Amplitude equation and long-range interactions in underwater sand ripples in one dimension.

We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form ht = - epsilon(h-h)+((hx)2-1)hxx-hxxxx+delta((hx)2)xx which, due to the first term, is neither completely local (it has long-range coupling through the average height h) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large delta (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.