A 1-d Ising model for ripple formation

TL;DR

This paper introduces a one-dimensional Ising model that qualitatively reproduces ripple formation dynamics, incorporating saltation effects and pair interactions, and provides a phase diagram and mean-field analysis.

Contribution

It presents a novel 1D Ising model capturing ripple dynamics, including metastability and a dynamical phase diagram, with a mean-field solution and a mapping to a 2D problem.

Findings

Ripple state is metastable, aligning with experiments and simulations.

A dynamical phase diagram is established.

Mean-field analysis relates ripple size to system parameters.

Abstract

A 1-d Ising model is shown to reproduce qualitatively the dynamics of ripple formation. Saltation effect is imposed using a Kawasaki dynamics and a pair interaction over some distance l. Within this model, the ripple state turns out to be metastable in agreement with cellular automata simulations as well as recent under water experiments. A dynamical phase diagram is obtained. A mean-field solution of the problem is given in terms of the ripple size. A mapping is then performed onto a 2-dimensional l x infinity static problem.