Amplitude equation for under water sand-ripples in one dimension.

Sand-ripples under oscillatory water flow form periodic patterns with wave lengths primarily controlled by the amplitude $d$ of the water motion. We present an amplitude equation for sand-ripples in one spatial dimension which captures the formation of the ripples as well as secondary bifurcations observed when the amplitude $d$ is suddenly varied. The equation has the form \[ h_t=- \epsilon(h-\bar{h})+\big((h_x)^2-1\big)h_{xx}- h_{xxxx}+ \delta((h_x)^2)_{xx} \] which, due to the first term, is neither completely local (it has long-range coupling through the average height $\bar{h}$) nor has local sand conservation. We discuss why this is reasonable and how this term (with $\epsilon \sim d^{-2}$) stops the coarsening process at a finite wavelength proportional to $d$. We compare our numerical results with experimental observations in a narrow channel.