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Linear stability of shallow morphodynamic flows

Published online by Cambridge University Press:  12 April 2021

Jake Langham Open the ORCID record for Jake Langham [Opens in a new window] ,
Mark J. Woodhouse Open the ORCID record for Mark J. Woodhouse [Opens in a new window] ,
Andrew J. Hogg Open the ORCID record for Andrew J. Hogg [Opens in a new window]  and
Jeremy C. Phillips Open the ORCID record for Jeremy C. Phillips [Opens in a new window]
Jake Langham*
Affiliation:
School of Mathematics, Fry Building, University of Bristol, BristolBS8 1UG, UK School of Earth Sciences, Wills Memorial Building, University of Bristol, BristolBS8 1RJ, UK
Mark J. Woodhouse
Affiliation:
School of Earth Sciences, Wills Memorial Building, University of Bristol, BristolBS8 1RJ, UK
Andrew J. Hogg
Affiliation:
School of Mathematics, Fry Building, University of Bristol, BristolBS8 1UG, UK
Jeremy C. Phillips
Affiliation:
School of Earth Sciences, Wills Memorial Building, University of Bristol, BristolBS8 1RJ, UK
*
†Email address for correspondence: [email protected]
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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 ${\textit {Fr}} = 1$ 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 ${\textit {Fr}}$, 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.

JFM classification

Geophysical and Geological Flows: Shallow water flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A31
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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