m s$^{-1}$ and $u_*=0.5$ m s$^{-1}$. The observed bulk fluxes are $\Phi_{dune}=454.5$ m$^2$s$^{-1}$ and $\Phi_{dune}=833.3$ m$^2$s$^{-1}$ with the corresponding constants $C=0.45$ m and $C=1.08$ m, respectively. The values are smaller than the bulk fluxes of isolated 2-dimensional dunes calculated by on plain ground without sand. Thus the velocities are also smaller than those observed for isolated transverse dunes on plain ground without sand. found less speed-up of the wind velocity over continuous sand dunes than over isolated transverse dunes. This agrees with the results in this model.

![Dune velocity $v_{dune}$ versus height $h$ for the shear stresses $u_*=0.4$ m s$^{-1}$ and $u_*=0.5$ m s$^{-1}$. The velocity decreases proportional to the reciprocal height.[]{data-label="fig:transv_scalv2"}](Figure12.eps){width="80.00000%"}

The model of two dimensional dunes with periodic boundary {#sec:trans_periodic}
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The simulations in this section have a periodic boundary condition in wind direction. So the parameter of sand influx that was used additionally before is no more available. Sand influx is set equal to the outflux. The avalanches flow down even if the slip face is divided by the boundary. Also the separation bubble follows the periodic boundary conditions. The calculations are made with dune fields of a length of two kilometers. The ground is completely filled 