P. H. Rutherford

Nonlinear growth of the tearing mode

The resistive tearing mode is analyzed in the nonlinear regime; nonlinearity is important principally in the singular layer around k·B = 0. In the case where the resistive skin time τs is much longer than the hydromagnetic time τ H, exponential growth of the field perturbation is replaced by algebraic growth like t2 at an amplitude of order (τ H / τ S )4/5. Application of the theory to the unstable tearing modes of a tokamak with a shrinking current channel yields good agreement with the observed amplitudes of the m ≥ 2 oscillations. The analysis excludes the very long wavelength mode, and m = 1 in the tokamak, for which the “constant‐Ψ” approximation is invalid.

Tearing mode in the cylindrical tokamak

Detailed computational results are presented on the stability and radial distribution of linear tearing modes in cylindrical tokamaks of various radial profiles. In the case of a skin‐current profile, a “double tearing mode”, with two points of discontinuity in the radial magnetic field perturbation is found. An analytical method is also derived for comparison of the stability of different radial profiles. It is further shown that the tearing mode can be driven by finite electron viscosity, as well as by the usual finite resistivity mechanism.

Drift Instabilities in General Magnetic Field Configurations

A theory of low‐frequency drift (universal) instabilities in a nonuniform collisionless plasma is developed for general magnetic field configurations including trapped particle effects, rather than the plane geometry which has previously received most attention. A type of energy principle shows that the special equilibrium distribution F(∈, μ), of interest in minimum‐B mirror configurations, is absolutely stable to these modes provided ∂F/∂∈ < 0 together with a second condition on ∂F/∂μ. For equilibrium distributions not of this special form, in particular for a Maxwell distribution with a density gradient, the case of axisymmetric toroidal configurations with closed poloidal field lines is considered in detail. Three unstable drift modes are found, a flute‐like mode, a drift‐ballooning mode local to the region of unfavorable curvature, and a drift‐universal mode. Stability criteria and growth rates for the modes are given. The equations also describe a recently discussed low‐frequency trapped‐particle in...

Nonlinear properties of the internal m = 1 kink instability in the cylindrical tokamak

An analysis is presented, in a cylindrical approximation, of the nonlinear behavior of the m = 1 magnetohydrodynamic kink instability that occurs in a tokamak when the “safety factor” q(r) = rBz / RB θ(r) falls below unity on axis. A kinked neighboring equilibrium is found, which is accessible from the initial straight equilibrium in the sense of satisfying the flux‐conservation constraints. Owing to the singular nature of the fundamental, all harmonics are excited in a singular region near where q (r) = 1. The nonlinear amplitude is moderate. It is shown that growing modes of this type should produce negative voltage spikes and inward shifts in major radius, as are seen in the experiments. The predicted magnitudes of these two effects are, however, much smaller than those observed.

Baldur: A one-dimensional plasma transport code

Abstract A version of the BALDUR plasma transport code which calculates the evolution of plasma parameters is documented. This version uses an MHD equilibrium which can be approximated by concentric circular flux surfaces. Transport of up to six species of ionized particles, of electron and ion energy, and of poloidal magnetic field is computed. A wide variety of source terms are calculated including those due to neutral gas, fusion and auxiliary heating. The code is primarily designed for modelling tokamak plasmas.

Collisional Diffusion in an Axisymmetric Torus

Collisional cross‐field diffusion of plasma in a general axisymmetric torus is calculated using the Fokker‐Planck collision term, and a variational principle is found for the diffusion coefficient. Anomalously large transport, as found by Galeev and Sagdeev, occurs where the effective collision frequency is of order or less than the “bounce” frequency of trapped particles. Only unlike‐particle collisions contribute, and the ion and electron diffusion coefficients are equal and independent of any radial electric field; these conclusions differ somewhat from those of Galeev and Sagdeev.

Destabilization of the trapped-electron mode by magnetic curvature drift resonances

Electron curvature drift resonances, ignored in earlier work on the trapped‐electron modes, are found to exert a strong destabilizing influence in the lower collision frequency range of these instabilities. Effects arising from ion temperature gradients, shear, and finite ion gyroradius are included with these ∇B drifts in the analysis, and the resultant eigenvalue equation is solved by numerical procedures rather than the commonly used perturbation techniques. For typical tokamak parameters the maximum growth rates are found to be increased over earlier estimates by roughly a factor of 4, and requirements on magnetic shear strength for stabilization are likewise more severe and very difficult to satisfy. For inverted density profiles, this new destabilizing effect is rendered ineffective, with the result that the modes can be stabilized for achievable values. of shear provided the temperature gradients are not too severe. Estimates of the particle and thermal energy transport are given for both normal an...

Impurity transport in the Pfirsch‐Schlüter regime

It is shown that the classical inward diffusion of high‐Z impurities in toroidal plasmas is enhanced by the Pfirsch‐Schluter effect. Numerical transport coefficients are evaluated. Typically, both density and temperature gradients are found to produce inward impurity diffusion.

Long‐wavelength kink instabilities in low‐pressure, uniform axial current, cylindrical plasmas with elliptic cross sections

The magnetohydrodynamic stability of a straight plasma column with elliptic cross section, carrying a uniform axial current, is investigated by extremizing the Lagrangian of the system using a natural coordinate system based on the magnetic field lines. Stability criteria are derived and growth rates are obtained analytically for systems with a uniform mass density inside the plasma. It is shown that the coupling between kink modes and Alfven waves produced by noncircularity is a destabilizing effect. A technique for solving the problem numerically is also discussed and used to demonstrate the effect of a spatially varying plasma density on the growth rate.

Overview of TFTR transport studies

A review of TFTR plasma transport studies is presented. Parallel transport and the confinement of suprathermal ions are found to be relatively well described by theory. Cross-field transport of the thermal plasma, however, is anomalous with the momentum diffusivity being comparable to the ion thermal diffusivity and larger than the electron thermal diffusivity in neutral beam heated discharges. Perturbative experiments have studied nonlinear dependencies in the transport coefficients and examined the role of possible nonlocal phenomena. The underlying turbulence has been studied using microwave scattering, beam emission spectroscopy and microwave reflectometry over a much broader range in k perpendicular to than previously possible. Results indicate the existence of large-wavelength fluctuations correlated with enhanced transport.