  We investigate the interplay of cosmic ray (CR) propagation and advection in
galaxy clusters. Propagation in form of CR diffusion and streaming tends to
drive the CR radial profiles towards being flat, with equal CR number density
everywhere. Advection of CR by the turbulent gas motions tends to produce
centrally enhanced profiles. We assume that the CR streaming velocity is of the
order of the sound velocity. This is motivated by plasma physical arguments.
The CR streaming is then usually larger than typical advection velocities and
becomes comparable or lower than this only for periods with trans- and
super-sonic cluster turbulence. As a consequence a bimodality of the CR spatial
distribution results. Strongly turbulent, merging clusters should have a more
centrally concentrated CR energy density profile with respect to relaxed ones
with very subsonic turbulence. This translates into a bimodality of the
expected diffuse radio and gamma-ray emission of clusters, since more centrally
concentrated CR will find higher target densities for hadronic CR proton
interactions, higher plasma wave energy densities for CR electron and proton
re-acceleration, and stronger magnetic fields. Thus, the observed bimodality of
cluster radio halos appears to be a natural consequence of the interplay of CR
transport processes, independent of the model of radio halo formation, be it
hadronic interactions of CR protons or re-acceleration of low-energy CR
electrons. Energy dependence of the CR propagation should lead to spectral
steepening of dying radio halos. Furthermore, we show that the interplay of CR
diffusion with advection implies first order CR re-acceleration in the
pressure-stratified atmospheres of galaxy clusters. Finally, we argue that CR
streaming could be important in turbulent cool cores of galaxy clusters since
it heats preferentially the central gas with highest cooling rate.
