Linear stability of shallow morphodynamic flows

We analyse the linear stability of uniform steady morphodynamic flows using an extended shallow-water model that permits material to be exchanged between a suspended sedimentary mixture and its underlying bed. Any physical closures are left as arbitrary functions of the flow variables, so that our conclusions apply to a wide class of models used in engineering and geosciences. The inclusion of morphodynamics modifies the usual threshold for roll-wave instability by introducing a singularity into the linearised system at the critical Froude number $Fr = 1$. This leads to unbounded growth of short-wave disturbances and corresponding ill-posedness of the governing equations, which may be traced to a resonance between stability modes associated with the flow and the bed. By incorporating a suitable physical regularisation, we show that ill-posedness may be removed without affecting the location of the underlying instability. Alternatively, the inclusion of a bed load flux layer, common in fluvial models, can be sufficient to avoid ill-posedness under modest constraints. Implications of our analyses are considered by employing simple closures, including a drag law that switches between fluid and granular characteristics, depending on the sediment concentration. Steady layers are shown to bifurcate into two states: dilute flows which are stable at low $Fr$ and concentrated flows which are always unstable to disturbances in concentration. Finally, properties of the morphodynamic instability and the effects of regularisation are examined in detail by computing growth rates of the linear modes across a wide region of parameter space.