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

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.