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In both the left and right panels, wave profiles have evolved significantly from the
initial harmonic wave shown by the dashed line. In spite of the fact that the evolution
occurred under very different wind-forcing conditions, and took very different times to
reach the breaking point, the magnitudes of the asymmetry (
A
s
=−
0
.
42), skewness (
S
k
=
0
36), as well as the profiles of the three
waves, are virtually identical. This highlights again the important role of the hydrody-
namic mechanism in redistributing the wave energy and forming the nonlinear wave pro-
file, whereas the wind here appears to serve merely as the source of energy to the wave
system.
The statistical properties obtained by means of the CS model for such incipient breaking
are shown in
Figure 4.9
. Note, again, that steepness/skewness and asymmetry are measured
at the different phases of the last prior-to-breaking oscillation. The estimates shown in
Figure 4.9
were obtained from a comprehensive set of numerical runs covering a range of
initial steepness IMS
.
84
,
0
.
82
,
0
.
83) and steepness (
=
0
.
34
,
0
.
36
,
0
.
=
0
.
10-0
.
30 and wind forcing
U
/
c
=
0-11. The steepness is shown
in terms of
kH
.
Since we have identified the steepness as a possible reason for wave collapse, it is most
instructive to investigate the limiting values of
kH
. The data points cluster (top subplots),
=
2
1
0
0.8
−0.2
0.6
−0.4
0.4
−0.6
0.2
−0.8
0
−1
0
0.5
1
0
0.5
1
kH
kH
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.5
1
skewness
Figure 4.9 Numerically simulated incipient breaking. (top left) skewness versus steepness; (top
right) asymmetry versus steepness; (bottom) asymmetry versus skewness
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