Environmental Engineering Reference
In-Depth Information
To describe the interplay between short-term temporal instability and spatial dy-
namics, we refer to the model
∂φ
∂
=
φ
+
φ
ξ
,
+
L
φ
,
f
(
)
g
(
)
m
(
r
t
)
D
[
]
(5.31)
t
obtained from general equation (
5.1
), with
F
(
t
) set to zero because this mechanism
does not require a time-dependent forcing, and without additive noise,
ξ
a
, in order to
isolate the role of multiplicative noise.
ξ
m
is a zero-average noise with intensity
s
m
.
We indicate by
φ
0
the stable homogeneous state of the system in the deterministic
case. Namely,
φ
(
r
,
t
)
=
φ
0
is a homogeneous solution of (
5.31
)when
s
m
=
0 [i.e.,
f
(
φ
0
)
=
0 because
L
[
φ
]
=
0 in homogeneous states]. Moreover, we consider cases
in which
g
(
0, so that the noise does not have the possibility of destabilizing
the homogeneous steady state (otherwise, the role of multiplicative noise would be
similar to that of additive noise).
The key features of pattern formation induced by multiplicative noise are that
for values of
s
m
lower than a critical value
s
c
, the state variable
φ
0
)
=
φ
(
x
,
t
) experiences
fluctuations about
0
but noise does not play any constructive role. At any point in
space, the evolution of the ensemble average
φ
φ
, starting from an initial condition
[
φ
(
t
=
0)
=
φ
0
], is similar to the one indicated in Fig.
5.17
by the dotted curve:
φ
decreases monotonically to the statistically steady state
φ
0
, i.e.,
d
φ
d
t
<
0
(5.32)
at any time (we use the total derivative because any point exhibits the same behavior).
Thus the system remains locked in the disordered phase and no pattern occurs. In
this case the spatial coupling does not play any significant role at steady state. Only
transiently, during the descent from
0
, would the spatial coupling be
able to induce a pattern. This transient occurrence is most probable when the descent
is slow and
D
is large, but patterns fade out as the system converges to its steady
state. The temporal behavior of
φ
(
t
=
0) to
φ
depicted in Fig.
5.17
remains qualitatively the
same regardless of the value of
D
. This means that the evolution of
φ
coincides
qualitatively with that of the correspondent zero-dimensional system, in which
D
φ
=
0,
ξ
m
.
The dynamics change when the noise intensity increases above a critical level (i.e.,
s
m
>
φ/
d
t
=
φ
+
g
(
φ
which evolves according to d
f
(
)
)
s
c
,
the multiplicative noise is able to induce a short-term instability of the equilibrium
state
s
c
). At first, let us consider only the zero-dimensional dynamics. When
s
m
>
φ
0
, i.e., with
d
φ
d
t
d
φ
d
t
lim
t
>
0
,
lim
≤
0
.
(5.33)
t
→∞
→
0
The first condition determines an initial instability: The dynamics initially tend to
move away from the basic state
φ
=
φ
0
; however, the second condition establishes
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