  The existence of the radius of marginal stability means that accretion flows
around black holes invariably undergo a transition from a MHD turbulent
disk-like flow to an inward plunging flow. We argue that the plunging inflow
can greatly enhance the trapping of large scale magnetic field on the black
hole, and therefore may increase the importance of the Blandford-Znajek (BZ)
effect relative to previous estimates that ignore the plunge region. We support
this hypothesis by constructing and analyzing a toy-model of the dragging and
trapping of a large scale field by a black hole disk, revealing a strong
dependence of this effect on the effective magnetic Prandtl number of the MHD
turbulent disk. Furthermore, we show that the enhancement of the BZ effect
depends on the geometric thickness of the accretion disk. This may be, at least
in part, the physical underpinnings of the empirical relation between the
inferred geometric thickness of a black hole disk and the presence of a radio
jet.
