calculation of the sand flux the steady state of the avalanche model is assumed to be reached instantaneously. In the dune model a certain amount of sand is transported over the brink to the slip face and in every iteration the sand grains are relaxed over the slip face by this avalanche model determining the steady state.

#### The time evolution of the surface

The calculation of the sand flux over a not stationary dune surface leads to changes by erosion and deposition of sand grains. The change of the surface profile can be expressed using the conservation of mass, $$\partial_t \rho + \nabla \Phi = 0 \, ,
   \label{eq:time_evol_h}$$ where $\rho$ is the sand density and $\Phi$ the sand flux per time unit and area. Both $\rho$ and $\Phi$ are now integrated over the vertical coordinate assuming that the dune has a constant density of $\rho_{\text{sand}}$, $$\label{eq:height_rho}
  h = \frac{1}{{\rho_{\text{sand}}}} \int \rho dz , \quad \quad
  {\boldsymbol{\rm q}} = \int {\boldsymbol{\rm \Phi}} dz.$$ Thus Equation (\[eq:time\_evol\_h\]) can be rewritten, as $$\label{eq:masscons_h}
  \frac{\partial h}{\partial t} = - \frac{1}{{\rho_{\text{sand}}}} \nabla{{\boldsymbol{\rm q}}}.$$ Finally, it is noted that Equation (\[eq:masscons\_h\]) is the only remaining time dependent equation and thus defines the characteristic time 