Initial formation of channels and shoals in a short tidal embayment

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.