Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 208

D

3

0

2

1

0

s gn

2.5 5 7.5 10

Figure 5.26. Curve of marginal instability for the VPT model according to the gen-
eralized mean-field technique ( k 0 =

1 and = 1). The dashed vertical line marks

the threshold s c = 0 . 5.

[i.e., the condition ω ( k ) = 0 is imposed], we obtain

d φ i

d t =− φ

2

i ) 2

i ) 2 i (1 + φ + (1 + φ ξ − D eff ( φ − φ ) . (5.56) gn i i

We can determine the steady-state pdf, p st ( ), for zero-dimensional model ( 5.56 )

by applying the methods described in the second chapter. Using the self-consistency
condition, we obtain the curve shown in Fig. 5.26 , which marks the boundary between
the region of parameter space in which the self-consistency equation has solutions
different from zero (region inside or above the curve) and the region outside the curve,
where only the solution

φ i ; φ i

0 exists. Therefore, under the simplifying assumptions
of the mean-field approach (see Box 5.3), patterns can occur only for the pairs
{

φ =

that fall above the region bounded by the curve. In particular, points of this
curve correspond to pairs of values of

s gn , D }

{ s gn , D } associated with wave numbers { k x , k y }

that satisfy the condition ω ( k ) = 0, i.e., pairs of wave numbers { k x , k y } that satisfy

k 0 −

k y )] 2

2

2 [2

− cos( k x ) − cos( = 0 . (5.57)

There are clearly infinite possible pairs of wave numbers satisfying condition ( 5.57 ),
as well as many other unstable modes corresponding to the region above or inside
the curve. The specific shape of the pattern emerging from the numerical simulations
of the model depends on the initial and boundary conditions. However, the threshold
s gn = 5 determined by short-term marginal stability has to be exceeded for

patterns to emerge (see dashed line in Fig. 5.26 ). Moreover, the transition to a patterned
state is reentrant, in that the stable state of the system switches back to the disordered
state,

s c = 0 .

φ = 0, for large values of s gn .Infact,Fig. 5.26 shows that the emergence of i