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to that simulated: the duration of the evolution to breaking, when the wind forcing was
doubled from
U
/
c
=
2
.
5to
U
/
c
=
5
.
0, was reduced from 32 to 9 wave periods (almost
four times).
A further doubling of the wind input, as shown in the bottom set of three panels,
led to another reduction of the evolution duration - from 9 to 3 periods. This is again
consistent with
Donelan
et al.
(
2006
) who showed that at very strong winds the rela-
tive wave growth slows down. The other patterns of nonlinear wave evolution appear
unaltered. The whole picture again points to the critical local steepness as the parameter
responsible for the onset of the water-surface collapse. The maximum values of steep-
ness
83 are almost the same as previously. These values
also demonstrate that the instantaneous effect of the wind on breaking onset is negli-
gible. The wind forcing of
U
=
0
.
36 and skewness
S
k
=
0
.
0 is now very strong, but this wind is still not
capable of pushing the wave over and reducing, even marginally, the critical steepness
at breaking.
Let us summarise observations made with this instructive
Figure 4.2
. Values of steep-
ness, skewness and asymmetry oscillate at a frequency half that of the carrier wave. While
the simulation begins with both skewness and zero asymmetry (sinusoidal wave), the sinu-
soidal wave immediately turns into a Stokes wave. It is, however, only conditionally a
Stokes wave. The shape of this wave oscillates, and it is only at a particular phase of these
oscillations that the wave shape is clearly that of the Stokes wave again: at the point of max-
imal skewness, when the wave is symmetric. As the skewness is decreasing or growing,
the asymmetry is also changing, that is the wave is tilting forward (negative asymmetry)
or backward (positive asymmetry). Values of
S
k
and
A
s
oscillate between their maximum
and minimum levels, but remain bounded, their maximum and minimum do not increase
in magnitude if the initial steepness is already large enough. In contrast, the oscillations
in steepness progressively grow in amplitude until a point is reached where breaking takes
place. It is therefore evident from
Figure 4.2
that it is the steepness which is the limiting
parameter for breaking to occur.
The coherence and phase relationships of steepness, skewness and asymmetry, outlined
qualitatively above, are analysed in
Figures 4.3
and
4.4
.
Figure 4.3
compares spectra of
running instantaneous steepness
/
c
=
10
.
(
1.1
, top subplot) and skewness
S
k
(
1.2
, second subplot),
their coherence (third subplot) and phase (bottom subplot). Since the time scale of the
simulations is dimensionless (i.e. presented in terms of wave periods), the frequency scale
is expressed in inverse wave periods. Therefore, as expected from visual examination of
Figure 4.2
, the peak of the steepness/skewness modulation occurs at twice the wave period
(0.5 of the inverse wave period). This frequency dominates the spectrum, in agreement with
theory (
Longuet-Higgins & Cokelet
,
1978
). The spectral density decreases very rapidly
away from the peak.
The peak is rather broad and covers a range of frequencies of 0.4-0.6 of inverse wave
periods. The coherence of the steepness and skewness oscillations in this range is 100%, as
could have been expected for numerical simulations of the theory with a model of such high
precision. The phase shift between the dominant oscillations of steepness and skewness is
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