A nonlinear evolution equation for sand ripples based on geometry and conservation

TL;DR

This paper derives nonlinear evolution equations for sand ripples based on geometry and conservation principles, revealing different behaviors under erosion and deposition conditions, including ripple coarsening and pattern formation.

Contribution

It introduces a new one-parameter nonlinear equation for sand ripples that accounts for coarsening dynamics during deposition-erosion balance.

Findings

Ripples obey the Benney equation under strong wind erosion.

Ripple structures undergo coarsening over time.

Growth of ripple wavelength slows down significantly during coarsening.

Abstract

From geometry and conservation we derive two nonlinear evolution equations for sand ripples. In the case of a strong wind leading to a net erosion of the sand bed, ripples obey the Benney equation. This leads either to order or disorder depending on whether dispersion is strong or weak. In the most frequent case where erosion is counterbalanced by deposition, we derive a new one-parameter nonlinear equation. It reveals ripple structures which then undergo a coarsening process at long times, a process which then slows down dramatically with the growth of the ripple wavelength.