\tau_{ph}$. For instance for a coronal loop long $L = 40,000~km$ and with an Alfvén velocity $v_{\mathcal A} = 2,000~km\, s^{-1}$ we obtain $\tau_{\mathcal A} = 20~s$, which is small compared to $\tau_{ph} \sim 5~m = 300~s$. This is the reason we carried out simulations with a photospheric forcing constant in time (see eq. (\[eq:bc\])), i.e. for which formally $\tau_{ph} = \infty$.

In the RMHD ordering the nonlinear timescale $\tau_{nl}$ is bigger than the Alfvén crossing time $\tau_{\mathcal A}$. As we shall see this ordering is maintained during our simulations and we will give analytical estimates of the value of $\tau_{nl}$ as a function of the characteristic parameters of the system.

An important feature of equations (\[eq:els1\])-(\[eq:els3\]) that we will use to generalize our results is that, apart from the Reynolds numbers, there is only one fundamental non-dimensional parameter: $$\label{eq:fsc}
f = \frac{\ell_c\, v_{\mathcal A}}{L\, u_{ph}}.$$ Hence all the physical quantities which result from the dynamical evolution, e.g. energy, Poynting flux, heating rate, timescales, etc., must depend on this single parameter $f$.

Energy Equation {#sec:ee}
---------------

From equations (\[eq:adim1\])-(\[eq:adim3\]), with $n=1$, and considering the Reynolds numbers equal, the following energy equation can be derived: $$\label{eq:poy}
\frac{\partial }{\partial t} \left( \frac{1}{2}  