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although the scatter is noticeable:
kH
85 with some outliers reaching values
above 0.9. The mean value of the limiting steepness is
=
0
.
75-0
.
4, which is very high, par-
ticularly considering that the value of the local slope near the crest is even higher than
this mean value.
Dyachenko & Zakharov
(
2005
) specifically investigated the shape of the
incipient breaker with a high-resolution version of a fully nonlinear model similar to that of
CS (
Dyachenko
et al.
,
1996
). They found a remarkable agreement of the limiting steepness
with the Stokes limiting-steepness criterion
(2.47)
. The shape of the wave, however, is dif-
ferent to that of the Stokes wave with the crest pointed at a 120
◦
angle (
Stokes
,
1880
). What
is most encouraging is that such an outcome of numerical simulations finds full experimen-
tal support (see
Section 5.1
). At such magnitudes of water surface slope, the surface may
simply collapse because of gravity, depending on the velocity field in the water. If so, the
role of the instability which led to the occurrence of these waves is not in generating wave
breaking as such, but rather in simply producing a very steep wave.
As indicated previously, skewness and asymmetry are not breaking criteria, but they
do have limiting values (see
Figure 4.2
). In
Figure 4.9
, the incipient-breaking skewness
is scattered in the range of
S
k
∼
0
.
=
0
.
7-1 (top left) and the asymmetry in the range from
5 (bottom). Within the scatter, there is no dependence of one property
on the other, except a possible negative correlation between the skewness and asymmetry
in the bottom panel. The latter result is supported by
Figure 4.2
, but is not necessarily a
feature of approaching breaking onset: a larger negative asymmetry is likely to be followed
by a greater skewness.
A
s
=−
0
.
35 to
−
0
.
4.1.3 Influence of wind and initial steepness
The role of the wind in wave breaking has already been mentioned several times throughout
the topic. It is apparently very important in growing the wave steepness (i.e.
Figures 4.2
,
4.5
,
4.7
). Once a wave is becoming steeper, the instability mechanism is leading it to
breaking sooner. If waves are initially below the threshold
10, they will not break
at all, despite modulations, unless wind forcing raises the steepness above the threshold.
In this regard, it has to be stressed that the threshold behaviour is also the most prominent
feature of breaking due to modulational instability in mechanically generated waves in two-
dimensional flumes (
Babanin
et al.
,
2007a
,
2010a
) and in directional (three-dimensional)
wind-forced field waves (
Banner
et al.
,
2000
;
Babanin
et al.
,
2001
), which fact most likely
links all the observations to the single physical course, the modulational instability (see
Chapter 5
).
In this section, we will discuss the capacity of the wind to instantaneously affect breaking
onset. That is: can the wind push a steep wave over and thus reduce the limiting steepness at
breaking? Because of the very large density difference between the water and the air, such
a possibility seems low. This conjecture is supported by
Figures 4.2
,
4.5
and
4.8
.Thelarge
scatter of the limiting-steepness values in
Figure 4.9
above, however, needs investigation.
In
Figure 4.10
, the nonlinear features of the incipient breaker are shown as a function
of IMS for a variety of wind-forcing conditions. Note that the simulation was run within a
=
0
.
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