Geography 423
Advanced Geomorphology

Models of Landscape Evolution

Summary: Intellectual structures are critical in geomorphology because

  1. landforms develop over longer timespans (usually much longer) than human lifespans
  2. there are different methodologies and objectives depending on the relevant theory
  3. by its vary nature, erosion eliminates evidence of landscape evolution
  4. the most profitable approach to landform studies is to derive hypotheses from process research and corroborate or falsify them with morphometric research
Thus far we have considered a number of specific concepts within the context of time and space in geomorphology. Now we visit concepts in the context of models of landscape evolution. Thorn (1988: 121) uses the analogy of fabric; the concepts are individual threads in the fabric of geomorphology. When the concepts are woven together, a pattern merges, but they can be woven in a variety of designs at the weaver's (modellers) discretion. Thus models are like fabric created by weaving together concepts in a pattern that embraces the various temporal and spatial scales of landform development.

W.M. Davis: The geographical cycle

W.M. Davis was geography professor at Harvard University. He wrote about his model from 1880-1938, travelled and spoke widely. Like his contemporaries in natural science he was strongly influenced by Charles Darwin (On the Origin of the Species) and Charles Lyell (Principles of Geology), although used evolution as a notion of history (inevitable progress or change over time) rather than a process and took a deterministic rather than probabilistic view of evolution like Darwin.

Thus Davis aspired to a deductive, theoretical, genetic model of landscape evolution. The concepts of structure, process and time were his theoretical framework:

Walter Penck: relating landforms to crustal movements

The best known manifestation of Penck's model is the retreating slope profile, where evolution of the profile is controlled by rate of output (river erosion) at the base and rate of uplift of the land. He was able to deduce various slope profiles for different combinations of river erosion, uplift and rock resistance, by assuming that stronger rock requires steeper slopes for the same rate of denudation. He also modelled stream longitudinal profile as controlled by uplift, rock type and stream discharge.

Another expression of his model was three categories of landform assemblage according to tectonic history (versus normal climate):

Penck also envisaged three landscapes resulting from slow, intermediate and rapid rates of uplift. Morphologically, they were similar to Davis' old, mature and young stages, but whereas Davis ascribed morphology to age (time-dependent), Penck's model was largely time-independent based on tectonic history.

Evaluation of Penck's model is hindered by its hurried writing, posthumous publication and confused representation in English, including misrepresentation by Davis who was defending his own ideas. Although there were important flaws and contradictions in Penck's work, and it was poorly translated or misrepresented especially by adherents to the Davisian school, it was the one comprehensive alternative to the cycle of erosion and thus was a focus for contrary ideas, such as emphasis on process rate (both endogenic and exogenic) and greater attention to slope retreat.

Lester King: global geomorphology during the era of process studies

King's model was based on a slope profile consisting of four segments, any one of which may be entirely absent:

  1. waxing slope: convex segment at the crest, dominated by soil creep of a weathered mantle; increasing slope angle is required to transport the greater quantity of slope debris with distance downslope
  2. free face: bedrock outcrop; retreats parallel with weathering and uniform removal; may be absent in areas of low relief
  3. debris slope: debris from the free face resting at the angle of repose; does not bury the free face but retreats with it
  4. waning: gentle concave profile controlled by sheetwash and transport of sediment over an eroded bedrock surface (pediment)

Systems modelling

Systems modelling swept though geomorphology in the 1960's with the shift to process and quantitative geomorphology. Although not directly related to process geomorphology, the two methodologies are so compatible that they are closely associated in the same manner as Davis'' cycle of erosion and denudation chronology.

The formal structure of systems modelling (inputs, outputs, throughputs), and its application either conceptually or quantitatively make it a popular framework for research and teaching (hydrological cycle, continuity equations).

a set of objects together with relationships between the objects and their attributes, or a set of objects and the common processes by which they interact
In geomorphology, objects are usually landscape elements at a particular scale, relationships are geomorphic processes and attributes are physical properties like slope gradient, soil texture or drainage density (also depending on scale).

The components of the system will depend on the relationships that the researcher or teacher considers important or relevant. A model is a simplification of a system that is assumed to exist in the real world. Since it will be based on selected parts of a system, the model will always be incorrect to some degree.

Scientists can agree on the definition of a particular system, but it can never be fully known. Conversely, a model can be simple and completely known, but always will be incomplete and useful only for specific purposes.

Types of systems:

  1. isolated: assumed to have boundaries that prevent the import or export of energy and mass; besides the universe, the only isolated systems are in labs or assumed for convenience
  2. closed: import or export of energy but not mass; the earth is closed to closed and some geomorphic systems can be modelled as if they are (e.g. an internal drainage basin)
  3. open: both energy and mass move freely across system boundaries, includes most of the natural world
The isolated system is the basis for the concept of maximum entropy where free energy decreases, as it becomes more evenly distributed and moves along increasing lower gradients. However in open systems, energy is input and output. Therefore natural systems tend to adjust by self-regulation thereby approaching or remaining near an equilibrium (e.g., circulation of the oceans and atmosphere in response to the global solar energy balance).

Types of system models according to complexity:

  1. qualitative data, verbal description, graphs to describe a system; does not permit true modelling but is a first stage
  2. identification and analysis of morphological variables (e.g. free face and talus); simplest model, only functional information is relative position of the components, but no inference or understanding of mass or energy exchange is possible
  3. identification and analysis of energy and mass transfer; cascading system (direction specified; energy and mass move from the free face to the talus and not vice versa)
  4. integration of form and process; unlike 3 these involve feedback, output from one part is input to another, opposing or slowing the general trend of the system (negative feedback) or magnifying the main trend (positive feedback); natural systems (except glacial systems) are characterized by negative feedback so that changes are resisted and tendency towards an equilibrium is maintained; thus change in relationships over time can be modelled, for example, the talus grows to bury the free face resulting in negative feedback and decreasing rates of mass wasting and talus slope growth over time
  5. control systems: conscious or deliberate human intervention in the system; useful for addressing engineering or planning issues (impact assessment)

Types of system models according to degree of understanding of the system:

  1. black box: nothing known except the relationship between input and output
  2. grey box: structure is known (subsystems considered), but no detailed knowledge or investigation
  3. white box: subsystems, storages and flows known and investigated in detail

Construction of a system model

Evaluation of systems modelling

John T. Hack: A time-independent model

The interactions of time and space add much complexity to modelling. Thus usually one theme is dominant and the other produces variability or noise. Traditionally time-dependent models were more readily accepted because they were compatible with the popular notion of evolution and slow change over time and qualitative descriptive investigation, and because the approach was advocated by W.M. Davis, the pre-eminent American geomorphologist. Davis, King and Penck (in that order) placed most, less and lesser emphasis on time, respectively. King also focused on process while Penck emphasized structure and process. These themes have a spatial component, although King and to a lesser extent Penck related landscapes to initiating (past) events (e.g. tectonic history). Davis' contemporary G.K. Gilbert had a methodology that was founded on principles of physics and engineering and was more suited to a time-independent approach. However Gilbert made no attempt to develop or champion a comprehensive model. With the quantitative revolution and shift to process geomorphology, there was renewed interest in Gilbert and attention to a time-independent perspective.

John T. Hack is the champion of a time-independent model where landscape variability due to age is not modelled, but rather considered a source of variability in landscape from related to contemporary process. This approach assumes a dynamic equilibrium between contemporary surficial processes and the surface upon which they are acting. Hack chose dynamic equilibrium as his conceptual and methodological framework. He derived this perspective directly from G.K. Gilbert who worked in the western US, where the dramatic semiarid landscape seems youthful and dynamic. Hack applied dynamic equilibrium to reinterpretation of the Appalachian Mountains, the landscape that lead Davis to think in terms of change over time.

The time-independent perspective of Gilbert is reflected in his "laws" of

These laws all reflect a perspective of spatial variation and dynamic equilibrium between forces and resistance.

The basic features of dynamic equilibrium as applied to spatial relations within a drainage basin by Hack:

Hack also maintained that his model is not comprehensive, that time can also be invoked to explain landscape features, but it does apply to the entire range of spatial scales of interest to geomorphologists. Under dynamic equilibrium, landscapes evolve without obvious change, unless there is a change in energy inputs (climatic change, tectonism) or surface resistance. Examples of the latter include the denudation of surface materials to expose harder or softer materials, or the accumulation of coarse materials in valley bottoms. The consequent adjustment to these changes represents a disequilibrium but does not conflict with the time-independent perspective. Hack argued like Penck that rates of uplift and erosion are linked, although he related erosion plus relief to uplift and rock resistance and had a thin database to support this relationship.

Evaluation of dynamic equilibrium:

  1. the past usually is poorly or only partly known, thus a model based on current conditions has a definite advantage
  2. mutual relationship with process geomorphology
  3. time-independent is an end-member of the distribution of systems and system models; these are fairly easily identified (e.g. and underfit stream is time-dependent relative to valley form but time-independent with respect to channel form)
  4. however, it is not usually this easy to resolve the complex forms that represent both time-independent and time-dependent behaviour; attention to spatial and temporal scale help, for example time-independent behaviour is more likely at more local scales, and the influence of past processes is proportional to their intensity and inversely proportional to time elapsed since the event
  5. dynamic equilibrium implies characteristic forms as opposed to relaxation forms
  6. situations where form is not maintained include uplift exceeding rates of erosion or increasing relief controlled by difference in rock resistance (e.g., inversion of topography)
  7. When small segments of landscape evolution are sampled it becomes difficult or impossible to distinguish between dynamic equilibrium (trending mean), steady state (constant mean) and dynamic metastable equilibrium (two scales of oscillations). Hack referred to both steady state and dynamic equilibrium, however a trending mean is much more likely in geomorphic systems.
  8. Dynamic equilibrium is more of a conceptual framework than a fully tested corroborated model, which will require much more extensive data bases. However, it is a very useful framework in that the reduced role of time is replaced with an expansion of spatial variability and the integration of parts of landscapes. In this respect it is tied to a systems approach and the notion that systems move toward equilibrium at a rate proportional to their distance from it. Thus those far from equilibrium change quickly (time-independent) and thus near equilibrium change slowly (time-dependent). This systems perspective unites both the time-dependent and time-independent viewpoints.

Mathematical Modelling

Mathematics allows ideas to be expressed with great precision but this also demands a great deal of knowledge about the phenomenon. With this precision and lack of ambiguity, mathematical models have internal logic (one step leads logically to another), and thus the primary source of error is external logic, i.e. the assumptions required so that the model can be expressed mathematically. In geomorphology, these assumption are often very constraining or even unrealistic given our incomplete understanding of geomorphic systems.

The use of mathematics provides the option of modelling which is deterministic (where one physical or chemical conditions leads uniquely from another) or probabilistic (a degree of randomness and therefore variability is introduced), with the recognition the natural world is deterministic but too complex to be modelled as such. Mathematical modelling has been out of the mainstream of geomorphological research with the emphasis on process observation and an inferential inductive methodology.

Following is a review of the mathematical models of two contemporary geomorphologists. The focus here is on methodology and logic rather than the mathematics.

Frank Ahnert: a mathematical model of slope evolution

Ahnert developed based on a central concept of equilibrium of debris supply and removal, rather than Hack's equilibrium between process and form. He modelled the spatially and temporally variable rates of waste production, transport and redeposition at points in the landscape.

Ahnert's model is discrete: comprised of a network of points, where computations are made individually and sequentially with small temporal and spatial increments. Waste thickness is related to form by modelling changes along slope profiles (2-D) or on a grid of points (3-D). He differentiated between direct removal (no redeposition), and point-to-point transfer (the norm) which results in accumulation of debris at the foot of the slope. The computations are of the change in waste cover thickness (mass balance) in response to mechanical and chemical weathering, slope wash, rain splash, soil creep and landsliding. Rational equations simulate each process.

The model user is able to simulate the influence of climate by varying the relative importance of mechanical and chemical weathering. The dynamic equilibrium between waste thickness and rates of includes decreasing weathering as waste thickens and increasing weathering as the waster cover thins, although a thin waste cover is most effective for chemical weathering and so rate of weathering initially increases with increasing waste cover until a threshold thickness beyond which weathering rates decline.

Splash transport is a non-linear function of slope gradient, while wash transport is a non-linear function of slope gradient, surface runoff and soil erodibility. Wash transport can be specified as point-to point or complete removal. Surface runoff is derived from a slope hydrology submodel based on the mass balance between precipitation at a point, infiltration capacity and runoff from upslope.

Ahnert modelled slow mass movement as either viscous (nonthreshold) or plastic (threshold), depending on slope gradient and fluidity of the material. In both cases, there is a critical slope angle but the value is zero for nonthreshold viscous mass wasting. Beyond a threshold slope, waste does not accumulate, so Ahnert has the material move to the next lowest point where slope gradient is less than the threshold. However this not a realistic description of rapid mass wasting, because 1) momentum usually carries the debris further than this model suggests and 2) the critical threshold for rapid mass wasting will vary from point to point as kinetic energy is converted to kinetic energy (i.e., in a moving mass the angle of dynamic friction is less than the angle of static friction).

Ahnert approach to modelling is numerical simulation, that is, the outcome depends not just on the mathematical functions but also the numerical values that the user enters to simulate, for example, rock structure (resistance) at a point, nature of base level change (reference data for iterations of the model), relative importance of weathering processes and dominant process of mechanical transport. By varying all these boundary conditions, Ahnert concluded that the strongest control on slope profile form was the mode of removal (point-to-point or direct).

The greatest strengths of his model are the use of waste thickness as a control and response with respect to slope processes and as a parameter than can be verified empirically. Also his model is geographic; it can accommodate spatial variability between point locations. The two main weaknesses are that he does not model feedback among processes and some relationships are suggested by observation and not physically based.

Michael J. Kirkby: deterministic modelling of process-response

Kirkby slope model is based on a mass balance equation formulated using differential calculus so that, unlike Ahnert, the slope is modelled simultaneously as a continuous profile. It is structured to accept submodels and the analytical use is based on vectors, although a co-ordinate system must be specified in order to solve the equations. Kirkby models change in elevation as a function of mechanical and chemical transport processes.

Kirkby's initial and preferred approach was analytical, that is, expressing theory symbolically and deriving new equations by manipulating the symbols; very much a deductive methodology. He eventually resorted to numerical modelling for the modelling of complex problems where the theory is inadequate, however, once empirical data (numbers) are inserted into equations, only numbers can be derived and not new equations.

Kirkby based the simulation of weathering effects on the concept of a soil deficit:

other important concepts:

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