scale of the model which is normally between 3–5 hours for every iteration.

#### The initial surface and boundary conditions:

As initial surface we take a plain sand bed of arbitrary sand height over the solid ground. The surface can additionally be disturbed by small Gaussian hills. An initial surface has to be smooth (at least under consideration of the separation bubble) and have slopes not larger than the angle of repose. The boundary conditions influence the surface height $h$ with its separation bubble, the sand flux $q$ and the height $R$ of the moving layer of the avalanche model. The boundary in both directions $x$ and $y$ with respect to the direction of the incoming wind are open or periodic. At open boundary in $x$–direction an additional parameter controls the sand influx $q_{in}$ into the simulated dune field. It is set constant along the lateral direction at $x=0$.

All calculations presented in the following sections are made with the conditions of a completely filled sand bed and unidirectional wind. All simulations model dune fields instead of single dunes.

The model of three dimensional dunes and translational invariance {#sec:trans_2dim}
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The main aim in this section is to justify why 