  We present an MHD simulation of the emergence of a radio minihalo in a galaxy
cluster core in a "secondary" model, where the source of the
synchrotron-emitting electrons is hadronic interactions between cosmic-ray
protons with the thermal intracluster gas, an alternative to the
"reacceleration model" where the cosmic ray electrons are reaccelerated by
turbulence induced by core sloshing, which we discussed in an earlier work. We
follow the evolution of cosmic-ray electron spectra and their radio emission
using passive tracer particles, taking into account the time-dependent
injection of electrons from hadronic interactions and their energy losses. We
find that secondary electrons in a sloshing cluster core can generate diffuse
synchrotron emission with luminosity and extent similar to observed radio
minihalos. However, we also find important differences with our previous work.
We find that the drop in radio emission at cold fronts is less prominent than
that in our reacceleration-based simulations, indicating that in this flavor of
the secondary model the emission is more spatially extended than in some
observed minihalos. We also explore the effect of rapid changes in the magnetic
field on the radio spectrum. While the resulting spectra in some regions are
steeper than expected from stationary conditions, the change is marginal, with
differences in the synchrotron spectral index of $\Delta\alpha$ < 0.15-0.25,
depending on the frequency band. This is a much narrower range than claimed in
the best-observed minihalos and produced in the reacceleration model. Our
results provide important suggestions to constrain these models with future
observations.
