- Email: [email protected]
accretion at the Gangneung Harbor, Korea
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Numerical investigation of the effect of wave diffraction on beach erosion/ accretion at the Gangneung Harbor, Korea Jong Dae Doa, Jae-Youll Jina, Sang Kwon Hyunb, Weon Mu Jeonga, Yeon S. Changa, a b
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Korea Institute of Ocean Science and Technology, Busan, Republic of Korea Sekwang Engineering Consultants CO., LTD, Seoul, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Telemac Coastal erosion Wave diffraction Shadow zone Breakwater
Since Gangneung Harbor breakwaters were built in the east coast of Korea from 1992 to 2002, the shoreline inside the harbor had severely changed as it was accreted in the area inside the harbor but eroded further south during the observation period (1979–2005). We investigated this process using Telemac-2DH model to calculate flows, waves, sediments and morphological changes in unstructured grids. The performance of the model was validated using the experiment data by Graven and Wang (2007). The model was run in two cases by turning on/ off the diffraction mode using the formula by Holthujisen et al. (2003). For this, we tuned the model by setting a shadow zone behind the breakwaters and by applying the diffraction formula only inside the shadow zone. The model results showed that wave heights significantly increased inside the shadow zone when the diffraction mode was on, compared to the case when the diffraction mode was off. This effect of diffraction was confirmed by the observations as the wave heights measured inside the shadow zone became 10–20% lower than those measured outside, which were nicely simulated by the model. In addition to the wave height, the wave-induced currents became stronger in the innermost area of the shadow zone with the diffraction mode. The model also successfully predicted the observed morphological change pattern because it simulated the shoreline erosion at the southern end of the shadow zone where the currents were bifurcated. Inside the shadow zone, sediment deposition occurred, corresponding to the observation, when the diffraction mode was on, whereas this deposition process was not simulated when the mode was off. The results support that the observed shoreline accretion in the harbor was mainly due to the reduced wave and current energy, which emphasizes the importance of accurate modelling of diffraction effect in the prediction of shoreline evolution.
1. Introduction Coastal structures such as breakwaters and groins are constructed to protect the shore from erosion by reducing the energy of approaching waves in the concerning area. If these structures are designed without sufficient considerations, however, it could lead to disastrous results causing shoreline erosions or accretions in unexpected places (Komar and McDougal, 1988). The side effect of groins is often observed when the sediments are eroded in the downdrift side of the structure while the shore is protected in the updrift side (Bakker et al., 1970). The side effects of breakwaters are not obvious because the damages may take place in wider area over longer period. One good example of the side effect of a breakwater is provided Fig. 1 that compares the shorelines of November 1979 and May 2005 in the south of the Gangneung Harbor located in the east coast of the Republic of Korea where waves normally approach the shore from the north or northeast. Two breakwaters were
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constructed around the harbor. The first breakwater (BW#1) was constructed during 1992–1997. After that, it was extended to southeast by constructing the BW#2 during 1997–2002. While these breakwaters have been successful in lowering the wave energy inside the harbor, they still have caused unexpected impacts on the shorelines behind the breakwaters. First, the shoreline has accreted at the south of the harbor (the accreted area is marked with red color in Fig. 1). Second, the coastline is eroded at further south where the area is marked with blue color. The erosion is significant in this region as the maximum retreat scale reaches up to ~100 m. It is likely that the accretion in the red colored area in Fig. 1 occurred when the sediments moved to north by the diffracted waves inside the shadow zone formed by the breakwaters. This sand movement to the north did not only cause the accretion but also cause deficit of sand at the southern end of the shadow zone in the blue colored area in the figure. It is because sediments outside the shadow zone kept
Corresponding author. E-mail address: [email protected] (Y.S. Chang).
https://doi.org/10.1016/j.jher.2019.11.003 Received 18 February 2019; Received in revised form 27 August 2019; Accepted 20 November 2019 1570-6443/ © 2019 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Please cite this article as: Jong Dae Do, et al., Journal of Hydro-environment Research, https://doi.org/10.1016/j.jher.2019.11.003
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Fig. 1. Comparison of shorelines measured at November 1979 (blue) and May 2005 (red) around the Gangneung Harbor.
Fig. 2. Layout of the laboratory experiment around the detached breakwater in the LSTF (Graven and Wang, 2007).
(Steady-state spectral WAVE model) (Smith et al., 2001) as they are based on spectral action balance equation. In spite of the weakness of these spectral wave models in calculating the diffraction effect, its performance has been improved by modifying the mild slope equation (e.g. Lin, 2013). One of the main purposes of this study is to investigate the performance of diffraction caused by the Gangneung Harbor breakwaters using Telemac-2D system. For this, we propose to examine two cases of wave propagations by turning on/off the diffraction effect of the model. Although the impact of wave diffraction on the coastal erosion behind coastal structures has been widely perceived, its process has not been closely investigated, which provides a motivation for this study. For the diffraction mode, we tuned the model to apply the Holthujisen’s formula only within a shadow zone that was predetermined before
moving south by the waves that were not influenced by the diffraction instead of providing sediments from the south, which caused the erosion in this area. To further investigate this process, we have employed the 2DH mode of TELEMAC-MASCARET system (Telemac-2D) developed by National d’ Hydraulique et Environnement (LNHE), part of the Research and Development Rirectorate of the French Electricity Board (Hervouet, 2007). Telemac-2D is coupled with the Tomawac wave model (Benoit et al., 1996). Tomawac is a phase-averaged spectral wave model, and it is able to consider the diffraction effect by modifying the wave number as proposed by Holthuijsen et al. (2003) in which the mild slope equation (Berkoff, 1972) is tuned with the energy propagation speed in geographic space. Tomawac is similar to SWAN (Simulating Wave Nearshore) wave model developed by Delft University of Technology (Booij et al., 1999; Ris et al., 1999) and STWAVE 2
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Fig. 3. Grid system in the computational domain with water depths in colored contours.
2. Model description
running the model. The predetermination of the shadow zone is necessary in order to save the computational cost and to reduce the modeling error outside the shadow zone. This error may occur when the diffraction formula is used in the area where the diffraction is not expected to occur. The predetermination of the shadow zone is possible in specifically this study because the direction of the approaching waves is also predetermined based on the observation. The second purpose of the present study is to investigate the littoral process that results in the shoreline accretion and erosion as observed in Fig. 1. For this, we calculate morphological changes based on the modeled hydrodynamics. As restricted in most of 2DH models, Telemac-2D cannot simulate the shoreline evolution directly because the coastline has to be fixed as a boundary during the model run. In Telemac-2D, however, it is still possible to estimate the shoreline process when minute change of morphology can be calculated in fine grids near the coast by employing unstructured grid system. Another benefit of the Telemac-2D system is that the Tomawac wave model is internally coupled into the flow module, which enables to simulate the wave-current interaction so that not only the wave-induced currents but also the impact of current on the wave field can be examined in the shadow zone of the harbor. The tides are not considered in this numerical study as the tidal ranges in the experimental site are generally not higher than 1 m.
Telemac-2D is a model package that consists of the flow, wave and sediment transport modules. One of the most characteristic features of the Telemac-2D is the unstructured grids. By using the finite element method, irregular shaped grids are applied to simulate hydrodynamic conditions in coastal regions with complex coastline. In the flow module, the shallow water equations are employed by solving the continuity and momentum equations (Asaro and Paris, 2000; Hervouet, 2000; Giardino et al., 2009; Robins and Davies, 2011) as
→ ∂h + u∙∇ (h) + h ∗ div (→ u ) = Sh ∂t
(1)
→ ∂u → → ∂Z 1 + u ∙∇ (u) = −g + Sx + div (hνt ∇ u) ∂t ∂x h
(2)
→ ∂v → → ∂Z 1 + v ∙∇ (v ) = −g + Sy + div (hνt ∇ v ) ∂t ∂y h
(3)
where h is the depth, νt is the momentum diffusion coefficient, z is the free surface height. Sh , Sx and Sy are the source/sink for the mass and momentum. In the wave module, Tomawac is internally coupled into Telemac-2, 3
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Fig. 4. Contours of the modeled wave heights around the detached breakwater.
equation as
which can generate the wave-induced currents and the littoral transport (Villaret et al., 2013). In the spectral action balance equation the wave action density, N , is conserved as
∂N ∂N ∂N ∂N ∂N + ẋ + ẏ + k ẋ + k ẏ = Q (x , y, k x , k y, t ) ∂t ∂x ∂y ∂x ∂y
∂hC ∂hUC ∂hVC ∂ ⎛ ∂C ⎞ ∂ ⎛ ∂C ⎞ + + = h∊ + h∊ +E−D ∂t ∂x ∂y ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ⎜
where k x and k y are the wave number vectors, and Q is the source/sink of wave action. The diffraction effect considered in Tomawac was proposed by Holthuijsen et al. (2003). The basic concept is that the wavelength changes in the presence of diffraction. The new wave number, k2 , is given as (5)
where k1 is the wave number without diffraction and δ is the diffraction parameter. δ is formulated by the mild slope equation (Berkoff, 1972) as
δ=
(1 − p)
(6)
where a is the wave amplitude, C is the celerity and Cg is the group velocity. The new wave number is used to solve the wave action conservation equation. The energy propagation speed in case of diffraction is given as
Cgd =
k2 Cg k1
In this section, the performance of Telemac-2D model is tested before applying to the real field observation data. For the validation, a dataset measured in the Large-scale Sediment Transport Facility (LSTF) is employed. LSTF is the laboratory facility developed by United States Army Corps of Engineers (USACE) for investigating nearshore sediment transport process (Hamailton and Ebersole, 2001). The facility consists of a 30-m-wide-, 50-m-long-, 1.4-m-deep-basin and includes wave generators, a sand beach and an instrumentation bridge with acoustic sensors and wave gages. Waves are generated using four synchronized unidirectional spectral wave generators. The dataset used to evaluate the model was obtained from the experiment by Graven and Wang (2007). In their experiment, a 4 m-long rubble-mound detached breakwater (DB) was located parallel to the shoreline at 4 m offshore from the coast as shown in Fig. 2 (the figure is captured from Graven and Wang, 2007). The wave height at the generator was 0.26 m and period was 1.5 s with wave direction of 10.0°. The sediment size was 0.15 mm. The model computational domain was constructed within LSTF (red rectangle in Fig. 2) with grid size of 0.25 m in both longshore and crossshore direction. Fig. 3 shows the grid system in the computational
where Cgd is the energy propagation speed in geographic space in case of diffraction. In the sediment transport module, the bedload transport is formulated for the non-cohesive sediments (Camenen and Larson, 2005) as
θ θ Φw = a w θcw, on +θcw, off θcw, m exp ⎛−b cr ⎞ Φc = ac θc θcw, m exp ⎛−b cr ⎞ ⎝ θcw ⎠ ⎝ θcw ⎠ ⎟
⎜
(10)
3. Model validation
(7)
⎜
∂Zb + ∇∙Qtot = 0 ∂t
where p is bed porosity. Zb is the seabed elevation, and Qtot is the total sediment transport obtained from Eqn. (8) and (9).
∇∙ (CCg ∇a) k12CCg a
(9)
where C is the suspended sediment concentration. h is the water depth and ∊ is the diffusion coefficient. E and D are the erosion and deposition rate, respectively. In Eqn. (9), E = ws Ceq where ws is settling velocity and Ceq is equilibrium near-bed concentration determined from an empirical formula, and D = ws Cref where Cref is the sediment concentration at the reference elevation. Therefore, E − D in Eqn. (9) is set proportional to the difference between Ceq and Cref . Once the sediment transport is calculated by solving Eqn. (8) and (9), the morphological change can be obtained as
(4)
|k2 |2 = k12 (1 + δ )
⎟
⎟
(8) where Φw and Φc are the normalized sediment transport by waves and current respectively. ac and b are empirically determined coefficients. θcw, m and θcw are the mean and maximum Shields parameter due to wave and current interaction, θcw, on and θcw, off is the maximum Shield parameter at onshore and offshore phase respectively, θc is the Shields parameter due to current, and θcr is the critical Shields parameter. The suspended sediment is formulated using the advection–diffusion 4
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Fig. 5. Comparison of the wave heights and currents between the model and observation data along the five selected cross-shore profiles at Y18, Y23, Y24, Y25, and Y34 as their locations are marked in Fig. 2. Black dots denote the measurements of the wave heights and longshore current velocities. Blue lines denote the model data.
simulated with high accuracy (> 90%). In case of the longshore current, the modeling accuracy becomes lower at ‘Y23′ and ‘Y24′ in the lee area of the DB while it is still sufficiently high at other locations. The errors in this area are likely due to the circulations produced in the edge of the DB, caused by the obliquely propagating waves because its turbulent eddy pattern is difficult to be accurately simulated by numerical models. Except for the complex turbulent flow motions near the DB edges, the Telemac-2D model data is in good agreement with the measured data, providing a verification of the model performance specifically on the effect of wave diffraction.
domain with water depths in colored contours. The location of the DB is marked in grey color. The incident wave conditions are based on the numerical experiment by Nam et al. (2009) as Texel-Marsen-Arsole (TMA) was used at the boundary to produce wave height of 0.23 m and period of 1.5 s. The model was run 185 min until a steady state reached. In Fig. 4, the modeled wave heights are contoured around the DB. The effect of wave diffraction is clearly observed as the wave height gradually decreased in the lee of the DB. The model results are also quantitatively evaluated when the wave height and currents are compared with measurements in Fig. 5, along the five selected cross-shore profiles from ‘Y18′ to ‘Y34′ as their locations are marked in Fig. 2. In the figure, black dots denote the wave heights and longshore current velocities measured along the instrumentation bridge, and the blue lines show the results computed by the Telemac-2D model. The model nicely represents the wave height field of the observation data not only offshore area of the DB but also in the lee area of it. Specifically, the magnitudes of reduced heights of the diffracted waves are successfully
4. Model setup in the Gangneung Harbor The wave conditions at Gangneung area vary seasonally. In summer when the weather is mild, the waves generally approach the shore in the normal direction (i.e. approach from NE) with wave heights < 1.5 m. In winter, the weather becomes severe as the extratropical 5
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Fig. 6. Rose diagram that shows the distribution of the wave heights and directions observed in Feb. 2015
zone, we could significantly save the computational cost to reduce the running time. Another important reason is that, if we applied the diffraction formula at every grid in the computational domain, the model became unstable outside the shadow zone where the diffraction was not expected to occur. Once the shadow zone was determined, the simulations were performed in two cases. In Case 1, we ran the model by turning the diffraction mode off, whereas the diffraction was on in Case 2. In both cases, we applied the wave-current interaction effect that is internally coupled in the Telemac-2D modeling system. The model was run for twelve hours by simulating a typical winter storm condition and by applying JONSWAP wave directional spectrum for the boundary condition (Hasselmann et al., 1973). We designed the wave directional spectrum to be narrow enough to generate the waves to mainly propagate from the north.
storms frequently develop in the East Sea (Jeong et al., 2007). Since the winter storms generally develop in the north of the Ganngneung area, the high waves approach the shore obliquely (i.e. approach from N), which produces the shading zone behind the breakwater. In Fig. 6, a rose diagram that shows the wave direction and height distribution that was observed in February 2015 in which 2–3 m height waves mainly approach the coast from the north. Based on this observation, we configured the model conditions by generating the waves to approach the breakwater from the north with the designed wave height 2 m and period 8 s at the model boundary to simulate a typical winter storm condition. In addition, this wave condition was specifically designed to produce a comparable wave height with the observed data measured during a storm period in the Gangneung Harbor, which will be discussed in Section 4. It should be also noted that the effects of tidal currents were not considered in this numerical experiments because tidal ranges are no greater than 1.0 m in this area and thus the tideinduced currents may not be significant. Fig. 7 shows the computational domain with unstructured grid system, which provides benefits in the simulations of this study. The complex coastlines around the Gangneung Harbor are nicely resolved with the unstructured grids: the grid size becomes finer near the structure to capture the minute changes in the coastline near the breakwaters. The computational domain is 4950 m × 2070 m. The grid size varies from a minimum of 10 m to a maximum of 60 m, by which 15,491 grid nodes are employed. Δt is set to 2 s and the model was run for 12 h in total. In order to investigate the wave diffraction effect, we divided the computational domain into two parts. In the ‘shadow zone’ that is marked with blue color in Fig. 7, we set a switch to turn on the diffraction mode by applying the Eqs. (1) and (3). The shadow zone was determined before running the model as we set the wave direction to the north. By applying the diffraction mode only inside the shadow
5. Field observation In this section, observation data sets measured near the Gangneung Harbor are used to investigate the wave diffraction effect inside the shadow zone. In Fig. 7, the three observational stations (S1, S2, and S3) are marked as S1 is located inside the shadow zone and S2 and S3 are located on the borderline and outside the shadow zone respectively. At these three stations, wave data were measured from 21th to 31st of March 2015 by the Korea Institute of Ocean Science and Technology. They were obtained at every half hour using bottom-mounted WTG (Wave and Tide Gauge) which measured the water pressure for 1024 s with frequency 2 Hz. This gave one power density spectrum (PDF) per data burst. The significant wave heights, Hs , as well as peak wave period, Tp , were then calculated from the PDF as its outcomes are shown in Fig. 8. During the period, severe wave conditions were observed two 6
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Fig. 7. The unstructured grids in the computational domain. The blue area denotes the shadow zone behind the Gangneung Harbor breakwaters. The shadow zone was determined by assuming the wave propagation from the north (N). W1, W2 and W3 are the locations where the modeled wave heights are compared between the two cases.
6. Results
times at March 23 and 26 at which HS were higher than 1.5 m and Tp were longer than 10 s. Although the pattern of Hs and Tp is generally similar between the three stations, meaningful discrepancies are also observed in the wave heights. At March 25–26, Hs rapidly increases likely due to local storm as it has a peak at March 26. At this time of peak, however, Hs are different between the stations as it is higher outside the shadow zone than inside. At S1, Hs ranges between 1.5 and 1.75 m while it ranges between 1.75 and 2.25 m at both of S2 and S3. This indicates that Hs has been reduced ~20% inside the shadow zone as the propagating waves are being diffracted. Although this lower Hs at S1 could be alternatively argued that the wave height becomes lower at S1 possibly due to the wave breaking, the similar water depth (h = 3.5, 3.9, and 8.1 m at S1, S2 and S3 respectively) between S1 and S2 excludes this postulate because of the higher Hs at S2. In addition, the similar wave heights between S2 and S3 supports that the waves were not breaking until they reached at S2. Furthermore, this pattern of Hs reduction inside the shadow zone not only occurs at March 26 but repeats at other times. At March 23, Hs ranges 1.5–1.6 m at S1, but it increases ~10% at S2 and S3 as Hs reaches up to 1.75 m at both of the stations. Even at March 31, Hs at S1 is ~10% lower than that at S3 although it is not clearly observed as their wave heights are only ~0.5 m. Compared to Hs , Tp is similar between the three stations and shows no clear discrepancy, which indicates that the likely reason for the wave height reduction inside the shadow zone is the wave diffraction. The model results that simulated the wave heights measured at March 26 will be discussed in the next section.
The pattern of the wave propagation calculated by the model is compared between the two cases in Fig. 9 in which the spatial distribution of model wave height, Hm0 , is contoured around the shadow zone. As expected, the Hm0 outside the shadow is almost identical between the two cases. Inside the shadow zone, however, it shows significant difference. In both cases, Hm0 decreases as the waves propagate inside the shadow zone from its southern boundary. When the diffraction mode is off in Case 1, however, Hm0 decreases more rapidly as the distance from the boundary increases. In Case 2, Hm0 inside the shadow zone is generally higher than that of Case 1, indicating that the waves propagate further inside the shadow zone due to the effect of the wave diffraction. A direct comparison of the modeled wave heights is shown in Fig. 10 in which the time series of Hm0 are compared between the two cases at the three observation stations (S1, S2, and S3). In the figure, the grey solid lines show the time variation of modeled Hm0 in Case 1 while the black dashed lines show the results of Case 2. It clearly shows that the wave heights are zero for the first couple of hours until the waves reach the stations. Once the waves arrive, Hm0 sharply increases higher than 1 m and gradually becomes stable at each of the stations at t > 1.02 d. At S1, H reaches ~1.5 m in both cases. It is interesting to note that this modeled Hm0 is similar to Hs measured in S1 at March 26 because the model conditions were designed to simulate the wave height of that storm period at the station. It is also interesting to find out that the pattern of Hm0 variation in S1 is different between the two 7
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Fig. 8. Observed data of significant wave heights (Hs ) and peak wave period (Tp ) from 21th to 31st of March 2015 using bottom-mounted WTG (Wave and Tide Gauge) measured at (a) ‘S1′, (b) ‘S2′ and (c) ‘S3′.
similar in both stations, S2 and S3. The similarity in the modeled wave heights indicates that the effect of wave diffraction diminishes outside the shadow zone, causing homogeneous wave field. When compared with the observation in which Hs ranges between 1.75 and 2.25 m at both S2 and S3 at March 26, the modeled wave height is within the comparable range as Hm0 ~ 1.7 m in both S2 and S3 for both cases. Therefore, Hm0 is reduced about 12% when the waves propagate into the shadow zone, which corresponds to the observation data. In order to investigate the diffraction effect inside the shadow zone,
cases. While the wave height is constant as Hm0 ~ 1.5 m at t > 1.01 d in Case 1, it shows a wiggled pattern in Case 2. This inconsistent Hm0 is probably because the wave diffraction causes inhomogeneous spatial distribution inside the shadow zone as shown in Fig. 9(b). Although it is not clearly observed, Hm0 in Case 2 is generally higher than that in Case 1 at S1 due to the diffracted waves that propagate into the shadow zone. This contribution of wave diffraction can be hardly observed outside the shadow zone. At S2 and S3, Hm0 in Case 1 is similar to that in Case 2 showing no wiggled pattern in both cases. In addition, Hm0 is even 8
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Fig. 9. Distribution of the modeled wave heights near the Gangneung Harbor. (a) Case 1: diffraction mode off; (b) Case 2: diffraction mode on.
flows southeast along the coast. This longshore current is generated when the waves approach the shore at an angle, producing wave-induced trust along the shore. The current pattern is similar between the two cases outside the shadow zone. It is because the wave conditions are same outside the shadow zone in both cases, not affected by the diffraction. However, the flow patterns are different at peeper part away from the shore (marked with red rectangle ‘D’) because circulations are observed when the diffraction mode is on. These flows were generated by the stronger wave-induced currents when the gradient of wave energy became greater due to diffraction. It should be noted that the magnitude of the wave-induced currents might be overestimated. Although tides or other conditions that could increase the current strength were not considered in the input conditions, the maximum current speed became as large as 0.5 m/s as shown in Fig. 12(c) and (d). Unfortunately, no measured data for currents are available for a direct comparison. However, the model results may be indirectly examined with the LSTF data. The input wave height and period for the laboratory experiment were 0.26 m and 1.5 s respectively, and both of the measured and modeled data in Fig. 5 show that the wave heights ranged at 0.1–0.2 m and the longshore current speed ranged at 0.1–0.2 m/s around the DB. Therefore, the magnitude of modeled currents near the Gangneung Harbor may still be within acceptable range, considering much larger input wave conditions (wave height = 2 m, wave period = 8 s). The morphological changes caused by the waves and currents inside the shadow zone were calculated using Eqs. (8) to (10). We increased the model’s morphology factor that determines the rate of morphological change 10 times larger than the default value. We employed this unusual value of the parameter in order to maximize the impact on the morphological change within the limited short time for model integration, which is justified when our purpose of the experiments is only to compare the pattern of the change between the two cases but not to exactly quantify the rate of changes. In Fig. 13, the morphological change calculated during the integration time is contoured. As expected, the seabeds in the nearshore region are generally under erosion near the southern boundary of the shadow zone (marked with red circle ‘D’) in both cases. It is interesting to find out that this erosional area roughly corresponds to the erosional area observed in Fig. 1. This indicates that the shoreline is vulnerable to the erosion in this area where the wave height sharply decreases in the shadow zone as shown in Fig. 9, which causes gradients in the sediment transport distribution along the coast. This location of the erosion also roughly corresponds to the circled areas ‘A’ and ‘B’ in Fig. 12 where the bifurcated currents carried sediments into different directions, which supports the erosion
three additional stations (W1, W2 and W3) are chosen as shown in Fig. 7. These three stations are located further inside the shadow zone compared to the observation stations. Specifically, we chose the location of W3 to be closest to the observation station S1 in order for the comparison with the modeled results at W1 and W2 where no observation data are available. Fig. 11 shows the time variation of H at W1, W2 and W3. As shown in Fig. 7, Hm0 in W3 is identical to that in S1 as it is also similar to the Hs measured at S1 at March 26. At this location, Hm0 shows small difference between the two cases as Hm0 in Case 2 is a bit higher than that in Case 1 when the diffraction mode is off. At W2, the difference in Hm0 increases between the two cases as Hm0 in Case 2 reaches ~1.0 m while it is only ~0.7 m in Case 1. The difference in the wave height can be more clearly observed at W1, the station located in the innermost inside the shadow zone, where Hm0 in Case 1 is only ~0.2 m while it is still ~1.0 m in Case 2. It is interesting to note that the wave height in W1 of Case 2 is similar to that in W2, which indicates that the waves successfully propagate inside the shadow zone when the diffraction mode is on. On the contrary, the waves in Case 1 do not reach to the innermost site due to the weak wave diffraction. It is also interesting to find out that Hm0 in Case 2 at W1 and W2 also shows a wiggled pattern due to the wave diffraction while Hm0 in Case 1 shows small variation in time. In Fig. 12, velocity vectors are shown to compare the flow patterns between the two cases. These wave-induced currents are produced when the waves are shoaled, refracted and diffracted over the complex topography in the shallow region around the harbor. In both cases, longshore currents strongly develop along the coast. It is also observed that the longshore currents are bifurcated near the southern end of the shadow zone (marked with red circle ‘A’) as a branch flows southeast outside the shadow zone while the other flows northwest into the shadow zone. The flow pattern of the northwest branch becomes complicated near the location marked with the red circle ‘B’ in Fig. 12 where a strong offshore current develops as the northwest branch is again bifurcated. It is likely that the offshore current develops due to the seabed topography. As shown in Fig. 12(c) and (d), the water depth became shallower at the location northwest of the circle ‘B’, which blocks the way of the current along the shore and divert it to the offshore. A significant discrepancy is found between the two cases at the innermost area just south of the Gangneung Harbor (marked with red circle ‘C’). While the branch of the longshore currents that flows northwest along the coast does not reach this area in Case 1, the current is strong enough to reach the area in Case 2 when the diffraction mode is on. Outside the shadow zone, the other branch of the bifurcated current
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Fig. 10. Time series of the modeled wave heights at the three observation stations (S1, S2 and S3) marked in Fig. 2. Grey solid lines: Case 1 (diffraction mode off); Black dashed lines: Case 2 (diffraction mode on).
was likely affected by the flow separations. Regardless of the similarity in the morphological change between the two cases, a significant difference is found in the innermost area of the shadow zone (marked with red rectangle ‘E’ in Fig. 13). While it is not observed in Case 1, the seabed close the coastline is accreted in Case 2 as it is likely due to the higher waves shown in Fig. 9 and also due to the stronger currents observed in Fig. 12. This deposition area corresponds to the area of the observed shoreline accretion (marked with red color in Fig. 1). This different result between the two cases indicates that the morphological changes inside the shadow zone are successfully simulated only if the diffraction mode is on. Another important question that arises on the sediment motions in the shadow zone is the impact of the wave-current interaction. As mentioned earlier, the Tomawac wave model is internally coupled into the flow module in Telemac-2D so that the currents may influence the wave field. In order to examine this, we performed additional model
run in which both of the diffraction and wave-current interaction modes were turned off. Fig. 14 compares Hm0 of this specific case with that of Case 1 (i.e. wave-current interaction is on but diffraction is off) at W1. Since the diffraction is not considered in both cases, Hm0 is low (~0.1 m). However, a difference between the two is clearly observed as H becomes higher if the wave-current interaction is considered. This difference in the wave height indicates that the current can influence on the wave field, though not significant, inside the shadow zone even without the diffraction. 7. Discussion In this section, we discuss the limitations of the present study in applying the phase-averaged wave model for the wave diffraction problem. In the spectral wave models, the diffraction effect has been implemented by modifying the wave number obtained from an energy 10
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Fig. 11. Time series of the modeled wave heights at the three model stations (W1, W2 and W3) marked in Fig. 2. Grey solid lines: Case 1 (diffraction mode off); Black dashed lines: Case 2 (diffraction mode on).
(graphics processing unit) instead of CPU (central processing unit) (Tavakkol and Lynett, 2017). However, these GPU-based phase-resolving models have not been coupled into the coastal engineering models that are designed to predict the morphological changes. Therefore, the scope of the present study has been limited to the phase-averaged modeling. The achievement of the present study is then the successful investigation of the impact of wave diffraction on the shoreline response in engineering-scale modeling. Another shortcoming of the present study is the simulation of input waves. Instead of producing the realistic wave scenarios, we used narrow wave spectrum to generate monotonous input wave conditions. In doing so, we produced the model wave height at W1 similar to the significant wave height measured at S1 in March 26, in order to clarify the effect of the wave diffraction. If broader directional spectrum was employed as an input, the wave diffraction effect might not be clearly identified because the waves at one-time step would advance into
balance equation (Booij et al., 1997; Rivero et al., 1997), however, the stability of these models was low (Mase, 2001). Holthuijsen et al. (2003) proposed the phase-decoupled method into SWAN by introducing the diffraction parameter, δ , as expressed in Eqn. (6). Since then, various efforts have been made to improve the diffraction effects in the spectral wave models (e.g. Lin, 2013). Its performance is still not satisfactory compared to the phase-resolving models. For example, the Boussinesq wave models such as FUNWAVE (Shi et al., 2012) have shown better performance in the wave diffraction effect. In addition, even three-dimensional models have been developed to simulate the wave propagations by directly solving the Navier-Stokes equations (Miquel et al., 2018). Although these phase-resolving models are more accurate in resolving the wave diffraction, their application to coastal modeling in engineering practices is still limited due to high computational cost. Recently, the performance of Boussinesq models has been significantly improved by solving the parallel codes using GPU 11
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Fig. 12. Comparison of flow vectors of the wave-induced currents. The shadow zone in Fig. 2 is marked with grey color. The three circles ‘A’, ‘B’ and ‘C’ denote the location where the flow pattern significantly changes. (a) Case 1: diffraction mode off; (b) Case 2: diffraction mode on; (c) magnified view around the circled area in (a) with bathymetry contours; (d) magnified view around the circled area in (b) with bathymetry contours.
We agree that an attempt to speculate on long-term beach morphological changes from short-term numerical calculations is not convincing because it inevitably includes errors inherent in the time-scale gap between the model results and the shoreline data, and thus application of the model results should be limited. Notwithstanding these limitations, this study provide meaningful results and indications on the shoreline process inside the shadow zone, which requires careful consideration in designing coastal structures to minimize the effect of diffraction that may cause unexpected erosions in the coastline.
broader directions in the next modeling step and some waves might propagate into the shadow zone regardless of the diffraction. By narrowing the input wave spectrum, therefore, we also narrowed the wave propagating directions outside the shadow zone so that the wave field inside the zone would be mainly influenced by the diffraction, which resulted in the clear discrepancies between the two cases of the model runs. Recognizing these shortcomings, the scope of the present study has been confined to examine the scenario of shoreline evolution process that might be caused by the wave diffraction in the shadow zone, not focusing to enhance the accuracy of simulations. For this, the shortterm numerical results from Telemac-2D phase averaged model was applied to understand the long-term process by extending its prediction.
8. Conclusions In this paper, we examined the role of wave diffraction on shoreline 12
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Fig. 13. Modeled morphological changes. (a) Case 1: diffraction mode off; (b) Case 2: diffraction mode on; (c) magnified view inside the rectangular region in (a); (d) magnified view inside the rectangular region in (b).
Fig. 14. Comparison of the wave heights modeled at the station W1 with diffraction mode off. Solid line: wave–current interaction off; dashed line: wave-current interaction on.
that was set behind the breakwaters by assuming the waves approaching from the north. The effect of wave diffraction was confirmed by the observations as the significant wave heights measured inside the shadow zone were 10–20% lower than those measured outside the shadow zone. This observational process was nicely simulated by the model because the modeled wave heights were also reduced with similar rate (~12%) inside the shadow zone, in case the diffraction mode was on. When the
evolution inside a shadow zone behind Gangneung Harbor breakwaters. We performed numerical experiments in the region because shoreline change occurred after two breakwaters were constructed around the harbor. We employed the Telemac-2D model in unstructured grids to simulate the waves, wave-induced currents, and the resulting morphological changes. We ran the model in two cases – 1) diffraction mode on, 2) diffraction mode off. When turning on the diffraction, we applied the formula by Holthuijsen et al. (2003) inside the shadow zone 13
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diffraction was off, the reduction of wave height inside the shadow zone become more serious, indicating that the waves did not successfully propagate into the shadow zone. In addition to the wave heights, the wave-induced currents were stronger inside the shadow zone when the diffraction mode was on. In this case, the more active hydrodynamic energy resulted in the sediment transport even to the innermost area of the shadow zone, which corresponds to the observation. The deposition process was not successfully simulated if the diffraction mode was off. In both cases, however, erosions occurred at the southern boundary of the shadow zone, which is likely due to the strong sediment transport gradient caused by the wave energy reduction at the zone boundary. The results of the present study indicate that the wave diffraction may play significant roles on shoreline evolutions behind coastal structures where the wave energy is expected to decrease. Therefore, it is important to improve the accuracy of the diffraction effect in modeling coastal processes. In this regards, future studies are required to test the phase-resolving wave models with higher accuracy, for the applications in engineering practices.
2004.10.019. Giardino, A., Ibrahim, E., Adam, S., Toorman, E.A., Monbaliu, J., 2009. Hydrodynamics and cohesive sediment transport in the ijzer estuary, belgium: case study. J. Waterway Port Coastal Ocean Eng. 4, 176–184. https://doi.org/10.1061/(ASCE) 0733-950X(2009) 135:4(176). Graven, M.B., Wang. P., 2007. Data Report: Laboratory Testing of Longshore Sand Transport by Waves and Currents; Morphology Change Behind Headland Structures. ERDC/CHL TR-07-8. US Army Corps of Engineers. Engineer Research and Development Center. Hamailton, D.G., Ebersole, B., 2001. Establishing uniform longshore currents in a largescale sediment transport facility. Coastal Eng. 42 (3), 199–218. https://doi.org/10. 1016/S0378-3839(00)00059-4. Hasselmann K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Mller, P., Olbers, D. J., Richter, K., Sell, W. Walden. H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)' Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe, A(8) (Nr. 12), p.95. Hervouet, J.-M., 2000. A high resolution 2-D dam-break model using parallelization. Hydrol. Process. 14, 2211–2230. https://doi.org/10.1002/1099-1085(200009) 14:13<2211::AID-HYP24>3.0.CO;2-8. Hervouet, J.M., 2007. Hydrodynamics of Free Surface Modelling with the Finite Element Method. Wiley, Newyork. Holthuijsen, L.H., Herman, A., Booij, N., 2003. Phase-decoupled refraction-diffraction for spectral wave models. Coastal Eng. 49 (4), 291–305. https://doi.org/10.1016/S03783839(03)00065-6. Jeong, W.M., Oh, S.H., Lee, D.Y., 2007. Abnormally high waves on the east coast. J. Korean Soc. Coastal Ocean Eng. 19 (4), 295–302 (In Korean). Komar, P.D., McDougal, W.G., 1988. Coastal erosion and engineering structures: The Oregon experience. J. Coastal Res. SI 4, 77–92. Lin, H.-G., 2013. An improvement of wave refraction-diffraction effect in SWAN. J. Mar. Sci. Technol. 21 (2), 198–208. https://doi.org/10.6119/JMST-012-1207-1. Mase, H., 2001. Mulit-directional random wave transformation model based on energy balance equation. Coast. Eng. J. 43 (4), 317–337. https://doi.org/10.1142/ S057856340 1000396. Miquel, A.M., Kamath, A., Chella, M.A., Archetti, R., Bihs, H., 2018. Analysis of different methods for wave generation and absorption in a CFD-based numerical wave tank. J. Mar. Sci. Eng. 6, 1–21. https://doi.org/10.3390/jmse6020073. Nam, P.T., Larson, M., Hanson, H., Hoan, L.X., 2009. A numerical model of nearshore waves, currents, and sediment transport. Coastal Eng. 56 (11–12), 1084–1096. https://doi.org/10.1016/j.coastaleng.2009.06.007. Ris, R.C., Holthuijsen, Booij, N., 1999. A third-generation wave model for coastal regions 2. Verification. J. Geophys. Res. 104 (C4), 7667–7681. https://doi.org/10.1029/ 1998JC900123. Rivero, F.J., Arcilla, A.S., Carci, E., 1997. Analysis of diffraction in spectral wave models. Proceedings of the 3rd International Symosium on Ocean Wave Measurement and Analysis, WAVES ’97, ASCE, New York, 431-445. Robins, P.E., Davies, A.G., 2011. Application of TELEMAC-2D and SISYPHE to complex estuarine regions to inform future management decisions. XVIIIth Telemac & Mascaret User Club. Chatou, France, October 19th–21th, 1-6. Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T., 2012. A high-order adaptive time-stepping TVD solver for Boussinesq modelling of breaking waves and coastal inundation. Ocean Modell. 43–44, 36–51. https://doi.org/10.1016/j.ocemod.2011. 12.004. Smith, J.M., Sherlock, A.R., Resio, D.T., 2001. STWAVE: Steady-state spectral wave model user’s manual for STWAVE, version 3.0. Coastal and Hydraulics Laboratory, US Army Corps of Engineers. Tavakkol, S., Lynett, P., 2017. Celeris: a GPU-accelerated open source software with a Boussinesq-type wave solver for real-time interactive simulation and visualization. Computer Phys. Comm. 217, 117–127. https://doi.org/10.1016/j.cpc.201 7.03.002. Villaret, C., Hervout, J.-M., Kopmann, R., Merkel, U., Davis, A.G., 2013. Morphodynamic modeling using the telemac finite-element system. Comput. Geosci. 53, 105–113. https://doi.org/10.1016/j.cageo.2011.10.004.
Acknowledgments This research was supported by the Korea Institute of Ocean Science and Technology for the projects titled ‘Development of application technologies for ocean energy and harbor and offshore structures [PE99731]’ and ‘Operation of KIOST open ocean research infrastructure and establishment of public utilization service system [PKA0017]’. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jher.2019.11.003. References Asaro, G., Paris, E., 2000. The effects induced by a new embankment at the confluence between two rivers: Telemac results compared with a physical model. Hydrol. Process. 14, 2345–2353. https://doi.org/10.1002/1099-1085(200009)14:13 <2345::AID-HYP33>3.0.CO;2-X. Bakker, W.T., Breteler, K., Roos, A., 1970. The dynamics of a coast with a groyne system. Proceedings of 12th conference on coastal engineering, Washington D.C. 1001-1020. Benoit, M., Marcos, F., Becq, F., 1996. Development of a third generation shallow water wave model with unstructured spatial meshing. Coastal Eng. Proc. 37, 465–478. Berkoff, J.C.W., 1972. Computation of combined refraction-diffraction. Proceedings of 13th international conference on coastal engineering. Vancouver. 471-490. Booij, N., Holthuijsen, L.H., Doorn, N., Kieftenburg, A.T.M.M., 1997. Diffraction in a spectral wave model. Proceedings of the 3rd International Symposium on Ocean Wave Measurement and Analysis, WAVES’ 97, ASCE, New York, 243-255. Booij, N., Ris, Holthuijsen, R.C., 1999. A third-generation wave model for coastal regions 1. Model description and validation. J. Geophys. Res. 104 (C4), 7649–7666. https:// doi.org/10.1029/98JC02622. Camenen, B., Larson, M., 2005. A general formula for non-cohesive bed load sediment transport. Estuar. Coastal Shelf Sci. 63, 249–260. https://doi.org/10.1016/j.ecss.
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Numerical investigation of the effect of wave diffraction on beach erosion/ accretion at the Gangneung Harbor, Korea Jong Dae Doa, Jae-Youll Jina, Sang Kwon Hyunb, Weon Mu Jeonga, Yeon S. Changa, a b
⁎
Korea Institute of Ocean Science and Technology, Busan, Republic of Korea Sekwang Engineering Consultants CO., LTD, Seoul, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Telemac Coastal erosion Wave diffraction Shadow zone Breakwater
Since Gangneung Harbor breakwaters were built in the east coast of Korea from 1992 to 2002, the shoreline inside the harbor had severely changed as it was accreted in the area inside the harbor but eroded further south during the observation period (1979–2005). We investigated this process using Telemac-2DH model to calculate flows, waves, sediments and morphological changes in unstructured grids. The performance of the model was validated using the experiment data by Graven and Wang (2007). The model was run in two cases by turning on/ off the diffraction mode using the formula by Holthujisen et al. (2003). For this, we tuned the model by setting a shadow zone behind the breakwaters and by applying the diffraction formula only inside the shadow zone. The model results showed that wave heights significantly increased inside the shadow zone when the diffraction mode was on, compared to the case when the diffraction mode was off. This effect of diffraction was confirmed by the observations as the wave heights measured inside the shadow zone became 10–20% lower than those measured outside, which were nicely simulated by the model. In addition to the wave height, the wave-induced currents became stronger in the innermost area of the shadow zone with the diffraction mode. The model also successfully predicted the observed morphological change pattern because it simulated the shoreline erosion at the southern end of the shadow zone where the currents were bifurcated. Inside the shadow zone, sediment deposition occurred, corresponding to the observation, when the diffraction mode was on, whereas this deposition process was not simulated when the mode was off. The results support that the observed shoreline accretion in the harbor was mainly due to the reduced wave and current energy, which emphasizes the importance of accurate modelling of diffraction effect in the prediction of shoreline evolution.
1. Introduction Coastal structures such as breakwaters and groins are constructed to protect the shore from erosion by reducing the energy of approaching waves in the concerning area. If these structures are designed without sufficient considerations, however, it could lead to disastrous results causing shoreline erosions or accretions in unexpected places (Komar and McDougal, 1988). The side effect of groins is often observed when the sediments are eroded in the downdrift side of the structure while the shore is protected in the updrift side (Bakker et al., 1970). The side effects of breakwaters are not obvious because the damages may take place in wider area over longer period. One good example of the side effect of a breakwater is provided Fig. 1 that compares the shorelines of November 1979 and May 2005 in the south of the Gangneung Harbor located in the east coast of the Republic of Korea where waves normally approach the shore from the north or northeast. Two breakwaters were
⁎
constructed around the harbor. The first breakwater (BW#1) was constructed during 1992–1997. After that, it was extended to southeast by constructing the BW#2 during 1997–2002. While these breakwaters have been successful in lowering the wave energy inside the harbor, they still have caused unexpected impacts on the shorelines behind the breakwaters. First, the shoreline has accreted at the south of the harbor (the accreted area is marked with red color in Fig. 1). Second, the coastline is eroded at further south where the area is marked with blue color. The erosion is significant in this region as the maximum retreat scale reaches up to ~100 m. It is likely that the accretion in the red colored area in Fig. 1 occurred when the sediments moved to north by the diffracted waves inside the shadow zone formed by the breakwaters. This sand movement to the north did not only cause the accretion but also cause deficit of sand at the southern end of the shadow zone in the blue colored area in the figure. It is because sediments outside the shadow zone kept
Corresponding author. E-mail address: [email protected] (Y.S. Chang).
https://doi.org/10.1016/j.jher.2019.11.003 Received 18 February 2019; Received in revised form 27 August 2019; Accepted 20 November 2019 1570-6443/ © 2019 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Please cite this article as: Jong Dae Do, et al., Journal of Hydro-environment Research, https://doi.org/10.1016/j.jher.2019.11.003
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Fig. 1. Comparison of shorelines measured at November 1979 (blue) and May 2005 (red) around the Gangneung Harbor.
Fig. 2. Layout of the laboratory experiment around the detached breakwater in the LSTF (Graven and Wang, 2007).
(Steady-state spectral WAVE model) (Smith et al., 2001) as they are based on spectral action balance equation. In spite of the weakness of these spectral wave models in calculating the diffraction effect, its performance has been improved by modifying the mild slope equation (e.g. Lin, 2013). One of the main purposes of this study is to investigate the performance of diffraction caused by the Gangneung Harbor breakwaters using Telemac-2D system. For this, we propose to examine two cases of wave propagations by turning on/off the diffraction effect of the model. Although the impact of wave diffraction on the coastal erosion behind coastal structures has been widely perceived, its process has not been closely investigated, which provides a motivation for this study. For the diffraction mode, we tuned the model to apply the Holthujisen’s formula only within a shadow zone that was predetermined before
moving south by the waves that were not influenced by the diffraction instead of providing sediments from the south, which caused the erosion in this area. To further investigate this process, we have employed the 2DH mode of TELEMAC-MASCARET system (Telemac-2D) developed by National d’ Hydraulique et Environnement (LNHE), part of the Research and Development Rirectorate of the French Electricity Board (Hervouet, 2007). Telemac-2D is coupled with the Tomawac wave model (Benoit et al., 1996). Tomawac is a phase-averaged spectral wave model, and it is able to consider the diffraction effect by modifying the wave number as proposed by Holthuijsen et al. (2003) in which the mild slope equation (Berkoff, 1972) is tuned with the energy propagation speed in geographic space. Tomawac is similar to SWAN (Simulating Wave Nearshore) wave model developed by Delft University of Technology (Booij et al., 1999; Ris et al., 1999) and STWAVE 2
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Fig. 3. Grid system in the computational domain with water depths in colored contours.
2. Model description
running the model. The predetermination of the shadow zone is necessary in order to save the computational cost and to reduce the modeling error outside the shadow zone. This error may occur when the diffraction formula is used in the area where the diffraction is not expected to occur. The predetermination of the shadow zone is possible in specifically this study because the direction of the approaching waves is also predetermined based on the observation. The second purpose of the present study is to investigate the littoral process that results in the shoreline accretion and erosion as observed in Fig. 1. For this, we calculate morphological changes based on the modeled hydrodynamics. As restricted in most of 2DH models, Telemac-2D cannot simulate the shoreline evolution directly because the coastline has to be fixed as a boundary during the model run. In Telemac-2D, however, it is still possible to estimate the shoreline process when minute change of morphology can be calculated in fine grids near the coast by employing unstructured grid system. Another benefit of the Telemac-2D system is that the Tomawac wave model is internally coupled into the flow module, which enables to simulate the wave-current interaction so that not only the wave-induced currents but also the impact of current on the wave field can be examined in the shadow zone of the harbor. The tides are not considered in this numerical study as the tidal ranges in the experimental site are generally not higher than 1 m.
Telemac-2D is a model package that consists of the flow, wave and sediment transport modules. One of the most characteristic features of the Telemac-2D is the unstructured grids. By using the finite element method, irregular shaped grids are applied to simulate hydrodynamic conditions in coastal regions with complex coastline. In the flow module, the shallow water equations are employed by solving the continuity and momentum equations (Asaro and Paris, 2000; Hervouet, 2000; Giardino et al., 2009; Robins and Davies, 2011) as
→ ∂h + u∙∇ (h) + h ∗ div (→ u ) = Sh ∂t
(1)
→ ∂u → → ∂Z 1 + u ∙∇ (u) = −g + Sx + div (hνt ∇ u) ∂t ∂x h
(2)
→ ∂v → → ∂Z 1 + v ∙∇ (v ) = −g + Sy + div (hνt ∇ v ) ∂t ∂y h
(3)
where h is the depth, νt is the momentum diffusion coefficient, z is the free surface height. Sh , Sx and Sy are the source/sink for the mass and momentum. In the wave module, Tomawac is internally coupled into Telemac-2, 3
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Fig. 4. Contours of the modeled wave heights around the detached breakwater.
equation as
which can generate the wave-induced currents and the littoral transport (Villaret et al., 2013). In the spectral action balance equation the wave action density, N , is conserved as
∂N ∂N ∂N ∂N ∂N + ẋ + ẏ + k ẋ + k ẏ = Q (x , y, k x , k y, t ) ∂t ∂x ∂y ∂x ∂y
∂hC ∂hUC ∂hVC ∂ ⎛ ∂C ⎞ ∂ ⎛ ∂C ⎞ + + = h∊ + h∊ +E−D ∂t ∂x ∂y ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ⎜
where k x and k y are the wave number vectors, and Q is the source/sink of wave action. The diffraction effect considered in Tomawac was proposed by Holthuijsen et al. (2003). The basic concept is that the wavelength changes in the presence of diffraction. The new wave number, k2 , is given as (5)
where k1 is the wave number without diffraction and δ is the diffraction parameter. δ is formulated by the mild slope equation (Berkoff, 1972) as
δ=
(1 − p)
(6)
where a is the wave amplitude, C is the celerity and Cg is the group velocity. The new wave number is used to solve the wave action conservation equation. The energy propagation speed in case of diffraction is given as
Cgd =
k2 Cg k1
In this section, the performance of Telemac-2D model is tested before applying to the real field observation data. For the validation, a dataset measured in the Large-scale Sediment Transport Facility (LSTF) is employed. LSTF is the laboratory facility developed by United States Army Corps of Engineers (USACE) for investigating nearshore sediment transport process (Hamailton and Ebersole, 2001). The facility consists of a 30-m-wide-, 50-m-long-, 1.4-m-deep-basin and includes wave generators, a sand beach and an instrumentation bridge with acoustic sensors and wave gages. Waves are generated using four synchronized unidirectional spectral wave generators. The dataset used to evaluate the model was obtained from the experiment by Graven and Wang (2007). In their experiment, a 4 m-long rubble-mound detached breakwater (DB) was located parallel to the shoreline at 4 m offshore from the coast as shown in Fig. 2 (the figure is captured from Graven and Wang, 2007). The wave height at the generator was 0.26 m and period was 1.5 s with wave direction of 10.0°. The sediment size was 0.15 mm. The model computational domain was constructed within LSTF (red rectangle in Fig. 2) with grid size of 0.25 m in both longshore and crossshore direction. Fig. 3 shows the grid system in the computational
where Cgd is the energy propagation speed in geographic space in case of diffraction. In the sediment transport module, the bedload transport is formulated for the non-cohesive sediments (Camenen and Larson, 2005) as
θ θ Φw = a w θcw, on +θcw, off θcw, m exp ⎛−b cr ⎞ Φc = ac θc θcw, m exp ⎛−b cr ⎞ ⎝ θcw ⎠ ⎝ θcw ⎠ ⎟
⎜
(10)
3. Model validation
(7)
⎜
∂Zb + ∇∙Qtot = 0 ∂t
where p is bed porosity. Zb is the seabed elevation, and Qtot is the total sediment transport obtained from Eqn. (8) and (9).
∇∙ (CCg ∇a) k12CCg a
(9)
where C is the suspended sediment concentration. h is the water depth and ∊ is the diffusion coefficient. E and D are the erosion and deposition rate, respectively. In Eqn. (9), E = ws Ceq where ws is settling velocity and Ceq is equilibrium near-bed concentration determined from an empirical formula, and D = ws Cref where Cref is the sediment concentration at the reference elevation. Therefore, E − D in Eqn. (9) is set proportional to the difference between Ceq and Cref . Once the sediment transport is calculated by solving Eqn. (8) and (9), the morphological change can be obtained as
(4)
|k2 |2 = k12 (1 + δ )
⎟
⎟
(8) where Φw and Φc are the normalized sediment transport by waves and current respectively. ac and b are empirically determined coefficients. θcw, m and θcw are the mean and maximum Shields parameter due to wave and current interaction, θcw, on and θcw, off is the maximum Shield parameter at onshore and offshore phase respectively, θc is the Shields parameter due to current, and θcr is the critical Shields parameter. The suspended sediment is formulated using the advection–diffusion 4
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Fig. 5. Comparison of the wave heights and currents between the model and observation data along the five selected cross-shore profiles at Y18, Y23, Y24, Y25, and Y34 as their locations are marked in Fig. 2. Black dots denote the measurements of the wave heights and longshore current velocities. Blue lines denote the model data.
simulated with high accuracy (> 90%). In case of the longshore current, the modeling accuracy becomes lower at ‘Y23′ and ‘Y24′ in the lee area of the DB while it is still sufficiently high at other locations. The errors in this area are likely due to the circulations produced in the edge of the DB, caused by the obliquely propagating waves because its turbulent eddy pattern is difficult to be accurately simulated by numerical models. Except for the complex turbulent flow motions near the DB edges, the Telemac-2D model data is in good agreement with the measured data, providing a verification of the model performance specifically on the effect of wave diffraction.
domain with water depths in colored contours. The location of the DB is marked in grey color. The incident wave conditions are based on the numerical experiment by Nam et al. (2009) as Texel-Marsen-Arsole (TMA) was used at the boundary to produce wave height of 0.23 m and period of 1.5 s. The model was run 185 min until a steady state reached. In Fig. 4, the modeled wave heights are contoured around the DB. The effect of wave diffraction is clearly observed as the wave height gradually decreased in the lee of the DB. The model results are also quantitatively evaluated when the wave height and currents are compared with measurements in Fig. 5, along the five selected cross-shore profiles from ‘Y18′ to ‘Y34′ as their locations are marked in Fig. 2. In the figure, black dots denote the wave heights and longshore current velocities measured along the instrumentation bridge, and the blue lines show the results computed by the Telemac-2D model. The model nicely represents the wave height field of the observation data not only offshore area of the DB but also in the lee area of it. Specifically, the magnitudes of reduced heights of the diffracted waves are successfully
4. Model setup in the Gangneung Harbor The wave conditions at Gangneung area vary seasonally. In summer when the weather is mild, the waves generally approach the shore in the normal direction (i.e. approach from NE) with wave heights < 1.5 m. In winter, the weather becomes severe as the extratropical 5
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Fig. 6. Rose diagram that shows the distribution of the wave heights and directions observed in Feb. 2015
zone, we could significantly save the computational cost to reduce the running time. Another important reason is that, if we applied the diffraction formula at every grid in the computational domain, the model became unstable outside the shadow zone where the diffraction was not expected to occur. Once the shadow zone was determined, the simulations were performed in two cases. In Case 1, we ran the model by turning the diffraction mode off, whereas the diffraction was on in Case 2. In both cases, we applied the wave-current interaction effect that is internally coupled in the Telemac-2D modeling system. The model was run for twelve hours by simulating a typical winter storm condition and by applying JONSWAP wave directional spectrum for the boundary condition (Hasselmann et al., 1973). We designed the wave directional spectrum to be narrow enough to generate the waves to mainly propagate from the north.
storms frequently develop in the East Sea (Jeong et al., 2007). Since the winter storms generally develop in the north of the Ganngneung area, the high waves approach the shore obliquely (i.e. approach from N), which produces the shading zone behind the breakwater. In Fig. 6, a rose diagram that shows the wave direction and height distribution that was observed in February 2015 in which 2–3 m height waves mainly approach the coast from the north. Based on this observation, we configured the model conditions by generating the waves to approach the breakwater from the north with the designed wave height 2 m and period 8 s at the model boundary to simulate a typical winter storm condition. In addition, this wave condition was specifically designed to produce a comparable wave height with the observed data measured during a storm period in the Gangneung Harbor, which will be discussed in Section 4. It should be also noted that the effects of tidal currents were not considered in this numerical experiments because tidal ranges are no greater than 1.0 m in this area and thus the tideinduced currents may not be significant. Fig. 7 shows the computational domain with unstructured grid system, which provides benefits in the simulations of this study. The complex coastlines around the Gangneung Harbor are nicely resolved with the unstructured grids: the grid size becomes finer near the structure to capture the minute changes in the coastline near the breakwaters. The computational domain is 4950 m × 2070 m. The grid size varies from a minimum of 10 m to a maximum of 60 m, by which 15,491 grid nodes are employed. Δt is set to 2 s and the model was run for 12 h in total. In order to investigate the wave diffraction effect, we divided the computational domain into two parts. In the ‘shadow zone’ that is marked with blue color in Fig. 7, we set a switch to turn on the diffraction mode by applying the Eqs. (1) and (3). The shadow zone was determined before running the model as we set the wave direction to the north. By applying the diffraction mode only inside the shadow
5. Field observation In this section, observation data sets measured near the Gangneung Harbor are used to investigate the wave diffraction effect inside the shadow zone. In Fig. 7, the three observational stations (S1, S2, and S3) are marked as S1 is located inside the shadow zone and S2 and S3 are located on the borderline and outside the shadow zone respectively. At these three stations, wave data were measured from 21th to 31st of March 2015 by the Korea Institute of Ocean Science and Technology. They were obtained at every half hour using bottom-mounted WTG (Wave and Tide Gauge) which measured the water pressure for 1024 s with frequency 2 Hz. This gave one power density spectrum (PDF) per data burst. The significant wave heights, Hs , as well as peak wave period, Tp , were then calculated from the PDF as its outcomes are shown in Fig. 8. During the period, severe wave conditions were observed two 6
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Fig. 7. The unstructured grids in the computational domain. The blue area denotes the shadow zone behind the Gangneung Harbor breakwaters. The shadow zone was determined by assuming the wave propagation from the north (N). W1, W2 and W3 are the locations where the modeled wave heights are compared between the two cases.
6. Results
times at March 23 and 26 at which HS were higher than 1.5 m and Tp were longer than 10 s. Although the pattern of Hs and Tp is generally similar between the three stations, meaningful discrepancies are also observed in the wave heights. At March 25–26, Hs rapidly increases likely due to local storm as it has a peak at March 26. At this time of peak, however, Hs are different between the stations as it is higher outside the shadow zone than inside. At S1, Hs ranges between 1.5 and 1.75 m while it ranges between 1.75 and 2.25 m at both of S2 and S3. This indicates that Hs has been reduced ~20% inside the shadow zone as the propagating waves are being diffracted. Although this lower Hs at S1 could be alternatively argued that the wave height becomes lower at S1 possibly due to the wave breaking, the similar water depth (h = 3.5, 3.9, and 8.1 m at S1, S2 and S3 respectively) between S1 and S2 excludes this postulate because of the higher Hs at S2. In addition, the similar wave heights between S2 and S3 supports that the waves were not breaking until they reached at S2. Furthermore, this pattern of Hs reduction inside the shadow zone not only occurs at March 26 but repeats at other times. At March 23, Hs ranges 1.5–1.6 m at S1, but it increases ~10% at S2 and S3 as Hs reaches up to 1.75 m at both of the stations. Even at March 31, Hs at S1 is ~10% lower than that at S3 although it is not clearly observed as their wave heights are only ~0.5 m. Compared to Hs , Tp is similar between the three stations and shows no clear discrepancy, which indicates that the likely reason for the wave height reduction inside the shadow zone is the wave diffraction. The model results that simulated the wave heights measured at March 26 will be discussed in the next section.
The pattern of the wave propagation calculated by the model is compared between the two cases in Fig. 9 in which the spatial distribution of model wave height, Hm0 , is contoured around the shadow zone. As expected, the Hm0 outside the shadow is almost identical between the two cases. Inside the shadow zone, however, it shows significant difference. In both cases, Hm0 decreases as the waves propagate inside the shadow zone from its southern boundary. When the diffraction mode is off in Case 1, however, Hm0 decreases more rapidly as the distance from the boundary increases. In Case 2, Hm0 inside the shadow zone is generally higher than that of Case 1, indicating that the waves propagate further inside the shadow zone due to the effect of the wave diffraction. A direct comparison of the modeled wave heights is shown in Fig. 10 in which the time series of Hm0 are compared between the two cases at the three observation stations (S1, S2, and S3). In the figure, the grey solid lines show the time variation of modeled Hm0 in Case 1 while the black dashed lines show the results of Case 2. It clearly shows that the wave heights are zero for the first couple of hours until the waves reach the stations. Once the waves arrive, Hm0 sharply increases higher than 1 m and gradually becomes stable at each of the stations at t > 1.02 d. At S1, H reaches ~1.5 m in both cases. It is interesting to note that this modeled Hm0 is similar to Hs measured in S1 at March 26 because the model conditions were designed to simulate the wave height of that storm period at the station. It is also interesting to find out that the pattern of Hm0 variation in S1 is different between the two 7
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Fig. 8. Observed data of significant wave heights (Hs ) and peak wave period (Tp ) from 21th to 31st of March 2015 using bottom-mounted WTG (Wave and Tide Gauge) measured at (a) ‘S1′, (b) ‘S2′ and (c) ‘S3′.
similar in both stations, S2 and S3. The similarity in the modeled wave heights indicates that the effect of wave diffraction diminishes outside the shadow zone, causing homogeneous wave field. When compared with the observation in which Hs ranges between 1.75 and 2.25 m at both S2 and S3 at March 26, the modeled wave height is within the comparable range as Hm0 ~ 1.7 m in both S2 and S3 for both cases. Therefore, Hm0 is reduced about 12% when the waves propagate into the shadow zone, which corresponds to the observation data. In order to investigate the diffraction effect inside the shadow zone,
cases. While the wave height is constant as Hm0 ~ 1.5 m at t > 1.01 d in Case 1, it shows a wiggled pattern in Case 2. This inconsistent Hm0 is probably because the wave diffraction causes inhomogeneous spatial distribution inside the shadow zone as shown in Fig. 9(b). Although it is not clearly observed, Hm0 in Case 2 is generally higher than that in Case 1 at S1 due to the diffracted waves that propagate into the shadow zone. This contribution of wave diffraction can be hardly observed outside the shadow zone. At S2 and S3, Hm0 in Case 1 is similar to that in Case 2 showing no wiggled pattern in both cases. In addition, Hm0 is even 8
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Fig. 9. Distribution of the modeled wave heights near the Gangneung Harbor. (a) Case 1: diffraction mode off; (b) Case 2: diffraction mode on.
flows southeast along the coast. This longshore current is generated when the waves approach the shore at an angle, producing wave-induced trust along the shore. The current pattern is similar between the two cases outside the shadow zone. It is because the wave conditions are same outside the shadow zone in both cases, not affected by the diffraction. However, the flow patterns are different at peeper part away from the shore (marked with red rectangle ‘D’) because circulations are observed when the diffraction mode is on. These flows were generated by the stronger wave-induced currents when the gradient of wave energy became greater due to diffraction. It should be noted that the magnitude of the wave-induced currents might be overestimated. Although tides or other conditions that could increase the current strength were not considered in the input conditions, the maximum current speed became as large as 0.5 m/s as shown in Fig. 12(c) and (d). Unfortunately, no measured data for currents are available for a direct comparison. However, the model results may be indirectly examined with the LSTF data. The input wave height and period for the laboratory experiment were 0.26 m and 1.5 s respectively, and both of the measured and modeled data in Fig. 5 show that the wave heights ranged at 0.1–0.2 m and the longshore current speed ranged at 0.1–0.2 m/s around the DB. Therefore, the magnitude of modeled currents near the Gangneung Harbor may still be within acceptable range, considering much larger input wave conditions (wave height = 2 m, wave period = 8 s). The morphological changes caused by the waves and currents inside the shadow zone were calculated using Eqs. (8) to (10). We increased the model’s morphology factor that determines the rate of morphological change 10 times larger than the default value. We employed this unusual value of the parameter in order to maximize the impact on the morphological change within the limited short time for model integration, which is justified when our purpose of the experiments is only to compare the pattern of the change between the two cases but not to exactly quantify the rate of changes. In Fig. 13, the morphological change calculated during the integration time is contoured. As expected, the seabeds in the nearshore region are generally under erosion near the southern boundary of the shadow zone (marked with red circle ‘D’) in both cases. It is interesting to find out that this erosional area roughly corresponds to the erosional area observed in Fig. 1. This indicates that the shoreline is vulnerable to the erosion in this area where the wave height sharply decreases in the shadow zone as shown in Fig. 9, which causes gradients in the sediment transport distribution along the coast. This location of the erosion also roughly corresponds to the circled areas ‘A’ and ‘B’ in Fig. 12 where the bifurcated currents carried sediments into different directions, which supports the erosion
three additional stations (W1, W2 and W3) are chosen as shown in Fig. 7. These three stations are located further inside the shadow zone compared to the observation stations. Specifically, we chose the location of W3 to be closest to the observation station S1 in order for the comparison with the modeled results at W1 and W2 where no observation data are available. Fig. 11 shows the time variation of H at W1, W2 and W3. As shown in Fig. 7, Hm0 in W3 is identical to that in S1 as it is also similar to the Hs measured at S1 at March 26. At this location, Hm0 shows small difference between the two cases as Hm0 in Case 2 is a bit higher than that in Case 1 when the diffraction mode is off. At W2, the difference in Hm0 increases between the two cases as Hm0 in Case 2 reaches ~1.0 m while it is only ~0.7 m in Case 1. The difference in the wave height can be more clearly observed at W1, the station located in the innermost inside the shadow zone, where Hm0 in Case 1 is only ~0.2 m while it is still ~1.0 m in Case 2. It is interesting to note that the wave height in W1 of Case 2 is similar to that in W2, which indicates that the waves successfully propagate inside the shadow zone when the diffraction mode is on. On the contrary, the waves in Case 1 do not reach to the innermost site due to the weak wave diffraction. It is also interesting to find out that Hm0 in Case 2 at W1 and W2 also shows a wiggled pattern due to the wave diffraction while Hm0 in Case 1 shows small variation in time. In Fig. 12, velocity vectors are shown to compare the flow patterns between the two cases. These wave-induced currents are produced when the waves are shoaled, refracted and diffracted over the complex topography in the shallow region around the harbor. In both cases, longshore currents strongly develop along the coast. It is also observed that the longshore currents are bifurcated near the southern end of the shadow zone (marked with red circle ‘A’) as a branch flows southeast outside the shadow zone while the other flows northwest into the shadow zone. The flow pattern of the northwest branch becomes complicated near the location marked with the red circle ‘B’ in Fig. 12 where a strong offshore current develops as the northwest branch is again bifurcated. It is likely that the offshore current develops due to the seabed topography. As shown in Fig. 12(c) and (d), the water depth became shallower at the location northwest of the circle ‘B’, which blocks the way of the current along the shore and divert it to the offshore. A significant discrepancy is found between the two cases at the innermost area just south of the Gangneung Harbor (marked with red circle ‘C’). While the branch of the longshore currents that flows northwest along the coast does not reach this area in Case 1, the current is strong enough to reach the area in Case 2 when the diffraction mode is on. Outside the shadow zone, the other branch of the bifurcated current
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Fig. 10. Time series of the modeled wave heights at the three observation stations (S1, S2 and S3) marked in Fig. 2. Grey solid lines: Case 1 (diffraction mode off); Black dashed lines: Case 2 (diffraction mode on).
was likely affected by the flow separations. Regardless of the similarity in the morphological change between the two cases, a significant difference is found in the innermost area of the shadow zone (marked with red rectangle ‘E’ in Fig. 13). While it is not observed in Case 1, the seabed close the coastline is accreted in Case 2 as it is likely due to the higher waves shown in Fig. 9 and also due to the stronger currents observed in Fig. 12. This deposition area corresponds to the area of the observed shoreline accretion (marked with red color in Fig. 1). This different result between the two cases indicates that the morphological changes inside the shadow zone are successfully simulated only if the diffraction mode is on. Another important question that arises on the sediment motions in the shadow zone is the impact of the wave-current interaction. As mentioned earlier, the Tomawac wave model is internally coupled into the flow module in Telemac-2D so that the currents may influence the wave field. In order to examine this, we performed additional model
run in which both of the diffraction and wave-current interaction modes were turned off. Fig. 14 compares Hm0 of this specific case with that of Case 1 (i.e. wave-current interaction is on but diffraction is off) at W1. Since the diffraction is not considered in both cases, Hm0 is low (~0.1 m). However, a difference between the two is clearly observed as H becomes higher if the wave-current interaction is considered. This difference in the wave height indicates that the current can influence on the wave field, though not significant, inside the shadow zone even without the diffraction. 7. Discussion In this section, we discuss the limitations of the present study in applying the phase-averaged wave model for the wave diffraction problem. In the spectral wave models, the diffraction effect has been implemented by modifying the wave number obtained from an energy 10
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Fig. 11. Time series of the modeled wave heights at the three model stations (W1, W2 and W3) marked in Fig. 2. Grey solid lines: Case 1 (diffraction mode off); Black dashed lines: Case 2 (diffraction mode on).
(graphics processing unit) instead of CPU (central processing unit) (Tavakkol and Lynett, 2017). However, these GPU-based phase-resolving models have not been coupled into the coastal engineering models that are designed to predict the morphological changes. Therefore, the scope of the present study has been limited to the phase-averaged modeling. The achievement of the present study is then the successful investigation of the impact of wave diffraction on the shoreline response in engineering-scale modeling. Another shortcoming of the present study is the simulation of input waves. Instead of producing the realistic wave scenarios, we used narrow wave spectrum to generate monotonous input wave conditions. In doing so, we produced the model wave height at W1 similar to the significant wave height measured at S1 in March 26, in order to clarify the effect of the wave diffraction. If broader directional spectrum was employed as an input, the wave diffraction effect might not be clearly identified because the waves at one-time step would advance into
balance equation (Booij et al., 1997; Rivero et al., 1997), however, the stability of these models was low (Mase, 2001). Holthuijsen et al. (2003) proposed the phase-decoupled method into SWAN by introducing the diffraction parameter, δ , as expressed in Eqn. (6). Since then, various efforts have been made to improve the diffraction effects in the spectral wave models (e.g. Lin, 2013). Its performance is still not satisfactory compared to the phase-resolving models. For example, the Boussinesq wave models such as FUNWAVE (Shi et al., 2012) have shown better performance in the wave diffraction effect. In addition, even three-dimensional models have been developed to simulate the wave propagations by directly solving the Navier-Stokes equations (Miquel et al., 2018). Although these phase-resolving models are more accurate in resolving the wave diffraction, their application to coastal modeling in engineering practices is still limited due to high computational cost. Recently, the performance of Boussinesq models has been significantly improved by solving the parallel codes using GPU 11
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Fig. 12. Comparison of flow vectors of the wave-induced currents. The shadow zone in Fig. 2 is marked with grey color. The three circles ‘A’, ‘B’ and ‘C’ denote the location where the flow pattern significantly changes. (a) Case 1: diffraction mode off; (b) Case 2: diffraction mode on; (c) magnified view around the circled area in (a) with bathymetry contours; (d) magnified view around the circled area in (b) with bathymetry contours.
We agree that an attempt to speculate on long-term beach morphological changes from short-term numerical calculations is not convincing because it inevitably includes errors inherent in the time-scale gap between the model results and the shoreline data, and thus application of the model results should be limited. Notwithstanding these limitations, this study provide meaningful results and indications on the shoreline process inside the shadow zone, which requires careful consideration in designing coastal structures to minimize the effect of diffraction that may cause unexpected erosions in the coastline.
broader directions in the next modeling step and some waves might propagate into the shadow zone regardless of the diffraction. By narrowing the input wave spectrum, therefore, we also narrowed the wave propagating directions outside the shadow zone so that the wave field inside the zone would be mainly influenced by the diffraction, which resulted in the clear discrepancies between the two cases of the model runs. Recognizing these shortcomings, the scope of the present study has been confined to examine the scenario of shoreline evolution process that might be caused by the wave diffraction in the shadow zone, not focusing to enhance the accuracy of simulations. For this, the shortterm numerical results from Telemac-2D phase averaged model was applied to understand the long-term process by extending its prediction.
8. Conclusions In this paper, we examined the role of wave diffraction on shoreline 12
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Fig. 13. Modeled morphological changes. (a) Case 1: diffraction mode off; (b) Case 2: diffraction mode on; (c) magnified view inside the rectangular region in (a); (d) magnified view inside the rectangular region in (b).
Fig. 14. Comparison of the wave heights modeled at the station W1 with diffraction mode off. Solid line: wave–current interaction off; dashed line: wave-current interaction on.
that was set behind the breakwaters by assuming the waves approaching from the north. The effect of wave diffraction was confirmed by the observations as the significant wave heights measured inside the shadow zone were 10–20% lower than those measured outside the shadow zone. This observational process was nicely simulated by the model because the modeled wave heights were also reduced with similar rate (~12%) inside the shadow zone, in case the diffraction mode was on. When the
evolution inside a shadow zone behind Gangneung Harbor breakwaters. We performed numerical experiments in the region because shoreline change occurred after two breakwaters were constructed around the harbor. We employed the Telemac-2D model in unstructured grids to simulate the waves, wave-induced currents, and the resulting morphological changes. We ran the model in two cases – 1) diffraction mode on, 2) diffraction mode off. When turning on the diffraction, we applied the formula by Holthuijsen et al. (2003) inside the shadow zone 13
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diffraction was off, the reduction of wave height inside the shadow zone become more serious, indicating that the waves did not successfully propagate into the shadow zone. In addition to the wave heights, the wave-induced currents were stronger inside the shadow zone when the diffraction mode was on. In this case, the more active hydrodynamic energy resulted in the sediment transport even to the innermost area of the shadow zone, which corresponds to the observation. The deposition process was not successfully simulated if the diffraction mode was off. In both cases, however, erosions occurred at the southern boundary of the shadow zone, which is likely due to the strong sediment transport gradient caused by the wave energy reduction at the zone boundary. The results of the present study indicate that the wave diffraction may play significant roles on shoreline evolutions behind coastal structures where the wave energy is expected to decrease. Therefore, it is important to improve the accuracy of the diffraction effect in modeling coastal processes. In this regards, future studies are required to test the phase-resolving wave models with higher accuracy, for the applications in engineering practices.
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Acknowledgments This research was supported by the Korea Institute of Ocean Science and Technology for the projects titled ‘Development of application technologies for ocean energy and harbor and offshore structures [PE99731]’ and ‘Operation of KIOST open ocean research infrastructure and establishment of public utilization service system [PKA0017]’. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jher.2019.11.003. References Asaro, G., Paris, E., 2000. The effects induced by a new embankment at the confluence between two rivers: Telemac results compared with a physical model. Hydrol. Process. 14, 2345–2353. https://doi.org/10.1002/1099-1085(200009)14:13 <2345::AID-HYP33>3.0.CO;2-X. Bakker, W.T., Breteler, K., Roos, A., 1970. The dynamics of a coast with a groyne system. Proceedings of 12th conference on coastal engineering, Washington D.C. 1001-1020. Benoit, M., Marcos, F., Becq, F., 1996. Development of a third generation shallow water wave model with unstructured spatial meshing. Coastal Eng. Proc. 37, 465–478. Berkoff, J.C.W., 1972. Computation of combined refraction-diffraction. Proceedings of 13th international conference on coastal engineering. Vancouver. 471-490. Booij, N., Holthuijsen, L.H., Doorn, N., Kieftenburg, A.T.M.M., 1997. Diffraction in a spectral wave model. Proceedings of the 3rd International Symposium on Ocean Wave Measurement and Analysis, WAVES’ 97, ASCE, New York, 243-255. Booij, N., Ris, Holthuijsen, R.C., 1999. A third-generation wave model for coastal regions 1. Model description and validation. J. Geophys. Res. 104 (C4), 7649–7666. https:// doi.org/10.1029/98JC02622. Camenen, B., Larson, M., 2005. A general formula for non-cohesive bed load sediment transport. Estuar. Coastal Shelf Sci. 63, 249–260. https://doi.org/10.1016/j.ecss.
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