Environmental Engineering Reference
In-Depth Information
theorem (see Box 3.2),
u
(
z
)
ξ
gn
=
s
gn
u
(
z
)
u
(
z
)
, i.e.,
ˆ
ξ
gn
,
m
(
t
)
=
t
)
ˆ
φ
,
φ
−
,
(
k
(
k
t
)
t
) and
ˆ
ξ
gn
,
a
(
t
)
=
s
gn
,
a
2
,
φ
±
,
s
gn
,
m
S
(
k
(
k
t
)
, where
s
gn
,
a
is the intensity of the additive
white Gaussian noise.
When the various terms are recomposed, Eq. (
B5.2-5
) becomes
∂
2
f
(
φ
0
)
s
gn
,
m
S
(
k
S
(
k
,
t
)
g
(
=
φ
0
)
+
Dh
L
(
k
)
+
,
t
)
+
s
gn
,
a
.
(B5.2-6)
∂
t
Therefore at steady state the structure function is
s
gn
,
a
2
f
(
φ
0
)
s
gn
,
m
.
S
st
(
k
)
=−
(B5.2-7)
φ
0
)
+
+
g
(
Dh
L
(
k
)
expressing the spatiotemporal evolution of
is known. In the following subsections
each of these methods is applied to specific examples and the skills and limitations
of each approach are discussed. It is worth mentioning that in general the wavelength
2
φ
k
max
, resulting from stability analysis, structure function, or generalized mean
field, will actually emerge only in an infinite domain. In this case, possible (small) dif-
ferences in wavelength - with respect to 2
π/
k
max
- observable in the numerical simu-
lations of the completemodel are due to approximations associatedwith themathemat-
ical techniques used in each of these prognostic methods. An example is the linear ap-
proximation involved in stability analysis. In contrast, when finite domains are consid-
ered (e.g., no-flux boundary conditions), wave numbers significatively different from
k
max
may emerge because the dominant wavelength has to be compatible with the size
of the domain. Generally the most unstable wavelength compatible with these bound-
ary conditions is selected in the process of pattern formation (see
Murray
(
2002
)).
When the expected spatial fields do not have a clear periodicity, i.e., in the case of
multiscale patterns, it is difficult to foresee the pattern shape by using these theoretical
analyses. Dispersion relation, steady-state structure function, and generalized mean-
field analysis can give some insight into pattern formation, but the definitive way to
assess noise-induced pattern formation is to numerically simulate the dynamics and
visually inspect the results. Such qualitative analysis can then be complemented by
quantitative evaluations, such as the analysis of transects and the investigation of the
statistical properties of coherence regions (e.g.,
Dale
,
1999
;
Kent et al.
,
2006
).
π/
5.1.2.4 Numerical simulation of random fields
Along with mathematical prognostic tools, numerical simulations of stochastic mod-
els are fundamental to verifying the analytical findings and characterizing patterns.
The typical approach is to discretize the continuous spatial domain by use of a regular
Cartesian lattice with spacing
x
=
y
=
. Original stochastic partial differential
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