4.1.1. Forcing Conditions

Wind velocity time series shows a diurnal variability associated to sea-breeze events (Figure 7a). A maximum wind speed of 16 m s<sup>−</sup>1, corresponding to the sea breeze event peak, was recorded during the experiment in the afternoon (1630 local time) of 27 May (Figure 7a). Offshore wave conditions, measured at 10 m water depth (Figure 7b), are highly correlated with local winds (Figure 7a). The wave height increased from *Hs* = 0.3 m measured at 10:00 to *Hs* = 1.0 m measured at 1800, with mean wave direction approaching from the NE (Figure 7b). The temporary structure was deployed during neap tides (light gray shade in Figure 7c) in order to decrease the influence of the tide on the effective length of the groin and hence restricting the swash zone width [16].

**Figure 7.** Measured (**a**) wind speed and wind direction (**b**) offshore significant wave height and wave direction at 10 m water depth and (**c**) mean sea level variation *η* from the Sisal gauge before, during (light gray shade) and after (dark gray shade) the groin deployment.

Sea breeze events increased the wave energy inside the surf zone (Figure 8a). The alongshore current velocity *Vy* becomes negligible during the land breeze (0300 h), whereas it reaches *Vy* > 0.3 m s−<sup>1</sup> near the sea breeze peak (1800 h) (Figure 8b). Thus, westward currents dominated the surf zone hydrodynamics during the measured period (red solid-line in Figure 8b). On the other hand, the mean cross-shore current *Vx* (blue solid-line in Figure 8b) shows negligible current intensity at this location/elevation during the 48 h period.

**Figure 8.** Surf zone conditions during and 24 h after the structure disturbance. (**a**) Significant wave height inside the surf zone and (**b**) cross- (blue line) and along- (red line) shore currents. Negative values in panel (**b**) indicate offshore/west-ward currents, while positive values indicate onshore/east-ward currents.

#### 4.1.2. Observed Morphology Evolution

Beach morphology changes are evaluated by analyzing the high spatial and temporal resolution RTK DGPS survey data (Table 1 and Figure 9). The measured beach survey before the structure deployment (0730 local time) shows the alongshore uniformity of the beach contour lines (Figure 9a). However, the deployment of a temporary impermeable structure (at *x* = 0 m) induces a shoreline perturbation owing to the sediment impoundment at the east side of the structure and erosion at the downdrift (west) side (a time-lapse video of the experiment is included as supplementary material). The beach survey conducted 24 h after the structure deployment (0842 of the 28 May 2015) shows significant changes for *h* > −0.6 m (Figure 9b). These observations are consistent with the westward alongshore current associated to sea breeze events (Figure 8b). The shoreline contour reaches a maximum advance of 6 m and maximum retreat of less than −4 m at the updrift and downdrift side, respectively. The calculated sediment volume impoundment, at the updrift side of the structure, reached 70 m3 in 24 h (2 to 6 m3/m), whereas the volume loss at the downdrift for the same period is less than 40 m3. Differences between up- and down- drift volume changes might be ascribed to the limited alongshore spatial coverage of the topographic measurements.

Alongshore uniformity for the submerged area (*h* < 0 m) was observed 24 h after the structure removal (0825 of the 29 May 2015). However, a clear perturbation in the subaerial beach profile was still present (Figure 9c). The disturbance smooths out during the following days, returning to the pre-disturbed condition (i.e., straight and parallel contours) 144 h after the structure removal (3 June 2015) (Figure 9d).

**Figure 9.** Beach survey (**a**) right before groin deployment, (**b**) right before groin removal (24 h later), (**c**) 24 h after groin removal, (**d**) 144 h after groin removal. The title of each subplot indicates the weekday and local time of each survey (W = Wednesday, T = Thursday, F = Friday).

The EOF analysis of the shoreline position allowed us to identify the spatial and temporal shoreline evolution before and after the structure removal. The first mode of variability, which represents more than 91% of the total variance, has been associated solely to the groin influence. The spatial function of the first mode (*e*1(*y*) in Figure 10a) describes the shoreline perturbation with positive/negative values associated to accretion/erosion at the updrift/downdrift side of the temporal groin. Furthermore, the temporal evolution (*c*1(*t*) in Figure 10b) shows an increase from zero to a maximum value in the first 24 h (before structure removal). After the structure removal, *c*1(*t*) shows slower decrease reaching a value close to zero at the end of the measured period. The second and third modes (not shown) represent 7.5% of the total variance and show spatial and temporal functions not associated to the groin presence. The beach recovery time can be calculated from the decay time of the perturbation described by the first temporal function (*c*1(*t*)) which is nearly 144 h.

**Figure 10.** Empirical orthogonal function analysis of the shoreline data during the experiment (27 May–3 June, 2015). Only the (**a**) spatial function and (**b**) the temporal evolution of the first mode are shown.

#### *4.2. Numerical Modelling*

The field data were employed to calibrate the GSb model. Thus, a 14.4 m long groin (10 m wet plus 4.4 m dry) was positioned at the center of the domain. The alongshore model computational domain was assumed equal to 41 m. Model grid cell resolution, *DX*, has been set equal to 1 m with a total number of cells, *NX*, equal to 41, whereas the model experiment has been simulated adopting a calculation time step, *DT*, equal to 0.005 h. For a direct comparison with field measurements, the recording time step of the output files has been set to 1 h. The median grain size, *D*50, has been set to 0.3 mm and the closure depth, *h*\*, 0.8 m. Hourly wave conditions recorded at the ADCP located 11 km offshore the beach have been adopted as input to the GSb model (Table 2).


 =


*Water* **2018** , *10*, 1806


**Table 2.** *Cont.*

Table 3 shows the values of the *RMSE* (Root Mean Square Error) from the comparison of the observed and calculated (for different KGSb values) shoreline positions, *yi*,*obs* and *yi*,*GSb*, respectively, at 23 h after the groin deployment; the *RMSE* is defined as,

$$RMSE = \sqrt{\frac{\sum\_{i=1}^{N} (y\_{i,GSb} - y\_{i,obs})^2}{N}} \tag{3}$$

where *N* is the number of transects along the considered shoreline. A KGSb = 0.01 value determined the lowest *RMSE*, showing a better agreement between the calculated and the observed shoreline positions.

**Table 3.** Calculated and observed shoreline positions at 14 transects along the shoreline and relative values of *RMSE* (Root Mean Square Error).


Numerical simulations were conducted for two different lateral boundary conditions: pinned or moving lateral boundary. Figure 11 shows the comparison of the simulated shoreline positions obtained after 23 h for the two different lateral boundaries; as expected, the shoreline shapes differ in vicinity of the lateral boundaries but overlap in vicinity of the groin. If a moving lateral boundary condition is selected, the boundary will move a specified distance over a certain time period. GSb lateral boundaries have been selected as moving boundaries.

**Figure 11.** Comparison of the simulated shoreline positions obtained after 23 h for the two different lateral boundaries.

The first beach survey was assumed as the initial shoreline position in the numerical model. In particular, at the down-drift (*x* = 0 m) and up-drift (*x* = 41 m) boundaries of the computational domain, the observed specific distances from the first beach survey, equal to −1.4 m and 1.8 m, respectively, over a period of 24 h, have been assumed. On the other hand, for the post structure removal condition the computation duration has been extended to 168 h taking as initial condition the survey at *t* = 24 h.

Figure 12c–g show that the calibrated numerical model satisfactorily predicts the downdrift shoreline evolution, whereas the model is not capable of fully reproducing the shoreline advance, at *t* = 21 h, in the updrift side (Figure 12h).

**Figure 12.** Measured and calculated shoreline position (**a**) 2 h, (**b**) 4 h, (**c**) 6 h, (**d**) 8 h, (**e**) 11 h, (**f**) 14 h, (**g**) 16 h and (**h**) 21 h after the structure deployment. Information on the brackets indicates the corresponding weekday (W = Wednesday, T = Thursday) and local time of field data and GSb simulation.

The model capability to predict the shoreline resilience after the structure removal was investigated (verification). The assumed initial shoreline position corresponds to the beach survey performed immediately before the structure removal and the numerical model is run without the structure using the daily mean conditions as measured for the following seven days. The numerical model calculated the drastic change occurring during the first 24 h after the groin removal (Figure 13a–e). Furthermore, it calculated the beach recovery occurring after approximately 7 days (Figure 13f). Therefore, within the framework of the field data gained in the investigation (sea-breeze conditions), the model can be considered as a reliable tool to conduct a numerical study on beach resistance and resilience for the adopted study area.

**Figure 13.** Measured and calculated shoreline position (**a**) 2 h, (**b**) 6 h, (**c**) 8 h, (**d**) 10 h, (**e**) 26 h and (**f**) 1 week after the structure removal. Information on the brackets indicates the corresponding weekday (T = Thursday, F = Friday) and local time of field data and GSb simulation.

#### **5. Discussion: Shoreline Stability**

Once the GSb model has been calibrated and verified, it has been used to investigate the beach resistance and resilience phenomena. The main limitation in the field data is related to the limited alongshore spatial coverage (40 m) of the topographic measurements. Therefore, for the numerical setup, we considered a 10 m long groin located in the middle of a 200 m long shoreline computational domain. For the parametric study we employed different cases (Table 4) encompassing constant (low-energy) wave conditions during a 720 h period. The latter allows us to assess the resistance and resilience sensitivity to different forcing conditions on a longer time-scale basis. For all cases, the shoreline position at the end of the simulation is shown in Figure 14a. A positive correlation between the shoreline distance change and the sediment transport was observed (Figure 14b) and hence *RS*(*t*0) decreases as the sediment transport rate *Q* [40,41] increases (Figure 14c). Values of Δ*S*(*t*0) were estimated for wave conditions in Table 4.

**Table 4.** GSb simulated cases to investigate beach resistance to a 10 m groin, for waves of *Tp* = 3.5 s, after 720 h (*t*0) of simulation.


**Figure 14.** (**a**) Shoreline position, owing to the 10 m long groin presence, after 720 h (**b**) the corresponding shoreline distance increases for different wave conditions listed in Table 4 and (**c**) resistance index temporal evolution for different wave conditions listed in Table 4.

*Water* **2018**, *10*, 1806

The numerical model was also used to simulate the beach recovery phenomena (i.e., resilience) after the structure removal. Thus, the numerical model is initialized with the shoreline from Test 6A at *t*<sup>0</sup> = 720 h (e.g., yellow line in Figure 14a), subjected to different wave forcing conditions (Table 5), without the presence of the structure. The shoreline recovery is significant during the first 24-h for all cases (Figure 15a), consistent with the field observations (Figure 9); afterward, it continues at a lower rate. The numerical results were used to compute the temporal evolution of *RL* using Equation (2) for each case (Figure 14b). Contrary to beach resistance, numerical results suggest that the beach resilience is not controlled by the longshore transport potential and it depends on the alongshore diffusivity *G* given by [40,41],

$$G = \frac{2\mu}{(h^\* + B)} H\_{1/50,b}^{5/2} \cos 2\theta\_b \tag{4}$$

where *μ* is assumed equal to 0.15 m1/2 s−1, *h*<sup>∗</sup> is the closure depth, *B* is the berm elevation, *H*1/50,*<sup>b</sup>* is the value of *H*1/50 at breaking [40,41], *θ<sup>b</sup>* is the wave angle breaking with respect to the mean rectilinear trend of the shoreline. The diffusivity is associated to the longshore spreading of a shoreline perturbation owing to its departure from equilibrium for the existing forcing. The numerical simulations show that, for a given value of *H*1/50, the beach resilience increases as the value of *G* increases (Figure 15b and Table 5). Therefore, alongshore diffusivity plays an important role on beach resilience in the study area. ° °

**Figure 15.** Beach resilience after structure removal: (**a**) shoreline change with respect to time after the structure removal and (**b**) resilience index temporal evolution.


**Table 5.** GSb simulated cases to investigate beach resilience associated to the groin removal. The initial condition for all simulations corresponds to the shoreline position for Test 6A at *t* = 720 h. *θo* is the deep-water wave angle.

### **6. Conclusions**

A field and numerical study of shoreline resistance and resilience was conducted on a sea-breeze dominated sandy beach. The following conclusions were found:


**Supplementary Materials:** A demo version of the GSb numerical model, for Mac and Windows systems, has been made available for the scientific community and can be downloaded at: www.scacr.eu and field measurements are available at http://ocse.mx/en/experimento/beach-resilience-to-coastal-structures-brics upon request. A time-lapse video of the beach perturbation due to the temporary groin disturbance is included as supplementary material.

**Author Contributions:** Conceptualization, G.M., A.T.-F., P.A.T.; Writing-Original Draft Preparation, G.M., A.T.-F., G.R.T., A.F.; Writing-Review & Editing, G.M., A.T.-F., G.R.T., A.F., L.L.; Field work, G.M., A.T.-F., J.L., L.P.-A.; Field data Analysis, G.M., Structure design, L.P.-A., J.L., P.A.T., Numerical modeling, G.R.T., A.F., L.L.; Numerical data Analysis, G.R.T., A.F., L.L.; Funding Acquisition, G.M., A.T.-F.

**Funding:** This research was funded by CONACYT through the Cátedras-CONACyT (Project 1146), Investigación Científica Básica (Project 284819) and the Laboratorio Nacional de Resiliencia Costera (Project LN 293354). Additional financial support was provided by PAPIIT DGAPA UNAM (IN101218) and Instituto de Ingeniería UNAM.

**Acknowledgments:** We acknowledge the field support provided by students and researchers at the Laboratorio de Ingeniería y Procesos Costeros at UNAM, especially from Gonzalo Uriel Martín Ruiz, David A. Gracia, Elena Ojeda, Tonatiuh Mendoza, José Carlos Pintado-Patiño, José Alberto Zamora, Daniel Toxtega, Martín Ezquivelzeta, Miguel Ángel Valencia, Paola Espadas, Luis Ángel Gallegos, Daniel Pastrana, Jesús Aragón, Pedro Cabañas, Alejandro Astorga, Enna López, Rafael Meza, Wilmer Rey and Marcos García. Special thanks to Elena Ojeda and David Gracia for providing the time-lapse video of the experiment.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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