Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 222

p φ

0.5

0.4

0.3

0.2

0.1

φ

4 4

Figure 5.37. Examples of pdf ( 5.77 ) in the case of Stratonovich interpretation of the
noise (i.e., ν = 1 / 2). The unimodal pdf corresponds to s gn = 0 . 8, and the bimodal

one refers to s gn = 10 (in both cases a = 1 and c = 1).

When s gn < a / 2 ν c the pdf has only one mode at φ = 0, whereas when the noise

intensity exceeds the threshold s c , t = c ) (the subscript t indicates that such

a th reshold refers to th e temporal dynamics), two noise-induced maxima occur at
±

a / (2 ν

( 2

c (see Fig. 5.37 ).
The generalized mean-field analysis allows us to explore whether patterns may
emerge from the dynamics of Eq. ( 5.73 ). If the most unstable wave number is con-
sidered - i.e., the condition

ν s gn c / a − 1) /

0 is assumed (see Box 5.3) - according to self-
consistency equation ( B5.3-5 ) patterns are expected to emerge when s gn >

ω ( k ) =

c , i.e.,
when the noise is able to induce bimodality in the corresponding temporal dynamics,
under both the Stratonovich and the Ito interpretation. On the other hand, the classical
mean-field analysis indicates that the order parameter m remains equal to zero for any
pair of values

a /

; thus, in this case, pattern formation is not associated with the
occurrence of a phase transition.
It is worth noticing that stability analysis by normal modes is unable to detect pattern
occurrence. In fact, the equation describing the linear spatiotemporal dynamics of the
ensemble average is

{ a , c }

=− a s gn c φ −

∂ φ
∂

D ( k 0 −∇

2 ) 2

+ φ , (5.79)

t

which gives the dispersion relation,

=− a s gn c − D ( k 0 − k 2 ) 2 γ ( k ) + , (5.80)

which shows that the growth factor γ is always negative for all values of noise-intensity

wave numbers.
Figures 5.38 and 5.39 show some examples of patterns resulting from the numerical
simulation of Eq. ( 5.73 ), under the Stratonovich or the Ito interpretation, respectively.
Well-defined patterns emerge with the dominant wave number dictated by the spatial