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RESULTS OF NUMERICAL EXPERIMENTS
Using this computational method a number of computational experiments of tsunami run-up on a shore of an arbitrary profile are carried out. The main goal of these computations was testing of this algorithm using results of other methods and investigation of the run-up height dependence from the shore profile.
The first series of test experiments was carried out for the uniformly sloping shore with following parameters:
| Spatial step of the grid: |
dx = 10 meters. |
| Initial number of computational grid-points: |
M = 400 |
| Time step : |
dt = 0.2 seconds |
| Length of an incident wave (expression 3): |
B = 0.05 |
| Initial wave height: |
h = 2 meters |
The wave was generated at the distance of 4 kilometers off the shore. The bottom profile was as following: from the water-edge point the bottom has the constant inclination angle (tg α=0.1) and increase linearly till the depth 100 m. Then, the depth is not varying until the left boundary. Table 1 shows the run-up heights of generated waves at different beach slopes:
Table 1. Calculated wave run-up height according to the beach slope angle.
| Beach slope angle tg(α) |
0.1 |
0.2 |
0.3 |
0.4 |
0.5 |
0.6 |
0.7 |
0.8 |
0.9 |
1 |
| Run-up height |
19.5 |
19.3 |
18.9 |
18.5 |
18.1 |
17.8 |
17.5 |
17.3 |
17.1 |
16.9 |
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During the way to the coast the wave height increases up to 8 meters. From
the table it is visible, that increasing of declination of a shore causes decreasing of the wave run-up
height, which corresponds to the results obtained earlier with the help of analytical ( Pelinovsky,
1982, 1985) and numerical methods ( Marchuk, 1982).
In Figure 3 the wave profile at the moment of maximum run-up is shown.
Figure 3. The moment of highest run-up of a wave to the uniform slope
Now we shall consider tsunami wave run-up on a shore with a more complicated profile. The second series of numerical experiments is carried out in order to investigate correlations between wave run-up height and shore profiles of submerging area. In this case the submerging area of the shore (up to 25 meters above mean sea level) has the profile, which is defined by the following expression:
Here the distance r measured from the initial water-edge point to the right, and r0 - is the width of a shore area with varied declination (in this series of calculations it was equal to 250 meters). All remaining parameters were the same, as in the previous case. In this series of computations the wave height near the shore was equal to 7.6 m. As the result of computations with the positive values of parameter C the following table was obtained:
Table 2. Run-up heights for positive curvature parameter values
| Curvature parameter C |
0.0 |
1.0 |
2.0 |
3.0 |
4.0 |
5.0 |
6.0 |
7.0 |
| Run-up height |
23.067 |
23.086 |
23.132 |
23.73 |
23.86 |
23.54 |
23.68 |
23.63 |
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The shore and the water surface profile when the curvature parameter value is equal to 4 are shown in the Figure 4.
Figure 4. Wave run-up on a shore with the positive curvature parameter (C
= 4)
Results of computations with the negative values of the curvature parameter C are shown in the Table 3.
Table 3. Run-up heights for negative curvature parameter values
| Curvature parameter C |
-1.0 |
-2.0 |
-3.0 |
-4.0 |
-5.0 |
-6.0 |
| Run-up height |
22.58 |
22.82 |
22.40 |
22.37 |
21.88 |
20.86 |
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The shore and the water surface profile when the curvature parameter value is equal to -6, are shown in Figure 5.
Figure 5. Wave run-up on a shore with the negative curvature parameter (C
= -6)
From Tables 2 and 3 it is seen that the maximum run-up height of the wave with
given initial parameters is observed when parameter C is equal to +4.
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