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6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
U/c
Figure 6.5 Modulation depth
R
as a function of wind forcing
U
/
c
for wave trains with three
different steepnesses of
1
=
0
.
17
,
2
=
0
.
005 (asterisks),
1
=
0
.
19
,
2
=
0
.
005 (diamonds) and
1
=
0
.
21
,
2
=
0
.
006 (circles). Here,
1
is steepness of primary waves,
2
is sideband steepness,
U
is characteristic wind speed in the model. Figure is reproduced from
Galchenko
et al.
(
2010
)
©
American Meteorological Society. Reprinted with permission
/
plotted versus wind forcing
U
c
where
U
is the characteristic wind speed in the CS model
which was used for the numerical simulations (see
Section 4.1.1
). Three different combi-
nations of primary and sideband steepnesses are shown:
1
=
0
.
17
,
2
=
0
.
005 (asterisks),
1
=
006 (circles).
At the very-strong-wind end, in
Figure 6.5
there is a hint of the modulational depth
increasing again. This is consistent with laboratory observations of
Waseda & Tulin
(
1999
)
who found that the wind forcing decreases sideband growth if weak and stimulates their
growth when strong. A slower sideband growth would mean, over the same wave fetch, a
less-developed modulation and its depth, and vice versa.
In general,
Waseda & Tulin
(
1999
) found that the imposed wind can have two effects
on modulational instability: alteration of the instability growth and change of the natural
bandwidth in
(2.12)
. The latter also depends on the wind speed relative to the wave phase
speed, that is, the wave age in the general sense. This interesting observation may mean that
relative spectral-peak width cannot be constant, as it is in the JONSWAP spectrum
(2.7)
.
That is, if the peak width for the waves with a continuous spectrum is determined by this
instability, wave-age-dependent parameterisations of the spectral peak (e.g.
Donelan
et al.
,
1985
) are more realistic, but that would have to be verified qualitatively and quantitatively.
0
.
19
,
2
=
0
.
005 (diamonds) and
1
=
0
.
21
,
2
=
0
.
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