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rather than being used as
)
=
F
(
f
)
S
ds
(
f
(7.22)
t
following its formulation in
(2.61)
. This is done to match the dimensions of dissipations
D
a
and
f
S
ds
(
df
.
Based on the data shown in
Figure 7.5
and using
(5.67)
, under the incipient dominant
breakers the dissipation rate per unit of area is
D
a
i
f
)
0003
m
3
s
3
, and under the broken
=−
0
.
/
s
3
. The latter, of course, should not be zero in
the absence of dominant breaking because dissipation due to breaking of waves of all other
scales persists. The difference
0002
m
3
=−
.
/
dominant waves it is
D
a
p
0
0001
m
3
s
3
D
a
i
−
D
a
p
=−
0
.
/
(7.23)
matches remarkably well the integral
(7.20)
:
F
00011
m
3
s
3
g
·
t
df
=−
0
.
/
.
(7.24)
f
Although this is a pleasing agreement, it should be remembered that both approaches yield
only an approximate estimate of the dissipation.
As has been mentioned above, the spectral difference in
Figure 7.3
is a lower-bound
estimate of dissipation due to the breaking, because some energy must have already been
lost before the spectra were measured. The difference
D
a
i
D
a
p
of the total dissipation
rates of kinetic energy is also likely to be a lower-bound estimate of the loss of energy from
the wave field due to breaking. As has also already been mentioned, some of the energy is
expended on work against buoyancy forces whilst entraining bubbles into the water, rather
than on generating the turbulence. The fraction of the wave energy dissipated, which is
expended on entraining the air, can be from 14% to 50% of the total (
Lammarre &Melville
,
1991
,
1992
;
Melville
et al.
,
1992
;
Blenkinsopp & Chaplin
,
2007
) and this defines the limits
of accuracy of the segmenting method. For the laboratory measurements of
Lammarre &
Melville
(
1991
,
1992
) and
Melville
et al.
(
1992
), some of the breaking was of a plunging
type. In contrast, most of the Lake George breaking was of the spilling type, which spends
less energy on entrainment. Therefore, we would expect that in our case the fraction of
energy expended this way is at the lower bound of 14% (
Blenkinsopp & Chaplin
,
2007
)or
similar. Nevertheless, the above comparison shows that the proposed method of estimation
of wave-breaking dissipation by considering the difference between the
incipient
and
post-
breaking
spectra provides a reasonable magnitude of the total dissipation.
Another interesting observation can also be made on the basis of
Figure 7.5
. Here, one
can see that the orbital velocities of the dominant waves did not change between the break-
ing and non-breaking segments. This implies that the energy, which is obviously gone from
the power spectra, must have been the potential energy, rather than kinetic, at least for the
−
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