set in.

A similar linear analysis was already performed by [@park88], who noted that if this is the mechanism responsible for coronal heating, then the energy flux $S_z \sim S / \ell^2$ must approach the value $S_z \sim 10^7\, erg\, cm^2\, s^{-1}$ necessary to sustain an active region before a saturating mechanism, magnetic reconnection of singular current sheets in Parker’s language, takes over.

In fact however the value reached by $S_z$ depends on the nonlinear dynamics, its value *self-consistently* determined by solving the *nonlinear problem*. An $S_z$ too small compared with observational constraints would then rule out the Parker model.

Effects of Diffusion {#sec:eod}
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The linear solution (\[eq:blin\])-(\[eq:vlin\]) has been obtained without taking into account the diffusive terms. This is justified, given the large value of the Reynolds numbers for the solar corona. But numerically it can be important. At very low resolution diffusion is so important that little or no nonlinear dynamics develop and the system reaches a balance between the photospheric forcing and diffusion of the large scale fields.

![Streamlines of the velocity field $\mathbf{u}_{\perp}^0$, the boundary forcing at the *bottom* plane $z=0$ for run A. In lighter vortices the velocity field is directed anti-clockwise while in 