Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 256

(a) (b)

Figure 6.6. Examples of model-generated patterns (trees or shrubs in black; grass or
bare soil in white). (a) “Pseudo-labyrinthine” pattern (in an 800

× 800 lattice, with

the same parameters as in Fig. 6.5, D = 0 . 3, uncorrelated random initial condition

with mean 0 350); (b) spot patterns [same

parameters as in Fig. 6.6 (a) but with mean initial condition

. 33, with simulation interrupted at time t =

= 0 . 28]. Figure taken

from D'Odorico et al. ( 2007b ).

6.6 Random switching between two deterministic dynamics
6.6.1 Swift-Hohenberg process driven by dichotomous noise
A recurrent process of pattern formation in environmental fluid dynamics is associated
with Rayleigh-Benard convection ( Chandrasekhar , 1981 ). This phenomenon leads to
the emergence of organization in atmospheric convection, which is often evidenced
by well-defined cloud patterns (Fig. 6.7 ). The organization results from (symmetry-
breaking) thermoconvective instability typically observed when a fluid overlies a hot
surface ( Chandrasekhar , 1981 ). To study the effect of hydrodynamic fluctuations in
systems exhibiting Rayleigh-Benard convective cells, Swift and Hohenberg ( 1977 )
developed a stochastic model based on a biharmonic equation forced by additive white
noise. The deterministic counterpart of this model is frequently used to investigate
deterministic patterns emerging from symmetry-breaking instability of biharmonic
dynamics ( Cross and Hohenberg , 1993 ).
In Chapter 5 we introduced the Swift-Hohenberg model and discussed the ability
of additive and multiplicative Gaussian noise to generate patterns in Swift-Hohenberg
dynamics. Here we consider the case of a Swift-Hohenberg system driven by DMN
and the emergence of patterns induced by the random switching between two de-
terministic Swift-Hohenberg dynamics. The two dynamics have the same (Swift-
Hohenberg) spatial coupling term and two different local kinetics, f 1 , 2 ( φ ):

∂φ ( r , t )

2

k 0 ) 2 = f 1 , 2 [ φ ( r , t )] − D ( ∇ + φ ( r , t ) . (6.11)

∂

t

Buceta and Lindenberg ( 2002a ) considered the case with

A 1 , 2 φ 1

3 2

f 1 , 2 ( φ ) =− ± φ − φ ∓ (6.12)