Non-neutral Plasma Column Research Articles

Nonlinear beat wave decay of Kelvin/diocotron waves on a two-dimensional vortex

We describe theory and experiments investigating nonlinear beat wave decay of diocotron modes on a nonneutral plasma column (or Kelvin waves on a vortex). Specifically, a Kelvin/diocotron pump wave varying as Ap exp [i(lpθ−ωpt)] decays into two waves: a Kelvin/diocotron daughter wave with exponentially growing amplitude Ad(t), mode number ld<lp, and frequency ωd; and an exponentially growing “beat wave” with mode number lb and frequency ωb. Nonlinear wave–wave coupling requires lb=lp−ld and ωb=ωp−ωd. The new theory simplifies and extends a previous weak-turbulence theory for the exponential growth rate of this instability, by instead using an eigenmode expansion to describe the beat wave as a wavepacket of continuum (Case/van Kampen) modes. The new theory predicts the growth rate, the nonlinear frequency shift (both proportional to Ap2), and the functional form of the beat wave, with amplitude proportional to ApAd*(t). Experiments observe beat wave decay on electron plasma columns for a range of mode numbers up to lp=5 and ld = 4, with results in quantitative agreement with the theory, including the ld = 1 case for which measured growth rates are negligible, as expected theoretically.

Linear Pinch Equilibrium of Non-Neutral Plasma Revisited: Phenomenological Consequences of a Numerical Accuracy Problem

Weibel in 1959 under considerations of a collisionless non-neutral cylindrical plasma column studied a linear pinch confinement equilibrium. As reported here, due to non-linearity of the ordinary differential equations obtained for the electrostatic and magnetostatic fields is possible to demonstrate that the confining features previously obtained are extremely dependent on the initial conditions, and the arrangement of two parameters (β - the ratio between ion and electron mass; M/KT - ratio between relativistic rest energy associated with the pair electron-ion and thermal energy kT ) related to the plasma column characteristics. We investigated in this paper the plasma column behavior (confining or non-confining) under modifications of that set of parameters. We detected a set of parameters values that imposes a confining configuration with an electronic skin effect on the plasma column, not yet reported or discussed in the literature.

Bernstein modes in a non-neutral plasma column

This paper presents theory and numerical calculations of electrostatic Bernstein modes in an inhomogeneous cylindrical plasma column. These modes rely on finite Larmor radius effects to propagate radially across the column until they are reflected when their frequency matches the upper hybrid frequency. This reflection sets up an internal normal mode on the column and also mode-couples to the electrostatic surface cyclotron wave (which allows the normal mode to be excited and observed using external electrodes). Numerical results predicting the mode spectra, using a novel linear Vlasov code on a cylindrical grid, are presented and compared to an analytical Wentzel Kramers Brillouin (WKB) theory. A previous version of the theory [D. H. E. Dubin, Phys. Plasmas 20(4), 042120 (2013)] expanded the plasma response in powers of 1/B, approximating the local upper hybrid frequency, and consequently, its frequency predictions are spuriously shifted with respect to the numerical results presented here. A new version of the WKB theory avoids this approximation using the exact cold fluid plasma response and does a better job of reproducing the numerical frequency spectrum. The effect of multiple ion species on the mode spectrum is also considered, to make contact with experiments that observe cyclotron modes in a multi-species pure ion plasma [M. Affolter et al., Phys. Plasmas 22(5), 055701 (2015)].

Collisionless and collisional effects on plasma waves from a partition squeeze

A simple 1D model is presented for the heating caused by cylindrically-symmetric plasma waves in a non-neutral plasma column due to the addition of a symmetric squeeze potential applied to the center of the column. We study this model by using analytical techniques and by using a numerical grids method solution, and we compare the results of this model to previous work (Ashourvan and Dubin (2014)). squeeze divides the plasma into passing and trapped particles; the latter cannot pass over the squeeze potential. In collisionless theory, enhanced heating is caused by additional bounce harmonics induced by the squeeze in the particle distribution, leading to Landau resonances at energies En for which the bounce frequency ωb(E) and wave frequency ωm satisfy ωm = nωb(En). As a result, heating is substantially higher than the case with no squeeze, even when ωm is much greater than the thermal bounce frequency ωb(T). Adding collisions to the theory creates a boundary layer at the separatrix between trapped and passing particles that further enhances the heating at small ωm/kmvs, where km is the axial wavenumber and vs is the velocity at the separatrix. However, at large ωm/vs, the heating from the separatrix boundary layer is only a small correction to the heating from collisionless resonances in the trapped particle distribution function.

Effect of squeeze on electrostatic Trivelpiece-Gould wave damping

We present a theory for increased damping of Trivelpiece-Gouid plasma modes on a nonneutral plasma column, due to application of a Debye shielded cylindrically symmetric squeeze potential φ1. We present two models of the effect this has on the plasma modes: a 1D model with only axial dependence, and a 2D model that also keeps radial dependence in the squeezed equilibrium and the mode. We study the models using both analytical and numerical methods. For our analytical studies, we assume that φ1/T≪1, and we treat the Debye shielded squeeze potential as a perturbation in the equilibrium Hamiltonian. Our numerical simulations solve the 1D Vlasov-Poisson system and obtain the frequency and damping rate for a self-consistent plasma mode, making no assumptions as to the size of the squeeze. In both the 1D and 2D models, damping of the mode is caused by Landau resonances at energies En for which the particle bounce frequency ωb(En) and the wave frequency ω satisfy ω=nωb(En). Particles experience a non-sinusoidal wave potential along their bounce orbits due to the squeeze potential. As a result, the squeeze induces bounce harmonics with n > 1 in the perturbed distribution. The harmonics allow resonances at energies En≤T that cause substantial damping, even when wave phase velocities are much larger than the thermal velocity. In the regime ω/k≫T/m (k is the wave number) and T≫φ1, the resonance damping rate has a |φ1|2 dependence. This dependence agrees with the simulations and experimental results.

Cyclotron waves in a non-neutral plasma column

A kinetic theory of linear electrostatic plasma waves with frequencies near the cyclotron frequency Ωcs of a given plasma species s is developed for a multispecies non-neutral plasma column with general radial density and electric field profiles. Terms in the perturbed distribution function up to O(1/Ωcs2) are kept, as are the effects of finite cyclotron radius rc up to O(rc2). At this order, the equilibrium distribution is not Maxwellian if the plasma temperature or rotation frequency is not uniform. For rc→0, the theory reproduces cold-fluid theory and predicts surface cyclotron waves propagating azimuthally. For finite rc, the wave equation predicts that the surface wave couples to radially and azimuthally propagating Bernstein waves, at locations where the wave frequency equals the local upper hybrid frequency. The equation also predicts a second set of Bernstein waves that do not couple to the surface wave, and therefore have no effect on the external potential. The wave equation is solved both numerically and analytically in the WKB approximation, and analytic dispersion relations for the waves are obtained. The theory predicts that both types of Bernstein wave are damped at resonances, which are locations where the Doppler-shifted wave frequency matches the local cyclotron frequency as seen in the rotating frame.

Stabilizing effect of a nonresonant radio frequency drive on the m=1 diocotron instability

It has been experimentally shown that the rotation radius of a non-neutral plasma column around the longitudinal axis of a Malmberg–Penning trap experiences a growth in amplitude (m=1 diocotron instability), leading to the loss of the plasma on the surface of the confining electrodes. A new stabilization mechanism has been investigated with the help of systematic experiments in the ELTRAP (ELectron TRAP) device where a high-frequency, low-amplitude drive has been applied on an azimuthally sectored electrode. An effective confining force is created, which reduces the offset of the column from the center. This interpretation and its theoretical analysis show a qualitative agreement with the experimental findings, where a net confinement effect is present for a wide range of drive amplitudes and frequencies.

Relativistic cold fluid radial equilibrium of an annular nonneutral plasma

The relativistic radial equilibrium of a nonneutral plasma confined in a cylindrical trap is considered for the general case in which a coaxial inner charged conductor is present. Both rigid and sheared modes of plasma rotation in crossed self-consistent electric and magnetic fields are analyzed. It is found that relativistic effects can limit the outer radius of an annular nonneutral plasma column for the fast sheared rotational mode. For the particular case of a uniform plasma density, the outer radius cannot exceed 3 r i c / | Ω | if the inner radius of the plasma column r i is smaller than 3 c / | Ω | , where Ω is the cyclotron frequency. On the contrary, radially unlimited equilibria are found when r i > 3 c / | Ω | . The Brillouin density limit is also analyzed and a class of sheared equilibria is found where the limit valid for the case of rigid rotation can be overcome.

The magnetron instability in a pulsar's cylindrical electrosphere

(abridged) The physics of the pulsar magnetosphere remains poorly constrained by observations. Little is known about their emission mechanism. Large vacuum gaps probably exist, and a non-neutral plasma partially fills the neutron star surroundings to form an electrosphere. We showed that the differentially rotating equatorial disk in the pulsar's electrosphere is diocotron unstable and that it tends to stabilise when relativistic effects are included. However, when approaching the light cylinder, particle inertia becomes significant and the electric drift approximation is violated. In this paper, we study the most general instability, i.e. by including particle inertia effects, as well as relativistic motions. This general non-neutral plasma instability is called the magnetron instability. We linearise the coupled relativistic cold-fluid and Maxwell equations. The non-linear eigenvalue problem for the perturbed azimuthal electric field component is solved numerically. The spectrum of the magnetron instability in a non-neutral plasma column confined between two cylindrically conducting walls is computed for several cylindrical configurations. For a pulsar electrosphere, no outer wall exists. In this case, we allow for electromagnetic wave emission propagating to infinity. When the self-field induced by the plasma becomes significant, it can first increase the growth rate of the magnetron instability. However, equilibrium solutions are only possible when the self-electric field, measured by the parameter $s_{\rm e}$ and tending to disrupt the plasma configuration, is bounded to an upper limit, $s_{\rm e,max}$. For $s_{\rm e}$ close to but smaller than this value $s_{\rm e,max}$, the instability becomes weaker or can be suppressed as was the case in the diocotron regime.

Relativistic stabilisation of the diocotron instability in a pulsar “cylindrical” electrosphere

Context. The physics of the pulsar inner magnetosphere remains poorly constrained by observations. Although about 2000 pulsars have been discovered to date, only a little is known about their emission mechanism. Large vacuum gaps exist in the magnetosphere, and a non-neutral plasma partially fills the neutron star surroundings to form an electrosphere. Aims. In a previous work, we showed that the differentially rotating equatorial disk in the pulsar's electrosphere is diocotron unstable in the non-relativistic regime. In this paper, we extend these results and study the relativistic and electromagnetic stabilisation effects by including the magnetic field perturbation and allowing for relativistic speeds of the guiding centre, in a self-consistent manner. We use the electric drift approximation, valid for low-density plasmas. Methods. We linearise the coupled relativistic cold-fluid and Maxwell equations in the electric drift approximation. The non-linear eigenvalue problem for the perturbed azimuthal electric field is solved numerically with standard techniques for boundary value problems like the shooting method. The spectrum of the relativistic diocotron instability is computed in a non-neutral plasma column confined between two cylindrically conducting walls. Results. For low-speed motions, we recover the eigenfunctions and eigenspectra of the non-relativistic diocotron instability. Our algorithm is also checked in the relativistic planar diode geometry for which an analytical expression of the dispersion relation is known. As expected, when the relativistic and electromagnetic effects become significant, the diocotron instability tends to stabilise. In cylindrical geometry, for some special rotation profiles, all azimuthal modes l are completely suppressed for sufficiently relativistic flows. However, for the profile relevant to the electrosphere, depending on the exact rotation curves, the growth rates can either significantly decrease till they vanish or persist for moderate l. Conclusions. The non-neutral plasma flowing in the pulsar electrosphere approaches the speed of light when reaching the light-cylinder. Therefore, relativistic and electromagnetic effects are important. They are able to completely suppress the diocotron instability. Nevertheless, results are sensitive to the tail of the rotation curves; therefore, particle diffusion across the magnetic field due to the diocotron instability only works efficiently close to the neutron star surface.

Diocotron instability in non-neutral plasmas with a stationary point in the rotation frequency profile

A new analytical solution to the problem of the l=1 diocotron mode instability in a hollow density non-neutral plasma column with finite lenght is presented. The starting point of the analysis is the paper of Finn et al. [J. M. Finn, D. del-Castillo-Negrete, and D. C. Barnes, Phys. Plasmas 6, 3744 (1999)], where the instability mechanism involves compression of the plasma in the direction parallel to the magnetic field, with conservation of its line-integrated density. In the limit of small curvature of the plasma end-fronts, the method presented here provides both eigenvalue and eigenfunction for the unstable l=1 “extreme mode.”

Trapped-particle diocotron modes

Recent experiments have characterized trapped-particle modes on a non-neutral plasma column [A. A. Kabantsev, C. F. Driscoll, T. J. Hilsabeck, T. M. O’Neil, and J. H. Yu, Phys. Rev. Lett. 87, 225002 (2001)], and in this paper we present a theoretical model of the modes. Theoretical predictions for the mode frequency, damping rate, and eigenmode structure are compared to experimental observation. The modes are excited on a non-neutral plasma column in which classes of trapped and passing particles have been created by the application of a potential barrier. The column resides in a Malmberg–Penning trap, and the barrier is created by applying a voltage to an azimuthally symmetric section of the wall near the axial mid-point of the column. Low energy particles near the edge of the column (where the barrier is strong) are trapped in one end or the other, while high energy particles near the center of the column transit the entire length. The modes have azimuthal variation l=1,2,…, and odd z-symmetry. The trapped particles on either side of the barrier execute E×B drift oscillations producing density perturbations that are 180° out of phase with each other, while passing particles run back and forth along the field lines attempting to Debye shield the perturbed charge density. The mode is damped by collisional scattering across the separatrix between trapped and passing particles. The damping rate is calculated using a boundary layer analysis of the Fokker–Planck equation. It is also shown that the damping is associated with the radial transport of plasma particles.

Non-neutral plasma column in an asymmetric trapping field.

A non-neutral plasma column in an asymmetric trapping field is considered in this paper. It is shown that nonlinearly interacting bulk plasma oscillations and quadrupole surface waves allow an exact analytical description. The absence of symmetry leads to nonintegrability and other nonlinear phenomena, such as passage through different resonances and phase locking. These phenomena should be observable in experiments involving elliptical traps and rotating walls and can affect the latters' applications.

Paul trap experiment to simulate intense nonneutral beam propagation through a periodic focusing field configuration

This paper describes the design concept for a compact Paul trap experimental configuration that fully simulates the collective processes and nonlinear transverse dynamics of an intense charged particle beam that propagates over large distances through a periodic quadrupole magnetic field. To summarize, a long nonneutral plasma column ( L⪢ r p) is confined axially by applied DC voltages V ̂ = const. on end cylinders at z=± L, and transverse confinement is provided by segmented cylindrical electrodes (at radius r w) with applied oscillatory voltages ± V 0( t) over 90° segments. Because the transverse focusing force is similar in waveform to that produced by a discrete set of periodic quadrupole magnets in a frame moving with the beam, the Paul trap configuration offers the possibility of simulating intense beam propagation in a compact experimental facility. The nominal operating parameters in the experimental design are: barium ions ( A=137); plasma column length 2L=2 m ; wall radius r w =10 cm ; plasma radius r p =1 cm ; maximum wall voltage V ̂ 0=400 V ; end electrode voltage up to V ̂ =500 V ; and voltage oscillation frequency f 0=1/T=60 kHz .

Finite length diocotron modes

Diocotron modes are discussed for a finite length nonneutral plasma column under the assumption of bounce averaged E×B 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., ∂δφ/∂z=0). One can think of the plasma as a collection of magnetic-field aligned rods that undergo E×B drift across the field and adjust their length so as to maintain the condition ∂δφ/∂z=0 inside the plasma. Using the Green function (for a region bounded by a conducting cylinder) to relate the perturbed charge density and the perturbed potential, imposing the constraint ∂δφ/∂z=0, and discretizing yields a matrix eigenvalue problem. The mode eigenvector δNl,ω(rj)≡∫dz δnl,ω(rj,z) is the lth azimuthal Fourier component of the z-integrated density perturbation, and the frequency ω 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 l=1 diocotron modes, long observed in experiments. The paper focuses on these two problems. To approach quantitative agreement with experiment for the l=1 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.

A Paul trap configuration to simulate intense non-neutral beam propagation over large distances through a periodic focusing quadrupole magnetic field

This paper considers an intense non-neutral charged particle beam propagating in the z-direction through a periodic focusing quadrupole magnetic field with transverse focusing force, −κq(s)[xêx−yêy], on the beam particles. Here, s=βbct is the axial coordinate, (γb−1)mbc2 is the directed axial kinetic energy of the beam particles, qb and mb are the charge and rest mass, respectively, of a beam particle, and the oscillatory lattice coefficient satisfies κq(s+S)=κq(s), where S is the axial periodicity length of the focusing field. The particle motion in the beam frame is assumed to be nonrelativistic, and the Vlasov-Maxwell equations are employed to describe the nonlinear evolution of the distribution function fb(x,y,x′,y′,s) and the (normalized) self-field potential ψ(x,y,s)=qbφ(x,y,s)/γb3mbβb2c2 in the transverse laboratory-frame phase space (x,y,x′,y′), assuming a thin beam with characteristic radius rb≪S. It is shown that collective processes and the nonlinear transverse beam dynamics can be simulated in a compact Paul trap configuration in which a long non-neutral plasma column (L≫rp) is confined axially by applied dc voltages V̂=const on end cylinders at z=±L, and transverse confinement in the x−y plane is provided by segmented cylindrical electrodes (at radius rw) with applied oscillatory voltages ±V0(t) over 90° segments. Here, V0(t+T)=V0(t), where T=const is the oscillation period, and the oscillatory quadrupole focusing force on a particle with charge q and mass m near the cylinder axis is −mκq(t)[xêx−yêy], where κq(t)≡8qV0(t)/πmrw2.

Single-parameter characterization of the thermal equilibrium density profile for intense non-neutral charged particle beams

The present analysis considers an intense non-neutral ion beam with characteristic axial velocity ${V}_{b}={\ensuremath{\beta}}_{b}c$ and directed kinetic energy $({\ensuremath{\gamma}}_{b}\ensuremath{-}1){m}_{b}{c}^{2}$ propagating in the $z$ direction through an applied focusing field which produces a transverse focusing force ${\mathbf{F}}_{\mathrm{foc}}=\ensuremath{-}{\ensuremath{\gamma}}_{b}{m}_{b}\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{2}(x{\stackrel{^}{\mathbf{e}}}_{x}+y{\stackrel{^}{\mathbf{e}}}_{y})$ on a beam ion (smooth focusing approximation). For a thermal equilibrium distribution function ${F}_{b}({H}_{\ensuremath{\perp}}^{0})\phantom{\rule{0ex}{0ex}}=\mathrm{const}\ifmmode\times\else\texttimes\fi{}\mathrm{exp}(\ensuremath{-}{H}_{\ensuremath{\perp}}^{0}/{T}_{b})$, it is shown that the normalized radial density profile $\ensuremath{\pi}{r}_{b}^{2}{n}_{b}^{0}(r)/{N}_{b}$, plotted as a function of $r/{r}_{b}$, depends on a single dimensionless parameter, ${\ensuremath{\delta}}_{b}={N}_{b}{Z}_{b}^{2}{e}^{2}/2{\ensuremath{\gamma}}_{b}^{2}{T}_{b}$, which is a measure of the normalized beam intensity. Here, ${N}_{b}\ensuremath{\equiv}\ensuremath{\int}\mathrm{dxdy}{n}_{b}^{0}(r)$ is the number of beam ions per unit axial length and ${r}_{b}^{2}\ensuremath{\equiv}{N}_{b}^{\ensuremath{-}1}\ensuremath{\int}\mathrm{dxdy}{r}^{2}{n}_{b}^{0}(r)$ is the mean-square beam radius. ``Universal'' profiles for the beam density are presented for a wide range of system parameters. The present results are readily extended to the case of a cylindrical non-neutral plasma column confined by a uniform axial magnetic field ${B}_{0}{\stackrel{^}{\mathbf{e}}}_{z}$.

Computation of the diocotron spectrum of a cylindrical non-neutral plasma

A finite difference method is developed and used to compute the diocotron spectrum (discrete plus continuum) of a non-neutral plasma column with an arbitrary profile of the number density. For step function density profiles the results agree with the analytical dispersion relation. In contrast to earlier results, it is found that for a continuously varying hollow density profile the mode with azimuthal mode number l=2 has a growth rate comparable to that for a step function hollow density profile. The results obtained have been verified using an alternative numerical method in which the density is assumed to be constant over small intervals in the radius.

Influence of profile shape on the diocotron instability in a non-neutral plasma column

In this paper we examine theoretically the influence of density profile shape on the diocotron instability in a cylindrical, low-density (ωpe2≪ωce2) non-neutral electron plasma column confined radially by a uniform axial magnetic field B0êz. The analysis assumes electrostatic flute perturbations (∂/∂z=0) about an axisymmetric equilibrium density profile ne0(r), where r=(x2+y2)1/2 is the radial distance from the column axis. Two classes of density profiles with inverted population in radius r are considered. These are the following: (a) a step-function density profile with uniform density n̂e⋅Δ in the column interior 0⩽r<rb−, and uniform density n̂e in an outer annular region rb−<r<rb+; and (b) a continuously-varying density profile of the form ne0(r)=n̂e(Δ+r2/rb2)(1−r2/rb2)2 over the interval 0⩽r<rb. Here, n̂e, rb−, rb+ and rb are positive constants, and the dimensionless parameter Δ measures the degree of “hollowness” of the equilibrium density profile ne0(r). Detailed linear stability properties are calculated for a wide range of system parameters, including values of the “filling factor” Δ, radial location rw of the cylindrical conducting wall, azimuthal mode number l, etc. As a general remark, in both cases, it is found that small increases in Δ from the value Δ=0 (corresponding to the strongest diocotron instability) can have a large effect on the growth rate and detailed properties of the instability. In addition, for the step-function density profile, the instability tends to be algebraic in nature and have a large growth rate in the unstable region of parameter space, whereas for the continuously-varying density profile, the instability is typically much weaker and involves a narrow class of resonant particles at radius r=rs satisfying the resonance condition ωr−lωE(rs)=0. Here, ωr=Re ω is the real oscillation frequency, and ωE(r)=−cEr0(r)/rB0 is the equilibrium E0×B0êz rotation velocity of the plasma column.

Kinetic description of cyclotron-range oscillations of a non-neutral plasma column

The kinetic analysis introduced by Prasad, Morales, and Fried [Prasad et al., Phys. Fluids 30, 3093 (1987)] is used to derive damping conditions and a differential equation for azimuthally propagating waves in a non-neutral plasma column in the limits rl/L≪1 and krl≪1 (where rl is the Larmor radius, k is the wave number, and L is the density scale length). The predictions of the kinetic analysis are verified using a two-dimensional particle-in-cell simulation of Bernstein modes in a thermal rigid-rotor equilibrium. Differences between modes in a strongly magnetized limit and near the Brillouin limit are studied in the simulation.