**1. Introduction**

The accurate prediction of three-dimensional (3D) beach changes on a coast with a large shoreline curvature, such as a coast with a sand spit, and a wave field that significantly changes in response to topographic changes, has been difficult to achieve in previous studies. As a result, regarding the beach changes around a sand spit, most previous studies have focused on the shoreline changes. Ashton et al. (2001) showed that when the incident angle of deep-water waves to the mean shoreline exceeds 45, shoreline instability occurs, resulting in the development of sand spits from a small perturbation of the shoreline. They successfully predicted the planar changes in a shoreline containing sand spits using infinitesimal meshes divided in the *x*- and *y*-directions. However in Ashton et al.'s model, only the longshore sand transport equation is employed as the sand transport equation instead of a two-dimensional (2D) sand transport equation in which both cross-shore and longshore sand transport are considered. Furthermore, in evaluating the wave field, wave conditions at the breaking point are transformed into deep-water values assuming a bathymetry with straight parallel contours and using Snell's law. Since the sand transport equation is expressed using these deep-water parameters, the effect of large 3D changes in topography on the wave field cannot be accurately evaluated. Furthermore, the finitedifference scheme used to evaluate the breaker angle and the method of calculating the wave field around the wave-shelter zone are altered depending on the calculation conditions, resulting in a complicated calculation method that requires special calculation techniques. Watanabe et al. (2004) developed a model for predicting the shoreline changes of a sand spit under the conditions that the sand spit significantly changes its configuration with changes in the wave field. They selected orthogonal curvilinear coordinates parallel and normal to the shoreline of the sand spit, and the seabed topography after various

© 2012 Uda et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

numbers of time steps was inversely determined from the time-dependent shoreline configuration given by the seabed slope. They also predicted the shoreline changes due to the spatial changes in longshore sand transport. Their model is also not a definitive model for predicting the 3D topographic changes of a sand spit.

BG Model Based on Bagnold's Concept and

(1)

is a unit vector in the wave

*c*

is the angle between the wave

is the seabed slope, tan

is the longshore gradient of

is the

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 341

are assumed to take place between the depth of closure *hc* and the berm height *hR*. A modified version of the BG model proposed by Serizawa et al. (2009a) was used to predict the formation of a sand spit. An additional term given by Ozasa & Brampton (1980) was also incorporated into the fundamental equation of the BG model to accurately evaluate the longshore sand transport due to the effect of the longshore gradient of the wave height. The

tan

*s*

*<sup>s</sup>* is the specific gravity of sand particles, *p* is the porosity of sand and *g*

3 for the instantaneous wave energy dissipation rate

is the bottom shear stress, *ut* is the

*<sup>m</sup> P u* (2)

*<sup>h</sup>* (3)

*s n s*

tan cos

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

tan sin tan

*c R h Zh*

2 *<sup>m</sup> <sup>H</sup> <sup>g</sup> <sup>u</sup>*

lines, respectively, *Z Zx Zy* , is the slope vector, *we*

the term given by Ozasa & Brampton (1980), *Hse H <sup>s</sup>*

*ut*=*Cfut*

*φt* to derive their sand transport equation, where

direction and the direction normal to the contour lines, tan *Z*

is a unit vector parallel to the contour lines,

Here, , *x y q qq* 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

is the equilibrium slope, tan , *<sup>s</sup> e Zy Zx* , *Ks* and *Kn* are the coefficients of longshore sand transport and cross-shore sand transport, respectively, *K*2 is the coefficient of

the wave height *H* measured parallel to the contour lines and tan is the characteristic slope of the breaker zone. In addition, *C*0 is the coefficient transforming the immersed-

is the acceleration due to gravity), *um* is the amplitude of the seabed velocity due to the orbital motion of waves given by Eq. (3), *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 *φ* based on the energetics approach of Bagnold (1963). In the model of Serizawa et al. (2006), *P* was formulated using the wave energy at the breaking point, but in this study, it is combined with the wave characteristics at a local point. Bailard & Inman

instantaneous velocity and *Cf* is the drag coefficient. We basically follow their study but assume that *φ* is proportional to the third power of the amplitude of the bottom oscillatory velocity *um* due to waves instead of the third power of the instantaneous velocity. The intensity of sand transport *P* is then given by Eq. (2), and its coefficient is assumed to be

weight expression into a volumetric expression (*C gp* <sup>0</sup> 1 1 *<sup>s</sup>* , where

3

<sup>0</sup> <sup>2</sup> <sup>c</sup>

*n cw*

fundamental equation is given by

direction, *se*

density of seawater,

(1981) used the relationship *φt*=

A sand spit is often formed by wave action at a location where the direction of the coastline abruptly changes. Uda & Yamamoto (1992) carried out a movable-bed experiment using a plane-wave basin to investigate the development of a sand spit. Two experiments were carried out: sand was deposited (1) on a shallow flat seabed and (2) on a coast with a steep slope. Their results showed that a slender sand spit extends along the marginal line between the shallow sea and offshore steep slope in Case 1, whereas a cuspate foreland is formed owing to the deposition of sand on the steep slope in Case 2, suggesting the importance of the effect caused by the difference in the depth of water where sand is deposited. We have developed a model for predicting beach changes based on Bagnold's concept (Serizawa et al., 2006) by applying the concept of the equilibrium slope introduced by Inman& Bagnold (1963) and the energetics approach of Bagnold (1963). Here, the BG model is used to simulate the extension of a sand spit on a shallow seabed and the formation of a cuspate foreland on a steep coast (Serizawa & Uda, 2011).

As another type of beach change due to waves on a coast with a shallow flat seabed, a tidal flat facing an inland sea is considered. On such a tidal flat subject to the action of waves with significant energy, a sandy beach may develop along the marginal line between the tidal flat and the land, and the sandy beach with a steep slope is clearly separated from the tidal flat along a line with a discontinuous change in the slope. On such beaches developing along the marginal line between the tidal flat and the land, longshore sand transport due to the oblique wave incidence to the shoreline and cross-shore sand transport during storm surges often occur. However, in addition to these sand transport phenomena, as part of the interaction between the tidal flat and the sandy beach, shoreward transport and the landward deposition of sand originally supplied from the offshore zone of the tidal flat, forming a slender sand bar, are often observed. Although this landward sand movement due to waves on the shallow tidal flat is considered to be part of the process by which sand transported offshore by river currents during floods returns to the shore, its mechanism has not yet been studied. These phenomena were observed on the Kutsuo coast, which has a very wide tidal flat and faces the Suo-nada Sea, part of the Seto Inland Sea, Japan. Here, the BG model was also used to predict the extension of a slender shore-normal sand bar observed on this coast (Serizawa et al., 2011). The observed phenomena were successfully explained by the results of the numerical simulation.
