.
This paper has examined the influence of the later
approximation for a model with an initial vertical field, and one with
an initial toroidal field.
Given the demands of three dimensional simulations, it is useful to
explore ways in which the global problem can be restricted without too
much loss of significance. One such simplification is the cylindrical
disk limit, and another is the restriction to a smaller domain in
.
This paper has examined the influence of the later
approximation for a model with an initial vertical field, and one with
an initial toroidal field.
The explicit comparison between simulations using the full 
and
ones that span only 
in angle suggests that the restricted
azimuthal domain is indeed a useful approximation. The vertical field
simulations are qualitatively very similar throughout. Although the
initial growth stage in the toroidal field cases are noticeably
different in the two different computational domains, the final
turbulent states are again very similar. With vertical fields the
magnetic azimuthal power spectrum has a significant break at relatively
high m wavenumbers, consistent with the fastest growing modes of the
vertical MRI. With toroidal fields the power spectra level out toward
low m and do not show the same break. Again, this is consistent with
the MRI: weak toroidal fields are unstable for all m less than
.
In either case, however, the power spectra of
the
and
simulations are very similar where their
wavenumbers overlap. Total stress and accretion rate values are
reduced by about 10% in the restricted domains, indicating that 10%
of the energy is to be found on the largest azimuthal scales. This is
not entirely negligible, but the qualitative difference
does not appear to be profound.
A significant factor in global simulations remains the grid resolution
which will always fall short of what one can achieve with a local
simulation. Here the average radial grid zone size is
.
The standard radial resolution in the local simulations of
HGB95 was almost a factor of 10 better. Because of the lack of
stratification, the vertical resolution in the cylindrical simulations
is comparable to the local model. A resolution comparison done by HK
found larger magnetic energies with better resolution, a result
consistent with more detailed resolution studies in the local shearing
box. Even with the grid resolution that is presently possible,
however, global simulations generate and maintain MHD turbulence with
physically significant stress levels.
Title Page
6. Conclusion