REDUCED COMPLEXITY MODELING OF SHORELINE RESPONSE BEHIND OFFSHORE BREAKWATERS

Prediction of the shoreline response behind offshore breakwaters is essential for coastal protection projects. Due to the complexity of the processes behind the breakwaters (e.g., wave diffraction, currents, longshore transport), detailed modelling needs high computational efforts. Therefore, simplifying the process effect in a simpler coastline model could be efficient. In this study, the coastline evolution model ShorelineS is used. A new routine was implemented in the model to adjust the wave heights and angles behind the offshore breakwaters. Two approaches from the literature and a newly introduced one were tested in this study. The model free grid system was used to simply track the breaker line; such an advantage also helped to form tombolo, which is not common for these types of models. The tests showed promising results for single and multi breakwaters systems; however, the newly introduced approach still needs further testing and refinement for better performance and less computational cost.


INTRODUCTION
Offshore breakwater schemes are applied as coastal protection measures in many places around the world. Predicting the response of the shoreline behind the breakwaters is, however, a great challenge for coastal engineers. Considerable changes in coastline curvature may take place behind the structure, especially when a tombolo develops. Detailed 2DH process-based models are often used to simulate the strongly curved local coastline features, but require an enormous computational effort to accurately compute the wave-driven transports. Using a coastline model would therefore be much more efficient, but the grids of the classical coastline models are typically not sufficiently flexible to cope with the curvature of tombolo's. The recently developed ShorelineS model (Roelvink et al., 2020) does, however, allow for flexible grid generation, which provides new opportunities. Here we add local wave shielding to the ShorelineS model to resolve the local coastline changes behind offshore breakwaters.

WAVE DIFFRACTION IN SHORELINE MODELS
In most of the existing shoreline models, the wave diffraction effect has been introduced; such a process should be simplified to avoid the high computational cost. Among the existing approaches; two were selected for this study: (the approaches are named by the studies first authors).

Dabees approach
The first selected approach was introduced by Dabees (2000). From the linear wave theory, to combine the refraction-diffraction behind the breakwaters, the wave heights are calculated as: The wave ray is assumed to follow a circular arc from the source point I to the point P, see Fig. 1 Where IP  is the angle of the straight line between I and P, 1  is the starting angle at I, P  is the wave angle at P, and  is the difference of the actual ray angle from the straight line angle. By Hurst approach Hurst et al. (2015) developed a shoreline model to investigate the behaviour of crenulate-shaped bays exposed to differing directional wave climates. The technique they used is to modify the wave angle and the wave power that approaches the coast, using this simple rule Where s  is the wave angle approached in the shadowed zone, ω is the angle between the shadow line and the shadowed cell and o  is the offshore wave approach angle.

WAVE DIFFRACTION IMPLEMENTATION IN SHORELINES
ShorelineS model computes shoreline changes behind offshore breakwaters as a result of the wavedriven transport. The computation of the effect of the breakwater on the breaking wave conditions (i.e., local wave height and angle) requires in the first place an accurate positioning of the breaker line, the methods to calculate these parameters are explained in this section.

Wave Height behind the Breakwater
For the wave height, Eq. 2,3, and 4 were used in the model. First, the d K is calculated from each tip, before combine both: for the waves coming from left and right.

Wave angle treatment behind the breakwater
Several approaches for treating the wave angle behind the breakwater have been tested in this study; three main approaches are explained in this section. Fig. 3-a illustrates definitions of parameters used in this study.

Modified Hurst approach.
Hurst approach (Hurst et al., 2015) was tested in this study after little modification: the diffracted wave angle is equal to 1.5 time the angle between the point of interest and the transition line (instead of the shadowed line). 3. Roelvink approach. A new approach to treat the wave angle is introduced in this study. The approach estimates the wave angle in the sheltered zone using a relative changing rate, that lies between the two selected approaches from the literature, see For all approaches, after calculating the diffracted angle from each tip; the final angle is combined based on the wave energy:

The breaker line behind the breakwater
One of the challenges in simulating the shoreline response behind the breakwater using only longshore transport formulas is the position of the breaker line (where the model calculates the corrected values of the wave height and angles).
In the existing models, it is common to assume a breaker line parallel to the shoreline, e.g., UNIBEST (Deltares, 2011).
In this study, a new approach has been introduced, taking into account the advantage of the model's free grid system. The approach assumes one separate perpendicular profile for each segment of the shoreline, the profile length follows the well-known Dean profile. During the simulation, to determine the location of the breaker line points; first, based on the breaking wave heights, the model estimates the breaking depths, second, it calculates the distances to the breaking depths from the shoreline. Then, based on these locations, the wave coefficients are calculated to correct the breaking wave heights. Finally, based on the calculated height, the model repeats the calculation to better estimate the locations, see Fig. 3. One of the advantages of the free grid is that the breaking points are updated during the simulation as the shoreline position changes, see the red dots in Fig. 4-d, and Fig. 5. The wave heights behind the breakwaters are lower than outside; therefore, the breaking depth points should be closer to the shoreline.

VERIFICATION TESTS
The previous approaches were implemented in the model; then, they were tested through a verification test.

Verification criteria
Several studies have been done on the shoreline response behind the breakwater, to validate the model at this stage, two studies were chosen. First, a study by Hsu and Silvester (1990) who developed a dimensionless relationship for the salient formation based on analysing data from prototypes Where B L is the breakwater length, B X is the distance between the breakwater and the original shoreline, and x is the distance between the new shoreline and the breakwater.
Second, a study by Khuong (2016) who used data for 93 projects with 1114 structures including physical conditions and shoreline measurements, and based on the analysis of such data, several empirical relationships were introduced regarding the salient or tombolo formation. For the salient, the study provided an empirical relation from the observation data: For the tombolo formation behind a single breakwater, the study provided a criterion /1 BB LX  which agrees with the laboratory study by Suh & Dalrymple (1987). Also, other empirical relations for tombolo width and the control points up and downdrift for the tombolo and salient formation were provided could be used to validate the model. In this study, the above criteria were chosen for the test; in summary, the shoreline should build a tombolo if the ratio between the length to the distance is more than one. While if the ratio is less than one, the shoreline should form a salient that should match with the empirical formulas above.
To perform the test, each approach was examined by running the model on 12 different situations that expect to cover all possible responses: limited response, salient, or tombolo. Different combinations of B X and B L were used to test the model for different expected responses, see Table 1.

Model setup
The test was performed nine times: the three wave angle approaches have been tested with three different methods to calculate the wave height (See Eq. 1) using r s d The modified version of Kamphuis formula (Kamphuis, 1991), presented by Dabees (2000), was used in the test, to include the wave height gradients alongshore The term sb dH ds was implemented so at each point (i), the model calculates the difference in wave height between (i-1, i+1) then divided by the distance. Other model inputs parameters were fixed for all simulations, see Table 2.

Model result
For Dabees approach, the overall trend is matched with the empirical relations for the salient. However, it did not allow to form a tombolo, see Fig. 4-a. For the modified Hurst and Roelvink approaches, the trend is also matched with the salient and both were able to form a tombolo, except the case when wave heights are calculated using r s d K K K , see Fig. 4-a and b. On the other hand using two coefficients sd KK or d K only, the results were acceptable, which has an advantage in terms of the simplicity of the calculations. Generally, both approaches the modified Hurst and Roelvink were acceptable in terms of forming tombolo. From another perspective which is the model behaviour, the Roelvink approach was preferred as the tombolo formation was smoother than in Hurst simulation. For wave heights, the method with

DISCUSSION AND CONCLUSIONS
Due to the offshore breakwater presence, the (breaking) wave height in the shadowed area is lower than in the exposed area. Such a difference leads to an alongshore gradient in the wave setup. Consequently, alongshore currents take place which leads to sand transport from the exposed to the shadowed area, so the equilibrium of the shoreline in the vicinity of the breakwater is mainly based on the gradient in breaking wave height and wave setup (Heerdink, 2003). Therefore, the breaker depth gradient plays a role in the equilibrium shape of the shoreline.
To implement the previous concept into the model, a mechanism was introduced to track the breaker line throughout the calculation in the shielded area. Such mechanism enhances the estimation of the breaker depths which affect the values of the breaking wave height and the angle for each segment before substituting into the longshore transport formula.
Three different approaches were tested to determine the wave angles behind the breakwaters: two from the literature (Dabees, 2000;Hurst et al., 2015) with little modification and a new approach was introduced. The new approach and the modification to Hurst approach have no quantified physical interpretation. The main concept behind them is to gradually change the wave orientation from the exposed area to the shielded area behind the breakwater. Such approaches keep the model preferential advantage, which is simplicity by simplifying the phenomena without complex calculations.
In some situations, when the breakwater length is larger than the offshore distance, there is a chance for a tombolo formation. These chances increase for higher waves. Tombolo formation is not always the preferred shoreline response behind the breakwater. If the breakwaters are built to reduce the erosion, a tombolo formation might increase the erosion downdrift. Moreover, it can be dangerous for beach users if the breakwater is built on a recreational beach. Therefore, predicting the beach response is very important in such situations (Heerdink, 2003). The new approaches introduced to the model in this research showed the ability to simulate tombolo formation, which is not common for such type of simple models.
Although the new approaches show a better prediction of the response in terms of the ability to form tombolo, Dabees approach showed better matching with the empirical formulas. Using Dabees approach, the shoreline was not able to form a tombolo because of the treated wave angles. Roelvink approach showed better behaviour than the modified Hurst approach in salient/tombolo formation. A couple of disadvantages of the new approaches: first, the treated wave directions behind breakwaters fluctuate for the same configuration based on the waves condition, second the shoreline builds up behind the breakwater faster than usual. The factors used in the new approaches should be tuned by comparing to physical or 2/3D numerical models.
During the validation test, only the ratio between breakwater length to offshore distance was used to test the shoreline response; however, more than 14 variables affect the shoreline response according to Hanson and Kraus (1990). Another limitation of this study is testing the shoreline response only by the final distance to the breakwater, which ignores the configuration of the final plan form and the up/downdrift areas. Both limitations should be considered to validate the model either by real-world case studies or by comparing to more empirical relations introduced in Khuong (2016). In addition, the longshore transport formula that includes the tidal current, introduced by Hanson et al. (2006), should be considered in future work.