It is demonstrated, by using a simple model, that bedforms in a short tidal embayment
can develop due to a positive feedback between tidal currents, sediment transport
and bedforms. The water motion is modelled by the depth integrated shallow water
equations. The system is forced by a prescribed free-surface elevation at the entrance
of the embayment. For the sediment dynamics a diffusively dominated suspended
load transport model is considered. Tidal averaging is used to obtain the bottom
profiles at the long morphological time scale.
The stability of a constantly sloping equilibrium bottom profile is studied for
various combinations of the model parameters. It turns out that without a mechanism
that generates vorticity this equilibrium profile is stable. In that case small-scale
perturbations can at most become marginally stable if no bedload term in the bottom
evolution equation is incorporated. If vorticity is generated, in our model by bottom
friction torques, the basic state is unstable. The spatial patterns of the unstable modes
and their growth rates depend, among other things, on the strength of the bottom
friction, the width of the embayment and the grain size: if the sediment under consideration
consists of large particles, the equilibrium will be more stable than when
smaller particles are considered. Without a diffusive term in the bed evolution equation,
small-scale perturbations become unstable. To avoid this physically unrealistic
behaviour bedload terms are included in the sediment transport. Furthermore, it is
shown that using an asymptotic expansion for the concentration as given in earlier
literature is only valid for small or moderate mode numbers and the technique is
extended to large mode numbers. A physical interpretation of the results is also given.