Deterministic chaos and historical geomorphology: A review and look forward
Introduction
The study of nonlinear dynamics in earth surface processes and landscapes has been dominated by the application of analytical techniques derived largely from mathematics, statistics, physics, computational science, and other fields characterized by experimental laboratory techniques and numerical models. While much of this has been quite fruitful, geomorphology is dominantly (and appropriately) a field-based discipline where ground truth is paramount. Further, many aspects of geomorphology and the earth sciences differ fundamentally from the experimental laboratory sciences (Frodeman, 1995, Bishop, 1998, Harrison, 1999, Baker, 2000). Confronting complex nonlinear dynamics in geomorphology (and other field-based earth and environmental sciences) ultimately requires that we go beyond importing methods and concepts from other fields, and problematize nonlinear dynamics from within a geomorphological (geological, pedological, ecological, geographical, etc.) context (Harrison, 1999, Phillips, 1999, Phillips, 2003a).
This paper focuses on a particular form of nonlinear complexity-deterministic chaos. One major purpose is to review the available techniques for detecting chaos in geomorphic systems, with an emphasis on those developed within geomorphology and related fields and specifically adapted to the type of data and research questions typical of the field. This work also seeks to broaden the conceptual framework for detecting chaos in geomorphology, and to explore the issue of unstable/nonequilibrium systems, vs. stable systems achieving a new equilibrium following a change in boundary conditions. Though the emphasis is on field-based studies, detection of chaos in field evidence is quite relevant to modelling studies, too, as it allows independent evaluation of model implications with respect to chaotic dynamics. This applies equally well to long-term landform evolution models, and to shorter-term dynamical simulations. Such models often have visualizations as output, some of which have properties or implications amenable to testing via the methods described or developed in this paper.
Geomorphic systems are dominantly nonlinear — i.e., outputs or responses are not proportional to inputs or forcings across the entire range of the latter. While there are situations where geomorphologists may restrict themselves to spatial or temporal scales where linearity obtains, in many geomorphic problems nonlinearity cannot be circumvented. The sources and causes of nonlinearity in geology, physical geography, and hydrology, and the prevalence of nonlinearity have been thoroughly reviewed in previous work, by both this author (Phillips, 1992, Phillips, 1999, Phillips, 2003a, Phillips, 2005) and others (Christofoletti, 1998, Werner, 1999, Dearing and Zolitschka, 1999, Sivakumar, 2000, Sivakumar, 2004, Thomas, 2001, Hergarten, 2002, Sparks, 2003, Zehe and Blöschl, 2004).
Nonlinear systems are not all, or always, complex, and even those which can be chaotic are not chaotic under all circumstances. Conversely, complexity can arise due to factors other than nonlinear dynamics. Given this, and the important implications of dynamical instability and chaos (see below), it is advantageous for geoscientists be able to detect chaos in geomorphic systems, to distinguish chaos from other types of irregularity, inconstancy, and complexity.
Section snippets
Instability and chaos
Nonlinearity admits the possibility of dynamical instability and chaos, the equivalence of which has been demonstrated elsewhere in both mathematical (Wiggins, 1990, Zaslavsky et al., 1991) and earth science (Phillips, 1992, Phillips, 1999) contexts. Chaos is common in earth surface systems, but such systems are not all, or always, chaotic. Indeed, many appear to have both stable, non-chaotic modes and unstable, chaotic modes (Phillips, 1999, Phillips, 2005).
Geomorphic systems are
Instability vs. new equilibria
Over the long temporal and broad spatial scales characteristic of many geomorphic problems, it may be difficult to distinguish between changes in system state which represent chaotic sensitivity to small perturbations in nonequilibrium systems, and relaxation to new equilibrium states. This distinction is discussed below.
The most fundamental environmental changes are not merely quantitative variations in specific parameters, but qualitative shifts in system state. The difference is typified by,
Chaos detection in temporal and spatial data
The intrinsic and exploratory values of models and abstract theory notwithstanding, to be useful in geomorphology any theoretical or methodological construct must ultimately answer questions arising from field observations or generate field-testable hypotheses. Thus methods for analyzing chaotic behavior applicable strictly to models, as opposed to empirical data or observations, are not considered. However, as noted earlier, many of the techniques described below are useful for field-based
Chaos in historical sequences
Rather than a long numerical time sequence, historical data in geomorphology more often consists of reconstructions (qualitative or quantitative) at relatively few (sometimes just two) points or slices of time. “Traditional” chaos detection methods based on temporal or spatial series are not applicable. However, by conceptualizing stable vs. unstable dynamics in geomorphically relevant terms such as convergent vs. divergent evolution, chaos can be detected in historical data.
Chaos detection
The techniques for detecting chaos described above are applicable to different types of data and problems — thus no single case study can illustrate them all. Geomorphic examples of applications of time series analysis, phase space plots, spatial signatures, variability within elementary areas, QASA, and landscape entropy are already available in the literature, as cited earlier (4.1 Time series analysis, 4.2 Phase space plots, 4.3 Spatial signatures, 4.4 Variability within elementary areas,
Discussion and conclusions
Though the illustrative examples are weighted toward cases suggesting instability and chaos, there is no intent here to make a case for the relative importance or prevalence of chaotic vs. non-chaotic systems-indeed, it is worth reiterating that many geomorphic systems have both chaotic and non-chaotic modes, and that (in)stability is emergent and scale-contingent. The point is that geomorphic systems can be chaotic, and in many cases it is important to be able to determine whether, or the
Acknowledgements
Constructive criticism by two anonymous reviewers on earlier drafts led to a much different and improved final version.
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