landscape modelling Because most landscapes change only very slowly, the erosion of valleys, for example, typically taking periods of 100 000 years or more, it is exceptional to be able to observe landscape evolution directly. Models provide an essential means of extrapolating measurements taken over a few years, or at most a century, to the geological time spans over which landscapes develop. There are difficulties in this extrapolation, because of progressive changes in the form of the land and substantial climate change over these periods, but the principles of landscape modelling are reasonably well established.
Models may be either qualitative or quantitative. W. M. Davis developed the concept of the geographical cycle of erosion in the 1890s, which provided a descriptive model of the sequence by which hillslopes and rivers decline in gradient over time in the absence of significant tectonic uplift, eventually becoming low-relief peneplains, which could later be uplifted to form erosion surfaces. Walter Penck in the 1920s and Lester King in the 1950s proposed that hillslopes normally evolved through parallel retreat of steep slopes, although Penck attributed the retreat primarily to a balance with tectonic activity, whereas King recognized parallel retreat as characteristic of steep semi-arid landscapes. John Hack in 1960 suggested that landscapes were generally in a balance between denudation and either tectonic uplift or fluvial downcutting, so that the landscape was essentially in a state of dynamic equilibrium. Although these models were essentially qualitative, they contain important concepts which are also relevant in the context of quantitative models, particularly the ideas of slope decline, slope retreat, and equilibrium.
One of the earliest quantitative models was set out in the Revd Osmund Fisher's 1866 paper ‘On the retreat of a chalk cliff’. G. K. Gilbert, working in the Henry Mountains of Utah in the 1870s, also formulated a set of principles for badland slopes, which are seen to be related to current slope models. In the 1950s, Bakker and de Heux made further progress on modelling the retreat of a cliff face to form a talus slope at its foot. In the 1950s too, Frank Ahnert was among the first to begin development of slope models in a recognized current form. Initially most slope models were for the evolution of a single slope profile and, before the widespread use of computers, many models were based on seeking mathematical solutions rather than using numerical simulation. Ahnert, in the 1970s, was again a leader in developing three-dimensional models of landscape evolution, and today slope profile and three-dimensional models coexist, both relying mainly on computer simulation, although there is also considerable value in scale models of landscapes, especially for fluvial and coastal geomorphology. Slope profile models allow more complex processes to be simulated, while three-dimensional models provide a basis for studying the integration of stream and slope processes to create a complete landscape.
Principles of current simulation models
Most, although not all, current slope models are built from four basic building blocks: (1) a framework consisting of a sediment budget equation; (2) one or more ‘process laws’; (3) boundary conditions linking the model to the remainder of the landscape; and (4) an assumed initial form from which the landscape evolves. These create a soluble system, which can be described formally by a set of partial differential equations.
The sediment budget equation essentially takes the form:Input − Output = Net increase in storage,
which is familiar in hydrology. For landscape models, the inputs and outputs are sediment and solute transport rates, and the net increase in storage represents an addition of sediment, that is deposition, and a net decrease (negative increase) erosion, or denudation (where relevant it should include any net addition or removal by wind transport). The sediment budget may be applied to many kinds of landscape unit, but is most commonly applied to sections of a slope profile (Fig. 1), although it may also be applied to parts of drainage basins (e.g. hillslopes and valley bottoms) or whole basins. A flow strip, which outlines a slope profile, lies between neighbouring lines of greatest slope (flow lines), and may have a variable width if there is convergence or divergence of the flow lines.
The sediment budget equation is undoubtedly the best established relationship in slope modelling. It not only plays a vital role in maintaining the integrity of models (by ensuring that all sediment is accounted for), but also provides the formal link between spatial rates of change (in sediment transport rates) and temporal rates of change (in elevation), which is at the methodological heart of slope modelling.
Process laws describe the observed empirical dependence of sediment and solute transport capacities (the maximum rate of steady sediment transport) on the landscape topography, usually through slope gradient,
s, and distance from the divide,
x (which can be generalized to flow strip area per unit strip width). Several process laws have been generalized to the form:Transport capacity ∝
xmsnThe distance (
x) term indirectly represents the effect of the collecting area for water at any point. The gradient (
s) term reflects the influence of gravity. For processes such as soil creep, rainsplash, or solifluction, the transport capacity is (approximately) directly proportional to gradient, and independent of flow (and therefore collecting area). The exponents
m and
n are therefore 0 and 1. For soil erosion (rillwash) processes, acting in rills or small gullies, the effect of overland flow is much more important, and the exponents are
m = 1−3;
n = 1−2. For a process such as solution, which depends directly on flow and hardly at all on gravity,
m = 1;
n = 0.
Actual sediment transport may not take place at the full sediment transport capacity, although in simple slope model, this is usually assumed to be the case. More generally, the rate of pick-up of sediment may be related to the ‘excess capacity’ principle. This states that where sediment is carried at less than its full capacity, the rate of pick-up is proportional to the excess capacity, that is the difference between the capacity and the actual sediment transport rate. This difference is particularly important in rivers, where different grain sizes are transported selectively, with different excess capacities for different grain size classes. Coarse material is normally carried at its full (rather low) capacity rate, while fine material may be carried at far below capacity, limited by the supply of suitable material upstream.
Boundary conditions may take many forms. For a simple slope profile, the upper boundary is most simply represented as a divide between two valleys, at which the sediment transport is zero. The base of a slope profile is usually at a river bank. The simplest way to represent this boundary is to define how its elevation changes over time: it may either be fixed (the simplest case) or gradually changing in response to the behaviour of the river, and/or its interaction with the sediment delivered by the hill slope.
The fourth and last essential component of a slope model is a set of initial conditions. For a slope profile, it is necessary to assume a profile form at the start of the period to be simulated. This may be chosen as an arbitrary form, such as a plateau or a slope of uniform gradient, it may be based on independent evidence of a previous landscape form, or it may use the surveyed form to begin the simulation from the present.
Figure 2 shows a sequence of slope profile forms simulated for a combination of rillwash and rainsplash processes, beginning from a plateau, with a divide at the top (left) and a fixed stream at the base. The compound process law used in this example is a sum of two terms for rainsplash and rillwash respectively.
The simulated slope forms are initially strongly convex, because of the incision of a ravine into the plateau but, over time, the profiles become convexo-concave. As time goes on, the influence of the initial form becomes progressively weaker, and the profile becomes largely characteristic of the processes forming it. This may be regarded as a useful property, in that direct evidence of past forms is generally rather speculative, but it also follows that it may not be possible to reconstruct former landscapes from the present. This property, for distinct initial profiles to evolve convergently, is known as ‘
equifinality’.
Calibration of the models can be partially achieved by comparing the process laws with short-term field measurements of process rates, but it must be recognized that climate has changed, and process rates with it, so that extrapolation of present rates further back than the mid-Holocene may be highly speculative. Formal validation of slope models is, therefore, largely impracticable at present, even though the compatibility of known processes and observed forms may be highly suggestive.
Among the problems encountered, there is considerable error in assessing many of the process rate parameters, and many of the processes are inherently non-linear. There is, therefore, the strong expectation that there are some situations of chaotic sensitivity to initial conditions, and an inherent unpredictability in the landscape evolution. Perhaps the best example is the difficulty in predicting exactly where channel heads will form, even though it is possible to predict the overall density of the drainage network.
Special cases
In a number of cases, simplified assumptions within slope models help to give an overview, and link back to the older qualitative models described above. For example it may be assumed that the landscape is uplifted tectonically at a rate exactly equal to the denudation rate. This matches Hack's steady ‘dynamic equilibrium’ assumptions, and it can be shown that for the simple single process law above the steady-state equation shows convexity for certain conditions and concavity otherwise. For the compound process law of Fig. 2 (for rainsplash plus rillwash), the steady-state profile is convexo-concave, and the convexity has a width corresponding exactly to the area where the first (rainsplash) term is dominant. Similarly the base of the slope is concave in the zone where the second (rillwash) term is dominant.
A second kind of simplifying assumption is to hold the bottom point of the slope profile at a fixed elevation (base level). For many process laws, including the examples given here, the slope profile can be shown to evolve towards a ‘characteristic form’ in which the denudation at every point is strictly proportional to its elevation above the base level. This assumption, and its consequences, correspond closely to the Davisian model, with the characteristic form corresponding to ‘maturity’ of the profiles, and the final form to a ‘peneplain’. Although less amenable to a simple analysis than the dynamic equilibrium above, it may be seen from Fig. 2 that, for the assumption of fixed base level, the profile due to rainsplash and rillwash is also convexo-concave, although the width of the convexity is slightly narrower than for dynamic equilibrium.
Developments in slope modelling
Beyond the simple slope models described here, there are many significant developments which are gradually reducing the limitations of the simple forms. The main growth areas are in three-dimensional models; in establishing explicit links between process rates and climate; in the enrichment of models with an understanding of soil and vegetation dynamics; and in attempting to reconcile models at different scales. These areas represent the current research frontiers in landscape modelling.
One important growth area is the extension of models from profiles to three-dimensional landscapes. The main difference in principle is that the distance from the divide (
x above) needs to be replaced with drainage area per unit flow strip width, and this changes dynamically as the landscape evolves, and must be re-computed periodically. Although three-dimensional models make greater computing demands, these can now be met, allowing investigation of the dynamic interaction between stream and hillslope processes. This determines not only stream long profiles, which determine the evolution of each slope base, but also the extension or contraction of the entire stream network, through the position of channel heads. Large-area three-dimensional models are also potentially valuable tools for exploring the response of landscapes to regional patterns of tectonic movements, and the isostatic uplift resulting from unloading due to erosion.
Although simple process laws are related to distance from divide (or area) and gradient, these laws are based on an understanding of the physical processes, and implicitly sum rates over the frequency distribution of rainfall and other climatic events. Increased knowledge of each process provides a means of building up the aggregate process laws from physical and chemical understanding of single events. This approach has the potential of making an explicit link from climate to process rates. As knowledge of past climates improves, it may become possible to simulate slope evolution reliably through periods of major past and future climate change.
One element in linking process rates and climate lies in a better understanding of the links with vegetation and soils. Vegetation cover and organic soil are highly effective in reducing erosion; stripping by erosion changes soil properties; where surface wash is important, there is strongly selective transport amongst the grain sizes moving on the surface; an understanding of these processes therefore plays another major part in mediating the impact of climate on process rates.
The last major growth area is in attempts to make workable models at a wide range of spatial scales and timescales. In assessing priorities for mitigating soil degradation, there is a need to understand area distribution at regional to continental scales. There is also a need to link slope evolution and erosion rates to the very broad scales of general circulation models for climate. Similarly, there is a need to understand processes down to the time scale of a single storm, and up to the Pleistocene or the Tertiary. In changing time and/or space scales, it is essential, although inherently difficult, to ensure that similar principles are applied at each scale, that models remain compatible as scales are changed, and that dominant process at each scale are properly represented.
M. J. Kirkby
