Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 188

Figure 5.12. Dynamics described by Ginzburg-Landau model ( 5.18 ) when a is pos-
itive ( a 1). The three panels correspond to t equal to 0, 150,

and 300 time units, and the gray-tone scale spans the interval [

=+ 0 . 1, D = 10, k 0 =

− 0 . 5 , 0 . 5]. Other

conditions are as in Fig. 5.10 .

nonlinear term comes into play. In this way, the linear term maintains the dynamics
away from the uniform state, 0, thereby allowing the spatial terms to remain

active (i.e., different from zero) and to generate patterns. Because of the nonlin-
ear term, the dynamics do not diverge but tend to a statistically steady state. Fig-
ure 5.12 shows an example of this type of dynamic. Notice that the dominant length
scale is again

φ =

k 0 , but the competition among the unstable modes close to
k 0 gives rise to a transition between a spots-and-gaps pattern to quasi-cylindrical
waves.
The occurrence of a periodic pattern when a

λ = 2 π/

>

0 can be easily assessed by the
analysis by normal modes of the homogeneous state

φ =

0, as described in Box 5.1.

Using Eq. ( B5.1-4 ) we obtain the dispersion relation

D ( k 0 − k 2 ) 2 γ ( k ) = a − , (5.19)

where k =| k | . The most unstable mode is k = k 0 , with γ ( k 0 ) = a . Therefore when

a > 0 the homogeneous state φ = 0 is unstable and periodic patterns emerge if

k 0 = ) appears in the linear-stability

analysis (i.e., stability analysis with respect to infinitesimal harmonic perturbations)
in that the nonlinear terms contribute with higher-order infinitesimals. Therefore the
linear-stability analysis is unable to assess whether the nonlinear terms can prevent
divergence. Moreover, the linear-stability analysis by normal modes is not able to
detect the transient occurrence of patterns when a

0. Notice that only the linear component of f ( φ

0, because such a technique
investigates only the long-term behavior of the system ( Trefethen et al. , 1993 ). For
a

<

< 0, dispersion relation ( 5.19 ) always gives γ< 0, indicating that no patterns exist

in the steady state of the system (i.e., for t ).

This analysis of deterministic model ( 5.17 ) shows that in this system periodic pat-
terns may always emerge because of the presence of pattern-forming spatial coupling.
If a

→∞

) tend to relax to a stable state. In this case patterns
are transient and disappear after some time. If in contrast a

< 0 the local dynamics f ( φ

> 0, the local component