A strategy to avoid ill-posedness in mixed sediment morphodynamics

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.