  Feedback processes by active galactic nuclei in the centres of galaxy
clusters appear to prevent large-scale cooling flows and impede star formation.
However, the detailed heating mechanism remains uncertain. One promising
heating scenario invokes the dissipation of Alfv\'en waves that are generated
by streaming cosmic rays (CRs). In order to study this idea, we use
three-dimensional magneto-hydrodynamical simulations with the AREPO code that
follow the evolution of jet-inflated bubbles that are filled with CRs in a
turbulent cluster atmosphere. We find that a single injection event produces
the CR distribution and heating rate required for a successful CR heating
model. As a bubble rises buoyantly, cluster magnetic fields drape around the
leading interface and are amplified to strengths that balance the ram pressure.
Together with helical magnetic fields in the bubble, this initially confines
the CRs and suppresses the formation of interface instabilities. But as the
bubble continues to rise, bubble-scale eddies significantly amplify radial
magnetic filaments in its wake and enable CR transport from the bubble to the
cooling intracluster medium. By varying the jet parameters, we obtain a rich
and diverse set of jet and bubble morphologies ranging from Fanaroff-Riley type
I-like (FRI) to FRII-like jets. We identify jet energy as the leading order
parameter (keeping the ambient density profiles fixed), whereas jet luminosity
is primarily responsible for setting the Mach numbers of shocks around
FRII-like sources. Our simulations also produce FRI-like jets that inflate
bubbles without detectable shocks and show morphologies consistent with cluster
observations.
