saltation flux $q$:

The time to reach the steady state of sand flux over a new surface is several orders of magnitude smaller than the time scale of the surface evolution. Hence, the steady state is assumed to be reached instantaneously. The length scale of the model is too large to include sand ripples. Nevertheless the kinetics and the characteristic length scale of saltation influence the calculation by breaking the scale invariance of dunes and by determining the minimal size of a barchan dune [@SauermannKroy2001]. A calculation of the saltation transport by the well known flux relations [@Bagnold41; @Lettau78; @Sorensen91] would restrict the model to saturated sand flux which is not the case for example at the foot of the windward side of a barchan dune due to little sand supply or at the end of the separation bubble in the interdune region between transverse dunes due to the vanishing shear stress in the separation bubble. The sand density $\rho(x,y)$ and the grain velocity ${\boldsymbol{\rm u}}(x,y)$ are integrated in vertical direction and calculated from mass and momentum conservation, respectively. We simplified the closed model of [@SauermannKroy2001] by neglecting the time dependent terms and the convective term of the grain velocity 