Stability of highly asymmetric non-neutral plasmas

When non-neutral plasma columns are subject to azimuthally asymmetric electrostatic potentials, they deform into azimuthally asymmetric shapes. A typical asymmetric shape is shown in Fig. 1. The companion paper presents methods of predicting the form of these shapes. Here we analyze the shapes’ stability. Stability is normally studied through modal analysis, in which nonlinear equilibria are calculated and subsequently perturbed. Eigenvalue collisions are often assumed to engender instability. However, as is well known in bifurcation theory, symmetries can stabilize eigenvalue collisions. We have two goals in this paper: ~1! to describe the various, often surprising, responses that the plasma makes to the applied perturbations, and ~2! to illustrate, in a realizable physical system, some of the complicated Hamiltonian bifurcation phenomenon that occur in systems which possess a high degree of symmetry. The non-neutral plasma columns are confined in Penning–Malmberg traps. A schematic of the system is shown in the companion paper. The plasma is held within a conducting cylindrical wall of radius R . Radial confinement is provided by a uniform axial magnetic field (B5B ẑ), while axial confinement is provided by an electrostatic well. More detailed descriptions of Penning–Malmberg traps can be found in the literature. We make the common assumption that the axially ( ẑ)-directed motion bounce averages out, and that the system can be described by the two-dimensional ~2D! drift Poisson equations. Then only motion in the (r ,u) plane perpendicular to B is important, and the particles follow E3B orbits, where the E field is determined by both the plasma itself and the imposed boundary condition at the cylindrical wall, V 5V(u). The 2D drift Poisson equations are isomorphic to the 2D Euler equations for incompressible, inviscid fluids. In this isomorphism, the plasma charge density corresponds to the fluid vorticity and the electrostatic potential corresponds to the stream function. Thus results obtained for nonneutral plasmas are also applicable to 2D fluid flows. In the case of a grounded confining wall @V(u)50# , the steady-state solution of the equations of motion is a centered, circular plasma. Breaking the symmetry @V(u)Þ0# deforms the plasma. While Chu et al. used perturbation techniques to prove that the plasma remains stable under small external potentials V(u), it seems intuitively plausible that sufficiently large potentials should lead to instability. How-