MODEL FOR PREDICTING BATHYMETRIC AND GRAIN SIZE CHANGES BASED ON BAGNOLD S CONCEPT AND EQUILIBRIUM SLOPE CORRESPONDING TO GRAIN SIZE COMPOSITION

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1 MODEL FOR PREDICTING BATHYMETRIC AND GRAIN SIZE CHANGES BASED ON BAGNOLD S CONCEPT AND EQUILIBRIUM SLOPE CORRESPONDING TO GRAIN SIZE COMPOSITION Yasuhito Noshi, Takaaki Uda 2, Masumi Serizawa 3, Takayuki Kumada 4 and Akio Kobayashi 5 A model for prediting bathymetri and grain size hanges onsidering the equilibrium slopes orresponding to eah grain size and its omposition was inorporated into the BG model proposed by Serizawa et al. (27). The model was applied to predit the beah hanges at Kemigawa Beah. The ross-shore sorting of sand, in whih the grain size gradually dereases with inreasing depth, and the formation of a gradually hanging longitudinal slope were aurately reprodued. The model was also applied to predit the seletive aumulation of fine sand in a navigation hannel after dredging and its impat on the nearby oast. Keywords: predition model; Bagnold s onept; equilibrium slope; navigation hannel; bathymetri hange; grain size hange INTRODUCTION In general, grain size sorting ours under wave ation; oarser materials aumulate near the shoreline, forming a steep slope, whereas fine sand tends to be deposited further offshore with a gentle slope. On the basis of these phenomena, Uda et al. (24) developed a ontour-line-hange model for prediting hanges in the longitudinal profile by onsidering the equilibrium slope orresponding to eah grain size. Kumada et al. (26) expanded the model, assuming that the longitudinal profile an be expressed as a ombination of several equilibrium slopes that are given by eah grain size population. The appliability of their model to pratial problems is high, and Kumada et al. (27) applied the model when designing the most effetive shape of an artifiial headland to prevent sand transport to Oharai Port. In their method, however, the equilibrium slope does not hange, even if the ontent of eah grain size in the population slightly hanges, sine eah equilibrium slope is assumed to be invariant, and therefore, it is diffiult to predit spatial hanges in the longitudinal profile assoiated with hanges in the omposition of eah grain size. In fat, the longitudinal slope gradually beomes gentle with inreasing ontent of fine sand, for example, in the wave-shelter zone, but these hanges in the loal slope in and around the wave-shelter zone due to this mehanism annot be predited by their model. Noshi et al. (29) improved the model using the ontour-line-hange model and taking the equilibrium slope orresponding to eah grain size and its omposition into aount. Although the appliability of this ontour-line-hange model to pratial problems is high, the handling of ompliated boundary onditions around various strutures is more diffiult ompared with the BG model proposed by Serizawa et al. (27), in whih the fundamental equations are solved in a mesh. In this study, Noshi et al. s modifiation of the ontour-line-hange model is inorporated into the BG model, whih has an advantage of being able to predit topographi hanges near the strutures with ompliated shape. To verify the appliability of the new model, it is applied to the bathymetri hanges of Kemigawa Beah and is used to investigate the mehanism of the seletive aumulation of fine sand into the navigation hannel of a port after dredging as well as its impat on the nearby oast. MODEL Fluid motion near the sea bottom under wave ation is osillatory, and therefore, the bak and forth motion of the bottom sediment is generated as a result of the fluid motion. Inman and Bagnold (963) assumed that the sand transport under wave ation is given by the sum of two omponents: shoreward sand transport aused by inoming waves and seaward sand transport aused by outgoing waves, and they derived a sand transport equation, whih expliitly inludes the seabed slope, based on the energy onservation law. By applying the onept of the equilibrium slope introdued by Inman and Bagnold (963) and the energetis approah of Bagnold (963), Serizawa et al. (26) proposed Eq. () for the Dr. Eng., ICOMNET, Ltd Aomi, Koto-ku, Tokyo 35-64, Japan 2 Dr. Eng., Exeutive Diretor, Publi Works Researh Center, -6-4 Taito, Taito, Tokyo -6, Japan 3 Coastal Engineering Laboratory Co., Ltd., Wakaba, Shinjuku, Tokyo 6-, Japan 4 Dr. Eng., Publi Works Researh Center, -6-4 Taito, Taito, Tokyo -6, Japan 5 Dr. Eng., Department of Oeani Arhiteture & Engineering, College of Siene & Tehnology, Nihon University, Narashinodai, Funabashi, Chiba , Japan

2 2 COASTAL ENGINEERING 2 transport flux of sand omposed of a single grain size using only the wave onditions at the breaking point. q G = tanβew Z tanβ 2 ( Z)( EC g ) os αb β G= C (2) Kε tan b Here, β is the equilibrium slope angle, z = ( z / x, z / y) is the gradient vetor of seabed elevation, e w is the unit vetor of wave diretion, K is the oeffiient of longshore sand transport, (EC g ) b is the energy flux at the breaking point, α b is the breaker angle and C is a oeffiient used to transform the immersed weight of sand into a volumetri expression. ε(z) is the depth distribution of the intensity of sediment transport and is given by the ubi equation proposed by Uda et al. (996). Serizawa et al. (27) expanded Eq. () so as to be appliable to the ase of sand with mixed grain size referring to Uda et al. (24), to develop a model for prediting hanges in beah topography and (k) grain size. Introduing the onept of a mixing layer proposed by Hirano (97), the ontent µ of eah grain size d (k) (k=, 2,, N) in the exhange layer is added to the variables to be solved. Sand transport equation for eah grain size is given by Eq. (3) by multiplying Eq. () by the ontent of eah grain size to evaluate its ontribution to wave energy dissipation while introduing the equilibrium slope of eah grain size. Furthermore, the longshore sand transport oeffiient for eah grain size (Eq. (5)) is assumed to be inversely proportional to the square root of the grain size referring to Kamphuis et al. (986). The beah hanges are determined by the sum of the omponents of the beah hanges for eah grain size, whih are alulated by the ontinuity equation for eah grain size (Eq. (6)). The ontent is alulated by solving the ontinuity equation in the exhange layer (Eq. (8)). () q G = tan β tanβ ew Z ( k =,2,..., N) (3) G ( k ) ( Z)( EC ) os 2 α tan ( k =,2,..., N) = µ C K ε β (4) K g b b A = ( k =,2,..., ) (5) d N z = q ( k =,2,..., N) (6) Z = N k= z (7) µ ( z B h = ( z B h k) k) µ µ B Z Z Z Z < ( k =,2,..., N) (8) Here N is the number of grain sizes, B h is the thikness of the exhange layer, µ : k =, 2,..., N is the ontent of eah grain size in the exhange layer, tanβ is the equilibrium slope of eah grain size, K (k) is the oeffiient of longshore sand transport orresponding to eah grain size, tan β : Z=-h C - h R is the mean slope of the initial ross setion between Z=-h and h R, and A is a onstant depending on the ondition of eah speifi oast. In addition, the grain size d (k) in Eq. (5) has a unit of mm.

3 COASTAL ENGINEERING 2 3 On a oast, not only the longshore sorting of the grain size of sand but also the ross-shore sorting of grain size an be simultaneously observed. Coarser material is deposited in the zone with a higher elevation on the foreshore due to the ross-shore sorting by waves, whereas fine sand tends to be deposited further offshore with inreasing depth, and as a result, the parallel ontours at the initial ondition deform, forming a seabed slope that gradually beomes flatter from the erosion zone to the aretion zone. Noshi et al. (26) obtained the empirial relationship given by Eq. (9) to determine tanβ using the ontent µ (k) of eah grain size and the equilibrium slope tanβ orresponding to eah grain size. tanβ = tanβ r+ tanβ ( r) (9) ln N ( i, k) ( tan ) = µ ln( tanβ ) ln k= β () ( tan ) = ln a+ b ln( d ) β () Then, Noshi et al. (29) proposed the ross-shore sand transport equation for the ontour-linehange model (Eq. (2)), using this method to alulate the loal longitudinal slope orresponding to the omposition of eah grain size. q z = µ ε ( z) γ K z { r (otβ / otβ ) + ( r) (otβ / otβ k =, 2, K, N r ( EC g ) b os 2 α sinβ bs )}; (2) Here, q z (k) ; k=, 2,, N is the ross-shore sand transport of eah grain size and ε(z) is the depth distribution of the intensity of sand transport, tanβ is the loal seabed slope. The total ross-shore sand transport is given by the weighted mean of these ross-shore sand transports with weight r. In this study, r is set to.5, assuming the same ontribution of eah term. In this study, the sand transport equations for eah grain size proposed by Serizawa et al. (Eqs. (3)- (5)) were expanded on the basis of Noshi et al. s onept. In formulating the sand transport equation, the oeffiients of ross-shore and longshore sand transport were separately inluded in the basi equations similar to that in Uda et al. (2). Gx Z qx = tan os w ( k =,2,..., N) tan β θ x β Gy Z qy = tan sin w ( k =,2,..., N) tan β θ y β (3a) (3b) G G x x y y = = K G, G = K G ( k,2,..., N) (4a, b) ( Z)( EC ) os 2 α tan ( k =,2,..., N) = µ C ε β (5) K g b b A = ( k,2,..., N) (6) d y = = tanβ = tanβ r+ tanβ ( r) ( k,2,..., N) (7) Here, q x is the ross-shore omponent of sediment transport flux (positive for shoreward), q y is the longshore omponent of sediment transport flux, w θ is the angle between the wave diretion and the x- axis, K x (k) and K y (k) are the oeffiients of ross-shore and longshore sand transport. The onservation of sand volume orresponding to eah grain size and the ontent of eah grain size in the exhange layer

4 4 COASTAL ENGINEERING 2 are the same as those in Serizawa et al. (27). When offshore breakwaters are onstruted, a waveshelter zone is formed behind these strutures. In suh ases, the wave diffration oeffiient and wave diretion in the viinity of the wave-shelter zone are alulated using the angular spreading method for irregular waves (Sakai et al., 26) and then, the wave height is redued by multiplying the breaker height without the offshore breakwaters by the wave diffration oeffiient. Furthermore, as the distribution of wave diretion, the diretion of diffrated waves is used. The additional longshore sand transport due to the longshore hange in breaker height was evaluated by the method given by Ozasa and Brampton (98). TOPOGRAPHIC CHANGES AT KEMIGAWA BEACH To verify the effetiveness of the proposed model, it was applied to predit the bathymetri hanges of Kemigawa Beah faing Tokyo Bay, as shown in Fig.. Figure 2 shows the bathymetry of this beah measured in 24. The straight, impermeable groins on both sides of the beah were extended by 23 m in 979, and a,3-m-wide poket beah with parallel ontours was produed by beah nourishment. Then, a urved part was added to the groins in 993, forming a large wave-shelter zone behind them. A Y-shaped groin was also onstruted in the entral part of the beah in 22. As a result, the shoreline advaned in the wave-shelter zones formed behind the urved and Y-shaped groins. This auses the deformation of the initial ontours as shown in Fig. 2. Tokyo Yokohama Chiba Kemigawa Beah 2km Figure. Loation of Kemigawa Beah faing Tokyo Bay. Calulation Curved groin (South) domain m 2 No. No. 2 No. 3 Figure 2. Bathymetry of Kemigawa Beah in 24. Gently sloping revetment -m N Curved groin (North) -9 Y-shaped groin m (unit : m) Figure 3 shows the longitudinal profiles along transet Nos., 2 and 3 shown in Fig. 2, where transet Nos. and 3 are respetively loated inside and outside the wave-shelter zone and No. 2 is loated in the middle of the beah. Figure 4 shows the orresponding ross-shore distribution of grain size omposition along the three transets.

5 COASTAL ENGINEERING 2 5 Elevation above MSL Z( m) No. No. 2 No. 3 H.W.L. L.W.L Offshore distane Y(m) Figure 3. Longitudinal profiles along three transets at Kemigawa Beah. Transet No d (mm) Offshore distane Y(m) Transet No d (mm) Offshore distane Y(m) Transet No.3 d (mm) Offshore distane Y(m) Figure 4. Cross-shore distribution of omposition of eah grain size along transet Nos., 2 and 3 (measured).

6 6 COASTAL ENGINEERING 2 Along transet No., a foreshore slope of /7 above the high water level (HWL) has formed, whereas the seabed slope is as gentle as /5 between the HWL and LWL. Beause of these differenes in slope, oarse sand has been deposited on the shore fae, and the offshore seabed with a gentle slope has been overed with fine and medium sand. Similar features an be seen along transet No. 2. Along transet No. 3, loated in the erosion zone, oarse sand is onentrated near the shoreline. Regarding the grain size omposition, the relatively steep slope is omposed of medium and oarse sand, whereas the gentle slope is mainly omposed of fine and medium sand without oarse sand. This is onsidered to be due to the effet of the longshore and ross-shore sorting of sand by waves; near the downoast boundary of the urved groins, a large volume of fine sand was deposited, whereas the ontent of oarse sand inreased outside the wave-shelter zone, beause fine sand was seletively transported away from the area due to the higher mobility of fine sand than that of oarse sand. APPLICATION TO KEMIGAWA BEACH Calulation onditions The alulation domain is a retangular region,3 m long and 45 m wide separated by the urved groins. For the alulation onditions, the beah slope in the zone shallower than -4 m is assumed to be /2, and that in the zone deeper than -4 m is /5, whih are the average measured slopes. The number of grain sizes is assumed to be three, and the typial grain sizes for the grain size ranges orresponding to fine sand and silt (d.25 mm), medium sand (.25 mm d.85 mm) and oarse sand (d.85 mm) are assumed to be.6,.425 and.85 mm, respetively. The ontents of the three grain sizes are 35% (.6 mm), 5% (.425 mm) and 5% (.85 mm) and the orresponding equilibrium slopes are assumed to be /6 (.6 mm), /4 (.425 mm) and /7 (.85 mm). The ontent of eah grain size was determined on the basis of measured values; the sieve analysis of 3 samples taken along transet Nos., 2 and 3 loated in the aretion, neutral and erosion zones was arried out to obtain the ontent of eah grain size range. The mean ontent of the beah material was determined as the average of the ontents of grain size along the three transets. As the initial ondition, parallel ontours were assumed in the alulation domain, and waves with a breaker height of H b = m, whih was determined to be the annual mean wave height, were inident to the initial ontours. The equilibrium slope tanβ orresponding to the grain size d (k) was determined from Eq. () given a=. and b=.93, whih were obtained by the least-squares method on the basis of the measured data for Kemigawa Beah. The other alulation onditions are shown in Table. Table. Calulation onditions. Cases Kemigawa Beah Dredging of navigation hannel Initial slope /5 /2 Initial grain size and ontent.6 mm, 33%.425 mm, 5%.85 mm, 5%.25 mm, 33%.75 mm, 33% 2. mm, 34% Equilibrium slope (grain size) /6 (.6 mm) /4 (.425 m) /7 (.85 mm) /5 (.25 mm) /5 (.75 mm) /3 (2. mm) Thikness of exhange layer B (m).54.5 Breaker height H b (m). 3. Breaker angle α (deg.) Water level Mean sea level Mean sea level Depth of losure h (m) 5.. Berm height h R (m) Coeffiient of sand transport A.3.3 Coeffiient of Ozasa and Brampton (98) Ratio of oeffiient of ross-shore transport to that of longshore sand transport K (k) (k) x /K y (k).62 K y (k).62 K y.2.2 Critial slope on land and seabed /2 and /2 /2 and /3 Calulation domain alongshore (m),3, Mesh size x (m) and y (m) 2 and 2 and 2 Time step t (hr).. Total number of steps 2, 6,

7 COASTAL ENGINEERING 2 7 Calulation onditions Figure 5 shows the predited bathymetry in 24 given the initial parallel ontours in 984. The advane of offshore ontours near the urved groins and the reession of nearshore ontours outside the wave-shelter zone with the formation of a sarp are in good agreement with the measured hanges. Furthermore, the beah slope near the shoreline beame steep and a gentle slope was formed in the offshore zone. Thus, the hanges in slope in the ross-shore diretion were aurately predited by the present model along with the longshore hanges in the foreshore slope. Figure 6 shows the predited longitudinal profiles along the three transets orresponding to the measured profiles in Fig. 3. In the profile along transet No. 3, loated in the erosion zone, a onave shape was formed beause of the steep slope near the shoreline. In ontrast, along transet No., loated in the aretion zone, the foreshore slope beame steep but a gentle slope was formed in the offshore zone. These features are in good agreement with the measured profiles. Figure 7 shows the results for the ontent of eah grain size along eah transet. It is larified that the ontent of fine sand inreases and the ontent of oarse sand dereases toward the wave-shelter zone. Furthermore, the predited results for the omposition of eah grain size and the longitudinal slope at eah point orrespond well to the measured results, as shown in Fig. 4. Thus, the measured results were explained by the present model. No. Curved groin (south) No. 2 No. 3 Curved groin (north) Y-shaped groin 2 Figure 5. Predited bathymetry.

8 8 COASTAL ENGINEERING 2 Elevation above MSL Z( m) No. No. 2 No. 3 H.W.L. L.W.L Offshore distane Y(m) Figure 6. Predited longitudinal profiles along transet Nos., 2 and 3. Transet No d (mm) Offshore distane Y(m) Transet No d (mm) Offshore distane Y(m) Transet No. 3 d (mm) Offshore distane Y(m) Figure 7. Cross-shore distribution of omposition of eah grain size along transet Nos., 2 and 3 (predited).

9 COASTAL ENGINEERING 2 9 PREDICTING DEPOSITION OF FINE SAND INTO NAVIGATION CHANNEL Calulation onditions Beause the present model is advantageous for prediting the transport of sand of eah grain size, the seletive aumulation of fine sand into a navigation hannel was predited using an imaginary ase. In the alulation, three grain sizes are onsidered, the ontents of whih are assumed to be µ =33% (d =.6 mm), µ 2 =33% (d 2 =.425 mm) and µ 3 =34% (d 3 =2 mm). The alulation domain is a retangle with m length and 5 m width. The other alulation onditions are shown in Table. Results When we assume that waves with breaker height H b =3. m are inident to the model beah from the diretion normal to the initial shoreline, the hanges in profile and grain size due to ross-shore sand transport after 2, steps are alulated as shown in Fig. 8. Coarse material aumulates near the foreshore forming a berm, whereas the offshore bed is overed with fine material. Furthermore, assuming that the initial bathymetry and grain size distribution before the installation of a breakwater and a groin have longshore uniformity with the ross-shore distribution, as shown in Fig. 8, then they are given by Figs. 9 and, respetively. 5 Elevation above M.S.L. Z (m) After 2,steps Initial plofile Y (m) mm.75 mm.25 mm 3 2 Figure 8. Predited longitudinal profile and ross-shore distribution of ontent of eah grain size.

10 COASTAL ENGINEERING Figure 9. Initial bathymetry before onstruting an offshore breakwater and a groin. Figure. Initial distribution of grain size before onstruting an offshore breakwater and a groin.

11 COASTAL ENGINEERING 2 Figure shows the predited bathymetry 2, steps after the installation of the offshore breakwater. Sand was transported around the tip of the groin from outside to inside the wave-shelter zone of the offshore breakwater. Immediately to the left of the groin, the shoreline loally reeded, and sand was transported further inward toward the port. Figure 2 shows the distribution of the median diameter of grain size orresponding to the bathymetry shown in Fig.. Fine sand was seletively transported from outside the wave-shelter zone, far from the breakwater, to near the groin and deposited near the navigation hannel offshore of the groin. Erosion ourred immediately to the left of the groin and sand was deposited deep in the wave-shelter zone. Beause of the aumulation of sand with oarser grain size near the shoreline, the foreshore slope beame steeper than that before the onstrution of the offshore breakwater. Figure 3 shows the bathymetry immediately after dredging the sand deposited in the navigation hannel. Waves were inident under these onditions, and the bathymetry after 2, steps is shown in Fig. 4. Sand was again transported while being deposited in the navigation hannel offshore of the groin. The orresponding distribution of the median diameter of grain size is shown in Fig. 5. It was found that fine sand was seletively transported and deposited offshore of the groin whereas the seabed on the nearby oast was overed with oarser material Figure. Predited bathymetry 2, steps after the onstrution of an offshore breakwater and a groin. Figure 2. Distribution of median diameter orresponding to Fig..

12 2 COASTAL ENGINEERING Figure 3. Bathymetry immediately after dredging of navigation hannel Figure 4. Predited bathymetry 2, steps after dredging. Figure 5. Distribution of median diameter orresponding to Fig. 4.

13 COASTAL ENGINEERING 2 3 CONCLUSIONS A model for prediting bathymetri and grain size hanges onsidering the equilibrium slopes orresponding to eah grain size and its ontent was inorporated into the BG model proposed by Serizawa et al. (27). The model was applied to predit the beah hanges at Kemigawa Beah. The ross-shore sorting of sand, in whih the grain size gradually dereases with inreasing depth, and the formation of a gradually hanging longitudinal slope were reprodued. The model was also applied to predit the seletive aumulation of fine sand in a navigation hannel. It was predited that fine sand would be seletively transported into the wave-shelter zone, ausing the deposition of sand in the navigation hannel, whereas the seabed on the nearby oast would be overed with oarser material. REFERENCES Bagnold, R. A Mehanis of marine sedimentation, in the Sea, M. N. Hill (ed.), 3, , Wiley, New York. Hirano, M. 97. River-bed degradation with armoring, Pro. JSCE, Vol. 95, (in Japanese) Inman, D. L. and Bagnold, R. A Littoral proesses, in The Sea, M. N. Hill (ed.), 3, , Wiley, New York. Kamphuis, J. W., Davies, M. H., Narim, R. B. and Sayao, O. J Calulation of littoral sand transport rate, Coastal Eng.,, -2. Kumada, T., Uda, T., Serizawa, M. and Noshi, Y. 26. Model for prediting hanges in grain size distribution of bed materials, Pro. 3 th ICCE, Kumada, T., Uda, T. and Serizawa, M. 27. Quantitative evaluation of ontrolling effet of headland on longshore sand transport using model for prediting hanges in ontour lines and grain size, Coastal Sediments 7, Noshi, Y., Kobayashi, A., Uda, T., Kumada, T. and Serizawa, M. 26. Relationship between loal seabed slope and grain size omposition of bed materials, Pro. 3 th ICCE, Noshi, Y., Kobayashi, A. and Uda, T. 29. Model for prediting bathymetri and grain size hanges onsidering equilibrium beah slopes orresponding to omposition of grain size and eah grain size, Jour. Coastal Res., 56, 8-2. Ozasa, H. and Brampton, A. H. 98. Model for prediting the shoreline evolution of beahes baked by seawalls, Coastal Eng., 4, Sakai, K., Uda, T., Serizawa, M., Kumada, T. and Kanda, Y. 26. Model for prediting threedimensional sea bottom topography of statially stable beah, Pro. 3 th ICCE, Serizawa, M., Uda, T., San-nami, T. and Furuike, K. 26. Three-dimensional model for prediting beah hanges based on Bagnold s onept, Pro. 3 th ICCE, Serizawa, M., Uda, T., San-nami, T., Furuike, K., Ishikawa, T. and Kumada, T. 27. Model for prediting beah hanges on oast with sand of mixed grain size based on Bagnold s onept, Coastal Sediments 7, Uda, T., Yamamoto, Y., Itabashi, N. and Yamaji, K Field observation of movement of sand body due to waves and verifiation of its mehanism by numerial model, Pro. 25 th ICCE, Uda, T., Kumada, T. and Serizawa, M. 24. Preditive model of hange in longitudinal profile in beah nourishment using sand of mixed grain size, Pro. 29 th ICCE, Uda, T., Serizawa, M., San-nami, T. and Ishikawa, T. 2. Predition of dynamially stable ebb tidal delta and measures for preventing offshore sand loss, Pro. 32 st ICCE. (in press)