Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 224

coupling, consistent with the results of the mean-field analysis. It is interesting to
notice that the pdf of the field is unimodal. This demonstrates that the spatial coupling
has also the effect of eliminating the bimodality intrinsic to the purely temporal
dynamics. No substantial differences emerge between the case with the Ito or the
Stratonovich interpretation of the noise term in that in both cases the power spectra
exhibit a clear maximum at k

=

k 0 ) however, in the case of the Ito interpretation,
patterns appear to be more regular than in the Stratonovich case, consistent with
similar observations reported for other spatiotemporal models ( Sagues et al. , 2007 ).

5.7.2 Prototype model with non-pattern-forming coupling
It is interesting to analyze what happens when the Swift-Hohenberg coupling in
model ( 5.73 ) is substituted with a non-pattern-forming coupling, e.g., the classical
diffusion D

2

∇ φ . In this case the model is

1
1

∂φ
∂

2

=− a φ + ξ gn ( t ) + D ∇ φ, (5.81)

t + c φ

2

and the dispersion relation is

=− a s gn c −

Dk 2 γ ( k ) + . (5.82)

Thus the stability analysis does not detect unstable modes for any noise intensity.
Moreover, the classical mean-field analysis shows that the order parameter m remains
equal to zero, indicating that no phase transitions occur. In spite of these analytical
results, numerical simulations of model ( 5.81 ) display some interesting features. An
example is shown in Fig. 5.40 for the case of the Ito interpretation (very similar patterns
emerge also in the case of the Stratonovich interpretation). Although the patterns are
less defined than in the other cases shown in this chapter, a fringed multiscale pattern
emerges. Similar to the case with Swift-Hohenberg spatial coupling, the pdf of the
field is unimodal.

0
In the previous subsection, we discussed the occurrence of (periodic or multiscale)
patterns as the result of the cooperation between a noise-induced temporal transition
and a spatial coupling. However, the specific form of g (

5.7.3 A case with g ( φ 0 ) =

φ ) used in model ( 5.73 ) sug-

gests another possible interpretation. In fact, g ( φ ) is formed by a constant component

(corresponding to additive noise) plus a -dependent component (corresponding to

purely multiplicative noise). Such two components are evident in the series expansion

φ