Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 143

Thus the coupling parameter α is treated as a Gaussian random variable with mean

α 0 and intensity s gn :

α = α + ξ .

(4.49) 0 gn

Inserting Eqs. ( 4.47 )-( 4.49 ) into Eq. ( 4.46 ), we obtain the analog of Eq. ( 4.26 )forthe
soil-moisture variable s :

d s
d t =

f ( s ) + g ( s ) ξ gn , (4.50)

where

+ α 0 s c )(1 s r ) bs c as c (1 s r )

f ( s ) = a (1 − − , g ( s ) = − , (4.51)

and a and b are the rates of advective precipitation and potential evapotranspir-
ation normalized with respect to the effective soil thickness nZ r , i.e., a = P a / ( nZ r )

and b = E max / ( nZ r ). Notice that the dynamics are intrinsically bounded between 0

and 1.
The solution of stochastic differential Eq. ( 4.50 ) leads to the steady-state probability
distribution p ( s ) of soil moisture. Rodriguez-Iturbe et al. ( 1991 ) calculated p ( s )by
using Ito's interpretation. To be consistent with the other results presented in this
chapter, here we use Stratonovich's interpretation and note that in this case the results
do not qualitatively depend on the interpretation as the system undergoes the same
noise-induced transitions regardless of the interpretation rule for the multiplicative-
noise term. Thus p ( s ) can be calculated with Eq. ( 2.83 ). An analytical expression
of p ( s ) can be determined, though it is not reported here as it is fairly long and
cumbersome. Rather, we show the results for the particular case in which r = 2and

c = 1:

as gn ( a − a ) − 1 (1 2 as gn ( 1 + a − a ) − 1

1 b 1 b

s ) − p ( s ) = Cs − (4.52)

2 as gn s +

s 2 )

2 as gn ( 1 − α 0 + a ) − 1 e −

1 b 1 b

× + ,

(1 s )

a (1 −

where C is the normalization constant. Figure 4.24 shows p ( s ) calculated with the
same parameters as in Rodriguez-Iturbe et al. ( 1991 ). The modes of the distribution
are obtained with Eq. ( 3.48 ). When r 1, Eq. ( 3.48 ) has five roots. In the

numerical example of Fig. 4.24 only one root is real for s gn =

= 2and c =

0, indicating that the
deterministic dynamics have only one stable state. As s gn increases above a critical
value (i.e., s gn > 475 in this example), Eq. ( 3.48 ) has three real roots corresponding

to one minimum and two maxima of p ( s ). In these conditions the distribution of s
becomes bimodal. For even larger values of s gn (i.e., s gn >

0 .

82), Eq. ( 3.48 )hastwo
more real roots; however, because both of them are negative (hence are outside the
domain [0, 1] of s ) their emergence is not associated with any other noise-induced
transition. Similar results can be obtained with Ito's interpretation of the multiplicative
noise, though in that case the transition occurs for a smaller value of s gn .

0 .