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and there is no standardisation amongst them. Besides, in terms of breaking rates some of
the statistics appear ambiguous. The same whitecap coverage, for example, can be achieved
at different frequencies of occurence of breaking events, depending on whether the events
(waves) are large or small and whether the severity is strong or weak.
Here, following
Babanin
(
1995
),
Banner
et al.
(
2000
) and
Babanin
et al.
(
2001
), we
define the breaking probability
b
T
for the dominant waves as the mean passage rate past
a fixed point of dominant wave-breaking events per dominant wave period
T
d
or, in other
words, the percentage of breaking crests
n
within a sequence of
N
wave crests:
nT
d
t
n
N
b
T
=
=
(2.3)
where
t
NT
d
is the duration of the wave record. Another definition of breaking probabil-
ity for dominant waves is also possible
(2.36)
, and is in fact of greater practical significance,
but it will be introduced and justified in
Section 2.7
because its meaning cannot be fully
appreciated before the issues of breaking strength are discussed.
The nondimensional quantity
b
T
is expressed in terms of the main temporal scale
T
d
of
the wave field, making it convenient for comparisons and analysis. The definition, how-
ever, is not as simple as it first seems. Because of the continuous-spectrum nature of
wind-generated waves, even the notion of dominant waves needs clarification. Otherwise,
any spectral distribution of breaking rates defined this way will be ambiguous, i.e.
b
T
(
=
f
)
dependence on frequency
f
:
n
(
f
)
T
n
(
f
)
b
T
(
f
)
=
=
)
.
(2.4)
t
N
(
f
Indeed, counting crests of waves for a specific period
T
=
1
/
f
makes no physical sense as
such a count will return a zero value.
Therefore, any spectral characteristic of breaking probability
b
T
(
, including breaking
of dominant waves with peak frequency
f
p
, from the very beginning implies the use of a
spectral band
f
f
)
f
such that wave crests with frequencies (periods) within this band,
i.e. individual waves with frequencies of
±
f
−
f
≤
f
≤
f
+
f
(2.5)
can be counted. In
Banner
et al.
(
2000
) and
Babanin
et al.
(
2001
), the dominant waves
were assumed to have frequencies within
±
30% of the vicinity of the spectral peak, i.e.
f
=
f
p
±
f
p
=
f
p
±
0
.
3
f
p
,
(2.6)
following the width of the peak enhancement region of the JONSWAP formulation for the
frequency spectrum
F
(
f
)
(
Hasselmann
et al.
,
1973
):
exp
π)
−
4
f
−
5
exp
−
4
f
f
p
2
(
f
−
f
p
)
5
4
−
2
f
p
g
2
2
σ
F
(
f
)
=
α
(
2
−
·
γ
.
(2.7)
Here,
α
is the level of equilibrium interval (tail) of the spectrum,
g
is the gravitational
constant,
γ
is the peak enhancement factor, and
σ
is the width of the spectral peak.
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