is transported to the following dune by saltation. When the dune has vanished once again the system approaches translational invariance. Hence, the effect of converging dunes perturbs the steady growth of transverse dunes in a field with a periodic boundary.

![Surface of a transverse dune field with periodic boundary conditions in both directions $1.19$ years after initiation. The shear velocity is $u_*=0.4 $ m s$^{-1}$. Height units are in meters, length and width units in two meters.[]{data-label="fig:transv_2d4"}](Figure5.eps){width="80.00000%"}

To get more information how a system of transverse dunes shows translational invariance a simulation of a dune fiel with periodic boundary in both horizontal directions is made. The field extends over $400$ m in length and width. As initial surface also small Gaussian hills are used. Figure \[fig:transv\_2d4\] shows the height profile after 5,000 iterations which corresponds to a time of $1.19$ years. The results of this simulation yield the same conclusions as the simulation with open lateral boundary. Also each merging of two dunes leads to a breaking of the symmetry. Figure \[fig:transv\_2d6\] shows the system at a state close to translational invariance. The final state of the calculation is reached when only one dune is left. A simulation with periodic 