boundaries in both directions shows no state where the system has a structure or oscillations in lateral direction.

![Surface of a transverse dune field with periodic boundary conditions in both directions $9.5$ years after initiation. The system reached a state close to translational invariance. The shear velocity is $u_*=0.4$ m s$^{-1}$. One unit corresponds to the length of $2$ m.[]{data-label="fig:transv_2d6"}](Figure6BW.eps){width="80.00000%"}

The conclusion from these calculations for three dimensional dunes would be that an open system of transverse sand dunes reaches translational invariance under ideal unidirectional wind. In the precedent simulations the periodic boundary inhibited the system to break the invariance. A calculation with open boundary conditions in both horizontal directions which is closer to real dune fields would give more information about this assumption but is rather complicated due to the fact that the dunes move out of the simulated area. More insight is given in the following section. Assuming translational invariance simulations of fields of two dimensional dunes are much more effective. They consume much less computional time and give the opportunity to simulate larger dune fields. In the following two sections we consider two situations of two dimensional dunes, a model with constant sand influx and a model 