On the requirement for mixing-heating to utilize bubble cosmic rays to
  heat the intracluster medium
  I conduct simple analytical estimates and conclude that mixing by vortices is
a more efficient process to transfer the cosmic ray energy of jet-inflated
bubbles to the intracluster medium (ICM) than streaming of cosmic rays along
magnetic field lines. Jets and the bubbles they inflate transfer heat to the
ambient gas in cooling flows in cluster of galaxies and in galaxies. The
internal energy of the jet-inflated bubbles is dominate by very hot thermal gas
and/or cosmic rays. Cosmic rays that stream along magnetic field lines that
connect the bubbles with the ICM heat the ICM as their energy is dissipated
there. I find that about half of the cosmic ray energy is dissipated in the
bubbles themselves. I also find that the ICM volume that the cosmic ray
streaming process heats is only about five times as large as the volume of the
bubbles. The outcome of heating by streaming only is that the cosmic rays form
a larger bubble filled with very hot thermal gas. Therefore, there is a
requirement for a more efficient process to transfer the internal energy of the
bubbles to the ICM. I suggest that this process is heating by mixing that
operates very well for both cosmic rays and the very hot thermal gas inside the
bubbles. This leaves mixing-heating to be the dominant heating process of
cooling flows.
