Countinuous and discrete models for abrasion processes

Abstract

The origin of the shapes of stones and other particles formed by water or wind has always attracted the attention of geologists and mathematicians. A classical model of abrasion due to W. J. Firey leads to a geometric partial differential equation representing the continuum limit of the process. This model predicts convergence to spheres from an arbitrary initial form; analogously, the two-dimensional version of the model predicts convergence to circles. The shapes of real stones are, however, not always round. Most notably, coastal pebbles tend to be smooth but somewhat flat, and ventifacts (e.g. pyramidal dreikanters) often have completely different shapes with sharp edges. Inspired by Firey´s results, a new PDE is derived in this paper, which not only appears to be a natural mathematical generalization of Firey´s PDE, but also represents the continuum limit of a genezalized abrasion model based on recurrent loss of material due to collisions of nearby pebbles. We also introduce a related, mezo-scale discre te random model which is ideally suited for analyzing wear processes in specific geometric scenarios. Preliminary results suggest that our model is capable to predict a broad range of limit shapes: polygonal shapes with sharp edges develop due to sand blasting (big stone surrounded by infinitesimally small particles), round stones emerge due to collisions with relatively big stones, and flat shapes are the typical outcome in the intermediate case. The results show nice agreement with real data despite the model´s simplicity.