Saturation regime of the beam-ion instability.

The growth rate of the beam-ion instability in linear approximation is high and the nonlinear theory of the instability in saturation regime is necessary to describe the experiment. Here we discuss how the theory of the beam-ion instability can be extended to the nonlinear regime and used for interpretation of the experimental results. 1 Basic equations Let us consider a train of bunches with the transverse rms dimensions x, y, x>> y, the number of particles Nb per bunch, and bunch spacing sb = cb. The n-th bunch is at distances zN = (N 1)sb, N = 1; 2::; nb from the head of the train. Position of the N-th bunch in the ring is sN(t) = (ct zN)mod(2R). Ions at the location s around the ring are described by the coordinates X;Y and have linear density dNi=ds. We use the dimensionless time =!0t, vertical y(), Y (; s) and horizontal coordinates x(), X(; s) of the bunches and ions, respectively, measured in units of the bunch rms y and x. The beam-ion interaction is just electrostatic Coulomb force. At the moment t, ions interact with the bunch with the bunch number N(; s) = 1+(c=!0 s)=sb. To simplify consideration, we assume that ions have independent of time Gaussian distribution in X with the average < X> = 0 and the rms x reducing the problem to 1D case. We also assume a Gaussian distribution for the beam in the vertical plane with the bunch centroid at yN(t). Equation of motion for the bunch centroid yN of the N-th bunch is obtained by aver-aging the force over the ion distribution in X and the bunch distribution, d2yN() d 2 + Q2y yN = Z dY i(Y; ; sN()) (yN() Y) [yN (t) Y