Environmental Engineering Reference
In-Depth Information
p s
2.5
2
1.5
1
______ s gn
0.1
0.5
s gn
1.0
s
0.2
0.4
0.6
0.8
1
Figure 4.24. The pdf of soil moisture [Eq. ( 4.52 )] calculatedwith c
=
1, r
=
2,
=
1,
nZ r =
0
.
10 m, E p =
1
.
6m/yr, P a =
0
.
8 m/yr, and
α =
0
.
637 (same parameters as
in Rodriguez-Iturbe et al. , 1991 ).
Thus environmental fluctuations acting on soil-moisture dynamics as multiplica-
tive noise are capable of inducing new preferential states that did not exist in the
deterministic counterpart of the process (see also Demaree and Nicolis , 1990 ).
The emergence of bimodality in the dynamics of soil moisture indicates that the
system has two preferred states corresponding to relatively dry and relatively wet
states, whereas intermediate conditions have low probability of occurrence. Notice
how, unlike the case presented in Subsection 4.3.4 , the emergence of bimodality in
Eq. ( 4.52 ) is completely induced by environmental fluctuations ( Rodriguez-Iturbe
et al. , 1991 ). The positive feedback between precipitation and soil moisture tends
to sustain and enhance anomalously dry and wet conditions, leading to the per-
sistence of these anomalies for relatively long times, whereas noise induces shifts
from a state to the other. This minimalist model has been used to explain the per-
sistence of climatic anomalies in low-latitude and midlatitude continental regions
(e.g., Nicholson , 2000 ).
4.5 Environmental systems forced by colored Gaussian noise
In the previous chapters we stressed how the emergence of noise-induced transi-
tions depends in general on both the noise intensity (i.e., the variance) and on its
autocorrelation. Exact expressions for the modes of stochastic processes driven by
autocorrelated (or colored ) noise exist in the case of DMN, whereas in the case of
autocorrelated Gaussian noise (or the Ornstein-Uhlenbeck process) only approximate
relations are available (see Chapter 2). In this section we show how the autocorrelation
of the random driver may affect the properties of environmental dynamics. To this end,
we consider the case of noise-induced bistability in coupled soil-moisture dynamics
described in the previous section and investigate how the results shown in Fig. 4.24
 
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