Environmental Engineering Reference

In-Depth Information

corresponding to the occurrence of a wave number with

γ
=

0, i.e., of conditions of

marginal stability. When
s
m

s
c
, positive growth factors emerge, and spatial patterns

(periodic or multiscale, depending on the value of
k
max
) are expected to occur.

Stability analysis may be a useful tool for pattern recognition, but it is not without

faults: Its effectiveness depends on the specific stochastic model under consideration.

We can also use other more complicated, prognostic tools based on the structure

function. The basic idea behind this method is that the presence of patterns will

eventually modify the correlation structure of the field. Instead of considering the

correlation function, this method analyzes its Fourier transform in space, which is

known as the
structure function
or power spectrum (see Appendix A):

>

e
−
i
k
·
r
G
(
r

S
(
k

,

t
)

=

,

t
)d
r

,

(5.9)

D

where
G
(
r

,

t
) is the correlation function,

M
(
t
)
2

A
2

1

A

(
r
,

(
r
+

t
)d
r
−

G
(
r

,

t
)

=

D
φ

t
)

φ

r

,

,

(5.10)

D

is the 2D spatial domain,
A
is its area, and

D
φ

=

,

.

M
(
t
)

(
r

t
)d
r

(5.11)

The structure function is useful in describing the spatial periodicity of the system,

similar to the way the power spectrum describes the temporal periodicity of a temporal

signal. The use of the structure function as a prognostic tool requires some methods

to determine the dynamics of
S
(
k

,

t
) from Eq. (
5.1
). These methods are explained in

Box 5.2.

Once the equation describing the time evolution of the structure function [i.e.,

Eq. (
B5.2-5
)] is obtained, we can determine the corresponding steady-state expression

[Eq. (
B5.2-7
)] and use it to assess whether periodic patterns - corresponding to the

existence of a maximum of the structure function for wave numbers
k
different from

zero - are expected to appear. Equation (
B5.2-7
) clearly shows that the additive

noise is fundamental to having a nonnull steady-state structure function. The role

of the multiplicative noise is instead fundamental in determining the wavelength of

the emerging patterns: In fact, the dominant wavelength
k
max
is found where the

denominator of Eq. (
B5.2-7
)ismaximum.
1

The third prognostic tool to foresee pattern formation is based on the generalized

mean-field theory, and it is presented in Box 5.3.

In Boxes 5.1, 5.2 and 5.3 we have described three analytical prognostic tools that

may be useful for assessing the occurrence of spatial patterns when the equation

1
Note that the denominator is rather similar to dispersion relation (
B5.1-4
): In particular, when
g
(

φ

=
φ

)

and the

multiplicative noise is white and Gaussian, we have that
g
S
(

f
(

φ

)

=

1. Thus

γ

(
k
)

=

φ
0
)

+
Dh
L
(
k
)

+
s
m
,asinthe

denominator of Eq. (
B5.2-7
).

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