Environmental Engineering Reference
In-Depth Information
Box 5.5: Short-term instability
We consider the model
∂φ
∂
t
=
f
(
φ
)
+
g
(
φ
)
ξ
m
(
r
,
t
)
+
D
L
[
φ
]
.
(B5.5-1)
The first steps of the stability analysis are the same as those described in Box 5.1 and
lead to the following equation for the evolution of the ensemble average
∂
φ
∂
=
f
(
φ
)
+
s
m
g
S
(
φ
)
+
D
L
[
φ
]
.
(B5.5-2)
t
If
f
and
g
are nonlinear functions of
φ
, their ensemble average involves all moments of
the pdf of
φ
. For example, for
f
(
φ
) the expansion is (
van Kampen
,
1992
)
d
2
f
(
1
2
φ
)
)
2
f
(
φ
)
=
f
(
φ
)
+
(
φ
−
φ
+···
.
(B5.5-3)
d
φ
2
Inserting Eq. (
B5.5-3
)into(
B5.5-2
), we find that the dynamics of
φ
do not depend on
only
. However, as we are
interested here in the initial evolution of small displacements from
φ
itself but also on the fluctuations of
φ
around
φ
φ
0
- i.e., we are
focusing on
t
0 - the nonlinear terms of the expansion can be neglected and the
zero-order Taylor's expansion,
→
f
(
φ
)
≈
f
(
φ
)
,
g
S
(
φ
)
≈
g
S
(
φ
)
,
(B5.5-4)
can be used. Similarly, we can also neglect the spatial gradients of the fluctuations and
assume that they are small in the short term. It follows that, for
t
→
0, Eq. (
B5.5-2
) can
be approximated at any point in space as (
Sagues et al.
,
2007
)
d
φ
d
t
d
V
(
φ
)
≈
f
(
φ
)
+
s
m
g
S
(
φ
)
=
f
eff
(
φ
)
=−
,
(B5.5-5)
φ
d
where
f
eff
and
are often indicated as the
effective kinetics
and the
effective
or
stochastic potential
(see Box 3.1), respectively. Notice that Eq. (
B5.5-5
) is identical to
the macroscopic equation obtained in the short term in a zero-dimensional dynamical
system with the same functions
f
and
g
. Therefore the short term stability analysis of
the state
V
φ
0
expressed by Eq. (
B5.5-5
) deals only with the temporal dynamics at any
given point of the domain.
Equation (
B5.5-5
) clearly shows that noise can destabilize the stable state
φ
0
of the
underlying deterministic dynamics. We consider two cases. In the first case the roots of
f
eff
(
0, and in the second case the state
φ
0
remains a zero of the right-hand side of Eq. (
B5.5-5
) - i.e.,
f
eff
(
φ
)
=
0 do not coincide with those of
f
(
φ
)
=
φ
0
)
=
0alsofor
s
m
>
0 - but sufficiently high noise intensities cause the instability of
φ
0
. In the first
case,
φ
0
is a stable state of the deterministic dynamics [i.e., for
s
m
=
0
f
(
φ
0
)
=
0 and
f
(
φ
0
)
<
0], whereas it is not an equilibrium point of the stochastic dynamics, i.e., for
s
m
>
0,
f
eff
(
φ
0
)
=
0. Multiplicative noise destabilizes the state
φ
0
if
d
φ
d
t
=
f
eff
(
φ
0
)
=
s
m
g
S
(
φ
0
)
>
0
.
(B5.5-6)
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