Revisiting the linear theory of sand dune formation

A linear stability theory for dune and antidune formation is presented that implements a rotational two-dimensional flow model. As in previous linear theories, the phase-lag between sediment transport and bed elevation remains the main mechanism driving instability. However, it is shown that this phase-lag varies significantly in a neighbourhood of the bed. Moreover, since the layer in which sediments are moving has a finite (though small) thickness, it is assumed that the perturbations of the fluid stress driving bedload transport should be evaluated at the top of the layer itself. It is shown that such an apparently minor modification of the classical approach alters remarkably the balance between stabilizing and destabilizing effects that drives the instability process. This also allows a clarification of the debated role of bedslope on the formation of dunes and antidunes. Following the above ideas, antidunes are shown to form in the absence of suspended sediment load and without any effect associated with sediment inertia. The present analysis ultimately allows a successful unification of the theories of dune and antidune formation.