Mathematics in the Geosciences, September 2001 - June 2002

The solid earth, oceans and atmospheres are profoundly nonlinear. While the oceans and atmosphere are well-described by first principles equations, many nonlinear processes in the solid earth lack such a description. Many geophysical problems possess an underlying discrete character, in contrast with a continuous one, or alternatively do not offer a well-posed PDE description but appear to be easy to characterize in a discrete fashion. Fracture of Earth materials provides a good example. Grains in a rock, approximately 1 mm in size, constitute the basic unit in this otherwise heterogeneous medium. These scenarios lend themselves in a natural way to a cellular automaton or a lattice gas formulation depending on whether the time dependence is intrinsically discrete or continuous, respectively. In an important subcategory of cellular automaton problems, the accessible states in the problems are discrete, and especially subject to delayed influences. Equations governing this class of problems are often called Boolean Delay Equations. Illustrative examples include percolation problems, with the attendant possibility of critical point behavior; earthquake and avalanche problems, including the possibility of self organized criticality and scaling; and the modeling of complex transport processes, which blend fluids with granular materials, and provide important insights in to complicated problems in the establishing of landforms and fluvial drainage patterns.

Keywords: earthquakes and avalanches, percolation, cellular automata, Boolean difference equations, sandpiles

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Abstracts      Material from Talks

Mathematics in the Geosciences, September 2001 - June 2002