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and
k
9
2
D
c
3
B
3
(
k
)
2
D
E
3
(
k
)
.
(13)
4
4
3
c
For a constant value of
Į
4
, as used in the original VIERS-1, this energy
balance does not lead to the desired flat
B
(
k
). Since
Į
4
is a parameter, one
can try to solve for an
Į
4
(
k
) that leads to a flat
B
(
k
); while it is possible to
find such an
Į
4
(
k
), its shape does not seem to be realistic.
In order to force the energy balance to result in a flat curvature spec-
trum, the dissipation term
-ȕkE
is replaced by a different one,
S
sd
, the
shape of which is chosen exactly so that it balances the energy input terms
if
B
(
k
) is flat. Although this is somewhat artificial, the same argument has
been used by Phillips (1985) in his attempt to obtain a
k
0.5
curvature spec-
trum. Reverting to a constant
Į
4
, we obtain (in the saturation range):
(
E
)
u
7
k
2
3
S
(
E
)
GZ
(
*
)
E
3
D
E
S
(
E
)
0
(14)
4
sd
c
c
3
with:
u
7
E
k
E
S
(
E
)
GZ
(
*
)
2
E
(
)
p
3
D
E
3
(
)
p
2
(15)
1
2
sd
4
c
E
c
3
E
sat
sat
In
this
'artificial'
dissipation
term,
the
extra
factors
(
E
/
E
)
p
(
B
/(
D
/
2
))
p
provide a generalisation; but for any
p
, clearly,
sat
Ph
when
E
=
E
sat
,
S
(
E
) = 0.
Justification for a dissipation term of this shape may be sought by noting
that, for
p
1
=
p
2
= 0, these terms conform to the general shape proposed by
Komen et al. (1984). This general shape (their equation 2.7) that has pa-
rameters
n
and
m
can be rewritten into:
&
&
0
.
24
S
(
k
)
c
(
3
1
n
(
)
m
F
n
m
1
u
n
1
g
1
n
/
2
k
n
/
2
E
(
k
)
.
(16)
dis
*
0
0081
Here, Komen et al.'s integral spectral parameters ˆ and Z have been
approximated by their Pierson-Moskowitz values of
ˆ
D
0
56
D
and
PM
Ph
Z
.
For
p
1
=
p
2
= 0, the dissipation term (Eq. 15) can be written as
1
3
Z
PM
p
