Allan N. Kaufman

Quasilinear Diffusion of an Axisymmetric Toroidal Plasma

In a toroidal plasma with axial symmetry, the three adiabatically invariant actions of a particle are the magnetic moment, the canonical angular momentum, and the toroidal flux enclosed by the drift surface. Resonant interactions between particles and the normal modes of collective oscillation produce mode growth or decay and random changes in the actions. This random walk is represented by a diffusion equation in action space. Both the diffusion tensor and the growth rate depend upon a coupling coefficient which represents the work done by a normal‐mode field eigenfunction on the current density of an unperturbed particle orbit. The diffusion of the plasma causes adiabatic changes in the electric and magnetic self‐consistent fields. Accordingly, energy is not conserved, but is exchanged with external currents.

Dissipative hamiltonian systems: A unifying principle

Abstract The concept of hamiltonian sysem is generalized to include a wide class of dissipative processes. Evolution of any observable is generated jointly by a hamiltonian, with an entropy-conserving Poisson bracket, and an entropy, with an energy-conserving dissipative bracket. This approach yields many of the standard kinetic equations, such as those representing particle collisions, three-wave interactions, and wave-particle resonances.

Ponderomotive effects in collisionless plasma: A Lie transform approach

A new method for the kinetic analysis of ponderomotive effects in collisionless plasma is presented. This method involves the application of the Lie‐transform perturbation technique to the Hamiltonian formulation of the Vlasov equation. Basically, a new system, in which the high frequency oscillations are absent, is found. In this system the distribution function evolves according to a ponderomotive Hamiltonian, which is the kinetic generalization of the ponderomotive potential. It is shown that the ponderomotive Hamiltonian can easily be determined from the well‐known linear susceptibility. This formalism is used to calculate several new results. Among these results are the general formula for the quasi‐static density perturbation produced by a hot magnetoplasma wave, a generalization of previous formulas for the laser‐generated quasi‐static magnetic field, and the general formula for the ponderomotive gyrofrequency shift produced by an electromagnetic wave propagating at an arbitrary angle.

Stochastic acceleration by an obliquely propagating wave‐An example of overlapping resonances

A simple problem exhibiting intrinsic stochasticity is treated: the motion of a charged particle in a uniform magnetic field and a single plane wave. Detailed studies of this wave‐particle interaction show the following features. An electrostatic wave propagating obliquely to the magnetic field causes stochastic motion if the wave amplitude exceeds a certain threshold. The overlap of cyclotron resonances then destroys a constant of the motion, allowing appreciable momentum transfer to the particles. A wave of large enough amplitude would thus suffer severe damping and lead to rapid heating of a particle distribution. The stochastic motion resembles a diffusion process even though the wave spectrum is monochromatic. The methods of this paper should be useful for other problems showing stochasticity such as superadiabaticity in mirror machines, destruction of magnetic surfaces in toroidal systems, and lower hybrid heating.

Upper-Hybrid Solitons

A nonlinear upper hybrid wave with negative dispersion propagates across a magnetic field as a soliton, when its pulse speed exceeds the magnetosonic speed.

Correlations of periodic, area-preserving maps

Abstract A simple analytical decay law for correlation functions of periodic, area-preserving maps is obtained. This law is compared with numerical experiments on the standard map. The agreement between experiment and theory is good when islands are absent, but poor when islands are present. When islands are present, the correlations have a long, slowly decaying tail.

Simulation of laser beat heating of a plasma

A new relativistic electromagnetic computer simulation code, with one spatial dimension, is described which explicitly follows right‐ and left‐going electromagnetic waves by integrating along the characteristics of Maxwell’s equations. To illustrate its suitability for the study of laser‐plasma interactions, simulations are discussed of the heating of plasma by two opposed lasers whose beat frequency drives a local plasma oscillation. Excellent agreement is obtained with analytic theory in the linear‐response regime.

The Darwin Model as a Tool for Electromagnetic Plasma Simulation

The Darwin model of electromagnetic interaction is presented as a self‐consistent theory, and is shown to be an excellent approximation to the Maxwell theory for slow electromagnetic waves.

Soluble theory of nonlinear beam-plasma interaction

A soluble theory of the post‐saturation portion of a beam‐plasma interaction is developed, concentrating on explaining the results of O’Neil, Winfrey, and Malmberg. Analytic progress is made possible by applying a certain constraint procedure, characterized by the ’’rotating‐bar’’ approximation, to a Hamiltonian formulation of the problem. The procedure yields, from the original N‐particle Hamiltonian H, a new, reduced Hamiltonian H, which has only two particle‐related degrees of freedom, and which maintains the conservation laws of energy and momentum possessed by H. The equations of motion coming from H still describe the self‐consistent interaction of a mode of the plasma with the beam particles, as opposed to previous work, and, because of the great reduction in the number of degrees of freedom, explicit expressions for the nonlinear frequency shift, and growth rate, of the mode can be obtained, which are in very good agreement with the simulation results of O’Neil, Winfrey, and Malmberg.

Congruent reduction in geometric optics and mode conversion

Standard eikonal theory reduces, to N=1, the order of the system of equations underlying wave propagation in inhomogeneous plasmas. The condition for this remarkable reducibility is that only one eigenvalue of the unreduced N×N dispersion matrix D(k,x) vanishes at a time. If, in contrast, two or more eigenvalues of D become simultaneously small, the geometric optics reduction scheme becomes singular. These regions are associated with linear mode conversion and are described by higher‐order systems. A new reduction scheme is developed based on congruent transformations of D, and it is shown that, in degenerate regions, a partial reduction of order is still possible. The method comprises a constructive step‐by‐step procedure, which, in the most frequent (doubly degenerate) case, yields a second‐order system, describing the pairwise mode conversion problem in four‐dimensional plasmas. This N=2 case is considered in detail, and dimensionality arguments are used in studying the characteristic ordering of the e...