scales, equations (\[eq:adim1\])-(\[eq:adim3\]) reduce to two simple wave equations. Coupled with the boundary conditions (\[eq:bc\]) the solution for times longer than the crossing time $\tau_{\mathcal{A}}$ reads:[ $$\begin{aligned}
& &\mathbf{b}_{\perp} (x,y,z,t) =
\left[ \mathbf{u}^{L} (x,y) - \mathbf{u}^{0} (x,y) \right]\
\frac{t}{\tau_{\mathcal A}}, \label{eq:blin}\\
& &\mathbf{u}_{\perp} (x,y,z,t) = \mathbf{u}^L (x,y) \frac{z}{L} 
+ \mathbf{u}^0 (x,y) \left( 1 - \frac{z}{L} \right). \label{eq:vlin}\end{aligned}$$ ]{} This shows that A) the loop velocity field is bounded by the imposed photospheric fields and b) the magnetic field grows linearly in time, uniform along the loop, while *mapping* the photospheric velocity field in the perpendicular planes. Therefore, for a generic set of velocities $\mathbf{u}^L$ and $\mathbf{u}^0$, the resulting magnetic fields (\[eq:blin\])-(\[eq:vlin\]) give rise to non-vanishing forces in the perpendicular planes which grow quadratically in time, becoming dynamically important after a certain interval [@buc07].

There exists a (singular) set of velocity forcing patterns, for which the generated coronal field has a vanishing Lorentz force. For simplicity consider $\mathbf{u}^L = 0$: in terms of potentials it follows that $\psi = -\varphi^0 \, t/\tau_{\mathcal A}$ and $\varphi = \varphi^0 \, (1-z/L)$ (where $\mathbf{u}^0_\perp = \mathbf{\nabla} \times
\left( \varphi^0\, \mathbf{e}_z \right)$). In this case both $\mathbf{b}_{\perp}$ and $\mathbf{u}_{\perp}$ are proportional to $\mathbf{\nabla}_{\perp} \times  ( 