Drift Cyclotron Loss Cone Instability Research Articles

Electrostatic instabilities in a mirror trap revisited

The conditions for the stabilization of the Drift-Cyclotron Loss-Cone (DCLC) and Double-Humped (DH) microinstabilities in a mirror trap are critically revisited assuming the plasma is confined in the kinetic regime, which is characterized by an empty loss cone. The temperature of warm ions, necessary for stabilization of the DH instability, is calculated. The fraction of warm ions necessary to stabilize the DCLC instability at a given radial density gradient is calculated. Assuming the wavelength is much shorter than the Larmor radius, a simple criterion for the stability of drift-cyclotron loss-cone oscillations is derived whose accuracy is verified by comparison with the solution of the exact dispersion equation and with known experimental data obtained in the past decades in PR-6, 2XII, 2XIIB, TMX, and TMX-U devices for plasma confinement.

Effect of general loss-cone distribution function on electrostatic ion-cyclotron instability in an inhomogeneous plasma: particle aspect analysis

The electrostatic ion-cyclotron instability (EICI) in low β (ratio of plasma to magnetic pressure), anisotropic, inhomogeneous plasma is studied by investigating the trajectories of the particles using the general loss-cone distribution function (Dory-Guest-Harris type) for the plasma ions. In particular, the role of the loss-cone feature as determined by the loss-cone indices, in driving the drift-cyclotron loss-cone (DCLC) instability is analysed. It is found that for both long and short wavelength DCLC mode the loss-cone indices and the perpendicular thermal velocity affect the dispersion equation and the growth rate of the wave by virtue of their occurrence in the temperature anisotropy. The dispersion relation for the DCLC mode derived here using the particle aspect analysis approach and the general loss-cone distribution function considers the ion diamagnetic drift and also includes the effects of the parallel propagation and the ion temperature anisotropy. It is also found that the diamagnetic drift velocity due to the density gradient of the plasma ions in the presence of the general loss-cone distribution acts as a source of free energy for the wave and leads to the generation of the DCLC instability with enhanced growth rate. The particle aspect analysis approach used to study the EICI in inhomogeneous plasma gives a fairly good explanation for the particle energisation, wave emission by the wave–particle interaction and the results obtained using this particle aspect analysis approach are in agreement with the previous theoretical findings using the kinetic approach.

Theoretical modeling of drift cyclotron loss-cone instability mode structures

Measurements of the drift cyclotron loss-cone instability mode structure have recently been performed on MIX [Phys. Fluids 29, 3439 (1986)], an axisymmetric mirror machine characterized by a small width Maxwellian cold stream and a larger width loss-cone plasma. To model these measurements the Fourier-space integral equation was solved, derived from the linearized Vlasov equation for electrostatic perturbations, with both a matrix version technique and a WKB approximation. It is found that the mode is prohibited from the cold stream region by the process of mode conversion, which explains experimental results quite well.

Detailed measurements of anisotropic mirror plasma ion energy distributions during drift cyclotron loss-cone instability

Using improved computer filtering and a reconstruction analysis we have been able to extract additional details on a mirror plasma’s midplane ion energy distribution from the current–voltage (I–V) curve of a voltage-swept electrostatic end loss analyzer (ELA). For the University of Maryland’s MIX-1 plasma [Phys. Fluids 23, 3439 (1976)], the diagnostic technique has provided us with important information on the ‘‘loss-cone region’’ of an anisotropic ion distribution during drift cyclotron loss-cone (DCLC) instability experiments. Observations include convincing verification of ion distribution modification during DCLC evolution and an enhancement of the loss-region boundary over that predicted by simplified trapping theory. The reconstruction analysis involves matching the differentiated I–V curves (i.e., dI/dV vs V) to theoretically predicted curves based on model midplane ion distribution functions. The theoretical curves are obtained from the model midplane distributions by a mapping transformation based on conservation of magnetic moment and total energy of each ion along lines of magnetic flux.

The drift cyclotron loss-cone instabilities in an inhomogeneous magnetic field

The effects of kinetic pole on drift cyclotron loss-cone instability are studied in detail, especially for the dtri-B-drift resonance contribution. It is found that the kinetic resonance acts as dissipative effect which can enhance the instability and broaden the unstable region of negative energy DCLC mode.

THE KINETIC DRIFT CYCLOTRON LOSS CONE INSTABILITIES IN AN INHOMOGENEOUS MEGNETIC FIELD

The kinetic effects of magnetic drift resonance on the drift cyclotron loss cone instability (DCLC) are studied in detail both analytically and numerically. It is found that the effects of inhomogeneous magnetic field in high β plasmas are important, the kinetic resonance act as a dissipative effect, which can enhance the instability and broaden the unstable region of negative energy DCLC mode. Finite β effects do not play the stabilization role predicted by the theories under the fluid approximation.

Three-dimensional mode structure of the drift cyclotron loss-cone instability in a mirror trap

The three-dimensional mode structure of the drift cyclotron loss-cone (DCLC) instability is characterized in an axisymmetric mirror machine. It is shown, for this experiment, that (1) the azimuthal mode structure does not assume the geometry associated with an azimuthal normal mode φ̃=f (r,z)exp(imθ) with a single integral value of m, (2) the presence of a cool-ion plasma stream does not, in general, stabilize the mode globally, and (3) the axial mode characteristics have finite structure.

Experimental observation of bounce-resonance Landau damping in an axisymmetric mirror plasma.

Suppression of the drift-cyclotron loss-cone instability (DCLC) in an axisymmetric mirror plasma has been observed for frequencies in the vicinity of the bounce frequency of electrostatically trapped thermal electrons. The location and width of the frequency region of suppression is in good agreement with calculations of bounce-resonance Landau damping.

Spatial localization of bursts of the drift cyclotron loss cone instability

Spatial localization of bursts of the drift cyclotron loss cone instability in a mirror machine is caused by asymmetric motion of the background plasma. The dynamic behavior of the plasma determines the appearance of the instability

The dependence of the drift cyclotron loss cone instability on the radial density gradient

Large-amplitude fluctuations (eφrms/kTe≲2) observed near the ion cyclotron frequency and its harmonics in a magnetic mirror are identified as the drift-cyclotron loss-cone (DCLC) instability through measurements of the frequency, wavenumber, propagation direction, and dependence on the loss cone ion velocity distribution and the radial density gradient. The mode amplitude is reduced when the plasma density and radial density gradient length are near the linear stability boundary, dropping by two orders of magnitude when the density gradient scale length is increased from 2 to 100 ion gyroradii. The mode is localized radially to a region of one ion gyrodiameter width near the location of the steepest density gradient. The bursting appearance of mode amplitude is accompanied by changes in its perpendicular wavelength as the plasma parameters change with time during its decay. Mode amplitude modulation which occurs simultaneously throughout certain flux volumes indicates that the fluctuations arise from an absolute instability. While oscillations at the fundamental of the ion-cyclotron frequency occur most frequently with the largest amplitude, a dominant mode is observed near the third harmonic of the ion cyclotron frequency localized at a steep gradient near the axis. One possible interpretation is that the wavelengths of lower harmonic DCLC modes are too long to satisfy the azimuthal quantization condition.

Effect of electron beam injection on DCLC instabilities in mirror machines

A model for the effect of an electron beam on the drift cyclotron loss cone (DCLC) instabilities in mirror machines is presented. It is shown that because of the ponderomotive force of the beamgenerated plasmons, the growth rates of the DCLC instability are modified substantially. The model explains some of the observations on the Constance mirror machine where the DCLC instability was found to be quenched by the injection of an electron beam.

Nonlinear saturation of the drift cyclotron loss-cone instability. II. Comparison with PR-6 data

The single-mode time-asymptotic nonlinear saturation of the drift-cyclotron loss-cone instability is considered. In this model, there is a uniform magnetic field as well as mirror losses. The linear stability boundary, and the saturation levels of unstable oscillations and floating potential near this boundary, are determined. The theory is then compared with observations on the PR-6 mirror experiment. Good agreement is obtained using a simple energy model for the slow time variation of the equilibrium distribution function.

Stabilization of Drift-Cyclotron Loss-Cone Instability of Plasmas by High Frequency Field

The drift-cyclotron loss-cone (DCLC) instability is driven by both the loss-cone velosity distribution of ions and the density gradient. The instability is most dangerous for MHD stabilized mirror plasma confinements. We investigate the stabilization mechanism by the high frequency field, which suppresses the end loss. The Fokker-Planck equation is numerically computed under the boundary conditions for the end stopper effect. The time varying distribution function deduced is used to calculate the dispersion equation of DCLC instability. It is estimated that the minimum energy of h.f. field to stabilize is 60% of the ion thermal energy.

The effect of end-cell stability on the confinement of the central-cell plasma in TMX

In the Tandem Mirror Experiment (TMX), the central-cell losses provide the warm unconfined plasma necessary to stabilize the drift-cyclotron loss-cone instability in the end cells. This places a theoretical limit on central-cell confinement, which is expressed as a limit on the end-cell to central-cell density ratio. As this density ratio increases in a TMX experiment, large increases of end-cell ion-cyclotron-frequency plasma fluctuations are observed. These fluctuations cause the central-cell confinement to decrease, in agreement with a theoretical model.

Feedback stabilization of drift cyclotron loss cone instability by modulated electron sources

It is shown that the drift cyclotron loss cone instability can be suppressed by modulating electron density within the plasma. With the feedback in +90° phase the critical density gradient needed for the onset of the drift cyclotron loss cone instability increases approximately linearly with the gain. Typically with the gain of −50Ω i the critical density gradient can be pushed up by as much as two orders of magnitude and minimum mirror plasma radius can be brought down in the same proportion.

Interaction of lower hybrid wave fields with drift-cyclotron loss-cone instability

The dispersion relation for the drift‐cyclotron loss‐cone mode in the presence of the lower hybrid wave is calculated using both electrostatic and finite β models. It is found that lower hybrid wave fields with frequency ω0 can stabilize the mode if ωlh<ω0<ω+, or ω0<ω−<ωlh, where ωlh = ωpi/(1+ω2pe/ω2ce)1/2, ω± = [±A+(A2+4ω2lh)1/2]/2, and A = ω2lhe/ωcik. If the plasma β is greater than a critical value βc, there is another stabilization region, namely, ω0≳δωlh, where δ is a numerical constant. Even though the stabilization effect is small in this region, the lower hybrid wave frequency for electron heating should be in this region in order to avoid enhancing the particle loss rate.

Excitation of Flute-Type Negative-Energy Ion-Cyclotron Waves due to Ion Gradient-B Drifts in High-β Mirror-Confined Plasmas

An excitation of flute-type negative-energy ion-cyclotron waves by means of an ion gradient-β drift is analyzed on the basis of the linearized local dispersion relation. The excitation mechanism of these negative-energy waves is found to be a kind of an ion-cyclotron damping. This instability occurs even in axially homogeneous plasmas, and is still not stabilized even at a radial scale length large enough to stabilize the DCLC (drift-cyclotron loss-cone) instability. This instability is stabilized in the same way as the DCLC instability by an addition of a warm plasma.

Nonlinear saturation of the drift cyclotron loss-cone instability

The nonlinear behavior of the drift cyclotron loss-cone instability close to a stability boundary is considered. The time-asymptotic small-amplitude saturation of a single mode is determined by a variation of the Bogoliubov method. A collisionless plasma model is used with no mirror losses or particle sources. Flute-like local perturbations are considered. The stability boundary in density, density-gradient space is determined for several different ion loss-cone distributions. The modified equilibrium ion distribution function, the electric field, and the amplitude of the fluctuating electrostatic potential are then obtained as a function of the fractional increase in density gradient above the critical value. The nonlinear frequency shift remains indeterminate. In all cases, stable saturation occurs and is due to a slight filling-in of the loss cone. Comparison with numerical simulation results show good agreement when the high thermal level of neighboring modes is artificially suppressed.

Stabilization of Drift-Cyclotron Loss-Cone Instability by R.F. Field

The drift-cyclotron loss-cone (DCLC) instability is driven by both the loss-cone velocity distribution of ions and the density gradient.The instability is most dangerous for mirror confinements. We investigate the stabilization mechanism by the radio frequency field, which suppressed the end loss. The Fokker-Planck equation is numerically computed under the boundary conditions for the end stopper effect. The distribution function deduced is used to calculate the dispersion equation of DCLC instability. It is estimated that the minimum energy of r.f. field to stabilize the instability is 60% of the ion thermal energy.

Interaction of a high frequency magnetic field with the drift-cyclotron loss-cone mirror instability

The effect of a small amplitude, high frequency field on drift-cyclotron loss-cone instability is investigated. With B1/B0=5% and ωce/ω0=80, the critical characteristic length of plasma density gradient reduces dramatically and is comparable to the finite β stabilization effect.