${\boldsymbol{\rm u}}(x,y)$. The expansion to two dimensions yields Equations (\[eq:3d\_rho\]) and (\[eq:3d\_u\]) where $\rho$ and ${\boldsymbol{\rm u}}$ are determined from the before obtained shear stress and the gradient of the actual surface, $$\label{eq:3d_rho}
  \text{div} \, (\rho \, {\boldsymbol{\rm u}}) 
  = \frac{1}{T_s} \rho \left( 1 - \frac{\rho}{\rho_s} \right) \; 
  \begin{cases}
    \Theta(h) & \rho < \rho_s\\
    1         & \rho \ge \rho_s
  \end{cases}
  ,$$ with $$\label{eq:rho_s_tau}
  \rho_s = \frac{2 \alpha}{g} \left( |{\boldsymbol{\rm \tau}}| - \tau_t \right) \quad \quad
  T_s = \frac{2 \alpha | {\boldsymbol{\rm u}}|}{g} \, \frac{\tau_t}{\gamma (|{\boldsymbol{\rm \tau}}| - \tau_t)}.$$ and $$\label{eq:3d_u}
   \frac{3}{4} \, C_d \frac{\rho_{\text{air}}}{\rho_{\text{quartz}}} d^{-1} \, ({\boldsymbol{\rm {v_{\text{eff}}}}} - {\boldsymbol{\rm u}})|{\boldsymbol{\rm {v_{\text{eff}}}}} - {\boldsymbol{\rm u}}|
    - \frac{g}{2 \alpha} \frac{{\boldsymbol{\rm u}}}{|{\boldsymbol{\rm u}}|}
    - g \, {\boldsymbol{\rm \nabla}} \, h
    = 0,$$ where $v_{\text{eff}}$ is the velocity of the grains in the saturated state,\
$${\boldsymbol{\rm v}}_{\text{eff}} =   \frac{2 {\boldsymbol{\rm u}}_*}{\kappa |{\boldsymbol{\rm u}}_*|} \cdot \nonumber$$ $$\label{eq:3d:v_eff_of_tau_g0}
\left( \sqrt{\frac{z_1}{z_m} u_*^2 + 
       \left( 1 - \frac{z_1}{z_m} \right)  
      \, u_{*t}^2} 
     + \left( 
     \ln{\frac{z_1}{z_0}} -2 \right) \, \frac{u_{*t}}{\kappa} 
  \right),$$ and $$\label{eq:ustar_s}
  u_* = \sqrt{\tau / {\rho_{\text{air}}}}$$ The constants and model parameters have been taken from [@SauermannKroy2001] and are summarized here: $g=9.81\,$m$\,$s$^{-2}$,$\kappa=0.4$, ${\rho_{\text{air}}}=1.225\,$kg$\,$m$^{-3}$, ${\rho_{\text{quartz}}}=2650\,$kg$\,$m$^{-3}$, $z_m=0.04\,$m, $z_0=2.5~10^{-5}\,$m, $D=d=250\,\mu$m, $C_d=3$, $u_{*t}=0.28\,$m$\,$s$^{-1}$, $\gamma=0.4$,$\alpha=0.35$ and $z_1=0.005\,$m. The sand density and the sand 