Ooid growth: Uniqueness of time-invariant, smooth shapes in 2D

TL;DR

This paper investigates the evolution of planar curves modeling ooid growth, demonstrating that physical parameters uniquely determine a symmetric, steady shape that aligns with natural observations.

Contribution

It establishes the uniqueness and symmetry of steady shapes in a nonlocal geometric model of ooid growth, linking mathematical results to geoscience observations.

Findings

Unique, time-invariant, convex shapes determined by environmental parameters

All steady solutions exhibit D2 symmetry

Model predictions align with natural ooid shapes

Abstract

Evolution of planar curves under a nonlocal geometric equation is investigated. It models the simultaneous contraction and growth of carbonate particles called ooids in geosciences. Using classical ODE results and a bijective mapping we demonstrate that the steady parameters associated with the physical environment determine a unique, time-invariant, compact shape among smooth, convex curves embedded in $R^2$. It is also revealed that any time-invariant solution possesses $D_2$ symmetry. The model predictions remarkably agree with ooid shapes observed in nature.