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10
0
10
−5
10
−10
10
−1
10
0
10
1
10
0
10
−5
10
−10
10
−1
10
0
10
1
1
0.5
0
10
−1
10
0
10
1
100
0
−100
10
−1
10
0
10
1
frequency
Figure 4.4 Numerical simulations (see
Figure 4.2
). Dimensionless wave period is 1. Co-spectra of
running skewness and asymmetry for waves of IMS
=
0
.
26
,
U
/
c
=
2
.
5. (top panel) Skewness
spectrum. (second top panel) Asymmetry spectrum. (second bottom panel) Coherence spectrum.
(bottom panel) Phase spectrum (in degrees), positive phase means asymmetry is leading. Dashed
line shows 90
◦
phase shift
breaker: he sees a very tall crest which begins to break with the water mass falling down
from the top and at the same time the front face is growing steeper as the wave is leaning
forward - obviously a very dangerous situation.
To briefly summarise the intermediate conclusions: we can speculate that a two-dimen-
sional nonlinear wave will break when, due to inherent modulations of its height, it reaches
some limiting steepness. The skewness and asymmetry also oscillate, in phase and in
quadrature with the steepness, respectively. In the simulations, however, they do not appear
to exhibit some specific limiting value at the point of breaking.
We shall now conduct a similar set of numerical simulations for a wave with initially
half the steepness IMS
=
0
.
13 (
Figures 4.5
and
4.6
). A very strong forcing of
U
/
c
=
10
.
0
is applied in order to achieve breaking in a reasonably short period of time.
The three panels in
Figure 4.5
show the time evolution of individual wave steepness
(top), skewness (middle) and asymmetry (bottom) as in
Figure 4.2
. The wave steepness in
the top subplot grows under the
U
0 wind forcing so rapidly that its oscillations are
barely visible over the strong mean trend. The value of steepness
/
c
=
10
.
=
0
.
33 at the point of
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