Tuesday, May 8, 2012
253 Rhodes Hall
Ever since the work of Aristotle, the shape and fate of pebbles have fascinated many people and have been in the focus of scientific interest. Starting with Lord Rayleigh, mathematicians developed partial differential equations describing the evolution of individual pebble shapes under abrasion. The combined effect of collisional abrasion is summarized in Bloore’s work. Geologists made valuable observations on global transport, i.e., how pebbles are moved on beaches by waves and currents. The two main geological observations about shingle beaches describe the emergence of dominant pebble size ratios as well as strong segregation of pebbles by maximal size. We propose a mathematical model which suggests that these two main geological observations about are interrelated. Our model is a based on a system of ODEs called the box equations, describing the evolution of pebble size ratios. We derive these ODEs as a heuristic approximation of Bloore's PDE describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. While representing a radical simplification of the latter, our system admits the study of a low-dimensional dynamical system and the inclusion of additional terms related to frictional abrasion. We show that nontrivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process. Our findings show a close connection with Aristotle's original model.