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τ
ν
=
ρ
a
C
V
U
10
, where
C
V
is the viscous drag
coefficient, along with
(3.7)
into
(7.37)
yields:
Banner & Peirson
(
1998
). Substituting their
τ
wave
=
ρ
a
U
10
(
C
D
−
C
V
).
(7.41)
Banner & Peirson
(
1998
) demonstrated a qualitative trend of the viscous stress as a
function of the wind speed, but did not present a quantitative dependence.
Tsagareli
et al.
(
2010
) digitised the data of
Banner & Peirson
(
1998
) and parameterised the viscous drag
as a function of wind speed
U
10
:
10
−
5
U
10
+
10
−
3
C
V
=−
5
·
1
.
1
·
.
(7.42)
Combined with a known dependence for sea drag
C
D
, this now provides an integral con-
straint which the wind-input spectral distribution
S
in
(
must satisfy.
A variety of parameterisations exist to choose from for the sea drag. Most common
are dependences for the drag coefficient
C
D
as a function of wind speed
U
10
;somealso
introduce the wind-forcing parameter
U
10
/
f
)
c
p
(see e.g.
Guan & Xie
,
2004
;
Babanin &
Makin
,
2008
,forareview).
Babanin &Makin
(
2008
) also showed that there are more than a
dozen other properties that the drag can depend on. As a result, the scatter of measurements
with respect to any of the
C
D
parameterisations is large, of the order of tens of percent,
but as
Tsagareli
et al.
(
2010
) demonstrated errors due to absence of a normalisation of the
wind input by the sea drag can be of the order of hundreds of percent or even more.
Specifically,
Tsagareli
et al.
(
2010
)used
C
D
parameterisation by
Guan & Xie
(
2004
),
which is an attempt to unify 25 relevant experimental dependences obtained by different
researchers in the period from 1958 to 2003. It is a
C
D
-versus-
U
10
expression which also
accommodates wave-age:
10
−
3
C
D
=
(
0
.
78
+
0
.
475
f
(δ)
U
10
)
·
(7.43)
where
85
B
A
1
/
2
δ
−
B
f
(δ)
=
0
.
.
(7.44)
The wave age dependence is included through the deep-water wave steepness
2
g
.
H
s
ω
δ
=
H
s
k
p
=
(7.45)
The empirical parameters
A
7 are chosen such that
C
D
in
(7.43)
is in
agreement with the latest results on this topic by
Drennan
et al.
(
2003
).
The computational range of operational spectral wave models is usually limited by a
relatively low-frequency upper cutoff in the vicinity of the spectral peak, and therefore in
order to verify the constraint
(7.37)
in operational models a parametric tail should be added
for the integration.
Tsagareli
et al.
(
2010
) argue that such a tail has to be
S
=
1
.
7 and
B
=−
1
.
f
−
5
,as
in JONSWAP parameterisation
(2.7)
. A similar parametric tail was also argued for before
by the authors of the WAM model (
Komen
et al.
,
1994
).
(
f
)
∼
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