Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 212

become unstable if s gn a coincides with the one obtained

in the short-term analysis. Second, the strength D of the spatial diffusive coupling
affects the range of unstable wave numbers. In particular, the unstable wave numbers
decrease when D increases, consistent with the fact that the diffusive coupling intro-
duces spatial coherence in the random field. However, D does not have an impact on
the occurrence of the instability, which depends only on the noise intensity. Therefore
these nonequilibrium transitions are induced only by noise. Third, the most (linearly)
unstable mode is always k max =

> − a . The threshold s gn =−

0 (provided that s gn > − a ) for any value of D .This
means that the linear component of the spatiotemporal dynamics tends to select an
infinite wavelength, which corresponds to a homogeneous state.
In the study of the interplay between additive noise and diffusive spatial coupling
presented in Section 5.3 , the analysis of the steady-state structure function led to a
result very similar to that shown in Fig. 5.29 (see Fig. 5.15 ). In fact, also in that case
the most unstable mode corresponded to k max = 0. However, in spite of the presence

of this well-defined peak at k 0, numerical simulations showed the emergence of

steady multiscale patterns (see Fig. 5.16 ). Because of the existence of a range of
unstable wave numbers (i.e., with positive growth factors

=

), many competing scales
are present in the pattern. The same behavior is also expected to emerge in the case
of multiplicative noise discussed in this section. However, numerical simulations of
( 5.58 ) lead to a different result. In fact, Fig. 5.30 shows that a pattern without a clear
periodicity occurs, but it evolves in time and tends to disappear in the long term. The
pdf of the field

γ

shows that patterns occur during a phase transition from the initial
basic state with order parameter m

φ

= φ = 0 to a new, substantially homogeneous 0

state with m = 0. In particular, in the case shown in Fig. 5.30 , numerical simulations

= . >

give m 100 time units.

The substantial difference with respect to the spatiotemporal model driven by
additive noise is that patterns are now transient. The explanation for this different
behavior can be found in the fact that the spatial coupling is unable to maintain the
system far from the homogeneous condition, in spite of the initial instability. Similar
to the case shown in Fig. 5.17 , the emergence of patterns sustained by multiplicative
noise requires a suitable spatial coupling, which is able to take advantage of the short-
term instability to maintain the system far from the homogeneous basic condition,
thereby creating a patterned state. However, this happens only when a pattern-forming
type of coupling is included in the model. In contrast, when other types of spatial
coupling are considered, they are unable to lock the system far from the homogeneous
state. In these cases, the spatial coupling interacts with the short-term instability
(which can be investigated through the dispersion relation 4 ), but this interaction lasts
only as long as the temporal dynamics are able to sustain the instability. As the initial

0 8for t

4 Recall that the stability analysis is performed on an equation that approximates only the first stages of the dynamics
of the ensemble average in the neighborhood of φ 0 .