Transverse coupled-bunch instability thresholds in the presence of a harmonic-cavity-flattened rf potential

A small vacuum chamber aperture is a present trend in the design of future synchrotron light sources. This leads to a large resistive-wall impedance that can drive coupled-bunch instabilities. Another trend is the use of passively driven cavities at a harmonic of the main radio frequency to lengthen the electron bunches in order to increase the Touschek lifetime and reduce emittance blowup due to intrabeam scattering. In some cases, the harmonic cavities may be tuned to fulfill the flat potential condition. With this condition met, it has been predicted in simulation that the threshold current for coupled-bunch resistive-wall instabilities is much higher than with no bunch lengthening at all. In this paper, the features of a bunch in the flat potential that would contribute toward this stabilization are identified and discussed. The threshold currents for these instabilities are estimated for the MAX IV 3 GeV storage ring at different values of chromaticity using macroparticle simulations in the time domain and, within the limits of the existing theory, frequency domain calculations. By comparing the results from these two methods and analyzing the spectra of the dominant head-tail modes, the impact of each of the distinguishing features of a bunch in the flat potential can be explained and quantified in terms of the change in threshold current. It is found that, above a certain chromaticity, the threshold current is determined by the radial structure of the zeroth-order head-tail mode. This happens at a lower chromaticity if the bunch length is longer.