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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