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In any case, however, the breaking strength exhibited by an individual wave in the Black
Sea is more like
s
=
99%
(2.31)
rather than 50% in the laboratory experiment
(2.27)
. While seemingly extreme, result
(2.31)
, however, is not unprecedented: for example, in his investigations
Bonmarin
(
1989
)
gave examples of almost complete dissipation of potential energy of plunging breakers, “as
if a wave crest had never existed at this location”.
This discussion brings us to the second reason why the
(2.27)
-like constant-fraction
estimate of severity coefficient
s
for a single breaking event should not be general. While
in the laboratory wave energy
(2.24)
can be treated in terms of a single wave which is
breaking, in the field the change of wave height of an individual wave is an ambiguous
property. As was mentioned in
Section 2.6
, the dominant waves exist as wave groups, and
individual waves propagate through the group at a relative speed
2
c
and correspondingly
change their height as they propagate.
Dominant waves usually break close to the peak of the group envelope (e.g.
Donelan
et al.
,
1972
;
Holthuijsen &Herbers
,
1986
;
Babanin
,
1995
). This means that once a wave has
started breaking at the top of a group, its height is decreasing due to both the breaking and
its moving towards the front face of the group. The latter reduction would occur regardless
of whether the wave was breaking or not.
Since the active phase of wave breaking lasts a fraction of a wave period (e.g.
Rapp &
Melville
,
1990
), and the wave group typically consists of 7-10waves (e.g.
Longuet-Higgins
,
1984
), then a wave propagating at relative speed
2
c
will start and finish breaking within
the group. It is logical and more accurate, therefore, to estimate the breaking strength in
terms of group breaking severity
s
g
, by integrating the wave energy loss over the group:
λ
g
8
λ
g
2
dx
E
sg
=
s
g
η(
x
)
.
(2.32)
0
Here,
E
sg
is the energy loss from the entire group,
λ
g
is the length of the group, and
the surface elevations
are integrated along a distance
x
within the group before the
breaking started. The factor of 8 is included in order to have wave energy expressed in terms
of wave height
H
, rather than amplitude
a
, for consistency with definitions
(2.23)
-
(2.30)
.
Note that there is no frequency scale
f
explicitly mentioned here, because the apparent
wave grouping is a property of dominant waves only. Therefore, in the spectral sense,
definition
(2.32)
is applicable to the frequency band of the spectral peak which according
to
(2.5)
-
(2.8)
and
(2.11)
is in the range
η(
x
)
f
p
=±
(
.
−
.
)
f
p
.
With such a definition, the whitecapping dissipation function should be estimated as
0
3
0
5
n
group
E
sg
t
S
ds
=
,
(2.33)
where
n
group
means the number of wave groups where breaking has occurred. Values of
severity
s
g
estimated this way will essentially be lower than the severity measured for
individual waves because the energy loss effect is smeared over all waves in the group.
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