with periodic boundary. Both models lead to new interesting conclusions.

The model of two dimensional dunes with constant sand influx {#sec:trans_const}
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The free parameters for this simulation are the sand influx $q_{in}$ and the shear velocity $u_*$. As initial surface we choose a plain ground filled with sand because the sand influx differs at least a little bit from the saturation flux on the dune field. Thus dune formation is initiated at the beginning of the dune field, i.e. where wind comes in.

Time evolution {#subsec:time_evol_2dc}
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The height profile of a dune field with a length of 4 km is presented at different times. The sand influx is set $q_{in}=0.017$ kg m$^{1}$s$^{-1}$ and the shear velocity $u_*=0.5$ m s$^{-1}$. A sand influx $q_{in}$ which is not equal to the sand flux of saltation transport over a plain surface lowers or raises the inlet of the dune field constantly. This initiates a small oscillatory structure which begins to move in wind direction (Figure \[fig:transv\_evo1\]) and generates more and more small dunes.

![Surface of a two dimensional simulation with constant sand influx after $4.56$ years in the left figure and $45.58$ years on the right. The shear velocity is $u_*=0.5$ 