Environmental Engineering Reference
In-Depth Information
r ) is a kernel function. Depending on the shape of the kernel func-
tion, this integral spatial coupling may belong to the pattern-forming or non-pattern-
forming category. Further details on these integral operators are provided in Chap-
ter 6 and in Appendix B, along with a description of the linkage between operators
based on derivatives and integrals.
Note that the presence of a pattern-forming coupling by itself does not mean
that a spatial pattern will necessarily emerge. In fact, in many cases the shape of
the function f (
where
ω
( r
) or the intensity of the coupling may not allow the occurrence of
steady patterns. In these cases, noise may have the key role of creating the dynamical
conditions for which the spatial coupling exerts its pattern-forming action even though
the deterministic counterpart of the dynamics is unable to generate stationary patterns.
Several examples are provided in this chapter.
φ
5.1.2.3 Prognostic tools for pattern detection
Unfortunately, the mathematical complexity of the spatiotemporal models expressed
by Eq. ( 5.1 ) hampers general analytical solutions. For this reason, several approxi-
mate analytical techniques have been developed for obtaining useful indications about
pattern formation. When the occurrence of a dominant wavelength is the main symp-
tom of pattern formation, the first available prognostic tool is provided by stability
analysis by normal modes. This analysis is based on the idea of disturbing the basic
state (i.e., the homogeneous equilibrium state) of the system with an infinitesimal per-
turbation and to assess whether the perturbation grows in time (in that case patterns
may emerge). The stability of the basic state is assessed with the dispersion equation,
which is the condition relating the growth factor
γ
of the perturbation to its wave
=
,
numbers k
k y ). We refer the interested reader to Box 5.1 for details on the way
the dispersion relation can be obtained.
Dispersion relation ( B5.1-4 ) can exhibit different scenarios: (i) the growth factor
( k x
γ
is negative for all wave numbers. In this case no wavelength is unstable and no pattern
may emerge; (ii) there is a range of wave numbers with positive growth factors, with
γ
exhibiting a maximum for a wave number k max that is finite and different from zero.
In this case, periodic patterns occur with a wavelength close to
λ =
2
π/
k max (the
exact wavelength will depend also on the boundary conditions); (iii)
γ
( k max )
>
0
but the most unstable mode is k max =
0. In this case multiscale patterns may
occur.
The dependence of the dispersion relation on the noise intensity s m is generally
studied to determine the transition from the first scenario (i.e., the case with no
unstable modes) and the other two (i.e., with unstable modes). In particular, from
Eq. ( B5.1-4 ) it is possible to obtain the threshold value of the noise intensity
f (
φ 0 )
+
Dh L ( k )
s c
=−
,
(5.8)
g S (
φ 0 )
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