Environmental Engineering Reference
In-Depth Information
Box 5.1: Stability analysis by normal modes
The use of stability analysis in stochastic models follows the same conceptual steps
described in Appendix B for deterministic systems, but here it is applied to the
spatiotemporal dynamics of the ensemble average of the field variable
. We consider
the case of systems whose dynamics are expressed by Eq. (
5.1
) with no time-dependent
forcing [i.e.,
h
(
φ
0]. The linear-stability analysis includes three steps. First, a
deterministic equation for the spatiotemporal dynamics of the ensemble average of the
field variable
φ
)
F
(
t
)
=
is obtained from (
5.1
) and the homogeneous basic state (i.e., the
homogeneous steady state) is found. In this step, the most complex aspect is the
evaluation of the ensemble average of the multiplicative random component
φ
.
As shown in Box 3.2, this term may be expressed as the product of the noise intensity,
s
m
, by a function
g
S
(
g
(
φ
)
ξ
m
φ
) of the state variable,
g
(
φ
)
ξ
m
=
s
m
g
S
(
φ
)
. The specific
φ
φ
=
function
g
S
(
) varies depending on the type of noise and its interpretation.
g
S
(
)
0
φ
=
when
g
(
const or the noise term in the Langevin equation is interpreted in Ito's
sense. In the case of Langevin equations with Gaussian white noise interpreted in the
Stratonovich sense, the Novikov theorem provides the result
)
)
g
(
g
(
φ
)
ξ
gn
=
s
gn
g
(
φ
φ
)
)
g
(
[see Eq.
B3.2-2
)], i.e.,
g
S
(
φ
)
=
g
(
φ
φ
). Thus, taking the ensemble average of Eq.
(
5.1
) [with
h
(
φ
)
F
(
t
)
=
0], we obtain
∂
φ
∂
=
f
(
φ
)
+
s
m
g
S
(
φ
)
+
D
L
[
φ
]
.
(B5.1-1)
t
The basic state,
φ
=
φ
0
=
const, is obtained as the zero of Eq. (
B5.1-1
) at steady state,
i.e.,
f
(
0.
In the second step of the stability analysis we look at the effect of small perturbations
of the basic state. To this end, Eq. (
B5.1-1
) is linearized, in that the effect of
nonlinearities becomes negligible in the case of infinitesimal perturbations. The Taylor's
expansion of Eq. (
B5.1-1
) around
φ
0
)
+
s
m
g
S
(
φ
0
)
=
φ
=
φ
0
, truncated to first order, leads to
∂
φ
∂
f
(
s
m
g
S
(
=
φ
0
)
φ
+
φ
0
)
φ
+
D
L
[
φ
]
,
(B5.1-2)
t
d
f
(
φ
)
d
g
S
(
φ
)
where
f
(
φ
|
φ
=
φ
0
and
g
S
(
φ
=
φ
0
is now
disturbed (third step) by adding an infinitesimal harmonic perturbation
φ
0
)
=
φ
0
)
=
φ
|
φ
=
φ
0
. The basic state
d
d
ˆ
e
γ
t
+
i
k
·
r
φ
=
φ
0
+
φ
,
(B5.1-3)
=
√
−
where
ˆ
φ
is the perturbation amplitude,
γ
is the growth factor,
i
1 is the imaginary
unit,
k
y
)isthe
coordinate vector. If (
B5.1-3
) is inserted into Eq. (
B5.1-2
), we obtain the so-called
dispersion relation:
=
(
k
x
,
k
y
) is the wave-number vector of the perturbation, and
r
=
(
x
,
f
(
s
m
g
S
(
γ
(
k
)
=
φ
0
)
+
φ
0
)
+
Dh
L
(
k
)
,
(B5.1-4)
L
φ
=|
|
where
h
L
(
k
), the Fourier transform of
(
), is a function of the wave number
k
k
L
and depends on the specific form of spatial coupling
considered. For example, if
L
(
φ
)
=∇
2
(
φ
),
h
L
(
k
)
=−
k
2
;if
L
(
φ
)
=∇
4
(
φ
),
h
L
(
k
)
=
k
4
.
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