Table of contents

Theory and computer simulation experiments have shown the potential importance of the fluid-like, modified two-stream instability as an ion heating mechanism in low-β plasmas. To take advantage of the strong ion heating which it can produce, we consider inducing the instability with large-amplitude waves in order to convert the energy in the wave fields into ion thermal energy. Since the instability is driven by relative electron-ion drifts across a magnetic field ₀, and has a frequency comparable to the lower hybrid frequency ω_LH, a natural choice for the driver wave is a magnetosonic wave which propagates across ₀ near the Alfvén speed. As a specific example, we demonstrate that for the parameters of the Princeton ST-Tokamak it is theoretically possible to achieve thermonuclear temperatures by anomalously absorbing these waves via the modified two-stream instability.

The condition for toroidal experiments to become energetically self-sustaining is examined and a simple relationship is given between the self-ignition temperature and the plasma current. The temperature is limited at low density by synchrotron radiation and at high density by β_θ-limitations. Tokamaks could, in principle, reach ignition conditions on Ohmic heating alone but will probably be limited by synchrotron radiation. A quite modest additional heating source allows ignition at higher density and the consequent examination of β_θ limitations. The power and energy requirements for neutral beam heating are examined and shown to be compatible with existing technology.

The authors consider the energy balance stability of a thermonuclear plasma in a system where energy release is assumed to be due solely to fusion reactions and energy losses are assumed to be governed by bremsstrahlung and the thermal conductivity of the plasma. It is also assumed that the temperature at the plasma boundary is much lower than the temperature at which the thermonuclear reactions are initiated and that the energy lifetime of the plasma is short compared with the charged-particle confinement time.

Expressions are obtained for the thermonuclear combustion stability criterion in respect of a fluctuation for which the temperature is either rising or falling at every point of the plasma. It follows from these expressions, in particular, that if the energy release is local and the coefficient of thermal conductivity is independent of the spatial derivatives of temperature, the energy balance of a deuterium-tritium plasma will be stable only when the temperature is greater than 30 keV.

The radial profile of a longitudinal plasma current is experimentally investigated. When the relative drift velocity between ions and electrons becomes larger than the ion thermal velocity, a temporal development of plasma parameters, such as electrical conductivity, electron temperature, longitudinal electrical field, and radial profile of the longitudinal current is observed. The effective collision frequency for electron heating is estimated. A set of initial and circuit conditions is obtained which assures that the final high-temperature plasma is produced without appreciable loss. It is concluded that the appearance of a restoring force towards plasma stability due to a sheared magnetic field may be a key point in attaining a final stable plasma with arbitrary plasma current.

The properties of hybrid stellarator fields with different values of ℓ are reviewed as related to the current configuration. The dependence of the rotational transform angle and the necessary helical current on the helical pitch parameter (ma/R), the radius of the separatrix (ma_s/R), the radial co-ordinates (mr/R), the aspect ratio (a/R) and the number ℓ is studied by using an average method to find the optimum parameters in the case of the linear configuration. These results are compared with the numerical ones in the toroidal stellarator configuration and the agreement is satisfactory in the case of medium aspect ratio (4 ∼ 5).

Variational procedures for neutron and photon transport calculations in CTR blanket studies are considered. The procedures can be readily implemented to yield accurate results in an efficient manner and will be especially useful in performing sensitivity studies and survey calculations. They are applicable to multidimensional problems and to the time-dependent, as well as the steady-state, case. The use of the dual-space approach for certain sensitivity studies is also considered. Numerical results are presented to demonstrate the efficacy of the procedures discussed.

The linear stability problem of a current flowing perpendicularly to a magnetic field is analysed. With the restriction that the growth rate should be larger than the ion cyclotron frequency, all the fast growing electrostatic modes are computed and compared. It is argued that the most significant modes of the problem (relevant to perpendicular collisionless shocks, pinches and the plasma focus) can be obtained without inclusion of effects due to gradients of the equilibrium quantities. The computations are divided into three broad categories: T_e ≫ T_i, T_e = T_i and T_i = 10 T_e. With the aid of computation and analysis an attempt has been made to cover the whole range of parameter space for this problem.

The non-linear longitudinal oscillations of a homogeneous plasma embedded in a weakly inhomogeneous magnetic field in the presence of an external high frequency field are investigated. The non-linearity of oscillations is manifested in two ways: (a) allowance for the terms (∇) in the equations of motion and (b) dependence of the plasma natural oscillation frequency on the displacement amplitude (or, otherwise, the loss of isochronity). It is demonstrated that the terms (∇) do not restrict the amplitude increase in the regions of hybrid resonances. The non-linear terms arising in the equations of motion governed by the inhomogeneity of ₀, item (a), restrict the oscillation amplitude in the resonant areas and, item (b), lead to a parametric build-up of oscillations. The conditions for the parametric instability and the growth rates of unstable oscillations are determined.

The effect of particle trapping on the conductivity of a toroidal plasma is investigated, taking account of like-particle collisions. The customary large-aspect-ratio approximation is not invoked, and results applicable to all aspect ratios are obtained. These agree with the previously known asymptotic results.

Several problems associated with the heating of toroidal plasmas by high-energy neutral beams are discussed. The high-energy particles slow down classically on the background plasma and the distribution function of these particles is evaluated as a function of time by solving a simplified Fokker-Planck equation. The stability of this distribution is then examined both for times before and after it reaches its equilibrium configuration. It is found that in both cases the high energy distribution is unstable unless the angle of injection is almost parallel to the field lines. Finally, the problem of the electric field which is produced by an injected neutral beam is discussed.

A solution of the Fokker-Planck equation is obtained for the distribution of trapped particles in the magnetic field ripples of an imperfectly axisymmetric Tokamak; the resulting particle flux and the ion heat flux arising from the toroidal drift of such particles are calculated, revising previous estimates which neglected the asymmetry of the ripple profile.

The determination of stability for a scalar-pressure toroidal plasma is formulated by means of an energy variational principle in which the geometric problem of variation of the flux-surface shapes is separated from the variation of the flux profile. By means of successive minimizations, holding the surface shapes and the flux profiles fixed, the potential energy is written as a one-dimensional integral over a function of pressure, differential flux, and the specific-inductance matrix. The latter depends only on the geometry of the surfaces. Stability conditions are derived by varying the flux profile and the geometry (specific inductance). Also, a non-linear stability condition is derived for the special case of a finite geometric distortion of the flux surfaces while holding the flux profiles fixed. Exact representations for the specific inductance are calculated for straight cylinders, axisymmetric toruses, and helically symmetric configurations, all with arbitrary cross sections.

It is shown that, in a plasma with a sharp boundary, surface absorption leads to a build-up of cyclotron oscillations with negative energy. As a result, the instability region expands. A detailed analysis is made of the processes giving rise to surface absorption. By employing the concept of modulated beams (Van Kampen waves) it is possible to construct a clear picture of the phenomenon, which is compared with other processes in inhomogeneous systems leading to an exchange of energy between oscillations and plasma particles (cyclotron and Cherenkov resonance).

By numerical simulation, we calculate the efficiency of a system to recover the energy carried away from a fusion reactor by the charged particles escaping through the magnetic mirrors. A beam of particles injected into a 2N collector electrostatic field configuration proposed by Post is studied, and the effect of varying the collector parameters (number of collector fins 2N, fin spacing, and electric field strength) as well as beam parameters (beam width, energy spread, and density at the entrance) is determined. We found that, for 2N ranging from 12 to 36, the optimum efficiency in the limit of low density, _T0, ranges from 0.82 to 0.88, respectively, for a beam with a factor of 2 in energy spread. Increasing the energy spread to a factor of 4, more typical of a reactor situation, decreased _T0 from 0.84 to 0.76 in the 24-fin system. We calculate that, for the same efficiency and applied electric fields, the 36-fin system can tolerate a maximum beam density greater by a factor of 2 than that for a 12-fin system. For a 24-fin system of reactor dimensions and energies this produces beam densities of 2.5 × l0⁶ particles/cm³ or, equivalently, 2.8MW/m².

The author draws attention to the fact that the theory of "drift" instabilities of toroidal systems has hitherto made no allowance for the type of instability associated with Alfvén waves. The presence of this type of instability is demonstrated for the case of an axisymmetric Tokamak with circular cross-section. A general method is developed for investigating Alfvén type instabilities and it is shown that the problem is essentially one of finding the "non-hydromagnetic" part of the perturbed plasma pressure. This kind of general approach is used for analysing plasma perturbations in conditions where the influence of finite conductivity and trapped particles is insignificant. It is shown that these perturbations can increase with time, if the plasma temperature gradient is non-zero. The perturbations investigated involve electrons and ions being displaced radially at an almost equal rate, so that the instabilities considered should lead not only to increased thermal conductivity but also to increased diffusion.

A detailed description of the time behaviour of a hydrogen discharge in the ST-Tokamak is based on measured radial electron temperature and density profiles at 12 different times, together with measurements of the Ohmic-heating current and voltage, the temporal, spatial, and spectral distributions of hydrogen light, the ion temperatures, and impurity concentrations. Early in the discharge the electron temperature profiles show evidence of a skin effect that develops on a time-scale of several milliseconds into a peaked profile of about 2.2 keV maximum. Thereafter the peak temperature stops growing and develops into a flat plateau, the width of which appears to be determined by the Kruskal-Shafranov limit. The average particle confinement time scales with , and reaches a maximum of 13-14 ms. The power balance is dominated by electron loss and re-cycling, rather than ion loss or radiation. The recycling process at the aperture limiter appears to involve sufficiently energetic neutral atoms to provide a fairly flat radial source function for particles, and hence to influence directly the development of the radial distribution of power input and energy balance.

The symposium was held in the L.B. Johnson School of Public Affairs at the University of Texas at Austin. There were 65 papers and about 140 participants from Belgium, France, Germany, Italy, Japan and the USA.

A translation of the abstracts of all the articles for this issue into French, Russian and Spanish.