Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 203

2 σ

1

30 t 10 20

Figure 5.21. Temporal behavior of the field standard deviation σ for s gn = 0 . 5 and

s gn = 2 . 5. The other parameters are a =− 1, D = 15, and k 0 = 1. The decreasing

curve corresponding to s gn = 1 is amplified by a factor of 10 3 .

in Fig. 5.20 seems to evolve from a labyrinthine to a striped shape. Third, a weak
bimodality may appear in the pdf of

φ

in the dynamics driven by multiplicative noise
(see Fig. 5.20 ). The presence of such bimodality depends on the model structure,
parameter values, and field size.
We conclude this section stressing that the presence of pattern-forming spatial
couplings is not sufficient for stable patterns to occur. The concomitant presence of
short-term instability is in fact necessary. For example, if prototype model ( 5.34 )
is simulated with a noise intensity lower than s c , no pattern emerges. Figure 5.21
shows the temporal evolution of the standard deviation of the field for two noise
intensities: When s gn > s c , the spatial variance increases and tends to the asymptotic

value corresponding to the patterned state shown in Fig. 5.20 , whereas when s m < s c

the spatial variance decreases to zero, namely the field tends to the homogeneous
state.

) terms
We now consider the effect of nonlinear terms on the multiplicative factor g (

5.4.2 The effect of nonlinear g ( φ

). To
this end, we consider the same mathematical structure as in Eq. ( 5.34 ), but with
g (

φ

2 , i.e.,

φ ) = φ

∂φ
∂

3 2

D ( k 0 +∇

2 ) 2

t =

a φ − φ + φ × ξ − φ. (5.42) gn