Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 218

a b

c d

Figure 5.34. Numerical simulation of stochastic models forced by dichotomous
noise: (a) corresponds to the model ( 5.69 ); the parameters are a =− 1, D = 10,

k 0 = 1, k = 0 . 5( τ c = 1), s dn = 2, t = 50 time units. (b) refers to model ( 5.70 )

with parameters a =− 1, D = 2 . 5, k = 0 . 5( τ c = 1), s dn = 20, t = 50 time units.

(c) shows the outcome of model ( 5.71 ) with parameters a =− 1, D = 15, k 0 = 1,

k = 0 . 5( τ c = 1), s dn = 200, t = 50 time units. (d) corresponds to model ( 5.72 ) with

parameters a =− 1, D = 5, k = 0 . 5( τ c = 1), s dn = 200, t = 50 time units.

Figure 5.34 shows an example of four fields resulting from the simulation of
models ( 5.69 )-( 5.72 ). We observe that patterns are qualitatively very similar to those
described in the previous sections, when the stochastic forcing was a white Gaussian
noise. This testifies that the temporal-noise correlation generally does not introduce
new pattern morphologies, which remain essentially dictated by the deterministic
components. However, the autocorrelation of the noise term can have two important
effects. The first effect concerns the pattern intensity: The noise correlation tends to
make patterns sharper and better defined and - in the case of periodic patterns - with
a more evident dominant wavelength. To quantify this effect, we use the ratio i p i p , 0

between the peak value of the power spectrum and the area subtended by the spectrum.

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