Application of envelope instability to high-intensity rings

The space-charge limit is imposed by beam response to low-order machine resonances. Here, the coherent response of the beam to such resonances is discussed, including the parametric resonance of collective beam modes with the periodic lattice, also known as the "envelope instability", when the second-order beam modes are considered. The relation of this parametric resonance to the coherent resonance condition of an integer type is explained. Practical application of such resonant responses to both structural and imperfection driven harmonics is addressed.