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effect, however, has been overlooked by the wave-forecast modelling community and it is
only very recently that the cumulative integral was explicitly suggested as an additional
breaking-dissipation term for spectral models (
Donelan
,
2001
;
Babanin & Young
,
2005
;
Young & Babanin
,
2006a
).
Formally speaking, this single expression for the cumulative term is also an oversimpli-
fication. Indeed, a number of different physics can lead to the cumulative dissipation, such
as the straining action of long waves on shorter waves (e.g.
Longuet-Higgins & Stewart
,
1960
;
Longuet-Higgins
,
1987
;
Donelan
,
2001
;
Donelan
et al.
,
2010
), the induced breaking
of shorter waves caused by larger breakers (e.g.
Manasseh
et al.
,
2006
;
Young & Babanin
,
2006a
) and the turbulent-viscosity damping of short waves in the wake of large break-
ing (e.g.
Banner
et al.
,
1989
). These mechanisms can be concurrent and tied together, or
can be disconnected as, for example, in the case of mature dominant waves which do not
break themselves, but still cause the strain and breaking of smaller-scale waves (see e.g.
Tsagareli
,
2009
;
Babanin
et al.
,
2011a
). Therefore, following the logic of
Babanin & van
der Westhuysen
(
2008
), the cumulative term itself may eventually need to be subdivided
into sub-terms
S
cum
(
f
)
=
S
straining
(
f
)
+
S
induced
breaking
(
f
)
+
S
turbulent
wake
+···
,
(7.27)
depending on the relative significance of the sub-terms and importance of consequences of
the separation of the cumulative effects in the physical space if that happens.
7.3.5 Whitecapping dissipation at extreme wind forcing
Many processes in the small-scale air-sea interactions run differently at extreme-wind con-
ditions. As a reference point for the extreme winds, the hurricane-scale classification can
perhaps be used: that is a tropical storm becomes a hurricane if the wind speed reaches
U
∼
33m
/
s
(
119 km
/
h
).
(7.28)
It is interesting to note that this is approximately the wind speed
U
10
at which the drag
coefficient
C
D
(3.8)
was found to saturate in the field observations by
Powell
et al.
(
2003
)
and
Jarosz
et al.
(
2007
) (see also
Section 9.1.3
).
A further number of experimental and theoretical studies followed and investigated this
conjecture (
Andreas
,
2004
;
Donelan
et al.
,
2004
,
2006
;
Barenblatt
et al.
,
2005
;
Makin
,
2005
;
Bye & Jenkins
,
2006
;
Kudryavtsev
,
2006
;
Kudryavtsev & Makin
,
2007
,
2009
,
2010
;
Black
et al.
,
2007
;
Stiassnie
et al.
,
2007
;
Vakhguelt
,
2007
;
Troitskaya & Rybushkina
,
2008
;
Rastigejev
et al.
,
2011
;
Soloviev & Lukas
,
2010
, among others). A detailed review of the
peculiarities of air-sea interactions in such conditions is far beyond the scope of the present
topic, but for a shorter review see
Section 9.1
. What should be emphasised here is that the
observed magnitude change of the sea drag and particularly the apparent change of the
C
D
-growth trend signifies a change of the surface roughness (see
Section 3.1
). And this
issue is certainly relevant for the wave-breaking and wave-dissipation topics in this topic.
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