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

**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 along the shoreline in the beginning. After 2 × 10<sup>4</sup> steps, the cuspate forelands merged with each 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 shown in **Figure 2**. After 4 × 10<sup>4</sup> steps, sand bars extended to the opposite shores, and the water body was about to separate into two lakes, and then the water body had separated into two completely independent lakes. Finally, two completely rounded lakes were formed.

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, however, strongly depends on the mean (*H*1/3) 5/2 flux averaged over a significantly long time [13]. **Figure 15** shows the mean (*H*1/3) 5/2 flux averaged over 10<sup>3</sup> steps at six stages between 1 × 103 and 1 × 105 steps. The arrows in the figure show the direction of the flux, and the color corresponds to the intensity of the flux. After 10<sup>3</sup> steps, outward flux was generated radially from the central part of the lake with a symmetric distribution, and the time-averaged flux at the central part was 0 because of the cancelation of the sum of the vectors. After 2 × 10<sup>4</sup> steps, the mean (*H*1/3) 5/2 flux was equivalent on both sides of the central cuspate foreland, facilitating the development of the cuspate foreland. After 4 × 10<sup>4</sup> steps, the cuspate forelands had further developed, and the direction of the mean (*H*1/3) 5/2 flux approached the direction normal to the shoreline. Finally, after 105 steps, its direction became normal to the shoreline of the rounded lake.

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

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

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

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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

steps in Case 2 [13].

that the probability of occurrence was given by an elliptic distribution.

**Figure 17** shows the predicted results of the lake averaged over 103

The mean sand transport flux after 4 × 10<sup>4</sup> and 5 × 104 steps in Case 1 can be drawn, as shown 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.
