**4.1. General conditions**

344 Numerical Simulation – From Theory to Industry

advantage of the BG model.

spit.

slope.

improved BG model.

prediction of topographic changes under all structural conditions, because the depth changes in the *x*-*y* plane are calculated, similarly to in the ordinary model for predicting 3D beach changes, and therefore the calculation can be carried out systematically. This is an

A movable-bed experiment was carried out using a plane-wave tank of 16 m width and 21 m length (Uda & Yamamoto, 1992). A model beach was made of sand with *d*50 = 0.28 mm. A sandy beach was established as the source of sand in the right half of the plane basin and conditions were set up such that leftward longshore sand transport developed. In Case 1, a shallow seabed with a water depth of 5 cm was formed in the left half of the wave basin and an offshore bed was formed with a steep slope of 1/5. In Case 2, a steep slope of 1/5 was produced instead of a shallow sea where the sand spit was formed. The angle between the direction normal to initial shoreline and the wave direction was 20° in order for sufficient longshore sand transport to occur. The elevation of the flat surface on the land was assumed to be 10 cm above mean sea level. Regular waves with *H*0' = 4.6 cm and *T* = 1.27 s incident to the model beach were generated for 8 hr. When a shallow sea exists, incident waves break immediately offshore of the shallow seabed, resulting in the rapid decay of waves on the shallow seabed. Because of this effect, sand is deposited near the marginal line between the shallow flat seabed and the steep offshore slope, resulting in the rapid elongation of a sand

Figures 1(a)-1(c) show the initial bathymetry and the beach topography after wave generation for 1 and 8 hr in Case 1, respectively. Here, the arrows in Figs. 1(a) and 1(d) show the breaking point (the tip of the arrows), the breaker height (the length of the arrows), and the wave directionat the breaking point (the direction of the arrows) measured immediately after the start of wave generation. Because the shoreline had a discontinuity due to a sudden change in the shoreline direction between the sand supply zone and the shallow seabedwhere sand was deposited, a straight sand spit extended from the boundary, and a slender sand spit extended along the marginal line between the shallow seabed and the steepoffshore slope over time. After 8 hr, the sand spit had reached the left boundary while forming a barrier island, and the width of the barrier island expanded upcoast from the left

Figures 1(d) and 1(e) show the initial bathymetry and the beach topography after wave generation for 8 hr in Case 2 with a steep offshore slope. The water depth in the zone where sand was deposited was large; thus, sand was deposited while forming a steep

Because this steep slope reaches a great depth, a cuspate foreland was formed without the development of a sand spit. These experimental results were used for validating the

**3. Movable-bed experiment on elongation of a sand spit** 

boundary because of the continuous sand supply.

The study area is the Kutsuo coast facing the Suo-nada Sea, part of the Seto Inland Sea, as shown in Fig. 2. Figure 3 shows an aerial photograph of the study area taken in 1999. The Harai River flows into the coast, which has a very wide tidal flat of approximately 1.5 km width offshore of the river. Although a river mouth bar extends on the north side of the Harai River, another slender sand bar has developed along the north side of the channel extending between the river mouth and the offshore tidal mud flat, and it intersects the river mouth bar perpendicular to the shoreline. The sand source for this slender sand bar is assumed to be the Harai River; sand transported offshore by flood currents is deposited

along both sides of the channel, and is then transported shoreward owing to the wave action.In this area, the slender sand bar as shown in Fig. 3 has been continuously developing. Figure 4 shows the bathymetry around the slender sand bar relative to the reference level (0.1m below mean sealevel) in 2008. Although the slender sand bar has moved slightly north compared with its position in Fig. 3, it extends in the cross-shore direction at *X* = 120 m in the central part of the river mouth bar, and the shoreline slightly protrudes near the connection point. The foreshore has been developing in the zone with elevation between 1.0 and 3.0 m. This foreshore is composed of coarse and medium-size sand, and its slope is as steep as 1/10. A tidal flat extends in the offshore zone, the elevation of which is lower than 1.0 m. In addition, there is a difference in the elevation of the tidal flat on both sides of the slender sand bar extending in the cross-shore direction with the ground elevation on the south side slightly higher than that on the north side.

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 347

**Figure 4.** Bathymetry around slender sand bar on tidal flat offshore of Kutsuo coast.

the cause of the difference in ground elevation on both sides of the sand bar.

the occurrence of shoreward sand transport.

During low tide on December 27, 2009, a field observation of the tidal flat offshore of the coast was carried out. The north end of the coast is separated by a vertical seawall protecting a park, as shown in Fig. 5. A sandy beach abruptly begins with a steep slope from the mud flat covered by cohesive materials, and the mud flat and sandy beach are clearly separated along a line where the slope changes abruptly. The tidal flat is composed of cohesive materials, whereas the sandy beach is composed of coarse sand and is well compacted.

Figure 6 shows a view of the entire slender sand bar. Although the sand bar is submerged during high tide, it is completely exposed during low tide, as shown in Fig. 6. The sand bar extends in the direction normal to the mean coastline. Comparing the elevations of the mud flat on both sides of the sand bar, the elevation on the right (south) side, which is next to theoffshore channel, is higher than that on the left (north) side. On the south side of the slender sand bar, the Harai River flows into the sea, and sand supplied to the tidal flat from the river mouth is considered to be transported and deposited on the surface of the tidal flat owing to the shoreward sand transport due to waves. In this case, sand supply to the area north of the slender sand bar is obstructed by the sand bar itself, and this is assumed to be

Figure 7 shows the tip of a branch separated from the offshore part of the main slender sand bar, as shown in Fig. 3. On this sand bar, decomposed granite, which was considered to be transported offshore by flood currents of the river, has been deposited. The elevation of the sand bar gradually increases shoreward, then at the landward end suddenly drops to the tidal flat with a steep slope approximately equal to the angle of repose of the sand, implying

Figure 8 shows the coastal conditions, looking landward from the tip of the slender sand bar. Although sand was deposited with a foreshore slope of 1/10 along the north side of the slender sand bar, the foreshore and flat tidal flat were clearly separated along the abrupt

**4.2. Field observation** 

**Figure 2.** Location of Kutsuo coast facing Suo-nada Sea, part of Seto Inland Sea.

**Figure 3.** Aerial photograph of Kutsuo coast.

BG Model Based on Bagnold's Concept and Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 347

**Figure 4.** Bathymetry around slender sand bar on tidal flat offshore of Kutsuo coast.

## **4.2. Field observation**

346 Numerical Simulation – From Theory to Industry

along both sides of the channel, and is then transported shoreward owing to the wave action.In this area, the slender sand bar as shown in Fig. 3 has been continuously developing. Figure 4 shows the bathymetry around the slender sand bar relative to the reference level (0.1m below mean sealevel) in 2008. Although the slender sand bar has moved slightly north compared with its position in Fig. 3, it extends in the cross-shore direction at *X* = 120 m in the central part of the river mouth bar, and the shoreline slightly protrudes near the connection point. The foreshore has been developing in the zone with elevation between 1.0 and 3.0 m. This foreshore is composed of coarse and medium-size sand, and its slope is as steep as 1/10. A tidal flat extends in the offshore zone, the elevation of which is lower than 1.0 m. In addition, there is a difference in the elevation of the tidal flat on both sides of the slender sand bar extending in the cross-shore direction with the ground

elevation on the south side slightly higher than that on the north side.

**Figure 2.** Location of Kutsuo coast facing Suo-nada Sea, part of Seto Inland Sea.

**Figure 3.** Aerial photograph of Kutsuo coast.

During low tide on December 27, 2009, a field observation of the tidal flat offshore of the coast was carried out. The north end of the coast is separated by a vertical seawall protecting a park, as shown in Fig. 5. A sandy beach abruptly begins with a steep slope from the mud flat covered by cohesive materials, and the mud flat and sandy beach are clearly separated along a line where the slope changes abruptly. The tidal flat is composed of cohesive materials, whereas the sandy beach is composed of coarse sand and is well compacted.

Figure 6 shows a view of the entire slender sand bar. Although the sand bar is submerged during high tide, it is completely exposed during low tide, as shown in Fig. 6. The sand bar extends in the direction normal to the mean coastline. Comparing the elevations of the mud flat on both sides of the sand bar, the elevation on the right (south) side, which is next to theoffshore channel, is higher than that on the left (north) side. On the south side of the slender sand bar, the Harai River flows into the sea, and sand supplied to the tidal flat from the river mouth is considered to be transported and deposited on the surface of the tidal flat owing to the shoreward sand transport due to waves. In this case, sand supply to the area north of the slender sand bar is obstructed by the sand bar itself, and this is assumed to be the cause of the difference in ground elevation on both sides of the sand bar.

Figure 7 shows the tip of a branch separated from the offshore part of the main slender sand bar, as shown in Fig. 3. On this sand bar, decomposed granite, which was considered to be transported offshore by flood currents of the river, has been deposited. The elevation of the sand bar gradually increases shoreward, then at the landward end suddenly drops to the tidal flat with a steep slope approximately equal to the angle of repose of the sand, implying the occurrence of shoreward sand transport.

Figure 8 shows the coastal conditions, looking landward from the tip of the slender sand bar. Although sand was deposited with a foreshore slope of 1/10 along the north side of the slender sand bar, the foreshore and flat tidal flat were clearly separated along the abrupt

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 349

considered to be due to the typical sand movement observed on only a very shallow tidal

**Figure 7.** Tip of branch separated from offshore part of main slender sand bar (December 27, 2009).

**Figure 8.** Coastal condition, while looking landward from tip of slender sand bar (December 27, 2009).

<sup>A</sup> <sup>B</sup>

**Figure 9.** Narrow neck of slender sand bar connected to land (December 27, 2009).

flat.

**Figure 5.** Beach separated by vertical seawall (December 27, 2009).

**Figure 6.** View of entire slender sand bar (December 27, 2009).

change in the slope passing through point B. By a sieve analysis of the beach material sampled at point A, the median diameter of the beach material was determined to be *d*50 = 1.50 mm at point A. Figure 9 shows the narrow neck of the slender sand bar connected to the land. Several lines showing high tide marks extend in the cross-shore direction on the surface of the sand bar, implying that the sand bar stably exists during tidal changes. Figure 10 shows the beach condition near the point connecting the land and the slender sand bar. The triangular high tide lines show that the contour lines are parallel to these high tide lines. Therefore, if waves are incident from the direction normal to the coastline, large shoreward sand transport may occur along these contour lines because of the large incident wave angle. However, the slender sand bar is stable without rapid beach changes. Taking into account the continuity condition of sand and the fact that the slender sand bar is stable, the materials forming the sand bar are considered to have been carried from the Harai River during floods. The observation results indicate that the sand transported offshore by flood currents returns to the beach owing to shoreward sand transport due to waves. On an exposed beach, the formation of a stable sand bar extending normal to the coastline, as observed on this coast, is difficult, and such a sand bar rapidly deforms under wave action. Taking this into account, the formation of a slender sand bar observed in this study is considered to be due to the typical sand movement observed on only a very shallow tidal flat.

348 Numerical Simulation – From Theory to Industry

**Figure 5.** Beach separated by vertical seawall (December 27, 2009).

**Figure 6.** View of entire slender sand bar (December 27, 2009).

change in the slope passing through point B. By a sieve analysis of the beach material sampled at point A, the median diameter of the beach material was determined to be *d*50 = 1.50 mm at point A. Figure 9 shows the narrow neck of the slender sand bar connected to the land. Several lines showing high tide marks extend in the cross-shore direction on the surface of the sand bar, implying that the sand bar stably exists during tidal changes. Figure 10 shows the beach condition near the point connecting the land and the slender sand bar. The triangular high tide lines show that the contour lines are parallel to these high tide lines. Therefore, if waves are incident from the direction normal to the coastline, large shoreward sand transport may occur along these contour lines because of the large incident wave angle. However, the slender sand bar is stable without rapid beach changes. Taking into account the continuity condition of sand and the fact that the slender sand bar is stable, the materials forming the sand bar are considered to have been carried from the Harai River during floods. The observation results indicate that the sand transported offshore by flood currents returns to the beach owing to shoreward sand transport due to waves. On an exposed beach, the formation of a stable sand bar extending normal to the coastline, as observed on this coast, is difficult, and such a sand bar rapidly deforms under wave action. Taking this into account, the formation of a slender sand bar observed in this study is

**Figure 7.** Tip of branch separated from offshore part of main slender sand bar (December 27, 2009).

**Figure 8.** Coastal condition, while looking landward from tip of slender sand bar (December 27, 2009).

**Figure 9.** Narrow neck of slender sand bar connected to land (December 27, 2009).

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 351

*<sup>I</sup>*=20relative to normal to initial shoreline

Wave conditions Incident waves: *HI*= 4.6 m (4.6 cm), *T* = 12.7 s (1.27 s), wave

Coefficient of longshore sand transport *Ks*=0.045

Coefficient of Ozasa & Brampton (1980) term *K*2 = 1.62*Ks* Coefficient of cross-shore sand transport *Kn* = 0.1*Ks*

Shoreward and landward ends: *qx* = 0,left and right boundaries: *qy* = 0

•Term of wave dissipation due to wave breaking: Dally et al. (1984)

•Total number of frequency components *NF* = 1 and number of

= 8

•Directional spreading parameter *S*max = 75 •Coefficient of wave breaking *K* = 0.17 and

Remarks Numbers in parentheses show experimental values. Space and time

equilibrium slope, because the seabed had an abrupt change in the slope along this marginal line along which sand was deposited, whereas the intervals between the contours became large in the offshore zone shallower than *hc*. The sand spit further extended along the marginal line with increasing time, and the length of the spit reached 3.5 m after 1 hr, as shown in Fig. 11(c). After 2 hr, the tip of the spit was connected to the left boundary and a barrier island had formed, enclosing a lagoon behind the barrier island (Fig. 11(d)). Although a slender, straight sand spit extended along the marginal line until 2 hr after the start of wave generation, sand started to be deposited upcoast of the left boundary after 4 hr

•Wave spectrum of incident waves: directional wave spectrum density

•Imaginary depth between minimum depth *h*0 and berm height *hR* : *h*0=

•Lower limit of *h* in terms of wave decay due to breaking: 0.7 m (0.7

scales in the calculation are 100- and 10-fold those in the experiment,

= 0.3

direction

Depth of closure *hc* = 2.5*H* (*H*: wave height)

tan*<sup>g</sup>*=1/2

*x* = 

model

2 m (2 cm)

respectively.

cm)

*<sup>c</sup>*=1/5

Berm height *hR*= 5 m (5 cm)

Equilibrium slope tan

Angle of repose

Coefficients of sand transport

Time intervals

Duration of calculation

Boundary conditions

Calculation of wave field

Mesh size

slope

*y* = 20 m

*t* = 0.001 hr (0.01 hr)

80 hr (8×104 steps) (8 hr)

obtained by Goda (1985)

directional subdivisions *N*

•Wave energy = 0 where *Z* ≥ *hR*

**Table 1.** Calculation conditions (numbers in parentheses: experimental conditions).

Energy balance equation (Mase, 2001)

**Figure 10.** Beach condition near point connecting land and slender sand bar (December 27, 2009).
