William A. Newcomb

Hydromagnetic stability of a diffuse linear pinch

Abstract The hydromagnetic energy principle is applied to the derivation of necessary and sufficient conditions for the hydromagnetic stability of a linear pinch with distributed plasma current (a diffuse linear pinch). The results are quite general in that the axial and azimuthal components of the magnetic field, which determine the structure of the pinch completely, are treated as arbitrary functions of distance from the axis. For purposes of illustration, the general results are applied to the limiting case of a pinch with the plasma current confined to an infinitely thin layer (a sharp pinch).

Convective Instability Induced by Gravity in a Plasma with a Frozen-In Magnetic Field

The convective instability induced by gravity in a compressible fluid layer is investigated in the special case of a plasma with a frozen‐in magnetic field B. The necessary and sufficient condition for stability, which is here derived from the hydromagnetic energy principle, is that the density gradient should exceed a certain critical value that is independent of B. Thus the rigidity given to the plasma by the frozen‐in field does not suffice to remove the instability but only to slow it down. The growth rates of the unstable displacements are calculated by means of a normal mode analysis and are shown to be inversely proportional to B when B is sufficiently large.

Magnetic Differential Equations

A necessary and sufficient condition is derived for a magnetic differential equation B·▿r = 0 to have a single‐valued solution r, where B is the field of a magnetohydrostatic equilibrium state or, more generally, and field with a system of toroidal magnetic surfaces.

Analytic equilibria with quadrupole symmetry in the paraxial limit

Mirror equilibria for arbitrary mirror ratio and flux‐tube eccentricity are obtained to leading order in the plasma pressure (beta expansion) in the paraxial limit (axial scale lengths long compared with radial scale lengths). The solutions are given in terms of quadratures over known functions. The theory is applied to a tandem‐mirror configuration.

Warm relativistic electron fluid

Equations of motion are derived for a warm relativistic collisionless electron fluid. In particular, the appropriate modification of the adiabatic equation of state is given. Use of the modified equation of state is illustrated by a calculation of the thermal correction to the zero‐temperature electron susceptibility tensor.

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.

Local instability near the vortex point of a field‐reversed mirror

A study is made of the curvature‐driven magnetohydrodynamic instability in the neighborhood of the vortex point of a simple field‐reversed magnetic‐mirror system, with a purely poloidal magnetic field. The linearized magnetohydrodynamic equation of motion for localized modes in that neighborhood is found to be completely solvable. Two independent sets of modes are found, axial and radial, with the same spectrum of eigenfrequencies. Each set of modes, axial and radial, includes exactly one exponentially growing mode, of which the growth rate is found to be Ω, where 2πΩ−1 is the time required for one complete traversal of a closed magnetic flux line by an Alfven‐wave signal.

Gyroscopic-quasielastic fluid systems☆

Abstract A simple model of gyroscopic-quasielastic fluid behavior in two dimensions is developed in general and is applied, in particular, to a guiding-center plasma under so-called “finite-orbit” conditions. Gyroscopic effects are associated (more or less directly) with a finite angular momentum of gyration, quasielastic effects (more or less indirectly) with differential drifts of guiding centers arising from gradients in the magnetic field. A systematic study is made of transformation properties under so-called “changes of representation”: changes in the definition of what is meant by the “same” fluid element at two successive times, with consequent redefinition of the fluid trajectories. (The requisite condition of invariance under exchange of equivalent elements is presupposed.) It is found that the gyroscopic and quasielastic terms transform covariantly under a change of representation. In particular, by a special choice of the representation, either term can be made to vanish identically.

Compressibility effect on instability growth rates

For an arbitrary magnetostatic equilibrium, instability growth rates are shown to be an increasing function of compressibility.

Magnetohydrodynamic wave drag

A calculation is made of the energy loss by emission of shear‐Alfven and slow magnetoacoustic waves by a perfectly conducting solid spherical projectile moving with uniform velocity through an ideal magnetoplasma.