  (Abridged) The relaxed cool-core Phoenix cluster (SPT-CL J2344-4243) features
an extremely strong cooling flow, as well as a mini-halo. Strong star-formation
in the brightest cluster galaxy indicates that AGN feedback has been unable to
inhibit this cooling flow. We have studied the strong cooling flow in the
Phoenix cluster by determining the radio properties of the AGN and its lobes.
In addition, we use spatially resolved observations to investigate the origin
of the mini-halo. We present new Very Large Array 1-12 GHz observations of the
Phoenix cluster which resolve the AGN and its lobes in all four frequency
bands, and resolve the mini-halo in L- and S-band. Using our L-band
observations, we measure the total flux density of the radio lobes at 1.5 GHz
to be $7.6\pm0.8$ mJy, and the flux density of the mini-halo to be $8.5\pm0.9$
mJy. Using L- and X-band images, we produce the first spectral index maps of
the lobes from the AGN and measure the spectral indices of the northern and
southern lobes to be $-1.35\pm0.07$ and $-1.30\pm0.12$, respectively.
Similarly, using L- and S-band data, we map the spectral index of the
mini-halo, and obtain an integrated spectral index of $\alpha=-0.95 \pm 0.10$.
We find that the mini-halo is most likely formed by turbulent re-acceleration
powered by sloshing in the cool core due to a recent merger. In addition, we
find that the feedback in the Phoenix cluster is consistent with the picture
that stronger cooling flows are to be expected for massive clusters like the
Phoenix cluster, as these may feature an underweight supermassive black hole
due to their merging history. Strong time variability of the AGN on
Myr-timescales may help explain the disconnection between the radio and the
X-ray properties of the system. Finally, a small amount of jet precession
likely contributes to the relatively low ICM re-heating efficiency of the
mechanical feedback.
