**3. Formation of sand spits and cuspate forelands with rhythmic shapes**

### **3.1. Calculation conditions**

Here, *fD* is the energy dissipation rate, *E* is the wave energy, *K* is a coefficient expressing the intensity of wave dissipation due to breaking, *h* is the water depth, *Γ* is the ratio of the critical wave height to the water depth on a flat bottom, and *γ* is the ratio of wave height to the water depth *H*/*h*. In addition, a lower limit was set for the water depth *h* in Eq. (3) similarly in [9].

In this method, the energy dissipation rate obtained from the calculation of the plane wave field including the effect of wave dissipation due to breaking was used for the calculation of sand transport. The same approach was employed in [15]. In the calculation of the wave field in the wave run-up zone, an imaginary depth was assumed as in [9]. Furthermore, the wave

In the numerical simulation of beach changes, the sand transport and continuity equations

the staggered mesh scheme. In estimating the intensity of sand transport near the berm top and at the depth of closure, the intensity of sand transport was linearly reduced to 0 near the berm height or the depth of closure to prevent sand from being deposited in the zone higher than the berm height and the beach from being eroded in the zone deeper than the depth of

Wave conditions Incident waves: *Hi* = 1 m, *T* = 4 s, wave direction θ*I* = 60° relative to direction normal to

Coefficient of Ozasa and Brampton term [11] *K2* = 1.62*Ks* Coefficient of cross-shore sand transport *Kn* = *Ks*

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

•Directional spreading parameter *Smax =*25 •Coefficient of wave breaking *K* = 0.17 and Γ = 0.3

•Wave energy = 0 where Z≥ h/R

•Term of wave dissipation due to wave breaking: Dally et al. model [14] •Wave spectrum of incident waves: directional wave spectrum density in [17] •Total number of frequency components *NF* = 1 and number of directional

•Imaginary depth between depth *h0* (0.5 m) and berm height *hR*

•Lower limit of *h* in terms of wave decay due to wave breaking: 0.5 m

<sup>→</sup> =0) were solved on the *x*-*y* plane by the explicit finite-difference method using

energy at locations with elevations higher than the berm height was set to 0.

initial shoreline

Coefficients of sand transport Coefficient of longshore sand transport *Ks* = 0.2

subdivisions *N*θ = 8

(∂*Z* / ∂*t* + ∇ •*q*

closure, similar to that in [16].

422 Computational and Numerical Simulations

Berm height *hR* = 1 m

Equilibrium slope tanβ*c* = 1/20 Angle of repose slope tanβ*g* = 1/2

Mesh size Δ*x* = Δ*y* = 20 m Time intervals Δ*t* = 0.5 h

**Table 1.** Calculation conditions.

Depth of closure *hc* = 4 m (still water depth)

Duration of calculation 2.75×104 h (5.5×104 steps)

Calculation of wave field Energy balance equation [13]

Ashton and Murray [3] showed that the generation of shoreline instability closely depends on the probability of occurrence of wave directions; sand spits develop in case that the probability of occurrence of a unidirectional waves is high, cuspate bumps develop in case that the probability of occurrence of waves incident from two directions is equivalent, and sand spits with hooked shoreline develop in case that waves are incident from two directions with different probabilities. The calculation conditions in this study were determined referring their results.

For the wave conditions, we assumed *Hi* = 1 m and *T* = 4 s, considering the formation of sand spits in a shallow lagoon. The wave direction was assumed to be obliquely incident from 60°, 50° and 40° counterclockwise or from the directions of ±60° with probabilities of 0.5:0.5 and 0.60:0.40, 0.65:0.35, 0.70:0.30, 0.75:0.25 and 0.80:0.20, while determining the direction from the probability distribution at each step. We considered a shallow lake with a flat solid bed, the depth of which was given by *Z* = -4 m, and a uniform beach with a slope of 1/20 and a berm height of *hR* = 1 m were considered on the landward end. At the initial stage, a small random perturbation with an amplitude of Δ*Z* = 0.5 m was applied to the slope. The calculation domain was a rectangle of 4 km length and 1.2 km width, and a periodic boundary condition was set at both ends. In addition, the depth of closure was assumed to be *h*c = 4 m. The equilibrium and repose slopes were 1/20 and 1/2, respectively. The coefficients of longshore and cross-shore sand transport were set to *Ks* = *Kn* = 0.2, respectively. The calculation domain was divided with a mesh size of Δ*x* = Δ*y* = 20 m, and Δ*t* was selected to be 0.5 h. The total number of calculation steps considered was 5.5×104 (2.75×104 h). The calculation of the wave field was carried out every 10 steps in the calculation of beach changes. Table 1 shows the calculation conditions.

### **3.2. Calculation results**

### *3.2.1. Oblique wave incidence from 60° counterclockwise*

Figure 2 shows the results of the calculations at eight stages starting from the initial straight shoreline with a slope of 1/20, to which a small random perturbation with an amplitude of Δ*Z* = 0.5 m was applied, up to 5.5×104 steps. The small perturbation applied to the slope at the initial stage developed into eleven cuspate forelands within 5×103 steps, and the shoreline projection increased with time while moving rightward owing to the wave incidence from the counterclockwise direction. Because of the periodic boundary conditions at both ends, the cuspate forelands that moved away through the right boundary reentered the calculation domain through the left boundary. After 1×104 steps, the shoreline protrusion had increased and had developed as slender sand spits. After 2×104 steps, the small-scale sand spits located adjacent to each other had merged into large-scale sand spits and disappeared, and finally six sand spits were formed.

Two reasons for these changes are considered [1, 3]. (1) Of the two sand spits of different scales, the small sand spit moves faster than the large sand spit in the absence of the wave-sheltering effect, and then the small sand spit catches up and merges with the large sand spit. (2) On the lee of sand spits with an elongated neck, a wave-shelter zone is formed and the velocity of sand spits is reduced in this zone because of wave calmness, resulting in the stoppage of the movement of the sand spits and in the merging of small sand spits with larger spits.

After 3×104 steps, the small sand spits located in the wave-shelter zone of the large-scale sand spits had stopped moving and merged into the large-scale sand spits, resulting in an increase in the interval between the sand spits and a decrease in the number of sand spits per finite length of the shoreline. After 4×104 steps, the number of sand spits had decreased to 2 and the tip of the sand spits approached closely to the original shoreline, permitting the downcoast

deposition zone immediately offshore of the shoreline was left intact at the base of the sand spit located at *y* = 2250 m but almost all parts had merged with the sand spits. Although two large-scale sand spits were formed from the straight shoreline within 5.5×104 steps, the offshore contour of -4 m depth obliquely extended and a gentle seabed slope was formed upcoast of the sand spit, whereas a very steep slope was formed at the tip of the sand spits. These features are in good agreement with those measured around sand spits in lake and bay [18]. At the downcoast base of the sand spit extending from *y* = 2400 m, part of the sand bar formed in the

recovered at the downcoast side of the sand spit on the basis of the present topography.

steps, because of the movement of sand spits sweeping rightward, part of the sand

steps was left intact, implying that historical changes could be

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passage of the sand of the sand spits.

previous process from 4×104

**Figure 3.** Change in wave field with development of sand spits.

After 5×104

Furthermore, the sand spits developed and protruded because (1) their tip is semicircular, meaning that the angle between the direction normal to the shoreline and the wave incident direction exceeds 45° at a point along the shoreline and the shoreline protrusion occurs at such a point owing to high-angle wave instability. (2) In a wave-shelter zone, sand transport is significantly reduced, whereas it is enhanced near the tip of the sand spits, and thus the derivative of the sand transport rate takes a maximum value near the boundary between the tip of the sand spits and the wave-shelter zone, inducing the protrusion of sand spits.

**Figure 2.** Development of sand spits from infinitesimal perturbation under wave conditions obliquely incident from 60° counterclockwise.

After 3×104 steps, the small sand spits located in the wave-shelter zone of the large-scale sand spits had stopped moving and merged into the large-scale sand spits, resulting in an increase in the interval between the sand spits and a decrease in the number of sand spits per finite length of the shoreline. After 4×104 steps, the number of sand spits had decreased to 2 and the tip of the sand spits approached closely to the original shoreline, permitting the downcoast passage of the sand of the sand spits.

Two reasons for these changes are considered [1, 3]. (1) Of the two sand spits of different scales, the small sand spit moves faster than the large sand spit in the absence of the wave-sheltering effect, and then the small sand spit catches up and merges with the large sand spit. (2) On the lee of sand spits with an elongated neck, a wave-shelter zone is formed and the velocity of sand spits is reduced in this zone because of wave calmness, resulting in the stoppage of the

Furthermore, the sand spits developed and protruded because (1) their tip is semicircular, meaning that the angle between the direction normal to the shoreline and the wave incident direction exceeds 45° at a point along the shoreline and the shoreline protrusion occurs at such a point owing to high-angle wave instability. (2) In a wave-shelter zone, sand transport is significantly reduced, whereas it is enhanced near the tip of the sand spits, and thus the derivative of the sand transport rate takes a maximum value near the boundary between the

movement of the sand spits and in the merging of small sand spits with larger spits.

424 Computational and Numerical Simulations

tip of the sand spits and the wave-shelter zone, inducing the protrusion of sand spits.

**Figure 2.** Development of sand spits from infinitesimal perturbation under wave conditions obliquely incident from

60° counterclockwise.

After 5×104 steps, because of the movement of sand spits sweeping rightward, part of the sand deposition zone immediately offshore of the shoreline was left intact at the base of the sand spit located at *y* = 2250 m but almost all parts had merged with the sand spits. Although two large-scale sand spits were formed from the straight shoreline within 5.5×104 steps, the offshore contour of -4 m depth obliquely extended and a gentle seabed slope was formed upcoast of the sand spit, whereas a very steep slope was formed at the tip of the sand spits. These features are in good agreement with those measured around sand spits in lake and bay [18]. At the downcoast base of the sand spit extending from *y* = 2400 m, part of the sand bar formed in the previous process from 4×104 steps was left intact, implying that historical changes could be recovered at the downcoast side of the sand spit on the basis of the present topography.

**Figure 3.** Change in wave field with development of sand spits.

Figure 3 shows the change in the wave field around the sand spits at each stage from the initial straight shoreline to the fully developed sand spits, as shown in Fig. 2. At the initial stage, waves are obliquely incident to the straight shoreline with uniform exposure to waves at all locations. With the development of the shoreline undulation over time, the wave-shelter zones were formed behind them. After 2×104 steps, the formation of sand spits was clear and the wave-shelter zone expanded downcoast, and the toe of the adjacent sand spit was included inside the wave-shelter zone. As a result, a marked reduction in wave height occurred, which in turn caused a reduction in sand transport. After 5.5×104 steps, long sand spits extended so that the toe of the slender sand spit was subject to the wave-sheltering effect of the upcoast sand spit. Owing to the development of sand spits over time, the entire original shoreline zone was included in the wave-shelter zone produced by sand spits, which is very similar to the calm wave zone protected by the extension of long port breakwaters.

of 60° except the wave incidence angle. Figure 5 shows the calculation results with an angle of 50°. The development of sand spits via the development of cuspate forelands owing to the instability mechanism was possible, but the scale of the sand spits was significantly reduced. However, no sand spits had developed at an angle of 40° and the shoreline undulations were smoothed out with time, as shown in Fig. 6, resulting that the shoreline undulations did not develop unless the wave incidence to the zone shallower than *hc* exceeds 45°. This result agrees with the conclusion in [19] that the shoreline instability develops only if the bathymetric changes related to shoreline perturbations extend to a depth where the wave angle is greater than the critical angle of 42° and the potential for coastline instability is therefore limited by

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the wave incidence angle at the depth of closure and not the angle at deep water.

**Figure 5.** Formation of sand spits under the condition of oblique wave incidence from 50°.

Figure 4 shows the sand transport flux after 5.5×104 steps. The longshore sand transport mainly develops along the outer margin of the sand spit with a maximum value at the tip of sand spit and then rapidly decreases. Examining the sand transport flux near the neck of the sand spit in Fig. 4, cross-shore sand transport flux from the exposed side to the lee of the sand spit is also observed at a location of *y* = 2750 m. Thus, the neck of the sand spits is gradually eroded and moves downcoast because of this cross-shore sand transport, caused by the small differ‐ ence between the given berm height *hR* and the actual crown height of the sandy beach comprising the neck. Sand can be directly transported from the exposed side to the lee side without the sand transport turning around the tip of the sand spits. This effect makes the movement of an entire sand spit possible.

**Figure 4.** Sand transport flux after 5.5×104 steps.

### *3.2.2. Oblique wave incidence from 50° and 40° counterclockwise*

In order to investigate the effect of the change in wave incidence angle to the beach changes, the calculations were carried out under the conditions of oblique wave incidence with an angle of 50° and 40°, while maintaining the same calculation conditions as in the case with an angle of 60° except the wave incidence angle. Figure 5 shows the calculation results with an angle of 50°. The development of sand spits via the development of cuspate forelands owing to the instability mechanism was possible, but the scale of the sand spits was significantly reduced. However, no sand spits had developed at an angle of 40° and the shoreline undulations were smoothed out with time, as shown in Fig. 6, resulting that the shoreline undulations did not develop unless the wave incidence to the zone shallower than *hc* exceeds 45°. This result agrees with the conclusion in [19] that the shoreline instability develops only if the bathymetric changes related to shoreline perturbations extend to a depth where the wave angle is greater than the critical angle of 42° and the potential for coastline instability is therefore limited by the wave incidence angle at the depth of closure and not the angle at deep water.

Figure 3 shows the change in the wave field around the sand spits at each stage from the initial straight shoreline to the fully developed sand spits, as shown in Fig. 2. At the initial stage, waves are obliquely incident to the straight shoreline with uniform exposure to waves at all locations. With the development of the shoreline undulation over time, the wave-shelter zones

wave-shelter zone expanded downcoast, and the toe of the adjacent sand spit was included inside the wave-shelter zone. As a result, a marked reduction in wave height occurred, which in turn caused a reduction in sand transport. After 5.5×104 steps, long sand spits extended so that the toe of the slender sand spit was subject to the wave-sheltering effect of the upcoast sand spit. Owing to the development of sand spits over time, the entire original shoreline zone was included in the wave-shelter zone produced by sand spits, which is very similar to the

develops along the outer margin of the sand spit with a maximum value at the tip of sand spit and then rapidly decreases. Examining the sand transport flux near the neck of the sand spit in Fig. 4, cross-shore sand transport flux from the exposed side to the lee of the sand spit is also observed at a location of *y* = 2750 m. Thus, the neck of the sand spits is gradually eroded and moves downcoast because of this cross-shore sand transport, caused by the small differ‐ ence between the given berm height *hR* and the actual crown height of the sandy beach comprising the neck. Sand can be directly transported from the exposed side to the lee side without the sand transport turning around the tip of the sand spits. This effect makes the

In order to investigate the effect of the change in wave incidence angle to the beach changes, the calculations were carried out under the conditions of oblique wave incidence with an angle of 50° and 40°, while maintaining the same calculation conditions as in the case with an angle

calm wave zone protected by the extension of long port breakwaters.

Figure 4 shows the sand transport flux after 5.5×104

movement of an entire sand spit possible.

**Figure 4.** Sand transport flux after 5.5×104 steps.

*3.2.2. Oblique wave incidence from 50° and 40° counterclockwise*

steps, the formation of sand spits was clear and the

steps. The longshore sand transport mainly

were formed behind them. After 2×104

426 Computational and Numerical Simulations

**Figure 5.** Formation of sand spits under the condition of oblique wave incidence from 50°.

Figures 8(a) and 8(b) show the wave height distribution around the cuspate forelands imme‐

incidence. The wave-sheltering effect due to the protruded cuspate forelands alternately extends to the bays. The importance of this effect to the development of shoreline undulations was pointed out in [3]; when the multiple cuspate forelands with a different size have developed, the effect of the high-angle wave instability becomes stronger at the tip of the forelands with a large size than that at the bays, so that the positive feedback will occur. In contrast, around the cuspate forelands with a small size, the effect of the high-angle wave instability is weakened by the wave-sheltering effect by the large cuspate forelands, and the cuspate forelands are gradually modified to a stable form. Furthermore, when a large cuspate foreland develops, sand composed of the small cuspate foreland is absorbed into the large forelands, resulting in the decline of the small cuspate forelands and the increase in size of the large cuspate forelands. Thus, the development of large cuspate forelands will continue while small cuspate forelands are gradually disappearing. This results in the decrease in the number

**Figure 7.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.50:0.50).

steps under the conditions of clockwise and counterclockwise wave

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diately after 4×104

of the cuspate forelands.

**Figure 6.** No shoreline instability under the condition of oblique wave incidence from 40°.

### *3.2.3. Oblique wave incidence from directions of ±60° with probabilities of 0.5:0.5*

A numerical simulation was carried out for the case that waves were obliquely incident from the directions of ±60° relative to the direction normal to the shoreline with probabilities of 0.5:0.5 on the initial straight coastline, given a small random perturbation at the initial stage. Figure 7 shows the bathymetric changes between the initial stage and 4×104 steps. After 1×104 steps, triangular cuspate forelands had developed and irregularly distributed. When waves were incident from one direction, asymmetric sand bars developed, as shown in Fig. 2. In contrast, when waves with the same probability were incident from the opposite direction, the cuspate forelands became symmetric, and small-scale cuspate forelands disappeared and merged with larger cuspate forelands. Because the probability of occurrence of both wave directions was the same and there was no net longshore sand transport, the unidirectional movement of the sand body did not occur.

After 2×104 steps, the number of triangular cuspate forelands had been reduced to five, and the development of triangular cuspate forelands further continued and small-scale cuspate forelands merged into larger cuspate forelands. Finally, after 4×104 steps, four large-scale cuspate forelands had developed. A steep slope was formed by successive sand deposition at the tip of cuspate forelands, whereas seabed with a gentle slope was formed in the bay. Thus, when waves were obliquely incident from the directions of ±60° relative to the direction normal to the shoreline with probabilities of 0.5:0.5, symmetric cuspate forelands were formed.

Figures 8(a) and 8(b) show the wave height distribution around the cuspate forelands imme‐ diately after 4×104 steps under the conditions of clockwise and counterclockwise wave incidence. The wave-sheltering effect due to the protruded cuspate forelands alternately extends to the bays. The importance of this effect to the development of shoreline undulations was pointed out in [3]; when the multiple cuspate forelands with a different size have developed, the effect of the high-angle wave instability becomes stronger at the tip of the forelands with a large size than that at the bays, so that the positive feedback will occur. In contrast, around the cuspate forelands with a small size, the effect of the high-angle wave instability is weakened by the wave-sheltering effect by the large cuspate forelands, and the cuspate forelands are gradually modified to a stable form. Furthermore, when a large cuspate foreland develops, sand composed of the small cuspate foreland is absorbed into the large forelands, resulting in the decline of the small cuspate forelands and the increase in size of the large cuspate forelands. Thus, the development of large cuspate forelands will continue while small cuspate forelands are gradually disappearing. This results in the decrease in the number of the cuspate forelands.

**Figure 6.** No shoreline instability under the condition of oblique wave incidence from 40°.

1×104

After 2×104

movement of the sand body did not occur.

428 Computational and Numerical Simulations

*3.2.3. Oblique wave incidence from directions of ±60° with probabilities of 0.5:0.5*

Figure 7 shows the bathymetric changes between the initial stage and 4×104

A numerical simulation was carried out for the case that waves were obliquely incident from the directions of ±60° relative to the direction normal to the shoreline with probabilities of 0.5:0.5 on the initial straight coastline, given a small random perturbation at the initial stage.

 steps, triangular cuspate forelands had developed and irregularly distributed. When waves were incident from one direction, asymmetric sand bars developed, as shown in Fig. 2. In contrast, when waves with the same probability were incident from the opposite direction, the cuspate forelands became symmetric, and small-scale cuspate forelands disappeared and merged with larger cuspate forelands. Because the probability of occurrence of both wave directions was the same and there was no net longshore sand transport, the unidirectional

steps, the number of triangular cuspate forelands had been reduced to five, and

the development of triangular cuspate forelands further continued and small-scale cuspate forelands merged into larger cuspate forelands. Finally, after 4×104 steps, four large-scale cuspate forelands had developed. A steep slope was formed by successive sand deposition at the tip of cuspate forelands, whereas seabed with a gentle slope was formed in the bay. Thus, when waves were obliquely incident from the directions of ±60° relative to the direction normal to the shoreline with probabilities of 0.5:0.5, symmetric cuspate forelands were formed.

steps. After

**Figure 7.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.50:0.50).

extended straight, whereas the shoreline curvature increased immediately right of the tip, forming a hooked shoreline. The contour of 4 m depth extended toward the tip of the forelands while obliquely intersecting with the shoreline left of the tip of the foreland, and then it

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extended parallel to the shoreline from the tip of the foreland with a large curvature.

**Figure 9.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.60:0.40).

increased after 2×104

Figure 10 shows the calculation results with probabilities of 0.65:0.35. Because probability of occurrence of wave incident from the left increased, the steepness of the cuspate forelands

steps and a hooked shoreline inclined rightward had formed. After

**Figure 8.** Wave field around cuspate forelands under oblique wave incidence from ±60° with probabilities of 0.50:0.50.

### *3.2.4. Oblique wave incidence from directions of ±60° with different probabilities*

To investigate the effect of the change in probabilities of occurrence of the oblique wave incidence to the development of sand spits and cuspate forelands, the calculation was made, while keeping oblique wave incidence from directions of ±60° relative to the direction normal to the shoreline, and changing probabilities among 0.60:0.40, 0.65:0.35, 0.70:0.30, 0.75:0.25 and 0.80:0.20, i.e., the condition that rightward longshore sand transport gradually increases with the change in probability. In each case, the results after 2×104 , 3×104 and 4×104 steps were compared.

Figure 9 shows the calculation results with probabilities of 0.60:0.40. Although symmetric cuspate forelands have developed with probability of 0.50:0.50, as shown in Fig. 7, asymmetric cuspate forelands that slightly inclined rightward have developed with probabilities of 0.60:0.40. Because the direction of net longshore sand transport was rightward, cuspate forelands developed while moving rightward. The shoreline left of the tip of cuspate forelands extended straight, whereas the shoreline curvature increased immediately right of the tip, forming a hooked shoreline. The contour of 4 m depth extended toward the tip of the forelands while obliquely intersecting with the shoreline left of the tip of the foreland, and then it extended parallel to the shoreline from the tip of the foreland with a large curvature.

**Figure 8.** Wave field around cuspate forelands under oblique wave incidence from ±60° with probabilities of

To investigate the effect of the change in probabilities of occurrence of the oblique wave incidence to the development of sand spits and cuspate forelands, the calculation was made, while keeping oblique wave incidence from directions of ±60° relative to the direction normal to the shoreline, and changing probabilities among 0.60:0.40, 0.65:0.35, 0.70:0.30, 0.75:0.25 and 0.80:0.20, i.e., the condition that rightward longshore sand transport gradually increases with

Figure 9 shows the calculation results with probabilities of 0.60:0.40. Although symmetric cuspate forelands have developed with probability of 0.50:0.50, as shown in Fig. 7, asymmetric cuspate forelands that slightly inclined rightward have developed with probabilities of 0.60:0.40. Because the direction of net longshore sand transport was rightward, cuspate forelands developed while moving rightward. The shoreline left of the tip of cuspate forelands

, 3×104

and 4×104

steps were

*3.2.4. Oblique wave incidence from directions of ±60° with different probabilities*

the change in probability. In each case, the results after 2×104

0.50:0.50.

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compared.

**Figure 9.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.60:0.40).

Figure 10 shows the calculation results with probabilities of 0.65:0.35. Because probability of occurrence of wave incident from the left increased, the steepness of the cuspate forelands increased after 2×104 steps and a hooked shoreline inclined rightward had formed. After

Similarly, the calculation results with probabilities of 0.75:0.25 are shown in Fig. 12. A slender

elongated rightward after 3×104 steps. After 4×104 steps, sand spits with a long, slender neck and a head extended approximately parallel to the original coastline had developed. Although the contours shallower than 3 m depth extended parallel to the shoreline, while forming the main body of the sand spits, the contour of 4 m depth had an embayment downcoast of sand spits. Finally, Fig. 13 shows the calculation results with probabilities of 0.80:0.20. Because the probability of occurrence of waves from the left markedly increased, many sand spits with a head extended parallel to the original shoreline were formed close to the coastline after

sand spits extended parallel to the coastline similar to the development of longshore sand bars.

**Figure 11.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.70:0.30).

steps, and the sand spit with a narrow neck had

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steps, the length of sand spits had further increased, but the head of

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sand spit started to develop after 2×104

steps. After 3×104

2×104

**Figure 10.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.65:0.35).

3×104 steps, sand spits were formed at the tip of the cuspate forelands and a shallow bay was formed between the apexes. After 4×104 steps, sand spits obliquely extended rightward from the tip of the cuspate forelands with a larger angle than that in Fig. 2. In particular, the calculation results obtained after 4×104 steps and the case in a lagoon facing Chukchi Sea shown in Fig. 1 are in good agreement.

Figure 11 shows the calculation results with probabilities of 0.70:0.30. Although sand spits had already developed after 2×104 steps, these sand spits further elongated downcoast after 3×104 steps, and after 4×104 steps sand spits with a narrow neck at the connecting point to the land and a long head were formed. The number of the sand spits formed per unit coastline length reduced to two from four in the case with probabilities of 0.65:0.35.

Similarly, the calculation results with probabilities of 0.75:0.25 are shown in Fig. 12. A slender sand spit started to develop after 2×104 steps, and the sand spit with a narrow neck had elongated rightward after 3×104 steps. After 4×104 steps, sand spits with a long, slender neck and a head extended approximately parallel to the original coastline had developed. Although the contours shallower than 3 m depth extended parallel to the shoreline, while forming the main body of the sand spits, the contour of 4 m depth had an embayment downcoast of sand spits. Finally, Fig. 13 shows the calculation results with probabilities of 0.80:0.20. Because the probability of occurrence of waves from the left markedly increased, many sand spits with a head extended parallel to the original shoreline were formed close to the coastline after 2×104 steps. After 3×104 steps, the length of sand spits had further increased, but the head of sand spits extended parallel to the coastline similar to the development of longshore sand bars.

**Figure 11.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.70:0.30).

3×104

calculation results obtained after 4×104

in Fig. 1 are in good agreement.

432 Computational and Numerical Simulations

already developed after 2×104

 steps, sand spits were formed at the tip of the cuspate forelands and a shallow bay was formed between the apexes. After 4×104 steps, sand spits obliquely extended rightward from the tip of the cuspate forelands with a larger angle than that in Fig. 2. In particular, the

**Figure 10.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.65:0.35).

Figure 11 shows the calculation results with probabilities of 0.70:0.30. Although sand spits had

steps, and after 4×104 steps sand spits with a narrow neck at the connecting point to the land and a long head were formed. The number of the sand spits formed per unit coastline length

reduced to two from four in the case with probabilities of 0.65:0.35.

steps and the case in a lagoon facing Chukchi Sea shown

steps, these sand spits further elongated downcoast after 3×104

**Figure 12.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.75:0.25).

**Figure 13.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.80:0.20).

Although the scale of the sand spits formed along the north shore of the Azov Sea, as shown in Fig. 1, is much larger than that of the calculation results, their geometrical configurations of the calculated results are in good agreement with the measured. The sand spits A, B, C and D, as shown in Fig. 1, have been formed mainly by the waves obliquely incident from the east. The sand spit D located at the west end has a long, slender neck and this feature agrees well with the calculation results of the sand spit formed under the incidence of a unidirectional waves, as shown in Fig. 2(h). Furthermore, the width and length of the neck of the sand spit becomes thick and short in the order of C, B and A, along with the development of a hooked shoreline behind the sand spit. These conditions are very similar to the development of the sand spits under the conditions that waves were incident from two directions with different

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**3.3. Discussion**

Thus, symmetric cuspate forelands were formed when waves were incident from the directions of ±60° and the probability of occurrence of waves is equivalent. With probabilities of 0.60:0.40, the asymmetry of cuspate forelands increased and sand spits started to form after 2×104 steps with probabilities of 0.65:0.35. Increasing probabilities of occurrence of waves from the left such as 0.75:0.25, sand spits with a head extending parallel to the original shoreline developed. In all the cases of the development of sand spits, a narrow neck was formed at the connecting point to the land; a general characteristic of the topography around a sand spit [1]. Thus, the mechanism based on the high-angle wave instability and the evolution of 3-D beach changes was explained well by the BG model. Using this model, not only the shoreline configuration but also the 3-D topographic changes around the sand spits and cuspate forelands could be predicted.

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**Figure 13.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.80:0.20).

### **3.3. Discussion**

steps

**Figure 12.** Formation of cuspate forelands (oblique wave incidence from ±60° with probabilities of 0.75:0.25).

predicted.

434 Computational and Numerical Simulations

Thus, symmetric cuspate forelands were formed when waves were incident from the directions of ±60° and the probability of occurrence of waves is equivalent. With probabilities of 0.60:0.40, the asymmetry of cuspate forelands increased and sand spits started to form after 2×104

with probabilities of 0.65:0.35. Increasing probabilities of occurrence of waves from the left such as 0.75:0.25, sand spits with a head extending parallel to the original shoreline developed. In all the cases of the development of sand spits, a narrow neck was formed at the connecting point to the land; a general characteristic of the topography around a sand spit [1]. Thus, the mechanism based on the high-angle wave instability and the evolution of 3-D beach changes was explained well by the BG model. Using this model, not only the shoreline configuration but also the 3-D topographic changes around the sand spits and cuspate forelands could be

Although the scale of the sand spits formed along the north shore of the Azov Sea, as shown in Fig. 1, is much larger than that of the calculation results, their geometrical configurations of the calculated results are in good agreement with the measured. The sand spits A, B, C and D, as shown in Fig. 1, have been formed mainly by the waves obliquely incident from the east. The sand spit D located at the west end has a long, slender neck and this feature agrees well with the calculation results of the sand spit formed under the incidence of a unidirectional waves, as shown in Fig. 2(h). Furthermore, the width and length of the neck of the sand spit becomes thick and short in the order of C, B and A, along with the development of a hooked shoreline behind the sand spit. These conditions are very similar to the development of the sand spits under the conditions that waves were incident from two directions with different probabilities. The shape of the sand spit A is very similar to that of the sand spit second from the right end in Fig. 10(c) calculated with probabilities of 0.65:0.35, and that of the sand spit C is similar to that located at right end in Fig. 11(b) calculated with probabilities of 0.70:0.30. On the north shore of the Azov Sea, easterly wind is considered to be predominant, and the sand spit D located at the west end could receive sufficiently large wave energy from the east because of long fetch, whereas wave action from the west is weak because of shorter fetch. As a result, wave action from the east became stronger and sand spit with a narrow, slender neck was considered to be formed. In contrast, in the sand spit A, the fetch from the east was short so that the wave action from the east was weakened, whereas wave action from the west was strengthened because of a long fetch. In addition, the increase in the fetch of the easterly wind was considered to cause the increase in scale of the sand spit. Zenkovich qualitatively ex‐ plained these features using a schematic diagram [1], but in this study these features observed in the field were successfully explained using the BG model.

wave field. The lengths of the groin and breakwater were determined, taking both the scale of sand spits and cuspate forelands and the wave diffraction effect of the structures into account.

obliquely incident from the direction of 60° and then a groin of 800 m length and 4 m point depth was installed across the central sand spit after the sand spits have fully developed owing to the shoreline instability (Fig. 14(a)). These sand spits have developed while moving rightward, and the sand spit that moved out of the right boundary enters again from the left boundary as it is because of the periodic boundary condition at both ends. Figures 14(b)-14(j)

After 2×103 steps, the sand spit located left of the groin connected to the groin with a lagoon inside, whereas erosion started right of the groin because rightward longshore sand transport was obstructed by the groin. After 4×103 steps, part of the sand blocked by the groin started to be transported to the right while turning around the tip of the groin. The same situation

around the tip of the groin up to 8×103 steps. Furthermore, as a result of sand discharge to the

decreased, and the location of the starting point P of sand bar approached the groin with time,

Until 1×104 steps, the sand spit formed at the tip of the groin elongated rightward along with the reduction in the scale of the sand bar left of the groin. After 1.5×104 steps, the sand spit extending from the tip of the groin became a flying spit [20, 21] because of the reduction in sand supply by longshore sand transport. Because the flying spit is an unstable topography, it rapidly disappeared until 2×104 steps. Then, because of the increased sand supply owing to the connection of another sand spit to the groin, a sand spit elongated obliquely from the tip

sand was deposited, forming a steep slope along the shoreline on the exposed side, but the water depth generally decreased in the offshore zone owing to the sweeping motion of the sand spit, causing offshore sand movement. In contrast, sandy beach with a gentle slope was formed in the lee of the sand spits and six branches were formed behind the sand spit. The

obliquely incident from the direction with an angle of 60° to the direction normal to the shoreline, and then a breakwater of 600 m length was installed offshore of sand spit A after

longshore sand transport was pushed seaward by the construction of a groin.

and 8×103

resulting in the decrease in the scale of the lagoon behind the sand bar.

steps were predicted after the installation of the structures.

Development of Sand Spits and Cuspate Forelands with Rhythmic Shapes and Their…

steps were calculated under the conditions that waves are

steps, and a sand spit was formed owing to the deposition of sand turning

steps. It was realized from the comparison of Figs. 14(a) and 14(j) that

steps were calculated under the conditions that waves were

steps, the volume of sand left of the groin

steps, and then the beach changes

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The calculation with no structures was carried out up to 3×104

up to an additional 3×104

**4.2. Calculation results**

show the results.

continued after 6×103

of the groin until 3×104

The beach changes until 3×104

area right of the groin between 4×103

*4.2.2. Effect of breakwater on formation of sand spits*

The beach changes until 3×104

*4.2.1. Effect of groin on formation of sand spits*

Falqués et al. [6] predicted the development of sand waves caused by high-angle wave instability using equations similar to that of our model. Their sand transport equation had the same stability mechanism as that in our model. However, because the calculation domain of the wave field was restricted between the offshore zone and the breaking point, they only predicted the development of sand waves but not the development of sand spits protruding offshore. In this study, wave decay in the breaker zone and the wave-sheltering effect by the sand spit themselves were evaluated, taking the local change in topography in the surf zone into account and using the energy balance equation for irregular waves.
