direction $z$, i.e.$$\mathbf{\nabla}_{\perp} = \mathbf{e}_{x}\, \frac{\partial}{\partial x} +
\mathbf{e}_{y}\, \frac{\partial}{\partial y},$$ while the dynamics in the planes is coupled to the axial direction through the linear terms $\propto \partial_z$.

To render the equation non dimensional magnetic fields have first been expressed in velocity units by dividing by $\sqrt{4\pi \rho_0}$ (where $\rho_0$ is a density supposed homogeneous and constant), i.e. considering the associated Alfvén velocities ($b \rightarrow b/\sqrt{4\pi \rho_0}$), and then both velocity and magnetic fields have been normalized to a typical photospheric velocity $u_{ph}$; lengths and times have been expressed in units of the perpendicular length of the computational box $\ell$ and its related “eddy turnover time” $t_{\perp} = \ell / u_{ph}$.

As a result, in equations (\[eq:adim1\])-(\[eq:adim2\]), the linear terms $\propto \partial_z$ are multiplied by the *dimensionless* Alfvén velocity $c_{\mathcal{A}} = v_{\mathcal A} / u_{ph} $, i.e the ratio between the Alfvén velocity associated with the axial magnetic field $v_{\mathcal A} = B_0 / \sqrt{4 \pi \rho_0}$, and the photospheric velocity $u_{ph}$.

The index $n$ is called *dissipativity*: the diffusive terms adopted in equations (\[eq:adim1\])-(\[eq:adim2\]) correspond to ordinary diffusion for $n=1$ and to so-called hyperdiffusion for $n > 1$. When $n=1$ the $\mathbf{\nabla}_{\perp}^2 / Re$ diffusive operator is recovered, 