A strategy to avoid ill-posedness in mixed sediment morphodynamics

Abstract

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.