Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 201

γ

1 s gn 2

k

1 1.5

0.5

s gn 1

1

s gn 0.1

2

Figure 5.19. Dispersion relation for prototype model ( 5.34 ), with spatial coupling a
la Swift-Hohenberg for three values of noise intensity ( a

=− 1, D = 2, and k 0 = 1).

stable state s c .

Mathematically, we can understand the short-term instability by observing that when
φ

φ

0 tend to disappear in spite of their initial amplification when s gn

>

is close to zero the disturbance effect that is due to the (multiplicative) noise tends
to prevail on the restoring effect of f . Conversely, as

3 of

φ grows, the leading term φ

0 .
Once the presence of a short-term instability has been detected, the capability of
spatiotemporal stochastic model ( 5.34 ) to generate patterns can be investigated, for
example, by use of the stability analysis by normal modes (see Box 5.1). In the case
of prototype model ( 5.34 ) we obtain from Eq. ( B5.1-4 ) a dispersion relation

f ( φ ) prevails over g and the deterministic local kinetics f tends to restore the state φ

D ( k 0 − k 2 ) 2 γ ( k ) = a + s gn − , (5.39)

which provides the same threshold s c as Eq. ( 5.36 ) for neutral stability; the maximum
amplification is for the wave number k = k 0 (see Fig. 5.19 ). It follows that statistically

steady periodic patterns, with wavelength λ = 2 π/ k 0 , emerge when the noise intensity

exceeds the threshold s c =− a .

The analysis of the structure function can be used as a prognostic tool to confirm
these results. The linear evolution of the structure function from Eq. ( B5.2-6 )is
∂

2 a k 2 ) 2 S ( k

S ( k , t )

D ( k 0 − = + s gn − , t ) . (5.40)

∂ t

Thus neutral stability corresponds to a critical value of the noise intensity:

D ( k 0 − k 2 ) 2 s c =− a + . (5.41)

When s gn < s c , the structure function tends to zero and no patterns occur. Conversely,

when threshold ( 5.41 ) is exceeded, the linear evolution of the structure function is
divergent and a range of wave numbers becomes unstable. Notice that the wave
number most prone to instability is k a . Therefore

these results are consistent with those obtained from the stability analysis.

= k 0 , which corresponds to s c =−