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Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water
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10.1063/1.4847035
/content/aip/journal/pof2/25/12/10.1063/1.4847035
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/12/10.1063/1.4847035
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(a) and (b) Bottom profiles and computational domains employed in the numerical simulations. The region ∈ [ , ] is the region actually used for calculating statistical information, while the left and right shaded regions are excluded due to startup effects and reflections from the boundary, respectively. The solid bottom profiles are piecewise linear and the dashed bottom profile has cosine shape.

Image of FIG. 2.
FIG. 2.

Left: Typical convergence of the ensemble-averaged kurtosis and freak wave probability with respect to the number of runs in the ensemble. Right: The error, defined by (5) , in the numerical solution as function of number of grid points per peak wavelength λ.

Image of FIG. 3.
FIG. 3.

Skewness and kurtosis for the experimental “case 1.” Crosses: Experimental results reported by Trulsen, Zeng, and Gramstad, 20 solid line: numerical simulations.

Image of FIG. 4.
FIG. 4.

Same as Figure 3 for the experimental “case 2.”

Image of FIG. 5.
FIG. 5.

Same as Figure 3 for the experimental “case 3.”

Image of FIG. 6.
FIG. 6.

Skewness, kurtosis, and probability of freak waves for the wave fields as functions of for the bathymetry in Figure 1(a) for slope length = 6 m. Three different depths of the shallow region: = 0.4 m (solid line), = 0.3 m (dashed line), and = 0.2 m (dotted line) are shown. The depth profile is shown in the right bottom pane.

Image of FIG. 7.
FIG. 7.

Same as Figure 6 , but for slope length = 18 m.

Image of FIG. 8.
FIG. 8.

Same as Figure 6 , but for slope length = 30 m.

Image of FIG. 9.
FIG. 9.

Probability distribution of the surface normalized by its standard deviation σ, at the location of maximum kurtosis (shallow end of the uphill slope) for the bathymetry in Figure 1(a) for = 18 m and (a) = 0.2 m, (b) = 0.3 m, and (c) = 0.4 m. Solid lines show the normal distribution.

Image of FIG. 10.
FIG. 10.

Wave spectra for four different positions for the cases with slope length = 18 m. Solid line: = 30 (before the first slope), dashed line: = 35 (after the first slope at the position of maximum kurtosis), dotted line: = 50 (some distance into the shallow region), dashed-dotted line: = 100 (some distance into the second shallow region).

Image of FIG. 11.
FIG. 11.

Skewness, kurtosis, and probability of freak waves for the wave fields as functions of for the piecewise linear bathymetry in Figure 1(b) for slope length = 18 m. Three different depths of the shallow region: = 0.4 m (solid line), = 0.3 m (dashed line), and = 0.2 m (dotted line) are shown. The depth profile is shown in the bottom right pane.

Image of FIG. 12.
FIG. 12.

Same as Figure 11 , but for the cosine-shaped bathymetry in Figure 1(b) .

Image of FIG. 13.
FIG. 13.

Wave spectra for four different positions for the cases with slope length = 18 m and cosine shaped bathymetry in Figure 1(b) . Solid line: = 30 (before the first slope), dashed line: = 35 (after the first slope at the position of maximum kurtosis), dotted line: = 50 (some distance into the second deeper region).

Image of FIG. 14.
FIG. 14.

Statistical properties for three different values of the significant wave-height of the incoming waves: = 0.04 m (solid line), = 0.06 m (dashed line), and = 0.08 m (dotted line).

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/content/aip/journal/pof2/25/12/10.1063/1.4847035
2013-12-17
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/12/10.1063/1.4847035
10.1063/1.4847035
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