**5.2. Segmentation of water body given elliptic distribution of probability (Case 2)**

In Case 2, wind blew from the direction of 45° with respect to the principal axis of the slender lake, that is, the probability of occurrence of the wind direction is given by an elliptic distribution [13]. Segmentation of Water Body and Lakeshore Changes behind an Island Owing to Wind Waves http://dx.doi.org/10.5772/intechopen.72550 61

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.

**5.1. Segmentation of water body given circular distribution of probability (Case 1)**

along the shoreline in the beginning. After 2 × 10<sup>4</sup>

60 Applications in Water Systems Management and Modeling

ever, strongly depends on the mean (*H*1/3)

shown in **Figure 2**. After 4 × 10<sup>4</sup>

**Figure 15** shows the mean (*H*1/3)

to the intensity of the flux. After 10<sup>3</sup>

of the cuspate foreland. After 4 × 10<sup>4</sup>

The mean sand transport flux after 4 × 10<sup>4</sup>

the direction of the mean (*H*1/3)

1 × 105

after 105

**Figure 14** shows the calculation results for the segmentation of a slender, rectangular water body with a longshore length of 4.5 km, and a width of 0.9 km (aspect ratio = 5), assuming that the probability of occurrence of wind direction was given by a circular distribution [13]. When wind waves were incident to the lakeshore, several cuspate forelands with irregular shapes developed

other, resulting in a reduction in their number, and sand bars with a hound's-tooth shape were formed. This development of cuspate forelands well explains the formation of the lakeshore, as

body was about to separate into two lakes, and then the water body had separated into two

The distributions of the wave height and longshore sand transport alter in response to the wind direction at each time. The formation of cuspate forelands and rounded lakes over time, how-

part of the lake with a symmetric distribution, and the time-averaged flux at the central part

flux was equivalent on both sides of the central cuspate foreland, facilitating the development

and 5 × 104

in **Figure 16** [13]. Intensive sand transport flux occurred along the shoreline of a cuspate foreland at the central part of the water body, enhancing further development of a cuspate foreland. Also, intensive sand transport took place near the right corner of the slender water body because of a large aspect ratio of the water body, which induced the formation of a circular lake.

In Case 2, wind blew from the direction of 45° with respect to the principal axis of the slender lake, that is, the probability of occurrence of the wind direction is given by an elliptic distribution [13].

steps. The arrows in the figure show the direction of the flux, and the color corresponds

completely independent lakes. Finally, two completely rounded lakes were formed.

5/2 flux averaged over 10<sup>3</sup>

steps, its direction became normal to the shoreline of the rounded lake.

**5.2. Segmentation of water body given elliptic distribution of probability (Case 2)**

was 0 because of the cancelation of the sum of the vectors. After 2 × 10<sup>4</sup>

steps, the cuspate forelands merged with each

5/2 flux averaged over a significantly long time [13].

steps, outward flux was generated radially from the central

steps, the cuspate forelands had further developed, and

5/2 flux approached the direction normal to the shoreline. Finally,

steps at six stages between 1 × 103

steps in Case 1 can be drawn, as shown

steps, the mean (*H*1/3)

and

5/2

steps, sand bars extended to the opposite shores, and the water

**5. Results**

**Figure 14.** Topographic changes in Case 1 under uniform distribution of occurrence of wind direction and intensity [13].

Uda et al. [8] predicted the formation of oriented lakes [15] using the BG model and showed that oriented lakes can develop when the probability of occurrence of the wind direction is given by an elliptic distribution. Here, the segmentation of a rectangular water body was predicted, assuming that the probability of occurrence was given by an elliptic distribution.

**Figure 17** shows the predicted results of the lake averaged over 103 steps in Case 2 [13]. Cuspate forelands with an asymmetric form had developed on both shores and inclined rightward (leftward) on lower (upper) shorelines in the beginning. Then, the cuspate forelands had merged to increase their size and moved rightward (leftward) on lower (upper) shorelines. These results are in good agreement with those obtained by Uda et al. [5] concerning the development of sand spits and cuspate forelands owing to the shoreline instability. Because the principal axis of the wind direction is at an angle of 45° relative to the shoreline, and the effect of wind blowing from the land to the lake can be neglected along lower shoreline, the oblique component of waves incident from the left had a higher probability than that of waves incident from the right. As a result, rightward sand transport predominantly

**Figure 15.** Distribution of mean (*H*1/3)5/2 flux in Case 1 [13].

caused the formation of a cuspate foreland with an asymmetric shape along the *y*-axis, and rightward movement of the cuspate foreland took place. The formation of a cuspate foreland with an asymmetric shape corresponds to the formation of a lagoon, as shown in **Figure 2**. Furthermore, the cuspate foreland markedly developed at the right (left) end on the lower (upper) shoreline because of the long fetch distance and large wave intensity after 2 × 104 steps. With time, the cuspate forelands near the end of the lake were connected to the ends and formed a barrier island, whereas the cuspate foreland in the central part markedly extended

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**Figure 17.** Topographic changes in Case 2 under elliptic distribution of occurrence of wind direction [13].

have an elliptic form. Finally, three segmented lakes with an elliptic shape were formed. The formation of the lakes with an elliptic shape with parallel principal axes explains the develop-

**Figure 18** shows the results of the calculation of the segmentation of a triangular water body, assuming that the probability of occurrence of wind direction was given by a circular

ment process of the elliptic lakes observed in Chukchi Sea shown in **Figure 1**.

**5.3. Segmentation of a triangular or crescent-shaped water body (Cases 3 and 4)**

steps, the water body on the left side was segmented to

to the opposite shore. After 5 × 10<sup>4</sup>

**Figure 16.** Mean sand transport flux in Case 1 [13].

Segmentation of Water Body and Lakeshore Changes behind an Island Owing to Wind Waves http://dx.doi.org/10.5772/intechopen.72550 63

**Figure 17.** Topographic changes in Case 2 under elliptic distribution of occurrence of wind direction [13].

caused the formation of a cuspate foreland with an asymmetric shape along the *y*-axis, and rightward movement of the cuspate foreland took place. The formation of a cuspate foreland with an asymmetric shape corresponds to the formation of a lagoon, as shown in **Figure 2**. Furthermore, the cuspate foreland markedly developed at the right (left) end on the lower (upper) shoreline because of the long fetch distance and large wave intensity after 2 × 104 steps. With time, the cuspate forelands near the end of the lake were connected to the ends and formed a barrier island, whereas the cuspate foreland in the central part markedly extended to the opposite shore. After 5 × 10<sup>4</sup> steps, the water body on the left side was segmented to have an elliptic form. Finally, three segmented lakes with an elliptic shape were formed. The formation of the lakes with an elliptic shape with parallel principal axes explains the development process of the elliptic lakes observed in Chukchi Sea shown in **Figure 1**.

#### **5.3. Segmentation of a triangular or crescent-shaped water body (Cases 3 and 4)**

**Figure 16.** Mean sand transport flux in Case 1 [13].

**Figure 15.** Distribution of mean (*H*1/3)5/2 flux in Case 1 [13].

62 Applications in Water Systems Management and Modeling

**Figure 18** shows the results of the calculation of the segmentation of a triangular water body, assuming that the probability of occurrence of wind direction was given by a circular

the elliptic lake that formed near *y* = 3.0 km became rounded and merged into a larger lake,

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

Similarly, **Figure 19** shows the results of the segmentation of a crescent-shaped water body with time. Rapid segmentation occurred in the vicinity of the both ends of the crescent water body in the beginning. In the area between *y* = 3.25 and 4.0 km, cuspate forelands that developed from both shores were alternately distributed on both shores, in contrast to the symmetric cuspate forelands in the central part. This explains the features observed in the water

hound's-tooth shape were formed in the area between *y* = 3.75 and 4.25 km. The segmentation

**Figure 20** shows the lakeshore changes in Case 5 with a rocky island in a circular lake, assuming that the probability of occurrence of wind direction was given by a circular distribution. Under

steps, five circular lakes were formed. The

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65

steps, sand bars with a

steps, circular lakes

steps well explains the development of the sand spits in

resulting in a decrease in the aspect ratio. Until 4 × 104

body facing the Chukchi Sea, as shown in **Figure 2**. After 1 × 10<sup>4</sup>

continued over time, and the lakes became rounded as a whole. After 10<sup>5</sup>

**5.4. Lakeshore changes in circular lake with a rocky or sandy island**

**Figure 20.** Lakeshore changes behind a rocky island in lake (Case 5).

with a radius corresponding to the initial lake width were formed and stabilized.

shape of the water body after 1 × 104

Lake Saroma shown in **Figure 5**.

**Figure 18.** Segmentation of a triangular water body with time.

distribution [7]. Although the results are similar to those in [7], numerical simulation was carried out with changing the size of the lake because of the revision in Eq. (2b). Segmentation rapidly occurred in the vicinity of the right end of the triangular water body, and elliptic lakes were formed in the area between *y* = 3.25 and 3.75 km in the beginning. Near the left end, the segmentation stage was delayed, and cuspate forelands extended from both shores. After 1 × 10<sup>4</sup> steps,

**Figure 19.** Segmentation of a crescent-shaped water body with time.

the elliptic lake that formed near *y* = 3.0 km became rounded and merged into a larger lake, resulting in a decrease in the aspect ratio. Until 4 × 104 steps, five circular lakes were formed. The shape of the water body after 1 × 104 steps well explains the development of the sand spits in Lake Saroma shown in **Figure 5**.

Similarly, **Figure 19** shows the results of the segmentation of a crescent-shaped water body with time. Rapid segmentation occurred in the vicinity of the both ends of the crescent water body in the beginning. In the area between *y* = 3.25 and 4.0 km, cuspate forelands that developed from both shores were alternately distributed on both shores, in contrast to the symmetric cuspate forelands in the central part. This explains the features observed in the water body facing the Chukchi Sea, as shown in **Figure 2**. After 1 × 10<sup>4</sup> steps, sand bars with a hound's-tooth shape were formed in the area between *y* = 3.75 and 4.25 km. The segmentation continued over time, and the lakes became rounded as a whole. After 10<sup>5</sup> steps, circular lakes with a radius corresponding to the initial lake width were formed and stabilized.

## **5.4. Lakeshore changes in circular lake with a rocky or sandy island**

distribution [7]. Although the results are similar to those in [7], numerical simulation was carried out with changing the size of the lake because of the revision in Eq. (2b). Segmentation rapidly occurred in the vicinity of the right end of the triangular water body, and elliptic lakes were formed in the area between *y* = 3.25 and 3.75 km in the beginning. Near the left end, the segmentation stage was delayed, and cuspate forelands extended from both shores. After 1 × 10<sup>4</sup>

**Figure 18.** Segmentation of a triangular water body with time.

64 Applications in Water Systems Management and Modeling

**Figure 19.** Segmentation of a crescent-shaped water body with time.

steps,

**Figure 20** shows the lakeshore changes in Case 5 with a rocky island in a circular lake, assuming that the probability of occurrence of wind direction was given by a circular distribution. Under

**Figure 20.** Lakeshore changes behind a rocky island in lake (Case 5).

the condition, a wave-shelter zone was primarily formed on the lee of the island against wind waves incident from *x*-axis. Sand was transported from the outside of the wave-shelter zone to the inside, and a symmetrical cuspate foreland started to form on the lee of the island. After 5 × 104 steps, the cuspate foreland connected to the island. Because sand was mainly transported from the opposite shore with a longer fetch distance to the lee of the island, the lakeshore on the opposite shore was eroded. Thus, when a rocky island is asymmetrically located at a location in a lake, the formation of a cuspate foreland and erosion on the opposite shore take place at the same time.

**Figure 21** shows the same results in Case 6. In this case, the wave-sheltering effect due to the island was strengthened than that in Case 5 because of the proximity of the island to the lakeshore, the cuspate foreland rapidly developed together with the formation of a large cuspate foreland. After 5 × 10<sup>4</sup> steps, a headland with a circular head was formed. Because the distance between the island and lakeshore decreased, the wave-sheltering effect increased, resulting in the greater cuspate foreland behind the island and erosion on the opposite shore.

The lakeshore changes in Case 7 with a sandy island in a circular lake are shown in **Figure 22**. When waves were incident to the sandy island, the island deformed by the action of waves incident from *x-*axis, which has the longest fetch distance, and slender sand bars extended

toward the *y*-axis. In the wave-shelter zone of this sand bars, double tombolo extended at first, which connected to the slender sand bars. With time, all sand comprised of the island

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67

amount of sand was deposited on the lee of the island, whereas the opposite shore was eroded. **Figure 23** shows the same results in Case 8. The initial circular island significantly deformed owing to the action of wind waves incident from the direction of *x*-axis, and sand bars extending to the direction of the *y*-axis were formed. Because of the short distance between the island and lakeshore, double tombolo quickly extended on the lee of the sandy island, while leaving a lagoon in the central part. With time, a barrier island was formed with a lagoon inside double tombolo, and the smaller lake behind the barrier island was rounded by wind waves

The mean sand transport fluxes averaged over 1000 steps between 1.9 × 10<sup>4</sup> + 1 and 2 × 104 steps in Cases 5 and 7 with the same arrangement of an island are shown in **Figure 24**. In Case 5 with a rocky island, the intensive sand transport flux occurred on both sides of the island with decreasing the intensity behind the island, whereas in Case 7, strong sand transport flux toward the tips of the sand bar occurred along the shoreline of sand bars. When setting point O at the center of the circular lake, and points **a** and **b** at both ends of the straight line through point O, as shown in **Figure 24**, the direction of sand transport flux is downward at points **a** and **b**. Out of waves incident to point **a**, waves incident from the upper half of the

steps, a large

were transported to the lakeshore and merged with the lakeshore. After 5 × 10<sup>4</sup>

**Figure 22.** Lakeshore changes around a sandy island in lake (Case 7).

in the closed water body. A large amount of sand was deposited behind the island.

**Figure 21.** Lakeshore changes behind a rocky island in lake (Case 6).

Segmentation of Water Body and Lakeshore Changes behind an Island Owing to Wind Waves http://dx.doi.org/10.5772/intechopen.72550 67

**Figure 22.** Lakeshore changes around a sandy island in lake (Case 7).

the condition, a wave-shelter zone was primarily formed on the lee of the island against wind waves incident from *x*-axis. Sand was transported from the outside of the wave-shelter zone to the inside, and a symmetrical cuspate foreland started to form on the lee of the island. After

**Figure 21** shows the same results in Case 6. In this case, the wave-sheltering effect due to the island was strengthened than that in Case 5 because of the proximity of the island to the lakeshore, the cuspate foreland rapidly developed together with the formation of a large cuspate

between the island and lakeshore decreased, the wave-sheltering effect increased, resulting in

The lakeshore changes in Case 7 with a sandy island in a circular lake are shown in **Figure 22**. When waves were incident to the sandy island, the island deformed by the action of waves incident from *x-*axis, which has the longest fetch distance, and slender sand bars extended

the greater cuspate foreland behind the island and erosion on the opposite shore.

**Figure 21.** Lakeshore changes behind a rocky island in lake (Case 6).

steps, a headland with a circular head was formed. Because the distance

 steps, the cuspate foreland connected to the island. Because sand was mainly transported from the opposite shore with a longer fetch distance to the lee of the island, the lakeshore on the opposite shore was eroded. Thus, when a rocky island is asymmetrically located at a location in a lake, the formation of a cuspate foreland and erosion on the opposite shore take place at the

5 × 104

same time.

foreland. After 5 × 10<sup>4</sup>

66 Applications in Water Systems Management and Modeling

toward the *y*-axis. In the wave-shelter zone of this sand bars, double tombolo extended at first, which connected to the slender sand bars. With time, all sand comprised of the island were transported to the lakeshore and merged with the lakeshore. After 5 × 10<sup>4</sup> steps, a large amount of sand was deposited on the lee of the island, whereas the opposite shore was eroded.

**Figure 23** shows the same results in Case 8. The initial circular island significantly deformed owing to the action of wind waves incident from the direction of *x*-axis, and sand bars extending to the direction of the *y*-axis were formed. Because of the short distance between the island and lakeshore, double tombolo quickly extended on the lee of the sandy island, while leaving a lagoon in the central part. With time, a barrier island was formed with a lagoon inside double tombolo, and the smaller lake behind the barrier island was rounded by wind waves in the closed water body. A large amount of sand was deposited behind the island.

The mean sand transport fluxes averaged over 1000 steps between 1.9 × 10<sup>4</sup> + 1 and 2 × 104 steps in Cases 5 and 7 with the same arrangement of an island are shown in **Figure 24**. In Case 5 with a rocky island, the intensive sand transport flux occurred on both sides of the island with decreasing the intensity behind the island, whereas in Case 7, strong sand transport flux toward the tips of the sand bar occurred along the shoreline of sand bars. When setting point O at the center of the circular lake, and points **a** and **b** at both ends of the straight line through point O, as shown in **Figure 24**, the direction of sand transport flux is downward at points **a** and **b**. Out of waves incident to point **a**, waves incident from the upper half of the

In the case of Lake Balkhash, part of sand comprised of the island was considered to be transported northwestward, forming a long slender sand bar. Such topographic changes can be explained by the mergence of a slender sand bar extended from the island and the sand bar

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

Specific geomorphological features associated with shoreline instability under a high-waveangle condition on the lakeshore, such as the development of sand spits, in several elongated water bodies were investigated, and the segmentation of a water body was numerically predicted using the BG model. It was concluded that a rectangular water body segmented into circular (elliptic) lakes when the probability of occurrence of the wind direction was given by a circular (elliptic) distribution. In each case, the wave-sheltering effect of the cuspate forelands played a primary role. Also, the mergence and segmentation of triangular and crescent-shaped slender water bodies were predicted using the BG model. It was further used for predicting the lakeshore changes when a rocky or sandy island exists in a circular lake. The deformation of a

sandy island and mergence of the sandy island to the lakeshore were predicted well.

\* and Shiho Miyahara2

1 Head, Shore Protection Research, Public Works Research Center, Taito, Tokyo, Japan

[1] Ashton A, Murray AB, Arnault O. Formation of coastline features by large–scale insta-

[2] Ashton A, Murray AB High-angle wave instability and emergent shoreline shapes: 1. Modeling of sand waves, flying spits, and capes. Journal of Geophysics Research.

[3] Zenkovich VP. Processes of Coastal Development. New York: Interscience Publishers;

[4] Ashton A, Murray AB, Littlewood R, Lewis DA, Hong P. Fetch limited self-organization

steps in Case 5 and

69

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

extended in the opposite direction, as shown in the results after 5 × 104

after 2 × 104

**6. Conclusions**

**Author details**

, Masumi Serizawa2

\*Address all correspondence to: coastseri@nifty.com

2006;**111**:F04011. DOI: 10.1029/2005JF000422

of elongate water bodies. Geology. 2009;**37**:187-190

2 Coastal Engineering Laboratory Co., Ltd., Shinjuku, Tokyo, Japan

bilities induced by high-angle waves. Nature. 2001;**414**:296-300

Takaaki Uda1

**References**

1967. p. 751

steps in Case 7.

**Figure 23.** Lakeshore changes around a sandy island in lake (Case 8).

lake causes downward longshore sand transport, and vice versa, when waves are incident from the lower half. Without an island, net sand transport at point A is 0 because of the symmetricity of the closed water body. With an island, however, the area of the water body in the lower half decreases than that in the upper half, resulting in weaker wave action. As a result, the direction of the net sand transport fluxes at points **a** and **b** became downward, enhancing sand transport from the upper half to the lower half, resulting in erosion in the upper half.

**Figure 24.** Mean sand transport flux averaged over 1000 steps in Cases 5 and 7.

In the case of Lake Balkhash, part of sand comprised of the island was considered to be transported northwestward, forming a long slender sand bar. Such topographic changes can be explained by the mergence of a slender sand bar extended from the island and the sand bar extended in the opposite direction, as shown in the results after 5 × 104 steps in Case 5 and after 2 × 104 steps in Case 7.
