Environmental Engineering Reference
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parameters of the dynamics, on
k
x
and
k
y
, and on
φ
i
, which remains unknown. To
determine
φ
i
, we observe that the mean of
p
st
(
φ
;
φ
i
,
k
x
,
k
y
) must coincide with
φ
i
.
Therefore the self-consistency condition,
+∞
F
φ
i
,
k
y
,
φ
i
=
φ
p
st
(
φ
;
φ
i
,
k
x
,
k
y
)d
φ
=
k
x
,
(B5.3-5)
−∞
can be used to obtain the unknown
φ
i
as a function of the wave numbers
k
x
and
k
y
.
Notice that when
k
x
=
k
y
=
φ
j
=
φ
i
0, Eq. (
B5.3-3
) becomes
. This corresponds to
the classic mean-field analysis, which is illustrated in Box 5.4.
The occurrence of solutions of Eq. (
B5.3-5
) different from
φ
0
is the
basic state of the system) corresponds to the loss of stability of the uniform basic state
with respect to periodic disturbances. It is expected that this loss of stability takes place
for only some specific value of the wave numbers
k
x
and
k
y
. More in detail, for each
form of the spatial coupling and of the equation governing the dynamics of
φ
i
=
φ
0
(where
φ
, a specific
wave number
k
max
will have the maximum destabilizing effect on
φ
0
. Spatial patterns
will likely evolve in these cases, with characteristic wavelength 2
π/
k
max
.
equation (
5.1
) is then transformed into a system of coupled stochastic ordinary dif-
ferential equations:
d
φ
i
d
t
=
f
(
φ
i
)
+
g
(
φ
i
)
ξ
i
(
t
)
+
Dl
(
φ
,φ
j
)
+
h
(
φ
i
)
F
(
t
)
+
ξ
i
(
t
)
,
(5.12)
m
,
i
a
,
φ
where
i
(
t
) is the value of the discretized state variable in the
i
th cell of the lattice,
l
(
φ
i
,φ
j
) is the discretized counterpart of the specific spatial coupling considered, and
j
runs over a suitable set of neighbors of cell
i
. As mentioned in Box 5.3, a general
expression for
l
(
φ
i
,φ
j
)is
φ
,φ
=
w
φ
+
nn
(
i
)
w
φ
,
l
(
j
)
(5.13)
i
i
i
j
j
j
∈
where the number of neighbors
nn
(
i
) and weighting factors
w
j
depend on
the specific finite-difference scheme adopted to numerically approximate the spatial
operator (
Strikwerda
,
2004
). For example, in the case of the Laplacian operator the
simplest (and most-used) scheme is
w
i
and
2
j
1
2
L
[
φ
]
=∇
φ
≈
l
(
φ
i
,φ
j
)
=
(
φ
j
−
φ
i
)
,
(5.14)
∈
nn
(
i
)
where the four nearest neighbors are involved,
1. This corre-
sponds to the set of weighting factors schematically reported in Fig.
5.7
. In the case
of the
w
i
=−
4, and
w
j
=
4
operator, the standard numerical approaches adopt the weighting factors
reported in Fig.
5.8
.
∇
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