Hazards: singularities in geomorphic systems

IMllMI ELSEVIER

Geomorphology I0 (1994) 19-25

Hazards: singularities in geomorphic systems A.E. Scheidegger Section of Geophysics, Technical University, Gusshausstrasse27-29-128/2, A-1040 Vienna,Austria Received November 20, 1993; revised March 8, 1994; accepted April 1, 1994

Abstract The development of landscapes is commonly interpreted as the outcome of the operation of a few simple phenomenological principles amongst which the Antagonism Principle is the most fundamental one. It states that there are two types of processes active in the formation of a landscape at the same time: the endogenic (tectonic) and the exogenic (climate-driven) processes. These two types of processes may more or less balance each other, in which case the actual aspect of a landscape corresponds to the instantaneous dynamic equilibrium between them. If one of the external parameters is changed, the equilibrium is supposed to be re-established by a corresponding continuous change of the other parameters (process-response theory). The dependences between various landscape parameters may become multivalued at junctions, cusps, etc. In that case, the system can jump from one branch of the process-response curve to another: a natural disaster occurs. However, modern views on the behavior of complex systems allow a somewhat different interpretation of the above phenomenological landscape interpretation, but also in the context of the Principle of Antagonism. However, a "stationary" landscape-state does now not correspond to a dynamic equilibrium, but to self-organized exogenic order at the edge of chaos. Under conditions of such self-organized criticality there are no natural space and time scales, so that fractal statistics are applicable; spatial and temporal correlations follow a power law. Extremely small accidental perturbations can cause the system to become unstable, leading to rapid changes in the landscape which are experienced as hazard-events. These follow a power law distribution for size versus frequency, provided the endogenic input occurs at a constant rate; they are therefore simply part and parcel of geomorphic systems.

1. Introduction and background: phenomenological landscape principles The development of landscapes can be described phenomenologically as the outcome of the operation of a few simple fundamental principles (Scheidegger, 1987): (i) the Antagonism Principle, (ii) the Instability Principle with (iii) the Catena Principle as corollary, (iv) the Selection Principle and (v) the Tectonic Predesign Principle. The Antagonism Principle is the most fundamental principle of landscape evolution. It refers to the fact that there are essentially two types of processes active in the formation of a landscape at the same time: the endogenic (tectonic) and the exogenic (climate0169-555x/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDIOI69-555X(94)O0014-1

driven) processes. Of fundamental importance is the fact that the two processes involved in the geomorphic antagonism have a fundamentally different stochastic nature: Endogenic processes are essentially non-random, exogenic ones essentially random (Scheidegger, 1979). The reason for this is that endogenic processes are connected with plate tectonic movements which are autocorrelated over thousands o f kilometers, whereas exogenic processes are connected with turbulence phenomena in liquids (water and air) which are essentially chaotic. Although the endogenic processes must be basically primary (without uplift there could be no degradation), the two processes are often assumed to more or less balance each other; thus the actual aspect of a landscape corresponds to the instantaneous

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A.E. Scheidegger/ Geomorphology 10 (1994) 19-25

dynamic equilibrium between them. If one parameter is changed by, say, tectonic processes or human activity, the others adjust themselves accordingly to reestablish equilibrium (Process-Response Theory - cf. Hack, 1960; Carson and Kirkby, 1972 Strahler, 1980; Terjung, 1982 and many others). This idea has to be modified by the observation that the "balance" is quite often unstable (Scheidegger, 1983). Apart from the fact that individual surface features tend to be impermanent, although their overall appearance may seem to be constant corresponding to the process-response theory, the dynamic equilibrium may actually be inherently unstable: any deviation from uniformity tends to grow; there is, so to speak, a positive feed-back between the size of the deviation that has already been reached and the rate by which this size increases. Although the dynamic equations of the system are unknown, the (presumably) nonlinear differential equations can be linearized for the purpose of stability analysis according to Taylor (1950), at least in the neighborhood of the deviation for a short time range. Then, for any parameter x(t) (this may be a vector) describing a landscape feature, the corresponding linearized differential equation is: ch~ t ) / d t = Ax

( 1)

where A is a parameter. Its solution is an exponential function of time; thus the growth-equation at the beginning has the form: xl t ) = C exp(At)

(2)

where C is a constant of integration. If A is positive, one has positive feedback and the condition is unstable, the deviation x ( t ) will grow exponentially and the value of A is a measure of the degree of the instability. The exponential function would increase infinitely so that it would lead to a catastrophic situation; however, the linearization holds true only for short time ranges so that an unstable state need not necessarly lead to a catastrophe: the growth process may eventually come to a stop when a saturation stage is reached. The instability model is evidently correct for many purely exogenic geomorphic features: the longitudinal profile of stepped valleys, the terraced transverse profiles of valleys, sequences of pools and riffles in river beds, the meandering of rivers, the formation of glacial cirques and certain dissolution phenomena (such as "onion-skin" weathering). Some of the above features

can also be described as a consequence of a Catena Principle, which is very closely related to the instability principle. In effect, such sequences are "chains" (Latin: catenae) of geomorphological elements that are repeated again and again; this is the consequence of instabilities that arise at certain points but reach a "saturation stage" so that the corresponding features are repeated in a sequence. One can express this as a separate principle, namely the Catenaprinciple (Scheidegger, 1986). The concept of a catena was originally invented by Milne ( 1935, 1947) in soil research who noted that certain definite sequences (catenas) of soil types recur on a slope; it was then adopted generally for self-repeating morphological features, such as eluvial-colluvial-alluvial zones on slopes (Scheidegger, 1986), slow (wide) - fast (narrow) - slow (wide) moving stretches in rivers (Scheidegger, 1986), flat (slow-moving) - steep (fast moving) - flat (slowmoving) sections on chronic clay slides (Gerber and Scheidegger, 1984) and fine-coarse-fine grained mass flow sequences on scree slopes (Gerber and Scheidegger, 1974). A further aspect of the exogenic processes is that they tend to occur in such a fashion that statically stable forms (with reference to the self-induced stresses) are preferentially "selected"; this fact is stated by the Selection Principle of Gerber (1969). Finally, the remaining landscape principle of those mentioned at the beginning states that many landscape features are designed by deep-seated tectonic processes ( Tectonic Predesign Principle - - Twidale, 1971; Morisawa, 1975; Scheidegger and Ai, 1986). This principle is really part of the Antagonism Principle; namely that part which refers to the endogenic processes.

2. Complexity theory Landscapes are very complex systems. It is therefore only natural that geomorpholgists have tried to apply general systems theory to them (e.g. Chorley, 1962). The term "system" was created by Von Bertalanffy (1950). To speak of a "system" one needs (i) a set of elements with some variable attribute of objects, (ii) a set of relationships between attributes and objects, and (iii) a set of relationships between those attributes of objects and the "environment", i.e., the region outside the system. Macroscopically, in geomorphology,

A.E. Scheidegger / Geomorphology 10 (1994) 19-25

the "objects" are landforms, the "attributes" are quantifiable properties of these landforms, and the "relationships" consist of exchanges of energy and mass between the landforms so that causal links are formed between the attributes. M i c r o s c o p i c a l l y , however, landscapes are m e c h a n i c a l systems, i.e., they consist of a great number of particles each of which has a position and a momentum. The state of such a system is represented by a point in phase space, i.e., in the multidimensional space which has one dimension for each coordinate and each momentum of each particle. The e v o l u t i o n of the state of the system is described by a t r a j e c t o r y in phase space. First, landscapes were treated as closed systems; i.e., systems with no interactions with any "outside". One can then immediately draw an analogy with equilibrium thermodynamics and define analogs of temperature and entropy. The conclusion is that such landscapes tend to develop towards an equalization of all relief at an average level. In a closed system in equilibrium, the phase trajectories over time are equally dense throughout the phase space (ergodic hypothesis). The condition that the entropy tends towards its possible maximum, allows one to calculate the most probable values of relevant variables in particular systems such as meander trains (Langbein and Leopold, 1966). More realistic is a view of landscapes as o p e n systems, i.e., systems which have an interaction with an "environment" (in this, the distinction between "system proper" and "environment" is by arbitrary fiat). The landscape is then considered as a system through which mass and energy flow or "cascade" from an input to an output (Terjung, 1982). The analogy with equilibrium thermodynamics is no longer possible; the macroscopic variables behave in a complex way (hence their study has been called the "Science of Complexity"), the phase trajectories are no longer ergodic; in fact, the latter tend towards certain "attractors" in phase space. If the attractors are fixed points, one has a trend towards stable equilibrium, if they are limit-cyclical or toroidal, one has certain stable evolution patterns, if they are otherwise (strange attractors), the behavior is generally unstable. Such systems that are nonlinear, nonequilibrium, deterministic, and incorporate randomness so that they are sensitive to initial conditions, and have strange attractors, are defined as c h a o t i c (definition after ~ambel, 1993, p.

21

14). In such systems, the entropy need not tend towards a maximum, in fact, it may well decrease (in contradiction to the second law of thermodynamics - - Prigogine and Stengers, 1984). Due to the form of the attractors some order can establish itself at the edge of chaos, at least for a certain length of time. Empirically, certain "laws" have been found to hold in self-ordered systems at the edge of chaos (Bak et al., 1988): The pertinent characteristic observables are spacially and temporally scale-invariant, i.e. they have been found to be fractal. In a fractal set of dimension D, there exists a power law of subsets: the number N of subsets of (linear) size D is proportional to L -o. In terms of size (M)-frequency (N number of events per unit time) distributions, the power law mentioned above is represented by a "Gutenberg-Richter (e.g. 1949) type" law: log N ( M ) = a - b M

(3)

which is a "power law" for N. Such power laws have been found everywhere in self-ordered natural systems. Since there are positive feedbacks in such systems, the process-response theory is no longer tenable.

3. Interpretation of landscape principles in the light of complexity theory The modern views on the behavior of complex systems sketched above allow a somewhat different interpretation of the phenomenological landscape principles than was common hitherto. The basis is evidently the Principle of Antagonism. The endogenic processes, and therewith also the Tectonic Predesign Principle, correspond to the input into a mechanical system originating from the "environment". At the other end of the spectrum are the exogenic processes based on atmospheric and aquatic turbulence whose input is essentially chaotic (see Section 1). The aspect of actual landscapes, then, does not correspond to a "dynamic equilibrium," but to self-organized order at the "edge of chaos" in an open, dissipative system. Orderly evolution is still subject to Taylor instabilities (positive feedback) for deviations from this order; these grow at the beginning exponentially as implied by Eq. (1); however, since the system is inherently nonlinear these deviations do not follow an unbridled growth equation of the the type of Eq. (1), but rather a "logistic equa-

22

A.E. Scheidegger/ Geomorphology 10 (1994) 19-25

t i o n " l i k e t h a t o f V e r h u l s t ( c £ the review by ~ambel, 1993): d ~ ( t ) / d t = rx(t) [ 1 - x ( t ) ]

(4)

where x is a normalized deviation-variable and r a system parameter. Although the Verhulst equation is nonlinear, it can still be integrated in closed form; its s~lution is: x(t) = 1/[1 + e x p ( - r t ) ]

(5)

This equation represents a sigmatoid growth curve for the deviation x(t) which tends towards a saturation t~alue of 1. Examples of such a saturation occurring are represented for instance by the fact that meander loops cannot grow indefinitely in size, slopes cannot become inlinitely high, pools and riffles in river beds cannot become arbitrarily large, etc. (Scheidegger, 1983). Steady-state landscapes correspond to reasonably stationary conditions of self-organized order at the edge of criticality with a power law (Eq. 3) for spacial and temporal correlations (Bak et al., 1988). In the critical state, there are no natural length and time scales, so that fractal statistics are applicable (Turcotte, 1992). Computer simulations have shown that such a self-organization can indeed establish itself in stochastic situations solely under the assumption of micro-scale relationships between individual elements of the system. Thus, Werner and Hallet (1993) have shown that solely the assumption of preferential growth patterns of individual ice-needles in stone-free regions during the freezing cycle on (periglacial) slopes automatically produces self-organized stone stripes in a corresponding numerical model. The results closely agree with the stone stripes found on such slopes in the glacial region of Mauna Kea, Hawaii. The basis of the process-response theory is that one presupposes small changes or perturbations in some of the external parameters describing the system to cause only small and continuous adjustments in the others so as to re-establish equilibrium: The pattern of the landscape development is described by a smooth, wellbehaved curve representing the dependence of one parameter on the values of the others; a model that hardly corresponds to the manifest facts.

4. Hazards

4.1. General remarks

Hazards are represented by the possibility that a reasonably stable orbit in system-phase space may change abruptly. In this, both, endogenic and exogenic processes can be involved: if the normally continuous relations between the parameters describing geomorphic variables (process-response relations) contain singularities, and if such a singularity is encountered during the geomorphic evolution (a perfectly "natural" possibility), an event appearing to an observer as a "catastrophe" occurs. First of all, it is possible that the primary input from the endogenic processes is not steady and continuous: The endogenic processes occur normally in a steadystate fashion so that they contribute to an orderly development of the Earth's surface; however, they, too, can become unstable and become chaotic which expresses itself to an observer in the form of earthquakes, volcanic eruptions, etc. However, our main concern in this paper is not with such "primary" instabilities in the endogenic input, but rather with the self-created instabilities occurring during the otherwise continuous and regular geomorphic evolution. Second, it is well known that even basically regular process-response dependences may become multivalued at junctions, cusps etc. (cf. the "catastrophe" theory of Thom, 1972); in such cases, small random perturbations can cause the system to jump from one branch of the process-response curve to another: a natural disaster occurs. Third, a general description of instability theory can be given in terms of attractors. The phase point describing a (temporarily) stable landscape system, as we have seen above, generally lies in a self-organized domain at the edge of chaos under the influence of a strange attractor: thus it should be clear that extremely small accidental perturbations can cause the system to become unstable leading to rapid changes in the landscape which are experienced as a disaster. Cases where minute changes in the initial conditions lead to a "catastrophic" long-term development occur under many "ordinary" conditions.

A.E. Scheidegger/ Geomorphology 10 (1994) 19-25

4.2. Slopes and landslides As a first example, we use denudational slope-development for which a general differential equation has been developed and discussed at length regarding its applicability by the author (Scheidegger, 1961): Oy/Ot= - ( Oy/Ox)~/1 + ( ,gy/ax) 2

(6)

where x is the abscissa (distance) and y the height of the slope. As it stands (and as it was used in 1961), it is assumed that the slope height y is monotonously increasing (or at least: not decreasing) with increasing x; if it is decreasing, (Oy/Ox) becomes negative and the slope would become higher by erosion, which is manifestly impossible under natural conditions. If Eq. (6) is to be applied to slopes decreasing with increasing x, it must be applied by replacing the partial differential by its absolute value: Oy/Ot = -

[Oy/Oxl~/1 +

(Oy/Ox) z

(7)

This is a highly nonlinear partial differential equation. When solving it on a computer by the method of finite differences, questions of mathematical stability are of utmost importance. Firstly, the conditions result.ing from the characteristics have to be observed (for background see e.g. Collatz, 1951); however, this is only a necessary, not a sufficient condition. Thus, it turns out, secondly, that, to ensure stability, the differentials have always to be taken in the direction of decreasing y. Thirdly, the iterations must only be continued to a point where the boundary conditions (y at x = 0, 100) do not make themselves felt. The solutions obtained in 1961 only involved slopes increasing (or at least not decreasing) in the direction of +x, so that Eq. (6) could be used. We have now modified the program to be applicable also to slopes decreasing in the direction of +x, observing the conditions mentioned above. All solutions obtained were mathematically stable, but an knick or hole in a smooth slope led now to a positive-feedback mechanism. As an example, we show here profiles of the regular non-hazardous development of a smooth slope bank (Fig. 1), and the development of the same bank with a ditch (through which all eroded material is thought to be removed) at the foot (Fig. 2). Figs. I and 2 differ only by the initial conditions (top curve); the units are normalized, i.e. 0 ~
23

taken). It is clear that the development of a slope without a ditch is regular (or stationary), if there is a ditch, the latter becomes rapidly deeper, until the critical height (Fellenius, 1940) of a stable slope bank (at least for a small angle of internal friction of the slope material) is inexorably (since the slope angle stays more or less the same) exceeded and a slide occurs. Thus, a minute change in the initial conditions (i.e. introducing a small ditch) causes a fundamentally different longterm behavior. This is characteristic for complex systems.

40.00

2

~ 20.00 0

000

20

40

60

80

Distance; T-steps 5.0 Fig. 1. Regular uniform recession of a smooth slope bank. Regarding units: see text.

60.00

40.00

~c-'

I

20.00

0.00

q) 00 -20.00

-40.00

-60.00

.........

o.oo

,. . . . . . . . .

,.........

,.........

20.00 4 0 . 0 o 60.00 Distonce; T-steps

~,,,

8o.oo "%o.oo"" 2.5

Fig. 2. Unstable recession of a slope bank with a ditch at the foot: the ditch becomes deeper. Regarding units: see text.

24

A.E. Scheidegger / Geomorphology 10 (1994) 19-25

The slope bank considered above has been assumed as somehow a priori given. Evidently, it is at the edge of criticality. An initial slope bank can also be the result of self-organized criticality (regarding this concept cf. Nicolis and Prigogine, 1977; Haken and Wunderlin, 1988 and Kauffman, 1993); the best known example in geomorphology is that of a sand or scree slope that has been formed by the steady addition of material from above (Bak and Chen 1991; see also Waldrop, 1992, p. 305). In this case the mass-movement is caused by gravity alone; the (temporary) stability is due to the interlocking of the sand or scree particles. Further additions of particles cause intermittent slides at irregular intervals; the number of slides in a given time interval (assuming the particle-addition rate as constant) as a function of size follows a power-law distribution (the average frequency of a slide of a given size is inversely proportional to some power of its size); as noted a feature of complex systems. It may be assumed to hold for the landslide size versus frequency distribution if the latter are due to a constant mass input from endogen ic (increase of height of relief due to tectonic uplift as a result of the tectonic plate motions) and consequent denudation by exogenic processes.

4.3. Planes and sink holes As a second example we refer to the development of a plain with a hole in it (without a hole, a horizontal plain does not denude at all; i.e. its profile remains a horizontal straight line forever). Fig. 3 shows the profile of such a plane with a small knick in it, using the usual slope-development equation (Eq. 7): this solution is for a linear case but is qualitatively the same for circular coordinates. The original knick can be assumed to have been caused by random fluctuations in the denudation processes; a good example would be the formation of a small solution-hole in a karst plateau which develops into a huge sink-hole. Such cases are well known from the limestone platforms of Yucatan and Florida: in Yucatan, deep "Cenotes" (water-filled holes) represent characteristic landscape features, and in Florida sinkholes with houses disappearing in them ( c f the Winter Park event: Foose, 1981 ) are notorious. Thus, again, small changes in the initial conditions cause large effects in the long run which are experienced as hazards.

50.00

% / / /

+//

++.oo C~ I 0.00

++

+

0..- I0.00 0

-30.00

V -50.00

................... , ................... , ................... , ................... , ................... , 0

20

40

60

80

100

Distance; T-steps 5.0 Fig. 3. D e e p e n i n g o f a hole in a plane.

5. Conclusions

The above analysis shows that geomorphic systems are principally open and highly nonlinear with the possibility of positive feedback. They cannot be treated by ordinary system theory; one has to use concepts from the science of complexity and non-equilibrium thermodynamics (i.e. its system-analog). "Stationary" landscapes are in reality self-organized patterns (cf. stone stripes) at the edge of chaos; small changes in the initial conditions can cause very different long-term behavior-patterns: thus the "process-response" concept is suspect in principle. "Hazards" are the result of sudden changes in long-term behavior caused by minute changes in the initial conditions. These can be the result of input from tectonic processes into the system and the resulting establishment of self-organized criticality. In this case, the hazard events follow generally a power-law distribution for size versus frequency, provided the input occurs at a constant rate. Hazards are therefore simply part and parcel of the normal geomorphic evolution: the steady-rate build-up of stress (input into the geomorphic system) by the plate-tectonic motions causes intermittent catastrophes with a power law (Eq. 3) characterizing the size-frequency distribution.

A.E. Scheidegger / Geomorphology 10 (1994) 19-25

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