Environmental Engineering Reference
In-Depth Information
As the most unstable (or least stable) modes correspond to the condition
0, it
is sufficient to investigate the self-consistency equation for the Langevin equation,
d
ω
(
k
)
=
φ
i
d
t
=
a
φ
i
−
D
eff
(
φ
i
−
φ
i
)
+
ξ
gn
,
i
,
(5.26)
where
D
4
k
0
2
4
D
eff
=
2
−
+
.
(5.27)
4
Equation (
5.26
) does not provide multiple solutions even when well-defined patterns
emerge (see Fig.
5.13
).
Equation (
5.23
) shows that the most unstable modes correspond to the condition
ω
0; see Eq. (
5.24
)], which can be satisfied by infinite pairs of
wave-number components,
k
x
(
k
)
=
0 [in fact,
ω
(
k
)
≥
,
k
y
, different fromzero. Thus stable patterns can emerge
from the interactions among several unstable modes. Notice that the condition with
zero wave number (i.e.,
k
x
=
k
y
=
0) gives a more restrictive condition of instability:
D
20
(
8
k
0
d
φ
i
d
t
=
Dk
0
φ
f
(
φ
i
)
+
g
(
φ
i
)
ξ
−
−
−
φ
−
m
)
+
ξ
.
(5.28)
i
i
i
gn
,
i
4
2
In other words, phase transitions (i.e.,
m
0) can occur only when many wave
numbers different from zero have already become unstable.
The classical mean-field analysis (see Box 5.4) correctly detects that the order
parameter
m
does not change, and then the periodic patterns sustained by additive
noise do not entail phase transitions. The pdf's of the field variable
=
φ
,shownin
Fig.
5.13
, confirm this theoretical finding. In fact, the pdf's of
remain unimodal,
symmetrical, and with zero mean even after the appearance of patterns.
Some additional comments should complete the picture drawn in this section on
the interplay between additive noise and pattern-forming spatial coupling:
φ
1. The nonlinear component of
f
(
) does not play any fundamental role in the mechanism's
inducing and sustaining spatial patterns. In fact, close to the deterministic stable state the
deterministic behavior is determined by the linear component of
f
(
φ
), as demonstrated by
the analysis of the structure function (and stability analysis) for the linearized models. Thus
the addition of the nonlinear term
φ
3
to model (
5.16
) does not substantially change any
of the previous results. The fine details of the patterns can change, but neither their stable
occurrence nor their dominant wavelength changes.
2. Pattern formation induced by additive noise is usually introduced in the science literature
as an example of a noisy precursor to deterministic pattern-forming bifurcation (
Sagues
et al.
,
2007
). According to this point of view, additive noise acts on a deterministic system
that exhibits a bifurcation point between a homogeneous stable state and a stable patterned
state [an example is Ginzburg-Landau model (
5.18
)]. In this case, the role of additive
noise is to anticipate the transition through the bifurcation point, inducing patterns even
when the deterministic system is in subcritical conditions. In other words, additive random
−
φ
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