Environmental Engineering Reference

In-Depth Information

ϕ

[
s
(
t
)] represents water losses that are

due to evapotranspiration
E
[
s
(
t
)] and leakage
L
[
s
(
t
)]. The infiltration rate

[
s
(
t
)

,

t
] is the rate of rainfall infiltration, and

χ

is a state-

dependent Poisson process because only a fraction of the rainfall amount infiltrates

the ground when the soil is close to saturation. Thus the probability distribution of

the depth
h
of infiltrating rainfall normalized with respect to
nZ
r
(i.e.,
h
=

ϕ

h

/

nZ
r
)

is (
Laio et al.
,
2001
)

s
)
∞

1

b
(
h
;
s
)

e
−
γ
h

(
h
−

e
−
γ
u
d
u

=
γ

+
δ

1

+

s
γ

,

(4.20)

−

γ
=

/α

δ

·

where

nZ
r

and

(

) is the Dirac delta function. Equation (
4.19
) can be rewritten

as

d
s

d
t
=
ϕ

(
s
(
t
)

,

t
)

−
ρ

(
s
)

,

(4.21)

nZ
r

where

nZ
r
is the normalized loss function.

Solutions of (
4.21
) for the case of

ρ

(
s
)

=
χ

(
s
)

/

independent of
s
(i.e., with no soil-moisture-

atmosphere feedbacks) were obtained by
Laio et al.
(
2001
). Feedbacks between soil

moisture and precipitation can be accounted for (
D'Odorico and Porporato
,
2004
)

through a state dependency on storm frequency (see Fig.
4.18
). We can determine the

pdf of soil moisture by solving (
4.21
) as shown in Chapter 2:

λ

(
s
)
exp

(
u
)
d
u

C

ρ

λ

(
u
)

p
(
s
)

=

−
γ

s

+

.

(4.22)

ρ

s

χ

[
s
(
t
)] is the sum of losses that are due

to evapotraspiration and leakage, both of which act as soil-drying processes. Daily

evapotraspiration is expressed as (
Laio et al.
,
2001
)

In the soil-water balance, the function

⎨

0

0

<

s
(
t
)

≤

s
h

s
(
t
)

−

s
h

E
w

s
h
<

s
(
t
)

≤

s
w

s
w
−

s
h

E
(
s
)

=

(4.23)

(
E
max
−
E
w
)
s
(
t
)
−
s
w

≤
s
∗

⎩

E
w
+

s
w
<
s
(
t
)

s
∗
−

s
w

s
∗
<

E
max

s
(
t
)

≤

1

,

where
s
∗
is the soil-moisture value below which plant transpiration is reduced by

stomatal closure,
s
w
is the soil-water content at the wilting point,
s
h
is soil moisture

at the hygroscopic point,
E
w
is the soil-evaporation rate, and
E
max
is the maximum

evapotranspiration reached at
s
∗
.
s
∗
and
s
w
depend on both soil and vegetation char-

acteristics. Thus, for values of soil moisture exceeding
s
∗
, evapotranspiration is not

limited by the soil-water content and occurs at a maximum rate
E
max
.As
s
decreases

below
s
∗
, plants undergo a state of water stress (
Porporato et al.
,
2001
) and linearly

reduce the rate of evapotranspiration, which becomes zero when
s
is at the wilting

point. For
s

s
w
, no transpiration occurs and all losses are due only to soil evapo-

ration, which becomes zero when
s
reaches the hygroscopic point
s
h
. Leakage losses

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