Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 194

S st i p

1.5

b

a

30

1

20

0.5

10

2.5 k

D

0.5 1 1.5 2 5 10 15 20

Figure 5.15. (a) Shape of the structure function at steady state for model ( 5.29 );
the upper curve corresponds to s gn = 3).

(b) Dependence of the peak intensity on the strength of the spatial coupling D ;
the solid, dotted, and dashed curves correspond to a

3, the lower one to s gn = 1( a =− 1 , D =

=− 0 . 01 , − 0 . 1, and − 1,

respectively.

If we consider again the prognostic tools to predict pattern occurrence, it is possible
to have a clue about the role of additive noise; in fact, the steady-state structure function
corresponding to stochastic model ( 5.29 ),

s gn
2( Dk 2

S ( k , t ) =

a ) ,

(5.30)

−

exhibits a maximum at k 0 (see Fig. 5.15 ). It follows that no strong periodicity is

selected, but a range of wave numbers close to zero competes to produce patterns
more complex than those discussed in the previous section. However, it is difficult
to anticipate the geometrical characteristics of these patterns from only the analysis
of the structure function. Therefore numerical simulations are also needed. An ex-
ample is shown in Fig. 5.16 , which shows patterns substantially different from those
observed when pattern-forming spatial couplings are present (see Fig. 5.13 ). As ex-
pected from the analysis of the structure function, no clear periodicity is detectable,
and many wavelengths are present; moreover, the boundaries of the coherence regions
are irregular. These spatial structures fall then in the class of multiscale fringed pat-
terns, which are especially relevant in the environmental sciences, in which there are
several instances of variables that exhibit a spatial distribution very similar to the one
shown in Fig. 5.16 ; a typical example is the distribution of vegetated sites in semiarid
environments; see Chapter 6.
Notice that in this case the other prognostic tools used to assess pattern occur-
rence - i.e., the dispersion equation and the generalized mean-field analysis - fail to
provide useful indications. Even though these patterns are very different from those
encountered in the previous section, the pdf of the field exhibits similar characteristics
(compare the second row of Fig. 5.16 with Fig. 5.13 ): The pdf is unimodal and its
mean coincides with the basic homogeneous stable state (i.e., m

=

0). There-
fore, in this case, no phase transitions in the mean occur, as indicated by the classical
mean-field analysis. This behavior follows the general notion that Gaussian additive

= φ = 0