**3. Model for predicting lakeshore changes**

**Figure 9.** Enlarged satellite image of Lake Balkhash.

**Figure 8.** Satellite image of Lake Balkhash in Kazakhstan.

54 Applications in Water Systems Management and Modeling

Lake Balkhash has 450 and 200 km lengths in the E-W and S-N directions, respectively (**Figure 8**). **Figure 9** shows an enlarged satellite image of the rectangular area in **Figure 8**. Island A is located at a location of 46°34′53.99"N and 78°50′17.47″E at the central part of the lake near the east end, and a cuspate foreland of 14 km length extends between island A and the lakeshore. On the shore opposite to island A, a triangular cuspate foreland B is formed with a barrier island. The sand bar extending northwestward from Island A is symmetric with respect to the centerline of the cuspate foreland, and the length of the sand bar is longer than the width of the island. From this, it is inferred that the cuspate foreland extended from the land to Island A by the sand supply from Island A and the land, and connected to Island A.

**2.4. Lake Balkhash**

For the calculation of the segmentation of a rectangular water body, the BG model employed for the calculation of oriented lakes [8] was used. Given a local fetch distance *F* at a given point (*g* is the acceleration due to gravity and *U* is the wind velocity), the significant wave height *H*1/3 was calculated using Wilson's formula [11, 12].

$$H\_{\rm n\beta} = f(F, \mathcal{U}) = 0.30 \left\{ 1 - \left[ 1 + 0.004 \left( gF/L^2 \right)^{1/2} \right]^2 \right\} \left( \mathcal{U}^2/\mathcal{g} \right) \tag{1}$$

In this calculation, a coordinate system (*xw*, *yw*) was set corresponding to the wave direction instead of a fixed coordinate system (*x*, *y*) for the calculation of beach changes with the rectangular calculation domain, ABCD, as shown in **Figure 10**, and the wave height was calculated in the rectangular domain A'B'C'D' including the domain ABCD. Neglecting the wave refraction effect, waves were assumed to propagate in the same direction as the wind. The fetch distance *F* was added from upwind to downwind along the *xw*-axis using Eq. (2) when the *xw*-axis was divided by mesh intervals Δ*x*w [13]. Here, the index *i* in Eq. (2a) is the mesh number along the *xw*-axis.

$$\mathbf{F}^{(\text{i+1})} = F^{(\text{i})} + \mathbf{r} \Delta \mathbf{x}\_w \tag{2a}$$

$$\mathbf{r} = \begin{cases} 1 & (Z \le 0) \\ 0 & (Z > 0) \end{cases} \tag{2b}$$

**Figure 10.** Selection of coordinate system (*x*, *y*) adopted for calculation of beach changes with rectangular calculation domain ABCD and another coordinate system (*xw*, *yw*) [13].

When a grid point was located on land and the downslope condition of *dZ/dxw* ≤ 0 was satisfied, the local fetch was reset as *F* = 0 (Eq. (3)).

$$F^{(0)} = 0 \text{ (if } Z \ge 0 \text{ and } dZ/dx\_w \le 0\text{)}\tag{3}$$

*<sup>C</sup>*<sup>1</sup> <sup>=</sup> *g*\_\_\_

**4. Calculation conditions**

*<sup>k</sup>*<sup>1</sup> √

beach from being eroded in the zone deeper than the depth of closure [14].

**Figure 11.** Probability distribution of occurrence of wind direction: (a) circular and (b) elliptic [13].

although the wind velocity was assumed to be constant.

\_\_\_\_

*g*/*γ* (*k*<sup>1</sup> = (4.004)2

Segmentation of Water Body and Lakeshore Changes behind an Island Owing to Wind Waves

When *F* and *H*1/3 are calculated using the coordinate system (*xw, yw*) according to the wave direction, the wave power *P* (Eq. (5)) can be calculated and assigned to each grid point on the coordinate system (*xw*, *yw*). The wave power *P* at each grid point in the calculation of beach changes was interpolated from this distribution of *P*. The mesh intervals (Δ*xw*, Δ*yw*) in the coordinate system (*xw*, *yw*) were taken to be the same as (Δ*x*, Δ*y*). Finally, the sand transport and continuity equations were solved on the *x*-*y* plane by the explicit finite-difference method using a staggered mesh scheme. In this study, the wind direction at each step in the calculation of beach changes was selected to be a value determined by random numbers so as to satisfy the probability distribution function of the occurrence of a certain wind direction,

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

Lakeshore changes in a rectangular water body with an aspect ratio of 5 owing to wind waves were first predicted when wind blew from all directions between 0 and 360° with the same probability of occurrence and intensity (Case 1) or blew at an angle of 45° relative to the principal axis of the rectangular water body with an elliptic probability of occurrence and intensity (Case 2), as shown in **Figure 11** [13]. Then, lakeshore changes in triangle- and crescent-shaped shallow water bodies with a flatbed were predicted in Cases 3 and 4, respectively. In all cases, the water depth of the flatbed, the berm height, and the initial beach slope were set to 3 m, 1 m, and 1/20, respectively. **Figure 12** shows the initial topography in each case. Random perturbations with

, *γ* = 0.8) (9b)

http://dx.doi.org/10.5772/intechopen.72550

57

When the grid point was located in the lake, *F* was recalculated. By this procedure, the wave height becomes 0 on the lee of the cuspate forelands, and the wave-sheltering effect alone can be evaluated.

For the sand transport equation, Eq. (4), which is expressed using the wave energy at the breaking point, was used [6].

$$\overrightarrow{q} = C\_0 \frac{K\_\circ P}{\tan \beta\_\varepsilon} \left\{ \tan \beta\_\varepsilon \overrightarrow{e\_w} - \lor \cos a \, |\, \overrightarrow{\nabla Z}\right\} \left( -h\_\varepsilon \le Z \le h\_\mathbf{g} \right) \tag{4}$$

$$P = \left. \varepsilon(Z) \left( \mathrm{EC}\_g \right)\_b \right\vert\_w \tan \beta\_w \tag{5}$$

$$
\tan \beta\_w = dZ / d\mathbf{x}\_w \left( \tan \beta\_w \ge 0 \right) \tag{6}
$$

Here, *q* <sup>→</sup> = (*qx* , *qy* ) is the net sand transport flux, *Z* (*x*, *y*, *t*) is the seabed elevation with reference to the still water level (*Z* = 0), ∇ <sup>⟶</sup>*<sup>Z</sup>* <sup>=</sup> (∂*Z*/∂*x*, <sup>∂</sup>*Z*/∂*y*) is the seabed slope vector, *<sup>e</sup>* → *<sup>w</sup>* the unit vector of the wave direction, *α* is the angle between the wave direction and the direction normal to the contour line, and |*cosα*| = | *e* →*w* ⋅ →∇*<sup>Z</sup>*|/|→∇*Z*|. tanβ<sup>c</sup> is the equilibrium slope of sand, and *K*<sup>s</sup> is the longshore and cross-shore sand transport coefficient. The *P* value in Eq. (5) is the wave dissipation ratio per unit area of the seabed and time between *Z* = −*h*<sup>c</sup> and *hR*, where sand movement occurs [6], and (*ECg* ) *b* is the wave energy flux at the breaking point. *xw* is the coordinate in the direction of wave propagation, and tan*βw* is set to 0 when tan*β<sup>w</sup>* < 0 is satisfied. tan*βw* is the seabed slope measured in the direction of wave propagation. In the calculation, the local beach slope measured along the wave ray was used for the beach slope in Eq. (5), as shown in Eq. (6). *hc* is the depth of closure, and *hR* is the berm height. *C₀* is the coefficient for transforming the immersed weight expression to the volumetric expression (*C*<sup>0</sup> <sup>=</sup> <sup>1</sup> /{(*ρ<sup>s</sup>* <sup>−</sup> *<sup>ρ</sup>*)*g*(1 <sup>−</sup> *<sup>p</sup>*)}; *ρ* is the seawater density, *ρ<sup>s</sup>* is the specific gravity of sand, *p* is the sand porosity, *g* is the acceleration due to gravity), *ε*(*Z*) in Eq. (5) is the depth distribution of sand transport and is defined so as to satisfy Eq. (7); in this study, a uniform distribution was employed (Eq. (8)).

$$f\_{\stackrel{h}{\rightarrow}} \varepsilon(Z)dZ = 1\tag{7}$$

$$
\varepsilon(Z) = 1/(h\_c + h\_n) \quad \text{( $-h\_c \le Z \le h\_n$ )}\tag{8}
$$

If *H*1/3 is approximately equal to the breaker height *H*<sup>b</sup> and *γ* is the ratio of the breaker height to water depth, the wave energy flux at the breaking point (*ECg* ) b in Eq. (5) can be written as Eq. (9a).

$$\left(\mathrm{E}\,\mathrm{C}\_{\mathrm{g}}\right)\_{\mathrm{b}} = \,\mathrm{C}\_{\mathrm{1}}\left(\mathrm{H}\_{\mathrm{b}}\right)^{\circ} \simeq \,\mathrm{C}\_{\mathrm{1}}\left(\mathrm{H}\_{\mathrm{1}/\mathrm{3}}\right)^{\circ}\tag{9a}$$

Segmentation of Water Body and Lakeshore Changes behind an Island Owing to Wind Waves http://dx.doi.org/10.5772/intechopen.72550 57

$$C\_1 = \frac{\rho g}{k\_1} \sqrt{\mathbf{g}/\gamma} \text{ (} k\_1 = \text{ (4.004)}^2\text{)}, \gamma = 0.8\text{)}\tag{9b}$$

When *F* and *H*1/3 are calculated using the coordinate system (*xw, yw*) according to the wave direction, the wave power *P* (Eq. (5)) can be calculated and assigned to each grid point on the coordinate system (*xw*, *yw*). The wave power *P* at each grid point in the calculation of beach changes was interpolated from this distribution of *P*. The mesh intervals (Δ*xw*, Δ*yw*) in the coordinate system (*xw*, *yw*) were taken to be the same as (Δ*x*, Δ*y*). Finally, the sand transport and continuity equations were solved on the *x*-*y* plane by the explicit finite-difference method using a staggered mesh scheme. In this study, the wind direction at each step in the calculation of beach changes was selected to be a value determined by random numbers so as to satisfy the probability distribution function of the occurrence of a certain wind direction, although the wind velocity was assumed to be constant.

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 [14].

## **4. Calculation conditions**

When a grid point was located on land and the downslope condition of *dZ/dxw* ≤ 0 was satis-

*F*(*i*) = 0 (if *Z* ≥ 0 and *dZ*/*dxw* ≤ 0) (3)

When the grid point was located in the lake, *F* was recalculated. By this procedure, the wave height becomes 0 on the lee of the cuspate forelands, and the wave-sheltering effect alone can

For the sand transport equation, Eq. (4), which is expressed using the wave energy at the

<sup>→</sup>*ew* <sup>−</sup> |cos*α*|<sup>∇</sup>

(*ECg*)*<sup>b</sup>*

, *qy* ) is the net sand transport flux, *Z* (*x*, *y*, *t*) is the seabed elevation with reference to the

is the wave energy flux at the breaking point. *xw* is the coordinate in the

tan*β<sup>w</sup>* = *dZ*/*dxw* (tan*β<sup>w</sup>* ≥ 0) (6)

<sup>⟶</sup>*<sup>Z</sup>* <sup>=</sup> (∂*Z*/∂*x*, <sup>∂</sup>*Z*/∂*y*) is the seabed slope vector, *<sup>e</sup>*

wave direction, *α* is the angle between the wave direction and the direction normal to

longshore and cross-shore sand transport coefficient. The *P* value in Eq. (5) is the wave dissipa-

direction of wave propagation, and tan*βw* is set to 0 when tan*β<sup>w</sup>* < 0 is satisfied. tan*βw* is the seabed slope measured in the direction of wave propagation. In the calculation, the local beach slope measured along the wave ray was used for the beach slope in Eq. (5), as shown in Eq. (6).

 is the depth of closure, and *hR* is the berm height. *C₀* is the coefficient for transforming the immersed weight expression to the volumetric expression (*C*<sup>0</sup> <sup>=</sup> <sup>1</sup> /{(*ρ<sup>s</sup>* <sup>−</sup> *<sup>ρ</sup>*)*g*(1 <sup>−</sup> *<sup>p</sup>*)}; *ρ* is the seawa-

gravity), *ε*(*Z*) in Eq. (5) is the depth distribution of sand transport and is defined so as to satisfy

ε(*Z*) = 1/(*hc* + *hR*) (−*hc* ≤ *Z* ≤ *hR*) (8)

= *C*<sup>1</sup> (*Hb*)

\_\_5

<sup>2</sup> ≈ *C*<sup>1</sup> (*H*1/3)

is the specific gravity of sand, *p* is the sand porosity, *g* is the acceleration due to

→∇*<sup>Z</sup>*|/|→∇*Z*|. tanβ<sup>c</sup>

<sup>⟶</sup>*Z*} (−*hc* <sup>≤</sup> *<sup>Z</sup>* <sup>≤</sup> *hR*) (4)

tan*β<sup>w</sup>* (5)

→

is the equilibrium slope of sand, and *K*<sup>s</sup>

*<sup>R</sup> ε*(*Z*)*dZ* = 1 (7)

)b

\_\_5

and *γ* is the ratio of the breaker height

in Eq. (5) can be written as

<sup>2</sup> (9a)

*<sup>w</sup>* the unit vector of the

and *hR*, where sand movement

is the

fied, the local fetch was reset as *F* = 0 (Eq. (3)).

56 Applications in Water Systems Management and Modeling

<sup>→</sup> = *C*<sup>0</sup>

*P* = ε(*Z*)

*Ks P* \_\_\_\_\_ tan*β<sup>c</sup>*

> →*w* ⋅

tion ratio per unit area of the seabed and time between *Z* = −*h*<sup>c</sup>

Eq. (7); in this study, a uniform distribution was employed (Eq. (8)).

−*hc h*

{tan*β<sup>c</sup>*

be evaluated.

Here, *q*

*hc*

<sup>→</sup> = (*qx*

occurs [6], and (*ECg*

ter density, *ρ<sup>s</sup>*

Eq. (9a).

still water level (*Z* = 0), ∇

the contour line, and |*cosα*| = | *e*

) *b*

∫

(*<sup>E</sup> Cg*)*<sup>b</sup>*

If *H*1/3 is approximately equal to the breaker height *H*<sup>b</sup>

to water depth, the wave energy flux at the breaking point (*ECg*

breaking point, was used [6].

*q*

Lakeshore changes in a rectangular water body with an aspect ratio of 5 owing to wind waves were first predicted when wind blew from all directions between 0 and 360° with the same probability of occurrence and intensity (Case 1) or blew at an angle of 45° relative to the principal axis of the rectangular water body with an elliptic probability of occurrence and intensity (Case 2), as shown in **Figure 11** [13]. Then, lakeshore changes in triangle- and crescent-shaped shallow water bodies with a flatbed were predicted in Cases 3 and 4, respectively. In all cases, the water depth of the flatbed, the berm height, and the initial beach slope were set to 3 m, 1 m, and 1/20, respectively. **Figure 12** shows the initial topography in each case. Random perturbations with

**Figure 11.** Probability distribution of occurrence of wind direction: (a) circular and (b) elliptic [13].

Wind velocity 20 m/s Berm height, *hR* 1 m Depth of closure, *hc* 3 m Equilibrium slope, tan*β<sup>c</sup>* 1/20 Coefficient of sand transport *Ks* = 0.2

Mesh size Δ*x* = Δ*y* = 20 m Time intervals Δ*t* = 10 h Duration of calculation 106 h (105

**Table 1.** Calculation conditions.

**Figure 13.** Arrangement of island in Cases 5–8.

Calculation cases Case 1: rectangular water body, circular probability distribution

5 × 105 h (5 × 104

Boundary conditions Shoreward and landward ends, *qx* = 0

Case 2: rectangular water body, elliptic probability distribution

Cases 5–8: topographic changes around an island located in a circular lake

http://dx.doi.org/10.5772/intechopen.72550

59

Case 3: segmentation of a triangular water body Case 4: segmentation of a crescent-shaped water body

Segmentation of Water Body and Lakeshore Changes behind an Island Owing to Wind Waves

steps) in Cases 5–8

steps) in Cases 1–4,

Right and left boundaries, *qy* = 0

**Figure 12.** Initial topographies in Cases 1–4.

the amplitude △*Z* = 0.1 m were added to the slope between *Z* = 1 and −3 m in the initial bathymetry. The wind velocity was 20 m/s. The calculation domain was discretized by △*x* = △*y* = 20 m with △*t* = 10 h. The depth distribution of sand transport was assumed to be a uniform distribution throughout the depth, and the equilibrium slope was 1/20. **Table 1** shows the calculation conditions for Cases 1–4. The wind velocity of 20 m/s is the value at which a significant wave height of approximately 1 m, the same as the berm height, could be generated, given the fetch distance of 4.6 km, being the distance along the diagonal of the initial rectangular water body in Cases 1 and 2. In Cases 3 and 4, the wind velocity was also assumed to be 20 m/s, and wind was assumed to blow from all directions with the same probability and intensity.

In predicting lakeshore changes when a rocky or sandy island is located in a closed water body, four calculations were carried out, as shown in **Figure 13**. In each case, a circular lake with a radius of 1000 m and a solid bottom of a constant depth of 3 m was set for the calculation domain. In this circular lake, a rocky or sandy island with a radius of 200 m was set at locations deviating from the center of the lake. The foreshore slope of the lakeshore was assumed to be 1/20. In the present study, the incident angle of waves to the mean shoreline exceeds 45° at


**Table 1.** Calculation conditions.

the amplitude △*Z* = 0.1 m were added to the slope between *Z* = 1 and −3 m in the initial bathymetry. The wind velocity was 20 m/s. The calculation domain was discretized by △*x* = △*y* = 20 m with △*t* = 10 h. The depth distribution of sand transport was assumed to be a uniform distribution throughout the depth, and the equilibrium slope was 1/20. **Table 1** shows the calculation conditions for Cases 1–4. The wind velocity of 20 m/s is the value at which a significant wave height of approximately 1 m, the same as the berm height, could be generated, given the fetch distance of 4.6 km, being the distance along the diagonal of the initial rectangular water body in Cases 1 and 2. In Cases 3 and 4, the wind velocity was also assumed to be 20 m/s, and wind was

In predicting lakeshore changes when a rocky or sandy island is located in a closed water body, four calculations were carried out, as shown in **Figure 13**. In each case, a circular lake with a radius of 1000 m and a solid bottom of a constant depth of 3 m was set for the calculation domain. In this circular lake, a rocky or sandy island with a radius of 200 m was set at locations deviating from the center of the lake. The foreshore slope of the lakeshore was assumed to be 1/20. In the present study, the incident angle of waves to the mean shoreline exceeds 45° at

assumed to blow from all directions with the same probability and intensity.

**Figure 12.** Initial topographies in Cases 1–4.

58 Applications in Water Systems Management and Modeling

**Figure 13.** Arrangement of island in Cases 5–8.

certain locations of the lakeshore, resulting in shoreline instability. Therefore, a small perturbation with the amplitude Δ*Z* = 0.1 m was added in the depth zone between *Z* = −3 and 1 m. In Cases 5 and 6, a rocky island was placed with its center deviating from the center of lake, and the wave-sheltering effect by the island was enhanced in Case 6, in which the island was set at a location closer to the lakeshore. In Cases 7 and 8, the arrangement of the island is the same as those in Cases 5 and 6, respectively, but the island is composed of sand. The other conditions are the same as those in Cases 1–4. **Table 1** shows the calculation conditions for Cases 5–8.
