l = 1 Diocotron Instability of Single Charged Plasmas

The linear stability analysis of the l = 1 diocotron perturbations in a low density single charged plasma confined in a cylindrical Penning trap is critically revisited. Particular attention is devoted to the instability due to the presence of one or more stationary points in the radial profile of the azimuthal rotation frequency. The asymptotic analysis of Smith and Rosenbluth for the case of a single-bounded plasma column (algebraic instability proportional to t[sup 1/2]) is generalized in a few respects. In particular, the existence of unperturbed density profiles that give rise to l = 1 algebraic instabilities growing with time proportionally to t[sup 1–1/m],m>=3 being the order of a stationary point in the rotation frequency profile, and even proportionally to t, is pointed out. It is also shown that smoothing the density jumps of a multistep density profile can convert algebraically growing perturbations into exponentially decaying modes. The relevant damping rates are computed. The asymptotic analysis (t-> (infinity)) of the fundamental diocotron perturbations is then generalized to the case of a cylindrical Penning trap with an additional coaxial inner conductor. It is shown that the algebraic instability found in the case of a single-bounded plasma column becomes exponential at longer times. The relevant linear growth rate is computed by a suitable inverse Laplace transform (contour integral in the complex plane). In the particular case of an uncharged inner conductor of radius a, the growth rate is shown to scale as a[sup 4/3] for a->0. The theoretical results are compared with the numerical solution of the linearized two-dimensional drift Poisson equations.