Environmental Engineering Reference

In-Depth Information

Thus the coupling parameter

α

is treated as a Gaussian random variable with mean

α

0
and intensity
s
gn
:

α
=
α

+
ξ

.

(4.49)

0

gn

Inserting Eqs. (
4.47
)-(
4.49
) into Eq. (
4.46
), we obtain the analog of Eq. (
4.26
)forthe

soil-moisture variable
s
:

d
s

d
t
=

f
(
s
)

+

g
(
s
)

ξ
gn
,

(4.50)

where

+
α
0
s
c
)(1

s
r
)

bs
c

as
c
(1

s
r
)

f
(
s
)

=

a
(1

−

−

,

g
(
s
)

=

−

,

(4.51)

and
a
and
b
are the rates of advective precipitation and potential evapotranspir-

ation normalized with respect to the effective soil thickness
nZ
r
, i.e.,
a

=

P
a
/

(
nZ
r
)

and
b

=

E
max
/

(
nZ
r
). Notice that the dynamics are intrinsically bounded between 0

and 1.

The solution of stochastic differential Eq. (
4.50
) leads to the steady-state probability

distribution
p
(
s
) of soil moisture.
Rodriguez-Iturbe et al.
(
1991
) calculated
p
(
s
)by

using Ito's interpretation. To be consistent with the other results presented in this

chapter, here we use Stratonovich's interpretation and note that in this case the results

do not qualitatively depend on the interpretation as the system undergoes the same

noise-induced transitions regardless of the interpretation rule for the multiplicative-

noise term. Thus
p
(
s
) can be calculated with Eq. (
2.83
). An analytical expression

of
p
(
s
) can be determined, though it is not reported here as it is fairly long and

cumbersome. Rather, we show the results for the particular case in which
r
=

2and

c
=

1:

as
gn
(
a
−

a
)
−
1
(1

2
as
gn
(
1
+
a
−

a
)
−
1

1

b

1

b

s
)
−

p
(
s
)

=

Cs

−

(4.52)

2
as
gn
s
+

s
2
)

2
as
gn
(
1
−
α
0
+

a
)
−
1
e
−

1

b

1

b

×

+

,

(1

s
)

a
(1

−

where
C
is the normalization constant. Figure
4.24
shows
p
(
s
) calculated with the

same parameters as in
Rodriguez-Iturbe et al.
(
1991
). The modes of the distribution

are obtained with Eq. (
3.48
). When
r

1, Eq. (
3.48
) has five roots. In the

numerical example of Fig.
4.24
only one root is real for
s
gn
=

=

2and
c

=

0, indicating that the

deterministic dynamics have only one stable state. As
s
gn
increases above a critical

value (i.e.,
s
gn
>

475 in this example), Eq. (
3.48
) has three real roots corresponding

to one minimum and two maxima of
p
(
s
). In these conditions the distribution of
s

becomes bimodal. For even larger values of
s
gn
(i.e.,
s
gn
>

0

.

82), Eq. (
3.48
)hastwo

more real roots; however, because both of them are negative (hence are outside the

domain [0, 1] of
s
) their emergence is not associated with any other noise-induced

transition. Similar results can be obtained with Ito's interpretation of the multiplicative

noise, though in that case the transition occurs for a smaller value of
s
gn
.

0

.

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