Trivelpiece-Gould Modes Research Articles

Electron Heating and Energy Transport Due to Nonlinear Landau Damping of Electrostatic Waves in an Electron Beam-Plasma System

Experimental and theoretical studies of electron heating and energy transport due to nonlinear Landau damping of electrostatic waves in an electron beam-plasma system immersed in a magnetic field were performed. Electron heating occurs by quasi-linear electron Landau damping of Trivelpiece-Gould mode induced explosively by nonlinear Landau damping of the slow space charge wave of the electron beam. It is predicted theoretically that the electron heating is caused by the energy transfer from the electron beam, and that the small axial drift of plasma electrons is generated by the momentum transfer from the electron beam. The energy transport by this drift can be balanced with the diffusion and the heat conduction, and this can maitain the large axial inhomogeneity of the electron temperature and density.

Trivelpiece-Gould modes in a corrugated plasma slab.

It is shown that the spectrum of electrostatic oscillations in a periodically corrugated plasma slab exhibits spatial mode locking, i.e., for each spectral branch there is an infinity of subbands with constant wave numbers. The widths of the largest subbands are evaluated. We also show that the eigenmodes are represented by a set of double layers. Under certain conditions, the initial problem for wave propagation is solved. The interaction of an electron beam with a plasma is studied and the corresponding instability growth rate is obtained.

Anisotropic self-defocusing of a Trivelpiece-Gould mode in a plasma

The paper presents a paraxial-ray theory of self-defocusing of obliquely propagating Trivelpiece-Gould (TG) modes in a homogeneous plasma. The TG mode with Gaussian distribution of intensity, in a direction perpendicular to its phase velocity, creates a ponderomotive force on electrons. The electrons are redistributed through ambipolar diffusion. The redistributed electrons are responsible for defocusing of the TG mode. The importance of our study is pointed out.

Parametric upconversion of TM and Trivelpiece-Gould (TG) modes to high frequency free electron laser (FEL)

A backward wave oscillator (BWO) filled with a strongly magnetized plasma supports TM and Trivelpiece-Gould (TG) modes. At large amplitudes these modes may act as wigglers for generating millimeter waves via free electron laser instability. The nonlinear coupling between the wiggler, the beam space charge mode, and the high frequency free electron laser wave is dominated by parallel motions. In the Raman regime the growth rate of instability goes as ∼ ω pb 1/2 /β 0 9/4 , where ωpb is the beam plasma frequency and β0 is the relativistic gamma factor.

Low-frequency instability in weakly ionized current-carrying magnetized plasmas. II. Behaviour in high-frequency wave field

The effect of the Trivelpiece-Gould mode on the behaviour of a low-frequency drift instability in a weakly ionized magnetized plasma is studied experimentally in a DC argon discharge. The ions are weakly magnetized, i.e. Omega ci approximately nu i, rho i approximately R ( Omega ci and rho i are the ion gyrofrequency and gyroradius, nu i is the ion-neutral collision frequency and R is the plasma radius). With increasing amplitude of the high-frequency field, the frequency and the amplitude of the instability decrease. Since the anomalously enhanced ambipolar field is considered as a very important factor in the mechanism driving the instability, special attention is paid to its variation with increasing high-frequency power. The nonlinear mechanisms of the dynamical control of the instability are discussed.

Experimental study of a plasma-filled backward wave oscillator

Experimental studies of a plasma-filled X-band backward-wave oscillator (BWO) are presented. Depending on the background gas pressure, microwave frequency upshifts of up to 1 GHz appeared along with an enhancement by a factor of 7 in the total microwave power emission. The bandwidth of the microwave emission increased from <or=0.5 GHz to 2 GHz when the BWO was working at the RF power enhancement pressure region. The RF power enhancement appeared over a much wider pressure range in a high beam current case (10-100 mT for 3 kA) than in a lower beam case (80-115 mT for 1.6 kA). The plasma-filled BWO has higher power output than the vacuum BWO over a broader region of magnetic guide field strength. Trivelpiece-Gould modes (T-G modes) are observed with frequencies up to the background plasma frequency in a plasma-filled BWO. Mode competition between the T-G modes and the X-band Tm/sub 01/ mode prevailed when the background plasma density was below 6*10/sup 11/ cm/sup -3/. At a critical background plasma density of approximately=8*10/sup 11/ cm/sup -3/ power enhancement appeared in both X-band and the T-G modes. Power enhancement of the S-band in this mode collaboration region reached up to 8 dB. Electric fields measured by the Stark-effect method were as high as 34 kV/cm while the BWO power level was 80 MW. These electric fields lasted throughout the high-power microwave pulse.<<ETX>>

Explosive instability in a backward wave oscillator

A plasma filled backward wave oscillator supports Trivelpiece-Gould (TG) and TM modes. The former can be driven unstable by a relativistic electron beam via Cerenkov interaction or by the plasma return current as two stream instability. This unstable TG mode can parametrically couple to a TM mode via a negative energy beam mode giving rise to an explosive instability. >

Observation of Trivelpiece-Gould modes in a plasma-filled backward wave oscillator.

Trivelpiece-Gould modes (TG modes) have been observed with frequencies up to the background plasma frequency in a plasma-filled X-band backward wave oscillator. The TG mode emission was broadband with observed frequencies ranging from 2.6 GHz, the lower limit of our frequency diagnostic, to ${\mathit{f}}_{\mathit{p}}$, the plasma frequency. Mode competition between the TG modes and the X-band ${\mathrm{TM}}_{01}$ mode prevailed when the background plasma density was below 6\ifmmode\times\else\texttimes\fi{}${10}^{11}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$ (${\mathit{f}}_{\mathit{p}}$\ensuremath{\sim}7 GHz). At a critical background plasma density of ${\mathit{n}}_{\mathrm{cr}}$\ensuremath{\sim}8\ifmmode\times\else\texttimes\fi{}${10}^{11}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$ (${\mathit{f}}_{\mathit{p}}$\ensuremath{\sim}8 GHz) a sudden power enhancement appeared simultaneously in both X band and the TG modes. Power enhancements in S and J bands were up to 8 and 3 dB, respectively, with the X band enhanced up to a factor of 6.

Plasma-filled dielectric Cherenkov maser

The effect of a background plasma on a dielectric Cherenkov maser is examined by solving the linearized, beam-plasma, dielectric-lined waveguide dispersion relation. The results indicate that introduction of the background plasma can produce a higher spatial growth rate for the beam-waveguide instability and can reduce the electric field at the dielectric surface when compared to the system with no background plasma present. It is also found that for some sets of waveguide parameters, the TM/sub 0n/ electromagnetic modes can propagate frequencies that are below the background plasma frequency in the system. These modes have a finite off frequency and are different from the Trivelpiece-Gould modes for the system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High-gain plasma Cerenkov maser

The linearized fluid and Maxwell's equations are used to calculate the dispersion relation for the transverse magnetic modes of a partially filled, plasma-lined, cylindrical waveguide interacting with a relativistic electron beam. Both the plasma liner and the electron beam are assumed to be cold, and the system is immersed in an infinite axial magnetic guide field. The dispersion relation is then used to calculate the growth rate for the instability between the slow-wave modes (Trivelpiece-Gould modes) of the plasma guide and the slow-space charge mode of the electron beam. The calculation is done in the high-gain, strong coupling limit. >

Nonlinear Landau Electron Heating by Electrostatic Waves in an Electron Beam-Plasma System

Electron heating due to nonlinear Landau damping of electrostatic waves in an electron beam-plasma system is studied theoretically. The fast and slow space charge waves of an electron beam and the Trivelpiece-Gould modes interact nonlinearly with plasma or beam electrons and cause an effective electron heating and a beam relaxation which leads to a saturation of electron heating. For nonlinear interaction with plasma electrons, the virtual (beat) wave produced by the two electrostatic waves can heat plasma electrons by its electron cyclotron damping. On the other hand, for nonlinear interaction with beam electrons, the Trivelpiece-Gould mode enhanced by nonlinear Landau damping can heat plasma electrons by its electron Landau damping. For both cases, the conservation of total energy flux and the existence of the energy transport are proved quantitatively.

Rf stabilization of universal drift mode based on plasma-maser theory.

The condition for stabilizing the universal drift mode by superimposing an external high-frequency Trivelpiece-Gould mode is investigated based on a new mode-coupling process (plasma-maser effect). The energy up-conversion from the universal drift mode to the high-frequency rf field stabilizes the low-frequency drift mode. Under suitable conditions, the new method based on plasma-maser interaction is effective at a smaller relative magnitude of the rf field than the one based on a ponderomotive force in a narrow sense.

Evolution of electron hole and electron wave pulses in a single-ended magnetoplasma

Formation and propagation of the electron phase-space hole and electron wave pulse (Trivelpiece–Gould mode) are experimentally investigated in a single-ended plasma in the very first stage of double layer formation. As soon as the rarefaction pulse excited at a collector terminating the plasma arrives at a plasma source, the electron holes are formed and propagate toward the collector with a speed comparable to the electron thermal speed. The electron energy distribution function dips at the hole and becomes broader in the upper stream region of hole propagation. The compression pulse formed in front of the plasma source just after rarefaction pulse arrival does not have a direct effect on the hole excitation.

Beam–plasma instabilities and the beam–plasma discharge

Using a new electron gun, a number of measurements bearing on the generation of beam–plasma discharge (BPD) in WOMBAT (waves on magnetized beams and turbulence) [R. W. Boswell and P. J. Kellogg, Geophys. Res. Lett. 10, 565 (1983)] have been made. A beam–plasma discharge is an rf discharge in which the rf fields are provided by instabilities [W. D. Getty and L. D. Smullin, J. Appl. Phys. 34, 3421 (1963)]. The new gun has a narrower divergence angle than the old, and comparison of the BPD thresholds for the two guns verifies that the BPD ignition current is proportional to the cross-sectional area of the plasma. The high-frequency instabilities, precursors to the BPD, are identified with the two Trivelpiece–Gould modes [A. W. Trivelpiece and R. W. Gould, J. Appl. Phys. 30, 1784 (1959)]. Which frequency appears depends on the neutral pressure. The measured frequencies are not consistent with the simple interpretation of the lower frequency as a Cerenkov resonance with the low-Trivelpiece–Gould mode; it must be a cyclotron resonance. As is generally true in such beam–plasma interaction experiments, strong low-frequency waves appear at currents far below those necessary for BPD ignition. These low-frequency waves are shown to control the onset of the high-frequency precursors to the BPD. A mechanism for this control is suggested, which involves the conversion of a convective instability to an absolute one by trapping of the unstable waves in the density perturbations of the low-frequency waves. This process greatly reduces the current necessary for BPD ignition.

Beam-Generated Waves in a Large Plasma Chamber

Plasma waves generated in a large vacuum chamber (20 × 30 m) by an electron beam of energy 0.5 to 2 keV, are measured. Both pre-beam plasma discharge (pre-BPD) and beam plasma discharge (BPD) states are investigated. The measured waves fall roughly into three categories: 1) a low-frequency spectrum with f << fee, fp; 2) a whistler mode spectrum; and 3) a high-frequency spectrum. The extremely low frequency waves are shown to be surface waves on a (nonneutral) plasma column but a higher frequency part of the low-frequency spectrum was not thoroughly investigated. It is speculated that it also represents surface waves on a nonneutral column. Its measured properties are consistent with the lower hybrid drift instability. The whistler mode spectrum seems to be a Cerenkov resonance with the lower Trivelpiece-Gould mode. It does not seem to play an important role in modifying the particle distributions. The high frequency spectrum is, as suggested by Rowland et al. [21], a Cerenkov resonance with the upper Trivelpiece-Gould mode. It is responsible for the major perturbation of the beam energy in BPD.

Growing and Damping of Space Charge Waves of an Electron Beam due to Nonlinear Landau Damping

We have observed experimentally damping and growing of fast and slow space charge waves of an electron beam due to nonlinear Landau damping in an electron beam-plasma system by using spectrum analysis. The former with positive energy damps and the latter with negative energy grows by nonlinear interaction with the Trivelpiece-Gould mode and the electron beam. The lowest and higher modes of the Trivelpiece-Gould modes with respect to a radial wavelength grow by this nonlinear wave-particle interaction satisfying the resonant condition.

Electrostatic and electromagnetic eigenmodes near the electron and ion gyrofrequencies in a cylindrical plasma

We show that under a wide range of conditions easily achievable in the laboratory, the dispersion of the radial eigenmodes of electromagnetic whistler mode waves in a cylindrical magneto-plasma is dominated by the electron inertia and becomes synonymous with the electrostatic Trivelpiece-Gould modes. A similar situation exists close to, but just below, the ion gyrofrequency.

Numerical calculation of Trivelpiece-Gould modes in a torus with circular cross-section

The Trivelpiece-Gould modes in a quasielectrostatic approximation are calculated for a torus with circular cross-section. The results obtained with a Galerkin method are compared with those obtained for a torus with square cross-section. The lowest modes and eigenvalues are shown for aspect ratios varying from 2 to 5.

Numerical Study of Ion Heating Due to Nonlinear Landau Damping in an Ion Beam-Plasma System

Ion heating due to nonlinear Landau damping in an ion beam-plamsa system is studied numerically by solving a set of equations representing the distortion of ion velocity distribution by this instability, where fast or slow space charge wave of an ion beam decays into Trivelpiece-Gould mode by nonlinear interaction with plasma ions and causes ion heating. A nonlinear Landau damping of a usual type by fast space charge wave shows that the ion temperature grows almost exponentially in initial time and saturates. A nonlinear Landau damping of an explosive type by slow space charge wave with negative energy exhibit an explosion of ion temperature and amplitude of all related waves at an explosion time. Considering effect of ion heating which makes nonlinear wave-particle coupling coefficients reduced, it is found that the explosion time of ion heating by slow space charge wave becomes several times larger than the original explosion time.

Toroidal Trivelpiece-Gould modes

Electron plasma waves are treated in quasi-electrostatic approximation in a toroidal cavity of rectangular cross-section in an infinitely strong azimuthal magnetic field. The differential equation for the electrostatic potential, derived from fluid equations, can be separated using cylindrical coordinates. The eigenvalue problem for the radial dependence is solved numerically by a shooting method. Eigenvalues are given for different aspect ratios. Comparison with appropriate modes of the straight geometry shows that the toroidal frequencies generally lie some percent above those for the straight case. Plots of the eigenfunctions demonstrate clearly the influence of toroidicity. The deviation from symmetry (which should appear for straight geometry) depends not only on the aspect ratio, but also strongly on the mode numbers.