John M. Greene

A method for determining a stochastic transition

A number of problems in physics can be reduced to the study of a measure‐preserving mapping of a plane onto itself. One example is a Hamiltonian system with two degrees of freedom, i.e., two coupled nonlinear oscillators. These are among the simplest deterministic systems that can have chaotic solutions. According to a theorem of Kolmogorov, Arnol’d, and Moser, these systems may also have more ordered orbits lying on curves that divide the plane. The existence of each of these orbit types depends sensitively on both the parameters of the problem and on the initial conditions. The problem addressed in this paper is that of finding when given KAM orbits exist. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of nearby periodic orbits. The relation between KAM surfaces and periodic orbits has been explored extensively here by the numerical computation of a particular mapping. An important part of this procedure is the introduct...

Resistive instabilities in general toroidal plasma configurations

Previous work by Johnson and Greene on resistive instabilities is extended to finite‐pressure configurations. The Mercier criterion for the stability of the ideal magnetohydrodynamic interchange mode is rederived, the generalization of the earlier stability criterion for the resistive interchange mode is obtained, and a relation between the two is noted. Conditions for tearing mode instability are recovered with the growth rate scaling with the resistivity in a more complicated manner than η3/5. Nyquist techniques are used to show that favorable average curvature can convert the tearing mode into an overstable mode and can often stabilize it.

Resistive instabilities in a diffuse linear pinch

The effects of small electrical resistivity, viscosity, and thermal conductivity on hydromagnetic instabilities for diffuse linear pinch configurations are considered. For finite-pressure systems, higher order equations must be solved than in previous work that was applicable to stellarator or slab geometries. A careful treatment of the problem of matching the perturbed solution across the boundaries that separate a resistive layer from outer hydromagnetic regions clarifies the relationship between resistive instabilities and the ordinary hydromagnetic instabilities that are obtained from the ideal equations. In a similar way consideration of various limiting cases brings out the connection between the different resistive modes. The manner in which thermal conductivity and viscosity assert themselves is exhibited, and for a special configuration it is shown explicitly how viscosity can exert a small stabilizing effect. Aside from this, instabilities are found whenever the pressure decrease is outward.

The second region of stability against ballooning modes

A new type of axisymmetric magnetohydrodynamic equilibrium is presented. It is characterized by a region of pressure and safety factor variation with a short scale length imposed as a perturbation. The equilibrium consistent with these profile variations can be calculated by means of an asymptotic expansion. The flexibility obtained by generating such equilibria allows for a close examination of the mechanisms that are relevant to ballooning instabilities – ideal-MHD modes with large toroidal mode number. The so-called first and second regions of stability against these modes are seen well within the limits of validity of the asymptotic expansion. It appears that the modes must be localized in regions with small values of the local shear of the magnetic field. The second region of stability occurs where the local shear is large throughout the range where the magnetic-field-line curvature is destabilizing.

Determination of Hydromagnetic Equilibria

Techniques for calculating hydromagnetic equilibria in toroidal systems which differ little from a uniform field are developed. The zeroth‐order magnetic surfaces in these systems differ appreciably from concentric circular toroids. Care is taken to match onto reasonable external fields at the plasma boundary. Expressions for various equilibrium properties including the rotational transform, the net current on each surface, and the magnetic lines of force are obtained. As an illustration of the theory it is shown that the application of a particular field perpendicular to the plane of the torus reduces the distortion associated with the introduction of pressure into the system.

Stability Criterion for Arbitrary Hydromagnetic Equilibria

A necessary and sufficient condition for the stability with respect to localized displacements is obtained for arbitrary bounded hydromagnetic equilibria. The use of a natural coordinate system which contains the important properties of the equilibrium configuration facilitates the understanding of the instability.

Two‐Dimensional Measure‐Preserving Mappings

A particular area‐preserving mapping of a plane onto itself has been studied in detail with the aid of a digital computer. A large number of fixed points, finite sets of points that transform into each other, were located and classified as elliptic or hyperbolic depending on the nature of the linearized mapping in the neighborhood. A quantity called the residue was calculated for each fixed point. This quantity can be used to predict whether other nearby fixed points are elliptic or hyperbolic. The results showed that there are considerable regions in which almost all the fixed points are hyperbolic. Further calculations were made to estimate the area enclosed by the invariant curves whose existence has been established by Moser. The boundary of this region appeared to coincide with the boundary of the region in which almost all the fixed points are hyperbolic.

Numerical determination of axisymmetric toroidal magnetohydrodynamic equilibria

Numerical schemes for the determination of stationary axisymmetric toroidal equilibria appropriate for modeling real experimental devices are given. Iterative schemes are used to solve the elliptic nonlinear partial differential equation for the poloidal flux function psi. The principal emphasis is on solving the free boundary (plasma-vacuum interface) equilibrium problem where external current-carrying toroidal coils support the plasma column, but fixed boundary (e.g., conducting shell) cases are also included. The toroidal current distribution is given by specifying the pressure and either the poloidal current or the safety factor profiles as functions of psi. Examples of the application of the codes to tokamak design at PPPL are given.

Interchange instabilities in ideal hydromagnetic theory

Interchange instabilities are examined using an ideal hydromagnetic model. The nature of the energy sources that can drive the instability is clarified.

One‐Dimensional Model of a Lorentz Plasma

A one‐dimensional electron plasma model, in which electron sheets can scatter in three dimensions in velocity space through small‐angle collisions with the stationary ions, has been constructed. Using the model, the dc conductivity of a Lorentz plasma is measured. The result agrees with the theoretical prediction, therefore confirming the validity of the model and exhibiting its capability for determining other transport coefficients. The decay rate of a standing wave when collisions are present is measured. The result verifies the theoretical result, in which the various decay increments are superimposed. The wavelength dependence of the collisional damping is observed.