Environmental Engineering Reference
In-Depth Information
k
γ
0.20
0.15
0.10
0.05
k
0.2
0.4
0.6
0.8
-0.05
-0.10
d
1
=
d
2
= 0
= 0
d
1
d
d
2
> 0
-0.15
d
1
> 0
2
> 0
-0.20
Figure B.5. Different dispersion relations for the case of differential-flow instability,
Eqs. (
B.27
). The parameters for the bold solid curve are
ϕ
=
0
.
72,
η
=
6
.
10,
δ
=
5
.
14,
p
=−
1
.
315,
d
1
=
0, and
d
2
=
1.
u
and vegetation density
v
, are used. The dynamics of the two variables are ex-
pressed as
∂
u
2
u
t
=
ϕ
(1
−
u
)
−
u
v
+
p
∇
u
+
d
1
∇
,
∂
∂v
∂
v
2
t
=
η
u
v
−
δ
+
v
+
d
2
∇
v.
(B.27)
1
The first term on the right-hand side of the first equation represents the rate of
increase in nutrient concentration, the second term accounts for the consumption of
nutrient by biomass, and the third term is the loss of nutrients by advection; the last
term models the spreading of
u
by diffusion. The first term on the right-hand side
of the second equation represents the nutrient-dependent rate of biomass growth; the
second term represents the state-dependent mortality rate, and the third term accounts
for the diffusion-like spatial spreading of biomass. The steady homogeneous state
[
u
0
)] is stable in the absence of drift
and diffusion when conditions (
B.5
) are met. Drift-induced instability occurs if the
drift term is able to destabilize the homogeneous state (
u
0
=
ηϕ
−
δ/
(
η
(
ϕ
−
1)),
v
=
ϕ
(
δ
−
η
)
/
(
ηϕ
−
δ
0
0
) evenwhen the Laplacian
terms are set equal to zero (see Section
B.4
). In this case dispersion relation (
B.26
)
provides the range of Fourier modes that are destabilized by drift (see Fig.
B.5
). As
noted by
Rovinsky and Menzinger
(
1992
), in the absence of a diffusion term the
interval of the unstable modes has no upper bound [Eq. (
B.26
)]. When a diffusion
term is added to the first equation (i.e.,
d
1
=
,v
0), the dispersion relation becomes
2
(
d
1
k
2
d
1
k
2
g
v
)
γ
+
−
f
u
−
g
v
−
i
pk
x
)
γ
+
(
f
u
g
v
−
f
v
g
u
+
i
pk
x
g
v
−
=
0
.
(B.28)
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