Nonnormality and transient behavior of the de Saint-Venant-Exner equations

Shallow-water flows on fixed and movable beds play an important role in both hydrology and river morphodynamics, and several efforts have been devoted to investigating their stability. The analysis is usually conducted focusing the asymptotic fate of small disturbances, using normal modes. In this work we instead have tackled another aspect of the stability problem studying the disturbance behavior at finite times. Recent studies have in fact shown that nonnormality of differential operators can induce a nonmonotonic decay of disturbances in asymptotically stable cases and the occurrence of transient growths. We have investigated these aspects for the de Saint-Venant-Exner equations and discussed the transient behavior of the energy of disturbances for both fixed and movable bed conditions. We show that the former case provides weak levels of nonnormality; while the latter can give rise to important transient growths. These growths result from uniquely linear mechanisms, are potentially able to trigger nonlinear instabilities, and are physically due to a transfer of potential energy between the water stream and the bed. The results encourage revisiting the study of morphodynamics from a nonmodal perspective.