H. L. Berk

Phase Space Hydrodynamics of Equivalent Nonlinear Systems: Experimental and Computational Observations

The one‐dimensional Vlasov equation describes the behavior of an incompressible self‐interacting classical fluid which moves in the (q, p) phase plane. This type of phase fluid occurs in many physical problems and its hydrodynamic properties can be examined from a general point of view. A characteristic feature with initially unstable spatially homogeneous configurations is the development of stable nonlinear phase structures. Such examples occur as the result of the gravitational Jeans instability, or the two‐stream and negative‐mass instabilities of charged‐particle beams. These structures can be related to one another by extending a duality principle due to Dory. The stable cavities in phase space which have been observed in numerical calculations on the two‐stream instability are compared with stable proton clusters which develop from the negative‐mass instability in the mirror experiment DCX‐1.

PLASMA WAVE REGENERATION IN INHOMOGENEOUS MEDIA

The one‐dimensional problem of electrons confined in a quadratic potential is solved in the WKB approximation. The unique feature of the problem is that, since particle bounce time is independent of energy, resonant particles that normally cause Landau damping periodically interact with the wave and regenerate all phase information. Because of the efficiency of the regeneration mechanism, the dielectric properties governing the system change and give rise to “regenerative modes” associated with harmonics at twice the bounce frequency. The eigenvalues calculated in this WKB approximation are in agreement with previous numerical calculations.

Stability of hot electron plasmas

The linear theory of a flute mode in a hot electron plasma is presented when the precessional drift frequency is comparable to or larger than the ion cyclotron frequency. The mode is found to stabilize when a sufficiently large component of cold electrons is present.

Stability of a Trapped Particle Equilibrium

A simple periodic Bernstein‐Greene‐Kruskal equilibrium is constructed, in which trapped electrons make a small contribution to the reactive part of the one‐wave dispersion relation. This contribution is found to have an important effect on stability of the wave against decay into sidebands. Enhanced growth rates and frequency shifts are calculated.

Finite Beta Stabilization of the Drift‐Cone Instability

An analytic expression giving the drift‐cone stability criterion of a finite‐beta plasma is derived using a mirror‐ratio‐dependent loss‐cone ion distribution. This is used to determine a beta limit for stability which is found to be insensitive to changes in mirror ratio and plasma temperature.

Stability of interchange modes in a hot electron plasma

A self‐consistent electromagnetic theory is presented, for arbitrary β, of the flute‐interchange mode of a hot‐electron plasma where the hot‐electron guiding‐center drift frequency may be comparable to or larger than the ion‐cyclotron frequency. In the presence of bad curvature, an interchange instability exists at small and large precessional frequency, and this instability is stabilized by cold electrons. Electromagnetic theory introduces finite‐β corrections that only modify the results of a simple electrostatic theory that includes only the external drifts.

Quasi-linear transport model for mirror machines

A quasi‐linear model to describe the evolution of a mirror plasma is presented. It is shown that, as in the 2XIIB mirror experiment, a plasma stream sufficiently reduces rf turbulence so that a high density steady state mirror plasma can be sustained by a neutral beam. The scaling law for this plasma configuration is derived. The energy lifetime is found to be the Spitzer electron‐ion drag time even when electron thermal conduction to the external plasma can be neglected.

Analytic field‐reversed equilibria

Several families of analytic equilibria for field‐reversed plasma are obtained for arbitrary internal pressure and toroidal field.

Plasma return currents in a magnetic field

Whereas plasma return currents along a magnetic field arise for an electromagnetic diffusion time, it is found that appreciable plasma return currents perpendicular to a magnetic field are not induced for built‐up times slow compared with an Alfven transit time.

Critical Length Criteria for Mirror Machines

Critical scale lengths for stability of mirror devices are determined. The analysis proceeds via a modified WKB technique which adds the effect of wave amplification due to the presence of singularities of the dielectric function. The approximate answers so determined compare favorably with the numerical solution of the appropriate equations. Results for large and small mirror ratios are presented. It is found that irrespective of the character of the wave in the infinite medium limit (e.g., drift‐cone or stable negative energy wave) stabilization is effected at roughly the same magnetic scale length for all but the longest axial wavelength modes.