In this paper we analyze the Hirano active layer model used in mixed sediment river morphodynamics concerning its ill-posedness. Ill-posedness causes the solution to be unstable to short-wave perturbations. This implies that the solution presents spurious oscillations, the amplitude of which depends on the domain discretization. Ill-posedness not only produces physically unrealistic results but may also cause failure of numerical simulations. By considering a two-fraction sediment mixture we obtain analytical expressions for the mathematical characterization of the model. Using these we show that the ill-posed domain is larger than what was found in previous analyses, not only comprising cases of bed degradation into a substrate finer than the active layer but also in aggradational cases. Furthermore, by analyzing a three-fraction model we observe ill-posedness under conditions of bed degradation into a coarse substrate. We observe that oscillations in the numerical solution of ill-posed simulations grow until the model becomes well-posed, as the spurious mixing of the active layer sediment and substrate sediment acts as a regularization mechanism. Finally we conduct an eigenstructure analysis of a simplified vertically continuous model for mixed sediment for which we show that ill-posedness occurs in a wider range of conditions than the active layer model.

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The active layer model (Hirano, 1971) is the most commonly used model to account for mixed-size sediment processes in modeling morphodynamics of rivers, coasts, and estuaries. In this model, only the sediment in the topmost part of the bed (the active layer, characterized by a certain thickness, and assumed to be fully mixed) interacts with the flow. The sediment in the active layer can be entrained and the transported sediment can be deposited in the active layer. The grain size distribution of the sediment below the active layer, the substrate, typically varies with elevation. There is a net flux of sediment between the active layer and the substrate if the bed aggrades or degrades. Due to the highly schematized treatment of the bed processes, the active layer model may present elliptic (rather than hyperbolic) behavior (Ribberink, 1987). A system of equations that models changes in time cannot be of an elliptic type. This is because in that case future conditions influence the present, which is physically unrealistic. Such a model is mathematically ill-posed. The solution of an ill-posed problem is unstable to short wave perturbations. Another example of an ill-posed problem is the twofluid model. Zanotti et al. (2007) developed a regularization strategy to restore the hyperbolic character when it becomes ill-posed. Our objective is to apply a similar concept to guarantee the hyperbolic character of the active layer model. @en

We present an accurate numerical approximation to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in one space dimension. Our solution procedure originates from the fully-unsteady matrix-vector formulation developed in [54]. The principal part of the problem is solved by an explicit Finite Volume upwind method of the path-conservative type, by which all the variables are updated simultaneously in a coupled fashion. The solution to the principal part is embedded into a splitting procedure for the treatment of frictional source terms. The numerical scheme is extended to second-order accuracy and includes a bookkeeping procedure for handling the evolution of size stratification in the substrate. We develop a concept of balancedness for the vertical mass flux between the substrate and active layer under bed degradation, which prevents the occurrence of non-physical oscillations in the grainsize distribution of the substrate. We suitably modify the numerical scheme to respect this principle. We finally verify the accuracy in our solution to the equations, and its ability to reproduce one-dimensional morphodynamics due to streamwise and vertical sorting, using three test cases. In detail, (i) we empirically assess the balancedness of vertical mass fluxes under degradation; (ii) we study the convergence to the analytical linearised solution for the propagation of infinitesimal-amplitude waves [54], which is here employed for the first time to assess a mixed-sediment model; (iii) we reproduce Ribberink's E8-E9 flume experiment [46].

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We present a new image analysis technique for measuring the grain size distribution (texture) of the bed surface during flow in a laboratory experiment. A camera and a floating device are connected to a carriage used to take images of the bed surface over the entire flume length. The image analysis technique, which is based on color segmentation, provides detailed data on spatial and temporal changes of the areal fraction content of each grain size at the bed surface. The technique was applied in a laboratory experiment conducted to examine a degradational reach composed of a well sorted two-fraction mixture of sand and gravel. The initial bed consisted of an upstream reach that was characterized by an imposed stepwise fining pattern (the bimodal reach) and a downstream sand reach. A lack of sediment supply and partial transport conditions led to the formation of a static armor in the bimodal reach, which resulted in a more abrupt spatial transition in the bed surface mean grain size. The associated spatial transition in slope led to a backwater effect over the bimodal reach, a streamwise reduction in sand mobility, and so a static armor that was governed by a downstream fining pattern. Although a morphodynamic equilibrium state under steady flow is generally characterized by normal flow, here the partial transport regime prevented the bed from adjusting toward normal flow conditions and the morphodynamic steady state was governed by a backwater. We applied a numerical morphodynamic sand-gravel model to reproduce the laboratory experiment. The numerical model captured the hydrodynamic and morphodynamic adjustment and the static armor well, yet the armoring occurred too slowly. Although the final configuration of the experiment shows features of a gravel-sand transition (i.e., a sudden transition in slope and mean grain size), we are hesitant to claim similarities between our results and the physical mechanisms governing a gravel-sand transition in the field.
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