Bifurcation patterns in dune and antidune instability

The conditions by which a uniform flow in an infinitely wide erodible channel loses stability towards a perturbed configuration are investigated. A fully coupled differential system based on a rotational flow model plus the Exner equation is adopted, so that no simplifying assumptions are made on the characteristic flow and bed timescales. At a linear level, the analysis shows the existence of three unstable eigenvalues in the parameter space, which can be associated to dune (subcritical flow), antidune and roll-wave (supercritical flow) instabilities, respectively. In particular, for values of the Froude number larger than two, antidunes and roll-waves are both unstable. A weakly nonlinear analysis is performed to investigate the bifurcation process associated to dune, antidune and roll-wave marginal instability. Two regions of interests in the parameter space are found, one related to the dune-plane bed-antidune transition, the second to the antidune-roll-wave competition. The analysis shows that both dune-plane and plane-antidune bifurcations can be either supercritical (forward) or subcritical (backward) depending on the values of the parameters. The presence of subcritical bifurcation can theoretically justify the hysteresis often observed in dune-plane-antidune transition, where different bed patterns are detected for the same values of the flow and the sediment parameters. As for the antidune-roll-wave competition, critical points exist where the marginal curves of both modes intersect. A weakly nonlinear expansion in the neighbourhood of these points leads to a system of coupled amplitude equations that describes their growth and mutual interactions. A variety of bifurcation patterns ultimately arises from the solution of the above system depending on the equation coefficients, which include secondary bifurcations of one or both solutions.