Laurence S. Hall

Three‐dimensional equilibrium of the anisotropic, finite‐pressure guiding‐center plasma: Theory of the magnetic plasma

Theoretical and numerical methods now give a complete solution to the problem of finite‐β plasma equilibrium in mirror magnetic wells and toroidal devices. The equilibria can be made consistent on all of the progressively longer time scales of the guiding‐center fluid model, including the particle magnetic drifts and the Coulomb scattering equilibrium of a neutral injected plasma. The theory of equilibrium in the guiding‐center fluid model of a finite‐β plasma with an arbitrary, anisotropic pressure tensor can be formulated as a classical magnetostatic system: ∇⋅B = 0, ∇×H = 0, B = H + 4πM(B). The plasma magnetization is found explicitly in terms of three physically distinct components related to the laws of conservation of magnetic moment, of longitudinal invariant, and of the sign of the velocity along B of particles that do not undergo mirror reflection. A condition is derived upon the field geometry whereby a large class of special equilibria can be found in which all particles on a given line drift o...

A theory of exact and approximate configurational invariants

Abstract A theory of integrable Hamiltonian systems in two dimensions is formulated and applied. The four-dimensional phase-space problem that is the vanishing of the Poisson bracket between another invariant and the Hamiltonian is here reduced to the solution of a series of two-dimensional configuration-space equations. (The theory is also applicable to more than two dimensions.) The constraints are found which admit to integrability of the orbits for magnetic or Coriolis forces as well as for forces derivable from a potential. When a system admits a given invariant, the invariant is found - for example, by quadrature. A number of examples including known and apparently previously unknown invariants are given. The theory of exact integrals of the motion also can be extended to the derivation of approximate invariants. The orbital structure of integrable or approximately integrable systems correlates with the degree (maximum power of the velocity) of a standardized invariant. The theory admits a variational principle, among other approximation techniques, for the computation of a “best” approximate invariant. The problem of the general cubic potential with one symmetric coordinate, V = 1 2 Ax 2 + 1 2 By 2 + Cx 2 y + 1 3 Dy 3 , (of which the well-studied Henon-Heiles potential is the special case for A = B and C = −D) is examined in detail.

Angular distribution function from axial density measurements in a mirror confinement experiment

The angular distribution of particle velocities in a magnetized plasma can be uniquely determined by measurement of the variation of density along a magnetic field line. From measurements on the 2XII mirror experiment, it is determined that the angular distribution of ions is monotonic and very closely approximates sin2aθ, where a = 2.25 and θ is the pitch angle at the center. Particles with sin θ < 0.72 communicate with the region outside the mirrors and comprise approximately 5% of the total density at the center.

Interaction Models, Negative Energy Waves, and Electrostatic Instabilities

It has been possible to introduce certain physical interpretations for the various contributions which a given species may make to the total dispersion relation for electrostatic instabilities of an anisotropic magnetized plasma. These interpretations are helpful in developing ones physical intuition regarding such instabilities. One appropriate specialization is the use of a physical model of the interaction of a distribution of particles with an imposed wave for the case in which the wave is set up primarily by a species other than that which drives the instability. Conversely, when the cooperative nature of the driver species is dominant, it is natural to describe its properties in terms of negative energy waves. The discussion shows the connection between these viewpoints as special cases of our more general description.

HIGHLY IONIZED, STEADY-STATE PLASMA SYSTEM

A system is described which produces a steady‐state helium plasma with greater than 95% ionization. The primary plasma is generated in a modified Phillips ion gauge discharge and diffuses along a magnetic field (B ∼ 1 kG) through a region in which neutral particles are preferentially removed. In the downstream region a plasma density of ∼2 × 1013 cm−3 and ion temperatures up to 10 eV have been achieved. The present paper describes the over‐all physical assembly, the operating characteristics of the system, and the nature of the plasma that is produced. In addition, a discussion of processes believed important in its operation is given.

Two‐Streaming Cyclotron Instability

A new electrostatic instability of counterstreaming ions occurring at cyclotron frequency is exhibited and discussed. It is conjectured that this instability may provide the mechanism for the enhanced trapping of the injected ion beam in DCX‐II. Stability criteria are developed and in the limit of high anisotropy of the ions and not‐too‐cold electrons, the criterion becomes approximately 12M ivs2<κTe/4 (unstable), where Mi is the ion mass, ±vs are the counterstreaming velocities, and Te is the electron temperature.

MAGNETOSTATIC EQUILIBRIA OF FINITE-PRESSURE MINIMUM-B PLASMA CONFIGURATIONS.

Magnetostatic equilibria for a broad class of plasma conditions are shown to correspond to solutions of the equations ∇·B = 0, ∇× H =0, where B = μ H. Magnetohydrodynamic stability then corresponds to satisfaction of the criteria H>0 and dH/dB>0, where B = |B | and H =μ−1B. Equilibria are studied for a variety of plasma distributions taken as functions of total energy and magnetic moment and in the most general cases, the coordinates of a line of magnetic flux. A new result is the determination of a relationship between well depth and the maximum pressure for magnetohydrodynamic stability which puts an upper limit on the well depth to achieve given pressure when the value of the maximum field strength is fixed. For so‐called collisional distributions, if p is the total pressure at the center and B0 is the maximum magnetic field, β = 8πp/B02<0.2, and the mirror ratio to achieve maximum β may not exceed 1.8. More general equilibria, where the pressures are functions of field lines as well, cannot improve ...

Thermoelectric Effects in a Joule‐Heated Plasma

The temperature and potential distributions in a fully ionized, Joule‐heated plasma have been computed for a one‐dimensional system in which the thermoelectric terms have been taken into account, and a comparison is made with the case in which the cross terms are neglected. It is found that the thermoelectric terms cause the temperature distribution to be significantly skewed. Moreover, if T0 is the maximum temperature, it is found that these terms cause the potential to drop sharply near the anode, in the sense opposing the current flow, by from 15 to 35% of kT0/e.

PLASMA MAGNETIZATION AND THE DETERMINATION OF HYDROMAGNETIC EQUILIBRIUM.

Equilibrium of a plasma and a magnetic field is determined through a plasma magnetization M with both local and nonlocal contributions. The equations of equilibrium are thus posed in soluble form, ∇ × H = 0 and ∇·B = 0 with B = H+4πM, possible previously only when pressures depended on | B | alone.

Bessel function J ν ( z ) of complex order and its zeros

Methods are developed for the computation of the complex zeros of (½ z ) −ν J ν ( z ) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the J v ( z ), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of J ν ( z ) for the rapid computation of Bessel functions of complex order.