\mathbf u_{\perp}^2
+ \frac{1}{2}  \mathbf b_{\perp}^2   \right)
= - \mathbf{\nabla} \cdot \mathbf{S}
  - \frac{1}{Re}  \left( \mathbf{ j }^2 + \mathbf{ \omega }^2 \right),$$ where $\mathbf{S} = \mathbf{B} \times (\mathbf{u} \times \mathbf{B})$ is the Poynting vector. As expected the energy balance of the system described by eq. (\[eq:poy\]) is due to the competition between the energy (Poynting) flux flowing into the computational box and the ohmic and viscous dissipation. Integrating eq (\[eq:poy\]) over the whole box the only relevant component of the Poynting vector is the component along the axial direction $z$, because in the $x$-$y$ plane periodic boundary conditions are used and their contribution to the Poynting flux is null. As $\mathbf{B} = c_{\mathcal{A}}\, \mathbf{e}_z + \mathbf{b}_{\perp}$ and $\mathbf{u} = \mathbf{u}_{\perp}$, this is given by $$\label{eq:sz}
S_z = \mathbf{S} \cdot \mathbf{e}_z = - c_{\mathcal{A}} 
\left( \mathbf{u}_{\perp} \cdot \mathbf{b}_{\perp} \right).$$ Considering that the velocity fields at the photospheric boundaries are given by $\mathbf{u}_{\perp}^0$ and $\mathbf{u}_{\perp}^L$, for the integrated energy flux we obtain $$\label{eq:tsz}
S = 
  c_{\mathcal A} \int_{z=L} \! \mathrm{d} a\, \left( \mathbf{u}_{\perp}^L 
\cdot \mathbf{b}_{\perp}  \right)
- c_{\mathcal A} \int_{z=0} \! \mathrm{d} a\, \left( \mathbf{u}_{\perp}^0 
\cdot \mathbf{b}_{\perp}  \right).$$ The injected energy flux therefore depends not only on the photospheric forcing and 