  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.
