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Nonlinear dynamical systems in geomorphology: revolution or evolution?
Geomorphology, 5 (1992) 219-229 Elsevier Science Publishers B.V., Amsterdam
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Nonlinear dynamical systems in geomorphology: revolution or evolution? Jonathan D. Phillips Department of Geography and Planning, East Carolina University, Greenville, NC 27858-4353, USA (Received August 28, 1991; revised November 12, 1991 ; accepted November 15, 1991 ).
ABSTRACT Phillips, J.D., 1992. Nonlinear dynamical systems in geomorphology: revolution or evolution?. In: J.D. Phillips and W. Renwick (Editors), Geomorphic Systems. Geomorphology, 5: 219-229. Geomorphic systems typically exhibit complex, apparently random behaviors and patterns in both spatial and temporal domains. This complexity can arise from the cumulative impacts of individual process-response mechanisms which are far too numerous to be accounted for in individual detail, or due to multiple controls over process-response relationships which operate over a range of spatial and temporal scales (stochastic complexity). Nonlinear dynamical systems (NDS) theory - - which includes specific techniques and concepts such as chaos, dissipative structures, bifurcation and catastrophe theory, and fractals - - shows that extreme complexity can also arise due to the nonlinear dynamics and couplings of relatively simple systems represented by relatively small equation systems (deterministic complexity ). While emphasis in earth science has been largel), on stochastic complexity, geomorphologists have long recognized the presence of deterministic complexity as well. Recent developments in the study of nonlinear dynamical systems have implications for the ability to make reliable long-term predictions and therefore applications of NDS in geomorphology may be revolutionary to some extent. However, virtually all the basic conceptual underpinnings of NDS theory can be mapped onto existing and even traditional theoretical concepts in geomorphology. In this sense, the use of NDS concepts in understanding earth surface processes and landforms is evolutionary. The link between traditional and NDS-based concepts is illustrated by analyzing a generalized geomorphic mass-flux system. It is shown that the system is in most cases unstable (consistent with chaos or deterministic complexity). However, the three attractor states of the system correspond exactly to the states of aggradation, degradation, and steady state.
Introduction
Geomorphologists have studied earth surface processes and landforms as dynamical systems in an explicit sense at least since systems theory was introduced into the discipline by Strahler ( 1950, 1952 ). The dynamical systems approach was further championed by Chorley (1962) and Chorley and Kennedy (1971 ). In the past two decades mathematicians, physicists, and chemists have revoluCorrespondence to: Dr. J.D. Phillips, Department of Geography and Planning, East Carolina University, Greenville, NC 27858-4353, USA.
tionized theoretical approaches to dynamical systems by focusing on the dynamics of nonlinear systems that are not in equilibrium, and on modeling the often abrupt changes of state in nonlinear systems. Specific concepts and mathematical techniques associated with nonlinear dynamical systems (NDS) theory have been widely touted (even in the popular press ) and applied in virtually every scientific discipline. These concepts include chaos, fractal geometry, and catastrophe theory. The utility of NDS concepts and techniques in geomorphology has been recognized for more than a decade (Thornes, 1981, 1983, 1985; Graf, 1979, 1988; Huggett,
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1980, 1985). In recent years the call for increasing application of NDS theory to geomorphology has become more insistent, in part because of the power of the ideas to solve geomorphic problems, and in part to link geomorphology more closely to the other physical sciences (Huggett, 1988; Culling, 1987, 1988; Montgomery, 1989; Slingerland, 1989; Malanson et al., 1990; Turcotte, 1990). The purpose of this paper is threefold. First, the evidence that earth surface processes and landforms behave as dissipative systems with deterministic complexity (chaos) is reviewed. Second, fundamental conceptual implications of NDS theory will be related to existing and traditional theoretical concepts of geomorphology. Finally, an attempt will be made to demonstrate the fundamental linkages between NDS theory and traditional concepts of landscape evolution.
Nonlinear dynamical geomorphic systems In general, the concepts and techniques of NDS theory will be applicable to geomorphic systems to the extent that such systems may be regarded as dissipative structures and to the extent that it is reasonable to suspect deterministic complexity, that is, chaotic behavior arising from relatively simple (nonlinear) mechanics and structures. Dissipative systems or structures are those in which energy is dissipated in maintaining order in states removed from equilibrium. A typical manifestation involves self-organizing processes which produce an orderly sequence of system configurations, each series of which is induced by a fluctuation. Huggett (1988) gives the geomorphological example of wave and fluvial bedform systems which pass through a series of transient forms associated with changing hydrodynamic regimes. Numerous other examples could be cited, as for instance the sequence of changes in sand beach morphology before, during, and after a storm. In these cases the basic characteristic of main-
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taining order away from equilibrium via energy dissipation is obvious and clearly supports treatment of geomorphic systems as dissipative structures. It can be argued more generally (see Huggett, 1988) that any systems which are open and which exhibit tfiresholds are dissipative. Even more specifically, if one starts with the fact that geomorphic systems are open (Strahlet, 1952; Chorley, 1962 ) and the assumption that some order is present, then the additional assumption of nonlinearities predetermines dissipative behavior. This arises because a rigorous definition of equilibrium implies a single-valued (linear) relationship between input and output (albeit a time-bound, perhaps transient relationship; Howard, 1988). These arguments imply that broader-scale geomorphic systems encompassing multiple inputs and outputs are also dissipative structures. Gu et al. (1987) and Montgomery ( 1989 ) have explicitly treated river systems as dissipative. Deterministic complexity is termed chaos in the NDS lexicon. Chaos describes irregular, apparently random behavior which arises deterministically due to nonlinear couplings in sometimes relatively simple systems. Chaos is distinguishable from stochastic complexity by extreme sensitivity to initial conditions and by increasing divergence of results or predictions with time. Highly irregular behavior is widely known in geomorphology, but the extent to which it is attributable to chaos, rather than to environmental heterogeneity or stochastic forcings, is unknown. One argument, typified by Culling ( 1987, 1988 ), is that chaos is to be expected in the landscape because many of the fundamental physical and chemical processes shaping the landscape (such as turbulent flow) are known to exhibit chaotic behavior. Turbulence is a well-known example of chaos (Eckmann, 1981 ). Chaotic behavior has also been found in climatic dynamics (Lorenz, 1963, 1964) and in data describing meteorological processes
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
such as rainfall (Rodriguez-Iturbe et al., 1989). Lorenz's work on predictability in atmospheric dynamics was indeed the pioneering work in the field of chaos. It should be noted that arguments supporting the presence of deterministic chaos in geomorphic systems do not preclude the presence of stochastic complexity instead of, or in addition to, chaotic dynamics (Huggett, 1988 ). Thus far, applications of chaos theory in earth sciences have been mainly pedagogical (Culling, 1987, 1988; Slingerland, 1989; Turcotte, 1990), but there is an accumulating literature supporting the presence of deterministic chaos in specific earth surface systems. Keilis-Borok (1990) has applied chaos to tectonics and earthquake prediction, and Newman and Turcotte (1990) developed a nonlinear dynamical model of fluvial erosion which produces a fractal landscape (characteristic of chaotic behavior). Wilcox et al. ( 1991 ) found no evidence of chaos in a complex time series of snowmelt runoff. This is consistent with Phillips (1992 ), who concluded that chaos is unlikely in most cases of surface runoff generation. Phillips also suggested, however, that chaotic behavior is possible with regard to infiltration-excess runoff generation and the hydraulic geometry of overland flow. Irregular, unpredictable behavior arising from simple deterministic systems has also been found. Phillips (1990a, b) discovered such behavior arising from a simple system based on the D'Arcy-Weisbach flow resistance equation for stream channels. Culling and Datko (1987) found that an irregular, fractal surface can arise from a Davisian downwasting regime descri]ged by a simple diffusion equation, and supported the finding with data describing soilcovered surfaces. Snow and Slingerland (1990) showed that complex, nonlinear behavior can arise from simple deterministic equations describing the evolution of the long profiles of rivers. Chaos is also linked to asymptotic instability. The latter is defined as exponential diver-
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gence from an equilibrium state following a perturbation. The concept of asymptotic stability is relevant to local stability and small perturbations-- in a geomorphic context these may be interpreted in terms of stability in response to perturbations which are not severe enough to destroy feedback mechanisms within the system. Asymptotic instability (which applies to perturbations of any magnitude) is a necessary condition for general chaos. (i.e., a system which typically exhibits chaotic behavior). Asymptotic instability is both neccessary and sufficient to indicate that a system c a n exhibit chaotic behavior under certain circumstances. The crux of the argument is that a system can be chaotic if at least one Lyapunov exponent is positive. The theory of Lyapunov exponents is a generalization of linear stability theory, and Lyapunov exponents are a standard tool for detecting and measuring chaos (Wiggins, 1990, p. 607). A dynamic system with n components will have n Lyapunov exponents, which determine the rate at which initially nearby points in the system phase space diverge or converge as the system evolves. The Lyapunov exponents are equivalent to the real parts of the eigenvalues of a Jacobian matrix of the equation system where matrix entries represent rates of change of each system variable as functions of the others (these concepts are covered in any text on NDS or chaos, see e.g. Thompson and Stewart, 1986. Briefer explanations in geoscience contexts are given by Fraedrich, 1987 and Phillips, 1992 ). If a system is asymptotically unstable then at least one Lyapunov exponent is positive and chaos is possible (Fraedrich, 1987; Wiggins, 1990). Several different types of geomorphic systems have been found to be asymptotically unstable in the sense above, including river hydraulic geometry (Slingerland, 1981; Phillips, 1990a), fluvial sediment budgets and sediment transport systems (Phillips, 1987; Rhoads, 1988), and the state factor model of soils (Phillips, 1989).
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Overall, the evidence points clearly to the legitimacy of applying NDS concepts and techniques to geomorphic systems, and that fundamental insights into landscape evolution might be gained by doing so.
Nonlinear dynamical systems and traditional geomorphology At the most basic conceptual level nonlinear dynamical systems theory has four characteristics which, while not necessarily unique to the NDS framework, in aggregate distinguish it from other approaches. First, the study of NDS is concerned with determining the evolution (trajectories) of systems between, or in relation to, equilibria. Second, the systems involved are dissipative structures. Third, system evolution is characterized by discontinuities which in mathematical terms are bifurcations or catastrophes. Fourth, there is at least the possibility of deterministic chaos. Fractal geometry is not intrinsically linked to NDS, but is a critical tool in the detection and description of chaos. Evidence that the four characteristics above are fundamental to the study of NDS is given by the fact that all are routinely included in textbooks on NDS analysis (e.g. Thompson and Stewart, 1986; Wiggins, 1990). The relationships between these four basic ideas of NDS theory and traditional geomorphic thought will be explored below.
Equilibrium and trajectories The concept of equilibrium is a fundamental one in geomorphology (see reviews by Howard, 1988; Montgomery, 1989), though the term has been variously and often poorly defined. The most fundamental geomorphic concept of equilibrium is that of Hack (1960) who held that the landscape is always in, or moving toward, a state of dynamic equilibrium where forms are adjusted to prevailing processes and
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environmental controls. NDS studies are typically concerned with the evolution of a physical system toward some stationary end-state, or with the trajectories of a system in state or phase space. These trajectories move away from points or restricted spaces which are repellors, and toward attractors. Repellors and attractors are, respectively, asymptotically unstable and stable equilibrium states (Thompson and Stewart, 1986). In an unstable equilibrium state, the response of the system to a perturbation will not return the system to, or nearly to, its pre-disturbance state. Rather, the system will remain in disequilibrium or move to a new, different equilibrium state. Semantic arguments with regard to geomorphic equilibrium are somewhat risky, due to the various definitions, some of which are inconsistent with each other and with definitions of other disciplines (Howard, 1988; Montgomery, 1989). Nonetheless, we proceed by noting that if an unstable equilibrium - - unlikely to persist in the face of inevitable environmental perturbations - - can be equated to nonequilibrium in classical geomorphology, then the basic concepts of phasespace trajectories in NDS theory and landscape evolution under dynamic equilibrium are entirely consistent. In the case outlined above the dynamic equilibrium landforms are analogous to attractors or stable equilibria. The evolution of landscapes between these dynamic equilibrium states over time is a trajectory on attractors. NDS theory allows and accounts for far richer phase space behavior than sequential evolution between successive attractors. However, it should be clear that classical dynamic equilibrium studies of landscape evolution can be interpreted in a NDS context, and are not inconsistent with a NDS-based approach. Reinterpretation of existing data and models in the dynamic equilibrium framework may provide a starting point for exploring the possibility of richer phase-space behavior.
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
Dissipative structures In at least one sense the idea of dissipative structures is clearly implicit in classical geomorphology. Dissipative structures will exist in any system characterized by open system behavior and irreversible processes. Most geomorphic processes are intuitively irreversible: debris does not move back upslope, and weathered material does not reconstitute. Irreversibility is also inherent in the application of the second law of thermodynamics to geomorphology. Open system conceptualization and application of the second law have a rich tradition in the application of entropy concepts in geomorphology (Leopold and Langbein, 1962; Langbein and Leopold, 1964; Scheidegger and Langbein, 1966; Yang, 1971; Chorley and Kennedy, 1971). Entropy concepts derived both from thermodynamic analogies and from information theory have been appplied; Zdenkovic and Scheidegger (1989) demonstrated their equivalence in a landscape context. In terms of recognizing and applying the basic characteristics of dissipative structures there is an extensive tradition in geomorphology. Although the validity of applying entropy concepts in geomorphology has been questioned (Davy and Davies, 1979 ), the point here is not to defend or justify such approaches per se, but to note their consistence with elements of NDS theory. While the characteristic of nonequilibrium as a source of order in dissipative system has not been explicitly explored in landscape studies to any great extent, adoption of the dissipative systems conceptual framework in geomorphology seems to be a logical evolutionary step.
Bifurcations Nonlinear dynamical systems often exhibit discontinuities in their evolution, called bifurcations. These discontinuities may represent a transition from one equilibrium state to an-
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other, or from regular or periodic behavior to chaos. Analytically, these discontinuities can be explored and described using bifurcation and stability analysis and catastrophe theory. Bifurcations in nonlinear dynamical systems are directly analagous to thresholds in geomorphic systems. It is equally accurate to reverse the linkage by defining thresholds as fundamental bifurcations in geomorphic systems. While explicit applications of catastrophe theory in geomorphology date back only to the late 1970s, recognition of discontinous evolution and the presence of thresholds is so fundamental to surficial process studies that it must (implicitly) date to the beginnings of the discipline. No attempt will be made to trace the history of the concept; well-known applications include the work by Hjulstrom (1935), Leopold and Wolman (1957), and Schumm (1972). Explicit consideration of the role of thresholds in geomorphic evolution dates at least to the work by Howard ( 1965 ). In general, it is the widespread recognition of the importance of thresholds/bifurcations in geomorphology that led to the earliest applications of NDS theory to landscape studies (Graf, 1979; Thornes, 1981, 1983). The NDS concept of bifurcations is thus entirely consistent with both prevailing and classical geomorphic thought.
Chaos The complicated, irregular behaviors and patterns often observed in geomorphic systems and landscapes have typically been interpreted in terms of stochastic complexity. That is, it has been widely believed that if one could somehow collect enough data or enough extremely detailed knowledge the apparent randomness could be resolved into deterministic components. Even those who model landscapes and surface processes stochastically have typically conceded that the stochastic behavior was largely or wholly apparent, resulting from the cumulative effects of innumera-
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ble deterministic events. But even authors such as Scheidegger and Langbein (1966), who provide a good example of this thinking, recognized at least a possibility of deterministic complexity. Scheidegger and Langbein ( 1966, pp. 1, 2) explicitly acknowledge that results indistinguishable from randomness can arise from even simple physical deterministic systems. Mann's (1970) discussion of randomness in nature from the geologist's perspective identified the need to determine whether randomness is inherent or apparent (i.e., deterministic or stochastic complexity) as a fundamental goal of earth science research. Application of the "random model" of stream network topology engendered a long-running debate as to the physical basis of the model, centered on the question of whether randomness can be considered a physical property of earth surface systems (see Abrahams, 1984). The very existence of such a debate implies that at least some geomorphologists recognized the possibility of deterministic complexity. The seeds of chaos are apparently sown in many physical systems. The presence and prevalence of thresholds/bifurcations in geomorphic systems is an especially compelling reason to suspect chaotic behavior. From a mathematical perspective every proposed typology of routes to chaos (i.e., mechanisms by which system behavior is transformed from regular or periodic to chaotic behavior) is based on bifurcations ( T h o m p s o n and Stewart, 1986, Wiggins, 1990). Consideration of deterministic complexity has not been the rule in geomorphology, but its presence has long been recognized. Adoption of a NDS framework and chaos theory would dramatically increase the attention paid to deterministic complexity and would eliminate the frequent implicit assumption that observes patterns and behavior indistinguishable from randomness are stochastic complexity arising from insufficiently detailed knowledge of the system.
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Mass flux model
The general consistency of traditional and NDS-based approaches to the study of geomorphic systems can be demonstrated by analyzing a generalized model based on mass fluxes. At the simplest level any geomorphic system can be depicted as a mass flux system including inputs (I), storage within the system (S), and outputs (O). Nonlinear dynamical systems studies typically taken an algorithmic approach, using numerical simulations of the equation systems or explorations of large timeseries data sets. However, insight into system dynamics can be obtained even for a generalized system where only qualitative relationships (whether a change in any component results in positive, negative, or negligible effects on any other component) are known. Solutions are obtained by linearizing the nonlinear equation system around an equilibrium condition (C) using Taylor analysis. This makes use of the fortuitous property that the stability characteristics of the linearized system are exactly the same as those of the nonlinear "parent" (Wiggins, 1990). The linearized equations are of the form:
dX/dt=AX
(1)
where X is the vector of variables describing the system components and A is an interaction matrix giving the components' influences on each other. Solutions for the behavior of a small perturbation x (t) over the short term are of the form:
x(t) =vCt exp (~.t)
(2)
where v are the eigenvectors and 2 the eigenvalues of A. Clearly, if the real part of any eigenvalue is positive the system diverges from the pre-disturbance equilibrium exponentially and the system is asymptotically unstable. The Routh-Hurwitz criterion can be used to determine the signs of the real parts of the eigenvalues even when only the qualitative nature of A is known (i.e., positive, negative, or zero signs
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
in each cell of the interaction matrix). RouthHurwitz analysis is discussed in detail by Puccia and Levins (1985); example geomorphic applications include work by Slingerland ( 1981 ) and Phillips (1990a). The real parts of the eigenvalues of the linearized system of eq. (2) are equivalent to the Lyapunov exponents of the full nonlinear system (Fraedrich, 1987; Wiggins, 1990, p. 607). Presence of any positive Lyapunov exponent indicates the possibility of chaos. Local stability near an equilibrium, as described above, is known as asymptotic stability. It is related to the behavior of nonlinear dynamical systems because repellors are asymptotically unstable while attractors are asymptotically stable (Thompson and Stewart, 1986, pp. 9, 204). The validity of the linearization approach for analyzing hyperbolic or asymptotic stability in chaotic systems is shown by Thompson and Stewart (1986, pp. 111, 112; 200-204) and Wiggins (1990, pp. 607-610). Linear systems can be asymptotically unstable, while chaos is a property of nonlinear systems. However, asymptotic instability of a nonlinear system is sufficient to indicate the possibility of sensitive dependence on initial conditions and increasing divergence over time which indicate chaos. If a system is determined to be asymptotically stable then the system configuration represents an attractor and the system can be expected to return to that state or configuration after a disturbance. Dynamics of the system trajectory as it returns may vary; discerning them requires numerical data. If a system is asymptotically unstable that configuration represents a repellor and must be a transient state. Conditions (perhaps hypothetical) under which the generalized system would be stable can be identified. These represent attracting states to which specific realizations of the generalized system would evolve under certain specified conditions. The influences o f / , S, and O on each other in the geomorphic mass flux system are easily
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determined from the mass balance relation, assuming input to be externally imposed and not directly influenced by S or O. This assumption is reasonable in the case of climatic and tectonic forcings in most cases:
OI/dt= OS/dt+ O0/dt
(3)
Even a simple system based on eq. (3) can be unstable and potentially chaotic, due to the absence of any self-stabilizing feedback and the "competitive" relationship between storage and output. However, a plausible stable geomorphic mass flux system can be envisaged by adding a selfstabilizing negative loop for storage, and by changing the link from storage to output from negative to positive. Mass storage can indeed be self-limiting. At high rates finite capacity limits storage. When storage is being depleted selective transport tends to limit removal as remaining material is progressively more difficult to entrain. A positive influence of storage on output could occur where sediment transport is selective and episodic. During lowenergy events heavier particles would be deposited in storage with lighter particles transported through the system. The accumulation of coarser material in storage provides a supply of transportable mass for high-energy events, thus increasing output for these events. Because mass either moves through the system or is removed from storage during high-energy events, and because the allocation of input to storage or output during low-energy periods on particle size rather than total input rates, the storage-output relationship is not competitive. A mass flux system with the structure outlined above has been shown to be asymptotically stable by Phillips ( 1991 ). It is clearly possible for a geomorphic mass flux system to be stable and non-chaotic. However, the formulation above assumes that no intrinsic thresholds are exceeded. The exceedence of such thresholds - - usually in the form of force/resistance or power/resistance ratios - - could provide feedbacks to input, storage,
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and output and dramatically influence system behavior We can add a power/resistance ratio as a fourth component of the system. Power/resistance thresholds (P/R) determine the allocation of input to storage or output. Power rather than the more familiar force/resistance terminology is used because the concern is with transport of mass in bulk rather than individual particles. Resistance in this context is defined by the critical power necessary to transport the available mass. IfP/R > 1 erosion and transport of mass out of the system will occur. IfP/R < 1 deposition and non-removal of storage will occur. Thus the P/R component of the system has a positive link to output and a negative link to storage. Because erosion within the system can augment external mass input when the threshold is exceeded, a positive link from P/R to input is established. Storage has a negative influence on the P/R threshold, since more or less stored material leads to higher or lower critical power thresholds for transport, and thus to lower or higher values of P/R. The interaction matrix linking the qualitative mutual influences of L S, O and P/R, (A in eq. 1 ) is shown in Table 1, where each entry aij represents the influence of the ith (row) component on the jth (column) component. The characteristic polynomial of A is: (4)
O~0~4 "~ OL123 "~ OL222 "Jr"OL32 Jr" O~4 = 0
The coefficient ~k of the characteristic polynomial is equivalent to the Feedback Fk at level k of the system (Puccia and Levins, 1985 ):
Fk= ~ (-1)m+~L(m,k)
(5)
TABLE 1 Interaction matrix for geomorphic mass flux system Parameter
1
Input
0
al2
a13
Storage
0
-- a22
- a23
S
0
P/R 0
-- a24
Output
0
- a32
0
0
Power/resistance
a4!
-a42
a43
0
where the right-hand side of eq. (5) is the determinant of A and L(m, k) is the product of rn disjunct loops with k system components. Feedback measures the influence of system components on each other. A disjunct loop is a sequence of one or more aij which have no component i o r j in common. Feedback at level 0 is defined as F0 = a0 = - 1. For the system in Table 1: F1 = - a 2 2
(6a)
F2 = ( - a 2 3 ) ( - a 3 2 ) q- ( - a 2 4 ) ( - a 4 2
(6b)
F 3 ( - a 2 4 a 4 3 a 3 2 ) -1 ( - a 2 4 a 4 1 a l 2 )
(6c)
F4 =a41 a 1 3 ( - a 3 2 ) ( - a 2 4 )
(6d)
The Routh-Hurwitz criterion (Puccia and Levins, 1985, pp. 167-172) states that the necessary and sufficient conditions for all real parts of the eigenvalues to be negative are: Fi<0
for all i and
(7a)
FtF2 +F3 < 0
for n = 3 or n = 4
(7b)
F~ and F3 are negative, but F2 and F4 must be positive. The Routh-Hurwitz criteria cannot be met. The system is unstable and potentially chaotic. While such a generalized model cannot represent all geomorphic systems, it is representative of many and suggests that the potential for chaos exists in many geomorphic systems. An obvious question is the extent to which the outcome of asymptotic instability and potential chaos depends on the assumptions as to the signs of the links in the model. This sensitivity can be explored in conjunction with the identification of the attractors of the system. Any system configuration where the feedback mechanisms identified above operate is a repellor for all trajectories in state space. The attracting states may be determined by identifying all feasible conditions where the RouthHurwitz criteria would be satisfied. These are alternate system configurations which are not valid for the long term or the general case, but which may exist in particular situations and
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
over restricted time periods. They also represent the necessary conditions for invalidating the instability outcome obtained above. For the system described above F4 > 0. For this coefficient to be negative the sign of one a~j would have to be reversed. It is not physically reasonable under any circumstance to reverse the signs of a32, representing the negative effect of output on storage; a4~, the positive effect of P/R on input (this could be zero, but would still result in an unstable system); or al 3, the positive influence of input on output. The influence of storage of the power/resistance ratio (a24) could be conceivably be positive in a situation where stored material is transportable, while the underlying surface is resistant. In this situation increasing storage could decrease resistance and raise the P/R ratio. Storage depletion would likewise increase resistance by exposing the resistant surface and reduce P/R. Reversing the sign of a24 from negative to positive would also make F2 negative as required by the Routh-Hurtz Criteria, if the storage-P/R relationship given by a24a42 >aE3a32. The latter represents outputstorage relationships. If a23>0 as discussed earlier, F2< 0 in any case. However, in this scenario F3, which was negative in the original formulation, becomes positive if a24 > 0, and violates the stability criteria. The mass flux system including a power/resistance threshold is thus unstable in any feasible configuration. The outcome of asymptotic instability and possible chaotic behavior is not sensitive to the assumptions about the links in the model.
Discussion The stable (strange) attractors of the mass flux system are those states where thresholds are not exceeded and the P/R component of the system is irrelevant. Storage must have a positive influence on output for stability to obtain. Due to this condition and the absence of thresholds in the attracting states, the latter
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must be associated with ongoing aggradation or degradation (or the transient steady state). Under steady state there would be no net storage change, so the positive influence of storage on output would be necessary to offset any additions or depletions to mass storage. Under a degradation regime of declining storage, any additions must stimulate increased output to maintain the regime. Likewise, in an aggrading system with increasing storage, any short-term depletion would need to be offset by a decline in output. Note that the strange attractors of the chaotic mass flux system correspond to the states of aggradation, degradation, and steady-state, thus linking the NDS-based analysis with classic - - and rather obvious - - geomorphic concepts. The linkage of instability and potential chaos with power/resistance thresholds is also fully consistent with prevailing geomorphic thought, which holds that complex response of the landscape and irregularities in landscape evolution are associated with thresholds. Chaos in geomorphic systems implies that long-term prediction is impossible and questions the concept of equifinality. The latter implies that different processes or different landscape histories produce similar landform results. In a chaotic system this is not possible, as the system state is sensitively dependent on initial conditions and differences are magnified over time. To the extent equifinality may be characteristic of chaotic geomorphic systems, it must occur at scales substantially broader than those at which chaos is observed. At these broader temporal and spatial scales equifinality is possible in a general sense with the similar end states corresponding to the strange attractors. Chaos also offers the promise that apparently random and exceedingly complex patterns and behaviors may have relatively simple deterministic explanations. While the proclamation that many (perhaps most) geomorphic systems may exhibit chaotic behavior may at first glance seem revolutionary, such is not necessarily the case.
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Asymptotic instability of the type discussed here or complex behavior arising from simple deterministic geomorphic models has already been found in a number of geomorphic systems (Slingerland, 1981; Phillips, 1987, 1989, 1990a, b; Thornes, 1985; Culling and Datko, 1987; Newman and Turcotte, 1990; and Snow and Slingerland, 1990; among others). It is important to recognize that an unstable system without positive self-enhancing feedback (i.e., a o > 0 ) will be chaotic only under certain circumstances. Further, instability and chaos do not preclude the existence of stable equilibria. It has already been noted that over short time periods (i.e., between thresholds) stability can exist. Further, over longer periods (or broader spatial scales ) undeniable regularities emerge which must represent some sort of stable or quasi-stable equilibria. At an intuitive level this can be conceptualized as viewing a phase diagram of a chaotic system from a steadily increasing distance. At some point the numerous trajectories would become indistinguishable, and the pattern would be dominated by the patterns of the attractors. Conclusions
Nonlinear dynamical systems theory represents an evolution in geomorphology, not a revolution. The basic conceptual tenets of NDS have long been recognized and applied to various extents. Many fundamental concepts and approaches in geomorphology can readily be reinterpreted by or cast in the framework of NDS theory. The fact that the stable attractors of a generalized mass-flux geomorphic system correspond to the well-known states of aggradation, degradation, and steady state both confirms the applicability of the NDS approach to the study of landforms and shows that adoption of the NDS viewpoint will not entail a paradigm shift (or at least that such a shift is evolutionary rather than revolutionary in nature). This is not to suggest that NDS theory is a
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passing trend or old wine in new bottles. While NDS concepts and techniques do not, in and of themselves, constitute a revolution in geomorphology, the ability to link NDS ideas and classical and contemporary geomorphic thought suggests that the NDS framework is a powerful tool for integrating and reinterpreting disparate approaches to the study of landforms. The very fact that the basic tenets of NDS can be mapped onto fundamental geomorphic ideas is a strong statement in this regard. It can thus the speculated that the evolutionary application of NDS concepts to geomorphology can potentially lead to revolutions in our understanding of geomorphic systems. References Abrahams, A.D., 1984. Channel networks: a geomorphological perspective. Water Resour. Res., 20:161-188. Chorley, R.J., 1962. Geomorphology and general systems theory. U.S.G.S. Prof. Pap., 500-B: 1-10. Chorley, R.J. and Kennedy, B.A., 1971. Physical Geography: A Systems Approach. Prentice-Hall, London. Culling, W.E.H., 1987. Equifinality: modem approaches to dynamical systems and their potential for geographical thought. Trans. Inst. Brit. Geogr., 12: 57-72. Culling, W.E.H., 1988. A new view of the landscape. Trans. Inst. Brit. Geogr., 13: 345-360. Culling, W.E.H. and Datko, M., 1987. The fractal geometry of the soil-covered landscape. Earth Surf. Proc. Landforms, 12: 369-385. Davy, B.W. and Davies, T.R.H., 1979. Entropy concepts in fluvial geomorphology: a reevaluation. Water Resour. Res., 15: 103-106. Eckmann, J.P., 1981. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys., 53:655-671. Fraedrich, K., 1987. Estimating weather and climate predictability on attractors. J. Atmos. Sci., 44: 722-729. Graf, W.L., 1979. Catastrophe theory as a model for change in fluvial systems. In: D.D. Rhodes and G.P. Williams (Editors), Adjustments of the Fluvial System. Kendall/Hunt, Dubuque, IA, pp. 13-32. Graf, W.L., 1988. Applications of catastrophe theory in fluvial geomorphology. In: M.G. Anderson (Editor), Modelling Geomorphological Systems. Wiley, Chichester, pp. 33-48. Gu, H., Chen, S., Qian, X., Ai, N. and Zan, R., 1987. Rivergeomorphologic processess and dissipative structure. In: V. Gardiner (Editor), International Geomorphology, Part II. Wiley, Chichester, pp. 211-224.
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Hack, J.T., 1960. Interpretation of erosional topography in humid temperate regions. Am. J. Sci., 258A: 80-97. Hjulstrom, F., 1935. Studies of the morphological activity of rivers as illustrated by the River Fyris. Bull. Geol. Inst. Univ. Uppsala, 25:221-527. Howard, A.D., 1965. Geomorphological systems: equilibrium and dynamics. Am. J. Sci., 263: 302-312. Howard, A.D., 1988. Equilibrium models in geomorphology. In: M.G. Anderson (Editor), Modelling Geomorphological Systems. Wiley, Chichester, pp. 49-72. Huggett, R.J., 1980. Systems Analysis in Geography. Clarendon, Oxford. Huggett, R.J., 1985. Earth Surface Systems. Springer, New York. Huggett, R.J., 1988. Dissipative systems: implications for geomorphology. Earth Surf. Proc. Landforms, 13: 4549. Keilis-Borok, V.I., 1990. The lithosphere of the earth as a nonlinear system with implications for earthquake prediction. Rev. Geophys., 28: 19-34. Langbein, W.B. and Leopold, L.B., 1964. Quasi-equilibrium states in channel morphology. Am. J. Sci., 262: 782-794. Leopold, L.B., and Langbein, W.B., 1962. The concept of entropy in landscape evolution. U.S.G.S. Prof. Pap., 500-A: 1-20. Leopold, L.B. and Wolman, M.G., 1957. River patterns: braided, meandering, and straight. U.S.G.S. Prof. Pap., 282-B: 39-85. Lorenz, E.N., 1963. Deterministic non-periodic flows. J. Atmos. Sci., 20: 130-141. Lorenz, E.N., 1964. The problem of deducing the climate from the governing equations. Tellus, 16:1-11. Malanson, G.P., Butler, D.R. and Walsh, S.J., 1990. Chaos theory in physical geography. Phys. Geogr., 11: 293304. Mann, G.J., 1970. Randomness in nature. Geol. Soc. Am. Bull., 81: 95-104. Montgomery, K., 1989. Concepts of equilibrium and evolution in geomorphology: the model of branch systems. Prog. Phys. Geogr., 13: 47-66. Newman, W.I. and Turcotte, D.L., 1990. Cascade model for fluvial geomorphology. Geophys. J. Int., 100: 433439. Phillips, J.D., 1987. Sediment budget stability in the Tar River basin, North Carolina. Am. J. Sci., 287: 780-794. Phillips, J.D., 1989. An evaluation of the state factor model for soil ecosystems. Ecol. Mod., 45:165-177. Phillips, J.D., 1990a. The instability of hydraulic geometry. Water Resour. Res., 26: 739-744. Phillips, J.D., 1990b. Multiple modes of adjustment in unstable river channel cross-section. J. Hydrol., 123: 39-49. Phillips, J.D., 1991. The human role in earth surface systems: Some theoretical considerations. Geogr. Anal., 23: 316-331.
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Phillips, J.D., 1992. Deterministic chaos in surface runoff. In: A.J. Parsons and A.D. Abrahams (Editors), Hydraulics and Erosion Mechanics of Overland Flow. University College of London Press, London. Puccia, C.J. and Levins, R., 1985. Qualitative Modeling of Complex Systems. Harvard Univ. Press, Cambridge, MA. Rhoads, B.L., 1988. Mutual adjustments between process and form in a desert mountain fluvial system. Ann. Assoc. Am. Geogr., 78:271-287. Rodriguez-Iturbe, I., Febres de Power, B., Sharifi, M.B. and Georgakakos, K.P., 1989. Chaos in rainfall. Water Resour. Res., 25: 1667-1675. Scheidegger, A.E. and Langbein, W.B., 1966. Probability concepts in geomorphology. U.S.G.S. Prof. Pap., 500C: 1-14. Schumm, S.A., 1972. Geomorphic thresholds and complex response of drainage systems. In: M. Morisawa (Editor), Fluvial Geomorphology. Publications in Geomorphology. SUNY, Binghamton, NY, pp. 299310. Slingerland, R., 1981. Qualitative stability analysis of geologic systems with an example from river hydraulic geometry. Geology, 9:491-493. Slingerland, R., 1989. Predictability and chaos in quantitative dynamic stratigraphy. In: T.A. Cross (Editors), Quantitative Dynamic Stratigraphy. Prentice-Hall, Englewood Cliffs, NJ, pp. 45-53. Snow, R.S., and Slingerland, R., 1990. Stream profile adjustment to crustal warping: Nonlinear results from a simple model. J. Geol., 98: 699-708. Strahler, A.N., 1950. Equilibrium theory of erosional slopes, approached by frequency distribution analysis. Am. J. Sci., 248: 673-696, 800-814. Strahler, A.N., 1952. Dynamic basis of geomorphoiogy. Geol. Soc. Am. Bull., 63: 923-938. Thompson, J.M.T. and Stewart, H.B., 1986. Nonlinear Dynamics and Chaos. Wiley, New York. Thornes, J.B., 1981. Structural instability and ephemeral channel behavior. Z. Geomorph., 26: 233-244. Thornes, J.B., 1983. Evolutionary geomorphology. Geography, 68: 225-235. Thornes, J.B., 1985. The ecology of erosion. Geography, 70: 222-235. Turcotte, D.L., 1990. Implications of chaos, scale-invariance, and fractal statistics in geology. Glob. Plan. Change, 89: 301-308. Wiggins, S., 1990. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York, 672 PP. Wilcox, B.P., Seyried, M.S. and Mattison, T.H., 1991. Searching for chaotic dynamics of snowmelt runoff. Water Resour. Res., 27: 1005-1010. Yang, C.T., 1971. Potential energy and stream morphology. Water Resour. Res., 7:311-322. Zdenkovic, M.L. and Scheidegger, A.E., 1989. Entropy of landscapes. Z. Geomorph., 33:361-371.
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Nonlinear dynamical systems in geomorphology: revolution or evolution? Jonathan D. Phillips Department of Geography and Planning, East Carolina University, Greenville, NC 27858-4353, USA (Received August 28, 1991; revised November 12, 1991 ; accepted November 15, 1991 ).
ABSTRACT Phillips, J.D., 1992. Nonlinear dynamical systems in geomorphology: revolution or evolution?. In: J.D. Phillips and W. Renwick (Editors), Geomorphic Systems. Geomorphology, 5: 219-229. Geomorphic systems typically exhibit complex, apparently random behaviors and patterns in both spatial and temporal domains. This complexity can arise from the cumulative impacts of individual process-response mechanisms which are far too numerous to be accounted for in individual detail, or due to multiple controls over process-response relationships which operate over a range of spatial and temporal scales (stochastic complexity). Nonlinear dynamical systems (NDS) theory - - which includes specific techniques and concepts such as chaos, dissipative structures, bifurcation and catastrophe theory, and fractals - - shows that extreme complexity can also arise due to the nonlinear dynamics and couplings of relatively simple systems represented by relatively small equation systems (deterministic complexity ). While emphasis in earth science has been largel), on stochastic complexity, geomorphologists have long recognized the presence of deterministic complexity as well. Recent developments in the study of nonlinear dynamical systems have implications for the ability to make reliable long-term predictions and therefore applications of NDS in geomorphology may be revolutionary to some extent. However, virtually all the basic conceptual underpinnings of NDS theory can be mapped onto existing and even traditional theoretical concepts in geomorphology. In this sense, the use of NDS concepts in understanding earth surface processes and landforms is evolutionary. The link between traditional and NDS-based concepts is illustrated by analyzing a generalized geomorphic mass-flux system. It is shown that the system is in most cases unstable (consistent with chaos or deterministic complexity). However, the three attractor states of the system correspond exactly to the states of aggradation, degradation, and steady state.
Introduction
Geomorphologists have studied earth surface processes and landforms as dynamical systems in an explicit sense at least since systems theory was introduced into the discipline by Strahler ( 1950, 1952 ). The dynamical systems approach was further championed by Chorley (1962) and Chorley and Kennedy (1971 ). In the past two decades mathematicians, physicists, and chemists have revoluCorrespondence to: Dr. J.D. Phillips, Department of Geography and Planning, East Carolina University, Greenville, NC 27858-4353, USA.
tionized theoretical approaches to dynamical systems by focusing on the dynamics of nonlinear systems that are not in equilibrium, and on modeling the often abrupt changes of state in nonlinear systems. Specific concepts and mathematical techniques associated with nonlinear dynamical systems (NDS) theory have been widely touted (even in the popular press ) and applied in virtually every scientific discipline. These concepts include chaos, fractal geometry, and catastrophe theory. The utility of NDS concepts and techniques in geomorphology has been recognized for more than a decade (Thornes, 1981, 1983, 1985; Graf, 1979, 1988; Huggett,
0169-555X/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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1980, 1985). In recent years the call for increasing application of NDS theory to geomorphology has become more insistent, in part because of the power of the ideas to solve geomorphic problems, and in part to link geomorphology more closely to the other physical sciences (Huggett, 1988; Culling, 1987, 1988; Montgomery, 1989; Slingerland, 1989; Malanson et al., 1990; Turcotte, 1990). The purpose of this paper is threefold. First, the evidence that earth surface processes and landforms behave as dissipative systems with deterministic complexity (chaos) is reviewed. Second, fundamental conceptual implications of NDS theory will be related to existing and traditional theoretical concepts of geomorphology. Finally, an attempt will be made to demonstrate the fundamental linkages between NDS theory and traditional concepts of landscape evolution.
Nonlinear dynamical geomorphic systems In general, the concepts and techniques of NDS theory will be applicable to geomorphic systems to the extent that such systems may be regarded as dissipative structures and to the extent that it is reasonable to suspect deterministic complexity, that is, chaotic behavior arising from relatively simple (nonlinear) mechanics and structures. Dissipative systems or structures are those in which energy is dissipated in maintaining order in states removed from equilibrium. A typical manifestation involves self-organizing processes which produce an orderly sequence of system configurations, each series of which is induced by a fluctuation. Huggett (1988) gives the geomorphological example of wave and fluvial bedform systems which pass through a series of transient forms associated with changing hydrodynamic regimes. Numerous other examples could be cited, as for instance the sequence of changes in sand beach morphology before, during, and after a storm. In these cases the basic characteristic of main-
J.D. PHILLIPS
taining order away from equilibrium via energy dissipation is obvious and clearly supports treatment of geomorphic systems as dissipative structures. It can be argued more generally (see Huggett, 1988) that any systems which are open and which exhibit tfiresholds are dissipative. Even more specifically, if one starts with the fact that geomorphic systems are open (Strahlet, 1952; Chorley, 1962 ) and the assumption that some order is present, then the additional assumption of nonlinearities predetermines dissipative behavior. This arises because a rigorous definition of equilibrium implies a single-valued (linear) relationship between input and output (albeit a time-bound, perhaps transient relationship; Howard, 1988). These arguments imply that broader-scale geomorphic systems encompassing multiple inputs and outputs are also dissipative structures. Gu et al. (1987) and Montgomery ( 1989 ) have explicitly treated river systems as dissipative. Deterministic complexity is termed chaos in the NDS lexicon. Chaos describes irregular, apparently random behavior which arises deterministically due to nonlinear couplings in sometimes relatively simple systems. Chaos is distinguishable from stochastic complexity by extreme sensitivity to initial conditions and by increasing divergence of results or predictions with time. Highly irregular behavior is widely known in geomorphology, but the extent to which it is attributable to chaos, rather than to environmental heterogeneity or stochastic forcings, is unknown. One argument, typified by Culling ( 1987, 1988 ), is that chaos is to be expected in the landscape because many of the fundamental physical and chemical processes shaping the landscape (such as turbulent flow) are known to exhibit chaotic behavior. Turbulence is a well-known example of chaos (Eckmann, 1981 ). Chaotic behavior has also been found in climatic dynamics (Lorenz, 1963, 1964) and in data describing meteorological processes
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
such as rainfall (Rodriguez-Iturbe et al., 1989). Lorenz's work on predictability in atmospheric dynamics was indeed the pioneering work in the field of chaos. It should be noted that arguments supporting the presence of deterministic chaos in geomorphic systems do not preclude the presence of stochastic complexity instead of, or in addition to, chaotic dynamics (Huggett, 1988 ). Thus far, applications of chaos theory in earth sciences have been mainly pedagogical (Culling, 1987, 1988; Slingerland, 1989; Turcotte, 1990), but there is an accumulating literature supporting the presence of deterministic chaos in specific earth surface systems. Keilis-Borok (1990) has applied chaos to tectonics and earthquake prediction, and Newman and Turcotte (1990) developed a nonlinear dynamical model of fluvial erosion which produces a fractal landscape (characteristic of chaotic behavior). Wilcox et al. ( 1991 ) found no evidence of chaos in a complex time series of snowmelt runoff. This is consistent with Phillips (1992 ), who concluded that chaos is unlikely in most cases of surface runoff generation. Phillips also suggested, however, that chaotic behavior is possible with regard to infiltration-excess runoff generation and the hydraulic geometry of overland flow. Irregular, unpredictable behavior arising from simple deterministic systems has also been found. Phillips (1990a, b) discovered such behavior arising from a simple system based on the D'Arcy-Weisbach flow resistance equation for stream channels. Culling and Datko (1987) found that an irregular, fractal surface can arise from a Davisian downwasting regime descri]ged by a simple diffusion equation, and supported the finding with data describing soilcovered surfaces. Snow and Slingerland (1990) showed that complex, nonlinear behavior can arise from simple deterministic equations describing the evolution of the long profiles of rivers. Chaos is also linked to asymptotic instability. The latter is defined as exponential diver-
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gence from an equilibrium state following a perturbation. The concept of asymptotic stability is relevant to local stability and small perturbations-- in a geomorphic context these may be interpreted in terms of stability in response to perturbations which are not severe enough to destroy feedback mechanisms within the system. Asymptotic instability (which applies to perturbations of any magnitude) is a necessary condition for general chaos. (i.e., a system which typically exhibits chaotic behavior). Asymptotic instability is both neccessary and sufficient to indicate that a system c a n exhibit chaotic behavior under certain circumstances. The crux of the argument is that a system can be chaotic if at least one Lyapunov exponent is positive. The theory of Lyapunov exponents is a generalization of linear stability theory, and Lyapunov exponents are a standard tool for detecting and measuring chaos (Wiggins, 1990, p. 607). A dynamic system with n components will have n Lyapunov exponents, which determine the rate at which initially nearby points in the system phase space diverge or converge as the system evolves. The Lyapunov exponents are equivalent to the real parts of the eigenvalues of a Jacobian matrix of the equation system where matrix entries represent rates of change of each system variable as functions of the others (these concepts are covered in any text on NDS or chaos, see e.g. Thompson and Stewart, 1986. Briefer explanations in geoscience contexts are given by Fraedrich, 1987 and Phillips, 1992 ). If a system is asymptotically unstable then at least one Lyapunov exponent is positive and chaos is possible (Fraedrich, 1987; Wiggins, 1990). Several different types of geomorphic systems have been found to be asymptotically unstable in the sense above, including river hydraulic geometry (Slingerland, 1981; Phillips, 1990a), fluvial sediment budgets and sediment transport systems (Phillips, 1987; Rhoads, 1988), and the state factor model of soils (Phillips, 1989).
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Overall, the evidence points clearly to the legitimacy of applying NDS concepts and techniques to geomorphic systems, and that fundamental insights into landscape evolution might be gained by doing so.
Nonlinear dynamical systems and traditional geomorphology At the most basic conceptual level nonlinear dynamical systems theory has four characteristics which, while not necessarily unique to the NDS framework, in aggregate distinguish it from other approaches. First, the study of NDS is concerned with determining the evolution (trajectories) of systems between, or in relation to, equilibria. Second, the systems involved are dissipative structures. Third, system evolution is characterized by discontinuities which in mathematical terms are bifurcations or catastrophes. Fourth, there is at least the possibility of deterministic chaos. Fractal geometry is not intrinsically linked to NDS, but is a critical tool in the detection and description of chaos. Evidence that the four characteristics above are fundamental to the study of NDS is given by the fact that all are routinely included in textbooks on NDS analysis (e.g. Thompson and Stewart, 1986; Wiggins, 1990). The relationships between these four basic ideas of NDS theory and traditional geomorphic thought will be explored below.
Equilibrium and trajectories The concept of equilibrium is a fundamental one in geomorphology (see reviews by Howard, 1988; Montgomery, 1989), though the term has been variously and often poorly defined. The most fundamental geomorphic concept of equilibrium is that of Hack (1960) who held that the landscape is always in, or moving toward, a state of dynamic equilibrium where forms are adjusted to prevailing processes and
J.D. PHILLIPS
environmental controls. NDS studies are typically concerned with the evolution of a physical system toward some stationary end-state, or with the trajectories of a system in state or phase space. These trajectories move away from points or restricted spaces which are repellors, and toward attractors. Repellors and attractors are, respectively, asymptotically unstable and stable equilibrium states (Thompson and Stewart, 1986). In an unstable equilibrium state, the response of the system to a perturbation will not return the system to, or nearly to, its pre-disturbance state. Rather, the system will remain in disequilibrium or move to a new, different equilibrium state. Semantic arguments with regard to geomorphic equilibrium are somewhat risky, due to the various definitions, some of which are inconsistent with each other and with definitions of other disciplines (Howard, 1988; Montgomery, 1989). Nonetheless, we proceed by noting that if an unstable equilibrium - - unlikely to persist in the face of inevitable environmental perturbations - - can be equated to nonequilibrium in classical geomorphology, then the basic concepts of phasespace trajectories in NDS theory and landscape evolution under dynamic equilibrium are entirely consistent. In the case outlined above the dynamic equilibrium landforms are analogous to attractors or stable equilibria. The evolution of landscapes between these dynamic equilibrium states over time is a trajectory on attractors. NDS theory allows and accounts for far richer phase space behavior than sequential evolution between successive attractors. However, it should be clear that classical dynamic equilibrium studies of landscape evolution can be interpreted in a NDS context, and are not inconsistent with a NDS-based approach. Reinterpretation of existing data and models in the dynamic equilibrium framework may provide a starting point for exploring the possibility of richer phase-space behavior.
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
Dissipative structures In at least one sense the idea of dissipative structures is clearly implicit in classical geomorphology. Dissipative structures will exist in any system characterized by open system behavior and irreversible processes. Most geomorphic processes are intuitively irreversible: debris does not move back upslope, and weathered material does not reconstitute. Irreversibility is also inherent in the application of the second law of thermodynamics to geomorphology. Open system conceptualization and application of the second law have a rich tradition in the application of entropy concepts in geomorphology (Leopold and Langbein, 1962; Langbein and Leopold, 1964; Scheidegger and Langbein, 1966; Yang, 1971; Chorley and Kennedy, 1971). Entropy concepts derived both from thermodynamic analogies and from information theory have been appplied; Zdenkovic and Scheidegger (1989) demonstrated their equivalence in a landscape context. In terms of recognizing and applying the basic characteristics of dissipative structures there is an extensive tradition in geomorphology. Although the validity of applying entropy concepts in geomorphology has been questioned (Davy and Davies, 1979 ), the point here is not to defend or justify such approaches per se, but to note their consistence with elements of NDS theory. While the characteristic of nonequilibrium as a source of order in dissipative system has not been explicitly explored in landscape studies to any great extent, adoption of the dissipative systems conceptual framework in geomorphology seems to be a logical evolutionary step.
Bifurcations Nonlinear dynamical systems often exhibit discontinuities in their evolution, called bifurcations. These discontinuities may represent a transition from one equilibrium state to an-
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other, or from regular or periodic behavior to chaos. Analytically, these discontinuities can be explored and described using bifurcation and stability analysis and catastrophe theory. Bifurcations in nonlinear dynamical systems are directly analagous to thresholds in geomorphic systems. It is equally accurate to reverse the linkage by defining thresholds as fundamental bifurcations in geomorphic systems. While explicit applications of catastrophe theory in geomorphology date back only to the late 1970s, recognition of discontinous evolution and the presence of thresholds is so fundamental to surficial process studies that it must (implicitly) date to the beginnings of the discipline. No attempt will be made to trace the history of the concept; well-known applications include the work by Hjulstrom (1935), Leopold and Wolman (1957), and Schumm (1972). Explicit consideration of the role of thresholds in geomorphic evolution dates at least to the work by Howard ( 1965 ). In general, it is the widespread recognition of the importance of thresholds/bifurcations in geomorphology that led to the earliest applications of NDS theory to landscape studies (Graf, 1979; Thornes, 1981, 1983). The NDS concept of bifurcations is thus entirely consistent with both prevailing and classical geomorphic thought.
Chaos The complicated, irregular behaviors and patterns often observed in geomorphic systems and landscapes have typically been interpreted in terms of stochastic complexity. That is, it has been widely believed that if one could somehow collect enough data or enough extremely detailed knowledge the apparent randomness could be resolved into deterministic components. Even those who model landscapes and surface processes stochastically have typically conceded that the stochastic behavior was largely or wholly apparent, resulting from the cumulative effects of innumera-
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ble deterministic events. But even authors such as Scheidegger and Langbein (1966), who provide a good example of this thinking, recognized at least a possibility of deterministic complexity. Scheidegger and Langbein ( 1966, pp. 1, 2) explicitly acknowledge that results indistinguishable from randomness can arise from even simple physical deterministic systems. Mann's (1970) discussion of randomness in nature from the geologist's perspective identified the need to determine whether randomness is inherent or apparent (i.e., deterministic or stochastic complexity) as a fundamental goal of earth science research. Application of the "random model" of stream network topology engendered a long-running debate as to the physical basis of the model, centered on the question of whether randomness can be considered a physical property of earth surface systems (see Abrahams, 1984). The very existence of such a debate implies that at least some geomorphologists recognized the possibility of deterministic complexity. The seeds of chaos are apparently sown in many physical systems. The presence and prevalence of thresholds/bifurcations in geomorphic systems is an especially compelling reason to suspect chaotic behavior. From a mathematical perspective every proposed typology of routes to chaos (i.e., mechanisms by which system behavior is transformed from regular or periodic to chaotic behavior) is based on bifurcations ( T h o m p s o n and Stewart, 1986, Wiggins, 1990). Consideration of deterministic complexity has not been the rule in geomorphology, but its presence has long been recognized. Adoption of a NDS framework and chaos theory would dramatically increase the attention paid to deterministic complexity and would eliminate the frequent implicit assumption that observes patterns and behavior indistinguishable from randomness are stochastic complexity arising from insufficiently detailed knowledge of the system.
J,D. PHILLIPS
Mass flux model
The general consistency of traditional and NDS-based approaches to the study of geomorphic systems can be demonstrated by analyzing a generalized model based on mass fluxes. At the simplest level any geomorphic system can be depicted as a mass flux system including inputs (I), storage within the system (S), and outputs (O). Nonlinear dynamical systems studies typically taken an algorithmic approach, using numerical simulations of the equation systems or explorations of large timeseries data sets. However, insight into system dynamics can be obtained even for a generalized system where only qualitative relationships (whether a change in any component results in positive, negative, or negligible effects on any other component) are known. Solutions are obtained by linearizing the nonlinear equation system around an equilibrium condition (C) using Taylor analysis. This makes use of the fortuitous property that the stability characteristics of the linearized system are exactly the same as those of the nonlinear "parent" (Wiggins, 1990). The linearized equations are of the form:
dX/dt=AX
(1)
where X is the vector of variables describing the system components and A is an interaction matrix giving the components' influences on each other. Solutions for the behavior of a small perturbation x (t) over the short term are of the form:
x(t) =vCt exp (~.t)
(2)
where v are the eigenvectors and 2 the eigenvalues of A. Clearly, if the real part of any eigenvalue is positive the system diverges from the pre-disturbance equilibrium exponentially and the system is asymptotically unstable. The Routh-Hurwitz criterion can be used to determine the signs of the real parts of the eigenvalues even when only the qualitative nature of A is known (i.e., positive, negative, or zero signs
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
in each cell of the interaction matrix). RouthHurwitz analysis is discussed in detail by Puccia and Levins (1985); example geomorphic applications include work by Slingerland ( 1981 ) and Phillips (1990a). The real parts of the eigenvalues of the linearized system of eq. (2) are equivalent to the Lyapunov exponents of the full nonlinear system (Fraedrich, 1987; Wiggins, 1990, p. 607). Presence of any positive Lyapunov exponent indicates the possibility of chaos. Local stability near an equilibrium, as described above, is known as asymptotic stability. It is related to the behavior of nonlinear dynamical systems because repellors are asymptotically unstable while attractors are asymptotically stable (Thompson and Stewart, 1986, pp. 9, 204). The validity of the linearization approach for analyzing hyperbolic or asymptotic stability in chaotic systems is shown by Thompson and Stewart (1986, pp. 111, 112; 200-204) and Wiggins (1990, pp. 607-610). Linear systems can be asymptotically unstable, while chaos is a property of nonlinear systems. However, asymptotic instability of a nonlinear system is sufficient to indicate the possibility of sensitive dependence on initial conditions and increasing divergence over time which indicate chaos. If a system is determined to be asymptotically stable then the system configuration represents an attractor and the system can be expected to return to that state or configuration after a disturbance. Dynamics of the system trajectory as it returns may vary; discerning them requires numerical data. If a system is asymptotically unstable that configuration represents a repellor and must be a transient state. Conditions (perhaps hypothetical) under which the generalized system would be stable can be identified. These represent attracting states to which specific realizations of the generalized system would evolve under certain specified conditions. The influences o f / , S, and O on each other in the geomorphic mass flux system are easily
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determined from the mass balance relation, assuming input to be externally imposed and not directly influenced by S or O. This assumption is reasonable in the case of climatic and tectonic forcings in most cases:
OI/dt= OS/dt+ O0/dt
(3)
Even a simple system based on eq. (3) can be unstable and potentially chaotic, due to the absence of any self-stabilizing feedback and the "competitive" relationship between storage and output. However, a plausible stable geomorphic mass flux system can be envisaged by adding a selfstabilizing negative loop for storage, and by changing the link from storage to output from negative to positive. Mass storage can indeed be self-limiting. At high rates finite capacity limits storage. When storage is being depleted selective transport tends to limit removal as remaining material is progressively more difficult to entrain. A positive influence of storage on output could occur where sediment transport is selective and episodic. During lowenergy events heavier particles would be deposited in storage with lighter particles transported through the system. The accumulation of coarser material in storage provides a supply of transportable mass for high-energy events, thus increasing output for these events. Because mass either moves through the system or is removed from storage during high-energy events, and because the allocation of input to storage or output during low-energy periods on particle size rather than total input rates, the storage-output relationship is not competitive. A mass flux system with the structure outlined above has been shown to be asymptotically stable by Phillips ( 1991 ). It is clearly possible for a geomorphic mass flux system to be stable and non-chaotic. However, the formulation above assumes that no intrinsic thresholds are exceeded. The exceedence of such thresholds - - usually in the form of force/resistance or power/resistance ratios - - could provide feedbacks to input, storage,
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J.D. PHILLIPS
and output and dramatically influence system behavior We can add a power/resistance ratio as a fourth component of the system. Power/resistance thresholds (P/R) determine the allocation of input to storage or output. Power rather than the more familiar force/resistance terminology is used because the concern is with transport of mass in bulk rather than individual particles. Resistance in this context is defined by the critical power necessary to transport the available mass. IfP/R > 1 erosion and transport of mass out of the system will occur. IfP/R < 1 deposition and non-removal of storage will occur. Thus the P/R component of the system has a positive link to output and a negative link to storage. Because erosion within the system can augment external mass input when the threshold is exceeded, a positive link from P/R to input is established. Storage has a negative influence on the P/R threshold, since more or less stored material leads to higher or lower critical power thresholds for transport, and thus to lower or higher values of P/R. The interaction matrix linking the qualitative mutual influences of L S, O and P/R, (A in eq. 1 ) is shown in Table 1, where each entry aij represents the influence of the ith (row) component on the jth (column) component. The characteristic polynomial of A is: (4)
O~0~4 "~ OL123 "~ OL222 "Jr"OL32 Jr" O~4 = 0
The coefficient ~k of the characteristic polynomial is equivalent to the Feedback Fk at level k of the system (Puccia and Levins, 1985 ):
Fk= ~ (-1)m+~L(m,k)
(5)
TABLE 1 Interaction matrix for geomorphic mass flux system Parameter
1
Input
0
al2
a13
Storage
0
-- a22
- a23
S
0
P/R 0
-- a24
Output
0
- a32
0
0
Power/resistance
a4!
-a42
a43
0
where the right-hand side of eq. (5) is the determinant of A and L(m, k) is the product of rn disjunct loops with k system components. Feedback measures the influence of system components on each other. A disjunct loop is a sequence of one or more aij which have no component i o r j in common. Feedback at level 0 is defined as F0 = a0 = - 1. For the system in Table 1: F1 = - a 2 2
(6a)
F2 = ( - a 2 3 ) ( - a 3 2 ) q- ( - a 2 4 ) ( - a 4 2
(6b)
F 3 ( - a 2 4 a 4 3 a 3 2 ) -1 ( - a 2 4 a 4 1 a l 2 )
(6c)
F4 =a41 a 1 3 ( - a 3 2 ) ( - a 2 4 )
(6d)
The Routh-Hurwitz criterion (Puccia and Levins, 1985, pp. 167-172) states that the necessary and sufficient conditions for all real parts of the eigenvalues to be negative are: Fi<0
for all i and
(7a)
FtF2 +F3 < 0
for n = 3 or n = 4
(7b)
F~ and F3 are negative, but F2 and F4 must be positive. The Routh-Hurwitz criteria cannot be met. The system is unstable and potentially chaotic. While such a generalized model cannot represent all geomorphic systems, it is representative of many and suggests that the potential for chaos exists in many geomorphic systems. An obvious question is the extent to which the outcome of asymptotic instability and potential chaos depends on the assumptions as to the signs of the links in the model. This sensitivity can be explored in conjunction with the identification of the attractors of the system. Any system configuration where the feedback mechanisms identified above operate is a repellor for all trajectories in state space. The attracting states may be determined by identifying all feasible conditions where the RouthHurwitz criteria would be satisfied. These are alternate system configurations which are not valid for the long term or the general case, but which may exist in particular situations and
NONLINEAR DYNAMICAL SYSTEMS IN GEOMORPHOLOGY: REVOLUTION OR EVOLUTION?
over restricted time periods. They also represent the necessary conditions for invalidating the instability outcome obtained above. For the system described above F4 > 0. For this coefficient to be negative the sign of one a~j would have to be reversed. It is not physically reasonable under any circumstance to reverse the signs of a32, representing the negative effect of output on storage; a4~, the positive effect of P/R on input (this could be zero, but would still result in an unstable system); or al 3, the positive influence of input on output. The influence of storage of the power/resistance ratio (a24) could be conceivably be positive in a situation where stored material is transportable, while the underlying surface is resistant. In this situation increasing storage could decrease resistance and raise the P/R ratio. Storage depletion would likewise increase resistance by exposing the resistant surface and reduce P/R. Reversing the sign of a24 from negative to positive would also make F2 negative as required by the Routh-Hurtz Criteria, if the storage-P/R relationship given by a24a42 >aE3a32. The latter represents outputstorage relationships. If a23>0 as discussed earlier, F2< 0 in any case. However, in this scenario F3, which was negative in the original formulation, becomes positive if a24 > 0, and violates the stability criteria. The mass flux system including a power/resistance threshold is thus unstable in any feasible configuration. The outcome of asymptotic instability and possible chaotic behavior is not sensitive to the assumptions about the links in the model.
Discussion The stable (strange) attractors of the mass flux system are those states where thresholds are not exceeded and the P/R component of the system is irrelevant. Storage must have a positive influence on output for stability to obtain. Due to this condition and the absence of thresholds in the attracting states, the latter
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must be associated with ongoing aggradation or degradation (or the transient steady state). Under steady state there would be no net storage change, so the positive influence of storage on output would be necessary to offset any additions or depletions to mass storage. Under a degradation regime of declining storage, any additions must stimulate increased output to maintain the regime. Likewise, in an aggrading system with increasing storage, any short-term depletion would need to be offset by a decline in output. Note that the strange attractors of the chaotic mass flux system correspond to the states of aggradation, degradation, and steady-state, thus linking the NDS-based analysis with classic - - and rather obvious - - geomorphic concepts. The linkage of instability and potential chaos with power/resistance thresholds is also fully consistent with prevailing geomorphic thought, which holds that complex response of the landscape and irregularities in landscape evolution are associated with thresholds. Chaos in geomorphic systems implies that long-term prediction is impossible and questions the concept of equifinality. The latter implies that different processes or different landscape histories produce similar landform results. In a chaotic system this is not possible, as the system state is sensitively dependent on initial conditions and differences are magnified over time. To the extent equifinality may be characteristic of chaotic geomorphic systems, it must occur at scales substantially broader than those at which chaos is observed. At these broader temporal and spatial scales equifinality is possible in a general sense with the similar end states corresponding to the strange attractors. Chaos also offers the promise that apparently random and exceedingly complex patterns and behaviors may have relatively simple deterministic explanations. While the proclamation that many (perhaps most) geomorphic systems may exhibit chaotic behavior may at first glance seem revolutionary, such is not necessarily the case.
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Asymptotic instability of the type discussed here or complex behavior arising from simple deterministic geomorphic models has already been found in a number of geomorphic systems (Slingerland, 1981; Phillips, 1987, 1989, 1990a, b; Thornes, 1985; Culling and Datko, 1987; Newman and Turcotte, 1990; and Snow and Slingerland, 1990; among others). It is important to recognize that an unstable system without positive self-enhancing feedback (i.e., a o > 0 ) will be chaotic only under certain circumstances. Further, instability and chaos do not preclude the existence of stable equilibria. It has already been noted that over short time periods (i.e., between thresholds) stability can exist. Further, over longer periods (or broader spatial scales ) undeniable regularities emerge which must represent some sort of stable or quasi-stable equilibria. At an intuitive level this can be conceptualized as viewing a phase diagram of a chaotic system from a steadily increasing distance. At some point the numerous trajectories would become indistinguishable, and the pattern would be dominated by the patterns of the attractors. Conclusions
Nonlinear dynamical systems theory represents an evolution in geomorphology, not a revolution. The basic conceptual tenets of NDS have long been recognized and applied to various extents. Many fundamental concepts and approaches in geomorphology can readily be reinterpreted by or cast in the framework of NDS theory. The fact that the stable attractors of a generalized mass-flux geomorphic system correspond to the well-known states of aggradation, degradation, and steady state both confirms the applicability of the NDS approach to the study of landforms and shows that adoption of the NDS viewpoint will not entail a paradigm shift (or at least that such a shift is evolutionary rather than revolutionary in nature). This is not to suggest that NDS theory is a
J.D. PHILLIPS
passing trend or old wine in new bottles. While NDS concepts and techniques do not, in and of themselves, constitute a revolution in geomorphology, the ability to link NDS ideas and classical and contemporary geomorphic thought suggests that the NDS framework is a powerful tool for integrating and reinterpreting disparate approaches to the study of landforms. The very fact that the basic tenets of NDS can be mapped onto fundamental geomorphic ideas is a strong statement in this regard. It can thus the speculated that the evolutionary application of NDS concepts to geomorphology can potentially lead to revolutions in our understanding of geomorphic systems. References Abrahams, A.D., 1984. Channel networks: a geomorphological perspective. Water Resour. Res., 20:161-188. Chorley, R.J., 1962. Geomorphology and general systems theory. U.S.G.S. Prof. Pap., 500-B: 1-10. Chorley, R.J. and Kennedy, B.A., 1971. Physical Geography: A Systems Approach. Prentice-Hall, London. Culling, W.E.H., 1987. Equifinality: modem approaches to dynamical systems and their potential for geographical thought. Trans. Inst. Brit. Geogr., 12: 57-72. Culling, W.E.H., 1988. A new view of the landscape. Trans. Inst. Brit. Geogr., 13: 345-360. Culling, W.E.H. and Datko, M., 1987. The fractal geometry of the soil-covered landscape. Earth Surf. Proc. Landforms, 12: 369-385. Davy, B.W. and Davies, T.R.H., 1979. Entropy concepts in fluvial geomorphology: a reevaluation. Water Resour. Res., 15: 103-106. Eckmann, J.P., 1981. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys., 53:655-671. Fraedrich, K., 1987. Estimating weather and climate predictability on attractors. J. Atmos. Sci., 44: 722-729. Graf, W.L., 1979. Catastrophe theory as a model for change in fluvial systems. In: D.D. Rhodes and G.P. Williams (Editors), Adjustments of the Fluvial System. Kendall/Hunt, Dubuque, IA, pp. 13-32. Graf, W.L., 1988. Applications of catastrophe theory in fluvial geomorphology. In: M.G. Anderson (Editor), Modelling Geomorphological Systems. Wiley, Chichester, pp. 33-48. Gu, H., Chen, S., Qian, X., Ai, N. and Zan, R., 1987. Rivergeomorphologic processess and dissipative structure. In: V. Gardiner (Editor), International Geomorphology, Part II. Wiley, Chichester, pp. 211-224.
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