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are shown
in
Figure 5.27
, bottom panel. At the spectral peak, these normalised breaking rates merge
together very clearly, but they stay separated both above and below the peak. Thus, if there
is a linear or quasi-linear dependence of
b
T
on
P
Breaking rates
b
T
(
f
)
normalised by their respective spectral densities
P
(
f
)
, it would only be applicable at the
spectral peak. Away from the peak, other influences make the dependence of
b
T
on
P
(
f
)
(
f
)
nonlinear or affect this dependence in another way.
This uncertainty was investigated, based on the Lake George measurements, by
Babanin & Young
(
2005
) and
Babanin
et al.
(
2007c
). They attempted to draw an analogy
with the findings of
Banner
et al.
(
2000
) and
Babanin
et al.
(
2001
) for the spectral peak and
sought a dependence of the breaking probability at different frequencies as a function of
the so-called saturation density at that frequency, a spectral analogue of the squared wave
steepness introduced by
Phillips
(
1958
,
1984
):
4
f
5
F
)
=
(
2
π)
(
f
)
σ
Phillips
(
f
.
(5.31)
2
g
2
In
Babanin & Young
(
2005
) and
Babanin
et al.
(
2007c
), the saturation
σ(
f
)
was nor-
malised by a directional spreading parameter:
σ(
f
)
=
σ
Phillips
(
f
)
A
(
f
).
(5.32)
Here,
A
is the integral characteristic of the inverse directional spectral width introduced
by
Babanin & Soloviev
(
1987
,
1998b
):
(
f
)
π
)
−
1
A
(
f
=
K
(
f
,θ)
d
θ
(5.33)
−
π
where
θ
is the wave direction, and
K
(
f
,θ)
is the normalised-by-its-maximum-value direc-
tional spectrum:
K
(
f
,θ
maximum
)
=
1
.
(5.34)
Normalisation by directional spreading was brought in by
Banner
et al.
(
2002
), who
also investigated breaking probability across the frequency as a function of the saturation
spectrum.
Banner
et al.
(
2002
) needed an additional normalisation in order to explain why
the wave-breaking threshold that they observed is not universal in terms of the saturation-
density values. As will be shown below, the results of
Banner
et al.
(
2002
)musthave
been influenced by the cumulative effect and the directional normalisation is in fact not
necessary. In
Babanin & Young
(
2005
) and
Babanin
et al.
(
2007c
), values of
A
were
A
(
f
)
≈
1
,
(5.35)
and therefore the normalisation did not impact the value of the universal threshold signifi-
cantly. When revisited, the threshold saturation level
(5.36)
did not change.
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