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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