Environmental Engineering Reference
In-Depth Information
5
Noise-induced pattern formation
5.1 Introduction
5.1.1 General aspects
A spatially extended dynamical system exhibits
geometrical patterns
when its state
variables show spatial coherence, i.e., their spatial distribution is neither uniform nor
random but exhibits organized spatial structures characterized by a certain regularity.
The occurrence of patterns is an important feature of spatiotemporal systems, as it is
a signature of the existence of order embedded in the underlying dynamics. This fact,
along with the beauty of some of their geometrical features, explains why patterns
have always drawn the attention of the science community.
Patterns are in fact ubiquitous in nature. In particular, a number of environmental
processes are known for their ability to develop highly organized spatial features (e.g.,
Greig-Smith
,
1979
). For example, remarkable degrees of organization can be found
in the spatial distribution of convective clouds (e.g.,
Lovejoy
,
1982
), dryland and
riparian vegetation (e.g.,
Macfadyen
,
1950
), sand ripples and dunes (e.g.,
Lancaster
,
1995
), river channels (e.g.,
Allen
,
1984
;
Ikeda and Parker
,
1989
;
Rodriguez-Iturbe
and Rinaldo
,
2001
), soil cracks (
Lachenbruch
,
1961
), coastlines (
Bird
,
2000
;
Ashton
et al.
,
2001
), peatlands (
Eppinga et al.
,
2008
, e.g.), and arctic hummocks and patterned
ground (
Gleason et al.
,
1986
).
These features are often undetectable on the ground, but have become visible
with the advent of aerial photography (e.g.,
Macfadyen
,
1950
). Figures
5.1
,
5.2
,
5.3
,
5.4
,and
5.5
show some examples of spectacular spatially periodic natural pat-
terns in a number of landscapes around the world. These patterns exhibit amazing
regular configurations of vegetation, landforms, or clouds. In some cases patterns
may spread over relatively large areas (up to several square kilometers) (
White
,
1971
;
Eddy et al.
,
1999
;
Valentin et al.
,
1999
;
Esteban and Fairen
,
2006
) and can be
found on different soils and with a broad variety of vegetation species and life-forms
167
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