Sorting and bad waves in unidirectional shallow-water flows

The stability of a uniform flow above an erodible bed composed of a bimodal mixture of sediments is investigated by means of linear analysis. Results show that, for any given set of the flow and sediment parameters, two distinct modes of instability arise, each one characterized by its own wave speed, growth rate and longitudinal wavelength, each one involving spatial variations of both grain size density and bed elevation. Although at a linear level no information on the amplitude of the perturbations is gathered, the analysis of the eigenvectors associated with the two modes of instability allows for an easy classification in terms of the relative amplitudes of the perturbations of bed elevation and size density. One eigenvalue is shown to be associated with the modifications of bed forms induced by the presence of the heterogeneous mixture, such as the local accumulation of finer and coarser material along the unit wavelength, the other with the formation of the low-amplitude sorting waves known as bedload sheets. In the present unidirectional shallow-water framework, only the sorting wave is found to be unstable, since dunes and antidunes, the relevant bed forms for this case, require a more refined rotational flow model in order to become unstable. On the other hand, the simple flow model adopted allows for the formulation of an algebraic eigenvalue problem that can be solved analytically, allowing for a deep insight into the mechanisms that drive both instabilities.