Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 130

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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