Three-dimensional river bed forms

The linear stability of a uniform flow in an infinitely wide erodible channel is investigated with respect to disturbances of the bed that are periodic in both the transverse and the longitudinal directions. A rotational flow and sediment transport model, originally developed to study the formation of two-dimensional dunes and antidunes, is straightforwardly extended to cover variations in the lateral direction. Sediment is assumed to be transported as bed load, disregarding the role of suspension. Following a standard linearization procedure, a dispersion relationship is obtained that expresses the growth rate and the celerity of the sand wave as a function of the streamwise and spanwise wavenumbers and of the relevant flow and sediment parameters. Regions of instabilities in the space of the parameters are found, which can be associated with bed forms of different kinds, spanning from dunes and antidunes to alternate bars. Therefore, the present theory allows for a unified view of the formation of two- and three-dimensional bed forms in rivers in terms of the relevant flow and sediment parameters.