  This article reports on three-dimensional (3-D) MHD simulations of
non-rotating and rapidly rotating black holes and the adjacent black hole
accretion disk magnetospheres. A particular emphasis is placed on the vertical
magnetic flux that is advected inward from large radii and threads the
equatorial plane near the event horizon. In both cases of non-rotating and
rotating black holes, the existence of a significant vertical magnetic field in
this region is like a switch that creates powerful jets. There are many
similarities in the vertical flux dynamics in these two cases in spite of the
tremendous enhancement of azimuthal twisting of the field lines and enhancement
of the jet power because of an "ergospheric disk" in the Kerr metric. A 3-D
approach is essential because two-dimensional axisymmetric flows are incapable
of revealing the nature of vertical flux near a black hole. Poloidal field
lines from the ergospheric accretion region have been visualized in 3-D and
much of the article is devoted to a formal classification of the different
manifestations of vertical flux in the Kerr case.
