1This paper traces the development of systems concepts in geomorphology. It submits that (1) they have closely followed developments in physics, chemistry, biology, and ecology; and (2) that they each represent a radically different way of thinking about geomorphic systems, all of which still have currency in geomorphological research.
2A ‘system’ is a whole compounded of many parts, or ‘a meaningful arrangement of things’ (Schumm 1977, p. 4). By this definition, a hillslope is a system consisting of ‘things’ (rock waste, organic matter, and so forth) arranged in a particular way, the arrangement being meaningful because it is understandable in terms of physical processes. Other definitions of systems refer to a set of objects, and attributes of the objects, standing together to form a regular and connected whole. Chorley and Kennedy wrote: “ A system is a structured set of objects and/or attributes. These objects and attributes consist of components or variables (i.e. phenomena which are free to assume variable magnitudes) that exhibit discernible relationships with one another and operate together as a complex whole, according to some observed pattern. ” (Chorley and Kennedy, 1971, p. 1–2).
3Geomorphology has inherited the idea of systems from physics, biology, and, to a lesser extent, chemistry. Physicists recognize three main kinds of system: simple systems, complex but disorganized systems, and complex and organized systems. The first two conceptions of systems have a long and illustrious history of study. Since at least the sixteenth-century revolution in science, astronomers have referred to a set of heavenly bodies connected together and acting on each other in accordance with certain laws as a ‘simple’ system: the solar system is the Sun and its planets; the Jovian system is the planet Jupiter and its moons. In geomorphology, a few boulders resting on a hillslope is a simple system. The conditions required to dislodge the boulders and their fate once they have been dislodged are predictable from mechanical laws involving forces, resistances, and equations of motion, in much the same way that the motion of planets around the Sun can be predicted from Newtonian laws.
4In the complex but disorganized conception of systems, a vast number of objects interact in a weak and haphazard manner. An example is gas in a jar. This system could consist of more than 1023 molecules colliding with each other. In the same way, the countless individual particles in a hillslope mantle are viewable as a complex but somewhat disorganized system, even if the hillslope mantle as a whole does have an organization. In both the gas and the hillslope mantle, the interactions are rather haphazard and far too numerous to study individually, so aggregate measures must be employed.
5In a more modern notion of systems, which has its roots in biology and ecology, objects interact strongly with one another to form systems with a complex and organized nature. Most biological systems and ecosystems are of this kind, but many other structures at the Earth’s surface display high degrees of regularity and rich connexions and may be thought of as complexly organized systems. Hillslopes, rivers, and beaches are examples. This kind of system, and its extension to non-equilibrium cases, has dominated systems thinking in geomorphology since the 1970s, although earlier systems ideas are still important.
6The first systems studied in science consisted of a few objects described by a few variables. The study of such simple systems dominated science until late in the nineteenth century. Deterministic equations of motion describe the motion of each system component as in simple mechanical system, say a few billiard balls moving on a billiard table or planets revolving around the Sun. These equations define the exact movement of the balls or planets with respect to the forces acting upon them. In the case of the billiard balls, just four variables are needed for each ball – position and velocity in a plane. Taken together, the equations for individual balls form a set of simultaneous differential equations which enable the prediction, for a given initial distribution of balls and known forces applied from outside by a billiard cue, of system changes, of the future location of all balls. The dynamics of virtually all simple systems, including balls rolling down inclined planes, levers, cogwheels, bodies colliding with each other, and billiard balls bouncing off cushions, can be studied and predicted using the classical methods of Newtonian mechanics (Waddington, 1977, p. 64).
7In geomorphology, the Newtonian conception of dynamics has proved a useful and powerful method of analysing simple systems. Its application to the study of the movement of material at the surface of the Earth has been particularly rewarding. Geomorphologists commonly regard erosion and weathering as the result of external forces or stresses applied at the Earth’s surface and internal forces resisting them. Strahler (1952) classified Earth materials according to their response to an applied stress caused by either gravity or by molecular processes. His groundbreaking work spawned a generation of Anglo-American geomorphologists who researched the small-scale erosion, transport, and deposition of sediments in a mechanistic framework (Martin and Church 2004). The literature dealing with the mechanics and dynamics of geomorphic systems is extensive. An early example is Nye’s (1951) application of plasticity theory to the flow of ice sheets and glaciers. Another case is the work of Bagnold, including his classic study of the physics of blown sand and desert dunes (Bagnold, 1954), his classic paper on the bedload stresses set up during the transport of cohesionless grains in fluids (Bagnold, 1956), and his approach to the problem of sediment transport from the viewpoint of general physics (Bagnold, 1966). Other examples are Yalin’s (1977) treatment of the mechanics of sediment transport and Graf’s (1971) account of the hydraulics of sediment transport.
8Nineteenth-century physicists invented classical thermodynamics, which is the study of heat transformation and exchange. Classical thermodynamics involves the study of heat (and so with the collision and interaction of particles) in large and closed systems in, or near, equilibrium states. A large system in classical thermodynamics contains a huge number of interacting particles (for example, colliding molecules in a gas or a huge number of billiard balls on a big billiard table for instance). Newtonian dynamics cannot tackle systems of this complexity because there are too many equations to handle. In the late nineteenth century, Boltzmann and Gibbs showed that systems that are complex but disorganized can be studied using a new kind of mathematics to find certain quantities of interest. The new mathematics was statistical averaging or, more technically, entropy maximizing (Wilson, 1981, p. 39). These methods cannot predict the behaviour of any one particle of gas, or one billiard ball among many. However, they can predict the distribution of particles among energy states and aggregate measures such as the pressure and temperature of a gas, and the distribution of velocities of billiard balls and the average rate at which a ball strikes a cushion. The predictions made by this branch of classical thermodynamics apply to closed system at, or very near to, equilibrium.
9Regarding geomorphic systems as systems of complex disorder opens up an interesting and powerful line of enquiry that applies principles of energy and materials accounting. Although the principles of accounting are simple, they can lead to sophisticated methods such as input–output analysis. They can also lead to statements of energy conservation and mass conservation that, in conjunction with principles of classical dynamics, assist in the study a variety of geomorphic systems. To be sure, the laws of Newtonian dynamics and thermodynamics are widely used in geomorphology. These laws cover four kinds of basic equations: balance equations, physical–chemical state equations, phenomenological equations, and entropy balance equations (Isermann, 1975).
These equations represent the laws of conservation. They indicate that what goes into a system must be stored, come out, or transform into something else; in other words, matter, energy, and momentum cannot suddenly appear or disappear in an unaccountable manner (Huggett, 1980, p. 93). For mass transactions, the principle of the conservation of mass applies, giving a mass or materials balance equation:
10where M is mass, t is time, Min is mass input, and Mout is mass output. Conservation of mass is a general principle of Nature and applies to all models of geomorphic systems that keep account of materials transactions. It applies to small-scale, short-lived systems such as deltas, to medium scale systems such as sedimentary basins, and to global-scale systems such as the sedimentary cycle and world biogeochemical cycles. Applied to hydrodynamic systems, the law of mass conservation is expressed as the continuity equation as first developed by Laplace. In rectangular co-ordinates and in differential form, the continuity equation is
11where is fluid density and u, v, and w are velocity components in the x, y, and z directions. Geomorphologists apply this ‘equation of continuity’ to all cases of mass balance, and not just fluid balances. Kirkby (1971, p. 15) used it in hillslope models. Applied to mass transactions on a hillslope, the continuity equation states that if more material enters a slope section than leaves it, then the difference must be represented by aggradation; conversely, if less material enters than leaves, then the difference must represent net erosion. The same principle of continuity of sediment transport applies to other geomorphic systems, including rivers (e.g. Dietrich and Perron, 2006).
12These equations express the dependence of one state variable upon another. A classic example is the gas law of Boyle and Gay-Lussac, which states that PV=RT, where P is pressure, V is volume, R is a gas constant, and T is absolute temperature. Some researchers have applied physical–chemical-state equations for heat to landscapes. Leopold and Langbein (1962) and Scheidegger (1964) drew an analogy between temperature and height and between change of heat and change of mass. The rationale behind this approach is a general analogy of the landscape with ideal isolated and closed systems (Chorley, 1962. Scheidegger, 1964; Karcz, 1980), the intrinsic randomness of geomorphic processes such as soil creep (Culling, 1963), and the aggregate randomness of complex landscape systems in which a large number of individually deterministic relationships interact (Leopold et al., 1962). Leopold and Langbein (1962) saw a direct analogy between landscape variables and thermodynamic variables. They equated height of the land surface above a base line with temperature, and equated mass with heat, which enabled them to equate change in landscape entropy with change in mass divided by height. Scheidegger (1964, 1967, 1991; Lechthaler-Zdenkovic and Scheidegger, 1989) elaborated this analogy, showing that for a steady state in a landscape, entropy production must be at a minimum. For such systems, this condition leads to a process–response concept, insofar as, if disturbed, an equilibrium system responds through the adjustment of the state variables to a new equilibrium configuration (Scheidegger, 2004, p. 13). It is probably fair to remark that the thermodynamic analogy for equilibrium systems has not contributed vastly to the understanding of landscape evolution. It suggested that the general direction of landscape development is always in the direction of relief equalization, as high areas lower and low areas build up. It also prompted Scheidegger (1992) to conclude that the systems approach in geomorphology is chiefly limited to purely exogenic processes, and cannot deal with underlying endogenic processes.
13These equations describe irreversible processes. The word ‘phenomenological’ refers to the fact that changes in observed macroscopic variables are described without trying to see what concomitant changes are occurring in atoms, molecules, or electrons. All phenomenological laws describe processes that tend to equalize the value of macroscopic variables throughout a system. They relate to the second law of thermodynamics, which is concerned with irreversible processes. The second law determines that heat will not flow from a colder to a hotter body without assistance. In other words, heat flow occurs spontaneously in a ‘downhill’ direction, but the performance of work must drive the process to make it flow ‘uphill’ from a cold body to a hot body. The phenomenological equation describing heat flow is Fourier’s law of heat conduction, that describing the flow of water in a porous medium is Darcy’s law, and that describing the flow of a solute is Fick’s first law of diffusion. In geomorphology, the first applications of homegrown phenomenological equations dealt with diffusion-like transport processes, such as soil creep, over hillslopes (e.g. Culling, 1960, 1963, 1965; Luke, 1972; Hirano, 1968, 1975). Scheidegger (2004, p. 13) pointed out that such transport equations are derivable from the analogy between thermodynamics and geomorphic systems being extended to non-equilibrium cases (see also Tomkoria and Scheidegger, 1967).
14The entropy equations simply keep account of energy changes – the entropy balance – caused by irreversible processes in a system.
15In part owing to the teachings of classical thermodynamics, equilibrium became a preceptive idea in geomorphology during the first half the twentieth century. Classical thermodynamics saw the Universe inexorably winding down to an ultimate ‘heat death’ (Brush, 1987). It is claimed that Davis’s theory of landscape development follows classical thermodynamic principles in that it ‘seems to operate in some respects like a closed system tending towards maximum entropy’ (Chorley, 1967, p. 87), although not all geomorphologists agree with this interpretation and counter that Davis in point of fact regarded landscapes as open systems (e.g. Ollier, 1968). Darwin’s (1859) evolutionary worldview also influenced it, providing the ‘developmental’ terms – youth, maturity, and senility – for the ‘geographical cycle’ (Chorley, 1965a; see also Stoddart, 1966). The word ‘developmental’ here implies a change toward a stable endpoint (Drury and Nisbet, 1971), as seen in the putative sequence of transformation from youthful landscape to senile peneplain.
16A closed system in classical thermodynamics does not exchange energy or material with it surroundings. Some authorities define a closed system as one that exchanges energy but not matter with its environment and an isolated system as one that exchanges neither energy nor matter with it environment. In closed systems, the total amount of energy, E, is always conserved (as stipulated by the first law of thermodynamics), the amount of available energy inevitably decreases to zero (as dictated by the second law of thermodynamics), and the entropy, S, of the system (the amount of unusable energy) increases to a maximum. Around the middle of the twentieth century, the theory of irreversible processes and open systems, which physicists, chemists, and biologists developed, led to a new thermodynamics. Open systems exchange energy or matter (or both) with their surroundings and can exhibit nonlinear behaviour. The immensely potent idea of open systems was the brainchild of von Bertalanffy (1932). From about 1932, von Bertalanffy explored the implications of viewing organisms as open systems, and, building on the groundbreaking work of Lotka (1924, 1954), which drew on chemical reaction theory, couched the dynamics of biological systems in terms of simultaneous differential equations (e.g. Bertalanffy, 1950). This work was the inspiration for the eventual injection of open systems concepts into geomorphology.
17Lotka had developed equations of ‘physical biology’ to describe ‘the mechanics of systems undergoing irreversible changes in the distribution of matter among several components of such systems’ (Lotka, 1924, p. 49). He had used the reaction equations:
18to define the kinetics of evolving systems (defined by variables – the Xs – and constrained by parameters – the Ps and Qs) (Lotka, 1924, p. 51). Lotka had noted three interesting properties of the equations. First, where velocities of change in such a system are zero, an equilibrium or, to be more exact a steady state, obtains. Second, where the parameters change slowly, a moving equilibrium (or what geomorphologists now call a dynamic equilibrium) results. Third, where a change in the system’s environment causes a displacement from equilibrium, Le Châtelier’s principle comes into operation, which means that the system equilibrium shifts to minimize the effects of the change; in other words, the system absorbs the change by making as few adjustments as possible.
19A note of caution is necessary here. The term ‘dynamic equilibrium’ is problematic. When first used in chemistry, it meant equilibrium between a solid and a solute maintained by solutional loss from the solid and precipitation from the solution running at equal rates. The word equilibrium captured that balance and the word dynamic captured the idea that, despite the equilibrium state, changes take place. In other words, the situation is a dynamic, and not a static, equilibrium. It could be argued that Gilbert used the term in this sense. Later geomorphologists have used the term to mean ‘balanced fluctuations about a constantly changing system condition which has a trajectory of unrepeated states through time’ (Chorley and Kennedy 1971, p. 203), which is similar to Lotka’s idea of moving equilibrium (cf. Ollier, 1968, 1981, p. 302–4).
20Building on Lotka’s foundations, von Bertalanffy established several important properties of open systems not possessed by closed systems. First, owing to the second law of thermodynamics, an open system will eventually attain time-independent equilibrium state – a steady state – in which the systems and its parts are unchanging, with maximum entropy and minimum free energy. In such a steady state, a system stays constant as a whole and in its parts, but material or energy continually passes through it. Second, as a rule, steady states are irreversible. Third, closed systems cannot behave equifinally (that is, arrive at equilibrium from different starting positions), but open systems, exchanging energy or matter with their environment, can do so with the steady state being independent of the starting conditions. An outcome of von Bertalanffy’s investigations was the emergence in the early 1950s of a general systems theory, which caught the attention of some geomorphologists (e.g. Chorley, 1962), at the core of which was the idea of open systems.
21Prigogine (1947) extended and generalized the thermodynamic theory of open systems. He famously wrote the total change in entropy, dS, in an open system as:
22where deS denotes transfer of entropy across the boundaries of the system (entropy exchange between the system and its environment), and diS denotes the production of entropy within the system due to irreversible processes (such as chemical reactions, diffusion, and heat transport). Only irreversible processes contribute to entropy production in a system and give time a sense of direction, sometimes metaphorically called ‘time’s arrow’.
23Starting with its introduction to ecology by Tansley (1935) and Lindeman (1942), an open systems approach found widespread applications within the Earth and life sciences, including geomorphology. Biological and ecological applications preceded geomorphological ones, but ideas from the two disciplines (and from pedology) began to cross-fertilize during the 1960s, when applications in the social sciences also emerged (e.g. Forrester 1961, 1969, 1971). The idea of equilibrium remained as dominant a focus as it had been in previous work, but it underwent reinterpretation in terms of steady state and dynamic equilibrium.
24Strahler (1950, 1952; see also 1980) introduced open systems theory to geomorphology, though Gilbert first mooted the idea of a system in the subject, and in doing so took an open systems approach in his concept of dynamic equilibrium: “ The tendency to equality of action, or the establishment of a dynamic equilibrium, has already been pointed out … but one of its most important results has not been noticed: … in each basin all lines of drainage unite in a main line, and a disturbance upon any line is communicated through it to the other main line and thence to every tributary. And as any member of the system may influence all others, so each member is influenced by every other .” (Gilbert, 1877, p. 123–4).
25Thus, Gilbert saw equilibrium landforms adjusting to geomorphic processes (Chorley, 1965b). Strahler’s low-key comments on geomorphic systems ushered in a revival of Gilbertian thinking in geomorphology. Melton (1958), Hack (1960), Chorley (1962), Howard (1965), Schumm (1977) and many others took up his call to action. For example, Hack (1960) abandoned the cyclic theory of landform development (as proposed by Davis) and instead adopted Gilbert’s concept of dynamic equilibrium as a philosophical base for interpreting erosional topography in the Central Appalachians, USA. In this conception, “ The landscape and the processes molding it are considered a part of an open system in a steady state of balance in which every slope and every form is adjusted to every other. Changes in topographic form take place as equilibrium conditions change, but it is not necessary to assume that the kind of evolutionary changes envisaged by Davis ever occur .” (Hack, 1960, p. 81). “ The concept [of dynamic equilibrium] requires a state of balance between opposing forces such that they operate at equal rates and their effects cancel each other to produce a steady state, in which energy is continually entering and leaving the system. The opposing forces might be of various kinds. For example, an alluvial fan would be in dynamic equilibrium if the debris shed from the mountain behind it were deposited on the fan at exactly the same rate as it was removed by erosion from the surface of the fan itself. Similarly a slope would be in equilibrium if the material washed down the face and removed from its summit were exactly balanced by erosion at the foot. “ (Hack, 1960, p. 86).
26Open systems thinking led to a new typology of systems, as first proposed by Chorley and Kennedy (1971), and adopted and adapted by Strahler (1980). According to these authors, there are several levels of systems: morphological (form) systems, cascading (flow) systems, and process–response (process–form) systems (the terms in parenthesis are Strahler’s). Morphological or form systems are conceived as sets of morphological variables, which are thought to interrelate in a meaningful way in terms of system origin or system function. An example is a hillslope represented by variables pertaining to hillslope geometry, such as slope angle, slope curvature, and slope length, and to hillslope composition, such as sand content, moisture content, and vegetation cover, all of which are assumed to form an interrelated set. Cascading systems or flow systems are conceived as ‘interconnected pathways of transport of energy or matter or both, together with such storages of energy and matter as may be required’ (Strahler, 1980). An example is a hillslope represented as a store of materials: the weathering of bedrock and wind deposition both add materials to the store, slope processes transfer materials through the store, and erosion by wind and fluvial erosion at the slope base remove materials from the store. Other examples of cascading systems include the water cycle, the biogeochemical cycle, and the sedimentary cycle, all of which may be identified at scales ranging from minor cascades in small segments of a landscape, through medium-scale cascades in drainage basins and seas, to mighty circulations involving the entire globe. Process–response systems or process–form systems are conceived as an energy flow system linked to a morphological system in such a way that system processes may alter the system form and, in turn, the changed system form alters the system processes. A hillslope may be viewed in this way with slope form variables and slope process variables interacting. Thus Small and Clark (1982) saw a hillslope as a natural system within which there are numerous complex linkages between ‘controlling’ factors, processes, and form.
27Some geomorphologists passionately and persuasively promulgated an open systems approach (e.g. Chorley and Kennedy, 1971). Other researchers applied the open system concept to soil–landscape units, so linking pedology and geomorphology (e.g. Walker and Ruhe, 1968; Ruhe and Walker, 1968; Huggett, 1975, 1982). However, the rise of a non-equilibrium systems paradigm ended the open systems’ period of celebrity in geomorphological research, although it persists as a teaching tool. Perhaps its chief legacy is the identification of negative and positive feedback links in geomorphic systems (e.g. Phillips, 2006).
28Widespread acceptance of the idea of non-equilibrium (or disequilibrium) started the third period of systems thinking in geomorphology. From the 1960s onward, some practitioners began questioning simplistic notions of equilibrium and steady state. Howard (1965) noted that geomorphic systems might possess thresholds that separate two rather different system economies. Schumm (1973, 1977) introduced the notions of metastable equilibrium and dynamic metastable equilibrium. He showed that thresholds within a fluvial system cause a shift in its mean state. The thresholds are not part of a change continuum, but show up as dramatic changes resulting from minor shifts in system dynamics, such as caused by a small disturbance. The thresholds may be extrinsic or intrinsic (see also Chappell, 1983). In metastable equilibrium, static states episodically shift when thresholds are crossed. In dynamic metastable equilibrium, thresholds trigger episodic changes in states of dynamic equilibrium.
29A system with constraints imposed upon it is driven away from equilibrium towards a new steady state. If the constraints are weak then the system will respond in a linear manner. If the constraints are strong then the system may change smoothly along a thermodynamic branch into non-equilibrium states in which the theorem of minimum entropy production still applies. At a certain distance from equilibrium, called the thermodynamic threshold, non-linear relationships emerge and the steady states along the branch are not of necessity stable. Beyond the threshold, the solutions of the equations governing the dynamics of the system may no longer be unique: the system may enter one of several new regimes. The threshold is therefore a bifurcation point. The path followed by the thermodynamic branch beyond the threshold may involve further thresholds and hence bifurcations. In passing through a bifurcation point, the system loses its structural stability and undergoes a sudden or catastrophic change to a new form.
30The implications of bifurcation theory are profound. Systems that possess bifurcations are describable by deterministic reaction–diffusion equations (similar in form to Lotka’s equations with diffusive terms included), but the presence of bifurcations implies that the dynamics of the system involves a chance element. When driven to some critical value the system can move in more than one way. The route taken is probably, in essence, a matter of chance fluctuation. The theory of bifurcations permits the same system to pass through a different series of states and, in a sense, introduces an historical dimension to system development (cf. Prigogine, 1980, p. 106).
31Some geomorphologists applied bifurcation theory to geomorphic systems in the late 1970s and early 1980s. They based their arguments on catastrophe theory, which is a special branch of bifurcation theory developed by René Thom (1975). Thom’s elegant treatment of bifurcations and structural stability applies only to systems with a finite number of degrees of freedom and described by ordinary, rather than partial, differential equations. So its application to geomorphic systems, where partial differential equations describing changes in space and time are of paramount importance, seemed unpromising (Karcz, 1980). Undeterred, geomorphologists tried to use Thom’s ideas (his cusp catastrophe proved a favourite) to explain certain processes at the Earth’s surface. Examples included Chappell’s (1978) cusp catastrophe model expressing relationships between wave energy, water table height relative to a beach surface, and erosion and accretion; Graf’s (1979, 1982) model of the condition at stream junctions in the northern Henry Mountains of Utah; and Thornes’s (1983) model of sediment transport in a river. Indeed, systems may possess multiple equilibrium points, connected by bifurcations. Thornes (1983, p. 234) saw catastrophe theory as promising to produce major new insights into historical problems such as river terrace formation, drainage initiation, and drainage basin evolution. The cusp catastrophe model still has currency, being used for example to explain the instability of a slip-buckling slope (Qin et al., 2001).
32All these intimations of complex dynamics and non-equilibrium within systems found a firm theoretical footing with the theory of nonlinear dynamics and chaotic systems that scientists from a range of disciplines developed, including geomorphology itself.
33Classical open systems research characteristically deals with linear relationships in systems near equilibrium. A fresh direction in thought and a deeper understanding came with the discovery of deterministic chaos by Lorenz in the 1963. Technically speaking, this was a rediscovery as Henri Poincaré had dealt with similar issues in nonlinear mechanics (e.g. Poincaré, 1881–1886). However, deterministic chaos did not make a grand entrance into the scientific mindset until the 1960s (for general reviews see Prigogine and Stengers, 1984; Gleick, 1988; Gribbin, 2004). The key change was the recognition of nonlinear relationships in systems. In geomorphology, nonlinearity means that system outputs (or responses) are not proportional to systems inputs (or forcings) across the full gamut of inputs (cf. Phillips, 2006).
34Nonlinear relationships produce rich and complex dynamics in systems far removed from equilibrium, which display periodic and chaotic behaviour. The most surprising feature of such systems is the generation of ‘order out of chaos’, with systems states unexpectedly moving to higher levels of organization under the driving power of internal entropy production and entropy dissipation. Systems of this kind, which dissipate energy in maintaining order in states removed from equilibrium, are dissipative systems. It is perhaps useful to distinguish ‘simple’ evolving system, such as planets, stars, and galaxies, from complex adaptive systems that learn or evolve by utilizing acquired information, as when a child learns his or her native language, a strain of bacteria becomes resistant to an antibiotic, and the scientific community tests new theories (Gell-Mann, 1994).
35In dissipative systems, non-equilibrium is the source of order, with spontaneous fluctuations growing into macroscopic patterns. The Bénard convective cell is an instructive example (Prigogine, 1980, p. 88). Imagine a horizontal layer of fluid at rest between two parallel planes. Warm the bottom plane and hold it a higher temperature than the top plane. When large enough, the temperature gradient between the two planes causes the state of rest to destabilize and convection begins. Entropy production increases because the convection is a new mechanism for heat transport. In more detail, while the fluid is at rest and below the threshold temperature gradient, small convection currents appear as fluctuations from the average state but they are damped and disappear. Above the critical temperature gradient, some of the fluctuations amplify to produce a macroscopic current. In effect, the fluctuations trigger an instability that the system accommodates by reorganizing itself. The macroscopic convective cell stabilizes by exchange of energy with the system’s environment. The general circulation of the atmosphere works on the same principle.
36The theory of complex dynamics predicts a new order of order, an order arising out of, and poised perilously at the edge of, chaos. It is a fractal order that evolves to form a hierarchy of spatial systems whose properties are holistic and irreducible to the laws of physics and chemistry. Geomorphic examples are flat or irregular beds of sand on streambeds or in deserts that self-organize themselves into regularly spaced forms – ripples and dunes – that are rather similar in size and shape (e.g. Baas, 2002). Conversely, some systems display the opposite tendency – that of non-self-organization – as when relief reduces to a plain. A central implication of chaotic dynamics for the natural world is that all Nature may contain fundamentally erratic, discontinuous, and inherently unpredictable elements. Nonetheless, nonlinear Nature is not all complex and chaotic. Phillips (2006) sagely noted that “ 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 ”.
37One of the most remarkable features of complex systems is their behaviour. Complex systems are sensitive to initial conditions, a notion popularized as the Butterfly Effect (in which a butterfly fluttering its wings in England causes a hurricane in Australia). They obey simple deterministic laws, but their behaviour is irregular. Indeed, it may be so irregular that it looks random. However, chaotic behaviour is not random; it is a cryptic, random-like pattern generated by simple deterministic laws. So, contrary to the traditional view that simple causes must produce simple effects (and the implied corollary that complex effects must have complex causes), chaos theory predicts that simple causes can create complex effects. Because of this, knowledge of the simple deterministic rules governing the behaviour of a complex system does not guarantee success in predicting the system’s future behaviour. However, it does mean that, for instance, landscape models do not need to become increasingly complex to give useful predictions (Favis-Mortlock and de Boer, 2003). Significantly, a system displaying chaotic behaviour through time usually displays spatial chaos, too. Thus, a landscape that starts with a few small perturbations here and there, if subject to chaotic evolution, displays increasing spatial variability as the perturbations grow (Phillips, 1999).
38Culling (1985, 1987) recognized the potential importance of nonlinear dynamics for geomorphic thinking (see also Huggett, 1988). Phillips is surely the most dogged and industrious proponent of nonlinear dynamics in Earth surface systems. From his studies, he drew up eleven principles of Earth surface systems, which illustrate the potency of the non-equilibrium paradigm (see also Huggett, 2003, p. 339). More recently, he has stressed the importance of confronting nonlinear complexity by ‘problematizing nonlinear dynamics from within a geomorphological context’, rather than applying analytical techniques derived from mathematics, statistics, physics, and other disciplines that use experimental laboratory techniques and numerical models (Phillips, 2006). To this end, and rooting his arguments in field-based studies, he discussed methods for detecting chaos in geomorphic systems, explored the idea of unstable and non-equilibrium systems versus stable system achieving a new equilibrium following a change in boundary conditions, and shed a new light on the question of space and time scales. Convergence versus divergence of a suitable system metric (elevation or regolith thickness for instance) is a hugely significant indicator of stability behaviour in a geomorphic system. In landscape evolution, convergence associates with downwasting and a reduction of relief, whilst divergence relates to dissection and an increase of relief. More fundamentally, convergence and divergence underpin developmental, ‘equilibrium’ conceptual frameworks, with a monotonic move to a unique endpoint (peneplain or other steady-state landform), as well as evolutionary, ‘non-equilibrium’ frameworks that engender historical happenstance, multiple potential pathways and end-states, and unstable states. The distinction between instability and new equilibria is critical to understanding the dynamics of actual geomorphic systems, and for a given scale of observation or investigation, it separates two conditions. On the one hand is a new steady-state equilibrium governed by stable equilibrium dynamics that develops after a change in boundary conditions or in external forcings. On the other hand, is a persistence of the disproportionate impacts of small disturbances associated with dynamic instability in a non-equilibrium system (or a system governed by unstable equilibrium dynamics) (Phillips, 2006). The distinction is critical because the establishment of a new steady-state equilibrium implies a consistent and predictable response throughout the system, predictable in the sense that the same changes in boundary conditions affecting the same system at a different place of time would produce the same outcome. In contrast, a dynamically unstable system possesses variable modes of system adjustment and inconsistent response, with different outcomes possible for identical or similar changes or disturbances. Several indicators potentially allow the identification of newly stable equilibria and dynamically unstable system states in field situations (tab. 1).
Table 1 – Criteria for distinguishing equilibrium and non-equilibrium system changes
Tableau 1 – Critères pour distinguer des changements d’état rattachés à une situation d’équilibre et de non-équilibre, d’après discussion in Phillips, 2006
based on the discussion in Phillips (2006)
39Scale is of crucial importance to an appreciation of systems dynamics (e.g. de Boer, 1992). As Phillips (2006) put it, “ Stability, chaos, and other manifestations of non-linear dynamical systems are emergent – that is, they appear or disappear as time frames, spatial resolutions, and levels of detail are changed ’ (tab. 2).
Table 2 – Stability–instability relationships in weathering systems, adapted from a diagram in Phillips, 2006.
Tableau 2 – Relations de stabilité–instabilité dans les systèmes d’altération, d’après un diagramme in Phillips, 2006
adapted from a diagram in Phillips (2006)
40At the conclusion of his review, Phillips (2006) argued that field-based investigations must inform studies on nonlinear dynamics in geomorphology, for only by relating system ideas to real-world landscape forms, processes, and histories, and by researching on-the-ground signs of chaos and other non-linear phenomena, can the geomorphologists test the systems theory. Therein lies a major challenge for future research into geomorphic systems.
41This paper has argued that ideas from physics, biology, and chemistry have strongly influenced geomorphological thinking. However, to conclude the essay, I would stress two points. First, I would argue that the impacts of the physical, biological, and chemical sciences have not inevitably been direct. Second, I would point out that those geomorphologists who have adopted the language and formalization of a systems approach, have fashioned their own ideas on system structure and function, devising applications befitting the geomorphic systems they study, some of which have been adopted by scientist in other disciplines. I would end by speculating a little on the future of the systems approach in geomorphology.
42It is probably the case that geomorphologists have imported many systems ideas from biology and evolutionary ecology, rather than directly through physics and chemistry. Strahler’s exposure to the open system concept, which set the systems bandwagon in geomorphology rolling, was through the writings of von Bertalanffy, a biologist. Interestingly, Strahler introduced the open systems model to geomorphology when its ramifications were being explored in many other sciences: Prigogine published his book on the thermodynamics of open system in 1947, and Denbigh published a book on the kinetics of open reaction systems in industrial chemistry in 1951. There again, the source of ideas about non-linear dynamics in geomorphology is more population ecology (e.g. May 1973) and meteorology (e.g. Lorenz 1963) than it is physics.
43Geomorphologists have borrowed some terms and concepts but they have adapted them to uniquely geomorphic settings to create original contributions to geomorphological enquiry. Scheidegger’s (1983) instability principle is a case in point. This principle rests on the idea that equilibrium in geomorphic systems is commonly unstable equilibrium and any deviation from the equilibrium state may be self-reinforcing, causing the deviation to grow. This principle ultimately links to the analytical framework of nonlinear partial differential equation systems, but field observations of increasing irregularity over time (such as cirques tending to grow, karst sinkholes tending to increase in size, valleys tending to develop steps) moved Scheidegger to propose it. The instability principle is an example of a homegrown systems idea in geomorphology. Another important example is the work that identifies networks of flows and interactions, representing them as box-and-arrow models or interaction matrixes and modelling them, which scientists in other disciplines have found useful.
44The above two points show that the evolution of systems ideas in geomorphology is complicated. As with most general developments in scientific thinking, parallel and overlapping lines of thought in many disciplines characterize the systems approach. Geomorphology has contributed to systems thinking in science, but equally, developments in physics, chemistry, biology, and ecology have strongly influenced systems thinking in geomorphology. So do systems ideas in geomorphology have a future? My feeling is that they do and that, ironically, their greatest contribution may be to help reunite the geomorphological traditions of process geomorphology and historical geomorphology. Undoubtedly, the systems approach has proved salutary in some branches of geomorphology, especially to those that prosecute a geomorphic process approach. Systems formulations have aided the research of many process geomorphologists and have armed them with a powerful teaching tool. On the other hand, most historical geomorphologists have deemed the systems approach an irrelevance. This may be partly because systems models deal with steady states and relatively short-term transient states – they do not have a long-term historical dimension that involves unique events. It may also partly result from the ‘Davis bashing’ practised by many of the systems pioneers in geomorphology, including Chorley and Hack, which undeservedly denigrated much of Davis’s work. With historical studies fast resurging, the future of the systems approach may seem uncertain. However, early discussions of non-linear dynamical systems intimated that some geomorphic systems contain deterministic elements and probabilistic elements. Deterministic elements derive from the universal and necessary operation of geomorphic laws, which apply in all landscapes at all times, though owing to thresholds, they may not operate in all landscapes at all times; probabilistic elements derive from historical happenstance and contingency (e.g. Huggett, 1988). The latest expansion of this idea suggests that because of their immanent deterministic–contingent duality, the study of non-linear geomorphic systems approach may help to bridge the gap between process and historical studies (Phillips, 2007). The argument is that geomorphic systems have multifarious environmental controls and forcings, which acting in concert can produce many different landscapes. What is more, some controls and forcings are causally contingent and specific to different times and places. Dynamical instability creates and enhances some of this contingency by encouraging the effects of small initial variations and local disturbances to persist and grow disproportionately big, as established in Scheidegger’s instability principle. Now, the combined probability of any particular set of global controls is low, and the probability of any set of local, contingent controls is even lower. As a result, the likelihood of any landscape or geomorphic system existing at a particular place and time is negligibly small – all landscapes are perfect, in the sense that they are an improbable coincidence of several different forces or factors (Phillips, 2007). This fascinating notion, which has much in common with Ollier’s (1981, p. 308–310; 1992) ‘evolutionary geomorphology’, dispenses with the view that all landscapes and landforms are the inevitable outcome of deterministic laws. In its stead, it offers a powerful and integrative new view that sees landscapes and landforms as circumstantial and contingent outcomes of deterministic laws operating in a specific environmental and historical context, with several outcomes possible for each set of processes and boundary conditions. It remains to be seen if this implication of dynamical systems theory can reconcile different geomorphological traditions. Should it do so, it would be a huge success for the systems approach.