Environmental Engineering Reference
In-Depth Information
pdf
pdf
p
d
f
60
0.5
0.5
30
0.0
φ
2.5
φ
2.5
φ
0.02
2.5
2.5
S
S
S
1
1000
1000
0.5
2
k
2
k
3
k
1
1
1
2
Figure 5.16. Numerical simulation of stochastic model (
5.29
) with
a
=−
0
.
1,
5 (the other conditions are as in Fig.
5.10
). The columns re-
fer to 0, 200, and 400 time units. The first row shows the field (the gray-tone scale
spans the interval [
D
=
2
.
5, and
s
gn
=
−
0
.
5
,
0
.
5]), the second row shows the pdf of the field variable
φ
,
and the third row shows the azimuth-averaged spectrum of the field.
noise is unable to give rise to phase transitions (i.e., changes of
m
), regardless of the
type of spatial coupling.
5.4 Multiplicative noise and pattern-forming spatial couplings
Similar to the case of additive noise, multiplicative noise and pattern-forming spatial
coupling can also cooperate to generate steady patterns. Such cooperation is, however,
more subtle than in the case of additive noise in that it is based on two key actions:
(i) the multiplicative random driver temporarily destabilizes the homogeneous stable
state
φ
0
of the underlying deterministic dynamics, and (ii) the spatial coupling acts
during this instability, generating and stabilizing a pattern. Notice that the instability
is induced by the multiplicative-noise term and not by the spatial coupling, as in the
case of the deterministic mechanism of Turing instability presented in Appendix B.
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