Environmental Engineering Reference
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resembles Turing's dynamics, in that it involves two diffusing species,
u
and
(ac-
tivator and inhibitor, respectively). However, unlike Turing's model, diffusion is not
important to the destabilization of the homogeneous state. In this case, one or both
species are subjected to advective flow (or “drift”), and instability emerges as a result
of the differential-flow rate of the two species (
Rovinsky and Menzinger
,
1992
). Al-
though diffusion is not fundamental to the emergence of differential-flow instability,
it plays a crucial role in imposing an upper bound to the range of unstable modes
k
and determines the wavelength of the most unstable mode (
Rovinsky and Menzinger
,
1992
). As a result of the drift, patterns generated by this process are not time indepen-
dent, as are those associated with Turing's instability. Rather, they exhibit traveling
waves in the flow direction. Self-organized patterns of this type have been observed in
nature, mainly in chemical systems (“the Belousov-Zhabotinsky reaction,”
Rovinsky
and Zhabotinsky
,
1984
). The same mechanism was also invoked to explain ecological
patterns subject to drift, including banded vegetation (
Klausmeier
,
1999
;
Okayasu
and Aizawa
,
2001
;
von Hardenberg et al.
,
2001
;
Shnerb et al.
,
2003
;
Sherrat
,
2005
).
We note that this mechanism of pattern formation induced by differential flow is often
classified as a Turing model in that in both models the dynamics can be expressed by
the same set of reaction-advection-diffusion equations. In the case of Turing models,
instability is induced by the Laplacian term, whereas in the case of differential-flow,
instability it is the gradient term that causes instability. For the sake of clarity here we
discuss the case of differential-flow instability separately.
We introduce themathematical model of differential-flow instability (e.g.,
Rovinsky
and Menzinger
,
1992
) assuming that only one of the two species undergoes a drift,
and we orient the
x
axis in the direction of the advective flow. The activator-inhibitor
dynamics can be expressed as
v
∂
u
+
p
∂
u
2
u
,
t
=
f
(
u
,v
)
x
+
d
1
∇
∂
∂
∂v
∂
2
t
=
g
(
u
,v
)
+
d
2
∇
v,
(B.22)
where
p
is the drift velocity, and
d
1
and
d
2
are the diffusivities of
u
and
v
, respectively.
Notice that when
p
=
0Eqs.(
B.22
) can be written in the same form as Eqs. (
B.1
).
0 the conditions on
d
1
and
d
2
for the emergence of patterns from Eqs.
(
B.22
) are less restrictive than those for Turing's instability. To stress the fact that
patterns emerge from the differential-flow rates of
u
and
When
p
=
, we first consider the
conditions leading to instability in the absence of diffusion and set
d
1
=
v
d
2
=
0. The
homogeneous steady state (
u
0
,v
0
), obtained as solution of the equation set
f
(
u
0
,v
0
)
=
g
(
u
0
0, is stable when conditions (
B.5
) are met. To determine the conditions
in which the differential flow destabilizes the state (
u
0
,v
0
)
=
,v
0
), we linearize
f
(
u
,v
)
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