  Previous axisymmetric investigations in two dimensions (2D) have shown that
rings and gaps develop naturally in non-ideal magnetohydrodynamic (MHD)
disk-wind systems, especially in the presence of ambipolar diffusion (AD). Here
we extend these 2D simulations to three dimensions (3D) and find that rings and
gaps are still formed in the presence of a moderately strong ambipolar
diffusion. The rings and gaps form from the same basic mechanism that was
identified in the 2D simulations, namely, the redistribution of the poloidal
magnetic flux relative to the disk material as a result of the reconnection of
a sharply pinched poloidal magnetic field lines. Thus, the less dense gaps are
more strongly magnetized with a large poloidal magnetic field compared to the
less magnetized (dense) rings. The rings and gaps start out rather smoothly in
3D simulations that have axisymmetric initial conditions. Non-axisymmetric
variations arise spontaneously at later times, but they do not grow to such an
extent as to disrupt the rings and gaps. These disk substructures persist to
the end of the simulations, lasting up to 3000 orbital periods at the inner
edge of the simulated disk. The longevity of the perturbed yet still coherent
rings make them attractive sites for trapping large grains that would otherwise
be lost to rapid radial migration due to gas drag. As the ambipolar diffusivity
decreases, both the disk and the wind become increasingly turbulent, driven by
the magnetorotational instability, with tightly-wound spiral arms becoming more
prominent in the disk.
