the axial Alfvén velocity (which have fixed values), but also on the value of the magnetic fields at the boundaries, which is determined by the dynamics of the system inside the computational box: *the injection of energy depends on the nonlinear dynamics which develops, and viceversa*.

The simplified topology investigated in this paper, i.e.  a strong axial magnetic field whose footpoints are dragged by 2D orthogonal motions applies to regions where emerging flux may be neglected. Consider the axial component of the velocity $u_z$ field carrying new magnetic field ($\mathbf{b}_{\perp}^{ef}$) into the corona. The associated Poynting flux is $$\label{eq:ef}
S_z^{ef} = \left( \mathbf{b}_{\perp}^{ef} \right)^2 u_z.$$ This flux component is negligible when $S_z^{ef} < S_z$, i.e., since all the components of the photospheric velocity fields are of the same order, $u_z \sim u_{ph}$, when

$$\label{eq:efc}
\left( b_{\perp}^{ef} \right)^2 <  B_0\, b_{\perp}^{turb}.$$

In § \[sec:sca\] we give an estimate of the value of the field $b_{\perp}^{turb}$ generated by the field-line dragging, and will be able to quantify for which value of $b_{\perp}^{ef}$ the emerging flux can be neglected.

Linear Stage {#sec:ls}
------------

For $t < \tau_{nl}$ nonlinear terms can be neglected. Neglecting also the diffusion terms, which play no role on large 