Geoscience Reference
In-Depth Information
f
−
4
tail is also a reality, both measured and predicted
theoretically (see
Sections 5.3.2
and
8.2
for this discussion). There are well-justified param-
eterisations of the wave spectrum based on such behaviour of the small spectral scales
(
Donelan
et al.
,
1985
). As concluded in
Tsagareli
et al.
(
2010
) and
Babanin
et al.
(
2010c
),
both subintervals may in fact exist in real spectra, and in the case of their co-existence the
f
−
5
tail is situated at higher frequencies which then lets the integral in
(7.37)
converge.
Co-existence of the two subintervals is also supported by measurements (e.g.
Forristall
,
1981
;
Evans & Kibblewhite
,
1990
;
Kahma & Calkoen
,
1992
;
Babanin & Soloviev
,
1998b
;
Resio
et al.
,
2004
).
In order to accommodate both types of tail behaviour for calibration purposes of the
source functions,
Tsagareli
(
2009
) and
Babanin
et al.
(
2010c
) suggested a combination of
the JONSWAP spectral parameterisation form
(2.7)
with that of
Donelan
et al.
(
1985
), so
that both subintervals of
f
−
4
and
f
−
5
are present in the equilibrium interval:
On the other hand, the
S
(
f
)
∼
⎧
⎨
f
−
4
exp
exp
−
(
f
−
f
p
)
2
2
5
4
f
f
−
4
2
f
p
g
2
π)
−
4
f
−
1
p
σ
α
(
2
−
·
γ
f
≤
f
t
,
p
F
(
f
)
=
(7.46)
f
−
5
exp
exp
−
(
f
−
f
p
)
2
2
σ
⎩
5
4
f
f
−
4
2
f
p
g
2
π)
−
4
f
t
f
−
1
α
(
2
−
·
γ
f
>
f
t
p
p
where
f
t
is the transition frequency. Similar spectral shapes, which can accommodate
either type of spectral tail, had been suggested before (e.g.
Young & Verhagen
,
1996
).
Here, they were unified and, following
Tsagareli
(
2009
), this combined spectral-shape
parameterisation is called the Combi spectrum.
The transition is typically located at
f
3
f
p
(5.42)
and has great importance for cal-
ibration of the dissipation function. As discussed in
Section 5.3.2
, it signifies transition
from the inherent-breaking spectral region to the part of the spectrum which is dominated
by the cumulative dissipation.
As far as the separate calibration of the dissipation term
S
ds
is concerned, the main
constraint states that the dissipation-function integral must not exceed the total wind input:
∼
R
D
f
∞
f
∞
S
ds
(
f
)
df
=
S
in
(
f
)
df
(7.47)
0
0
where ratio
R
D
≤
1. The ratio of the two integrals as a function of the wave development
stage
U
10
/
c
p
is known experimentally (e.g.
Donelan
,
1998
), and therefore the dissipation
term can also be studied and tuned individually.
According to
Donelan
(
1998
), this ratio
R
D
stays within the range of 95-100% for most
stages of wave development, reaching 100% at the Pierson-Moscowitz full-development
limit (
Pierson & Moskowitz
,
1964
). It is only at the very early stages that the total wind
input can be significantly larger than the total dissipation.
Search WWH ::
Custom Search