Table of contents

Radiation and ionization losses due to impurities present in a high-temperature plasma have been calculated for a light element (oxygen), which is completely stripped in the core of existing tokamak discharges, and a heavy one (iron), which is only partially stripped. Two extreme cases have been treated: a) corona equilibrium is attained; the radiated power is then equal to the product of electron density n_e, impurity density n_imp, and a function of the electron temperature, T_e; b) impurities re-cycle with a constant radial velocity v₀ (either inwards or outwards) in a background plasma (represented by the radial profiles of n_e and T_e). In this case, radiation and ionization losses are proportional to the impurity flux ψ_imp and are a decreasing function of the diffusion velocity v₀. The results presented can be used to evaluate losses in a practical case.

A straight column of fusioning plasma, confined in the radial direction by a magnetic field and axially by cold massive plugs or tampers, is considered and equilibrium profiles are calculated. Numerical values and approximations used are appropriate to liner compression fusion concepts. Alpha-particle slowing by the plasma is treated non-locally. It is argued that expanding buffer plasmas will be present from the material vaporized from the ends, and this phenomenon is included qualitatively in the model.

Key reactor parameters of intensely beam-driven toroidal plasmas with 50: 50 D-T composition maintained by neutral-beam injection of both D and T, together with plasma re-cycling, are determined. The deuterons and tritons are injected with equal intensity and velocity. This mode of operation is most appropriate for high-duty-factor, high-power-density operation, in the absence of pellet injection.

The velocity distributions of energetic D and T ions are calculated from a relatively simple slowing-down model, but include a tail above the injection velocity that results from energy diffusion. Fusion power multiplication Q and fractional tritium burn-up are determined from fusion reactivities calculated for beam-target, hot-ion, and thermonuclear reactions. For conditions where Q ∼ 1, beam-target reactions are dominant, but reactions among the hot ions contribute substantially to the total reaction rate when n_hot/n_e ≳ 0.2. To maintain the bulk-plasma temperature at progressively smaller n_eτ_E, it is necessary to continuously increase n_hot/n_e, so that reactions among energetic ions serve to maximize the Q-value and power density in low-n_eτ_E-driven operation.

A time-dependent explicit finite-difference code is developed for computing growth rates of gross magnetohydrodynamic modes in helically symmetric high-β, ℓ = 1 equilibria. The effect of numerical dispersion on current- and pressure-driven modes and the convergence of the growth rate with the grid size for modes in screw-pinch, θ-pinch, and helical equilibria are discussed. In the case of diffuse MHD equilibria with a long helical period length, the eigenvalues computed by this code are in agreement with those of a δW-analysis and of experiments and are a factor of about two smaller than predicted by the surface current theory. The growth rates o f m ≥ 2 modes are given as functions of the helical periodicity number, the helical amplitude of the magnetic axis, beta, and the longitudinal wave number k.

The question of maintaining plasma equilibrium in a three-turn stellarator by means of an external transverse field H_⊥ is investigated in an ideal magnetohydrodynamic model as a first approximation with respect to β, where β is the ratio of the gas-kinetic pressure of the plasma to the magnetic pressure. An "island" structure of the paraxial magnetic surfaces is observed under conditions of precise compensation and slight undercompensation of the homogeneous component of the plasma magnetic field by the field H_⊥. The optimum conditions are found for controlling the position of the magnetic axis with an external transverse field. Relations are obtained for estimating the maximum equilibrium pressure of the plasma in a three turn stellarator in the presence of an external transverse magnetic field.

A collisional drift wave instability driven by electron temperature gradients is described which appears to predict the main features of the precursors of the internal disruptions observed in tokamaks, i.e. the calculated frequency and growth rate are in quantitative agreement with the observed ones, and the predicted profiles are, at least qualitatively, consistent. In these experiments the plasma appears, despite all the uncertainties of the parameters, to be close to the theoretical threshold. It is noted that this instability can appear either if η_e ≡ d ln T_e/d ln N > 0 or η_e < 0. It is thus suggested that the theory might also be relevant to the anomalous current penetration observed in all tokamaks. Finally, a collisionless instability also occurs when η_e < 0.

The process of non-uniform laser-driven DT plasma burning caused by the thermonuclear burn wave produced and propagating in plasma is investigated theoretically. The energy transfer from the burning plasma region to the remaining cold portion is assumed to be realized either by α-particles or by free electrons. A similarity solution of this problem has been obtained and, deriving from this solution, the conditions for "firing-up" non-uniform thermonuclear targets are defined.

Consideration is given to some of the scaling problems involved in the application of turbulent heating to much larger toroidal systems than those employed up to now. Single-pulse heating is studied in a model in which the time variation of the skin depth is allowed for and energy transport from the skin to the interior of the plasma maintains the poloidal beta factor, β_p, at a constant value. With the total heating current limited to a value I_m (for a safety factor q > 1), the ion acoustic instability can be generated only as long as the skin depth δ satisfies the condition δ < I_m/(nec_s2πa) where n is the plasma density, c_s the ion sound speed and a is the minor radius. Consequently, depending on the efficiency of the energy transport, it is not certain that single-pulse heating can raise the plasma temperature to that required for ignition of a fusion reactor, and some form of multiple-pulse-heating may be required.

The non-linear saturation of the dissipative trapped-ion mode is analysed. The basic mechanism considered is the process whereby energy in long-wavelength unstable modes is non-linearly coupled via × convection to shortwavelength modes stabilized by Landau damping due to both circulating and trapped ions. In the usual limit of the mode frequency being small relative to the effective electron collision frequency, a one-dimensional non-linear partial differential equation for the potential can be derived, as was first shown by LaQuey, Mahajan, Tang, and Rutherford. The stability and accessibility of the possible equilibria for this equation are examined in detail, both analytically and numerically. The equilibrium emphasized by LaQuey et al. is shown to be unstable. However, a class of non-linear saturated states which are stable to linear perturbations is found. Included in the analysis are the effects of both ion collisions and dispersion due to finite-ion-banana-width effects. Cross-field transport is estimated and the scaling of the results is considered for tokamak parameters (specifically those for the Princeton Large Torus). It is concluded that the anomalous cross-field transport can be much lower than the estimate of Kadomtsev and Pogutse, for relevant parameters near marginal stability for the linear modes.

Recent numerical and analytical investigations of finite-ion-gyroradius effects in the context of the Vlasovfluid model are in good agreement with results of high-beta stellarator experiments. MHD-unstable m = 2 modes in near-theta-pinch devices, such as Scyllac, appear to be stabilized at an ion temperature at least four times smaller than believed previously. This conclusion makes wall stabilization of m = 1 modes more feasible.

Current research on long-wavelength laser interaction with magnetized plasmas is summarized. Special attention is given to laser beam guiding in plasma density minima, to plasma heating and to investigations of laser-produced plasmas in strong magnetic fields. The basic theoretical and experimental background aad research are discussed. Reactor studies based on lassr heated plasma systems are reviewed. Some areas of possible future research are mentioned.

In a recent paper [Nucl. Fusion 1976 16 31], Cohn, Parker, and Jassby discuss the interesting possibility of achieving ignition at low plasma temperatures (_e ≃ _i ≃ 6 keV, where _e and _i are the average electron and ion temperatures, respectively) in tokamaks operating inside the collisional regime (C = ν_eiA/ω_be > 0.1, where ν_ei is the electron-ion collision frequency, ω_be is the electron bounce frequency, and A is the plasma column aspect ratio). Assuming that the linear scaling of the energy confinement time, τ, with density, n, which is observed in present-day devices, will remain valid in larger ones operating within the collisional regime, Cohn, Parker,and Jassby conclude that ignition at the above temperature is possible if the cross-section of the plasma column is elongated. The elongation is necessary to increase the poloidal flux which, in turn, allows higher values of the plasma density. However, their results are optimistic because the effects of the elongation on the MHD equilibrium and stability constraints are neglected. In this note, these effects are taken into account. The new values of the elongation which are needed for ignition in the proposed regime are so high that they are difficult to obtain in practice.

According to Galvão, the .values chosen for limiter safety factor (q_a = 2.5) and poloidal beta (β_p = 0.5 R₀/a_p) in our illustrative examples of compact, high-density, high-field tokamak operation [Nucl. Fusion 1976 16 31] may be too optimistic for plasmas of non-circular cross-section. Galvão's treatment, however, applies only to elliptical plasmas. In fact, MHD-stability analysis shows that plasmas of D-shaped, doublet, and rectangular cross-sections are stable at lower values of q than are available to elliptical plasmas of the same shape factor, S.

Some 65 of the 150 papers presented at this excellently organized and planned Conference reported new experimental information on the magnetic confinement of high-temperature matter. Contrary to some fears that the reduction to two years of the interval between Conferences would lead to a paucity of new material, these papers ' reported important new advances in several areas; and the Conference was both timely and fruitful.