up with sand. The initial surface consists of small Gaussian hills which disturb a plain surface.

Time evolution {#time-evolution}
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The situation is similar to the simulations of three dimensional transverse dune fields. In the beginning many dunes of different size grow in the system but after some time the dunes approach same heights. As in the three dimensional case all the dunes keep growing so that the number of dunes has to decrease. A more detailed description of the process of merging dunes is given in the following section. The periodic boundary forces a decrease of the number of dunes and this process makes the evolution rather complex. Figures \[fig:transv\_1dp1\], \[fig:transv\_1dp2\] and \[fig:transv\_1dp3\] show the height profile of a dune field of a length of two kilometers at three time steps. Figure \[fig:transv\_1dp1\] depicts a surface with dunes of different sizes which seem to interact strongly with each other. The regular state of the system in Figure \[fig:transv\_1dp2\] is disturbed in Figure \[fig:transv\_1dp3\]. The number of dunes decreases quite regularly versus time (Figure \[fig:transv\_nu\_d\]). This process is very slow and the dunes grow much slower than in the model with open boundary.

![Surface of a dune field with a 