- \mathbf{u}^0$, with $\tau_{\mathcal{R}} = {\ell_{c}^2 Re}/ \left( 2 \pi  \right)^2$ the diffusive time-scale and with $\tau_{\mathcal A} = L / c_{\mathcal A}$ the Alfvén crossing time, the solution is given by: $$\label{eq:bdiff}
\boldsymbol{b}_{\perp} \left( x, y, t \right) =
\boldsymbol{u}_{ph} \left( x, y \right)
\frac{\tau_{\mathcal R}}{\tau_{\mathcal A}}
\left[ 1 - \exp  \left( - \frac{t}{\tau_{\mathcal R}} \right) \right],$$ $$\begin{gathered}
 \label{eq:jdiff}
\left| j \left( x, y, t \right) \right| =
\left| \boldsymbol{u}_{ph} \left( x, y \right) \right|
\left( \frac{2 \pi}{\ell_{c}} \right)  
\frac{\tau_{\mathcal R}}{\tau_{\mathcal A}} \times \\
\times \left[ 1 - \exp  \left( - \frac{t}{\tau_{\mathcal R}} \right) \right].\end{gathered}$$ So that the magnetic energy $E_M$

![Streamlines of the velocity field $\mathbf{u}_{\perp}^L$, the boundary forcing at the *top* plane $z=L$ for run A. The numerical grid has 512x512 points in the x-y planes, with a linear resolution of $\sim 8\, km$.\
\[fig:vL\]](f2.eps){width="47.00000%"}

and the ohmic dissipation rate $J$ are given by $$\begin{gathered}
 \label{eq:emdiff}
E_M =  \frac{1}{2} \int_V \mathrm{d}^3 \boldsymbol{x} \ \boldsymbol{b}_{\perp}^2 = \\
= \frac{1}{2} \ell^2\, L\, u^2_{ph}\,
\left( \frac{\tau_{\mathcal R}}{\tau_{\mathcal A}} \right)^2
\left[ 1 - \exp  \left( - \frac{t}{\tau_{\mathcal R}} \right) \right]^2,\end{gathered}$$ $$\begin{gathered}
 \label{eq:hrdiff}
J =  \frac{1}{Re} \int_V \mathrm{d}^3 \boldsymbol{x} \ j^2 =  \\
= \ell^2\, L\, u^2_{ph}\,
\frac{\tau_{\mathcal R}}{\tau_{\mathcal A}^2}
\left[ 1 