Abstract
A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.
Highlights
Modelling of fluvial morphodynamic processes is a powerful tool to predict the future state of a river after, for instance, an intervention or a change in the discharge regime (Blom et al 2017), and as a source of understanding of theV
We have studied a two-dimensional system of equations used to model mixed-size river morphodynamics as regards to its well posedness
In particular we have focused on modelling of the secondary flow induced by flow curvature and the effect of the bed slope on the sediment transport direction, which causes particles to deviate from the direction of the bed shear stress
Summary
Modelling of fluvial morphodynamic processes is a powerful tool to predict the future state of a river after, for instance, an intervention or a change in the discharge regime (Blom et al 2017), and as a source of understanding of theV. Modelling of fluvial morphodynamic processes is a powerful tool to predict the future state of a river after, for instance, an intervention or a change in the discharge regime (Blom et al 2017), and as a source of understanding of the. A framework for modelling the morphodynamic development of alluvial rivers is composed of a system of partial differential equations for modelling the flow, change in bed elevation and change in the bed surface texture. The Saint-Venant (1871) equations account for conservation of water mass and momentum and enable modelling processes with a characteristic length scale significantly longer than the flow depth in one-dimensional cases. The shallow water equations describe the depth-averaged flow in two-dimensional cases. Conservation of unisize bed sediment is typically modelled using the Exner (1920) equation and, under mixed-size sediment conditions, the active layer model (Hirano 1971) accounts for mass conservation of bed sediment of each grain size
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