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dissipation as a function of wave spectrum, i.e. an expression in the standard form
(7.1)
employed in relevant applications of this function, or some other formulation. Therefore,
the authors tested a number of empirical parameterisations in the search for a dissipation
term capable of accommodating the observed difference.
They demonstrated that standard dissipation terms used in wave-forecast models, i.e.
WAM (
Gunther
et al.
,
1992
;
Komen
et al.
,
1994
), essentially overestimate the rates of
spectral energy loss in wave fields with moderate background steepness. In order to achieve
a reasonable quantitative agreement between the kinetic and dynamic approaches, a power
function as strong as
F
12
had to be used. The authors concluded that this result supports
the experimental observations of the threshold behaviour of wave breaking by
Banner
et al.
(
2000
) and
Babanin
et al.
(
2001
).
We fully agree with this interpretation. The new method appears to be a powerful ana-
lytical means for fundamental investigations of the spectral wave dissipation, capable of
investigating and verifying both qualitative and even quantitative properties of the dissi-
pation term. For the last part, however, it remains dependent on the choice of empirical
function to be substituted as this term, and this choice needs to be made carefully.
Indeed,
S
ds
(
(
f
)
12
seems unrealistic. Such a function may help to eliminate the
dissipation below the threshold, where the whitecapping contribution is in fact expected to
be zero or negligible, but it will provide extremely strong dissipation rates above the thresh-
old where no other empirical or theoretical approach has suggested anything like
n
f
)
∼
F
(
f
)
5. In
our view, a dissipation function which directly incorporates the threshold behaviour, such
as that in
(5.40)
, can be tested in this regard. It would be interesting to see whether the
'moderate steepness' of
Zakharov
et al.
(
2007
) corresponds to the experimental threshold
of
Babanin
et al.
(
2007d
,
2010c
) described in
Section 5.3.2
.
>
7.2 Simulating the wave dissipation in phase-resolvent models
From the point of view of wave-energy dissipation, the phase-resolvent models can be
subdivided into two large groups. The first group, the models that simulate the water
and air sides of the wavy surface separately, such as the study of water-wave evolution
by
Zakharov
et al.
(
2007
) described in
Section 7.1.2
above, or an air-sea model which
includes the full set of wind-wave interactions (
Chalikov & Rainchik
,
2011
)-haveto
incorporate the dissipation implicitly. In order to be able to describe the breaking dissi-
pation explicitly, the second group have to be two-phase models, such that they allow for
the dynamics of air bubbles injected into the water side and of water droplets ejected into
the air side. A significant number of such models are available now (
Abadie
et al.
,
1998
;
Zhao & Tanimoto
,
1998
;
Chen
et al.
,
1999
;
Watanabe & Saeki
,
1999
;
Mutsuda & Yasuda
,
2000
;
Christensen & Deigaard
,
2001
;
Grilli
et al.
,
2001
;
Guignard
et al.
,
2001
;
Tulin &
Landrini
,
2001
;
Hieu
et al.
,
2004
;
Song & Sirviente
,
2004
;
Zhao
et al.
,
2004
;
Iafrati &
Campana
,
2005
;
Dalrymple & Rogers
,
2006
;
Lubin
et al.
,
2006
;
Liovic & Lakehal
,
2007
;
Iafrati
,
2009
;
Dao
et al.
,
2010
;
Janssen & Krafczyk
,
2010
;
Lakehal & Liovic
,
2011
, among
others).
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