Model of Erosion Pattern Evolution: Dynamics of Elementary Parcels

Victor G. Mossotti
U.S. Geological Survey

Deposition and erosion patterns are observed across an extremely wide range of scale on surfaces of geological and ecological interest, as well as on surfaces of cultural importance. The purpose of this work is to craft a framework for the design of automata-based simulations of deposition and erosion processes. The surfaces studied in this work, mainly calcareous stone buildings and monuments, served as surrogate substrates for a wide variety of naturally occurring catchments found in geological settings, albeit on a lesser scale.

In natural settings, depositional and erosional processes generally evolve over periods that include many wet-dry cycles. Under the varying conditions of wet-dry cycling, the rate-limiting path in a system of connected processes can switch with time among a set of alternative processes. The modeling challenge is compounded by the fact that the physicochemical landscape continuously changes throughout wet-dry cycling in response to the integrated consequences of ongoing depositional and erosional processes. We have adopted a modeling approach based on the observation that all types of depositional systems can exhibit spatially fractal zones within their full-scale patterns. Such observations support the assumption that control of deposition and erosion dynamics is exercised at the smallest scale of the measurement. In this model, we assume that local interactions at the scale defined by the measurement resolution drive the evolution of the observed depositional and erosional dynamics at all scales. It follows that there are two closely coupled parts to modeling the dynamics of such a system. The first part, which is discussed in this paper, is to model depositional and erosional dynamics accommodated by separate resolution elements; the second part, and probably the most challenging, is to model interactions between the parcels of material forming the pattern mosaic.

The mosaic model provides perspective on links between primary delivery processes, variables endogenous to the substrate, and deposition and release processes, as well as on conditions controlling short-range and long-range correlations in patterns. As a consequence of feedback loops and competitive processes, the model predicts that 1) relationships between measurable properties of the system are generally nonlinear, and 2) soiling pattern evolution may follow a trajectory that is hypersensitive to the system variables. In general, the mosaic model shows that short-term experiments or observations on isolated system components generally should be interpreted with caution when used for the prediction of long term phenomena. This is because significant feedback loops may be decoupled from the system when observations are made over a limited range of scale in space and time. In addition, nonlinear controls on pattern evolution need to be taken into account in modeling future states of the system, whether the decisive variable is the sedimentation load in a benthic system or the time-to-remediation for a cultural monument. Lastly, the sensitive dependence of pattern evolution on the initial state of the system may result in outcomes that are not reproducible under virtually identical conditions.

Created by Mathematica  (April 20, 2004)