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=- ε(h-mean(h))+((h_x)^2-1)h_(xx)- h_(xxxx)+ δ((h_x)^2)_(xx) which, due to the first term, is neither completely local (it has long-range coupling through the average height mean(h)) nor has local sand conservation. We argue that this is reasonable and that this term (with ε = 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.