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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