**1. Introduction**

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Zenkovich showed that multiple sand spits with rhythmic shapes may develop in a shallow water body such as the Azov Sea and a lagoon facing Chukchi Sea, as shown in Fig. 1, and called them as the spits of Azov type [1]. Zenkovich concluded that under oblique wave incidence with the angle between the direction normal to the shoreline and the wave direction being larger than 45°, shoreline instability may develop, and during the development of sand spits, the wave-sheltering effect due to the sand spits themselves plays an important role. Ashton et al. [2] adopted this mechanism in their model and successfully modeled this shoreline instability using the upwind scheme in their finite difference method to prevent the numerical instability on the basis of the conventional longshore sand transport formula. Furthermore, this mechanism was called high-angle wave instability in [3]. Littlewood et al. [4] predicted the shoreline of log-spiral bays using their model. Serizawa et al. [5] predicted the development of sand spits and cuspate forelands under oblique wave incidence with the angle between the direction normal to the shoreline and the wave direction being larger than 45°, given a small perturbation in the initial topography, and showed that the three-dimen‐ sional (3-D) beach changes of sand spits and cuspate forelands with rhythmic shapes can be predicted using the BG model (a 3-D model for predicting beach changes based on Bagnold's concept). Falqués et al. [6] also predicted the development of sand waves caused by high-angle wave instability using equations similar to that of our model, but not the development of sand spits and cuspate forelands protruding offshore. The sand transport equation of the BG model was derived by applying the concept of the equilibrium slope in [7] and the energetics approach of Bagnold [8]. In the fundamental equation of the BG model in [9], the sand transport flux was assumed to be proportional to the wave energy dissipation rate instead of the third power

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

transport due to the effect of the longshore gradient of wave height. The fundamental equation

( )

tan sin tan

<sup>=</sup> í ý ì ü ï ï +- - í ý î þ î þ

tan cos

*Ke Z P q C <sup>K</sup> <sup>H</sup> K K <sup>e</sup>*

2

b ¶

*s*

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

 b

<sup>→</sup> =(−∂*Z* / ∂ *y*, ∂*Z* / ∂ *x*). Moreover, *Ks* and *Kn* are the

*P* =Φ*all* (2)

é ù <sup>=</sup> =- ³ ê ú ë û (3)

<sup>→</sup> ⋅∇*<sup>H</sup>*→ is the longshore gradient of the wave

uur (1)

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

421

<sup>→</sup> is the unit vector of the wave

¶

tan

<sup>→</sup> =(*qx*, *qy*) is the net sand transport flux, *Z* (*x*, *y*, *t*) is the elevation, *n* and *s* are the local

<sup>→</sup> is the unit vector parallel to the contour lines, *α* is the angle between the wave

*<sup>c</sup> s n <sup>s</sup>*

a

coordinates taken along the directions normal (shoreward) and parallel to the contour lines,

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,

coefficients of longshore and cross-shore sand transport, respectively, *K*<sup>2</sup> is the coefficient of

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

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

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

> ( ) ( ) <sup>2</sup> 1 0 *Φ f E K g h Γ E f all D <sup>D</sup>* g

 a

uur uuuur

ì ü - Ñ ï ï

( )


 *h Zh* 

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*

gravity), *hc* is the depth of closure, and *hR* is the berm height.

of the amplitude of the bottom oscillatory velocity *um* due to waves.

sum of the energy dissipation of each component wave due to breaking.

b *n cw*

b

( )

*c R*

of sand transport is given as follows.

uur

Here, *q*

direction, *es*

0

tan*βc* is the equilibrium slope, and *tanβ es*

the Ozasa and Brampton term [11], ∂*H* / ∂*s* =*es*

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