is implemented. Time is discretized with a third-order Runge-Kutta method.

We use a computational box with an aspect ratio of 10, which spans $$0 \le x, y \le 1, \qquad 0 \le z \le 10.$$

Boundary and Initial Conditions {#sec:bc}
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As boundary conditions at the photospheric surfaces ($z=0,\ L$) we impose two independent velocity patterns, intended to mimic photospheric motions, made up of large spatial scale projected convection cell flow patterns constant in time. The velocity potential at each boundary is given by: $$\begin{gathered}
 \label{eq:bc}
\varphi (x,y) = \frac{1}{\sqrt{ \sum_{mn} \alpha_{mn}^2}} \ \sum_{k,l}
            \frac{\ell\, \alpha_{kl}}{2\pi \sqrt{k^2+l^2}} \   \\ 
            \sin \left[ \frac{2 \pi}{\ell} \left( kx+ly \right) + 2\pi \, \xi_{kl} \right]\end{gathered}$$ We excite all the wave number values $(k,l) \in \mathbb{Z}^2$ included in the range $3 \le \left( k^2 + l^2 \right)^{1/2} \le 4$, so that the resulting average injection wavenumber is $k_{c} \sim 3.4$, and the average injection scale $\ell_{c}$, the convection cell scale, is given by $\ell_c = \ell  / k_{c}$. $\alpha_{kl}$ and $\xi_{kl}$ are two sets of random numbers whose values range between $0$ and $1$, and are independently chosen for the two boundary surfaces. The normalization adopted in eq. (\[eq:bc\]) sets the value of the corresponding 