**2. Numerical model**

We use Cartesian coordinates (*x*, *y*) and consider that the elevation at a point *Z* (*x*, *y*, *t*) is a variable to be solved, where *t* is the time. The beach changes are assumed to occur between the depth of closure *hc* and the berm height *hR*. The BG model [9] was used to predict the beach changes. An additional term given by Ozasa and Brampton [11] was incorporated into the fundamental equation of sand transport in the BG model to evaluate the longshore sand transport due to the effect of the longshore gradient of wave height. The fundamental equation of sand transport is given as follows.

of the amplitude of the bottom oscillatory velocity due to waves, and the wave energy dissipation rate was given by that due to wave breaking at each point determined in the calculation of the wave field. Here, the development of sand spits and cuspate forelands with rhythmic shapes were predicted first using this numerical model, and then the effects of the construction of a groin and a breakwater on the development of sand spits and cuspate

**Figure 1.** Multiple sand spits with rhythmic shapes developed in Azov-type shallow water body facing Chukchi Sea in

We use Cartesian coordinates (*x*, *y*) and consider that the elevation at a point *Z* (*x*, *y*, *t*) is a variable to be solved, where *t* is the time. The beach changes are assumed to occur between the depth of closure *hc* and the berm height *hR*. The BG model [9] was used to predict the beach changes. An additional term given by Ozasa and Brampton [11] was incorporated into the fundamental equation of sand transport in the BG model to evaluate the longshore sand

Russia [1].

**2. Numerical model**

420 Computational and Numerical Simulations

forelands with rhythmic shapes were investigated using the same model [5, 10].

$$\begin{aligned} \overset{\circ}{q} = \text{C}\_{0} \frac{P}{\tan \beta \epsilon} \begin{cases} K\_{n} \left( \tan \beta \epsilon \stackrel{\leftrightarrow}{e\_{w}} - \left| \cos \alpha \right| \stackrel{\leftrightarrow}{\nabla Z} \right) \\ + \left\{ (K\_{s} - K\_{n}) \sin \alpha - \frac{K\_{2}}{\tan \beta \overline{\beta}} \frac{\partial H}{\partial s} \right\} \tan \beta \; \; e\_{s} \right) \\ \left( -h\_{c} \le Z \le h\_{R} \right) \end{cases} \tag{1}$$

Here, *q* <sup>→</sup> =(*qx*, *qy*) is the net sand transport flux, *Z* (*x*, *y*, *t*) is the elevation, *n* and *s* are the local coordinates taken along the directions normal (shoreward) and parallel to the contour lines, respectively, ∇*<sup>Z</sup>* <sup>→</sup>=(∂*<sup>Z</sup>* / <sup>∂</sup> *<sup>x</sup>*, <sup>∂</sup>*<sup>Z</sup>* / <sup>∂</sup> *<sup>y</sup>*) is the slope vector, *ew* <sup>→</sup> is the unit vector of the wave direction, *es* <sup>→</sup> is the unit vector parallel to the contour lines, *α* is the angle between the wave direction and the direction normal to the contour lines, *tan<sup>β</sup>* <sup>=</sup> <sup>|</sup>∇*<sup>Z</sup>* <sup>→</sup><sup>|</sup> is the seabed slope, tan*βc* is the equilibrium slope, and *tanβ es* <sup>→</sup> =(−∂*Z* / ∂ *y*, ∂*Z* / ∂ *x*). Moreover, *Ks* and *Kn* are the coefficients of longshore and cross-shore sand transport, respectively, *K*<sup>2</sup> is the coefficient of the Ozasa and Brampton term [11], ∂*H* / ∂*s* =*es* <sup>→</sup> ⋅∇*<sup>H</sup>*→ is the longshore gradient of the wave height *H* measured parallel to the contour lines, and *tanβ*¯ is the characteristic slope of the breaker zone. In addition, *C*<sup>0</sup> is the coefficient transforming the immersed weight expression into a volumetric expression (*C*<sup>0</sup> =1 /{(*ρ<sup>s</sup>* −*ρ*)*g*(1− *p*)}, where *ρ* is the density of seawater, *ρ*s is the specific gravity of sand particles, *p* is the porosity of sand and *g* is the acceleration due to gravity), *hc* is the depth of closure, and *hR* is the berm height.

The intensity of sand transport *P* in Eq. (1) is assumed to be proportional to the wave energy dissipation rate [9], on the basis of the energetics approach of Bagnold [8]. *P* is given by the wave energy dissipation rate due to wave breaking at a local point *Φall* (Eq. (2)) in accordance with the BG model in [12], in which the intensity of sand transport is proportional to the wave energy at the breaking point, instead of the assumption that it is proportional to the third power of the amplitude of the bottom oscillatory velocity *um* due to waves.

$$P = \mathbb{Q}\_{all} \tag{2}$$

For the calculation of the wave field, the numerical simulation method using the energy balance equation [13], in which the directional spectrum of irregular waves is the variable to be solved, was employed with an additional term of energy dissipation due to wave breaking [14], similarly to that in [9]. *Φall* in Eq. (2) was calculated from Eq. (3), which defines the total sum of the energy dissipation of each component wave due to breaking.

$$\boldsymbol{\Phi}\_{\text{all}} = \boldsymbol{f}\_{\text{D}} \boldsymbol{E} = \mathbf{K} \sqrt{\boldsymbol{g}/\hbar} \left[ 1 - \left( \boldsymbol{\Gamma} / \boldsymbol{\gamma} \right)^{2} \right] \boldsymbol{E} \qquad \left( \boldsymbol{f}\_{\text{D}} \ge \mathbf{0} \right) \tag{3}$$

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].

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

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

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

every 10 steps in the calculation of beach changes. Table 1 shows the calculation conditions.

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*

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

adjacent to each other had merged into large-scale sand spits and disappeared, and finally six

= 1 m and *T* = 4 s, considering the formation of sand

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

http://dx.doi.org/10.5772/57043

423

h). The calculation of the wave field was carried out

steps. The small perturbation applied to the slope at the

steps, the small-scale sand spits located

**3.1. Calculation conditions**

For the wave conditions, we assumed *Hi*

steps considered was 5.5×104 (2.75×104

= 0.5 m was applied, up to 5.5×104

*3.2.1. Oblique wave incidence from 60° counterclockwise*

and had developed as slender sand spits. After 2×104

**3.2. Calculation results**

sand spits were formed.

results.

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

In the numerical simulation of beach changes, the sand transport and continuity equations (∂*Z* / ∂*t* + ∇ •*q* <sup>→</sup> =0) were solved on the *x*-*y* plane by the explicit finite-difference method using 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 closure, similar to that in [16].


**Table 1.** Calculation conditions.
