The sagging shape of shoreline formed on downdrift side of the structures due to seasonal oblique wave incidence

Downdrift coastal erosion has occurred at natural or man-made groynes on Korea’s eastern coast, caused by oblique high waves in winter months. The resulting shoreline planform has a sagging shape with a maximum indentation point within 10 the eroded shoreline. This study focused on solving the frequent and severe coastal erosion problem of this type at the Jeongdongjin review of wave data over 40 years from the National Oceanic and Atmospheric Administration (NOAA), as well as analyzing shoreline monitoring images for identifying the yielding line of maximum indentation points. An analytical method was developed to verify the eroding shoreline in a sagging shape and its maximum indentation by applying the conservation principle of sediment transport and the empirical model of equilibrium shoreline. To examine how well the 15 empirical formula is suitable for the Jeongdongjin coast, the annual directional spectrum of the observed wave data was applied to the simple diffraction wave model for the gamma breakwater, and satisfactory agreement was obtained by comparing it with the shoreline results. Breaking wave height and angle, duration, longshore sediment transport coefficient, and protruding length of the groyne were the inputs. The theoretical results are in good agreement with those of the shoreline monitoring program. The factors mitigating downdrift coastal erosion of this type were identified by applying the obtained theoretical solution, and 20 the engineering solutions were examined via factor analysis.

al. (2021) applied the parabolic bay shape equation (PBSE or parabolic model) of Hsu and Evans (1989) to indirectly reflect 55 the effect of wave diffraction caused by coastal structures. High-resolution numerical models composed of waves, currents, and topography change modes have also been developed to simulate topographic changes around structures (for example, Xbeach and Sbeach). These laboratory experiments and numerical modeling contributed to solving scientific questions about downdrift erosion by providing a similar sagging shape. However, engineering countermeasures to reduce coastal erosion damage on the downdrift side of structures still lack reliable factor analysis. 60 This study developed a theoretical approach, using the PBSE, that can predict the critical point (i.e., maximum indentation) in a sagging shoreline planform on the downdrift side of an LST barrier. The validity of this approach was examined by comparing its results with those from a shoreline change model that converges to the equilibrium shoreline of the SEP. Further, as the method was verified using the video monitoring data at Jeongdongjin Beach, where sagging shoreline curves have formed frequently by seasonally changing waves at the downdrift of a group of large irregular natural rocks, as shown in Fig. 2. 65 Section 2 describes the analysis of seasonal wave distribution from NOAA's wave data and the longshore sediment transport equation. Section 3 presents the well-known parabolic model (Hsu and Evans, 1989) for a stable bay shape and its 70 approximation for the theoretical analysis of the eroded shoreline planform, which appears as a sagging shape with a maximum 2.2 Analysis of seasonal longshore sediment transport 100 Komar and Inman (1970) conducted field experiments on the energy flux in the longshore direction, , and the longshore sediment transport (LST) rate, , and reported the following relationship: where , , , and are the seawater density, sediment density, sediment porosity (typical approximately 0.3 to 0.4), and acceleration of gravity, respectively. is the immersed weight of the sediment transport rate. is a dimensionless coefficient 105 (e.g., CERC coefficient) that depends on seabed property and significant wave height, which can be taken as 0.39 (USACE, 1984; but was taken as 0.77 in Komar and Inman). The alongshore component of the energy flux per unit length of beach is defined as where subscript b denotes the condition at wave breaking, ( ) is the wave energy flux at breaking, and is the breaking 110 wave angle between the shoreline and wave crest line. According to the USACE (1984), sandy longshore sediment serves as a function of wave direction and wave breaking height ( ), as shown in Eq. (4). Here, the unit for the LST rate, , is 3 /s. ) has a value of 0.0847 for most types of sand. In Eq. (4), the LST coefficient K = 0.39, the acceleration of gravity g = 9.81 /s 2 , the spilling wave breaking index κ = 0.78, the sediment specific gravity s = 2.57, and the porosity p 115 = 0.35 for most types of sand.
In Eq. (4), the LST rate calculated at wave breaking can be expressed by using the deep-water wave data shown in Eq. (5), assuming that the isobath of the seabed is parallel to the straight shoreline. To compensate for the variation in the incident where and are the significant wave height and wave period, respectively. The wave direction in the deep water is measured between the outward normal to the shoreline and wave orthogonal or from the true north direction , as shown in Fig. 4. Thus, = 2 − + , with a positive amount of sediment pointing south and a negative pointing north. Furthermore, is a factor that reflects the characteristics of the sediment and waves, including specific gravity, porosity, breaking index, 125 and wave angle. where has a value of 0.0719 approximately for most sands, assuming that the effect of is negligible.  By using NOAA wave data to plot the monthly wave factor for the LST in terms of ⁄ , the results shown in Fig. 6 reveal a strong seasonal-dependent trend in the direction of the LST, highlighting southward transport in winter months (November 140 to February) and northward in summer (July to September). Thus, we can expect coastal erosion with a sagging curve to reach a maximum around February at the end of winter and September at the end of summer. However, if the seasonal LST bypasses the beach without being intercepted by the natural pillar rocks (such as the groyne in Fig. 2) at different water levels and wave conditions, moderate or severe sagging could result.   ⁄ as a function of the oblique wave direction group , of 2° intervals at the wave breaking in winter months (November-February) from NOAA wave data. We determined the oblique angle based on the N38°E as = 0 and calculated the wave height and breaking angle by N38°E ± 50° in deep water. 5 2 ⁄ shows a tendency of increase or decrease within −7.5° to +12.5°, whereas the number of occurrences exceeded 5000 ranges within −2.5° to +7.5°. 150 These imply that high waves in winter may cause shoreline sagging arrives from the sector within N38°E −2.5° to N38°E +7.5°. The LST can be calculated by using Eq. (5) from the waves approaching the shoreline obliquely. However, coastal structures 155 (e.g., natural or artificial groynes) and retention at the updrift coast control the amount of the LST available to a beach. Fig. 8 shows a conceptual sketch of the change in the LST encountering a groyne on a straight shoreline. Before the sediment bypassing the groyne in Fig. 8(a), shoreline advance occurred on the updrift side, with retreat at downdrift. As soon as the sediment bypasses the tip of the groyne, long-term shoreline equilibrium may reach both sides of the groyne. However, during this process in field conditions, the locations of A and A' may fluctuate, with the former gradually advancing updrift, while 160 the latter slowly shifts downdrift.

Parabolic bay shape equation
The PBSE proposed by Hsu and Evans (1989) for embayed coasts in static equilibrium is provided by ( Fig. 9) 165 in which represents the radial distance from the parabolic focus to the point on the equilibrium shoreline; is the distance between the wave crest line (wave crest base line) and the line that passes through the downdrift control point parallel to the shore base line; is the angle between the wave crest base line and the line joining the focus and the downdrift control point; 170 is between the wave crest base line and the radius R for the point on the equilibrium shoreline, and 0 , 1 , and 2 are the coefficients derived from regression analysis for static bay shapes (Hsu and Evans, 1989). At the downdrift control point, the boundary condition 0 + 1 + 2 = 0 is satisfied to endure a common tangent at = .

175
When the downdrift straight section of an embayment is long, Eq. (7a) can be approximated as Within this stretch, waves break at an angle , which can be expressed by Eq. (9) from the approximation of Eq. (8). β absent from in Eq. (9) implies that can be solely determined from the wave angle at wave breaking, or vice versa. Table 1 and Fig. 10 show the relationship between and in Eq. (9) for the equilibrium shoreline.

Analysis of shoreline change caused by oblique waves 185
In this section, we consider a simple littoral cell, in which the shoreline and depth contour are initially straight and parallel, and the LST is blocked within the groynes. In the first case, we ignored the effect of diffracted waves because of the presence of a short groyne on the left side in Fig. 11. However, with groynes of moderate length and no LST from the updrift coast, wave diffraction occurs, causing erosion in the lee of the shadow zone with a crenulated bay shape because of oblique waves.
As the wave action continues, the transition point (for example, C.P. marked as A, B, and C in Fig. 11) at which no erosion 190 occurs on the original shoreline would shift downward with time, while coastal erosion to its left widens. Also in Fig. 11, 1 6 ⁄ , 1 3 ⁄ and 1 2 ⁄ denote the times when the control point reaches the points that represent 1/6, 1/3, and 1/2 of the beach length L between the groynes, respectively. At t ≥ 1/2 , the planform remains in equilibrium.
Here, is defined as 2 ⁄ . Additionally, the time to reach static equilibrium, 1 2 ⁄ , when = 2 ⁄ , is provided by to long groyne with protruding length , which is located at = 2 ⁄ by using the parallel shoreline approximation of Hsu and Evans (1989) in Eq. (8). Hence, the shoreline advance width, 2 ⁄ , of the parallel shoreline is expressed as where it is assumed that converges to zero, 0 and 2 are zero, and 1 is unity.
The cross mark × in Fig. 12 indicates the maximum indentation position on each sagging shoreline at different time steps. As shown in Fig. 12, we assume the downdrift shoreline orientation to be the same as the wave breaking angle ( ) for applying the PBSE based on the downdrift control point (A, B, and C, respectively) at each time step. Fig. 13 illustrates an example.

220
The location of the maximum indentation point ( , ) shown in Fig. 13 can be determined by using the PBSE approximation given by where angle for locating the maximum indentation can be obtained from Eq. (9) for the wave direction at wave breaking. 225 In addition, is the distance measured from the groyne in the direction of the initial (mean) shoreline, and ( ) is a timevariant function of , which can be obtained by Eq. (15) by using the approximation of the PBSE in Eq. (8): Applying Eq. (15) to Eqs. (14a) and (14b) results in the following alternative expressions for and : where ′ = / . Fig. 14 shows the locations of ′ = / and ′ = / as a function of dimensionless ′ = / (0 to 4 with increments of 0.5) for from 1° to 25° with increments of 1°. Fig. 14 indicates that the erosion width increases with an increase in several parameters (e.g., the protruding length of the groyne , , , and ). 240  where (x, y) are the Cartesian coordinates with x-axis positive pointing seaward, y-axis alongshore, and origin at the MSL, while ℎ and ℎ denote the berm height and closure depth, respectively. is the LST calculated by using the CERC formula 250 (USACE, 1984), and represents the cross-shore sediment transport per unit width of the shoreline (Lee and Hsu, 2017).
Within the region of wave diffraction or the vicinity of a coastal structure, an alternative expression is applied to the LST.
where is the wave angle within the diffraction zone, which can be determined by using the PBSE (Lim et al., 2021). In the numerical calculations, we calculated or assigned the quantity of the LST at each grid. For example, we used = 0 for the 255 eroding shoreline along the boundary of the groyne. Taking the most appropriate wave angle of = 10° at the downdrift of the sagging section, we ran the numerical model for a range of time durations (from 6 h to 4 weeks) by using the prototype data for Jeongdongjin Beach-natural groyne with protruding length = 80 m, beach length L = 850 m, winter high waves of 2.11 m, and LST coefficient C = 0.0847 in Eq. 265 (4). We found the results to be in good agreement with the field observations, as shown in Fig. 16, which reveals that the shoreline adjacent to the groyne advances seaward by the 6th hour before being eroded (landward) afterward.

Results of comparison with Jeongdongjin monitoring data
As shown in Fig. 17, wave data observed near Jeongdongjin Beach were analyzed to examine whether it is appropriate to apply the PBSE to this coastal waters. Wave data were observed with an AWAC wave meter at a depth of 32.4 m for 3 years from 275 September 27th, 2013 to November 21st, 2016. And the distribution of the annual mean wave direction obtained from the data is shown in Fig. 18. As shown in Fig. 18, the average shoreline formed by the change of the direction of the breaking wave under the influence of the breakwater after the construction of the gamma-type breakwater and the result of equilibrium shoreline using PBSE were compared (Lim et al., 2019). Although a simple diffraction wave model was applied, it was concluded that the application was appropriate to the characteristics of the incident wave observed in the central eastern coast   scientific data accumulation and analysis; at Jeongdongjin, a video monitoring program that used four cameras commenced in February 2014, covering 3,280 m (97.3 %) of the local shoreline within a total of 3,370 m (Fig. 19). 290 In this study, the continuously changing shoreline caused by the seasonal waves enables overlapping and averaging of the pixel values from the 180 instant images taken at 1-s intervals for 3 min. To extract the average pixel value from the images, we divided the cumulative sum of the attribute values of every pixel by the number of captured images, from which we 295 determined the coordinates of the ground control points and changing shoreline. To rectify the plane coordinate system for the shoreline image, we applied the geometric transformation equation of Lippmann and Holman (1989), which transforms the image coordinates to ground coordinates as follows: where and are the Cartesian coordinates in the photographic images; , , and are the coordinates of the actual ground control point position corresponding to the and of the photographic image; is the focal length of the camera; and is the tilt of the camera. By using this method, we analyzed the images of critical points taken twice a day from December 6-30, 2015, at Jeongdongjin Beach, as shown in Fig. 20, and compared them with the theoretical solution. Nevertheless, note that the location of the critical points on these images might not include that of the maximum indentation. Therefore, the actual 305 extent of shoreline retreat may be larger than that presented. https://doi.org/10.5194/esurf-2021-71 Preprint. Discussion started: 13 October 2021 c Author(s) 2021. CC BY 4.0 License.
As shown in Fig. 20, our results of the video monitoring data agree well with those of the theoretical solution for the critical points (i.e., maximum indentation) that used the PBSE approximation, the LST coefficient C = 0.0847, and (ℎ + ℎ ) = 8 m for the Jeongdongjin Beach on Korea's eastern coast. In Fig. 20, video monitoring data indicate the location of the critical points caused by seasonal oblique high waves in November and December 2015, while we obtained the theoretical results 310 from the analysis of the NOAA wave data within the same period of time. Further analysis using the NOAA wave data over 40 years, as shown in Fig. 7(a), indicates that the occurrence of values varying from 0° to 30° may last between zero and 4 months, as shown in Fig. 21, and their corresponding maximum indentations are marked in Fig. 20 for comparison. Because our results exclude shoreline retreat because of cross-shore sediment transport, the theoretical solution for the coordinates of the maximum indentation from Eqs. (16a) and (16b) with the LST could be underestimated. In addition, both theoretical equations neglect sediment bypassing from updrift. Therefore, the solutions must be modified if bypassing occurs. 320 By limiting the alongshore distance of the downdrift control point (at shown in Fig. 13) with the breaking wave angle in relation to the protruding length of the groyne (as shown in Fig. 20), the following relationship can be assumed: Here, the subscript l denotes the limiting value. Table 2 compares the variables and , obtained from Eqs. (10) and (23), respectively. If obtained for each is greater than obtained for a given , in Eqs. (16a) and (16b) should be replaced 325 by because bypassing might not occur between 10° and 17.5°. As shown in Fig. 21, the monitoring data in December 2015 support the theoretical solution that the use of = 10° for calculating . In Table 2, (=√ ) indicates no effect of protruding length or beach length L (= 850 m), because either the LST is small or the wave duration is too short. However, the erosion width may be reduced as bypassing occurs when the protruding length is short. Thus, the limit in in relative to L is expected to be within one-half of the beach length, being 425 m (= L/2, where L = 850 m), which is within the range of 330 beach length (approximately 600 m) covered by the video monitoring equipment.

Discussion: Engineering countermeasures for mitigating seasonal erosion
Although the characteristics of the seasonal changes in incident waves cannot be modified, the extent of erosion can be reduced 335 either by artificially nourishing the beach to advance the shoreline, or by placing a short groyne to promote sediment accretion within the potentially eroding section. Fig. 22 compares the reduction in potential coastal erosion and its maximum depth https://doi.org/10.5194/esurf-2021-71 Preprint. Discussion started: 13 October 2021 c Author(s) 2021. CC BY 4.0 License.

Conclusions
In winter on Korea's eastern coast, severe downdrift erosion often occurs around structures blocking LST because of the inflow of oblique waves. When structures such as groynes and headlands block or delay the LST, erosion damage occurs on the opposite shore. Our study presented a theoretical approach to control downdrift erosion caused by seasonal oblique wave incidence and to protect the facilities behind it and a solution for coastal engineering. First, by analyzing the 40-year wave data 355 of the NOAA Jeongdongjin coast, we analyzed the wave incidence characteristics by season and theoretically presented the analysis solution of the maximum erosion width by the seasonal oblique wave incidence by analyzing the control characteristics of longshore sediment transport at the study site by using natural rocks. Here, the PBSE proposed by Hsu and Evans (1989) indirectly reflected the diffraction effect of structures with long protrusions. The applicability of PBSE to the Jeongdongjin coast was verified by comparison with the surrounding equilibrium shoreline and the mean shoreline results obtained by 360 applying the directional wave spectrum to a simple diffraction wave model. Seasonal erosion on the downdrift side of the structures is mainly related to structural specifications (e.g., protruding length of the structures), wave environment (e.g., duration of wave action as well as wave breaking height and angle), and sand particle size information (e.g., LST coefficient, porosity, and density). The influence varies depending on the location of the control point, which controls the shape of the shoreline sag. The length unit variables that determine the control point include the 365 beach length, longshore drift length, and protrusion length of the structure. We found the results to be in good agreement with https://doi.org/10.5194/esurf-2021-71 Preprint. Discussion started: 13 October 2021 c Author(s) 2021. CC BY 4.0 License. the yielding line of the maximum erosion width obtained by closed-circuit television (CCTV) in winter in Jeongdongjin. The most critical breaking wave angle of = 10 °. was found for a given groyne length of = 80 m.
Engineering countermeasures are needed to reduce seasonal hot spots in areas where oblique wave incidence occurs frequently.
Reducing the downdrift erosion is impossible because of the seasonal change characteristics of the incident wave or the 370 surrounding sand environment as natural factors. Therefore, in this study, we proposed two major methods to mitigate erosion artificially. The first is to reduce erosion by easing the protrusion length of natural rocks by installing a beach nourishment facility near the hot spot to advance the shoreline. The second is to construct a structure such as a small-scale groyne that acts as a control point to prevent the critical point of the sagging shoreline to retreat to the shore. The engineering approach of this study is expected to aid the design of a groyne to reduce the potential coastal erosion on the downdrift coast, as well as the 375 strategy of nourishing an eroding beach where necessary.

Data availability
Not applicable.