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 =
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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