Coherent and incoherent space charge resonance e↵ects

The question of interplay of coherent and incoherent space charge driven resonances and of their Landau damping has found some interest in beam dynamics of modern high-intensity synchrotrons. We revisit the theoretical and simulation models describing coherent half-integer parametric resonances, analyze their Landau damping in 2D beams on the basis of simulated tune spectra and conclude that above second order (envelope modes) they play no role in realistic, Gaussian-like beam models. We also analyze incoherent resonance e↵ects in the beam core regions and find that their role has been underestimated in part of the literature, in particular with regard to the very long-term beam evolution as in synchrotrons. We conclude that for such time scales more careful analysis of realistic simulation models is needed to support synchrotron design and evaluation of experiments.