  Rotating magnetized compact objects and their accretion discs can generate
strong toroidal magnetic fields driving highly magnetized plasmas into
relativistic jets. Of significant concern, however, has been that a strong
toroidal field in the jet should be highly unstable to the non-axisymmetric
helical kink (screw) $m=1$ mode leading to rapid disruption. In addition, a
recent concern has been that the jet formation process itself may be unstable
due to the accretion of non-dipolar magnetic fields. We describe large-scale
fully three-dimensional global general relativistic magnetohydrodynamic
simulations of rapidly rotating, accreting black holes producing jets. We study
both the stability of the jet as it propagates and the stability of the jet
formation process during accretion of dipolar and quadrupolar fields. For our
dipolar model, despite strong non-axisymmetric disc turbulence, the jet reaches
Lorentz factors of $\Gamma\sim 10$ with opening half-angle $\theta_j\sim
5^\circ$ at $10^3$ gravitational radii without significant disruption or
dissipation with only mild substructure dominated by the $m=1$ mode. On the
contrary, our quadrupolar model does not produce a steady relativistic
($\Gamma\gtrsim 3$) jet due to mass-loading of the polar regions caused by
unstable polar fields. Thus, if produced, relativistic jets are roughly stable
structures and may reach up to an external shock with strong magnetic fields.
We discuss the astrophysical implications of the accreted magnetic geometry
playing such a significant role in relativistic jet formation, and we outline
avenues for future work.
