Flute Oscillations in Bounded Plasmas and the Effect of Incoherently

Flute oscillations in bounded, non-uniform plasmas are considered from the guiding centre point of view. The effects of field curvature are replaced by generalized forces that are proportional to the parallel and perpendicular components of the particle energy. A perturbation analysis, covering all density regimes and including the finite gyro-radius effects, is carried out in cylindrical co-ordinates. It is shown that the final equation can always be solved by a variational method. Explicit solutions in terms of confluent hypergeometric functions are also presented for a special class of density distributions. For a monoenergetic plasma, in which all ions precess coherently, there can be stable regions at both very low and very high densities. If the ions precess incoherently, due to a spread in the energy, then a new over-stability emerges in the above stable regions. This over-stability, whose growth rate is very small compared to its oscillation frequency, is shown to be in agreement with the observations made in neutral injection experiments. 1. I N T R O D U C T I O N PLASMAS in a simple mirror magnetic field are known to be subject to flute or interchange instability. Simple hydromagnetic theory predicts a growth rate of y’[mkT/ (mirR,)] for an isotropic, cylindrical plasma with a sharply defined vacuum-plasma interface, where m is the azimuthal mode number, and RCl/r is the radius of curvature of the field lines. Various experiments, however, show that this kind of catastrophic instability does not always occur in a mirror machine. There have been several theoretical explanations of the discrepancy. Among them, an important one was presented by ROSENBLUTH et al. (1962) attributing the stabilization of flutes to finite gyro-radius effects. ROSENBLUTH et al. however, considered only plasmas of very high densities (min/c0B2 > 1) which, unfortunately, are not often realized in the laboratory. A different analysis, valid for all density regimes, was subsequently published by MIKHAILOVSKII (1963) for an unbounded planar plasma. A similar result was also obtained by POST (1963) through a simple derivation based on the continuity equation and the guiding centre drift equation with finite gyro-radius corrections. The same approach was employed by DAMM et al. (1963; 1965) and KUO et al. (1964) in their analyses of flute oscillations in bounded, low density (min/coB2 < 1) plasmas in connexion with neutral injection experiments. Excellent agreement was obtained between theory and experiments as regards the limits of stability and the frequency of oscillations. All these papers, including more general treatments by ROSENBLUTH and SIMON (1965) and KRALL (1965) use an artificial gravitational field to simulate the effects of magnetic field curvatures. In such a gravitational model, all ions at a given point in the system have exactly the same gravitational drift velocity irrespective of the particle energy, Whereas, in a true mirror machine, ions with different energies precess around the axis at different velocities. It appears, therefore, that the adequacy of the gravitational model is questionable, unless the plasma has monoenergetic ions. * The research reported herein was supported wholly by the U.S. Atomic Energy Commission under Contract AT(l1-1)-34, Project 128. 215