*5.2.2. Changes in wave field*

358 Numerical Simulation – From Theory to Industry

**Figure 15.** Changes in longitudinal profiles (Case 1: flat shallow seabed).

Figures 18(a)-18(c) show the distributions of the calculated wave height after wave generation for 0, 1 and 8 hr. At the initial stage, the longshore change in the wave height is large near the location with an abrupt change in the coastline orientation. Although a semicircular cuspate foreland developed until 1 hr, a marked decay in the wave height occurred over the short distance along the protruding shoreline, because the wave height was extremely low behind the protruded shoreline. The change in longshore sand transport due to this decay in wave height was successfully taken into account by the additional term given by Ozasa & Brampton (1980). The spatial change in longshore sand transport is large in this area because of the abrupt decrease in wave height, meaning that sand was rapidly deposited. However, after 8 hr, the area where the wave height abruptly decreased had disappeared and the wave height was smoothly distributed alongshore.

## *5.2.3. Sand transport flux*

Figure 19 shows the sand transport flux after wave generation for 0, 1 and 8 hr. Although the sand transport flux was initially large to the right of *X* = 8 m, which is the sand supply area, as in Case 1, after 1 hr the area with a large sand transport flux had moved leftward with the elongation of the sand spit. Moreover, the sand transport flux was reduced because the angle between the direction normal to the contour lines and the direction of incident waves decreased near the right boundary. After 8 hr, the sand spit had reached the left boundary, and the absolute value of sand transport flux decreased because the angle between the direction normal to the contour lines and the wave incidence direction had been reduced near both boundaries.

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 361

**Figure 17.** Bird's-eye view of topographic changes (Case 2: steep slope and deep seabed).


**Figure 16.** Predicted results for development of cuspate foreland on a coast with abrupt change in coastline orientation (Case 2: steep slope and deep seabed).

been reduced near both boundaries.

between the direction normal to the contour lines and the wave incidence direction had

**Figure 16.** Predicted results for development of cuspate foreland on a coast with abrupt change in

coastline orientation (Case 2: steep slope and deep seabed).

**Figure 17.** Bird's-eye view of topographic changes (Case 2: steep slope and deep seabed).

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 363

**Figure 19.** Sand transport flux (Case 2: steep slope and deep seabed).

**Figure 18.** Changes in wave height (Case 2: steep slope and deep seabed).

**Figure 19.** Sand transport flux (Case 2: steep slope and deep seabed).

**Figure 18.** Changes in wave height (Case 2: steep slope and deep seabed).

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 365

*y* = 5 m intervals in the cross-shore and longshore

**6. Simulation of formation of slender sand bar on Kutsuo coast** 

*x* = 

calculation, the wave energy was set to 0.

calculation domain was divided by

the shoreline of a tombolo connected to the slender island.

using time intervals of

**6.1. Bathymetric changes**

To model the accumulation of sand on the tidal flat due to wave action and the formation of a slender sand bar, a point source in a single mesh (5 m×5 m) was assumed. The intensity of the point source was determined by trial and error, and it was set to 3.75×104 m3/yr at the point (*x*, *y*) = (200 m, 0 m). In addition, an upper limit of 0.5 m was assumed as the elevation of the sandy island. When the elevation of the island reached this height during the

Given the simplified initial topography and the conditions of the annual energy-mean waves of the Kutsuo coast (significant wave height *Hi* of 0.4 m and wave period *T* of 4 s), and assuming wave incidence from the direction normal to the coastline, the beach changes were predicted. The depth of closure was given by *hc* = 2.5*H* (*H*: wave height at a local point). The water depth of the initial bottom of the tidal flat was assumed to be 2 m. This initial flat bottom was assumed to be a solid bed, and a sandy beach with a slope of 1/10 was set at the landward end of the flat bottom. The berm height was assumed to be 0.5 m, and angles of the equilibrium slope and repose slope were set as 1/10 and 1/2, respectively. The

directions, respectively, and a calculation for up to 5000 hr (5×104 steps) was carried out

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 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

*t* = 0.1 hr. Table 2 shows the calculation conditions.

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

#### *5.2.4. Changes in longitudinal profiles*

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