Circular, stationary profiles emerging in unidirectional abrasion

TL;DR

This paper introduces a PDE model for bedrock abrasion by unidirectional impacts, revealing the emergence of stable, circular arc profiles that can evolve from various initial shapes, with implications for geological formations.

Contribution

It demonstrates the existence and stability of circular stationary profiles in a PDE model of bedrock abrasion, extending previous geometrical PDE models.

Findings

Circular profiles are stable stationary solutions.

Profiles can evolve from diverse initial shapes.

Simulations suggest geological relevance of transient profiles.

Abstract

We describe a PDE model of bedrock abrasion by impact of moving particles and show that by assuming unidirectional impacts the modification of a geometrical PDE due to Bloore exhibits circular arcs as solitary profiles. We demonstrate the existence and stability of these stationary, travelling shapes by numerical experiments based on finite difference approximations. Our simulations show that, depending on initial profile shape and other parameters, these circular profiles may evolve via long transients which, in a geological setting, may appear as non-circular stationary profiles.