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continues to decrease gradually, but unlike
a
1
does not become zero. Even at these stages of
development, the induced dissipation persists, even though it slows down as the dominant
waves become less steep. This is due to, for example, stretching and compressing of short
waves by underlying longer waves, thus causing these shorter waves to break (e.g.
Longuet-
Higgins & Stewart
,
1960
;
Phillips
,
1963
;
Longuet-Higgins
,
1987
;
Donelan
,
2001
;
Donelan
et al.
,
2010
).
Figure 7.13
also compares the coefficient
a
1
obtained by means of the constraint
(7.47)
and experimental data (bold line). The only experimental estimate of
a
1
=
0065 is avail-
able from
Young & Babanin
(
2006a
) based on their analysis of a single wave record when
wind forcing was quite extreme:
U
10
/
0
.
c
p
∼
6
.
5. In
Figure 7.13
,thevalueof
a
exp
=
0
.
0065
is achieved at
U
10
/
6, and at higher values of wind forcing the magnitude is
somewhat greater. Qualitatively, this is consistent with the experiment.
Young & Babanin
(
2006a
) stressed that their estimate is a lower-bound approximation of the actual value
since they measured the dissipation by comparing the difference in energy of wave trains
which were already breaking to wave trains which had completed breaking and were once
again gaining energy from the wind. By definition, this approach will underestimate the
energy loss. In this regard, the quantitative agreement between the calibrated values of
a
1
and the measurement is encouraging.
Figure 7.14
shows the dissipation source function
S
ds
(
c
p
=
2
.
(5.40)
based on the coeffi-
cients
a
1
and
a
2
, computed for the Combi spectrum
(7.46)
with
U
10
/
f
)
c
p
=
2
.
7 and wind
speed
U
10
=
10m
/
s. In the figure, the corresponding wind energy-input source function
is also plotted, computed for the same wind-forcing conditions as described above
in this section (see also
Tsagareli
et al.
,
2010
, for more detail). The integrals of the two
source functions are consistent according to the physical constraint
(7.47)
, but the shape of
the dissipation function at high frequencies raises further questions about the calibration
of
S
ds
(
S
in
(
f
)
)
.
The most striking feature of the comparison of
S
ds
(
f
)
and
S
in
(
)
f
f
is the difference at
high frequencies, where the cumulative term
T
2
(
dominates, by up to two orders of
magnitude. Mathematically, this feature is apparent if coefficients
a
1
and
a
2
in
(5.40)
are
frequency independent, i.e. only vary as a function of wave age, as shown in
Figure 7.13
.
Physically, however, such a difference is difficult to justify because, in order to maintain
the high dissipation rates, a very strong energy flux from the lower-frequency part of the
spectrum by means of, for example, the nonlinear interaction term
S
nl
in
(2.61)
would be
necessary.
This raises a question as to whether the coefficient
a
2
is scale independent. It is likely
that a frequency-dependent correction is required to ensure that the magnitude of the wave
dissipation
S
ds
(
f
)
.
To define the frequency-dependent form for coefficient
a
2
, it was decided to apply a
dimensionless correction function
Z
f
)
remains comparable with the wind-input function
S
in
(
f
)
. Note that
this correction is only approximate as the nonlinear transfer
S
in
should also play some role
(e.g.
Young & van Vledder
,
1993
). It should be pointed out that such a correction does not
affect the principal physical constraint
(7.47)
as it is applied in the frequency region where
(
f
)
such that as a result
S
ds
(
f
)
∼
S
in
(
f
)
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