velocity define the sand flux over a surface element ${\boldsymbol{\rm q}}(x,y)={\boldsymbol{\rm u}}(x,y) \rho(x,y)$.

#### Avalanches:

Surfaces with slopes which exceed the maximal stable angle of a sand surface, the called [*angle of repose*]{} $\Theta \approx 34^o$, produce avalanches which slide down in the direction of the steepest descent. The unstable surface relaxes to a somewhat smaller angle. For the study of dune formation two global properties are of interest. These are the sand transport downhill due to gravity and the maintenance of the angle of repose. To determine the new surface after the relaxation by avalanches the model proposed by is used, $$\label{eq:dhdt_a}
  \frac{\partial h}{\partial t} = -
  C_a R \left( \left| \nabla h \right| - \tan \Theta \right)$$ $$\label{eq:drho_adt}
  \frac{\partial R}{\partial t} 
  + \nabla \left(R {\boldsymbol{\rm u}}_a \right) =
  C_a R \left( \left| \nabla h \right| - \tan \Theta \right),$$ where $h$ denotes the height of the sand bed, $R$ the height of the moving layer, $C_a$ is a model parameter and the velocity ${\boldsymbol{\rm u}}_a$ of the sand grains in the moving layer is obtained by, $$\label{eq:3d:ua}
  {\boldsymbol{\rm u}}_a = - u_a \frac{\nabla h}{\tan \Theta},$$ where $u_a$ is the velocity at the angle of repose. Like in the 