**4. Model for predicting topographic changes**

Topographic changes including the accumulation of sand in the shoreward zones of the artificial reefs caused by waves and the landward transport of windblown sand from the salients after the installation of the artificial reefs were predicted using a model for predicting 3-D beach changes under combined actions of waves and wind [21]. With this model, the beach changes caused by waves occurring in the depth zone between the depth of closure (*h*c) and the berm height (*h*R) are predicted using the BG model [22], whereas topographic changes caused by windblown sand in the area landward of a berm top are predicted by a cellular automaton method.

The BG model is based on following concepts: (1) the contour line becomes orthogonal to the wave direction at any point at the final stage, similarly, (2) the local beach slope coincides with the equilibrium slope at any point, and (3) a restoring force is generated in response to the deviation from the statically stable condition, and sand transport occurs owing to this restoring force [22].

In the calculation of windblown sand by the cellular automaton method, the two most important processes, saltation and avalanche, are taken into account [20]. Twodimensional meshes were taken on Cartesian coordinates (*x*, *y*), and the elevation at the mesh point was set as *z* (*x*, *y*, *t*). The mesh size is assumed to be sufficiently larger than the size of the sand particles. The saltation distance *L*<sup>s</sup> was defined using Eq. (1) on the basis of the observation results obtained by Andreotti et al. [30] and is the simplest polynomial expression that can be used to evaluate the obtained results of the sand flux on a sand dune including multiphase flow.

$$L\_S = L\left[1.0 + b\_1\left(\frac{Z}{h}\right) - b\_2\left(\frac{Z}{h}\right)^2\right] \tag{1}$$

*b*<sup>1</sup> and *b*<sup>2</sup> are the coefficients to control sand transport flux, expressed by the product of *L*<sup>s</sup> defined using Eq. (1) and the mass of moving sand. *L* is a reference distance (here, we choose 1.0 m), and *h* is the reference height (here, the berm height).

Eq. (1) shows that the higher the elevation where sand particles are deposited, the longer the distance that the sand particles are transported by wind, but *L*<sup>s</sup> has a limit and the sand flux after the maximum value is reached is regarded as a constant, and the decreasing functional form is not employed. When there is an obstacle in the field, saltation is assumed not to occur, because a vortex is formed behind an obstacle owing to the separation of the flow [31]. Originally, the sand flux is given by the product of the mass of moving sand and the saltation distance, and the sand flux can be expressed by Eq. (1) when the wind velocity is constant, assuming that the mass of moving sand is constant. When the wind velocity changes, the coefficient of Eq. (1) can be changed depending on the wind velocity.

#### **Figure 16.**

*Schematic view of condition for windblown sand to occur. (a) No windblown sand (*B *<* w*'). (b) Windblown sand (0 <* w *< 10* m*). (c) Windblown sand (10* m *<* w*).*

To combine the BG model and the cellular automaton method, the calculation domains were separated at the location of the berm, assigning the landward region of the berm as the domain of windblown sand. The rate of windblown sand is assumed to attain equilibrium at a location distant from the starting point for the approach run in the downwind direction. Here, the condition for the windblown sand to occur was defined, as shown in **Figure 16**, depending on the backshore width, assuming that the minimum approach run is 10 m [32]. No windblown sand is transported when the beach width *B* is smaller than the foreshore width *w*'; the mass of moving sand (*q*) attributable to windblown sand was given by the value multiplied by the coefficient *μ* shown by Eq. (2) when the backshore width *w* is smaller than 10 m. When *w* is greater than 10 m, *μ* is unity.

$$\mu = \frac{1}{2} \left[ \cos \left( \frac{\pi}{10} b \right) + 1 \right] \tag{2}$$
