Table of contents

The author solves numerically the problem of using an uncompensated space charge to stabilize the flute instability in a dilute plasma (, where ω_pi and Ω_i are the ion plasma and ion cyclotron frequencies, respectively). For typical devices, this condition is fulfilled when the plasma density n < 10¹⁰ cm⁻³. It is shown that, if the space charge is sufficiently high, the flute instability is completely stabilized. The critical plasma decompensation (n_i0 – n_e0)/n_e0) needed to stabilize different oscillation modes and the dependence of that critical value on plasma density, the ion Larmor radius and the plasma-to-wall gap are established.

To understand plasma phenomena which will occur in a wall-confined, shock-heated fusion system, experiments and computer simulations have been carried out. The thermonuclear fusion cycle, energy transfer from a β > 1 hot plasma to a wall, plasma confinement by means of walls and a magnetic dam, and some new strong shock wave physics results are described and reviewed.

The use of the compressional hydromagnetic mode (also called the magnetosonic or, simply, the fast wave) is examined in some detail with respect to the heating of a tritium plasma containing a few percent deuterium. Efficient absorption of wave energy by the deuteron component is found when ω = ω_C (deuterons), with Q_wave ⪆ 100. Reasonable efficiencies are found also for electron heating, but coherence effects between transit-time and Landau damping for electrons reduce the total absorption for both processes to one-half of the transit-time power, calculated separately.

The fusion output of a two-component neutral-injected plasma can be enhanced by selective heating of the injected deuterons. Also, selective deuteron absorption may be used for ion-tail creation by radiofrequency excitation alone, as an alternative to neutral injection. The dominant behaviour of the high-energy deuteron distribution function is found to be f(v) ∼ exp[(3/2)∫^vdv<Δv>/<(Δv_⊥)²>], where <Δv> is the Chandrasekhar-Spitzer drag coefficient, and <(Δv_⊥)²> is the Kennel-Engelmann quasi-linear diffusion coefficient for wave-particle interaction at the deuteron cyclotron frequency. An analytic solution to the one-dimensional Fokker-Planck equation, with r.f.-induced diffusion, is developed, and using this solution together with Duane's fit to the D-T fusion cross-section, it is found that the nuclear-fusion power output from an r.f.-produced two-component plasma can significantly exceed the incremental (radiofrequency) power input.

Calculations of Q including the effects of velocity diffusion, finite ion temperature, electric fields and charge exchange are presented. It is found that if all charge-exchange fast ions are lost then Q is severely reduced for a neutral fraction, n_n/n_e⪆7 × 10⁻⁶. These losses are reduced by re-ionization of the charge-exchanged fast ions. Calculations for the case of a two-component tokamak show that the effective charge-exchange rate is typically reduced by 2.5−4 times owing to re-ionization. With allowance for this, it is found that, in a stationary system, the plasma loss rate nτ_p must be greater than 10¹² cm⁻³ to achieve a Q greater than unity.

The paper gives sufficient conditions for a plasma to be only simply connected. An example is given of the case where a plasma must have more than one component of connectivity or, as we say in what follows, must be a multicomponent plasma. The conditions under which a plasma can be both simply connected and multicomponent are also described.

The r.f. preferential plugging of multi-ion-species plasma is studied theoretically and experimentally in a magnetic-cusp plasma. In a He⁺, N⁺ and plasma He⁺ is effectively plugged in, while N⁺ and are allowed to escape freely from a container by the applied r.f. field. In a H⁺, and plasma, all ion species are plugged in the container. These experiments can be explained by theory.

Unignited, and therefore externally driven, tokamak plasmas can be the core for economic tokamak power reactors over a significant range of machine sizes where the plasma amplification factor Q exceeds 15. In such driven systems, high-Z impurity levels from 2 to 4 times that in ignited plasmas can be tolerated. Large driven tokamaks with plasma currents greater than 4 MA achieve the highest Q-value by operating with a 50-50 D-T plasma using the minimum amount of injected power to maintain the plasma in energy equilibrium. Driving large tokamaks with injection can also be used to extend the reactor burn time should impurities accumulate. For smaller tokamaks, the optimum Q is obtained by injecting deuterium neutral beams at maximum power onto a pure tritium plasma and operating the device in the two-energy-component mode described by Dawson, Furth and Tenney. A natural connection between these two operating modes is thus obtained. The potential benefits of low-Z liners in tokamaks are also studied and it is found that the tolerable impurity level of a contaminant like carbon can be an order of magnitude larger than the tolerable level of impurities like Fe or Mo.

Stability against flute modes. localized inside the plasma column, of magnetic confinement configurations which are relevant for the realization of poloidal divertors, is studied for a bell-shaped current-density profile. The ideal MHD-equilibrium is described by using the Herrnegger-Maschke solutions of the equilibrium equations. It is shown that there is little difference between the stability conditions for toroidal diffuse pinches with the shape of the cross-section resembling a normal or inverted D, and that these two shapes of the plasma cross-section are preferable to a rectangular one if the plasma column is very elongated in the vertical direction. A comparison is made with the results based upon a flat-current-density equilibrium model.

Neutral gas which penetrates into a hot plasma body, of characteristic dimension L_b, average density and temperature T, consists mainly of a slow and a fast component, of densities n_ns, n_nf and temperatures T_ns≪T, T_nf≃T, respectively. These components have the penetration lengths L_ns = l/σ_cs and L_nf= 1/σ_cf, corresponding to the critical densities n_cs=1/σ_csL_b and n_cf=l/σ_cfL_b where l/σ_cf≃lOO/σ_cs≃5 × 10¹⁸ m⁻² for hydrogen in the range 10⁶ < T < 10⁷ K. Thus, hot plasmas can be divided into the classes of permeable dilute, permeable non-dilute, and impermeable systems defined by ⪅n_cs≪n_cf, n_cs≪⪅n_cf, and n_cs ≪ n_cf ≪ , respectively.

Concerning tokamaks, an analysis of the plasma-neutral gas interaction leads to the following conclusions:

1. So far, all reported tokamak experiments have been conducted in the permeable density range and have been limited by instabilities in the transition region close to = n_cf. Full-scale tokamak reactors should, on the other hand, operate far inside the impermeable density range.

2. The driving force of instabilities due to the pressure gradient is expected to reach its maximum value when approachesthe density limit ncf from below, i.e. at the transition to the impermeable state.

3. The experimentally observed parameter values at the instability limit agree with those calculated from the theory on collisionless ballooning modes driven by the pressure gradient.

It is shown that the density dependence of the most favorable frequency of the r.f. electric field to confine a cusp plasma is consistent with that suggested by a theory taking account of collective response of a plasma. Based on this dependence, automatic tuning of the frequency to the optimum value for the decaying plasma is demonstrated using a feedback system piloted by an ion density measurement.

In studying the non-linear evolution of non-resonant trapped-particle instabilities, it is desirable to exploit the fact that their frequencies are smaller than the bounce frequency by employing a kinetic equation for the bounce-averaged distribution function. Such an equation for trapped particles is derived in axisymmetric toroidal geometry by transforming the familiar drift-kinetic equation to toroidal co-ordinates and employing an annihilation operator. The "bounce-averaged drift-kinetic equation" facilitates the application of Dupree's strong turbulence formalism in toroidal geometry, under the assumption that turbulent scattering of particles out of the magnetic well is unimportant. On the basis of this formalism, the trapped-particle interchange mode is analysed. The saturation amplitude of the instability and the corresponding turbulent transport coefficients are obtained.

Measurements of impurity ion concentrations and energy distributions in the 2XII mirror device are desaibed. The measurements were made using a fast-pulsed gas valve and a charge-exchange analyser that employs both electrostatic and magnetic separation to obtain energy spectra of separate particle species. The predominant high-Z material detected wascarbon, having a concentration of ∼2% and a mean energy of 3-6 keV. Smaller amounts of oxygen were the only materialother than hydrogen detected. These energetic impurity levels are determined to be contained against self-scattering into themirror loss cones and to have only minor influences on plasma life-time and on mean ion and electron energy.

The microinstability theory of the two-energy-component toroidal system is developed and applied to systems typical of those in proposed fusion break-even experiments. The energetic ion beam distributions F_b(v_||, v_⊥, t) obtained from approximate solutions of the Fokker-Planck equation are studied for high-frequency microinstabilities arising from or in both transient and collisional equilibrium states. For typical beam-plasma parameters, the system is shown to be stable to high-frequency modes during the parallel injection of a continuous, monoenergetic source.

Investigations of the feasibility of the hypothesis that the disruptive instability in tokamaks is caused by the break-up of magnetic surfaces are presented. The envisioned break-up is caused by the interaction of m = 2, n = 1 and m = 3, n = 1 helical perturbations or by the interaction of the m = 2, n = 1 perturbation and the m = 1, n = 0 toroidal field (with R/a = 5). Both numerical and analytic work has been done for two MHD-equilibrium current profiles: parabolic [q ∼ (1 – r²/2a²)⁻¹] and the peaked model (q ∼ 1 + λr²/a²). For the parabolic profile, perturbation strengths of 1.5% are required to break up surfaces when m = 2 and m = 3 perturbations are present. When the m = 2 perturbation alone interacts with the toroidal field, perturbation strengths greater than 4% are required. For the peaked model with q(0) = 1.1, q(a) = 3.7, perturbation strengths of 0.5% were found to be sufficient to break the surfaces when both m = 2 and m = 3 perturbations are present, whereas strengths of 1.5% are required when only the m = 2 perturbation interacts with the toroidal field. These perturbation amplitudes are in reasonable agreement with experimental measurements. The peaked model allows break-up of surfaces for smaller perturbation strengths, primarily because of its greater shear. This dependence on shear permits speculation on a possible mechanism for the disruptive instability.

The paper summarizes experimental results obtained in the Garching Belt Pinch I. Here, toroidal high- β- plasmas are produced by shock heating, the plasma cross-section being highly elongated in the direction of the major torus axis. The plasma can be confined for about 100 μs until the toroidal plasma current is Ohmically damped out. With various diagnostic techniques, it is shown that the plasma reaches a quasi-stationary equilibrium in shape, temperature and density after about 25 μs. This equilibrium is investigated and compared with theoretical predictions. In the experiment, the plasma displays macroscopically stable behaviour for low plasma currents. If the plasma current is increased gradually, the first instability that occurs is an m = 1, n = 1 kink mode. This mode is excited artificially by a suitable current pulse in an external coil in order to determine the stability limit. For a plasma with highly elongated cross-section area (axis ratio ε = 5 − 6), the stability limit q_crit = 2.4 is found.

In the CIRCE experiment, the plasma is obtained by the trapping of a plasma injected into a magnetic bottle by electron heating at cyclotron resonance. The plasma density lies between 5×10¹¹cm⁻³ and 10¹²cm⁻³, the electron temperature is about 100 keV and the ion temperature is in the range of few hundred electronvolts. Gross instabilities are not observed. The ratio of the plasma density to the neutral-gas density inside the plasma is higher than 100. A few kilowatts of r.f. power at 8 GHz are sufficient to obtain these results, a fact which looks encouraging as far as the creation of a more effective fast-neutraltarget plasma using the CIRCE-experiment concept is concerned.

An anisotropic, finite-β plasma trapped in a mirror well is subject to local hydromagnetic instabilities that depend strongly upon the shape of the angular distribution function of the contained particles. Moreover, the angular distribution is a solution of a diffusion equation that can be significantly modified by appropriate changes in the relative positions and amplitudes of injected beams used to create or maintain the plasma. The question of how to inject beams so that the greatest overall pressure is maintained without the plasma becoming unstable and without inordinate demands upon the technology of beam injection is an application of the theory of continuous games. We examine this problem, and the optimal strategies for a number of games of interest are determined and described

Different magnetically confined fusion reactor concepts, like the tokamak, the reversed-field pinch, the high beta stellarator and the linear theta pinch, are discussed by means of a few quantities prescribed by technical and physical requirements. These quantities are: the thermal reactor power, the wall power loading, the first-wall-to-plasma-radius ratio, the blanket thickness, the toroidal aspect ratio and the plasma beta. Pulsed operation was included by introducing a dimensionless time τ, the ratio of the burning time to the pulse cycle time. Other reactor and plasma parameters are derived from these quantities. Under certain assumptions, the systems are compared on a simplified basis in different respects including the energy balance, the question of Ohmic power losses when normal conducting magnets are used, cost structure and fast power supplies. Finally, the main features and advantages of each concept are summarized.

The purpose of this paper is to study the influence of an external magnetic field on the self-compression of a plasma column of finite electrical conductivity. The effect of this axial magnetic field on interesting plasma parameters such as density, hydrodynamic velocity, and ion and electron temperatures is examined. The presence of an external magnetic field leads to higher parameter values and to faster plasma accumulation towards the column axis. A numerical double-step Lax-Wendorff scheme was used in the framework of the two-component MHD-model for a deuterium plasma.

There are four large next-generation tokamak devices presently being designed. These are: (1) the JET (Joint European Torus) project sponsored by the Commission of the European Communities in association with EURATOM, (2) the TCT (Two-Component Torus) project underway in the United States of America, (3) the JT-60 (Japan-Tokamak) project of Japan and the T-20 tokamak project of the Union of Soviet Socialist Republics. The meeting held in Dubna provided an opportunity for leading members of the various projects to discuss the scientific objectives and detailed design features of each device, with the hope that an informal exchange of information on what each team was trying to do and how it was going about it could lead to some useful co-ordination of effort and avoidance of duplicate effort. The timing of the meeting was such that any resulting design modifications could yet be incorporated into present designs, even in the most advanced of the projects.