The evolution of pebble size and shape in space and time

TL;DR

This paper introduces a simplified mathematical model using ODEs to explain how pebble size ratios and segregation patterns in shingle beaches evolve over time, emphasizing the role of friction.

Contribution

It develops a novel ODE-based model linking pebble size evolution and segregation, extending Bloore's PDE with frictional effects and analyzing stability of size ratios.

Findings

Predominant pebble size ratios emerge only with friction.

Attractors representing size ratios are stabilized by segregation.

Model simulations show ongoing segregation maintains size ratio stability.

Abstract

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size are interrelated. Our model is a based on a system of ODEs called the box equations, describing the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's PDE describing collisional abrasion. While representing a radical simplification of the latter, our system admits 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 stabilized by the ongoing segregation process.