Environmental Engineering Reference
In-Depth Information
to Chapter 5 for more details on order formation in stochastic spatiotemporal systems;
deterministic theories are reviewed in Appendix B.
We need to stress that most of these deterministic and stochastic theories have not
been experimentally validated in the field. A typical example is the case of vegeta-
tion: We are not aware of any conclusive experimental evidence that the vegetation
patterns observed in many regions of the world are actually induced by mechanisms
of symmetry-breaking instability. It has been noticed ( Borgogno et al. , 2009 )that
it is really difficult to test these theories just by comparing model simulations with
observed patterns, because the same types of patterns can be obtained with differ-
ent models. This is because the amplitude equations - which determine important
properties of pattern geometry - belong to only a few major classes (see Cross and
Hohenberg , 1993 ; Leppanen , 2005 ). Thus the claim of relating patterns to processes
by developing process-based models capable of reproducing the observed patterns
is probably too ambitious. Because most models differ in the processes invoked
as key mechanisms of pattern formation (see for example Section 6.3 ), their val-
idation should be based on the assessment of the significance of these processes
( Barbier et al. , 2008 ).
Field validations become even more difficult for patterns induced by noise. In this
case patterns would emerge as an effect of the temporal variability of environmental
drivers. However, long-term observations of patterned landscapes are rare. It is even
harder to envision a field experiment in which we can compare a system undergoing
environmental fluctuations with a “control system” with similar properties except for
being subjected to time-invariant environmental conditions. Thus the view emerging
from the study of noise-induced environmental patterns is that this research field is
still relatively new. Theoretical frameworks are far frombeing experimentally testable,
and the application of existing theories of noise-induced pattern formation has just
started to appear in the environmental sciences. In fact, most of the literature on self-
organized pattern formation is based on the deterministic mechanisms discussed in
Appendix B, and only a handful of studies have investigated the possible emergence
of environmental patterns as a noise-induced effect.
In this chapter we show that new interesting questions exist on the role of ran-
dom drivers in the self-organization of spatially extended systems. To this end, we
provide some examples of applications to the biogeosciences of relatively recent the-
ories of noise-induced pattern formation. The few existing studies invoke stochastic
mechanisms of pattern formation based on noise-induced stabilization of short-lived
coherent structures emerging in the underlying deterministic dynamics (Section 6.4 );
the effect of multiplicative noise in systems undergoing nonequilibrium phase transi-
tions (Section 6.5 ); repeated random switching between two deterministic dynamics
(Section 6.6 ); spatiotemporal stochastic resonance (Section 6.7 ); spatiotemporal
stochastic coherence (Section 6.8 ). After a brief discussion on the typical mathematical
representation of spatial interactions in environmental systems, in the following
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