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Granular media are frequently found in nature and in industry and theirtransport by a fluid flow is of great importance to human activities. One caseof particular interest is the transport of sand in open-channel and riverflows. In many instances, the shear stresses exerted by the fluid flow arebounded to certain limits and some grains are entrained as bed-load: a mobilelayer which stays in contact with the fixed part of the granular bed. Underthese conditions, an initially flat granular bed may be unstable, generatingripples and dunes such as those observed on the bed of rivers. In free-surfacewater flows, dunes are bedforms that scale with the flow depth, while ripplesdo not scale with it. This article presents a model for the formation ofripples and dunes based on the proposition that ripples are primary linearinstabilities and that dunes are secondary instabilities formed from thecompetition between the coalescence of ripples and free surface effects.Although simple, the model is able to explain the growth of ripples, theirsaturation (not explained in previous models) and the evolution from ripples todunes, presenting a complete picture for the formation of dunes.

Linear and nonlinear instabilities of a granular bed: determination of the scales of ripples and dunes in rivers