Abstract
In the transition process of dune formation, a hysteresis is observed between flow depth and discharge, which is analogous to the multiple-valued relations between the total Shields stress and the effective Shields stress found in experiments. In this study, a weakly nonlinear stability analysis is performed in order to provide a theoretical explanation for the hysteresis observed in the transition of dunes. The growth rate expansion method is used to derive a nonlinear amplitude equation, which shows that when the average bed slope is large and the effect of the local bed inclination is small, the transition of dune formation is characterized by the subcritical bifurcation. This implies that the hysteresis is caused by the subcritical bifurcation.
