘ N, 3.38∘ E (Fig. 3) and a resolution of 149 m × 119 m in the x and y directions, respectively. The size of the finer grid is 9.5 km × 9.5 km, with its lowest left vertex at 39.6∘ N, 3.38∘ E and a resolution of 60 m. The coarse grid in Playa de Palma covers a domain of 21.5 km × 27.7 km, with its lowest left vertex at 39.31∘ N, 2.5∘ E (Fig. 4) and a resolution of 100 m × 100 m in x and y directions. The domain of the finer grid is 13 km × 10.8 km starting at 39.47∘ N, 2.58∘ E, with a resolution of 50 m × 50 m. In all cases, the SWAN output consisted of the 2-D variance energy density spectrum and the spectral parameters of propagated wave conditions. Each output SWAN spectra corresponded to 1 h of simulation and were used as the input wave conditions of SWASH.

SWASH simulations in Cala Millor have been performed on a 1.5 km × 3.2 km rectangular grid, with its lowest left vertex at 39.57∘ N, 3.38∘ E, a resolution of 3 m × 3 m (Fig. 3) and a maximum depth at 17 m. A larger SWASH domain was required in Playa de Palma, so a 3 m × 3 m grid covering a domain of 3 km × 7 km starting at 39.47∘ N, 2.75∘ E and tilted 45∘ in order to orient the wave maker boundary parallel to the beach, at 15 m depth, was used. The SWASH simulations lasted for 30 min, with a time step of 0.05 s to keep the Courant number between 0.01 and 0.5.

The initial wave conditions imposed at the eastern boundary in Cala Millor and at the southwestern boundary in Playa de Palma corresponded to the 2-D variance energy density spectrum field provided by the corresponding SWAN simulations.

The simulations of SWAN and SWASH were performed with the same over-land extent as the width of the beaches, that is, 35–40 m in Cala Millor and 30–50 m in Playa de Palma. The SWASH model requires a rectangular computational grid, so dummy values were used behind the measured topography in order to complete the computational grid, given that the beach width is not uniform.

The final output of the model combination consists of instantaneous water level elevations in the whole domain and the position of the coastline at each time step. The shoreline simulated is obtained as a mask of wet–dry points on the computational grid, providing the limit of flooding at each time step and mesh position. More precisely, the SWASH model obtains the moving shoreline ensuring non-negative water depths for a one-dimensional case (|u|Δt)/(Δx) ≤ 1. Flooding never happens faster than one grid size per time step, which is physically correct. Thus, the calculation of the dry areas does not need any special feature.

Finally, the PETRA model is used to evaluate the changes in the beach profile under the different sea level and wave conditions, in order to assess the limitations of the assumption of the unchanged profile. PETRA is a cross-shore beach model that simulates the sediment transport along a single beach profile. It takes into account both the hydrodynamic conditions and the conservation of sand to study the response in 1-D shape. The wave conditions are computed as a phase-averaged model and the sediment transport is calculated using different formulations. The reader is referred to González et al. (2007) for detailed information of this model.

The SWAN–SWASH model set-up described in Sect. 2.4 has been run under present-day and future climate conditions in both domains. The first step aims at validating the model performance, for which the present-day runs, forced with realistic offshore waves, have been compared against measured nearshore wave parameters. In the present-day runs deep water conditions were retrieved from the SIMAR database (Pilar et al., 2008), which is a 58-year wave reanalysis generated with the WAM model (WAMDI GROUP, 1988). The reanalysis, which is freely distributed by Puertos del Estado, covers the western Mediterranean and provides 3-hourly wave data up to 2011 and hourly data since then. The two closest SIMAR grid points to each of the domains were selected to force the SWAN model for the periods of validation (as detailed later). Although this data set has already been evaluated against observations (Pilar et al., 2008; Martínez-Asensio et al., 2013, 2015), we have further compared the output with the offshore waves observed at Capdepera buoy in order to ensure the reliability of the forcing in the particular periods and locations studied here (Sect. 3.1).

The sea level rise is included simply by indicating the still water level corresponding to the sea level in the projections by 2100.

Once validated, the model set-up has been forced under future climate conditions. To so do, projected sea level rise together with changes in the wave climate have been run. A summary of the values used for sea level and waves is presented in Table 1. Regarding sea level, projections by 2100 have been estimated following Slangen et al. (2014), who provided the regional distribution of the different contributors to sea level change under two climate change scenarios, namely representative concentration pathway (RCP)45 and RCP85 (Moss et al., 2010). These are representative of moderate and large emission scenarios, respectively. Slangen et al. (2014) used an ensemble of 21 atmosphere–ocean coupled general circulation models (AOGCMs) from the Coupled Model Intercomparison Project Phase 5 (CMIP5) archive to estimate changes in ocean circulation and heat uptake contribution, atmospheric loading, land ice contribution (including all glaciers, ice caps and ice sheets on Greenland and Antarctica), groundwater depletion and mass load redistribution worldwide, together with the associated uncertainties for each term. As the regional distribution of each component was provided, we selected the Mediterranean region and averaged the sum of the components as the input for projected sea level rise. The results lead to a regional sea level rise of 48 ± 23 cm and 67 ± 31 cm by 2100 for RCP45 and RCP85, respectively. Uncertainties quoted correspond to 1σ deviation from the ensemble mean (Slangen et al., 2014). Such values are thus consistent with the widely adopted values of sea level rise and the definition of the future climate scenarios (Brunel and Sabatier, 2009; Tamisea and Mitrovica, 2011; Church et al., 2011; IPCC, 2013)

Changes in the wave climate during the 21st century have been obtained from regional wave projections over the western Mediterranean (Puertos del Estado et al., 2016). These projections were carried out using the WAM model with a spatial resolution of 1/6∘ (over the same grid as the SIMAR database) and forced with a set of dynamically-downscaled surface wind fields from AOGCMs. A total of six simulations were used, five corresponding to the A1B scenario and one to the A2 scenarios (IPCC SRES, 2000). Each projection was accompanied by a control simulation representing the climate of the last four decades of the 20th century, as it is usual practice. As the regional wave projections were computed before the adoption of the new set of RCP scenarios, for the purposes of the work, it is assumed here that the A1B (A2) scenario is equivalent to RCP45 (RCP85). One of these simulations is exemplarily represented in Fig. 5, in which the evolution of Hs under the A2 scenario is depicted for the mean regimen (Fig. 5a) and for the extremes (Fig. 5b). Changes in the mean and extreme wave regimes have been assessed by computing the differences between the values averaged over the period 2080–2100 (from the future projections, in blue in Fig. 5) and those averaged over 1980–2000 (from the control simulations, in black in Fig. 5). These differences reach 0.2 m under calm conditions and up to 0.3 during an extreme event. The obtained differences were then added to the hindcasted values from the reanalysis, which represent the best approach to the actual present-day climate (in red in Fig. 5). For each beach, the closest grid points (the same location as for the SIMAR database) were selected to simulate the future wave climate. At the point representative of the deep water wave regime of Cala Millor, the resulting values for mean Hs were 1.20 and 0.95 m for the A2 and A1B scenarios, respectively, while in Playa de Palma the values were 0.63 m and 0.65, respectively. The storm events have been assessed computing the 10-year return periods by fitting a generalized Pareto distribution to each time series. The values obtained were 4.5 and 4.2 m under the A2 and A1B climate change scenarios in Cala Millor and 4.3 and 4.4 m in Playa de Palma. Given the similarities between the two wave climate change scenarios, a single (average) value for the simulations has been used (see Table 1). Regarding the wave direction, the changes are negligible and remain unchanged in the future simulations.

In summary, six wave simulations have been carried out for each site to predict the shoreline changes under mean conditions: one for each of the two sea level rise scenarios (RCP45 and RCP85) and one for their respective upper and lower uncertainty limits (i.e. plus 1σ and minus 1σ). Please note that, for the sake of simplicity, hereinafter we will refer to ±1σ as our upper and lower uncertainty limits, respectively. In addition, four simulations have been performed for extreme conditions; due to computational constraints, we focused on the two highest sea levels for each scenario, that is, the occurrence of the 10-year return level storm occurring over the two sea level scenarios and their upper limit (i.e. for the mean value and the mean value +1σ).

As described above, the SIMAR wave reanalysis has been taken as representative of the offshore wave conditions and used to force the numerical model set-up. To illustrate its reliability, the time series at the closest grid point in Cala Millor has been compared against observations from the nearby Capdepera buoy. The time series and scatter plots of the measured and modelled statistical wave parameters (Hs, Tp, θ) are shown in Figs. 6 and 7 for a 3-month period (January–March 2014). The root mean square error (RMSE) and the correlation coefficient (ρ) between observed and modelled parameters are quoted in the figures. Results show that the hindcast agrees well with the observed Hs and Tp with correlations over 0.8 and a small RMSE. For wave direction, however, the correlation decreases down to 0.5, mostly due to the fact that the WAM resolution cannot properly resolve the coastal topography near the SIMAR location. A closer look at Fig. 6 (bottom panel) reveals that SIMAR contains waves from the NW (315∘) which are not recorded by the buoy. However, waves from the dominant directions (i.e. from N, 0∘, to SE, 135∘) are not affected, and, therefore, Hs and Tp have enough accuracy to represent the wave climate of this offshore area.

Despite the differences found in the wave direction, the advantages of using reanalysed data instead of observations for the input wave in SWAN are evident: first, the modelled time series are complete, while observations are often gappy; and second, the deep water waves can be propagated over large domains, thus providing values close to our two areas of study. Although the validation of the numerical hindcast is limited to a single grid point close to Cala Millor, previous assessments (e.g. Martínez-Asensio et al., 2013) also validated this hindcast, and it is therefore assumed that the reanalysis is equally valid for Playa de Palma.

The output of the SWAN model has been validated against observations in the two beaches. In Cala Millor the results of SWAN forced with SIMAR data were compared with nearshore wave observations during the period from 14 March to 14 April 2014 (i.e. a total of 755 h of simulation). The closest grid points of the SWAN model to each of the three directional wave ADCPs were selected. Resulting correlations, RMSEs and biases are listed in Table 2 for the three ADCPs and for the three wave parameters. Overall, the statistical parameters show good agreement between measurements and the model output, with correlations over 0.9 for Hs and Tp and over 0.7 for the wave direction. To further illustrate the model performance, observed and modelled time series are plotted in Figs. 8 and 9. Both reflect the ability of the model to capture the magnitude and variability of nearshore waves. Nevertheless, during the storm events recorded (as in 28 March), the model underestimates the observed Hs by up to 30 cm.

In Playa de Palma, the simulated waves have been compared with the observations from a buoy moored at 23 m depth and with an ADCP at 17 m depth for the period from 1 to 30 September 2015 (i.e. a total of 720 h of simulation). The results are summarized in Table 3 and the time series are plotted in Figs. 10 and 11. Like in Cala Millor, there is a good agreement in Hs with correlations over 0.9. For Tp, however, observations display higher variability than modelled data, which makes the correlations drop to 0.32–0.34 and the bias reach 0.4–0.6 s for the buoy ADCP, respectively (see Fig. 11). Possible reasons for this discrepancy are the instrumental noise in measurements and/or the influence of local wind within the SWAN domain. The differences between observed and modelled wave directions are also larger than in Cala Millor, with non-significant correlations. The reason for the discrepancies in wave direction is probably the inability of the model to accurately represent the wave diffraction occurring at the SE of the bay of the Playa de Palma, where the buoy and the ADCP are located. This area is protected by a headland (see Fig. 4) that may cause worse results in wave direction.

A total of four and three simulations have been carried out with the SWASH model for the Cala Millor and Playa de Palma beaches, respectively, in order to validate the model results with measurements of shoreline positions. The dates chosen for the validation correspond to dates in which the video monitoring provided good quality images, with them being also close to the dates when the bathymetry surveys were performed (they are listed in Tables 4 and 5). Wave makers were defined at the eastern boundary of the SWASH model domain in Cala Millor and at the southwestern boundary in Playa de Palma, in both cases with the SWAN wave conditions. These input wave conditions for the validation process are specified in Tables 4 and 5 for the indicated dates.

Observed and modelled shoreline changes for each case study have been compared in Figs. 12 and 13 along the two beaches. Results show that the modelled shorelines line up with observations in all cases. In Cala Millor the agreement is better in the central part of the beach, while some differences are found in the northern and southernmost sector. It is important to remark that images obtained from the beach cameras are increasingly uncertain with the distance from the cameras (Sect. 2.3 for details). Therefore, part of the difference between the measured and simulated shoreline at the ends may come from this error in measurements. In Playa de Palma only the area between 39.51 and 39.53∘ N is used for the comparison as this is the stretch of the shoreline where the video system has the requested quality. We will also restrict the discussion on future projections to this sector.

The RMSEs and biases between observations and model results have been calculated for each case and are listed in Tables 4 and 5. These statistics must be set in a proper context in order to evaluate how good the model performance is. To do so, the temporal variability of the shoreline position has been estimated as the standard deviation (cross-shore) at each alongshore position for which 10 coastlines measured from video monitoring have been used. In Cala Millor, higher variability is observed, calculated between April and May 2014, in the central part of the beach (mean value of 8.4 m) and lower towards the ends, with a mean value along the entire beach of 5.5 m. Figure 14 shows the shorelines simulated for the case studies (red lines), the corresponding measured shorelines (blue lines) and the variability of the shoreline (grey area), zoomed in around an area at the centre of the beach. In the case of Playa de Palma, the shoreline displays a cross-shore variability of 6 m in the area around the centre of the beach and lower at the extremes, with a mean value of 3 m, as calculated with observations between August and October 2014. The results are plotted in Fig. 15 in which again the central area has been zoomed in, in order to highlight the differences. Notably, the modelled shorelines are very similar to each other, because the forcing is also similar in the three case studies.