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frequency the stronger will be the accumulating overall dissipation due to the lengthening
lower-frequency band of breaking waves.
Mathematically, the cumulative dependence of the spectral dissipation is described by
an integral of the wave spectrum rather than by an algebraic function of this spectrum. One
such formulation for the dissipation cumulative term, the original expression by
Young &
Babanin
(
2006a
), is given in
(5.40)
, and for the cumulative behaviour of the breaking
probability at higher frequencies in
(5.41)
. Graphically,
this effect
is illustrated in
Figure 5.30
.
It should be pointed out that the cumulative dissipation does not deny or cancel the
inherent dissipation, that is the dissipation due to non-forced breaking at scales higher
than the spectral peak. The inherent term is present in the
S
ds
(
formulation
(5.40)
, and
at each frequency it is a function of the local-in-frequency spectrum
F
f
)
as with any
other dissipation term. That is, the breaking dissipation should consist of two terms, the
inherent-dissipation term
S
inh
and the cumulative-dissipation term
S
cum
:
S
br
(
(
f
)
)
=
S
inh
(
)
+
S
cum
(
).
f
f
f
(7.25)
is the only breaking-dissipation mechanism at the peak and below the peak, but
for shorter waves of
f
S
inh
(
f
)
>
f
p
its role becomes progressively less important as the strength
of
S
cum
(
3
f
p
(see
Babanin & Young
,
2005
;
Babanin
et al.
,
2007c
, and discussions in
Section 5.3.2
).
The dissipation terms presently employed in operational wave-forecast models (see
Section 7.4
for a dedicated discussion of this topic), effectively only include the inherent
term
S
inh
(
f
)
accumulates, to the point of being negligible at some scales of
f
>
. Various weighting coefficients
have been proposed over the years to help the associated apparent dissipation bias across
the spectrum, but since the two terms are functionally different, no amount of tuning can
compensate for the missing physics in all circumstances.
It is relevant to mention here that in reality, the wave attenuation consists of very many
physical processes, independent or interdependent, and not all of them even relate to the
wave breaking. As
Babanin & van der Westhuysen
(
2008
) suggested, the overall dissipation
should be best described by a sum of those as
f
)
and disregard the cumulative term
S
cum
(
f
)
S
br
(
f
)
=
S
inh
(
f
)
+
S
cum
(
f
)
+
S
turb
+
S
visc
+
S
wind
+ยทยทยท
(7.26)
where
S
turb
is dissipation due to turbulent viscosity,
S
visc
is due to molecular viscosity,
S
wind
is due to interaction of waves with adverse winds etc. This way, each dissipation
term may have a different formulation as dictated by the relevant physics, and any of them
may turn to zero as necessary while the wave evolution and the wave energy dissipation
will still proceed.
Now, we should step back and point out that the induced breaking/dissipation is not a
newly discovered feature which still awaits further experimental and theoretical support
and confirmation. It has been observed and elaborated analytically for a substantial period
of time (e.g.
Longuet-Higgins & Stewart
,
1960
;
Phillips
,
1963
;
Longuet-Higgins
,
1987
;
Banner
et al.
,
1989
;
Donelan
,
2001
;
Melville
et al.
,
2002
;
Manasseh
et al.
,
2006
). This
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