Mathematical analysis of the Saint-Venant-Hirano model for mixed-sediment morphodynamics

Sediment of different size are transported in rivers under the action of flow. The first and still most popular sediment continuity model able to deal with mixed sediment is the so-called active layer model proposed by Hirano (1971, 1972). In this paper, we consider the one-dimensional hydromorphodynamic model given by the Saint-Venant equations for free-surface flow coupled with the active layer model. We perform a mathematical analysis of this model, extending the previous analysis by Ribberink (1987), including full unsteadiness and grainsize selectivity of the transported load by explicitly considering multiple sediment fractions. The presence of multiple fractions gives rise to distinct waves traveling in the downstream direction, for which we provide an analytical approximation of propagation velocity under any Froude regime. We finally investigate the role of different waves in advecting morphodynamic changes through the domain. To this aim, we implement an analytical linearized solver to analyze the propagation of small-amplitude perturbations of the bed elevation and grainsize distribution of the active layer as described by the system of governing equations. We find that initial gradients in the grainsize distribution of the active layer are able to trigger significant bed variations, which propagate in the downstream direction at faster pace than the bed wave arising from the unisize-sediment Saint-Venant-Exner model. We also verify that multiple sorting waves carry multiple associated bed perturbations, traveling at different speeds.