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 )      Search WWH ::

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