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Equilibrium shoreline modelling of a high-energy meso-macrotidal multiple-barred beach
Marine Geology 347 (2014) 85–94
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Marine Geology journal homepage: www.elsevier.com/locate/margeo
Equilibrium shoreline modelling of a high-energy meso-macrotidal multiple-barred beach Bruno Castelle ⁎, Vincent Marieu, Stéphane Bujan, Sophie Ferreira, Jean-Paul Parisot, Sylvain Capo, Nadia Sénéchal, Thomas Chouzenoux CNRS, UMR EPOC, Université Bordeaux 1, France
a r t i c l e
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Article history: Received 1 August 2013 Received in revised form 18 October 2013 Accepted 4 November 2013 Available online 14 November 2013 Communicated by J.T. Wells Keywords: empirical shoreline model system memory shoreline proxy meso-macrotidal beach
a b s t r a c t 8-year time series of incident wave energy and monthly alongshore-averaged beach surveys at the high-energy meso-macrotidal multiple-barred Truc Vert beach are analysed. We apply two behaviour-oriented equilibrium shoreline models that relate the rate of cross-shore shoreline displacement to the wave energy and the wave energy disequilibrium between the wave energy and the equilibrium wave energy that would cause no change to the present shoreline location. The two models show similar skill. Results show that the equilibrium shoreline model concept works on meso-macrotidal multiple-barred beaches, with similar skill when applied to nonbarred and/or micro-mesotidal beaches, provided that a relevant shoreline proxy is used. Simulations show that Truc Vert beach responds predominantly at seasonal timescales rather than at individual storm frequency. The first winter storms drive the most pronounced erosion events because both the wave energy disequilibrium and erosion change potential are large. The best shoreline proxy at Truc Vert beach is the mean high water level, where the inner-bar and berm dynamics have little influence on the shoreline cross-shore displacement. Implications for shoreline monitoring through video imagery on this type of beach are discussed. Results also reveal that the equilibrium shoreline concept can be extended to an equilibrium beach profile concept pending further improvements. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Shoreline position along wave-dominated sandy coasts varies over a wide range of temporal and spatial scales in response to a variety of processes (Stive et al., 2002). Along non-engineered sandy coasts, longterm climate and sea level change, underlying geological settings and sediment budget are key parameters influencing changes in shoreline position on timescales of decades to millennia (Stive et al., 1990). On shorter timescales, from years and decades down to days and even hours for single storms, changes in wave energy arriving at the coast are the dominant process impacting shoreline change with both crossshore and alongshore surf and swash sediment transport processes dictating changes in shoreline position. On most of open coasts, alongshore processes typically act on longer timescales than cross-shore processes and do not dominate the annual shoreline variability (e.g. Davidson and Turner, 2009; Yates et al., 2009; Hansen and Barnard, 2010). Over the last decades, a number of complex process-based models have been developed to simulate and further predict beach changes (e.g. De Vriend et al., 1993; Roelvink and Broker, 1993; Nicholson et al., 1997; Roelvink et al., 2009). Yet, computational costs and misspecifications of the physics and boundary conditions prevent them from properly predicting shoreline evolution on long timescales ⁎ Corresponding author. E-mail address: [email protected] (B. Castelle). 0025-3227/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.margeo.2013.11.003
(i.e. years to decades). For instance, sheet flow sediment transport processes (e.g. Calantoni and Puleo, 2006), sand stirring accounting for breaking-wave-induced turbulence as a surface boundary conditions (e.g. Grasso et al., 2012) or swash zone sediment transport processes (Masselink and Puleo, 2006) are not properly considered in coastalevolution models. These misspecifications of the physics typically cascade up through the scales resulting in an inescapable build-up of errors and unreliable simulations. Only recently, coastal profile models succeeded in simulating surfzone beach profile evolution on timescales of days to months and sometimes years with fair accuracy (e.g. Ruessink et al., 2007; Kuriyama et al., 2012; Walstra et al., 2012). Yet, these models do not simulate the evolution of the beach face, and therefore the shoreline, as they ignore swash zone processes. Instead, empirically-based behaviour-oriented models can lead to more reliable long-term evolution than do parameterizations of much smaller-scale processes in process-based models, as evidenced in many geomorphological systems (Murray, 2007). Equilibrium beach response concepts have been used to model the evolution of beach profiles (Larson and Kraus, 1989) and the longterm evolution of cross-shore sandbar location (Plant et al., 1999) as well as that of the alongshore variability of the sandbar (Splinter et al., 2011). The fundamental assumption is that the present beach state is determined by the recent history of both the wave field and the beach morphology (Wright and Short, 1984). Using this concept, a wealth of equilibrium-based empirical shoreline models have recently been
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developed that are capable of simulating shoreline behaviour on timescales of days to years at a number of cross-shore dominated sandy beaches (Miller and Dean, 2004; Davidson and Turner, 2009; Frazer et al., 2009; Yates et al., 2009; Davidson et al., 2010; Yates et al., 2011; Kuriyama et al., 2012; Long and Plant, 2012; Davidson et al., 2013; Splinter et al., 2013). In essence, in all models the rate of cross-shore shoreline change is both a function of the disequilibrium (the difference between the present and equilibrium condition) and the magnitude of forcing available to move the sand with a time-varying forcing term outside the disequilibrium term limiting non-realistic sediment transport during small waves (for an overview of existing model formulations, see Splinter et al., 2013). These models can be forced by the incident wave energy (e.g. Yates et al., 2011), the incident wave power, the Dean parameter (Gourlay, 1968) thus also accounting for sediment size and wave period (e.g. Davidson et al., 2013) or the cross-shore component of the radiation stress (Yates et al., 2009). Overall, provided accurate surveys spanning a duration of at least two years sampled monthly, equilibrium shoreline models were found to predict shoreline changes on timescales of years with good accuracy (e.g. Splinter et al., 2013), therefore opening new perspectives of application (Davidson et al., 2013). To date, empirical shoreline models have been tested on micromesotidal sandy beaches, but their validity on meso-macrotidal multiple-barred beach has never been addressed. As an idealized definition of shoreline is that it coincides with the physical interface of land and water, the intersection of a tidal datum with the coastal profile is commonly used as a proxy of shoreline position (see Boak and Turner, 2005, for a review on shoreline definition and detection). On micromesotidal beaches, the intersection with Mean Sea Level (MSL) is commonly defined as the shoreline (Yates et al., 2009; Davidson et al., 2013). Alternatively, mean high water level (MHWL) is chosen when the MSL contour is underwater and therefore inaccessible during a significant number of beach surveys (Stockdon et al., 2002; Yates et al., 2011). Detailed assessment of the best shoreline proxy to use was not performed in previous behaviour-oriented shoreline modelling studies. On meso-macrotidal beaches where the cross-shore length of the intertidal domain is typically of the order of hundreds of metres (see for instance Masselink, 2004), shoreline definition is in practice even more blurred. In this paper, we analysed (Section 2) 8 years of data of beach morphology and iso-contour evolution at the meso-macrotidal high-energy multiple-barred beach of Truc Vert (SW France). We use two simple equilibrium shoreline models (Section 3) to explore shoreline response at Truc Vert Beach. Results are presented in Section 4 and further discussed in Section 5 before conclusions are drawn. The modelling effort provides insight into the overall shoreline response at Truc Vert. Results also show that the equilibrium shoreline concept works on a high-energy meso-macrotidal multiple-barred beach. Testing most of the existing shoreline definitions, we find that the best shoreline proxy at Truc Vert beach is roughly MHWL. Results also reveal that the equilibrium shoreline concept can be extended to an equilibrium beach profile concept. 2. Study site and data 2.1. Study site Truc Vert beach is located in SW France (Fig. 1). Truc Vert beach is meso-macrotidal with a maximum spring tide range of 4.8 m. The wave climate is energetic, strongly seasonally modulated, with an annual mean significant wave height Hs of 1.36 m, a mean period around 6.5 s and a dominant WNW incidence (Butel et al., 2002). Overall, summer conditions are dominated by short waves from the NW whilst winter conditions are characterized by long-period higher waves coming from the W and WNW sectors. Significant wave height occasionally exceeds 10 m during severe storms with peak wave periods often larger
Fig. 1. Map of SW France showing locations of Truc Vert beach and that of (grey circle) the Candhis directional wave buoy and (black) the WW3 virtual buoy.
than 15 s. The sediment consists of fine to medium quartz sand with mean grain of about 350–400 μm with large spatial variations (200– 700 μm over very short distances of 5 m) linked to the presence of various bedforms (megaripples, cusps and rip channels, see Gallagher et al., 2011). The morphology of the beach is complex, mostly double-barred, three-dimensional and highly dynamic itepalmar10 with ubiquitous rips (e.g. Bruneau et al., 2011). The intertidal inner bar has most of the time a transverse bar/rip morphology (Sénéchal et al., 2009). The subtidal outer bar is most of the time crescentic with an average wavelength of about 700 m (Castelle et al., 2007). Both sandbars migrate southward as a result of a southerly longshore drift of about 4 × 105 m3/year (Idier et al., 2013). 2.2. Wave data We use about 11.5 years of continuous wave outputs gathered from the Wavewatch III model (Tolman, 1991) at every 3 h since January 2002 at the grid point 1° 30′ W, 44° 30′ N in about 70-m depth (WW3 point in Fig. 1). Knowledge of wave conditions prior to the first topographic survey was required to drive one of the behaviour-oriented models that needs the antecedent wave forcing to account for beach history. To improve the accuracy of the wave forcing, we use in-situ measurements collected by the Candhis buoy located further onshore and closer to Truc Vert beach in 54-m depth (1° 26.8′ W, 44° 39.15′ N in Fig. 1). Because of a number of buoy malfunctions, wave data were collected intermittently since January 2002 until April 2013 resulting in a total of about 5 years of wave data. Fig. 2a shows the comparison between the modelled (HsWW3) and measured (HsCandhis) significant
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Fig. 2. (a) Scatter diagram of measured significant wave height at the Candhis buoy HsCandhis versus 3-hour modelled significant wave height HsWW3, with the solid black line indicating the linear regression giving HsCandhis = −0.01526 + 0.9052HsWW3. (b) Resulting scattered directional plot of significant wave height HsCandhis for the entire dataset. The solid and dashed black lines indicate mean shoreline orientation and shore-normal incidence at Truc Vert, respectively. The dashed grey lines indicate groundswell shadowing regions to the North and to the South behind Brittany and Galicia, respectively (see Fig. 1). In all panels colorbar indicates peak wave period in seconds.
wave heights. Using linear regression, we use the corrected significant wave height Hs = −0.01526 + 0.9052HsWW3. The overall directional wave climate at Truc Vert is given in Fig. 2b, showing wind seas at a wide range of direction and high-energy swells from the W–NW sector. 2.3. Sand level surveys and equilibrium change observations Monthly or bimonthly topographic surveys have been acquired from September 2003 until April 2013, with a 1-year gap in 2008. Topographic surveys are conducted at low tide during spring tides using a DGPS (Trimble 5700) with an accuracy of about 2.5 cm in the horizontal and 10 cm in the vertical. Sand level elevation is referenced to benchmarks of the French National Geodesic Service (NGF-IGN 69) with the 0-contour corresponding to MSL. The distance between each transect varies overall from 20 to 40 m. Most of the time, the inner bar was not entirely surveyed in the cross-shore direction as the water level was too high (wave set-up) and/or the bar crest was too deep. The alongshore coverage of the topographic surveys was about 350 m from 2003 to 2008, 750 m from 2008 to 2012 and about 1200 m from 2012 to present (see Fig. 3). Based on a detailed inspection of all the available topographic surveys, data collected prior to April 2005 were disregarded either because of vertical referencing issues, ubiquity of spurious data points or malfunction of the DGPS. Since April 2005, 8 additional surveys were removed from the dataset for the same reasons. In total, beach morphology was sampled 121 times between April 8, 2005 and March 23, 2013. This equates to, on average, a beach topography surveyed nearly once every month, although the interval between successive observations is irregular with a notable 1-year gap in 2008. Spatial interpolation of the scattered DGPS measurements of sand level is required to both generate a digital elevation model on a given grid and further compute the alongshore-averaged beach profile. After a detailed assessment of the commonly used interpolation methods on our dataset, anisotropic kriging was chosen. Isotropic methods typically resulted in the formation of cuspate features, with a spacing equalling to that of the transects, which were increasingly developed for increasing cross-shorebeach slope. This ultimately resulted in a slight smoothing of the bar and berm features when computing the alongshore-averaged beach profile. Anisotropic kriging was also preferred to prevent spurious interpolations for occasional highly irregular survey transects (in both the cross-shore and alongshore directions) due to strongly alongshore non-uniform sandbar patterns (see for instance Fig. 4a). Interpolation was performed on a regular grid with an alongshore and cross-shore mesh size of 20 m and 2 m, respectively, with an anisotropy ratio and angle of 10 and 90° in beach coordinates, respectively.
Fig. 3. Truc Vert Beach bathymetry surveyed on April 6, 2008. The 0-m iso-contour (thick black line) stands for the Mean Sea Level (MSL). The black dashed box shows the area of interest with indication of the typical alongshore extension of the topographic surveys during the periods 2004–2008, 2008–2012 and 2012–2013.
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Fig. 4. (a,b,c) Examples of typical beach surveys at Truc Vert Beach with colorbar indicating elevation in metres and (d,e,f) corresponding alongshore-averaged beach profile with ±1 standard deviation of iso-contours at a 1-m interval (denoted by the horizontal error bar). (a,d): Low tide terrace morphology with a weakly-developed alongshore-uniform berm, a runnel at x ≈ 170 m and two narrow shallow rips with the small white points indicating the survey points, (b, e) bar and rip morphology with a well-developed cuspate berm and two deep rip channels, (c, f): reasonably alongshore-uniform concave beach face shape profile typical of a longshore bar trough beach state. In the latter situation the inner sandbar is deep and located farther offshore and was therefore not surveyed. In (d, e, f) the Lowest Astronomical Tide (LAT) level, Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level are indicated.
Fig. 4 shows examples of three representative beach topographies of Truc Vert beach states with: (1) a low tide terrace, typically found in summer after a long period of fair weather conditions, with an alongshore intertidal bar attached to the beach cut by shallow and narrow rip channels (Fig. 4a); (2) a bar and rip morphology, which is the modal beach state at Truc Vert (58% of the time, Sénéchal et al., 2009), with deep rip channels and a well-developed alongshore non-uniform berm due to protruding megacusp horns attaching to the bar (Fig. 4b); (3) an alongshore-uniform concave beachface shape profile typical of a longshore bar trough beach state following a storm event with the bar located too far off the beach and too deep to be surveyed (Fig. 4c). The corresponding mean beach profiles are given in the bottom panels of Fig. 4 with superimposed ± 1 standard deviation of isocontours at a 1-m interval denoted by the horizontal error bar. The latter gives a measure of how representative the mean cross-shore position of a given iso-contour is. Fig. 5 shows the superimposed mean beach profiles surveyed at Truc Vert Beach since April 2005 revealing a large variability, particularly at 100 m b x b 150 m where berm formation and dynamics are preferably observed. To address what the best shoreline proxy z is at Truc Vert beach, we computed the time evolution of the mean cross-shore position of all the iso-contours from −1 m to 8 m at a 0.1 m interval. Fig. 6 shows example of time series of measured shoreline cross-shore position for the proxies z = 0, 2 m, 4 m and 6 m with corresponding time series of corrected significant wave height Hs. A visual inspection of the time series indicates that Hs is highly variable over seasonal timescales with
moderate energy waves during summer (Hs b 2 m) and common high-energy storm waves (Hs N 4 m) during autumn and winter (Fig. 6a). Seasonal cycles are also clearly seen in the shoreline position for proxies z = 2 m and 4 m with typical amplitude of about 50 m and 30 m, respectively. In contrast, the iso-contour z = 6 m barely
Fig. 5. Superimposed mean beach profiles surveyed at Truc Vert Beach from April 2005 to April 2013 with indication of the Lowest Astronomical Tide (LAT) level, Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level.
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Fig. 6. Time series of (a) corrected significant wave height Hs and (b) measured mean shoreline cross-shore position for the proxies z = 0, 2 m, 4 m and 6 m with ±1 cross-shore standard deviation indicated by the vertical error bars.
evolves throughout the seasons whilst iso-contour z = 0 shows large variations although no clear cycles are seen with anomalous changes not associated with corresponding fluctuations in incoming wave height. Time series of shoreline cross-shore positions S(z) and Hs were used to constrain the equilibrium shoreline model. 3. Equilibrium models Alongshore gradients in sediment transport around Truc Vert beach are small (Idier et al., 2013). Accordingly, it can be assumed that shoreline variability due to gradients in alongshore transport is small compared to the variability associated with cross-shore processes. Given that improving existing equilibrium shoreline models is not the purpose of this paper, here we use 2 existing model concepts that are briefly described below. Wave energy E is used to force the 2 models as it was found to give slightly better results than other wave characteristics (Yates et al., 2009).
assumes that the equilibrium wave energy is a linear function of the current shoreline position S: Eeq ðSÞ ¼ aS þ b
where a and b are 2 free model parameters, meaning that the equilibrium wave energy that would cause no change in shoreline position essentially depends on the current shoreline position. The second approach is based on the work of Davidson et al. (2013), Splinter et al. (2013) and is hereafter referred to as the DA13 model. The major difference with YA09 model is that, instead of computing Eeq as a function of the current shoreline position S, DA13 models computes Eeq from the antecedent wave forcing, thus explicitly reflecting the degree of observed memory of the system. To be consistent with YA09 approach, we use E both instead of the dimensionless fall velocity Ω and incident wave power P in Davidson et al. (2013), Splinter et al. (2013). Accordingly, the equilibrium wave energy reads: 2Φ=Δt X
3.1. Model formulations Behaviour-oriented equilibrium shoreline models relate the rate of cross-shore shoreline displacement dS/dt to the, typically hourly, wave energy E and the wave energy disequilibrium ΔE between E and the equilibrium wave energy Eeq that would cause no change to the present shoreline location. The shoreline displacement rate is typically assumed proportional to both E and ΔE multiplied by a constant change rate coefficient: dS 1=2 ¼ C E ΔE dt
ð1Þ
where, for a given shoreline S(z), C± are change rate coefficients for accretion (C+ for ΔE b 0) and erosion (C− for ΔE N 0), with the energy disequilibrium ΔE = E − Eeq. From this point, 2 types of approaches are used. The first one is based on the work of Yates et al. (2009, 2011), Long and Plant (2012) and is hereafter referred to as YA09 model. This simple and efficient model
ð2Þ
Eeq ðt Þ ¼
− jΔt=Φ
E j 10
j¼0 2Φ=Δt X
ð3Þ j¼0
j¼0
with j the number of data points in the wave forcing time series prior to the calculation point (j = 0); Δt the sampling interval in days and Φ the memory decay of the system a free model parameters in days meaning that Φ is the number of days in the past when the exponentially decaying weighting factor decreases to 10%, with 2Φ taken as a limit for Eeq computations (Davidson et al., 2013). 3.2. Model fit and skill For each shoreline proxy z, the temporal mean of shoreline position S was removed, yielding time series of shoreline position fluctuations about the mean. YA09 and DA13 models have 4 and 3 free model parameters, respectively. For each simulation, we also added one free
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parameter, d, to allow for S(t = 0) not to inevitably equals to the initially measured cross-shore distance of the shoreline proxy. This was a necessary requirement as the initial profile exhibited a berm, hence limiting the validity of the shoreline position for proxies within the upper profile. In addition, setting d = 0 artificially gives more weight to the initial survey in the model hindcast. This is an issue herein also because the alongshore coverage of the first topographic survey was short (350 m), thus resulting in a less precise alongshore-averaged beach profile than for the subsequent surveys. For each model and each shoreline proxy z, we used a probabilistic optimization method, the so-called Simulated Annealing algorithm (Bertsimas and Tsitsiklis, 1993), to solve for the 4 or 5 free model parameters that minimize the root-meansquare error (RMSE) between the model and observations. Simulated annealing, overcoming local RMSE minima in this nonlinear system, searched within a realistic bounded range of the parameters: for YA09 model − 0.1 m2/m b a b 0; 0 b b b 1 m2; − 2 m·hr− 1/m3 b C+ b 0; − 2 m·hr− 1/m3 b C− b 0; and for DA13 model: 0 b Φ b 1000 days; −50 m·hr−1/m3 b C+ b 0; −50 m·hr−1/m3 b C− b 0; and −15 m b d b 15 m for both YA09 and DA13 models. For each simulation, the model skill was addressed through the R-squared (coefficient of determination) denoted R2 between modelled and observed shoreline position. Given that the same method was systematically performed for all shoreline proxies (−1 m b z b 8 m at a 1-m interval), we obtained the best fit free model parameters a(z), b(z), C+(z) and C−(z) for YA09 model and Φ(z), C+(z) and C−(z) for DA13 model. 4. Results Fig. 7 shows the time evolution of modelled beach profiles using YA09 model (Fig. 7b) and DA13 model (Fig. 7c) together with the time evolution of the shoreline for 4 proxies z = 0, 2 m, 4 m and 6 m. The modelled position of all shoreline proxies shows strong seasonal variation, with slow accretion for long periods of low-energy waves,
and faster erosion during episodic, high-energy wave events. Consistent with the overall measured beach profile shapes (Fig. 5), steep upper and gentle sloping lower beach profiles are simulated. Interestingly enough, the model successfully reproduces the overall erosion trend from 2005 to 2009 and the subsequent overall accretion trend from 2010 onwards, suggesting that this long-term erosion/accretion sequence has been essentially driven by cross-shore processes. Addressing shoreline evolution for the proxy z = 2 m shows that the shoreline accretes 86% and 70% of the time using YA09 and DA13 model, respectively. This difference is due to the modelled equilibrium wave energy Eeq that is about twice larger for YA09 than for DA13 model (Fig. 7a). Surprisingly enough, despite the fact that Eeq is computed using current shoreline position and antecedent wave forcing in the YA09 and DA13 models, respectively, Eeq in YA09 and Eeq in DA13 are very similar in pattern (Fig. 7a). In line with previous works (e.g. Yates et al., 2009) the first winter storms, typically in December/January at Truc Vert, drive the most pronounced erosion events because the wave energy disequilibrium ΔE and erosion change potential E are large (Fig. 7). Even if Truc Vert beach is often exposed to severe storms in February/March, they do not significantly erode the beach as ΔE is smaller due to larger Eeq values, which results in a smaller change potential |E1/2ΔE|. Instead, even if wave energy E is often elevated through early spring, overall accretion typically occurs. A notable exception in this 8-year time series is the winter 2011 that was characterized by the absence of severe storm (Fig. 7), thus triggering the overall accretion trend from 2010 onwards. There is large variability in the ability of the model to simulate shoreline evolution depending on the shoreline proxy z. This is further emphasized in Fig. 8, which shows both the root-mean-square error (RMSE) and the squared correlation R2 for both models as a function of the considered shoreline proxy. Results show that YA09 and DA13 models have very similar skills with, for instance, RMSE ≈ 8 m and R2 ≈ 0.55–0.65 at 1 m b z b 4 m. YA09 model explains slightly more of the observed shoreline variability for z b 1 m than DA13 model.
Fig. 7. (a) Observed wave energy E (thin black line) and modelled equilibrium wave energy Eeq (thick lines) for the shoreline proxy z = 2 m for the YA09 model (grey) and the DA13 model (black). Modelled shoreline positions for all shoreline proxies −1 m b z b 8 m at a 0.1-m interval for (b) YA09 model and (c) DA13 model. In (b,c) the time evolution of the shoreline position for the 4 proxies z = 0, 2 m, 4 m and 6 m and corresponding measurements are superimposed as a black line.
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Fig. 8. R-squared R2 (black) and root-mean-square error RMSE (grey) between the observed shoreline position and that modelled with YA09 model (squares) and DA13 model (circles) as a function of the shoreline proxy z. The black dotted lines indicate the Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level. The horizontal grey dotted line indicates for which shoreline proxy the models give the best agreement with field data.
DA13 and YA09 model skill rapidly degrades for both z N 4 m and z b 0 (grey areas visually determined in Fig. 8). This variability in model skill is further emphasized below. The observed and modelled (using optimal model parameters) shoreline position for 5 representative shoreline proxies (z = 0, 0.4 m, 1.5 m, 3 m and 6 m) are shown in Fig. 9. These 5 elevations were selected because they are commonly used shoreline proxies (z = 0 is MSL, z = 0.4 m is the lowest high tide level commonly used through video imagery to get daily shoreline data and z = 1.5 m is MHWL) or because they correspond to a particular compartment of the beach (z = 3 m is the typical berm elevation and the iso-contour z = 6 m is close to the dune foot and barely moves). These isocontours described below are representative of the surrounding isocontours as well. Using the proxy z = 0 corresponding to MSL (Fig. 9a), measured shoreline position shows large-amplitude variations which are not always readily related to changes in incident wave energy. This is because this part of the profile is both very gently sloping and strongly influenced by the inner-bar dynamics. Given that the crossshore inner-bar dynamics at Truc Vert beach is influenced largely by tide-range variations (Almar et al., 2010), this is not captured by the model that is forced by the time-varying incident wave conditions only. This results in large model errors. A similar comment can be drawn for the proxy z = 0.4 m (Fig. 9b), corresponding to the lowest high tide level, which is a commonly used proxy to address shoreline position through video imagery on meso-macrotidal beaches (e.g. Birrien et al., 2013), although seasonal cycles are relatively clearer in the measured shoreline positions than for z = 0. RMSE exceeds 10 m with both models, with DA13 model explaining less than 35% of the observed shoreline variability. As shown in Fig. 8, best model results are obtained using roughly the proxy z = 1.5 m, corresponding to MHWL. For this proxy, the model simulates shoreline evolution with fair accuracy (Fig. 9c) as it explains 66% and 62% of the observed shoreline variability with a RMSE of 7.5 m and 8 m using YA09 and DA13 approaches, respectively. Model behaviour is slightly different for z = 3 m corresponding to the typical berm elevation (Fig. 9d) as spurious rapid accretion sometimes occur (see for instance the autumns of 2005 and 2011 in Fig. 9d). These anomalous changes not associated with
Fig. 9. Time series of the observed (circles with ±1 cross-shore standard deviation indicated by the vertical error bars) and modelled shoreline position using YA09 (thick grey line) and DA13 (thick black line) models for representative shoreline proxies: (a) z = 0, (b) z = 0.4 m, (c) z = 1.5 m, (d) z = 3 m and (e) z = 6 m.
corresponding fluctuations in wave energy, and therefore not captured by the equilibrium model, are related to rapid berm formation during abnormally low-energy autumn waves. For the proxy z = 6 m close to the dune foot (Fig. 9e), model fluctuations drastically decrease down to less than 5 m.
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5. Discussion and concluding remarks Results show that the equilibrium shoreline model concept works on a high-energy meso-macrotidal multiple-barred beach. Although Eeq is computed using current shoreline position and antecedent wave forcing in the YA09 and DA13 models, respectively, the differences between the 2 models are rather subtle (Figs. 7 and 9). A notable difference is that with the DA13 model the shoreline accretion tends to slightly level off at the end of the summers whilst the shoreline continues to accrete at a steady rate with YA09 model (see Fig. 9). This is because Eeq is larger for YA09 model than for DA13 model (see Fig. 7a for z = 2 m). This, in turn, makes the shoreline (z = 2 m) accrete 86% and 70% of the time using YA09 and DA13 models, respectively, resulting in a slight levelling off of shoreline accretion with DA13 model. Detailed high-frequency topographic surveys in summer would help in determining which model gives the best estimation of erosion/accretion proportions. For the best shoreline proxy z = 1.5 m, a RMSE and a R2 of about 8 m and 0.65 are obtained, respectively, for both YA09 and DA13 models. Model skill is therefore similar to that obtained both at Narrabeen and the Gold Coast in Australia (Davidson et al., 2013) but slightly lower than that on California beaches (Yates et al., 2009, 2011). A more detailed inspection of model skill shows that R2 increases up to (decreases down to) 0.75 (0.5) if addressing model performance after (before) 2009 only. This suggests that the 350-m alongshore coverage of the topographic surveys prior to 2009 was too short to get rid of the influence of the inner bar and rip alongshore variability on the mean profile characteristics. Since 2009, the alongshore coverage exceeding two rip spacings results in much more accurate alongshoreaveraged shoreline determination. It is also expected that model skill would have been further improved if performing model hindcast since 2009 only. Results suggest that both YA09 and DA13 models predict the highest percentage of the total measured shoreline variability using roughly the shoreline proxy z = 1.5 m, corresponding to MHWL at Truc Vert. Assuming that shoreline evolution at Truc Vert beach is essentially driven by cross-shore processes which are captured by the model, this suggests that the intersection of the coastal profile with the vertical elevation MHWL should be used as shoreline proxy at Truc Vert beach. It remains
unknown to what extent this statement applies for other mesomacrotidal beaches. Further work in similar wave-dominated environments is necessary to test the generality of our findings. This can have major implication for video monitoring of meso-macrotidal beaches as the lowest high tide level is most of the time used in order to have daily shoreline measurements (e.g. Birrien et al., 2013). Instead, time exposure images at MHWL should be used to have an accurate proxy of shoreline position. Despite time-exposure video images at MHWL will not give daily shoreline information, it is enough to accurately drive equilibrium shoreline models (Splinter et al., 2013). The variability in the best-fit model parameters as a function of the proxy z is addressed in Figs. 10 and 11 for YA09 and DA13 models, respectively. Fig. 10 shows that there is a high but coherent vertical variability in the YA09 model optimal free parameters at 0 b z b 4 m. As indicated in Yates et al. (2009), a and C± are related and can have compensating physical effects, whilst b (Fig. 9b) depends on the temporal mean removed from the shoreline time series and has therefore limited physical meaning. a increases with increasing z (Fig. 9a) meaning that, when going up along the profile, beach width decreasingly influences the rate at which the beach erodes or accretes. |C−| decreases with increasing z (Fig. 9d) meaning that, for a given wave energy, shoreline erodes less rapidly in the upper profile. Whilst the evolutions of a and C− are not surprising, that of C+ is more intriguing with |C+| peaking at z ≈ 3 m (Fig. 9c). This corresponds to the typical berm elevation (Fig. 3). The maximized |C+| at z ≈ 3 m therefore characterizes the berm formation that builds up rapidly during prolonged period of fair weather conditions. Both |C+| and |C−| decreasing for increasing z for z N 3 m is the signature of the decreasing residence time of swash and surf processes when going up along the profile during the tide. Fig. 11 shows the same plot but for DA13 model. Along most of the beach profile, large Φ values are found with systematically Φ N 200 days (Fig. 11a), meaning that Eeq is suppressed to a weak seasonal signal value close to the time-series mean. This suggests that Truc Vert beach responds predominantly at seasonal timescales rather than at individual storm frequency, that is, a similar behaviour to that observed on the Gold Coast (Davidson et al., 2013). This matches the dominant seasonal mode of observed shoreline variability at Truc Vert (Fig. 6b). We hypothesize that the local minimum of Φ at z ≈ 3 m is the signature of the berm dynamics that was captured in the YA09
Fig. 10. Optimal model free parameters as a function of the considered shoreline proxy z for the YA09 model: (a) equilibrium slope a and (b) b the y intercept of the equilibrium wave energy linear function, (c) accretion rate coefficient C+, and (c) erosion rate coefficient C−. In all panels the black dotted lines indicate the Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level.
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Fig. 11. Optimal model free parameters as a function of the considered shoreline proxy z for the DA13 model: (a) Φ the memory decay in days, (b) accretion rate coefficient C+, and (c) erosion rate coefficient C−.
model (Fig. 10b). C+ (Fig. 11b) and C− (Fig. 11c) show very similar patterns with an order of magnitude of difference. The large variability in the best-fit model parameters depending on the considered elevation contour has important implications. For instance, Yates et al. (2011) compare shoreline model coefficients obtained at 4 Californian beaches, suggesting that the differences are related to the grain size and that the coefficients can be transportable between sites with similar grain size and different wave climates. They suspect that the reduced mobility of the coarser sand at Ocean Beach explains the lower erosion rate coefficients |C−| found at this site. Yet, MHWL is used at Ocean Beach whilst MSL is used for the 3 other Californian beaches. Fig. 10b shows that |C−| decreases with increasing isocontour elevation, thus suggesting that the differences in |C−| in Yates et al. (2011) can be attributed to the considered shoreline proxy instead of the grain size. These differences are more likely related to a combination of both. Using the same shoreline proxy will be a necessary requirement to accurately perform inter-site shoreline model coefficient comparisons and further relate the differences to characteristics (wave climate, sediment size, inherited geology) at the individual sites. As discussed in previous studies (e.g. Yates et al., 2009; Davidson et al., 2013), this type of model has a number of limitations. Ignoring alongshore processes is one of them. Simulations with one additional free model parameter including the estimated longshore drift were performed and did not show any significant improvement. This is in line with the estimated negligible alongshore variability in longshore drift around Truc Vert beach (Idier et al., 2013). Other quantities such as the Dean parameter, wave height and power were tested in both models and gave similar results although model skill was slightly lower than with E. An exception is using DA13 model combining wave power and Dean parameter (Davidson et al., 2013; Splinter et al., 2013) that slightly improves model skill for most of the shoreline proxies. Inclusion of tide in the model will likely improve model skill in the upper profile as, at Truc Vert, the high tide mark ranges from +0.4 m to +2.4 m above MSL. A given severe storm during a high spring tide will erode much more the subaerial beach than a storm during neap tide (see for instance Coco et al., 2013) , which is not taken into account
in the present model. Masselink et al. (2006) show that tidal water level variations, and the resulting variability in the residence times for swash, surf and shoaling wave processes, are crucial to the morphological response of the intertidal domain. Therefore, residence times should be considered. For instance, equilibrium shoreline models on mesomacrotidal beaches could be improved by, at each time step, addressing if the considered beach elevation is wet or dry accounting for tidal elevation and an estimation of both the wave set-up and storm surge. At a given time step, a given dry iso-contour would not be affected by wave forcing. This will likely improve the ability of shoreline models to hindcast single storm response as well as berm formation and destruction that are strongly affected by water level variations (Masselink et al., 2006). The nearshore sandbar state can also influence the rate at which the shoreline evolves. For instance, a decayed sandbar therefore providing no protection to the beach during a sequence of storms can result in anomalously severe shoreline erosion (e.g. Castelle et al., 2008). Dynamically coupling behaviour-oriented shoreline (e.g. Yates et al., 2009; Davidson et al., 2013) and sandbar (e.g. Plant et al., 2006; Splinter et al., 2011) models will likely improve our ability to predict shoreline change. The significant variability of the best-fit parameter as a function of z (Figs. 10 and 11) suggests that the equilibrium shoreline concept can be extended to an equilibrium profile concept. Given that the model gives a cross-shore position for a given sand level elevation, beaches exhibiting pronounced bar (or berm) and trough morphologies, where a given sand elevation can be found at different cross-shore locations, are not suitable for such an application. To conclude, results show that the equilibrium shoreline model concept works on meso-macrotidal multiple-barred beaches provided a relevant shoreline proxy is used. Testing most of the existing shoreline definitions (Boak and Turner, 2005) on a 8-year time series of monthly alongshore-averaged beach surveys, we find that the best shoreline proxy at Truc Vert beach is the MHWL iso-contour. Results also reveal that the equilibrium shoreline concept can be extended to an equilibrium beach profile concept, suggesting that the approach is suitable for broader application which is in line with (Davidson et al., 2013).
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Acknowledgements This work was done within the framework of the project BARBEC (ANR N2010 JCJC 602 01). Financial support from the SOLAQUI (OASU) and the Region Aquitaine is acknowledged.
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Larson, M., Kraus, N.C., 1989. SBEACH: numerical model for simulating storm-induced beach change. Technical Report CERC-89-9. U.S. Army Corps of Engineers, Vicksburg, Mississipi. Long, J.W., Plant, N.G., 2012. Extended Kalman Filter framework for forecasting shoreline evolution. Geophysical Research Letters 39 (L13603). Masselink, G., 2004. Formation and evolution of multiple intertidal bars on macrotidal beaches: application of a morphodynamic model. Coastal Engineering 51, 713–730. Masselink, G., Puleo, J.A., 2006. Swash-zone morphodynamics. Continental Shelf Research 26, 661–680. Masselink, G., Kroon, A., Davidson-Arnott, R.G.D., 2006. Morphodynamics of intertidal bars in wave-dominated coastal settings — a review. Geomorphology 73, 33–49. Miller, J.K., Dean, R.G., 2004. A simple new shoreline change model. Coastal Engineering 51, 531–556. Murray, A., 2007. Reducing model complexity for explanation and prediction. Geomorphology 90, 178–191. Nicholson, J., Broker, I., Roelvink, J.A., Price, D., Tanguy, J.M., Moreno, L., 1997. Intercomparison of coastal area morphodynamic models. Coastal Engineering 31, 97–123. Plant, N.G., Holman, R.A., Freilich, M.H., 1999. A simple model for interannual sandbar behavior. Journal of Geophysical Research 104 (C7), 15755–15776. Plant, N.G., Holland, K.T., Holman, R.A., 2006. A dynamical attractor governs beach response to storms. Geophysical Research Letters 33 (L17697). Roelvink, J.A., Broker, I., 1993. Cross-shore profile models. Coastal Engineering 21, 163–191. Roelvink, J.A., Reniers, A.J.H.M., van Dongeren, A., de Vries, J.V., McCall, R., Lescinski, J., 2009. Modelling storm impacts on beaches, dunes and barrier islands. Coastal Engineering 56, 1133–1152. Ruessink, B.G., Kuriyama, Y., Reniers, A.J.H.M., Roelvink, J.A., 2007. Modeling cross-shore sandbar behavior on the timescales of weeks. Journal of Geophysical Research 112 (F03010). Sénéchal, N., Gouriou, T., Castelle, B., Parisot, J.-P., Capo, S., Bujan, S., Howa, H., 2009. Morphodynamic response of a meso- to macro-tidal intermediate beach based on a long-term dataset. Geomorphology 107, 263–274. Splinter, K.D., Holman, R.A., Plant, N.G., 2011. A behavior-oriented dynamic model for sand bar migration and 2DH evolution. Journal of Geophysical Research 116. Splinter, K.D., Turner, I.L., Davisdson, M.A., 2013. How much data is enough? The importance of morphological sampling and duration for calibration of empirical shoreline models. Coastal Engineering 77, 14–27. Stive, M.J.F., Roelvink, J.A., De Vriend, H.J., 1990. Large-scale coastal evolution concept. The Dutch Coast. Proc. 22nd Int. Conf. on Coast. Eng., vol. 9. ASCE, New York, pp. 1962–1974. Stive, M.J.F., Aarninkhof, S.G.J., Hamm, L., Hanson, H., Larson, M., Wijnberg, K.M., Nicholls, R.J., Capobianco, M., 2002. Variability of shore and shoreline evolution. Coastal Engineering 47, 211–235. Stockdon, H., Sallenger Jr., A., List, J., Holman, R.A., 2002. Estimation of shoreline position and change using airborne topographic lidar data. Journal of Coastal Research 18, 502–513. Tolman, H.L., 1991. A third generation model for wind waves on slowly varying, unsteady and inhomogeneous depths and currents. Journal of Physical Oceanography 21, 782–797. Walstra, D.J.R., Reniers, A.J.H.M., Ranasinghe, R., Roelvink, J.A., Ruessink, G.G., 2012. On bar growth and decay during interannual net offshore migration. Coastal Engineering 60, 190–200. Wright, L.D., Short, A.D., 1984. Morphodynamic variability of surf zones and beaches: a synthesis. Marine Geology 56, 93–118. Yates, M.L., Guza, R.T., O'Reilly, W.C., 2009. Equilibrium shoreline response: observations and modeling. Journal of Geophysical Research 114 (C09014). Yates, M.L., Guza, R.T., O'Reilly, W.C., Hansen, J.E., Barnard, P.L., 2011. Equilibrium shoreline response of a high wave energy beach. Journal of Geophysical Research 116 (C04014).
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Marine Geology journal homepage: www.elsevier.com/locate/margeo
Equilibrium shoreline modelling of a high-energy meso-macrotidal multiple-barred beach Bruno Castelle ⁎, Vincent Marieu, Stéphane Bujan, Sophie Ferreira, Jean-Paul Parisot, Sylvain Capo, Nadia Sénéchal, Thomas Chouzenoux CNRS, UMR EPOC, Université Bordeaux 1, France
a r t i c l e
i n f o
Article history: Received 1 August 2013 Received in revised form 18 October 2013 Accepted 4 November 2013 Available online 14 November 2013 Communicated by J.T. Wells Keywords: empirical shoreline model system memory shoreline proxy meso-macrotidal beach
a b s t r a c t 8-year time series of incident wave energy and monthly alongshore-averaged beach surveys at the high-energy meso-macrotidal multiple-barred Truc Vert beach are analysed. We apply two behaviour-oriented equilibrium shoreline models that relate the rate of cross-shore shoreline displacement to the wave energy and the wave energy disequilibrium between the wave energy and the equilibrium wave energy that would cause no change to the present shoreline location. The two models show similar skill. Results show that the equilibrium shoreline model concept works on meso-macrotidal multiple-barred beaches, with similar skill when applied to nonbarred and/or micro-mesotidal beaches, provided that a relevant shoreline proxy is used. Simulations show that Truc Vert beach responds predominantly at seasonal timescales rather than at individual storm frequency. The first winter storms drive the most pronounced erosion events because both the wave energy disequilibrium and erosion change potential are large. The best shoreline proxy at Truc Vert beach is the mean high water level, where the inner-bar and berm dynamics have little influence on the shoreline cross-shore displacement. Implications for shoreline monitoring through video imagery on this type of beach are discussed. Results also reveal that the equilibrium shoreline concept can be extended to an equilibrium beach profile concept pending further improvements. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Shoreline position along wave-dominated sandy coasts varies over a wide range of temporal and spatial scales in response to a variety of processes (Stive et al., 2002). Along non-engineered sandy coasts, longterm climate and sea level change, underlying geological settings and sediment budget are key parameters influencing changes in shoreline position on timescales of decades to millennia (Stive et al., 1990). On shorter timescales, from years and decades down to days and even hours for single storms, changes in wave energy arriving at the coast are the dominant process impacting shoreline change with both crossshore and alongshore surf and swash sediment transport processes dictating changes in shoreline position. On most of open coasts, alongshore processes typically act on longer timescales than cross-shore processes and do not dominate the annual shoreline variability (e.g. Davidson and Turner, 2009; Yates et al., 2009; Hansen and Barnard, 2010). Over the last decades, a number of complex process-based models have been developed to simulate and further predict beach changes (e.g. De Vriend et al., 1993; Roelvink and Broker, 1993; Nicholson et al., 1997; Roelvink et al., 2009). Yet, computational costs and misspecifications of the physics and boundary conditions prevent them from properly predicting shoreline evolution on long timescales ⁎ Corresponding author. E-mail address: [email protected] (B. Castelle). 0025-3227/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.margeo.2013.11.003
(i.e. years to decades). For instance, sheet flow sediment transport processes (e.g. Calantoni and Puleo, 2006), sand stirring accounting for breaking-wave-induced turbulence as a surface boundary conditions (e.g. Grasso et al., 2012) or swash zone sediment transport processes (Masselink and Puleo, 2006) are not properly considered in coastalevolution models. These misspecifications of the physics typically cascade up through the scales resulting in an inescapable build-up of errors and unreliable simulations. Only recently, coastal profile models succeeded in simulating surfzone beach profile evolution on timescales of days to months and sometimes years with fair accuracy (e.g. Ruessink et al., 2007; Kuriyama et al., 2012; Walstra et al., 2012). Yet, these models do not simulate the evolution of the beach face, and therefore the shoreline, as they ignore swash zone processes. Instead, empirically-based behaviour-oriented models can lead to more reliable long-term evolution than do parameterizations of much smaller-scale processes in process-based models, as evidenced in many geomorphological systems (Murray, 2007). Equilibrium beach response concepts have been used to model the evolution of beach profiles (Larson and Kraus, 1989) and the longterm evolution of cross-shore sandbar location (Plant et al., 1999) as well as that of the alongshore variability of the sandbar (Splinter et al., 2011). The fundamental assumption is that the present beach state is determined by the recent history of both the wave field and the beach morphology (Wright and Short, 1984). Using this concept, a wealth of equilibrium-based empirical shoreline models have recently been
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developed that are capable of simulating shoreline behaviour on timescales of days to years at a number of cross-shore dominated sandy beaches (Miller and Dean, 2004; Davidson and Turner, 2009; Frazer et al., 2009; Yates et al., 2009; Davidson et al., 2010; Yates et al., 2011; Kuriyama et al., 2012; Long and Plant, 2012; Davidson et al., 2013; Splinter et al., 2013). In essence, in all models the rate of cross-shore shoreline change is both a function of the disequilibrium (the difference between the present and equilibrium condition) and the magnitude of forcing available to move the sand with a time-varying forcing term outside the disequilibrium term limiting non-realistic sediment transport during small waves (for an overview of existing model formulations, see Splinter et al., 2013). These models can be forced by the incident wave energy (e.g. Yates et al., 2011), the incident wave power, the Dean parameter (Gourlay, 1968) thus also accounting for sediment size and wave period (e.g. Davidson et al., 2013) or the cross-shore component of the radiation stress (Yates et al., 2009). Overall, provided accurate surveys spanning a duration of at least two years sampled monthly, equilibrium shoreline models were found to predict shoreline changes on timescales of years with good accuracy (e.g. Splinter et al., 2013), therefore opening new perspectives of application (Davidson et al., 2013). To date, empirical shoreline models have been tested on micromesotidal sandy beaches, but their validity on meso-macrotidal multiple-barred beach has never been addressed. As an idealized definition of shoreline is that it coincides with the physical interface of land and water, the intersection of a tidal datum with the coastal profile is commonly used as a proxy of shoreline position (see Boak and Turner, 2005, for a review on shoreline definition and detection). On micromesotidal beaches, the intersection with Mean Sea Level (MSL) is commonly defined as the shoreline (Yates et al., 2009; Davidson et al., 2013). Alternatively, mean high water level (MHWL) is chosen when the MSL contour is underwater and therefore inaccessible during a significant number of beach surveys (Stockdon et al., 2002; Yates et al., 2011). Detailed assessment of the best shoreline proxy to use was not performed in previous behaviour-oriented shoreline modelling studies. On meso-macrotidal beaches where the cross-shore length of the intertidal domain is typically of the order of hundreds of metres (see for instance Masselink, 2004), shoreline definition is in practice even more blurred. In this paper, we analysed (Section 2) 8 years of data of beach morphology and iso-contour evolution at the meso-macrotidal high-energy multiple-barred beach of Truc Vert (SW France). We use two simple equilibrium shoreline models (Section 3) to explore shoreline response at Truc Vert Beach. Results are presented in Section 4 and further discussed in Section 5 before conclusions are drawn. The modelling effort provides insight into the overall shoreline response at Truc Vert. Results also show that the equilibrium shoreline concept works on a high-energy meso-macrotidal multiple-barred beach. Testing most of the existing shoreline definitions, we find that the best shoreline proxy at Truc Vert beach is roughly MHWL. Results also reveal that the equilibrium shoreline concept can be extended to an equilibrium beach profile concept. 2. Study site and data 2.1. Study site Truc Vert beach is located in SW France (Fig. 1). Truc Vert beach is meso-macrotidal with a maximum spring tide range of 4.8 m. The wave climate is energetic, strongly seasonally modulated, with an annual mean significant wave height Hs of 1.36 m, a mean period around 6.5 s and a dominant WNW incidence (Butel et al., 2002). Overall, summer conditions are dominated by short waves from the NW whilst winter conditions are characterized by long-period higher waves coming from the W and WNW sectors. Significant wave height occasionally exceeds 10 m during severe storms with peak wave periods often larger
Fig. 1. Map of SW France showing locations of Truc Vert beach and that of (grey circle) the Candhis directional wave buoy and (black) the WW3 virtual buoy.
than 15 s. The sediment consists of fine to medium quartz sand with mean grain of about 350–400 μm with large spatial variations (200– 700 μm over very short distances of 5 m) linked to the presence of various bedforms (megaripples, cusps and rip channels, see Gallagher et al., 2011). The morphology of the beach is complex, mostly double-barred, three-dimensional and highly dynamic itepalmar10 with ubiquitous rips (e.g. Bruneau et al., 2011). The intertidal inner bar has most of the time a transverse bar/rip morphology (Sénéchal et al., 2009). The subtidal outer bar is most of the time crescentic with an average wavelength of about 700 m (Castelle et al., 2007). Both sandbars migrate southward as a result of a southerly longshore drift of about 4 × 105 m3/year (Idier et al., 2013). 2.2. Wave data We use about 11.5 years of continuous wave outputs gathered from the Wavewatch III model (Tolman, 1991) at every 3 h since January 2002 at the grid point 1° 30′ W, 44° 30′ N in about 70-m depth (WW3 point in Fig. 1). Knowledge of wave conditions prior to the first topographic survey was required to drive one of the behaviour-oriented models that needs the antecedent wave forcing to account for beach history. To improve the accuracy of the wave forcing, we use in-situ measurements collected by the Candhis buoy located further onshore and closer to Truc Vert beach in 54-m depth (1° 26.8′ W, 44° 39.15′ N in Fig. 1). Because of a number of buoy malfunctions, wave data were collected intermittently since January 2002 until April 2013 resulting in a total of about 5 years of wave data. Fig. 2a shows the comparison between the modelled (HsWW3) and measured (HsCandhis) significant
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Fig. 2. (a) Scatter diagram of measured significant wave height at the Candhis buoy HsCandhis versus 3-hour modelled significant wave height HsWW3, with the solid black line indicating the linear regression giving HsCandhis = −0.01526 + 0.9052HsWW3. (b) Resulting scattered directional plot of significant wave height HsCandhis for the entire dataset. The solid and dashed black lines indicate mean shoreline orientation and shore-normal incidence at Truc Vert, respectively. The dashed grey lines indicate groundswell shadowing regions to the North and to the South behind Brittany and Galicia, respectively (see Fig. 1). In all panels colorbar indicates peak wave period in seconds.
wave heights. Using linear regression, we use the corrected significant wave height Hs = −0.01526 + 0.9052HsWW3. The overall directional wave climate at Truc Vert is given in Fig. 2b, showing wind seas at a wide range of direction and high-energy swells from the W–NW sector. 2.3. Sand level surveys and equilibrium change observations Monthly or bimonthly topographic surveys have been acquired from September 2003 until April 2013, with a 1-year gap in 2008. Topographic surveys are conducted at low tide during spring tides using a DGPS (Trimble 5700) with an accuracy of about 2.5 cm in the horizontal and 10 cm in the vertical. Sand level elevation is referenced to benchmarks of the French National Geodesic Service (NGF-IGN 69) with the 0-contour corresponding to MSL. The distance between each transect varies overall from 20 to 40 m. Most of the time, the inner bar was not entirely surveyed in the cross-shore direction as the water level was too high (wave set-up) and/or the bar crest was too deep. The alongshore coverage of the topographic surveys was about 350 m from 2003 to 2008, 750 m from 2008 to 2012 and about 1200 m from 2012 to present (see Fig. 3). Based on a detailed inspection of all the available topographic surveys, data collected prior to April 2005 were disregarded either because of vertical referencing issues, ubiquity of spurious data points or malfunction of the DGPS. Since April 2005, 8 additional surveys were removed from the dataset for the same reasons. In total, beach morphology was sampled 121 times between April 8, 2005 and March 23, 2013. This equates to, on average, a beach topography surveyed nearly once every month, although the interval between successive observations is irregular with a notable 1-year gap in 2008. Spatial interpolation of the scattered DGPS measurements of sand level is required to both generate a digital elevation model on a given grid and further compute the alongshore-averaged beach profile. After a detailed assessment of the commonly used interpolation methods on our dataset, anisotropic kriging was chosen. Isotropic methods typically resulted in the formation of cuspate features, with a spacing equalling to that of the transects, which were increasingly developed for increasing cross-shorebeach slope. This ultimately resulted in a slight smoothing of the bar and berm features when computing the alongshore-averaged beach profile. Anisotropic kriging was also preferred to prevent spurious interpolations for occasional highly irregular survey transects (in both the cross-shore and alongshore directions) due to strongly alongshore non-uniform sandbar patterns (see for instance Fig. 4a). Interpolation was performed on a regular grid with an alongshore and cross-shore mesh size of 20 m and 2 m, respectively, with an anisotropy ratio and angle of 10 and 90° in beach coordinates, respectively.
Fig. 3. Truc Vert Beach bathymetry surveyed on April 6, 2008. The 0-m iso-contour (thick black line) stands for the Mean Sea Level (MSL). The black dashed box shows the area of interest with indication of the typical alongshore extension of the topographic surveys during the periods 2004–2008, 2008–2012 and 2012–2013.
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Fig. 4. (a,b,c) Examples of typical beach surveys at Truc Vert Beach with colorbar indicating elevation in metres and (d,e,f) corresponding alongshore-averaged beach profile with ±1 standard deviation of iso-contours at a 1-m interval (denoted by the horizontal error bar). (a,d): Low tide terrace morphology with a weakly-developed alongshore-uniform berm, a runnel at x ≈ 170 m and two narrow shallow rips with the small white points indicating the survey points, (b, e) bar and rip morphology with a well-developed cuspate berm and two deep rip channels, (c, f): reasonably alongshore-uniform concave beach face shape profile typical of a longshore bar trough beach state. In the latter situation the inner sandbar is deep and located farther offshore and was therefore not surveyed. In (d, e, f) the Lowest Astronomical Tide (LAT) level, Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level are indicated.
Fig. 4 shows examples of three representative beach topographies of Truc Vert beach states with: (1) a low tide terrace, typically found in summer after a long period of fair weather conditions, with an alongshore intertidal bar attached to the beach cut by shallow and narrow rip channels (Fig. 4a); (2) a bar and rip morphology, which is the modal beach state at Truc Vert (58% of the time, Sénéchal et al., 2009), with deep rip channels and a well-developed alongshore non-uniform berm due to protruding megacusp horns attaching to the bar (Fig. 4b); (3) an alongshore-uniform concave beachface shape profile typical of a longshore bar trough beach state following a storm event with the bar located too far off the beach and too deep to be surveyed (Fig. 4c). The corresponding mean beach profiles are given in the bottom panels of Fig. 4 with superimposed ± 1 standard deviation of isocontours at a 1-m interval denoted by the horizontal error bar. The latter gives a measure of how representative the mean cross-shore position of a given iso-contour is. Fig. 5 shows the superimposed mean beach profiles surveyed at Truc Vert Beach since April 2005 revealing a large variability, particularly at 100 m b x b 150 m where berm formation and dynamics are preferably observed. To address what the best shoreline proxy z is at Truc Vert beach, we computed the time evolution of the mean cross-shore position of all the iso-contours from −1 m to 8 m at a 0.1 m interval. Fig. 6 shows example of time series of measured shoreline cross-shore position for the proxies z = 0, 2 m, 4 m and 6 m with corresponding time series of corrected significant wave height Hs. A visual inspection of the time series indicates that Hs is highly variable over seasonal timescales with
moderate energy waves during summer (Hs b 2 m) and common high-energy storm waves (Hs N 4 m) during autumn and winter (Fig. 6a). Seasonal cycles are also clearly seen in the shoreline position for proxies z = 2 m and 4 m with typical amplitude of about 50 m and 30 m, respectively. In contrast, the iso-contour z = 6 m barely
Fig. 5. Superimposed mean beach profiles surveyed at Truc Vert Beach from April 2005 to April 2013 with indication of the Lowest Astronomical Tide (LAT) level, Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level.
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Fig. 6. Time series of (a) corrected significant wave height Hs and (b) measured mean shoreline cross-shore position for the proxies z = 0, 2 m, 4 m and 6 m with ±1 cross-shore standard deviation indicated by the vertical error bars.
evolves throughout the seasons whilst iso-contour z = 0 shows large variations although no clear cycles are seen with anomalous changes not associated with corresponding fluctuations in incoming wave height. Time series of shoreline cross-shore positions S(z) and Hs were used to constrain the equilibrium shoreline model. 3. Equilibrium models Alongshore gradients in sediment transport around Truc Vert beach are small (Idier et al., 2013). Accordingly, it can be assumed that shoreline variability due to gradients in alongshore transport is small compared to the variability associated with cross-shore processes. Given that improving existing equilibrium shoreline models is not the purpose of this paper, here we use 2 existing model concepts that are briefly described below. Wave energy E is used to force the 2 models as it was found to give slightly better results than other wave characteristics (Yates et al., 2009).
assumes that the equilibrium wave energy is a linear function of the current shoreline position S: Eeq ðSÞ ¼ aS þ b
where a and b are 2 free model parameters, meaning that the equilibrium wave energy that would cause no change in shoreline position essentially depends on the current shoreline position. The second approach is based on the work of Davidson et al. (2013), Splinter et al. (2013) and is hereafter referred to as the DA13 model. The major difference with YA09 model is that, instead of computing Eeq as a function of the current shoreline position S, DA13 models computes Eeq from the antecedent wave forcing, thus explicitly reflecting the degree of observed memory of the system. To be consistent with YA09 approach, we use E both instead of the dimensionless fall velocity Ω and incident wave power P in Davidson et al. (2013), Splinter et al. (2013). Accordingly, the equilibrium wave energy reads: 2Φ=Δt X
3.1. Model formulations Behaviour-oriented equilibrium shoreline models relate the rate of cross-shore shoreline displacement dS/dt to the, typically hourly, wave energy E and the wave energy disequilibrium ΔE between E and the equilibrium wave energy Eeq that would cause no change to the present shoreline location. The shoreline displacement rate is typically assumed proportional to both E and ΔE multiplied by a constant change rate coefficient: dS 1=2 ¼ C E ΔE dt
ð1Þ
where, for a given shoreline S(z), C± are change rate coefficients for accretion (C+ for ΔE b 0) and erosion (C− for ΔE N 0), with the energy disequilibrium ΔE = E − Eeq. From this point, 2 types of approaches are used. The first one is based on the work of Yates et al. (2009, 2011), Long and Plant (2012) and is hereafter referred to as YA09 model. This simple and efficient model
ð2Þ
Eeq ðt Þ ¼
− jΔt=Φ
E j 10
j¼0 2Φ=Δt X
ð3Þ j¼0
j¼0
with j the number of data points in the wave forcing time series prior to the calculation point (j = 0); Δt the sampling interval in days and Φ the memory decay of the system a free model parameters in days meaning that Φ is the number of days in the past when the exponentially decaying weighting factor decreases to 10%, with 2Φ taken as a limit for Eeq computations (Davidson et al., 2013). 3.2. Model fit and skill For each shoreline proxy z, the temporal mean of shoreline position S was removed, yielding time series of shoreline position fluctuations about the mean. YA09 and DA13 models have 4 and 3 free model parameters, respectively. For each simulation, we also added one free
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parameter, d, to allow for S(t = 0) not to inevitably equals to the initially measured cross-shore distance of the shoreline proxy. This was a necessary requirement as the initial profile exhibited a berm, hence limiting the validity of the shoreline position for proxies within the upper profile. In addition, setting d = 0 artificially gives more weight to the initial survey in the model hindcast. This is an issue herein also because the alongshore coverage of the first topographic survey was short (350 m), thus resulting in a less precise alongshore-averaged beach profile than for the subsequent surveys. For each model and each shoreline proxy z, we used a probabilistic optimization method, the so-called Simulated Annealing algorithm (Bertsimas and Tsitsiklis, 1993), to solve for the 4 or 5 free model parameters that minimize the root-meansquare error (RMSE) between the model and observations. Simulated annealing, overcoming local RMSE minima in this nonlinear system, searched within a realistic bounded range of the parameters: for YA09 model − 0.1 m2/m b a b 0; 0 b b b 1 m2; − 2 m·hr− 1/m3 b C+ b 0; − 2 m·hr− 1/m3 b C− b 0; and for DA13 model: 0 b Φ b 1000 days; −50 m·hr−1/m3 b C+ b 0; −50 m·hr−1/m3 b C− b 0; and −15 m b d b 15 m for both YA09 and DA13 models. For each simulation, the model skill was addressed through the R-squared (coefficient of determination) denoted R2 between modelled and observed shoreline position. Given that the same method was systematically performed for all shoreline proxies (−1 m b z b 8 m at a 1-m interval), we obtained the best fit free model parameters a(z), b(z), C+(z) and C−(z) for YA09 model and Φ(z), C+(z) and C−(z) for DA13 model. 4. Results Fig. 7 shows the time evolution of modelled beach profiles using YA09 model (Fig. 7b) and DA13 model (Fig. 7c) together with the time evolution of the shoreline for 4 proxies z = 0, 2 m, 4 m and 6 m. The modelled position of all shoreline proxies shows strong seasonal variation, with slow accretion for long periods of low-energy waves,
and faster erosion during episodic, high-energy wave events. Consistent with the overall measured beach profile shapes (Fig. 5), steep upper and gentle sloping lower beach profiles are simulated. Interestingly enough, the model successfully reproduces the overall erosion trend from 2005 to 2009 and the subsequent overall accretion trend from 2010 onwards, suggesting that this long-term erosion/accretion sequence has been essentially driven by cross-shore processes. Addressing shoreline evolution for the proxy z = 2 m shows that the shoreline accretes 86% and 70% of the time using YA09 and DA13 model, respectively. This difference is due to the modelled equilibrium wave energy Eeq that is about twice larger for YA09 than for DA13 model (Fig. 7a). Surprisingly enough, despite the fact that Eeq is computed using current shoreline position and antecedent wave forcing in the YA09 and DA13 models, respectively, Eeq in YA09 and Eeq in DA13 are very similar in pattern (Fig. 7a). In line with previous works (e.g. Yates et al., 2009) the first winter storms, typically in December/January at Truc Vert, drive the most pronounced erosion events because the wave energy disequilibrium ΔE and erosion change potential E are large (Fig. 7). Even if Truc Vert beach is often exposed to severe storms in February/March, they do not significantly erode the beach as ΔE is smaller due to larger Eeq values, which results in a smaller change potential |E1/2ΔE|. Instead, even if wave energy E is often elevated through early spring, overall accretion typically occurs. A notable exception in this 8-year time series is the winter 2011 that was characterized by the absence of severe storm (Fig. 7), thus triggering the overall accretion trend from 2010 onwards. There is large variability in the ability of the model to simulate shoreline evolution depending on the shoreline proxy z. This is further emphasized in Fig. 8, which shows both the root-mean-square error (RMSE) and the squared correlation R2 for both models as a function of the considered shoreline proxy. Results show that YA09 and DA13 models have very similar skills with, for instance, RMSE ≈ 8 m and R2 ≈ 0.55–0.65 at 1 m b z b 4 m. YA09 model explains slightly more of the observed shoreline variability for z b 1 m than DA13 model.
Fig. 7. (a) Observed wave energy E (thin black line) and modelled equilibrium wave energy Eeq (thick lines) for the shoreline proxy z = 2 m for the YA09 model (grey) and the DA13 model (black). Modelled shoreline positions for all shoreline proxies −1 m b z b 8 m at a 0.1-m interval for (b) YA09 model and (c) DA13 model. In (b,c) the time evolution of the shoreline position for the 4 proxies z = 0, 2 m, 4 m and 6 m and corresponding measurements are superimposed as a black line.
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Fig. 8. R-squared R2 (black) and root-mean-square error RMSE (grey) between the observed shoreline position and that modelled with YA09 model (squares) and DA13 model (circles) as a function of the shoreline proxy z. The black dotted lines indicate the Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level. The horizontal grey dotted line indicates for which shoreline proxy the models give the best agreement with field data.
DA13 and YA09 model skill rapidly degrades for both z N 4 m and z b 0 (grey areas visually determined in Fig. 8). This variability in model skill is further emphasized below. The observed and modelled (using optimal model parameters) shoreline position for 5 representative shoreline proxies (z = 0, 0.4 m, 1.5 m, 3 m and 6 m) are shown in Fig. 9. These 5 elevations were selected because they are commonly used shoreline proxies (z = 0 is MSL, z = 0.4 m is the lowest high tide level commonly used through video imagery to get daily shoreline data and z = 1.5 m is MHWL) or because they correspond to a particular compartment of the beach (z = 3 m is the typical berm elevation and the iso-contour z = 6 m is close to the dune foot and barely moves). These isocontours described below are representative of the surrounding isocontours as well. Using the proxy z = 0 corresponding to MSL (Fig. 9a), measured shoreline position shows large-amplitude variations which are not always readily related to changes in incident wave energy. This is because this part of the profile is both very gently sloping and strongly influenced by the inner-bar dynamics. Given that the crossshore inner-bar dynamics at Truc Vert beach is influenced largely by tide-range variations (Almar et al., 2010), this is not captured by the model that is forced by the time-varying incident wave conditions only. This results in large model errors. A similar comment can be drawn for the proxy z = 0.4 m (Fig. 9b), corresponding to the lowest high tide level, which is a commonly used proxy to address shoreline position through video imagery on meso-macrotidal beaches (e.g. Birrien et al., 2013), although seasonal cycles are relatively clearer in the measured shoreline positions than for z = 0. RMSE exceeds 10 m with both models, with DA13 model explaining less than 35% of the observed shoreline variability. As shown in Fig. 8, best model results are obtained using roughly the proxy z = 1.5 m, corresponding to MHWL. For this proxy, the model simulates shoreline evolution with fair accuracy (Fig. 9c) as it explains 66% and 62% of the observed shoreline variability with a RMSE of 7.5 m and 8 m using YA09 and DA13 approaches, respectively. Model behaviour is slightly different for z = 3 m corresponding to the typical berm elevation (Fig. 9d) as spurious rapid accretion sometimes occur (see for instance the autumns of 2005 and 2011 in Fig. 9d). These anomalous changes not associated with
Fig. 9. Time series of the observed (circles with ±1 cross-shore standard deviation indicated by the vertical error bars) and modelled shoreline position using YA09 (thick grey line) and DA13 (thick black line) models for representative shoreline proxies: (a) z = 0, (b) z = 0.4 m, (c) z = 1.5 m, (d) z = 3 m and (e) z = 6 m.
corresponding fluctuations in wave energy, and therefore not captured by the equilibrium model, are related to rapid berm formation during abnormally low-energy autumn waves. For the proxy z = 6 m close to the dune foot (Fig. 9e), model fluctuations drastically decrease down to less than 5 m.
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5. Discussion and concluding remarks Results show that the equilibrium shoreline model concept works on a high-energy meso-macrotidal multiple-barred beach. Although Eeq is computed using current shoreline position and antecedent wave forcing in the YA09 and DA13 models, respectively, the differences between the 2 models are rather subtle (Figs. 7 and 9). A notable difference is that with the DA13 model the shoreline accretion tends to slightly level off at the end of the summers whilst the shoreline continues to accrete at a steady rate with YA09 model (see Fig. 9). This is because Eeq is larger for YA09 model than for DA13 model (see Fig. 7a for z = 2 m). This, in turn, makes the shoreline (z = 2 m) accrete 86% and 70% of the time using YA09 and DA13 models, respectively, resulting in a slight levelling off of shoreline accretion with DA13 model. Detailed high-frequency topographic surveys in summer would help in determining which model gives the best estimation of erosion/accretion proportions. For the best shoreline proxy z = 1.5 m, a RMSE and a R2 of about 8 m and 0.65 are obtained, respectively, for both YA09 and DA13 models. Model skill is therefore similar to that obtained both at Narrabeen and the Gold Coast in Australia (Davidson et al., 2013) but slightly lower than that on California beaches (Yates et al., 2009, 2011). A more detailed inspection of model skill shows that R2 increases up to (decreases down to) 0.75 (0.5) if addressing model performance after (before) 2009 only. This suggests that the 350-m alongshore coverage of the topographic surveys prior to 2009 was too short to get rid of the influence of the inner bar and rip alongshore variability on the mean profile characteristics. Since 2009, the alongshore coverage exceeding two rip spacings results in much more accurate alongshoreaveraged shoreline determination. It is also expected that model skill would have been further improved if performing model hindcast since 2009 only. Results suggest that both YA09 and DA13 models predict the highest percentage of the total measured shoreline variability using roughly the shoreline proxy z = 1.5 m, corresponding to MHWL at Truc Vert. Assuming that shoreline evolution at Truc Vert beach is essentially driven by cross-shore processes which are captured by the model, this suggests that the intersection of the coastal profile with the vertical elevation MHWL should be used as shoreline proxy at Truc Vert beach. It remains
unknown to what extent this statement applies for other mesomacrotidal beaches. Further work in similar wave-dominated environments is necessary to test the generality of our findings. This can have major implication for video monitoring of meso-macrotidal beaches as the lowest high tide level is most of the time used in order to have daily shoreline measurements (e.g. Birrien et al., 2013). Instead, time exposure images at MHWL should be used to have an accurate proxy of shoreline position. Despite time-exposure video images at MHWL will not give daily shoreline information, it is enough to accurately drive equilibrium shoreline models (Splinter et al., 2013). The variability in the best-fit model parameters as a function of the proxy z is addressed in Figs. 10 and 11 for YA09 and DA13 models, respectively. Fig. 10 shows that there is a high but coherent vertical variability in the YA09 model optimal free parameters at 0 b z b 4 m. As indicated in Yates et al. (2009), a and C± are related and can have compensating physical effects, whilst b (Fig. 9b) depends on the temporal mean removed from the shoreline time series and has therefore limited physical meaning. a increases with increasing z (Fig. 9a) meaning that, when going up along the profile, beach width decreasingly influences the rate at which the beach erodes or accretes. |C−| decreases with increasing z (Fig. 9d) meaning that, for a given wave energy, shoreline erodes less rapidly in the upper profile. Whilst the evolutions of a and C− are not surprising, that of C+ is more intriguing with |C+| peaking at z ≈ 3 m (Fig. 9c). This corresponds to the typical berm elevation (Fig. 3). The maximized |C+| at z ≈ 3 m therefore characterizes the berm formation that builds up rapidly during prolonged period of fair weather conditions. Both |C+| and |C−| decreasing for increasing z for z N 3 m is the signature of the decreasing residence time of swash and surf processes when going up along the profile during the tide. Fig. 11 shows the same plot but for DA13 model. Along most of the beach profile, large Φ values are found with systematically Φ N 200 days (Fig. 11a), meaning that Eeq is suppressed to a weak seasonal signal value close to the time-series mean. This suggests that Truc Vert beach responds predominantly at seasonal timescales rather than at individual storm frequency, that is, a similar behaviour to that observed on the Gold Coast (Davidson et al., 2013). This matches the dominant seasonal mode of observed shoreline variability at Truc Vert (Fig. 6b). We hypothesize that the local minimum of Φ at z ≈ 3 m is the signature of the berm dynamics that was captured in the YA09
Fig. 10. Optimal model free parameters as a function of the considered shoreline proxy z for the YA09 model: (a) equilibrium slope a and (b) b the y intercept of the equilibrium wave energy linear function, (c) accretion rate coefficient C+, and (c) erosion rate coefficient C−. In all panels the black dotted lines indicate the Mean Sea Level (MSL) and Highest Astronomical Tide (HAT) level.
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Fig. 11. Optimal model free parameters as a function of the considered shoreline proxy z for the DA13 model: (a) Φ the memory decay in days, (b) accretion rate coefficient C+, and (c) erosion rate coefficient C−.
model (Fig. 10b). C+ (Fig. 11b) and C− (Fig. 11c) show very similar patterns with an order of magnitude of difference. The large variability in the best-fit model parameters depending on the considered elevation contour has important implications. For instance, Yates et al. (2011) compare shoreline model coefficients obtained at 4 Californian beaches, suggesting that the differences are related to the grain size and that the coefficients can be transportable between sites with similar grain size and different wave climates. They suspect that the reduced mobility of the coarser sand at Ocean Beach explains the lower erosion rate coefficients |C−| found at this site. Yet, MHWL is used at Ocean Beach whilst MSL is used for the 3 other Californian beaches. Fig. 10b shows that |C−| decreases with increasing isocontour elevation, thus suggesting that the differences in |C−| in Yates et al. (2011) can be attributed to the considered shoreline proxy instead of the grain size. These differences are more likely related to a combination of both. Using the same shoreline proxy will be a necessary requirement to accurately perform inter-site shoreline model coefficient comparisons and further relate the differences to characteristics (wave climate, sediment size, inherited geology) at the individual sites. As discussed in previous studies (e.g. Yates et al., 2009; Davidson et al., 2013), this type of model has a number of limitations. Ignoring alongshore processes is one of them. Simulations with one additional free model parameter including the estimated longshore drift were performed and did not show any significant improvement. This is in line with the estimated negligible alongshore variability in longshore drift around Truc Vert beach (Idier et al., 2013). Other quantities such as the Dean parameter, wave height and power were tested in both models and gave similar results although model skill was slightly lower than with E. An exception is using DA13 model combining wave power and Dean parameter (Davidson et al., 2013; Splinter et al., 2013) that slightly improves model skill for most of the shoreline proxies. Inclusion of tide in the model will likely improve model skill in the upper profile as, at Truc Vert, the high tide mark ranges from +0.4 m to +2.4 m above MSL. A given severe storm during a high spring tide will erode much more the subaerial beach than a storm during neap tide (see for instance Coco et al., 2013) , which is not taken into account
in the present model. Masselink et al. (2006) show that tidal water level variations, and the resulting variability in the residence times for swash, surf and shoaling wave processes, are crucial to the morphological response of the intertidal domain. Therefore, residence times should be considered. For instance, equilibrium shoreline models on mesomacrotidal beaches could be improved by, at each time step, addressing if the considered beach elevation is wet or dry accounting for tidal elevation and an estimation of both the wave set-up and storm surge. At a given time step, a given dry iso-contour would not be affected by wave forcing. This will likely improve the ability of shoreline models to hindcast single storm response as well as berm formation and destruction that are strongly affected by water level variations (Masselink et al., 2006). The nearshore sandbar state can also influence the rate at which the shoreline evolves. For instance, a decayed sandbar therefore providing no protection to the beach during a sequence of storms can result in anomalously severe shoreline erosion (e.g. Castelle et al., 2008). Dynamically coupling behaviour-oriented shoreline (e.g. Yates et al., 2009; Davidson et al., 2013) and sandbar (e.g. Plant et al., 2006; Splinter et al., 2011) models will likely improve our ability to predict shoreline change. The significant variability of the best-fit parameter as a function of z (Figs. 10 and 11) suggests that the equilibrium shoreline concept can be extended to an equilibrium profile concept. Given that the model gives a cross-shore position for a given sand level elevation, beaches exhibiting pronounced bar (or berm) and trough morphologies, where a given sand elevation can be found at different cross-shore locations, are not suitable for such an application. To conclude, results show that the equilibrium shoreline model concept works on meso-macrotidal multiple-barred beaches provided a relevant shoreline proxy is used. Testing most of the existing shoreline definitions (Boak and Turner, 2005) on a 8-year time series of monthly alongshore-averaged beach surveys, we find that the best shoreline proxy at Truc Vert beach is the MHWL iso-contour. Results also reveal that the equilibrium shoreline concept can be extended to an equilibrium beach profile concept, suggesting that the approach is suitable for broader application which is in line with (Davidson et al., 2013).
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Acknowledgements This work was done within the framework of the project BARBEC (ANR N2010 JCJC 602 01). Financial support from the SOLAQUI (OASU) and the Region Aquitaine is acknowledged.
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