Exact axially symmetric galactic dynamos
  We give a selection of exact dynamos in axial symmetry on a galactic scale.
These include some steady examples, at least one of which is wholly analytic in
terms of simple functions and has been discussed elsewhere. Most solutions are
found in terms of special functions, such as associated Lagrange or
hypergeometric functions. They may be considered exact in the sense that they
are known to any desired accuracy in principle. The new aspect developed here
is to present scale invariant solutions with zero resistivity that are
self-similar in time. The time dependence is either a power law or an
exponential factor, but since the geometry of the solution is self-similar in
time we do not need to fix a time to study it. Several examples are discussed.
Our results demonstrate (without the need to invoke any other mechanisms)
X-shaped magnetic fields and (axially symmetric) magnetic spiral arms (both of
which are well observed and documented) and predict reversing rotation measures
in galaxy halos (now observed in the CHANG-ES sample) as well as the fact that
planar magnetic spirals are lifted into the galactic halo.
