The damping of transverse coherent instabilities by harmonic cavities

At nonzero chromaticity, the threshold current due to transverse coupled bunch instabilities in an electron storage ring is defined by intrabunch head-tail motion of higher than zeroth order. Multibunch tracking simulations predict that this threshold can be increased to several times its original value through the introduction of bunch lengthening harmonic cavities. One previously suggested explanation is the narrower spectra of the elongated bunches but reliable estimates for the threshold currents are not obtainable for anything other than rigid beam motion since the usual Sacherer formalism is not directly applicable to beams in a non-harmonic potential. A new scheme has been developed in which the decay time of a higher than zeroth order transverse head-tail mode may be estimated by taking into account the synchrotron tune spread generated by the harmonic cavity potential. This scheme is presented along with the results of numerical simulations performed in order to confirm the analytical predictions and justify the assumptions made. The extension of the scheme to more complex scenarios is also discussed.