*5.1.1. Bathymetric changes*

Figures 11(a)-11(f) show the results for the predicted development of a sand spit on a shallow flat seabed given the same conditions as those in the experiment. A slender sand spit with a length of approximately 2 m was formed until 0.5 hr because of the deposition of sand supplied from upcoast along the marginal line between the flat shallow seabed and the steep offshore bottom, as shown in Fig. 11(b). Rapid shoreward sand transport also occurred owing to the restoration effect of the beach slope corresponding to the deviation from the


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

At locations with elevations higher than the berm height, the wave energy was set to 0. For convenience, the space scale in the calculation was set to 100-fold that in the experiment, and then the calculated results were reduced 100-fold. Although the movable-bed experiment was carried out under regular wave conditions, the wave field was calculated using the energy balance equation for irregular waves while regarding regular waves in the experiment as irregular waves, because repeated calculations were necessary owing to the

Given the same initial topography and wave conditions as those in the experiment (regular waves with *H*0' = 4.6 cm and *T* = 1.27 s obliquely incident to the model beach with an angle of 20), the beach changes after 8 hr were predicted. The depth of closure was given by *hc* = 2.5*H* (*H*: wave height at a local point). The berm height was assumed to be 5 cm, and the angles of the equilibrium slope and repose slope were set as 1/5 and 1/2, respectively, on the

intervals in the cross-shore and longshore directions, respectively, and the calculation for up

Figures 11(a)-11(f) show the results for the predicted development of a sand spit on a shallow flat seabed given the same conditions as those in the experiment. A slender sand spit with a length of approximately 2 m was formed until 0.5 hr because of the deposition of sand supplied from upcoast along the marginal line between the flat shallow seabed and the steep offshore bottom, as shown in Fig. 11(b). Rapid shoreward sand transport also occurred owing to the restoration effect of the beach slope corresponding to the deviation from the

*x* = 

*t* = 1×10-3 hr. Table 1 shows the

*y* = 20 cm

**5. Application to movable-bed experiment** 

**5.1. Formation of barrier island on flat shallow seabed** 

to 8 hr (8×104 steps) was carried out using time intervals of

basis of the experimental results. The calculation domain was divided by

bathymetric changes in this calculation.

calculation conditions.

*5.1.1. Bathymetric changes* 

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

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

because of the blockage of longshore sand transport by the left boundary. Offshore sand transport in the deep zone also occurred, as shown in Fig. 11(e). During this process, the shoreline advanced and the width of the barrier island formed by the extension of the sand spit gradually increased from the left boundary. After 8 hr, the effect of the blockage of longshore sand transport by the left boundary had reached the upcoast and the width of the barrier island had also increased at *X* = 9 m, where the sand spit first developed, as shown in Fig. 11(f).

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 353

**Figure 11.** Results for predicted development of sand spit on a coast with abrupt change in coastline

orientation (Case 1: flat shallow seabed).

We compared the experimental results after 1 hr with the calculated results, as shown in Figs. 1(b) and 11(c), respectively. Both sets of results indicated that a sand spit extended from the location with a sudden change in the coastline orientation along the marginal line on the flat shallow seabed and were in good agreement. However, there was some discrepancy in the location of the tip of the sand spit. Similarly, both experimental and calculated results after 8 hr, shown in Figs. 1(c) and 11(f), respectively, are in good agreement in that the width of the barrier island was increased by the blockage of longshore sand transport by the left boundary and that a gentle slope was formed at a depth of approximately 8 cm owing to erosion along with the formation of a scarp near the right boundary. With regard to the experimental results for the extension of the sand spit reported by Uda & Yamamoto (1992), Watanabe et al. (2004) successfully predicted the shoreline changes related to the extension of the sand spit using a one-line model with a curvilinear coordinate system. However, in the present study, we were able to predict the development of a barrier island after the extension of the sand spit.

Figures 12(a)-12(f) show bird's-eye view of the extension of the barrier island in Case 1, looking upcoast from above the downcoast. Although a simple sand spit extended from the boundary between the sand supply and accretion zones on the flat shallow seabed, sand had already been transported shoreward, forming a subsurface sand bar along the marginal line between the steep offshore slope and flat shallow seabed, until 0.5 hr before the extension of the sand spit, implying the generation of rapid shoreward sand transport at the abrupt change in the slope. Furthermore, sand was deposited over the steep offshore slope between 4 and 8 hr after the start of wave generation and an extremely steep slope was formed near the downcoast boundary. In contrast, a wave-cut gentle slope was formed offshore of the erosion zone located upcoast.

## *5.1.2. Changes in wave field*

Significant changes in the wave field occurred on the shallow flat seabed with the extension of the barrier island, as shown in Fig. 13. Initially, the wave height was reduced by up to approximately 1.5 cm because of wave breaking along the marginal line of the shallow flat seabed. However, the extension of the sand spit owing to the shoreward sand transport was very rapid, and the wave height was markedly reduced on the shallow flat seabed after wave generation for 1 hr. After 8 hr, a calm wave zone extended in the entire area behind the barrier because of the rapid development of the barrier, and the wave height had a uniform distribution.

erosion zone located upcoast.

*5.1.2. Changes in wave field*

uniform distribution.

Fig. 11(f).

because of the blockage of longshore sand transport by the left boundary. Offshore sand transport in the deep zone also occurred, as shown in Fig. 11(e). During this process, the shoreline advanced and the width of the barrier island formed by the extension of the sand spit gradually increased from the left boundary. After 8 hr, the effect of the blockage of longshore sand transport by the left boundary had reached the upcoast and the width of the barrier island had also increased at *X* = 9 m, where the sand spit first developed, as shown in

We compared the experimental results after 1 hr with the calculated results, as shown in Figs. 1(b) and 11(c), respectively. Both sets of results indicated that a sand spit extended from the location with a sudden change in the coastline orientation along the marginal line on the flat shallow seabed and were in good agreement. However, there was some discrepancy in the location of the tip of the sand spit. Similarly, both experimental and calculated results after 8 hr, shown in Figs. 1(c) and 11(f), respectively, are in good agreement in that the width of the barrier island was increased by the blockage of longshore sand transport by the left boundary and that a gentle slope was formed at a depth of approximately 8 cm owing to erosion along with the formation of a scarp near the right boundary. With regard to the experimental results for the extension of the sand spit reported by Uda & Yamamoto (1992), Watanabe et al. (2004) successfully predicted the shoreline changes related to the extension of the sand spit using a one-line model with a curvilinear coordinate system. However, in the present study, we were able to predict the

Figures 12(a)-12(f) show bird's-eye view of the extension of the barrier island in Case 1, looking upcoast from above the downcoast. Although a simple sand spit extended from the boundary between the sand supply and accretion zones on the flat shallow seabed, sand had already been transported shoreward, forming a subsurface sand bar along the marginal line between the steep offshore slope and flat shallow seabed, until 0.5 hr before the extension of the sand spit, implying the generation of rapid shoreward sand transport at the abrupt change in the slope. Furthermore, sand was deposited over the steep offshore slope between 4 and 8 hr after the start of wave generation and an extremely steep slope was formed near the downcoast boundary. In contrast, a wave-cut gentle slope was formed offshore of the

Significant changes in the wave field occurred on the shallow flat seabed with the extension of the barrier island, as shown in Fig. 13. Initially, the wave height was reduced by up to approximately 1.5 cm because of wave breaking along the marginal line of the shallow flat seabed. However, the extension of the sand spit owing to the shoreward sand transport was very rapid, and the wave height was markedly reduced on the shallow flat seabed after wave generation for 1 hr. After 8 hr, a calm wave zone extended in the entire area behind the barrier because of the rapid development of the barrier, and the wave height had a

development of a barrier island after the extension of the sand spit.

**Figure 11.** Results for predicted development of sand spit on a coast with abrupt change in coastline orientation (Case 1: flat shallow seabed).

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 355

Figure 14 shows the sand transport flux 0, 1 and 8 hr after the start of wave generation. Although the initial sand transport flux was large in the area to the right of *X* = 8 m, where the sand source was located, the area with a large sand transport flux had moved left with the extension of the sand spit after 1 hr. In contrast, the sand transport flux decreased in magnitude near the right boundary, because the angle between the direction normal to the contour lines and the direction of incident waves was reduced. After 8 hr, the sand spit had reached the left boundary and the sand transport flux had significantly decreased, and the

Figures 15(a)-15(d) show the experimental and predicted changes in longitudinal profiles along transect *X* = 0 m located at the right boundary, and transects *X* = 9, 12 and 14 m crossing the flat shallow seabed, respectively. Along transect *X* = 0 m, although the experimental and predicted results, which indicated that the depth of closure was -12 cm, are in agreement, as shown in Fig. 15(a), the eroded volume in the calculation was overestimated in the nearshore zone, where there was less scarp erosion. However, the sand budget in the cross section was approximately maintained and the parallel recession of the cross section while maintaining a constant slope was accurately predicted in the calculation. Along transect *X* = 9 m, the development of a berm of 5 cm height after 1 hr, as shown in Fig. 15(b), was observed in both the experiment and simulation, but the location of the berm was slightly seaward in the simulation. After 8 hr, however, the berm location had moved landward and a stable barrier island had formed. These experimental and calculated changes are in good agreement. Furthermore, no beach changes occurred on the flat shallow seabed because the elongation of the sand spit was too rapid to permit wave intrusion into the flat seabed. Along transect *X* = 12 m, the elongation of the sand spit was limited until 1 hr, as shown in Fig. 15(c), and there was little development of the berm. However, a substantial berm had developed after 8 hr. The experimental and calculated results are in good agreement regarding these points. Along transect *X* = 14 m near the left boundary, although a sand bar did not develop until 1 hr, a large amount of sand had been deposited after 8 hr, forming a barrier island with a 1.2 m width, as shown in Fig. 15(d). In this case, the seabed slope gradually steepened because of the continuous deposition of sand along the offshore steep slope, resulting in the deposition of sand up to a depth of -23 cm, which is

*5.1.3. Sand transport flux*

area with a large sand transport flux had become small.

approximately two fold larger than the depth of closure of -12 cm.

Figures 16(a)-16(f) show the results for the calculation of the development of a cuspate foreland on a steep coast *t* = 0, 0.5, 1, 2, 4, and 8 hr after the start of wave generation given the same conditions as those in the experiment. The contour lines that extended parallel to

**5.2. Formation of cuspate foreland on steep coast** 

*5.2.1. Bathymetric changes*

*5.1.4. Changes in longitudinal profiles*

**Figure 12.** Bird's-eye view of topographic changes (Case 1: flat shallow seabed).
