Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 189

) is able to sustain the dynamics, the patterns remain stable in time, and a steady
state is reached if a suitable nonlinear component is also present. However, it is im-
portant to stress that such a nonlinear term is not crucial to pattern formation, in that
its role is only to stabilize the field

f ( φ

that would otherwise diverge.
This example suggests a general rule: When a spatial coupling able to induce peri-
odic patterns is present in a spatiotemporal dynamical system, patterns may emerge,
provided that the local dynamics maintain the system away from the homogeneous
steady state. This happens when the homogeneous steady state is unstable, i.e., when
d f (

φ

φ 0 >

φ )

0 . (5.20)

φ

d

On the contrary, when the derivative in ( 5.20 ) is negative, patterns can occur, but only
transiently.

5.2.2 The role of the additive noise
These analyses of the deterministic dynamics underlying stochastic model ( 5.16 )
are fundamental to understanding the role of additive noise and its ability to induce
pattern formation. In fact, the key idea is that additive noise ξ a is able to keep the

dynamics away from their homogeneous steady state even when this state is stable.
Although in this case the deterministic system can exhibit only transient patterns that
vanish as t , in the presence of additive noise, patterns may persist, sustained

by the noise. To demonstrate this pattern-inducing role of the additive noise

→∞

ξ a ,
Fig. 5.13 shows the results of numerical simulations of stochastic model ( 5.16 ).
The results confirm that a very clear and stable pattern occurs in spite of a being
negative: The noise component maintains the dynamics away from the deterministic
steady state 0, thereby allowing the spatial differential terms to drive the system

into the patterned state with wavelength 2

φ 0 =

k 0 . In this sense, the pattern is noise
induced; in fact, if the noise variance is set to zero, patterns disappear, and the system
converges to its homogeneous stable state

π/

0. Finally, we notice that the patterns in
Fig. 5.13 are similar to those shown in Figs. 5.10 and 5.11 , because the main geometric
characteristic of these patterns (i.e., the dominant periodicity) is dictated by spatial
coupling

φ =

], which remains the same in all cases. The most obvious difference
between deterministic and stochastic patterns induced by additive noise (beside the
fact that deterministic patterns are transient or unsteady) is the irregularity of the
contours in the stochastic case (i.e., patterns are fringed ), which is due to the local
disturbance caused by the noise.
This pattern-inducing role of the additive noise can be detected by the structure
function, as defined in Box 5.2. Using Eq. ( B5.2-7 ) we obtain

L [ φ

s gn

=

2 D ( a .

S st ( k ) (5.21)

k 0 ) 2

− k 2 + −