Comparison between theory and simulations for longitudinal instabilities of coasting beams

By using the linear kinetic theory and the particle-in-cell code SCOP-RZ/PATRIC the longitudinal instability observed in the ESR has been investigated for different values of the longitudinal impedance of the machine. The numerical investigations show that initially the instability grows exponentially as predicted by the linear theory. When the perturbation reaches large values, significant high-order harmonics are produced: steepening of the density profile as well as saturation of the instability growth appear first, and turbulence later on. Analytical and numerical studies with a small RF voltage applied at the detuned cavity gap have also been carried out in order to meet the conditions in which actual measurements were made at the ESR.