Also the magnitude of the turbulent magnetic stresses decrease as the grid resolution increases (decreasing the numerical diffusivity accordingly), with no convergence in sight (Fromang & Papaloizou 2007). It may well be, however, that this problem is related to an insufficient size of the simulation domain, and that bigger, stratified shearing boxes will show better convergence (Regev & Umurhan 2008).

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

See http://www.nordita.dk/data/brandenb/pencil-code/

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Code

In these magnetised, low-density regions the field is almost force-free, and no dynamical Alfvén waves are excited.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... resolution

This is because our numerical resistivity is always orders of magnitude greater than in realistic accretion discs.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... wavelength

It was shown by Foglizzo & Tagger (1995,1994) that in the $\beta=1$ case the azimuthal field MRI should have a slightly higher growth rate than the Parker instability. However, their linear stability analysis assumed a constant gravity field, rather than a linear field which is the more relevant for Keplerian discs (Kim et al. 1997). Hence the growth rate of the Parker instability several scale heights from the midplane may be underestimated in Foglizzo & Tagger (1994).

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... box

We exclude the vertical field from the top and bottom six grid zones, to avoid conflict with the zero-vertical-field boundary condition.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.