Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 202

p d f p d f

p d f

0.5 0.5

60

30

0.0 φ

2.5 φ 2.5 φ

0.02

2.5 2.5

S S S

1

800 800

0.5

3 k

3 k 3 k 1 2 1 2

1 2

Figure 5.20. Results of the numerical simulation of prototype model ( 5.34 ). The
parameters are a

5 (the other conditions are as
in Fig. 5.10 ). The three panels correspond to t equal to 0, 10, and 100 time units. The
gray-tone scale spans the interval [

=− 1, D = 15, k 0 = 1, and s gn = 2 .

− 2 , 2].

The numerical simulation of stochastic model ( 5.34 ) confirms these theoretical
findings. Figure 5.20 shows an example of patterns emerging in the simulation. They
have the same basic characteristics as those observed in the case of additive noise
(see Fig. 5.13 ): They are statistically stable and exhibit a clear dominant wavelength
corresponding to k 0 . Moreover, in this case the pdf of the field is unimodal with the
mean at

0, demonstrating that no phase transitions occur (i.e., m remains the same
as in the disordered case). However, there are some differences with respect to the
case with additive noise. First, the edges of the stripes or other geometrical features
existing in the pattern are more regular when patterns are induced by multiplicative
noise. This regularity is consistent with the fact that the power spectrum of the field
exhibits a sharper peak at k 0 in the case of multiplicative noise (compare the third
rows of Figs. 5.13 and 5.20 ). This difference is due to the fact that multiplicative noise
is modulated by the local value of

φ =

φ , which has the effect of enhancing the spatial

coherence of the pattern because is spatially correlated (from the definition itself

of a patterned state). Second, patterns induced by additive noise exhibit more stable
shapes than those induced by multiplicative noise. For example, the pattern shown

φ