the outer spine following the general direction of the open field. In the present parametric study, the inclination angle $\theta = 10^\circ$ is the same for all simulations. As in PDD15, energy is injected in the closed domain through line-tied twisting motions at the bottom boundary. The imposed velocity field is given by Equation 3 of PDD15.

We follow the evolution of the free magnetic energy in the system, taken as the difference, $E_{mag}$, between the total magnetic energy $E_m$ and its initial value: $E_{mag}(t)=E_m(t)-E_m(t=0)$. This is only a proxy of the actual free magnetic energy, defined as the difference between the total magnetic energy in the system and the magnetic energy of the potential field having the same distribution of the normal magnetic field across the six boundaries. Although the system is initially potential and $B_z$ at the bottom boundary is constant in time, since the magnetic flux distribution changes slightly on the other boundaries, the potential field and its energy change in time. @Pariat15b showed in one case (their Fig. 2) that the difference between $E_{mag}$ and the real free energy is at most $20\%$, and that the two values are monotonically related.

The domain is filled with 