Selection of dune shapes and velocities - Part 2: A two-dimensional modelling

We present in this paper a simplification of the dune model proposed by Sauermann et al. which keeps the basic mechanisms but allows analytical and parametric studies. Two kinds of purely propagative two dimensional solutions are exhibited: dunes and domes. The latter, by contrast to the former, do not present a slip face. Their shape and velocity can be predicted as a function of their size. We recover that dune profiles are not scale invariant (small dunes are flatter than the large ones), and that the inverse of the velocity grows almost linearly with the dune size. We furthermore get the existence of a critical mass below which no dune solution exists. It rises the problem of dune nucleation: how can dunes appear if any bump below this minimal mass gets eroded and disappears? The linear stability analysis of a flat sand bed shows that it is unstable at large wavelengths: dune can in fact nucleate from a small sand mass if the proto-dune is sufficiently long.