velocity rms (see eq. (\[eq:potfi\])) to $1/\sqrt{2}$, i.e. $$\label{eq:rmscon}
\int_0^{\ell} \! \! \! \int_0^{\ell} \mathrm{d}x\,  \mathrm{d}y \
\left( u_x^2 + u_y^2 \right)  = \frac{1}{2}$$

At time $t=0$ no perturbation is imposed inside the computational box, i.e. $\mathbf{b}_{\perp} = \mathbf{u}_{\perp} = 0$, and only the axial magnetic field $B_0$ is present: the subsequent dynamics are then the effect of the photospheric forcing (\[eq:bc\]) on the system, as described in the following sections.

ANALYSIS {#sec:an}
========

In order to clarify aspects of the linear and nonlinear properties of the RMHD system, we provide an equivalent form of the equations (\[eq:adim1\])-(\[eq:adim3\]). In terms of the Elsässer variables $\mathbf{z}^{\pm} = \mathbf{u}_{\perp} \pm \mathbf{b}_{\perp}$, which bring out the basic symmetry of the equations in terms of parallel and anti-parallel propagating Alfvén waves, they can be written as $$\begin{gathered}
\frac{\partial \boldsymbol z^+}{\partial t}  =
- \left(  \boldsymbol z^- \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{z}^+
+ c_\mathcal{A} \, \frac{\partial \boldsymbol z^+}{\partial z} \\
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp} \boldsymbol z^+
 - \boldsymbol{\nabla}_{\perp} P, \label{eq:els1}\end{gathered}$$ $$\begin{gathered}
\frac{\partial \boldsymbol z^-}{\partial t}  =
- \left(  \boldsymbol z^+ \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{z}^-
- c_\mathcal{A} \, \frac{\partial \boldsymbol z^-}{\partial z} \\
+ \frac{(-1)^{n+1}}{Re_n}   \boldsymbol{\nabla}^{2n}_{\perp} \boldsymbol z^-
 - \boldsymbol{\nabla}_{\perp} P, \label{eq:els2}\end{gathered}$$ $$\boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{z}^{\pm} = 0,  \qquad 