The fluid motion of Eq. (3) is written in a form that inadvertently omits a contribution from the rise in the trapped (and the fall in the untrapped) particle density at the interface between trapped and untrapped particles (i.e., at r r s ). The equation should readwhere the trapped and untrapped densities are written in terms of Heaviside step functions Q as follows:and n u ͑r͒ n 0 ͑r͒Q͑r s 2 r͒ .The dispersion relation of Eq. (4) for top hat density profiles [n 0 ͑r͒ n 0 Q͑R p 2 r͒] then becomes µ R p r swhere I m ϵ I m ͑r s ͞l D ͒ are modified Bessel functions of the first kind and f 0 ϵ m 2 f͞f E . This new dispersion relation has two solutions, with only the lower frequency solution being relevant here. The remaining discussion and results in the paper are not affected by this correction, since the two dispersion relations are equivalent in the limit of l D ! 0, and the correct equation was used in the kinetic treatment for comparison to experiments.

Recent experiments have characterized trapped-particle modes on a non-neutral plasma column ͓A.

The damping mechanism of a recently discovered trapped-particle mode is identified as collisional velocity scattering of marginally trapped particles. The mode exists on non-neutral plasma columns that are partially divided by an electrostatic potential. This damping mechanism is similar to that responsible for damping of the dissipative trapped-ion mode. The damping rate is calculated using a Fokker-Planck analysis and agrees with measurement to within 50%. Also, an experimental signature confirms a causal relation between scattering of marginally trapped particles and damping.

Plasmas with peaked radial density profiles have been generated in the world’s largest helicon device, with plasma diameters of over 70 cm. The density profiles can be manipulated by controlling the phase of the current in each strap of two multistrap antenna arrays. Phase settings that excite long axial wavelengths create hollow density profiles, whereas settings that excite short axial wavelengths create peaked density profiles. This change in density profile is consistent with the cold-plasma dispersion relation for helicon modes, which predicts a strong increase in the effective skin depth of the rf fields as the wavelength decreases. Scaling of the density with magnetic field, gas pressure, and rf power is also presented.

Diocotron modes are discussed for a finite length nonneutral plasma column under the assumption of bounce averaged EϫB drift dynamics and small Debye length. In this regime, which is common to experiments, Debye shielding forces the mode potential to be constant along field lines within the plasma ͑i.e., ‫ץ/␦ץ‬zϭ0). One can think of the plasma as a collection of magnetic-field aligned rods that undergo EϫB drift across the field and adjust their length so as to maintain the condition ‫ץ/␦ץ‬zϭ0 inside the plasma. Using the Green function ͑for a region bounded by a conducting cylinder͒ to relate the perturbed charge density and the perturbed potential, imposing the constraint ‫ץ/␦ץ‬zϭ0, and discretizing yields a matrix eigenvalue problem. The mode eigenvector ␦N l, (r j )ϵ͐dz ␦n l, (r j ,z) is the l th azimuthal Fourier component of the z-integrated density perturbation, and the frequency is the eigenvalue. The solutions include the full continuum and discrete stable and unstable diocotron modes. Finite column length introduces a new set of discrete diocotron-like modes. Also, finite column length makes possible the exponential growth of lϭ1 diocotron modes, long observed in experiments. The paper focuses on these two problems. To approach quantitative agreement with experiment for the lϭ1 instabilities, the model is extended to include the dependence of a particle's bounce averaged rotation frequency on its axial energy. For certain distributions of axial energies, this dependence can substantially affect the instability. D ϭv / p , where v is the thermal velocity. The frequency ordering and the smallness of the Debye length justify a reduced description of the plasma. In this Zero Debye Length Reduced Description, the plasma cannot tolerate ͑shields out͒ any electric field ‫)‪z‬ץ/ץ(‬ along the magnetic field. The plasma density, n(z,,r,t), is constant along z within the plasma and drops abruptly to zero at the plasma ends ͑on the scale D →0). Along each field line, the plasma is characterized by a well-defined length 2L (,r,t). For convenience, we use L(,r,t) for the half-length.The plasma can be thought of as a collection of magnetic-field aligned rods that move across the field through EϫB drift motion and adjust their length so that ‫‪z‬ץ/ץ‬ vanishes everywhere inside the plasma. We will see that this constraint is satisfied if ‫‪z‬ץ/ץ‬ vanishes on the plasma surface ͓i.e., for ͉z͉ϭL(,r,t)].With L(,r,t) determined in this way, the plasma state is specified by the two-dimensional z-integrated density distribution N͑,r,t ͒ϭ ͵ Ϫϱ ϩϱ dz n͑z,,r,t ͒ϭ2 n͑,r,t ͒L͑ ,r,t͒, ͑1͒ a͒ Electronic