Nonnormality in morphodynamics

The early theoretical works on  morphodynamic  focused on determining which kind of bed form will develop under given flow condition and sediment characteristics. To this aim a linear stability approach was used to determine  the asymptotic fate of the morphological instability of the water-sediment interface.

It is important to keep in mind that the main feature of such an approach is to focus on the asymptotic temporal fate of the disturbances , and that  the main results concerns the dispersion relation, and its dependence on the parameters of the physical problem. However, no information is gained on the behavior of the system at finite times.

An example of the importance of considering the transient behaviour of a dynamical system is shown in panel (a) of figure 1. Despites systems A and B are asymptotically stable (the initial perturbation decays to zero after very long time ), curves A and B exhibit very different behavior for finite times: while the perturbation in system A decays monotonically to zero,  it undergoes  a transient growth in system B. The physical reason of this non-monotonic behavior lies in the nonnormality of the differential  operator which governs the perturbation temporal evolution. 
 WLevolutionFigure 2

Consider for instance the simple two-dimensional algebraic problem reported in panel  (b) of figure 1, the non-orthogonality of the eigenvector set entails that, although all the eigenvalues are negative and single eigenvectors decay monotonically in time, their resultant experiences a transient growth. 
Although the long-term asymptotic fate of the system is correctly driven by the eigenvalue analysis, the transient behavior requires more sophisticated analysis  when the eigenvectors set is not orthogonal. 
One of the most fascinating and unexplored aspects in morphodynamic concerns the transient dynamics of bed forms, i.e., from the very first  stages of bed instability to the emergence of their typical wavelengths. Experiments have rarely focused on this aspect, but the few available data depict non trivial behavior, showing, generally, the initial growth of short wave bed forms that are dominant during the initial stage of bed form  formation, and that then tend to decay and merge with longer waves. An example of such evolution, studied in the case of alternate bar, is reported in figure 2. Our current researches focus on the wavelength selection mechanisms affecting the early-stage morphodynamic pattern. In particular we focus on bar and dune patterns as well as long sand waves. We pursue this research both from an experimental and a theoretical point of view.  
The experimental branch of our research is mainly concentrated on alternate bar dynamics. It is devoted to increase the knowledge of the inception phase of the alternate bar patterns, and it is conducted through the interpretation of the photographical and topographical data collected in flume experiments. Care is taken in describing the temporal evolution of the average pattern wavelengths and describing the merging and amalgamation processes. A beautiful amalgamation sequence obtained in a flume experiment performed in a 0.5 m wide channel is reported in figure 3. The arrows with a white head indicate a slowly moving front, while the arrows with a black head indicate a fast moving front. The flow is from left to right. A vertical black line painted on the right side wall marks a reference point useful to observe how the bed evolves while time is running.

The theoretical branch our research, instead,  is devoted to analyze the bed form dynamics from a nonmodal point of view. In particular we demonstrate the existence of transient energy exchange between fluid kinetic energy, water potential energy and bed potential energy, which, ultimately, are a key factor for selecting different wavelengths at different times. In figure 4, for instance, we report the most amplified wavelengths of the alternate bars pattern at different times, while in figure 5 we report the temporal evolution of the wavelength of a dune pattern.

barsFigure 4  ddunennnormFigure 5

Selected papers

Nonnormality and transient behavior of the de Saint-Venant-Exner equations. C. Camporeale and L. Ridolfi, Water Resources Res. 45, W08418 (2009), DOI:10.1029/2008WR007587

Modal versus nonmodal linear stability analysis of river dunes C. Camporeale and L. Ridolfi, Phys. Fluids 23, 104102 (2011), DOI:10.1063/1.3644673