Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 186

5.2 Additive noise and pattern-forming spatial couplings
Consider the stochastic model
∂φ
∂ t =

a φ + D L [ φ ] + ξ , (5.16) gn

where ξ gn is zero-mean Gaussian white

(in space and time) noise with intensity s gn . Equation ( 5.16 ) is the prototype model to
illustrate how patterns may occur when both multiplicative noise and time-dependent
components are not present in general model ( 5.1 ) [i.e., F ( t )

φ ( r , t ) is the scalar field, a is a parameter, and

= 0and g ( φ ) = 0]. In

this section we study the case in which ] is a pattern-forming spatial coupling; the

case with non-pattern-forming coupling is considered in the next section.
In more detail,

L [ φ

] is modeled here as a Swift-Hohenberg coupling [see
Eq. ( 5.6 )]. As mentioned in Subsection 5.1.2.2 , the Swift-Hohenberg coupling can
be decomposed into a drift term

L [ φ

k 0 φ

and a pattern-forming spatial coupling given
by Eq. ( 5.4 ). The Swift-Hohenberg coupling is one of the simplest couplings able to
form patterns in a deterministic setting, and for this reason we adopt it as an exem-
plifying case. In the sixth chapter we discuss in detail the physical meaning of the
biharmonic operator and the link between the mathematical structure

−

2

k 0 ) 2

− D ( ∇ +

L φ

and the integral form of the spatial coupling ]. This will help us understand why

such coupling is frequently used in a number of applications ( Cross and Hohenberg ,
1993 ). Here we recall only that structure ( 5.6 ) was introduced by Swift and Hohen-
berg ( 1977 ) to study the effect of hydrodynamic fluctuations in systems exhibiting
Rayleigh-Benard convection ( Chandrasekhar , 1981 ). This phenomenon leads to the
emergence of organization in atmospheric convection, which is often evidenced by
well-organized cloud patterns. The organization results from (symmetry-breaking)
thermoconvective instability typically observed when a fluid overlies a hot surface
( Chandrasekhar , 1981 ).

[

5.2.1 Analysis of the deterministic dynamics
We first study the following deterministic dynamics associated with model ( 5.16 ):
∂φ
∂

2

k 0 ) 2

= φ + L φ = φ − ∇ + φ.

f ( ) D [ ] a D ( (5.17)

t

When a < 0, the asymptotic steady state is the homogeneous field φ = 0, regardless

of the initial conditions (i.e., 0 is a stable state). However, if the transient between

the initial condition and the steady state is sufficiently long and D

φ =

a sufficiently large,
the spatial coupling may be able to cause the emergence of transient spatial patterns.
Figure 5.10 shows an example of this behavior: If weak spatial gradients existing
in the initial condition activate spatial interactions, during the transient the spatial
coupling term [i.e.,

/

2

k 0 ) 2

− D ( ∇ + φ ] is able to form well-developed patterns that