  We derive general depth-integrated model equations for overland flows
featuring the evolution of suspended sediment that may be eroded from or
deposited onto the underlying topography ('morphodynamics'). The resulting
equations include geometric corrections that account for large variations in
slope angle. These are often non-negligible for Earth-surface flows and may
consequently be important for simulating natural hazards. We also show how to
adapt existing finite volume schemes for the classical shallow water equations,
to simulate our new equations in a way that preserves uniform steady states and
exactly conserves the combined mass of the flow and bed. Finally, to
demonstrate our formulation, we present computations using simple example model
closures, fed by point flux sources. On initially constant slopes, flows
exhibit different behaviours depending on the grade. Shallow slopes lead to
weakly morphodynamic spreading flows that agree well with analytical similarity
solutions. On more severe slopes, rapid erosion occurs, leading to
self-channelising flows and ultimately a 'super-erosive' state, in which
sediment entrainment and gravitational acceleration perpetually reinforce each
other.
