Finite Orbit Effects Research Articles

Interchange, rotational, and ballooning stability of long-thin axisymmetric systems with finite-orbit effects

The stability of axisymmetric tandem mirror plasmas with respect to interchange, rotational, and ballooning modes is investigated in the paraxial approximation. The stabilizing effects of finite orbits, rigid energetic fast-drifting electrons, nearby conducting walls, and line-tying by a cold plasma halo are incorporated. Numerical calculations are performed to construct equilibria with finite plasma pressure and to determine linear stability by integrating a two-dimensional, initial-value equation. These numerical calculations support and extend analytical results. Analytical and numerical stability criteria are obtained.

The anisorrhopic guiding-center fluid

A study is made of so-called “finite-orbit effects” in a two-dimensional guiding-center plasma. The macroscopic mass motion of the plasma is represented on the basis of a simple incompressible one-fluid model (so-called “representative fluid”), and the guiding-center motions of single particles are then referred to a Lagrangian coordinate network comoving with the representative fluid. The fluid motion defines the network motion. It turns out, however, to have no effect on the guiding-center motion relative to the network (autonomy theorem). It is found, in other words, that the relative trajectories of guiding centers are determinable in advance independently of the network motion (or the fluid motion), and this provides the necessary information to determine all the state parameters of the representative fluid (density of mass, density of gyrational angular momentum, etc.) as functions of the time, t, at any given point of the network. Once this information is available, the fluid motion is then completely determined by the remaining hydrodynamic equations (equation of motion, equation of incompressibility). The so-called “finite-orbit effects” take the form of gyroscopic-quasielastic forces in the equation of motion. No special isorrhopy condition is assumed. (This refers to a special initial condition assumed in an earlier work, for the sake of analytical simplicity. Here, the special initial condition is dropped.) Much attention is devoted to problems of wave propagation and stability. There are two independent sets of wave modes (if a nonvanishing anisorrhopy is allowed): so-called fluid modes, and so-called drift modes, respectively defined as first-order perturbations in the network motion (or the fluid motion) relative to the fixed coordinate frame, and in the guiding-center motion relative to the network. The stability conditions against both sets of modes are found to be quite stringent, much more so than in the earlier isorrhopic case. Nonetheless, a reasonably extensive class of stable solutions is shown to exist.

Stability of the field-reversed mirror

The stability of a field-reversed mirror plasma configuration is studied with an energyprinciple derived from the Vlasov equation. Because of finite orbit effects, the stability properties of a fieldreversedmirror are different from the stability properties of similar magnetohydrodynamic equilibria. TheVlasov energy principle developed here is applied to a computer simulation of an axisymmetric field-reversedmirror state. Possible instabilities with ℓ = 1, 2 are found at low growth rate. A simple physical model for thisinstability is outlined.

Rotational instability in a linear theta pinch

The m=1 ’’wobble’’ instability of the plasma column in a 5-m linear theta pinch has been studied using an axial array of orthogonally viewing position detectors to resolve the wavelength and frequency of the column motion. The experimental results are compared with recent theoretical predictions that include finite Larmor orbit effects. The frequency and wavelength characteristics at saturation agree with the predicted dispersion relation for a plasma rotating faster than the diamagnetic drift speed. Measurements of the magnetic fields at the ends of the pinch establish the existence of currents flowing in such a way that they short out the radial electric fields in the plasma column. The magnitude of rotation, the observed delay in the onset of m=1 motion, and the magnitude of end-shorting currents can all be understood in terms of the torsional Alfvén waves that communicate to the central plasma column the information that the ends have been shorted. The same waves are responsible for the torque which rotates the plasma and leads to the observed m=1 instability. Observations of the plasma in the presence of solid end plugs indicate a stabilization of high-m number modes and a reduction of the m=1 amplitude.

Beam shear-Alfven instability in two-component tokamak

The authors consider from a theoretical standpoint the excitation of shear-Alfven waves in a tokamak in which the plasma contains a fast ion group. It is assumed that the fast ions are formed as a result of fast neutral atoms injected into the tokamak. An expression is derived for the growth rate of the shear-Alfven oscillations, with an arbitrary relationship between the transverse wave length and the dimension of the Larmor and drift orbits. The authors analyse the part played by the finite orbit effect in the case of a longitudinal beam with a small Maxwellian velocity spread and an arbitrary ratio between the directed velocityV0 and the Alfven velocitycA. It is demonstrated that at a finite ratioV0/cA two physically different types of oscillations may build up, corresponding to the two signs of the ratioΩ/k¦ (Ω is the oscillation frequency,k¦ is the longitudinal wave number). The growth rate for perturbations with Ω/k¦ 0.

Stabilization of interchange modes in multiple-mirror confined plasmas

Interchange modes were investigated experimentally in a multiple-mirror field in which the field line curvature was adjustable both by varying the mirror ratio and by adding linked-quadrupole fields. Experiments were performed both with a thermal lithium plasma and a hot hydrogen plasma. The m=1 flute was observed growing at a rate which decreased strongly with decreasing magnetic field curvature. Higher order modes were observed for lower values of field curvature. These modes, which were calculated to be stabilized by finite Larmor orbit effects in the absence of line-tying, were found to be driven unstable by resistive effects from the line-tying, in qualitative agreement with observations. The quadrupole fields were observed to have a stabilizing effect on the higher order modes.

The role of finite–orbit effects in the theory of short–wavelength non-potential oscillations in a tokamak plasma

The paper investigates the non-potential (non-electrostatic) perturbations of a plasma contained in a toroidal magnetic trap of the tokamak type. The transverse wavelengths are assumed to be small in comparison with the ion Larmor radius. It considers the case of perturbations considerably extended along the lines of force of a magnetic field k‖qR < 1 (where k‖ is the longitudinal wave number, q the tokamak safety factor and R the radius of curvature of the magnetic axis). It is shown that in this case the dispersion relation for short-wavelength non-potential perturbations is affected substantially by the effects of the finite orbits of the transit ions. The dispersion relation for such perturbations is obtained; it differs substantially from that for similar perturbations in a straight magnetic field. It is pointed out that these perturbations can grow as a result of electron dissipation.

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.

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.