Environmental Engineering Reference
In-Depth Information
4.4.4 Noise-induced bistability in climate dynamics:
Effect of land-atmosphere interactions
The interaction of multiplicative Gaussian noise with the nonlinear dynamics of
the soil-water balance was investigated by
Rodriguez-Iturbe et al.
(
1991
) through a
minimalist model of soil-moisture dynamics at the regional-to-semicontinental scale:
nZ
r
d
s
d
t
=
P
(
s
)
−
E
(
s
)
,
(4.46)
where
n
is the soil porosity,
Z
r
is the depth of the root zone (i.e., of the surface layer
of soil that is active in water exchanges with the atmosphere),
s
is the relative soil
moisture (0
(
s
) is an infiltration function expressing
the fraction of
P
that infiltrates the ground, and
E
(
s
) is the evapotranspiration rate.
E
depends on the soil-water content and decreases from its maximum value
E
max
to
zero as
s
tends to zero. Known as potential evapotranspiration,
E
max
represents the
maximum rate of evapotranspiration when soil moisture is not limiting and depends on
atmospheric conditions and solar irradiance. The infiltration function
<
s
≤
1),
P
is the rainfall rate,
(
s
) accounts
for the increase in infiltration capacity with increasing soil storage capacity. Thus
(
s
) is a decreasing function of soil moisture.
Rodriguez-Iturbe et al.
(
1991
) modeled
E
(
s
)and
(
s
)as
E
max
s
c
s
r
=
,
=
−
,
E
(
s
)
(
s
)
1
(4.47)
where
c
depends on the land cover, whereas
and
r
are nonnegative constant param-
eters. Interesting dynamics emerge from Eq. (
4.46
) when the rainfall rate is expressed
as a function of soil moisture, i.e.,
P
=
P
(
s
), to account for the state dependence
of precipitation induced by land-atmosphere interactions. This dependence would be
due to the fact that in a given region a fraction
P
m
of
P
is contributed by locally
recycled moisture, i.e., by moisture from local evapotranspiration, and the remaining
precipitation
P
a
is from atmospheric moisture advected from outside of the region
(e.g.,
Salati et al.
,
1979
;
Eltahir and Bras
,
1996
;
Trenberth
,
1998
). Thus the precip-
itation
P
m
from local recycling is a (linear) function of the evapotranspiration rate
P
m
=
kE
,where
k
is a proportionality constant that depends on the region's size,
the mean wind velocity, and precipitable moisture. Because the evapotranspiration
rate depends on soil moisture [Eqs. (
4.47
)], precipitation recycling induces a state
dependency in the overall rainfall regime:
s
c
)
P
(
s
)
=
P
a
+
P
m
=
P
a
(1
+
α
,
(4.48)
where
kE
max
expresses the strength of the feedback. The rainfall rate is affected
by climate fluctuations.
Rodriguez-Iturbe et al.
(
1991
) noted that, because advected
precipitation is controlled by evaporation from the oceans, it has a relatively small
CV, whereas precipitation recycling undergoes stronger environmental fluctuations.
α
=
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