Modal versus nonmodal linear stability analysis of river dunes

When a free surface turbulent shear flow interacts with a deformable cohesionless sediment bottom, dune patterns can arise and perform a complex time evolution. These bed forms are very widespread in fluvial environments and have catalyzed an intense research activity of both an applicative and theoretical nature. This work investigates the non-normality of the linearized mathematical operator which rules the initial value problem, in order to detect possible transient growths and make a comparison with the picture described by the outcome of the eigenvalue problem. The dune dynamics have resulted to be heavily non-normal in large regions of the parameter space and to be able to develop important transient growths, even for asymptotically stable wavenumbers. The effects on progressive wavelength elongation that has been observed in some experiments are discussed, and an explanation, based on purely linearly mechanisms, is proposed and compared with some experimental data. A discussion about the saturation length concept in the sediment transport modeling is also proposed.