**2. Numerical model (BG model)**

With the elongation of a sand spit or a sand bar, the shape of the wave-shelter zone behind the spit or the sand bar changes, and therefore, the repeated calculation of the wave field and topographic changes is required. We use Cartesian coordinates (*x*, *y*) and consider the elevation at a point *Z* (*x*, *y*, *t*) as a variable to be solved, where *t* is time. The beach changes 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 fundamental equation is given by

340 Numerical Simulation – From Theory to Industry

for predicting the 3D topographic changes of a sand spit.

foreland on a steep coast (Serizawa & Uda, 2011).

explained by the results of the numerical simulation.

**2. Numerical model (BG model)** 

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

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

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

With the elongation of a sand spit or a sand bar, the shape of the wave-shelter zone behind the spit or the sand bar changes, and therefore, the repeated calculation of the wave field and topographic changes is required. We use Cartesian coordinates (*x*, *y*) and consider the elevation at a point *Z* (*x*, *y*, *t*) as a variable to be solved, where *t* is time. The beach changes

$$\overline{q} = \mathbb{C}\_{0} \frac{P}{\tan \mathfrak{R}\_{c}} \begin{bmatrix} K\_{n} (\tan \mathfrak{R}\_{c} \overline{e\_{w}} - |\cos \mathfrak{a}| \, \, \overline{\nabla Z}) \\ + \left\{ (K\_{s} - K\_{n}) \sin \alpha - \frac{K\_{2}}{\tan \overline{\beta}} \frac{\partial H}{\partial \mathbf{s}} \right\} \tan \beta \overline{e\_{s}} \end{bmatrix} \tag{1}$$
 
$$\left( -h\_{c} \le Z \le h\_{R} \right)$$

$$P = \mathfrak{p} \text{ } \mathfrak{u}^3\_m \text{\$} \tag{2}$$

$$
\mu\_m = \frac{H}{2} \sqrt{\frac{\text{g}}{h}} \tag{3}
$$

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 lines, respectively, *Z Zx Zy* , is the slope vector, *we* is a unit vector in the wave direction, *se* is a unit vector parallel to the contour lines, is the angle between the wave direction and the direction normal to the contour lines, tan *Z* is the seabed slope, tan*c* 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 term given by Ozasa & Brampton (1980), *Hse H <sup>s</sup>* is the longshore gradient 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 immersedweight expression into a volumetric expression (*C gp* <sup>0</sup> 1 1 *<sup>s</sup>* , where is the density of seawater, *<sup>s</sup>* is the specific gravity of sand particles, *p* is the porosity of sand and *g* 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 (1981) used the relationship *φt*=*ut*=*Cfut* 3 for the instantaneous wave energy dissipation rate *φt* to derive their sand transport equation, where is the bottom shear stress, *ut* is the 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

included in the coefficients of longshore and cross-shore sand transport, *Ks* and *Kn*, respectively. *um* can be calculated by small-amplitude wave theory in shallow water using the wave height *H* at a local point (Eq. (3)), which can be obtained by the numerical calculation of the plane-wave field. The depth of closure *hc* is assumed to be proportional to the wave height *H* at a local point and is given by Eq. (4), referring to the relationship given by Uda & Kawano (1996).

$$h\_c \text{=KH} \qquad \text{(K=2.5)} \tag{4}$$

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 343

in Eq. (1) is separately expressed as Eq. (8) when

c

cos tan

 (9)

c c

(10)

<sup>0</sup> sin tan tan *s s <sup>c</sup> q CKP* (11)

*s*

tan tan sin <sup>1</sup> tan tan *n*

are taken, Eqs. (9) and (10) are derived for the

(8)

In addition, at locations whose elevation is higher than the berm height, the wave energy was set to 0. The calculation of the wave field was carried out every 10 steps in the

Equation (1) shows that the sand transport flux can be expressed as the sum of the component along the wave direction and the components due to the effect of gravity normal to the contours and the effect of longshore currents parallel to the contours. To investigate

neglecting the additional term given by Ozasa & Brampton (1980), and when the inner

cross-shore and longshore components of sand transport, *qn* and *qs*, respectively. Furthermore, under the condition that the seabed slope is equal to the equilibrium slope, Eq.

*nn ss q qe qe*

cos cos tan *nn n q e q CKP* 

In Eq. (9), the cross-shore sand transport *qn* becomes 0 when the local seabed slope is equal to the equilibrium slope, and the longshore sand transport *qs* becomes 0 when the wave direction coincides with the normal to the contour lines, as shown in Eqs. (10) and (11). When a discrepancy from these conditions arises, sand transport is generated by the same stabilization mechanism as in the contour-line-change model (Uda & Serizawa,

Taking the above into account, the first term in the parentheses in Eq. (1) gives the sand transport in the case that the rates of longshore and cross-shore sand transport are equal (*Ks* = *Kn*), and the second term is the additional longshore sand transport in the case that the rates are different (*Ks*>*Kn*). The physical meaning of the second term is that longshore sand transport is generated by the small angular shift that occurs when the wave direction is incompletely reversed in the oscillatory movement due to waves, and the second term also models the additional longshore sand transport due to the effect of longshore currents, the

Although the applicability of the contour-line-change model to the prediction of beach changes is limited when the shape of coastal structures is complicated because it tracks the movement of lines with specific characteristics, the BG model can be applied to the

 and *q* 

0

*<sup>K</sup> q e q CKP <sup>K</sup>*

0

*ss s*

effect of which is only partially included in the first term.

and of *se*

calculation of topographic changes.

the physical meaning of Eq. (1), *q*

 and *q* 

(10) reduces to Eq. (11).

products of *ne*

2010).

In the numerical simulation of beach changes, the sand transport equation and the continuity equation are solved on the *x*-*y* plane by the explicit finite-difference methodemploying staggered mesh scheme. In the estimation of sand transport near the berm top and the depth of closure, sand transport was linearly reduced to 0 near the berm height or the depth of closure to prevent sand from depositing in the area higher than the berm height and beach erosion in the zone deeper than the depth of closure.

The wave field was calculated using the energy balance equation given by Mase (2001), in which the directional spectrum *D* (*f*, ) of the irregular waves varies with the energy dissipation term due to wave breaking (Dally et al., 1984). Here, *f* and are the frequency and wave direction, respectively. In this method, wave refraction, wave breaking and wave diffraction in the wave-shelter zone can be calculated with a small calculation load. The energy dissipation term due to wave breaking *φ* (Dally et al., 1984), which is incorporated into the energy balance equation (Eq. (5)), is given by

$$\frac{\partial}{\partial \mathbf{x}} (DV\_x) + \frac{\partial}{\partial y} (DV\_y) + \frac{\partial}{\partial \Theta} (DV\_0) = F - \phi \tag{5}$$

$$\phi = \left(\mathbf{K} / h\right) \mathbf{D} \mathbf{C}\_{\mathcal{S}} \left[\mathbf{1} - \left(\boldsymbol{\varGamma} / \boldsymbol{\upgamma}\right)^{2}\right] \qquad \qquad \left(\phi \ge 0\right) \tag{6}$$

Here, D is the directional spectrum, (*Vx*, *Vy*, *V*) is the energy transport velocity in the (*x*, *y*, ) space, *F* is the wave diffraction term given by Mase (2001), *K* is the coefficient of the wavebreaking intensity, *h* is the water depth, *Cg* is the wave group velocity ( *C gh <sup>g</sup>* in the approximation in shallow-water wave theory), is the ratio of the critical breaker height to the water depth on the horizontal bed and is the ratio of the wave height to water depth. To prevent the location where the berm develops from being excessively seaward compared with that observed in the experiment or the field, a lower limit was considered for *h* in Eq. (6). As a result of this procedure, wave decay near the berm top was reduced, resulting in a higher landward sand transport rate. In the calculation of the wave field on land, the imaginary depth *h*' between the minimum depth *h*0 and berm height *hR* was considered, as given by Eq. (7), similarly to in the ordinary 3D model (Shimizu et al., 1996).

$$h' = \left(\frac{h\_R - Z}{h\_R + h\_0}\right)^r h\_0 \qquad \left(r = 1\right) \qquad \left(-h\_0 \le Z \le h\_R\right) \tag{7}$$

In addition, at locations whose elevation is higher than the berm height, the wave energy was set to 0. The calculation of the wave field was carried out every 10 steps in the calculation of topographic changes.

342 Numerical Simulation – From Theory to Industry

which the directional spectrum *D* (*f*,

into the energy balance equation (Eq. (5)), is given by

Here, D is the directional spectrum, (*Vx*, *Vy*, *V*

approximation in shallow-water wave theory),

the water depth on the horizontal bed and

by Uda & Kawano (1996).

included in the coefficients of longshore and cross-shore sand transport, *Ks* and *Kn*, respectively. *um* can be calculated by small-amplitude wave theory in shallow water using the wave height *H* at a local point (Eq. (3)), which can be obtained by the numerical calculation of the plane-wave field. The depth of closure *hc* is assumed to be proportional to the wave height *H* at a local point and is given by Eq. (4), referring to the relationship given

In the numerical simulation of beach changes, the sand transport equation and the continuity equation are solved on the *x*-*y* plane by the explicit finite-difference methodemploying staggered mesh scheme. In the estimation of sand transport near the berm top and the depth of closure, sand transport was linearly reduced to 0 near the berm height or the depth of closure to prevent sand from depositing in the area higher than the

The wave field was calculated using the energy balance equation given by Mase (2001), in

and wave direction, respectively. In this method, wave refraction, wave breaking and wave diffraction in the wave-shelter zone can be calculated with a small calculation load. The energy dissipation term due to wave breaking *φ* (Dally et al., 1984), which is incorporated

*DV DV DV F x y x y*

=1

*h Z <sup>h</sup> h r h Zh*

given by Eq. (7), similarly to in the ordinary 3D model (Shimizu et al., 1996).

*r*

0

*R*

*h h*

*R*

To prevent the location where the berm develops from being excessively seaward compared with that observed in the experiment or the field, a lower limit was considered for *h* in Eq. (6). As a result of this procedure, wave decay near the berm top was reduced, resulting in a higher landward sand transport rate. In the calculation of the wave field on land, the imaginary depth *h*' between the minimum depth *h*0 and berm height *hR* was considered, as

space, *F* is the wave diffraction term given by Mase (2001), *K* is the coefficient of the wavebreaking intensity, *h* is the water depth, *Cg* is the wave group velocity ( *C gh <sup>g</sup>* in the

0 0

<sup>2</sup> 1 0 *K h DCg*

berm height and beach erosion in the zone deeper than the depth of closure.

dissipation term due to wave breaking (Dally et al., 1984). Here, *f* and

= ( =2.5) *<sup>c</sup> h KH K* (4)

) of the irregular waves varies with the energy

) is the energy transport velocity in the (*x*, *y*,

is the ratio of the critical breaker height to

is the ratio of the wave height to water depth.

*R*

(7)

(5)

(6)

are the frequency

) Equation (1) shows that the sand transport flux can be expressed as the sum of the component along the wave direction and the components due to the effect of gravity normal to the contours and the effect of longshore currents parallel to the contours. To investigate the physical meaning of Eq. (1), *q* in Eq. (1) is separately expressed as Eq. (8) when neglecting the additional term given by Ozasa & Brampton (1980), and when the inner products of *ne* and *q* and of *se* and *q* are taken, Eqs. (9) and (10) are derived for the cross-shore and longshore components of sand transport, *qn* and *qs*, respectively. Furthermore, under the condition that the seabed slope is equal to the equilibrium slope, Eq. (10) reduces to Eq. (11).

$$
\overrightarrow{dq}^{\prime} = q\_n \overrightarrow{e\_n} + q\_s \overrightarrow{e\_s} \tag{8}
$$

$$q\_n = \overrightarrow{e\_n} \cdot \overrightarrow{q} \quad = \mathbb{C}\_0 \mathbb{K}\_n P \left| \cos \alpha \right| \left( \frac{\cos \alpha}{\left| \cos \alpha \right|} - \frac{\tan \beta}{\tan \beta\_c} \right) \tag{9}$$

$$q\_s = \overrightarrow{e\_s} \cdot \overrightarrow{q} = \mathbb{C}\_0 K\_s P \sin \alpha \left\{ \frac{\tan \mathfrak{B}}{\tan \mathfrak{B}\_c} + \frac{K\_n}{K\_s} \left( 1 - \frac{\tan \mathfrak{B}}{\tan \mathfrak{B}\_c} \right) \right\} \tag{10}$$

$$q\_s \approx \mathbb{C}\_0 K\_s P \sin a \qquad \left( \because \tan \mathfrak{B} \approx \tan \mathfrak{B}\_c \right) \tag{11}$$

In Eq. (9), the cross-shore sand transport *qn* becomes 0 when the local seabed slope is equal to the equilibrium slope, and the longshore sand transport *qs* becomes 0 when the wave direction coincides with the normal to the contour lines, as shown in Eqs. (10) and (11). When a discrepancy from these conditions arises, sand transport is generated by the same stabilization mechanism as in the contour-line-change model (Uda & Serizawa, 2010).

Taking the above into account, the first term in the parentheses in Eq. (1) gives the sand transport in the case that the rates of longshore and cross-shore sand transport are equal (*Ks* = *Kn*), and the second term is the additional longshore sand transport in the case that the rates are different (*Ks*>*Kn*). The physical meaning of the second term is that longshore sand transport is generated by the small angular shift that occurs when the wave direction is incompletely reversed in the oscillatory movement due to waves, and the second term also models the additional longshore sand transport due to the effect of longshore currents, the effect of which is only partially included in the first term.

Although the applicability of the contour-line-change model to the prediction of beach changes is limited when the shape of coastal structures is complicated because it tracks the movement of lines with specific characteristics, the BG model can be applied to the prediction of topographic changes under all structural conditions, because the depth changes in the *x*-*y* plane are calculated, similarly to in the ordinary model for predicting 3D beach changes, and therefore the calculation can be carried out systematically. This is an advantage of the BG model.

BG Model Based on Bagnold's Concept and

Its Application to Analysis of Elongation of Sand Spit and Shore – Normal Sand Bar 345

**Figure 1.** Experimental results for development of sand spit on a coast with abrupt change in coastline

The study area is the Kutsuo coast facing the Suo-nada Sea, part of the Seto Inland Sea, as shown in Fig. 2. Figure 3 shows an aerial photograph of the study area taken in 1999. The Harai River flows into the coast, which has a very wide tidal flat of approximately 1.5 km width offshore of the river. Although a river mouth bar extends on the north side of the Harai River, another slender sand bar has developed along the north side of the channel extending between the river mouth and the offshore tidal mud flat, and it intersects the river mouth bar perpendicular to the shoreline. The sand source for this slender sand bar is assumed to be the Harai River; sand transported offshore by flood currents is deposited

**4. Field observation of formation of slender shore-normal sand bar** 

orientation (Uda & Yamamoto, 1992).

**4.1. General conditions** 
