Environmental Engineering Reference
In-Depth Information
Box 5.2: Structure function
The structure function is defined in Fourier space as
ˆ
t
)
ˆ
S
(
k
,
t
)
=
φ
(
k
,
φ
(
−
k
,
t
)
,
(B5.2-1)
where
ˆ
t
).
The first-order temporal derivative of the structure function is
φ
(
−
k
,
t
) is the Fourier transform of
φ
(
r
,
∂
S
(
k
,
t
)
∂
∂
ˆ
t
)
ˆ
=
t
φ
(
k
,
φ
(
−
k
,
t
)
∂
t
t
)
t
)
ˆ
ˆ
∂
φ
(
k
,
t
)
∂
φ
(
−
k
,
t
)
ˆ
ˆ
=
φ
(
−
k
,
+
φ
(
k
,
.
(B5.2-2)
∂
t
∂
t
The terms on the right-hand side of Eq. (
B5.2-2
) depend on the first-order temporal
derivative of the Fourier transform of
t
), which we can calculate by taking the
Fourier transform of the linearized version of Eq. (
5.1
) [with
F
(
t
)
φ
(
r
,
=
0], applying a
procedure similar to the one used for Eq. (
B5.1-2
):
∞
ˆ
f
(
∂
φ
,
(
k
t
)
g
(
=
φ
0
)
φ
,
+
φ
0
)
φ
,
ξ
m
(
r
,
+
ξ
a
(
r
,
(
r
t
)
(
r
t
)
t
)
t
)
∂
t
0
t
)]
e
−
i
k
·
r
d
r
.
+
D
L
[
φ
(
r
,
(B5.2-3)
Equation (
B5.2-3
) can be substantially simplified if the noise terms are uncorrelated (or
white) in space. In this case Eq. (
B5.2-3
) can be rewritten as
ˆ
∂
φ
(
k
,
t
)
φ
0
)
ˆ
φ
0
)
ˆ
Dh
L
(
k
)
ˆ
f
(
g
(
=
φ
(
k
,
t
)
+
φ
(
k
,
t
)
ξ
m
(
t
)
+
ξ
a
(
t
)
+
φ
(
k
,
t
)
,
(B5.2-4)
∂
t
where
h
L
(
k
) is the same operator already defined in Box 5.1. Using Eq. (
B5.2-4
)in
(
B5.2-2
) we obtain
2
f
(
Dh
L
(
k
)
∂
∂
ˆ
t
)
ˆ
ˆ
t
)
ˆ
t
φ
(
k
,
φ
(
−
k
,
t
)
=
φ
0
)
+
φ
(
k
,
φ
(
−
k
,
t
)
φ
0
)
ˆ
ξ
m
(
t
)
2
g
(
t
)
ˆ
+
φ
(
k
,
φ
(
−
k
,
t
)
+
ˆ
ξ
a
(
t
)
+
ˆ
ξ
a
(
t
)
.
φ
(
k
,
t
)
φ
(
−
k
,
t
)
(B5.2-5)
ˆ
t
)
ˆ
φ
,
φ
−
,
=
,
We can now recognize that
(
k
(
k
t
)
S
(
k
t
) by definition; the remaining
terms involve products of
ˆ
(or
ˆ
φ
φ
2
) and noise, and can be treated as follows. If we
ˆ
t
)
ˆ
define
z
=
φ
(
k
,
φ
(
−
k
,
t
), i.e.,
S
(
k
,
t
)
=
z
, the second and third addenda in
Eq. (
B5.2-5
) can be writt
e
n in the form
u
(
z
)
ξ
, with
u
(
z
)
=
z
in the second
=
√
z
in the third addendum. The expected value of the product
of a function of a stochastic variable, and a noise term assumes different values
depending on the considered type of noise. In the case of Gaussian white noise in a
Langevin equation interpreted in the Stratonovich sense we can apply Novikov's
addendum and
u
(
z
)
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