Linear stability of shallow morphodynamic flows

TL;DR

This paper analyzes the linear stability of shallow morphodynamic flow models, identifying conditions leading to ill-posed equations and proposing physical processes that restore well-posedness, with implications for engineering and geoscience modeling.

Contribution

It provides a comprehensive stability analysis of morphodynamic models, highlighting issues of ill-posedness and demonstrating how physical processes can ensure well-posed equations.

Findings

Naive morphodynamic models are ill-posed at Froude number 1.

Turbulent diffusion and bed load transport restore well-posedness.

Dilute flows are stable at low Froude numbers, while concentrated flows are always unstable.

Abstract

It is increasingly common for models of shallow-layer overland flows to include equations for the evolution of the underlying bed (morphodynamics) and the motion of an associated sedimentary phase. We investigate the linear stability properties of these systems in considerable generality. Naive formulations of the morphodynamics, featuring exchange of sediment between a well-mixed suspended load and the bed, lead to mathematically ill-posed governing equations. This is traced to a singularity in the linearised system at Froude number $Fr = 1$ that causes unbounded unstable growth of short-wavelength disturbances. The inclusion of neglected physical processes can restore well posedness. Turbulent momentum diffusion (eddy viscosity) and a suitably parametrised bed load sediment transport are shown separately to be sufficient in this regard. However, we demonstrate that such models typically inherit an associated instability that is absent from non-morphodynamic settings. Implications of our analyses are considered for simple generic closures, including a drag law that switches between fluid and granular behaviour, depending on the sediment concentration. Steady morphodynamic flows bifurcate into two states: dilute flows, which are stable at low $Fr$, and concentrated flows which are always unstable to disturbances in concentration. By computing the growth rates of linear modes across a wide region of parameter space, we examine in detail the effects of specific model parameters including the choices of sediment erodibility, eddy viscosity and bed load flux. These analyses may be used to inform the ongoing development of operational models in engineering and geosciences.