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, which, by contrast to the former, do not show avalanche slip face. Their shape and velocity can be predicted as a function of their size. We recover in particular 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 stable dune exists. However, the linear stability analysis of a flat sand sheet shows that it is unstable at large wavelengths and suggests a mechanism of dune initiation. 
Bruno Andreotti and Philippe Claudin