J B Taylor

Plasma rotation and toroidal drift modes

This paper discusses the structure of drift waves in a rotating toroidal plasma. The rotation destroys an underlying symmetry that is the basis for the conventional ballooning representations of perturbations in a torus and alternative descriptions are needed. One such description exploits the residual symmetry that persists despite rotation. It shows that sheared rotation annuls the toroidal coupling between perturbations associated with different magnetic surfaces, so that cylinder criteria rather than toroidal `ballooning criteria again become relevant. As expected, sheared rotation reduces the radial mode width, and presumably, therefore, the anomalous transport. It can also alter the scaling of anomalous transport with magnetic field from Bohm to gyro-Bohm. Another description of perturbations leads, as is well known, not to eigenmodes but to perturbations with a Floquet-like time dependence on a magnetic surface. We show that this Floquet solution actually conceals an arbitrary time dependence of the perturbation! At the usual leading order in a high mode number expansion, the Floquet form and the eigenmode form are equivalent and are equally valid descriptions. However, in a more accurate theory only the eigenmode form persists. The Floquet form, and its short-term growth rate, should be regarded as transients associated with particular starting conditions and with the use of an idealized (linear) velocity profile.

Structure of short-wavelength drift modes and transport in a toroidal plasma

Short-wavelength fluctuations, such as electron and ion drift waves, may be one cause of anomalous transport in toroidal plasmas. The purpose of this paper is to establish the radial structure of these modes and to clarify some discrepancies in the literature. The conventional form of toroidal drift modes can occur only at isolated plasma radii and are unlikely to be the source of a universal transport. However, a more recently described class of electron and ion drift modes can occur at all plasma radii. They have a structure consistent with an anomalous transport exhibiting the Bohm scaling with magnetic field and decreasing with plasma rotation.

Structure and damping of toroidal drift waves (and their implications for anomalous transport)

The conventional theory of high-n toroidal drift waves, based on the ballooning representation, indicates that shear-damping is generally reduced in a torus compared to its plane-slab value. It therefore describes the most unstable class of toroidal drift waves. However, modes of this type occur only if the diamagnetic frequency omega *(r) has a maximum in r, and they affect only a small fraction, O(1/n1/2), of the plasma radius around this maximum. Consequently they may produce little anomalous transport. Within the ballooning description, there is another class of toroidal drift waves with very different properties to the conventional ones. The new modes have greater shear-damping (closer to that in a plane-slab) than the conventional ones and so have a higher instability threshold. However, they occur for any plasma profile and at all radii, and they have larger radial extent. Consequently they may produce much greater anomalous transport than the possibly benign conventional modes. This suggests a picture of anomalous transport in which the plasma profile is determined by marginal stability, but marginal to the new class of modes not to the conventional ones.

Turbulence in two-dimensional plasmas and fluids

In two dimensions (2D), a guiding-centre plasma and an inviscid fluid can be described by a continuum model or by quasi-particles (filaments) with Coulomb interaction. Other 2D continuum models are equivalent to quasi-particles with a screened Coulomb interaction. Such 2D systems of quasi-particles have negative temperature equilibria, characterized by large fluctuations. In the continuum models this corresponds to the spontaneous appearance of macroscopic clumps of charge or vorticity and may represent a stage of 2D turbulence. However, this raises the basic question; can a limited number of particles ever represent the behaviour of a continuous fluid? At first sight the two models are incompatible. The particles are conservative, have finite degrees of freedom and few isolating invariants. The fluid has infinite degrees of freedom and is conservative only if dissipation is ignored - when it has an infinity of invariants. Despite these fundamental differences, the two systems may be reconciled if the fluid is viscous and the quasi-particles are chosen appropriately. Roughly speaking, a small viscosity destroys invariants of an ideal fluid and limits its degrees of freedom, while preserving its essentially conservative behaviour.

Relaxation and magnetic reconnection in laboratory plasmas

The author studies the concept of plasma relaxation by magnetic reconnection which has been applied to many laboratory plasmas, but whose origin lies in observations on the Toroidal Pinch. This is one of the simplest of plasma confinement systems. In essence, it involves only a toroidal vessel in which a toroidal magnetic field B0 is first created by external coils then, after a suitable ionizing process, a toroidal current I is induced. It is this current which is responsible for plasma heating, compression, and confinement.

Stability of toroidal plasmas: the influence of magnetic shear, periodicity and rotation

This paper describes work on an important class of plasma instabilities in toroidal confinement systems. These are instabilities with large mode number in the toroidal direction but long wavelength parallel to the magnetic field. After a brief historical introduction the now standard method—the ballooning representation—for calculating such modes is described. This method is remarkably successful for most stationary plasmas, but breaks down for configurations with low magnetic shear or with significant sheared rotational flow. These are two areas of great current interest because of their association with transport barriers in tokamaks. Some extensions of ballooning theory for dealing with these situations are described and some preliminary results, in particular showing the stabilizing effect of sheared rotation, are presented.

Plasma containment and stability theory

Models of the behaviour of high temperature plasma are applied to the problem of plasma confinement in magnetic traps. A wide variety of possible instabilities is disclosed. In magnetic mirror traps the low frequency instabilties can be overcome by design of the magnetic field. The high frequency instabilities, particularly those associated with the loss-cone character of the equilibrium distribution function, are more persistent and appear to impose severe restrictions on the dimensions of the plasma. Consequently toroidal traps seem to offer a better prospect for long-term containment but at present they are subject to low frequency instabilities which persist even when conditions for hydromagnetic stability have been met. These instabilities may be due to small resistive effects or to an unstable drift wave. The resistive instabilities should disappear at high temperature and the drift-wave instability should be overcome by increased shear in the magnetic field.

RELAXATION AND TOPOLOGY IN PLASMA EXPERIMENTS

Topological factors control the behavior of magnetised turbulent plasma in two ways. One is through constraints on the magnetic field. This leads to the concept of the relaxed state. The other is through the topology of the mechanical and magnetic boundaries of the plasma. Consequently, while the constraints are universal, the relaxed states themselves differ greatly from one situation to another. In a toroidal system there are two classes of relaxed state, one exhibiting field reversal and another exhibiting current saturation. The relaxed states of spherical systems depend on whether both the mechanical and magnetic boundaries, or only the mechanical boundary, is topologically spherical. In the first case the relaxed state is unique, in the other it depends on the properties of the boundary. These differences are discussed in connection with a number of experiments and the theoretical predictions are compared with the experimental data.

Nonlinear Plasma Theory

R Z Sagdeev and A A Galeev New York: W A Benjamin 1969 pp xii + 122 price £5 4s 2d This monograph is based on a series of lectures given by the Soviet authors at the International Centre for Theoretical Physics in Trieste during 1966. As nonlinear theory has been a speciality of Soviet theorists the original lectures aroused much interest and this permanent record will be of considerable value.