A model for ripple instabilities in granular media

TL;DR

This paper extends a granular flow model to include wind effects, showing that wind-induced ripple formations on granular beds are linearly unstable under certain conditions, with ripple size related to saltation length.

Contribution

It introduces a phenomenological model incorporating wind effects into granular flow and analyzes the linear stability leading to ripple formation.

Findings

Flat beds are linearly unstable against ripple formation.

Critical wave vector is approximately the saltation length.

Instability occurs in specific parameter regions.

Abstract

We extend the model of surface granular flow proposed in \cite{bcre} to account for the effect of an external `wind', which acts as to dislodge particles from the static bed, such that a stationary state of flowing grains is reached. We discuss in detail how this mechanism can be described in a phenomenological way, and show that a flat bed is linearly unstable against ripple formation in a certain region of parameter space. We focus in particular on the (realistic) case where the migration velocity of the instability is much smaller than the grains' velocity. In this limit, the full dispersion relation can be established. We find that the critical wave vector is of the order of the saltation length. We provide an intuitive interpretation of the instability.