Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 200

p φ

0

0.3

a

4

b

0.2

3

2

0.1

1

t

0.4 φ

1 5 10 0.1 0.2 0.3

φ − φ 0 obtained as an ensemble average of 10 6 real-
izations of model ( 5.37 ) with a

Figure 5.18. (a) Behavior of

=− 1. The initial condition is φ = 0 . 1, and s gn = 0 . 5

and s gn = 5 for the lower and the upper curve, respectively. (b) Example of the

steady-state pdf of model ( 5.38 )for a =− 1, s gn = 5, and positive initial condition.

φ =

In this case f eff ( 0 ) 0 for any value of s gn . The condition for marginal stability

is then

φ 0 =

d f eff
d

a + s gn = 0 , (5.36)

φ

which provides the critical noise intensity s c =− a . From condition ( B5.5-7 ), it fol-

lows that φ 0 is stable for s gn < − a and becomes unstable for s gn > − a . In the former

case, small disturbances around the steady state φ

0 are dimmed whereas in the latter

they tend to grow exponentially for t 0. However, this amplification is transient if

the spatial coupling is absent (see Fig. 5.17 ): Fig. 5.18 (a) shows the temporal behav-
ior of the ensemble average of a number of numerically evaluated realizations of the
zero-dimensional stochastic model obtained when the spatial component is eliminated
from Eq. ( 5.34 ):

→

d d t =

3

f ( φ ) + g ( φ ) ξ gn ( t ) = a φ − φ + φξ gn ( t ) . (5.37)

The results in Fig. 5.18 (a) show that the growth phase appears only when s gn > s c and

at the beginning of simulations (i.e., at short term), whereas the effect of the initial
perturbation disappears in the long term.
The steady-state pdf corresponding to model ( 5.37 ) can be found with the methods
described in Chapter 2:

2
2 s gn

φ

a
s gn

Ce −

− 1

p ( φ ) = φ , (5.38)

where C is a normalization constant and the domain of p ( φ ) is either the positive or

the negative semiaxis of , depending on whether the initial condition is positive or

negative valued. In fact, the condition g (

φ

0 be a natural
boundary of the dynamics [an example of pdf is shown in Fig. 5.18 (b)]. Equation
( 5.38 ) shows that the mode of the steady-state distribution is always at

φ = 0) = 0 entails that φ =

0forany
noise intensity. This confirms that in the long run perturbations of the deterministic

φ =