influx on large perpendicular scales (see eq. (\[eq:bc\])) from the boundaries into the loop. The nonlinear terms $\left(  \mathbf z^{\mp} \cdot \mathbf{\nabla}_{\perp} \right) \mathbf{z}^{\pm}$ are then responsible for the transport of this energy from the large scales toward the small scales, where energy is finally dissipated, i.e. converted to heat and/or particle acceleration.

A well-known important feature of the nonlinear terms in equations (\[eq:els1\])-(\[eq:els3\]) is the absence of self-coupling, i.e.  only counterpropagating waves interact non-linearly, and if one of the two fields $\mathbf{z}^{\pm}$ is zero, there are no nonlinear interactions at all. This fact, i.e.  that counter-propagating wave-packets may interact only while they are crossing each other, lies at the basis of the so-called Alfvén effect [@iro64; @kra65], which ultimately renders the nonlinear timescales longer and slows down the dynamics.

From this description three different timescales arise naturally: $\tau_{\mathcal A}$, $\tau_{ph}$ and $\tau_{nl}$. $\tau_{\mathcal A} = L/v_{\mathcal A}$ is the crossing time of the Alfvén waves along the axial direction $z$, i.e.the time it takes for an Alfvén wave to cover the loop length $L$. $\tau_{ph} \sim 5~m$ is the characteristic time associated with photospheric motions, while $\tau_{nl}$ is the nonlinear timescale.

For a typical coronal loop $\tau_{\mathcal A} \ll 