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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
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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