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
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
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