Trapped Particle Instability Research Articles

On the Theory of Trapped Particle Instability. II. Stability of a Small-Amplitude B-G-K Wave

In a preceding report, (J. Phys. Soc. Japan 30 (1971) 1722) a general theory is developed to investigate the stability of a collisionless, one-dimensional plasma which sustains a large-amplitude, monochromatic electron plasma wave. In this paper, this theory is used to investigate further details of the instability by assuming a specific from of the unperturbed distribution, such that it is consistent with the small-amplitude limit of a Bernstein-Greene-Kruskal distribution. The properties of the dispersion relation are characterised by three parameters, the wavenumber of the carrier, its amplitude and the excess trapped-electron density Δ N t (compared with the value expected from the linear theory). In the weak-coupling limit, stable and unstable regions are shown in the plane spanned by Δ N t and bounce frequency. A numerical calculation for a moderately strong-coupling case shows that even in this case growing waves are separated into several distinct wavenumber regions.

Trapped-Ion Instabilities in Ion-Acoustic Wave

Growth of two types of sideband waves has been observed experimentally when a relatively large-amplitude ion-acoustic wave propagates in a collisionless plasma having the electron-to-ion temperature ratio Te/Ti≈20. The observed frequency spectrum and growth rate, as well as the frequency and amplitude ranges of the large-amplitude wave for which the sideband wave growth is observed are in good agreement with the theory of trapped particle instabilities. The sidebands extract the energy from the large-amplitude wave and cause it to be heavily damped.

Localization of Trapped-Particle Instabilities

By considering the evolution of an initially localized disturbance, we show that collisionless trapped-particle modes in a Tokamak geometry remain highly localized in the radial direction up to the point of nonlinear saturation. The inferred localization dimension of only a few "banana" widths is considerably less than the normal-mode picture has indicated. We discuss the implications of our results on estimates of the turbulent diffusion coefficient.

Nonlinear Saturation of the Collisional Trapped-Particle Instability

The detrapping effects of fluctuating electrostatic potentials on trapped particles in a Tokamak geometry are investigated. It is shown that nonlinear detrapping saturates the collisional trapped-particle instability at a fluctuating potential $\ensuremath{\phi}$ such that $e\ensuremath{\phi}\ensuremath{\ll}\ensuremath{\epsilon}T$, where $\ensuremath{\epsilon}$ is the inverse aspect ratio.

Nonlocal Theory of Trapped Particle Instability

A nonlocal theory for the low-frequency trapped particle instability in toroidal systems has been developed in a model field. The stability of the trapped particle mode is studied by using normal mode analyses with radial boundary conditions. In the absence of magnetic shear, the finite-orbit effect is shown to be incapable of stabilizing the mode; large magnetic shear is needed for stabilization. The ratio of the localization width of the mode to the characteristic density length is of the order of (Λ/r;)1/2ε1/4 (Λ is the charactistic “banana” size, r is the characteristic density length, and ε is the aspect ratio). An estimate of the nonlinear diffusion coefficient for the mode is given.

Effect of Detrapping on Trapped-Particle Instabilities in Tokamaks

The effect of weak detrapping of magnetically trapped particles on low-frequency, trapped-particle modes in sheared magnetic configurations is considered; such detrapping can be imposed by oscillation of the magnetic configuration, e.g., by major-radius compression. Detrapping has a dynamical effect analogous to collisions. In particular, it is shown that trapped-particle instabilities in tokamaks can be dynamically stabilized by weak oscillation of the aspect ratio with frequency Ω and relative amplitude λ, thus imitating ion collisions with ion detrapping. In the intermediate regime of the collisional trappedparticle instability, where νi/e≪ω∼ε1/2ω*≪νe/ε, stabilization is found if ω≪Ω<λνe/ε, where ε = r/R. The stabilizing effect (surprisingly of order λ) arises from oscillation of the field strength near its maximum, where detrapping occurs. These results are for simple models of the geometry.

Finite-β Stabilization of the Collisionless Trapped Particle Instability

A particularly dangerous mode of instability in confined plasmas is the trapped particle mode which is highly insensitive to magnetic shear and average favorable curvature. The study of the collisionless trapped particle mode is extended to include the effects of finite β = 8πnT / B2. It is found that the “magnetic well” dug by plasma diamagnetism is highly effective in stabilizing the collisionless trapped-particle mode. It is found that for magnetohydrodynamically stable systems the collisionless trapped-particle mode becomes stable when the diamagnetic well is sufficient to give all particles average favorable drift. It would appear from this result that, in general, the diamagnetic well should be an effective stabilizing agent against all nonflutelike microinstabilities.

Radial Dependence of Trapped-Particle Instabilities

The radial localization of trapped-particle instabilities is studied for a peaked diamagnetic-frequency profile and for a realistic profile. In the later case it is shown that modes can be excited only with a high radial wave number. The correspondingly reduced coefficient of turbulent diffusion is computed for the collisionless instability.

On the Theory of Trapped Particle Instability. I. General Formulation and Analysis for Weak Coupling Limit

A general formulation is developed to investigate the stability of a collisionless, one-dimensional plasma which sustains a large-amplitude, monochromatic electron plasma wave. A nonlinear dispersion relation is derived which governs the stability of a small-amplitude test wave. It is shown that untrapped electrons can produce a significant modification to the Kruer-Dawson-Sudan dispersion relation which was derived by considering only the effect of trapped electrons on the test wave. As an application a weak-coupling limit is considered. It is shown that in this limit waves can grow in the distinct wavenumber regions given by \(|\varDelta k|{=}\sqrt{2n+1}k_{B}(n{=}0,1,2,\cdots)\), and | Δ k |≪ k B where Δ k is the difference of the wavenumbers of the test wave and the original large-amplitude wave and k B the bounce frequency divided by the phase velocity of the large-amplitude wave. Various characteristics of the growing waves are obtained.

Stabilizing effect of a high-frequency magnetic field on the trapped-particle instability

The stabilizing effect which an oscillating magnetic field of frequency Ω perpendicular to the main toroidal field has on the collisionless trapped-particle instability is studied for frequencies in the range ωbe < Ω < Ωi, where ωbe is the electron bounce frequency and Ωi is the ion cyclotron frequency. It is found that the fastest-growing modes are stabilized when ε3/8 k⊥ v̄eB1/ΩB0 > 1, where k⊥ the wave number perpendicular to the main field B⃗, v̄e is the electron thermal velocity, B1 is the amplitude of the oscillating field, and ε−1= R0/r, the aspect ratio. The extension of the calculation to other modes is discussed, and an additional example is given.

Sideband Instability

Computer experiments on one-dimensional sheet plasmas show that large amplitude electron plasma oscillations with trapped particles can be unstable. This behavior is consistent with both a recent experiment and a recent theory predicting a trapped particle instability.

Electro magnetic trapped particle modes in multipoles

We discuss the effect of a finite plasma pressure on the trapped particle instability and we show that the critical β may be more severe than the usual ballooning limit.

Stabilization of the trapped particle mode by a cool plasma component

Abstract The stabilizing effect of a relatively cool plasma component on the trapped particle instability is investigated. For Tokamak parameters with two temperature components (Th ≈ 1 keV, Tc ≈ 100 eV) of comparable density, the trapped particle mode is effectively stabilized.

Trapped particle instabilities in toroidal systems

The paper is a continuation of the study of collisionless instabilities due to particle trapping. Particular attention is paid to the effects which promote stabilization of this instability. The author considers in detail the influence of: 1. unequal ion and electron temperatures; 2. finite orbits; 3. circulating particles; 4. temperature gradient. It is found that it is extremely difficult to achieve complete stabilization of ‘trapped-particle’ instabilities. However, the various stabilizing factors affect the instability most strongly when the particles have different temperatures

Stabilization of trapped particle instability by finite orbit effect

A stabilization criterion is obtained for trapped particle instability. It occurs due to periodic deviation of the particle guiding centres from the magnetic surfaces.

Comments on “Influence of Static, Radial Electric Fields on Trapped Particle Instabilities in Toroidal Systems”

First Page

Energy Principle for Plasma Stability against Low-Frequency Interchanges

An energy principle is constructed for the stability of a plasma equilibrium F ≡ F(μ, J, K) against low-frequency interchanges, including trapped particle instabilities.

Influence of Static, Radial, Electric Fields on Trapped Particle Instabilities in Toroidal Systems

The drift motion of trapped particles in toroidal systems leads to a new branch of gravitational instability. This is stabilized by radial electric fields. However, a new unstable mode then appears, which is similar to that due to ion impurity density gradients.