used. In the following the different steps at every iteration are explained.

#### The air shear stress $\tau$ at the ground:

The shear stress perturbation over a single dune or over a dune field is calculated using the algorithm of . The $\tau_x$-component points in wind direction and the $\tau_y$-component denotes the lateral direction. The calculation is made in Fourier space, where $k_x$ and $k_y$ denote the wave numbers, $$\begin{aligned}
  \label{eq:tau_x}
  \hat\tau_x(k_x,k_y) = 
    \frac{h(k_x,k_y) k_x^2}{|k|} 
  \frac{2}{U^2(l)} \cdot  \nonumber \\
  \left( 1 +   
   \frac{2 \ln L|k_x| + 4 \gamma + 1 +
      i \, \text{sign}(k_x) \pi}{\ln l/z_0} \right),\end{aligned}$$ and $$\label{eq:tau_y}
  \hat\tau_y(k_x,k_y) = \frac{h(k_x,k_y) k_x k_y}{|k|}
  \frac{2}{U^2(l)},$$ where $|k| = \sqrt{k_x^2 + k_y^2}$ and $\gamma=0.577216$ (Euler’s constant). $U(l)$ is the normalized velocity of the undisturbed logarithmic profile at the height of the inner region $l$ [@SauermannPhD2001] defined in . The roughness length $z_0$ is set to $0.0025$ m and the so called characteristic length $L$ to $10$ m.

Equations (\[eq:tau\_x\]) and (\[eq:tau\_y\]) are calculated in Fourier space and have to be multiplied with the logarithmic velocity profile in real space in order to obtain the total shear stress. The surface is assumed to be instantaneously rigid and the effect of sediment transport 