A two-mode model to study the effect of space charge on TMCI in the "long-bunch" regime

Using a two-mode approach for the Transverse Mode-Coupling Instability (TMCI) in the “short-bunch” regime (where the mode-coupling takes place between the modes 0 and – 1, such as in the CERN LHC), both a reactive damper (ReaD) and Space Charge (SC) are expected to be beneficial: the ReaD would shift the mode 0 up while SC would shift the mode – 1 down, but in both cases the coupling (and related instability) would occur at higher intensities. However, the situation is more involved in the “long-bunch” regime (where the mode-coupling takes place between higher-order modes, such as in the CERN SPS). As the ReaD modifies only the (main) mode 0 and not the others, it is expected to have no effect for the main mode-coupling. As concerns SC, it modifies all the modes except the mode 0, and the result has been a subject of discussion for two decades. A two-mode approach is discussed in detail in this contribution for the case of a single bunch interacting with a broad-band resonator impedance in the “long-bunch” regime. This model reveals in particular that in the presence of SC, the intensity threshold can only be similar to or lower than the no-SC case.