Numerical Modeling of Shoreline Change due to Structure-Induced Wave Diffraction

Journal of Coastal Research SI 56 78-82 ICS2009 (Proceedings) Portugal ISSN 0749-0258 Numerical Modeling of Shoreline Change due to Structure-Induced Wave Diffraction I.H. Kim and J.L. Lee Department of Constructional Disaster Prevention Engineering Kangwon National University, Samchuck 245-711, Korea kimih@kangwon.ac.kr Department of Civil and Environment Engineering Sungkyunkwan University, Suwon 440-746, Korea jllee@skku.edu ABSTRACT KIM, I.H. and LEE, J.L., 2009. A fast forecasting system of typhoon-generated waves. Journal of Coastal Research, SI 56 (Proceedings of the 10th International Coastal Symposium), 78 82. Lisbon, Portugal, ISSN 0749-0258. In this study, we develop a new shoreline change model modified using a logarithmic spiral bay equation for the reliable prediction of shoreline configurations such as crenulated-shaped bays and salients etc. The longshore transport rate in the sheltered zone is obtained in terms of empirical coefficients using the concept of a logarithmic spiral bay in order to yield the crenulated bay features due to the wave diffraction effect. The simulated shoreline results are well fitted to a parabolic bay shape as well as the logarithmic spiral bay shape because the effect of primary longshore sand transport rate usually taken in the model tends to stretch out the downcoast shoreline parallel to the local prevailing wave crest direction. The present model successfully produces salient and crenulated-shaped bay beaches. A comparison between shoreline data prior to and after construction of several coastal structures reveals that the present approach is considered acceptable for practical application. ADITIONAL INDEX WORDS: Spiral bay, salient, headland, shoreline change model, wave diffraction INTRODUCTION A coastline is rarely straight; some segments may curve gently in plan, while others are more indented, or even of a more intricate configuration with various shapes and sizes. These planforms are composed of various configurations, depending on their indentation and on the magnitude of the water area enclosed. A number of accretional landforms along the east coast of Korea are depicted in Figure 1a, the bays and barrier beaches of which require special attention because of their close link with human activities. Under the influence of global wind systems, most coasts receive persistent swell waves from a particular oblique direction. Therefore, beaches tend to be built up transverse to the prevailing direction of approach of the incident waves. Such features that are usually formed between headlands on the coast can be observed on maps and aerial photographs, and are universal. The east coast of Korea from Jukbyun to Pusan extends in an almost continuous series of asymmetrical curved bays joining one headland to the next. In Figure 1b example is shown on the east coast of Korea showing a series of bays due to the NE swell. Over the last few decades many beaches along the east coast of Korea have narrowed or been completely lost to coastal erosion. Coastal development and sea-level rise due to climate change threaten the remaining beaches. One of the main problems in the design of these coastal structures is the prediction of the shoreline response. Two approaches, numerical and empirical, have been undertaken in order to study the effects of man-made coastal structures. Several numerical models have been developed to solve the governing equations behind an offshore breakwater. Among them, shoreline change modeling has evolved considerablely over several decades. Unfortunately, these approaches have not been the most efficient way to produce reliable long-term predictions of crenulated-shaped bay beaches, which are common features found all around the world. The difficulty in the numerical modeling is caused by the inevitable approximations in the solution of combined refraction and diffraction in water of changing depths, non-linear wave effects, and sediment transport in wave-current Figure 1. Accretional landforms along the east coast of Korea; various configurations of shoreline features and a series of crenulated shaped bays 78

Shoreline Changes due to Wave Diffraction flow, etc. The second approach requires an a priori assumption of the shape of the shoreline. A number of researchers have carried out empirical analysis based on beach equilibrium concepts (e.g. McCormick 1993, Hsu and Silvester 1990) and on small scale models and field observations (see Rosati 1990 and ASCE 1994 as general references). For the more practical applications, the empirical bay shape equations have been modified recently to assess the beach stability for natural and man-made bay beaches, prior to and after structural installation, as well as for the planning of headland-bay beach in static equilibrium for recreation and tourism. In the present study, we propose a shoreline change model that has been developed by assuming that the shoreline curvature derived from a logarithmic spiral relationship is applicable to all structure-induced shorelines in equilibrium state. BAY SHAPE STUDY Geologists and geographers were the first to be interested in the existence of such bay shapes (Halligan 1906), but the shape as a stable physiographic feature was first recognized by Jennings (1955), without the full knowledge of the waves involved. Davies (1958) realized the importance of wave refraction, while Yasso (1965) measured the planforms of a number of prototype bays in the United States of America and showed that they were equivalent to a logarithmic spiral. This empirical relationship had been accepted for almost 25 years until Hsu and Evans (1989) developed a more universal relationship. The shapes of these bays have been given a variety of names in the literature, such as zeta bays (Halligan 1906, Silvester et al. 1980), half-heart bay (Silvester 1960), crenulate shaped bays (Ho 1971, Silvester and Ho 1972), spiral beaches (Krumbein 1944, Le Blond 1972), curved or hooked beaches (Rea and Komar 1975), headland bay beaches (Le Blond 1979, Wong 1981, Phillips 1985), and pocket beaches (Silvester et al. 1980). It is generally thought that these bays erode according to alongshore-transport gradients controlled by the influence of the upwind headland, or pinning point, on the angle of waves reaching the beach. Therefore, the orientation of these bays is an excellent indicator of the direction of net sediment movement along the coast, indicating nature s method of balancing wave energy and load of sediment transport. By observing these shapes along large reaches of shoreline it can be shown that littoral drift is in the same direction for many kilometers or even hundreds of kilometers. This knowledge can help in the prediction of future trends in shoreline movement over long periods of time. SHORELINE CURVATURE DUE TO STRUCTURE-INDUCED DIFFRACTION If the shore-attached multiple breakwaters are built, they usually change the wave approaching pattern due to wave refraction and/or diffraction, and eventually produce a spiral bay decreasing in curvature away from the pinning point at the center of the spiral. Several authors have predicted that refraction and/or diffraction ought to produce bays of this general form. The logarithmic spiral is an elegant hypothesis, although Komar and others (e.g. Phillips, 1985) argue that the theoretical ideal shape is less important than more general descriptors such as a downdrift decrease in curvature. Headland-bay beaches have been fitted to other curves, sometimes more successfully than to logarithmic spirals (e.g. Moreno and Kraus 1999), but no mechanisms are generally accepted for these correspondences. A definition sketch of a logarithmic spiral is given in Figure 2 of which the equation is: R R 2 1 exp( cot ) (1) where θ is the angle between radii R 2 and R 1 (where R 2 >R 1 ) and α is the constant angle between either radius and its tangent to the curve. There were specific values for these variables for the stable condition that only varied with the wave obliquity β as depicted in Figure 3. Thus there was only one value of R 2 /R 1 or α of the logarithmic spiral relationship for an equilibrium stable bay for any given wave obliquity β. Differentiating a series form of Eq. (1) by θ and taking a leading term yields R cot R (2) Figure 2. Definition sketch of logarithmic spiral Figure 3. Relationship between α versus wave obliquity β in Eq. (1) 79

Kim and Lee equlibrium shoreline given by Eq. (2). Therefore, the shoreline will reach equilibrium when the orientation of the shoreline parallels the diffracted wave crests approaching from the dominant direction, leading to a zero-alongshore-flux equilibrium configuration. This one-line model of coastal evolution assumes that the shoreface maintains a constant shape as the coastline position translates seaward or landward, and that gradients of alongshore sediment transport within the surf zone control long-term coastal evolution. One line of beach cells is therefore sufficient to specify the location of the shoreline. Figure 4. Comparison between solutions of differential equation (Eq. (2)) and logarithmic bay (Eq. (1)) where α is the constant along the coast, but probably varies with incident wave conditions, R is the distance from the tip of the structure causing wave diffraction to the shoreline position, and θ is the angle measured from the main layout line of the coastal structure. In Figure 3, a comparison is shown between Eq. (1) and its approximate equation (2). In the present study, we assume that the shoreline curvature derived from the logarithmic spiral relationship is applicable to all structure-induced shorelines in an equilibrium state. PERFORMANCE TEST Modeling results are presented that suggest a fundamentally new methodology applicable for all kinds of various coastline features. Using a simple one-line numerical shoreline model, we generate crenulated bays and salient beaches that owe their existence to the shape of diffracted or refracted wave crests. Crenulated Bay Shape The crenulated bays are produced by a series of shore-attached multiple breakwaters built along the shoreline designed in a planform. It is the pattern obtained by wave refraction and/or diffraction as waves that approach from a dominant direction interact with the headland, which results in the bay decreasing in SHORELINE CHANGE MODEL The governing equation of the present shoreline change model is derived on the basis of conservation of sand mass, similarly to other common one-line models often used in coastal studies (Hanson and Kraus 1989). By modeling the shoreline position changes we are taking into account the underlying coastal erosion processes because all of the changes in the geometry of the shoreline are the end result of erosion phenomena. This equation is mathematically expressed as follows: xs 1 Q q 0 t D* y (3) where, x s is the shoreline position from the y-axis, y is the alongshore distance, Q is the alongshore transport rate, D * is the depth to which erosion and accumulation extend, and q is the line source or sink of sand. On a sandy coastline, alongshore gradients in the sediment flux, Q, tend to cause changes in the shoreline position. Q is comprised of Q s by wave-driven alongshore transport and Q d by structureinduced transport. Wave-driven alongshore transport can be related to the deepwater wave approach angle relative to the global shoreline orientation with the commonly used CERC equation (Komar, 1998): Q s = KH b 5/2 sin( b - )cos( b - ) (4) where H b is the breaking-wave height, b is the breaking-wave angle, is the local shoreline angle relative to the global shoreline orientation, and K is an empirical constant. The structure-induced transport would be dependent on the difference between the actual shoreline curvature and that for an Figure 5. Model results for spiral shape by shore-attached breakwater and parabolic shape by upcoast headland and downcoast control curvature away from the pinning point at the center of the spiral. Figure 5a shows an example of a sample model run that produces a hook-shaped bay as the shape of the prototype bay in a static equilibrium. The model result for the parabolic shape between upcoast headland and downcoast control is shown in Figure 5b. 80

Shoreline Changes due to Wave Diffraction Salient Formation Offshore breakwaters are generally shore-parallel structures that effectively reduce the amount of wave energy reaching a protected stretch of shoreline. Sand accumulation occurs in the lee of a breakwater placed offshore due to wave diffraction and refraction, aided by nearshore current circulation during its formation. Figure 6. Model results for salient formation by single and double detached breakwaters A single breakwater parallel to the coast with normally incident waves, as shown in Figure 6a, is in good agreement for α value of 80 with an empirical relationship given by Silvester and Hsu (1993). The empirical relationship is depicted in dimensionless functions divided by wavelength. Figure 6b shows the results for multiple detached breakwaters. MODEL APPLICATION TO BUGU COAST For the verification of the present model, we extracted past shoreline positions for Bugu coast in Uljin from recent and historic aerial photography. The years used for the shoreline change analysis included 1980, 1997, and 2005. Mean tide range along the Bugu coast is about 30cm. Kim et al. (2001) represented the downdrift pattern of net sediment due to the predominant wave direction from the northwest. The shoreline positions from aerial photos are depicted in Figures 7 and 8 as symbols and the results obtained by the present model are compared as bars. Figure 7 shows results for shoreline changes due to Nagok beach groin and Nagok Harbor breakwater construction and Figure 8 shows results due to to Wonjun construction. CONCLUSIONS In the present study, a new shoreline change model has been developed using a logarithmic spiral bay equation for the reliable prediction of the shoreline configurations such as crenulatedshaped bays, and salients, etc. The longshore transport rate in the sheltered zone is obtained in terms of empirical coefficients using the concept of a logarithmic spiral bay in order to yield the Figure 7. Shoreline changes due to Nagok beach groin and Nagok Harbor breakwater construction; aerial analysis (square: 2005-1980; circle: 1997-1980) and predicted (bars) Figure 8. Shoreline changes due to Wonjun construction; aerial analysis (square: 2005-1980; circle: 1997-1980) and predicted (bars) 81

Kim and Lee crenulated bay features due to the wave diffraction effect. The simulated shoreline results are well fitted to the parabolic bay shape as well as the logarithmic spiral bay shape because the effect of the primary longshore sand transport rate usually taken in the model tends to stretch out the downcoast shoreline parallel to the local prevailing wave crest direction. The present model successfully produces salient and crenulated-shaped bay beaches. A comparison of aerial photos prior to and after construction of several coastal structures reveals that the present approach is considered to be acceptable for practical application. LITERATURE CITED American Society of Civil Engineers ASCE, 1994. Coastal groins and nearshore breakwaters. Technical Engineering and design guides as adapted from the US Army of Engineers, No. 6. DAVIES, J.L., 1958. Wave refraction and the evolution of shoreline curves. Geogr. Stud. 5:1-14. HALLIGAN, G.H., 1906. Sand movement on the New South Wales coast. Proc. Limn. Soc. N.S.W. 31:619-40. HANSON, H. and KRAUS, N.C., 1989. GENESIS:Generalized Model for Simulating Shoreline Change. Tech. Rep. CERC- 89-19, U.S. Army Engr. Waterways Experiment Station, Coastal Engrg. Res. Center, Vicksburg, MS. HO, S.K., 1971. Crenulate shaped bays. Asian Inst. Tech., Master Eng. Thesis, No. 346. HSU, J.R.C., and EVANS, C., 1989. Parabolic bay shapes and their application. Proc. Instn. Civel Engrs. 87: 557-70. HSU, J.R.C., and SILVESTER, R., 1990. Accretion behind single offshore breakwater. J. Waterway, Port, Coastal & Ocean Eng., 143-48. JENNINGS, J.N., 1955. The influence of wave action on coastal outline in plan. Aust. Geogr. 6:36-44. KIM, A, LEE, J.L., and CHOI, B.H., 2001. Analysis of wave data and Estimation of Littoral Drifts for the Eastern Coast of Korea, J. KSCOE, 13, 1:18-34. KOMAR, P.D., 1998. Beach Processes and Sedimentation. Simon & Schuster: Upper Saddle River, New Jersey. KRUMBEIN, W.C., 1944. Shore Processes and Sedimentation. Englewood Cliffs, N.J.: Prentice Hall. LE BLOND, P.H., 1972. On the formation of spiral beaches. Proc. 13 th Inter. Conf. Coastal Eng., ASCE 2: 1331-45. LE BLOND, P.H., 1979. An explanation of the logarithmic spiral plan shape of headland-bay beaches,. Journal of Sedimentary Petrology 49(4), 1093-1100. MCCORMICK, M.E., 1993. Equilibrium shoreline response to breakwaters. Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 119, No. 6, November/Dec., 657-670. MORENO, L.J. and KRAUS, N.C., 1999. Equilibrium Shape of Headland-Bay Beaches for Engineering Design, Proceedings of Coastal Sediments 99, ASCE, 1, 860-875. PHILLIPS, D.A., 1985. Headland-bay beaches revisited:an example from Sandy Hook, New Jersey. Marine Geology 65: 21-31. REA, C.C, and KOMAR, P.D., 1975. Computer simulation models of a hooked beaches shoreline configuration. J. Sed. Petrology 45:866-72. ROSATI, J. D., 1990. Functional Design of Breakwaters for Shore Protection: Empirical Methods, Technical Report CERC-90-15, US Army Engineer Waterways Experiment Station, Vicksburg, MS. SILVESTER, R., 1960. Stabilization of sedimentary coastlines. Nature 188, Paper 4749, 467-69. SILVESTER, R. and HO, S.K., 1972. Use of crenulate shaped bays to stabilize coasts. Proc. 13 th Inter. Conf. Coastal Eng., ASCE 2:1347-65. SILVESTER, R., TARCHIVA, Y. and SHIBANO, Y., 1980. Zeta bays, pocket beaches and headland control. Proc. 17 th Inter. Conf. Coastal Eng., ASCE 2:1306-19. SILVESTER, R. and HSU, J.R.C., 1993. Coastal Stabilization: Innovative Concepts. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. WONG, P.P., 1981, Beach evolution between headland breakwaters. Shore and Beach 49(3):3-12. YASSO, W.E., 1965. Plan Geometry of Headland Bay Beaches. Journal of Geology, 73, 702-714. ACKNOWLEDGMENTS This study was partially supported by a project of the Advanced Coastal Technology and Environment Center of the Ministry of Land, Transport and Maritime Affairs. 82