**6.1. Bathymetric changes**

364 Numerical Simulation – From Theory to Industry

**Figure 20.** Changes in longitudinal profiles (Case 2: steep slope and deep seabed).

also in good agreement along transect *X* = 10 m, as shown in Fig. 20(c).

Figures 20(a)-20(c) show the experimental and predicted changes in longitudinal profiles along transect *X* = 0 m located at the right boundary, transect *X* = 9 m near the location where the coastline orientation abruptly changes, and transect *X* = 10 m, respectively. Along transect *X* = 0 m, the parallel recession of the cross section is accurately reproduced in the calculation, as shown in Fig. 20(a). Along transect *X* = 9 m, a slope that slightly inclined landward had formed in the experiment after 8 hr, whereas a flat surface was predicted in the calculation. With the exception of these points, the experimental and predicted results are in good agreement, as shown in Fig. 20(b). The experimental and calculated results are

*5.2.4. Changes in longitudinal profiles*

Figure 21 shows the calculation results obtained after every 104 steps. Initially, only a flat bottom extended offshore of the sandy beach with a straight shoreline. The solid circle in Fig. 21(a) shows the location of the sand source, and a sandy beach with a slope of 1/10 extended along the marginal line between the tidal flat and the land, as observed on the Kutsuo coast. Owing to the wave action under these conditions, a slender submerged sand bar started to form after 104 steps, as shown in Fig. 21(b), as a result of the deposition of sand supplied from the sand source. The landward end of the slender sand bar was sharp and similar to a comet tail formed on the lee of an island. On the other side of the slender sand bar, longshore sand transport toward the lee of the slender sand bar was induced from the nearby coast, resulting in the formation of a cuspate foreland because of the wave-sheltering effect of the sand bar. After 2 × 104 steps, the submerged sand bar had become a sandy island because of its continuous development (Fig. 21(c)). After 2 × 104 steps, the cuspate foreland behind the sandy island was more developed than that after 104 steps. The beach width in the zone between *x* = 50 and 100 m was very small and a neck was formed. After 3 × 104 steps, the widths of the sandy island and the neck behind the island had increased, and sand that had originally been supplied from the offshore point source had reached the beach, resulting in the connection of the sandy island to the beach (Fig. 21(d)). Finally, the island developed a spoonlike shape with the shoreline of a tombolo connected to the slender island.

The development of the sandy island continued up to 5×104 steps, and the widths of the sandy island and the neck between the island and the tombolo continued to increase (Figs. 21(e) and 21(f)). The numerical results for the development of a slender sand bar and the

resultant sandy island successfully explain the results observed on the Kutsuo coast, as shown in Fig. 3, and the fact that the slender sand bar has a neck near the landward end, as shown in Figs. 8-10.

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 367

**Figure 21.** Calculation results observed after every 104 steps.

Figures 22(a)-22(f) show bird's-eye view of the development of a slender sand bar developed on a flat seabed. Sand supplied from a sand source at (*x*, *y*) = (200 m, 0 m) was deposited to form an island with the gradual shoreward movement of sand. After 2×104 steps, a slender island connected to the land extended with the formation of a tombolo because of the wave-sheltering effect of the island itself. Because of the continuous supply of sand, the width of the island increased and the scale of the tombolo increased with increasing number of steps. The final configuration of the slender bar was very similar to that observed on the Kutsuo coast, as shown in Fig. 9.


**Table 2.** Calculation conditions.

**Figure 21.** Calculation results observed after every 104 steps.

that observed on the Kutsuo coast, as shown in Fig. 9.

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

*x* = *y* = 5 m

0

Duration of calculation 5000 hr (5 × 104 steps)

*t* = 0.1 hr

(1984) model

*hR*:*h*0= 0.5 m

Remark Point source in single mesh (3.75×104 m3/yr)

c= 1/10

g= 1/2

Tide condition H.W.L. = +2.0 m Berm height *hR* = 0.5 m

Equilibrium slope tan

Angle of repose slope tan

Coefficients of sand

Calculation of wave

**Table 2.** Calculation conditions.

transport

Mesh size

field

Time intervals

Wave conditions Incident waves: *HI* = 0.4 m, *T* = 3 s, wave direction

initial shoreline

shown in Figs. 8-10.

resultant sandy island successfully explain the results observed on the Kutsuo coast, as shown in Fig. 3, and the fact that the slender sand bar has a neck near the landward end, as

Figures 22(a)-22(f) show bird's-eye view of the development of a slender sand bar developed on a flat seabed. Sand supplied from a sand source at (*x*, *y*) = (200 m, 0 m) was deposited to form an island with the gradual shoreward movement of sand. After 2×104 steps, a slender island connected to the land extended with the formation of a tombolo because of the wave-sheltering effect of the island itself. Because of the continuous supply of sand, the width of the island increased and the scale of the tombolo increased with increasing number of steps. The final configuration of the slender bar was very similar to

Coefficient of longshore sand transport *Ks*= 0.05

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

Energy balance equation (Mase, 2001)

density obtained by Goda (1985)

•Wave energy = 0 where *Z* ≥ *hR*

directional subdivisions *N*

Coefficient of Ozasa & Brampton (1980) term *K*2= 1.62 *Ks* Coefficient of cross-shore sand transport *Kn*= 0.2 *Ks*

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

•Wave spectrum of incident waves: directional wave spectrum

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

•Imaginary depth between minimum depth *h*0 and berm height

•Lower limit of *h* in term of wave decay due to breaking: 0.2 m

= 0.3

= 8

•Directional spreading parameter *S*max = 10 •Coefficient of wave breaking *K* = 0.17 and

*<sup>I</sup>* = 0, normal to

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 369

Figure 23 shows the changes in the longitudinal profile along transect *y* = 0 m, which passes through the center of the slender sandy island, as shown in Fig. 21(f). Because the development of the sandy island along this transect was very rapid, the results obtained after not only 1 ×104 steps but also 5×103 and 1.5×104 steps are also shown. Although the sand bar gradually extended landward, the wave height along the side slopes of the sand bar (or the resultant sandy island) was reduced owing to the wave-sheltering effect induced by the sand bar itself, resulting in a decrease in the depth of closure along the side slope. For example, a steep slope with an angle of repose of 1/2 had formed in the zone deeper than *Z* = -0.3 m after 1×104 steps. The elevation of the beach connecting the land and the sandy island

Figure 24 shows the changes in the cross section along transect *x* = 100 m, which passes through the neck of the slender sandy island, and transect *x* = 50 m near the boundary between the sandy beach and the tidal flat under the initial conditions, as shown in Fig. 21(f). After 104 steps, an island with a sharp top and a side slope with a steep angle of repose on both sides of the island had formed along transect *x* = 100 m. The increase in the width of the sandy beach over time was slow. In contrast, because of the increase in the wavesheltering effect owing to the development of the sandy island, the cuspate foreland was more developed along transect *x* = 50 m than along transect *x* = 100 m, and its width

**Figure 23.** Changes in longitudinal profile along transect y = 0 m passing through center of slender

**6.2. Changes in longitudinal and transverse profiles**

increased to reach the berm height of *hR* = 0.5 m.

increased over time.

sandy island.

**Figure 22.** Bird's-eye view of topographic changes.

## **6.2. Changes in longitudinal and transverse profiles**

368 Numerical Simulation – From Theory to Industry

**Figure 22.** Bird's-eye view of topographic changes.

Figure 23 shows the changes in the longitudinal profile along transect *y* = 0 m, which passes through the center of the slender sandy island, as shown in Fig. 21(f). Because the development of the sandy island along this transect was very rapid, the results obtained after not only 1 ×104 steps but also 5×103 and 1.5×104 steps are also shown. Although the sand bar gradually extended landward, the wave height along the side slopes of the sand bar (or the resultant sandy island) was reduced owing to the wave-sheltering effect induced by the sand bar itself, resulting in a decrease in the depth of closure along the side slope. For example, a steep slope with an angle of repose of 1/2 had formed in the zone deeper than *Z* = -0.3 m after 1×104 steps. The elevation of the beach connecting the land and the sandy island increased to reach the berm height of *hR* = 0.5 m.

Figure 24 shows the changes in the cross section along transect *x* = 100 m, which passes through the neck of the slender sandy island, and transect *x* = 50 m near the boundary between the sandy beach and the tidal flat under the initial conditions, as shown in Fig. 21(f). After 104 steps, an island with a sharp top and a side slope with a steep angle of repose on both sides of the island had formed along transect *x* = 100 m. The increase in the width of the sandy beach over time was slow. In contrast, because of the increase in the wavesheltering effect owing to the development of the sandy island, the cuspate foreland was more developed along transect *x* = 50 m than along transect *x* = 100 m, and its width increased over time.

**Figure 23.** Changes in longitudinal profile along transect y = 0 m passing through center of slender sandy island.

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 371

**Figure 25.** Wave fields corresponding to beach changes.

**Figure 24.** Changes in cross section along transect *x* = 100 m passing through neck of slender sandy island and transect *x* = 50 m near boundary between sandy beach and tidal flat.

#### **6.3. Wave height and direction**

The wave fields corresponding to the beach changes are shown in Fig. 25. The wave height on the tidal flat was initially uniform (0 step) and uniform wave decay due to wave breaking occurred on the sandy beach extending between the land and the tidal flat. After 1×104 steps, a submerged sand bar had been formed by the accumulation of sand supplied from a point source. This sand bar induced a change in the wave field; the wave height was reduced along the side slope of the slender sand bar, and oblique wave incidence occurred on both sides of the sand bar owing to wave diffraction by the sand bar itself. This oblique

#### BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 371

**Figure 25.** Wave fields corresponding to beach changes.

370 Numerical Simulation – From Theory to Industry

**Figure 24.** Changes in cross section along transect *x* = 100 m passing through neck of slender sandy

The wave fields corresponding to the beach changes are shown in Fig. 25. The wave height on the tidal flat was initially uniform (0 step) and uniform wave decay due to wave breaking occurred on the sandy beach extending between the land and the tidal flat. After 1×104 steps, a submerged sand bar had been formed by the accumulation of sand supplied from a point source. This sand bar induced a change in the wave field; the wave height was reduced along the side slope of the slender sand bar, and oblique wave incidence occurred on both sides of the sand bar owing to wave diffraction by the sand bar itself. This oblique

island and transect *x* = 50 m near boundary between sandy beach and tidal flat.

**6.3. Wave height and direction**

wave incidence caused shoreward sand transport flux along the contours of the sand bar. On the other hand, on the lee of the submerged sand bar, the wave height was reduced, similarly to in the case of a detached breakwater, inducing longshore sand transport toward the lee of the sand bar from outside the wave-shelter zone. After 2×104 steps, as a result of the landward extension of the slender sand bar and the decrease in water depth, wave breaking occurred there, causing a marked reduction in wave height. At the same time, the oblique wave incidence continued in this area and landward sand transport from the island to the land continuously occurred. After 4×104 and 5×104 steps, the same beach changes as those until 2×104 steps continued.

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 373

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