A non-Markovian model of rill erosion

A non-Markovian model of rill erosion Michael Damron∗ C.L. Winter† December 2008 Abstract We introduce a new model for rill erosion. We start with a network similar to that in the Dynamical Discrete Web** and instantiate a dy- namics which makes the process highly non-Markovian. The behavior of nodes in the streams is similar to the behavior of Polya urns with time-dependent input. In this paper we use a combination of rigorous arguments and simulation results to show that the model exhibits many properties of rill erosion; in particular, nodes which are deeper in the network tend to switch less quickly. 1 Introduction 1.1 Reinforcement and Rill Erosion Stochastic processes with reinforcement are inherently non-Markovian and there- fore may model some real phenomena more accurately than can their Markovian counterparts. Reinforcement is a mechanism that provides a bias to a system, making it more likely to occupy states the more often those states are visited. Some well-studied examples include variations on the urn of P´olya (the original introduced in [4] and this and subsequent models studied, for example, in [1] and [9]) and reinforced random walks ([3, 15]). The infinite memory exhibited in these examples can force a system to spend most (or almost all) of its time in a small subset of its state space. Many natural phenomena exhibit similar behavior; for instance, the overall pattern of erosion on a hillslope is relatively stable once it is established, although small details of the pattern may change frequently and catastrophes that permanently alter it may occasionally occur. ∗Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012, USA. Email: damron@cims.nyu.edu; Research supported in part by NSF grant number OISE- 0730136. †National Center for Atmospheric Research, 1850 Table Mesa Dr., Boulder, CO 80305, USA. Email: lwinter@ucar.edu 0MSC2000: Primary: 60K35, 82D99 Secondary: 60G09 0** See arXiv:0704.2706 and arXiv:math/0702542 0Keywords: erosion; rill erosion; P´olya urn; exchangeability; dynamical discrete web 1 arXiv:0810.1483v2 [math.PR] 13 Dec 2008 We investigate a discrete time, infinite-memory random process defined on the nodes and edges of an oriented diagonal lattice (Figure 1) that we propose as a simple model of hillslope erosion. The lattice starts out smooth in the sense that it has no edges initially, but it sprouts edges everywhere the instant the process starts, much as rain can start soil erosion everywhere on a hillslope at once. Edges may connect an interior node to two, one, or neither of the two nodes directly above it. Exactly one edge descends from each interior node, and it points either left or right. At every node and at every time step a simple two parameter reinforcing law, based on the entire history of the network above a given interior node, randomly determines the direction of the nodes descending edge and then is updated. Obvious modifications of these statements apply to nodes at the top or bottom (if one exists) of the lattice. The current pattern of connections among nodes represents the present state of the process, and the patterns stability – measured by the tendency of the same state, or one similar to it, to occur on subsequent iterations of the process – represents the patterns strength as a memory. The degree of reinforcement is set by tuning two parameters, r and α. At any given moment the current pattern is a collection of dendritic networks that appears similar to drainage networks found in nature; indeed, lattice models have often been used to investigate the morphology of natural drainage networks (e.g. [17]). We focus on the surficial dynamics of rill networks [10], rather than their morphology. Put in terms of erosion, we are more interested in the process of erosion than we are in the result. The analogy between our model and erosion, specifically rill erosion, is straightforward: r can be interpreted as a rainfall rate (or equivalently, as the rate of sediment generation) and α−1 as the resistance of soil to erosion, while the reinforcement dynamics correspond to the overland flow of water and sedi- ment down a hill. Rills are small, ephemeral channels that transport sediment down hillslopes when it rains [18]. They form when rainfall and runoffdislodge particles from the soil surface and transport them along flow paths governed by variations in the surface roughness of soils and the soil’s ability to resist erosion. Flow depths in rills are typically on the order of a few centimeters or less, while the longest channels in rill networks can be several meters long. Pro- cesses affecting rill erosion take place over timescales ranging from milliseconds to hours. The topology of rill networks is relatively unstable when compared to larger scale natural drainage systems (of which rills may be a part) like gulley systems and river basins. Rill networks are most unstable at their tops where bound- aries between rills and inter-rill areas

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