Ira B. Bernstein

An Energy Principle for Hydromagnetic Stability Problems

The problem of the stability of static, highly conducting, fully ionized plasmas is investigated by means of an energy principle developed from one introduced by Lundquist. The derivation of the principle and the conditions under which it applies are given. The method is applied to find complete stability criteria for two types of equilibrium situations. The first concerns plasmas which are completely separated from the magnetic field by an interface. The second is the general axisymmetric system.

THEORY OF ELECTROSTATIC PROBES IN A LOW DENSITY PLASMA

The theory of sperical and cylindrical probes immersed in plasmas of such low density that collisions can be neglected is formulated. The appropriate Boltzmann equation is solved, yielding the particle density and flux as functionals of the electrostatic potential, the situation in the body of the plasma, and the properties of the probe. This information when inserted in Poissons equation serves to determine the potential, and hence the probe characteristic. No a priori separation into sheath and plasma regions is required. Though amenable to a determination of the full probe characteristic, the method is applied in detail and numerical results are presented only for the collection of monoenergetic ions, for the case of negligible electron current. These results indicate that the potential is not so insensitive to ion energy as has been believed, and that if the probe radius is sufficiently small, there enters the possibility of a class of ions which are trapped near the probe in troughs of the effective...

Quasi-linear theory of plasma waves

The quasi‐linear theory of plasma waves in a collision‐free plasma is formulated in such a way that corrections to the adiabatic assumption and those due to nonlinear effects can be readily estimated in terms of the properties of the initial spatially homogeneous part of the distribution function. Decaying Fourier components of the electric field are treated in a uniform way with unstable Fourier components. It is shown that the case of a three‐dimensional plasma is qualitatively different from a one‐dimensional plasma, the former leading to a decay to zero of the fluctuating electric field.

Runaway Electrons in an Ideal Lorentz Plasma

The phenomenon of electron runaway under a constant electric field is treated in the simplified model of a Lorentz gas, in which the electrons experience Coulomb collisions with infinitely massive ions only. This model permits a rigorous systematic analysis of the problem, devoid of arbitrary elements, in the limit of weak electric field. The analysis leads to a decomposition of velocity space into three regions. In the first of these, the low velocity domain, the electron distribution function is dominated by collisions and hence almost isotropic; it obeys a diffusion equation driven by the applied field. The second region, one of intermediate velocity, is characterized by quasi‐steady flow in velocity space, for which the low velocity region provides the source. Lastly there is a high velocity region, fed by the intermediate region, in which the electrons run away almost freely under the action of the electric field, with only an exceedingly weak diffusion due to collisions. The problem is solved explic...

Ion Wave Instabilities

The dispersion relation for electrostatic oscillations in a magnetic field is derived on the basis of the Boltzmann equation for arbitrary velocity distributions and for propagation in an arbitrary direction under the following restrictions: (1) The thermal velocities of the particles and the phase velocity of the wave are small compared to those of light; (2) the component of the wave vector k perpendicular to the magnetic field is small compared to the reciprocal of the gyration radius of an ion at the larger of the mean ion, and electron thermal, energies; (3) the magnetic field B is uniform. The dispersion relation is formally identical with that for electrostatic oscillations in the absence of a magnetic field. The dispersion relation is examined for stability under the further restrictions that: (4) The mean thermal energy of the ions is small compared to that of the electrons; (5) the electron distribution function for the component of the velocity v along B has a single maximum. It is found that a...

Condensed Presentation of Transport Coefficients in a Fully Ionized Plasma

A presentation of transport coefficients in a fully ionized plasma in terms of effective collision, gyration, and oscillation frequencies is given. It has the virtue of reducing the task of presenting the data to the tabulation of certain dimensionless functions of restricted variation. Moreover the quantities involved are directly related to the reciprocals of the coefficients usually considered, for instance to the resistivity rather than the conductivity.

On the Ionization and Ohmic Heating of a Helium Plasma

A method which is used for the ionization and heating of helium plasmas in various stellarator models consists of inducing an approximately constant electric field along the main axial confining magnetic field. In this paper we present the details and results of various calculations pertaining to this method. The gas is assumed initially to be 10% ionized and at a temperature of a few electron volts.The major approximations on which the calculations are based are (1) Maxwellian velocity distributions for the particles and (2) negligible charged particle loss across the confining magnetic field. The equations for the power balance and number balance were integrated in time numerically for a variety of initial conditions. The results indicate that temperatures of a few hundred electron volts are attainable within milliseconds with an applied field of the order 0.1 volt per cm at densities of 3 × 1013 particles per cm3. It was found that the maximum temperature is limited by power loss in bremsstrahlung radi...

Stability of a Current‐Carrying Plasma

The critical current is obtained for the onset of ion wave instability in the experimentally interesting case of a current‐carrying plasma with an electron distribution function given by the conductivity theory of Spitzer and Harm. The results are presented in a form suitable for comparison with experiment.

Effect of Charge Exchange on the Critical Current for the Onset of Ion‐Wave Instabilities

Recent experimental results on the afterglow in the stellarator indicate that upon passage of an electrical current through the plasma rapid diffusion sets in at a certain critical current. This critical current can be computed theoretically and is found to depend in significant fashion on the tail of the ion distribution function. This tail may well be non‐Maxwellian due to charge exchange with neutrals. The ion distribution function is computed by solving the Fokker‐Planck equation, and it is found that the tail may be suppressed if there are sufficient neutrals present. This distorted distribution is then used to predict a new critical current for ion‐wave instabilities. The calculations are presented in a form which admits of easy comparison with experiment.