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.
Bibliographical noteFunding Information:
Acknowledgements. We thank C.G. Johnson, University of Manchester, for useful discussions concerning shallow-layer models and analysis, and L.T. Jenkins for comments on the manuscript. The main results of this paper were obtained as part of the Newton Fund grant ‘Quantitative Lahar Impact and Loss Assessment under Changing Land Use and Climate Scenarios’: NE/S00274X/1. Initial investigations were conducted during the ‘Strengthening Resilience in Volcanic Areas’ (STREVA) project, funded by the Natural Environment Research Council (NERC) and the Economic and Social Research Council (ESRC): NE/J020052/1. M.J.W. acknowledges funding from the NERC award ‘VolcTools – enhancing ease of use and uptake of tools to improve prediction and preparedness of volcanic hazards’: NE/R003890/1; A.J.H. acknowledges an APEX fellowship from the Royal Society, UK: APX/R1/180148; and J.C.P. acknowledges support from a University of Bristol Research Fellowship.
© The Author(s), 2021. Published by Cambridge University Press.
- shallow water flows
- particle/fluid flow