  X-ray images of galaxy clusters often display underdense bubbles which are
apparently inflated by AGN outflow. I consider the evolution of the magnetic
field inside such a bubble, using a mixture of analytic and numerical methods.
It is found that the field relaxes into an equilibrium filling the entire
volume of the bubble. The timescale on which this happens depends critically on
the magnetisation and helicity of the outflow as well as on properties of the
surrounding ICM. If the outflow is strongly magnetised, the magnetic field
undergoes reconnection on a short timescale, magnetic energy being converted
into heat whilst the characteristic length scale of the field rises; this
process stops when a global equilibrium is reached. The strength of the
equilibrium field is determined by the magnetic helicity injected into the
bubble by the AGN: if the outflow has a consistent net flux and consequently a
large helicity then a global equilibrium will be reached on a short timescale,
whereas a low-helicity outflow results in no global equilibrium being reached
and at the time of observation reconnection will be ongoing. However, localised
flux-tube equilibria will form. If, on the other hand, the outflow is very
weakly magnetised, no reconnection occurs and the magnetic field inside the
bubble remains small-scale and passive. These results have implications for the
internal composition of the bubbles, their interaction with ICM -- in
particular to explain how bubbles could move a large distance through the ICM
without breaking up -- as well as for the cooling flow problem in general. In
addition, reconnection sites in a bubble could be a convenient source of
energetic particles, circumventing the problem of synchrotron emitters having a
shorter lifetime than the age of the bubble they inhabit.
