Finite-amplitude river dunes

The linear and weakly nonlinear stability of a uniform flow in an infinitely wide open channel with erodible bottom is studied. Under suitable conditions the flow is found to be unstable, leading to the formation of dunes and antidunes. At a linear level, the corresponding regions of existence are presented and compared with experimental data. A weakly nonlinear analysis is then performed in a neighbourhood of the critical conditions for dune and antidune formation. The analysis shows that, for values of the ratio of the shear velocity to the depth-averaged velocity of practical interest, dune bifurcation is supercritical, whereas antidune bifurcation is subcritical. The latter result suggests a possible interpretation of the plane-antidune transition, where plane bed and antidune configurations are observed to coexist for the same values of the flow and the sediment parameters. The supercritical behaviour of the dune bifurcation allows for the prediction of an equilibrium amplitude that successfully compares with the amplitudes measured in laboratory experiments.