Environmental Engineering Reference

In-Depth Information

1

E
s

L s

cm d

0.8

0.7

______
a

0.03

b

0.50

............
a

0.03

b

0.60

0.6

____
a

0.03

b

0.30

0.5

0.4

0.3

0.2

0.1

s

0.2

0.4

0.6

0.8

1

Figure 4.20. Stable and unstable states of the deterministic process shown as in-

tersections of the loss curve (thick curve),
E
(
s
)

+

L
(
s
), with the average input rate

α
∗
=

α
∗

λ

(
s
)

(
a

+

bs
)

for different values of
b
. The parameters are the same as in

Fig.
4.19
.

λ

bs
. The solid-curve plot shows very well-defined bimodal behavior. The

emergence of these two preferential states is important from a climatologic perspective

because it suggests that, because of soil-moisture-precipitation feedbacks, the system

is more likely to be in either “dry” or “wet” conditions, whereas long-term average

climatic conditions have the lowest probability of occurrence. Thus in some regions

the common use of long-term average parameters to characterize the hydroclimatic

regime would not provide meaningful information on soil-moisture dynamics.

In Chapter 3 we have also stressed that, because of the dependency of

(
s
)

=

a

+

on
s
,

the WSN in Eq. (
4.9
) is multiplicative. Because multiplicative random drivers are

known for their ability to cause noise-induced transitions (see Chapter 3) we could

wonder whether the bistability (i.e., bimodality) found in
p
(
s
)(Fig.
4.19
)emerges

as a noise-induced effect or as a result of the soil-moisture-precipitation feedbacks.

To address this point, we study the deterministic counterpart of Eq. (
4.19
), which we

obtain by replacing the noise term with its mean value

λ

α
∗
:

λ

(
s
)

α
∗

nZ
r
λ

d
s

d
t
=

(
s
)

−
ρ

(
s
)

,

(4.25)

α
∗
is the mean of
h

nZ
r
h
[from Eq. (
4.20
)], which in general might differ

where

=

from

1 imposes a bound to the amount of water that

can infiltrate the ground. The equilibria of the deterministic dynamics are given

by the condition

α

because the condition
s

≤

=
α
∗
/

ρ

(
s
)

nZ
r
λ

(
s
), i.e., by intersections of the
E
(
s
)

+

L
(
s
)curve

α
∗
λ

=
α
∗
(
a

(Fig.
4.20
) with the line

(
s
)

+

bs
)(Fig.
4.20
, thin straight line). In the

absence of feedbacks (
b

0) this line is horizontal and only one intersection (stable

state) exists. With suitable slopes of the

=

(
s
) relation, multiple intersections may exist,

as shown in Fig.
4.20
(dotted line): In this case the deterministic dynamics already

exhibit three equilibria corresponding to two stable states separated by an unstable

λ

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