Environmental Engineering Reference
In-Depth Information
H
max
and investigate the dynamics of the normalized variable,
H
=
H
max
.To
simplify the notation, in what follows we drop the superscript and refer to
H
as the
normalized soil depth. The soil-production function
f
(
H
) decreases with the soil
thickness because the soil mantle protects the bedrock from weathering agents (e.g.,
Ahnert
,
1988
;
Dietrich et al.
,
1995
;
Heimsath et al.
,
1997
).
Carson and Kirkby
(
1972
)
speculate that
f
(
H
) has a maximum and that for relatively shallow soils the rate of soil
production decreases with decreasing values of
H
because the mechanical weathering
by roots is weaker in shallow soils, where only sparse vegetation is able to grow. Thus
we model
f
(
H
)as
H
/
=
ρ
r
ρ
f
(
H
)
s
w
b
(
H
+
b
)(1
−
H
)
,
(4.16)
where
ρ
r
and
ρ
s
are the densities of the parent rock and of the soil, respectively,
w
b
is the rate of bedrock weathering, and
b
is a parameter determining the rate of soil
production when the bedrock is at the surface. Notice that, if
b
=
0, the dynamics
wouldtendto
H
=
0 and remain locked in this state because on the denuded slope
the rate of soil production would be zero. In this case the steady-state probability
distribution of
H
is
p
(
H
)
=
δ
(
H
). We consider the case
b
>
0 and note that the
dynamics have a boundary at
H
1. We can use the theory presented in Chapter 2
to determine the steady-state probability distribution of
H
:
=
H
α
(1
λ
a
(1
λ
−
1
(
H
b
)
−
−
1
p
(
H
)
=
Ce
−
H
)
+
,
(4.17)
+
b
)
a
+
(1
+
b
)
=
w
ρ
/ρ
where
a
s
and
C
is the normalization constant. Figure
4.15
shows how the
shape of
p
(
H
) dramatically changes with different values of the noise parameters
b
r
λ
and
. Depending on these parameters, the system may have only one preferential
state contained between 0 and 1 [Fig.
4.15
(a), dashed curve], or at the boundaries
H
α
=
0[Fig.
4.15
(a), dotted curve] and
H
=
1[Fig.
4.15
(b), dotted curve] of the [0,1]
interval. Thus relatively high values of
are associated with high erosion rates -
i.e., with weathering-limited systems - whereas with relatively low values of these
parameters, the system develops thicker soil deposits (transport-limited dynamics).
Bimodality may emerge in intermediate conditions [U-shaped distributions in Fig.
4.15
(a) and Fig.
4.15
(b), solid curve], indicating that the system has two preferential
states (either no soil mantle or relatively deep soil deposits), whereas intermediate
conditions have a low probability of occurrence. These systems have a high likelihood
to be in either transport-limited or weathering-limited conditions as suggested by
D'Odorico
(
2000
), who investigated the same dynamics of soil development with a
more complex form of the function
f
(
H
) obtaining qualitatively similar results. The
transition times between the modes of the bistable soil dynamics was investigated by
D'Odorico et al.
(
2001
) in terms of mean-first-passage times.
λ
and
α
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