Table of contents

On the basis of an analysis of the ITER L-mode energy confinement database, two new scaling expressions for tokamak L-mode energy confinement are proposed, namely a power law scaling and an offset-linear scaling. The analysis indicates that the present multiplicity of scaling expressions for the energy confinement time τ_E in tokamaks (Goldston, Kaye, Odajima-Shimomura, Rebut-Lallia, etc.) is due both to the lack of variation of a key parameter combination in the database, f_s = 0.32 R a^−0.75 k^0.5 ∼ A a^0.25k^0.5, and to variations in the dependence of τ_E on the physical parameters among the different tokamaks in the database. By combining multiples of f_s and another factor, f_q = 1.56 a² kB/RI_p = q_eng/3.2, which partially reflects the tokamak to tokamak variation of the dependence of τ_E on q and therefore implicitly the dependence of τ_E on I_p and n_e, the two proposed confinement scaling expressions can be transformed to forms very close to most of the common scaling expressions. To reduce the multiplicity of the scalings for energy confinement, the database must be improved by adding new data with significant variations in f_s, and the physical reasons for the tokamak to tokamak variation of some of the dependences of the energy confinement time on tokamak parameters must be clarified.

Long time dissipative MHD simulations have been done of high beta, large aspect ratio tokamaks with bean shaped cross-section, resembling the PBX experiment. A disruption free path from the first to the second regimes is followed explicitly. The amplitudes of long wavelength resistive MHD modes increase with beta up to a critical value. As beta increases above this value, the mode amplitudes decrease. If the pressure and current profiles are not broad enough, long wavelength modes produce a disruption at moderate beta, which rapidly flattens the pressure profile. In the computational model, time dependent equilibria are produced by a balance of MHD quasilinear effects, transport, a pressure source and the toroidal electric field. Dissipation is provided by Spitzer resistivity and perpendicular thermal conduction. A rigid conducting wall boundary condition is used, but there is a highly resistive edge layer, which simulates a thin vacuum region. The numerical method makes use of conformal coordinates, which are generated by prescribing a set of external coil currents.

Plasma shape and position control in elongated tokamaks is analysed, using the TSC code. The paper presents a new algorithm which allows the shape evolution of a discharge to be entirely preprogrammed. The algorithm computes poloidal field coil voltages as functions of time, using magnetic measurements made close to the vessel wall. No preprogrammed coil current wave forms are required. By simulating the start-up phase of typical TCV tokamak discharges, it is shown that the actual shape evolution follows closely the preprogrammed one. X-points can be specified at arbitrary positions. Active stabilization of the dominant vertical mode is achieved by a conventional PD feedback loop. The optimization of feedback coefficients, the trade-off between shape accuracy and power dissipation in the poloidal field coils, as well as the computing power requirements for implementing the algorithm in a real experiment are discussed.

On DIII-D, periods of improved particle and energy confinement have been observed in low q, low B_T divertor discharges with Ohmic heating alone. The Ohmic H-mode has characteristics similar to those of the H-mode produced by auxiliary heating. In the Ohmic H-mode the energy confinement time can have values near those predicted by Neo-ALCATOR scaling and a factor of two above those of non-H-mode Ohmic discharges at similar densities. These observations indicate that the H-mode is not limited to an improvement in confinement over that of the L-mode in auxiliary heating discharges.

The energy confinement properties of low density, high ion temperature L- and H-mode plasmas are investigated. For L-mode plasmas it is shown that, although the global confinement is independent of density, the energy confinement in the central region is significantly better at low densities than at higher densities. The improved confinement appears to be associated with the steepness of the density gradient. For the H-mode phase, although the confinement at the edge is dramatically improved, which is once again associated with the steep density gradient in the edge region, the central confinement properties are essentially the same as for the standard L-mode. The results are compared in a qualitative manner with the predictions of the ion temperature gradient instability theory and appear to be in disagreement with some aspects of this theory.

Numerical ray tracing analysis of slow mode lower hybrid (LH) waves is used to obtain a qualitative description of LH heating in ℓ = 2 torsatrons. The dependences of LH wave propagation and absorption on the plasma density and temperature, the frequency, the parallel wave number spectrum, and the antenna location are determined. The results obtained using a single harmonic model torsatron magnetic field are in qualitative agreement with those obtained for a more realistic field calculated using the Biot-Savart law.

Small quantities of medium-Z and high-Z impurities were injected into JET plasmas and their progression was followed by two soft X-ray cameras with good spatial and temporal resolution. The results show that the transport of impurities is much slower in the central region of the plasma than outside of it and that it is greatly enhanced during sawtooth crashes. Simulation of the evolution of the emissivity profiles using an impurity transport code yielded values for the radially dependent transport coefficients; in particular, these values were compared for cases of ohmically heated plasmas and RF heated plasmas.

Effects of both electron and ion temperature gradients on the toroidicity induced drift mode have been investigated, taking into account full electron dynamics (resonant and non-resonant). The instability persists well above the critical temperature gradient (η ⪆ 1.5 for stability) previously found on the basis of the electron temperature gradient alone.

Three-dimensional magnetohydrodynamic (MHD) simulations are carried out in order to reveal the fundamental mechanism of the self-sustainment process in the reversed field pinch (RFP) plasma. It is confirmed that the RFP configuration is sustained in a cyclic process, where the MHD relaxation phase and the resistive diffusion phase appear cyclically and alternatively. In the MHD relaxation process, the RFP plasma approaches a Taylor's minimum energy state, but departs from there in the diffusion process. In other words, since MHD relaxation processes periodically release excess magnetic energy accumulated in the resistive diffusion phase, the RFP plasma can stay in the neighbourhood of the minimum energy state. The mechanism of this cyclic process is disclosed. Specifically, when at least two ideal kink (m = 1) modes become unstable, MHD relaxation can take place. This is because the MHD relaxation progresses through non-linear reconnection of the m = 0 mode, which is driven by non-linear coupling between the unstable kink modes. Therefore, self-sustainment processes can be achieved by the non-linear effects of, essentially, m = 0 and m = 1 modes. The quantitative dependence of the relaxation-diffusion cycle on the aspect ratio of the device is considered, along with its dependence on the magnetic Reynolds number. These results are consistent with recent experiments and indicate that a coherent oscillation, which is often observed in experiments, is necessary for self-sustainment. The influence of self-sustainment processes on particle confinement is briefly discussed.

Ballooning flute modes in tokamaks in the presence of trapped high energy particles are investigated. It is shown that a stability window occurs when a finite population of high energy particles exists. In appropriate conditions, high energy particles may lead to stable values of pressure that are considerably higher than those predicted by ideal MHD theory. It is found that the stability window is very sensitive to the main plasma component parameters and to the value of the hot particle pressure. It is shown that perturbations with characteristic frequencies ω_* and _D are increased out of the stability region; here, ω_* is the ion drift frequency and _D is the average magnetic drift frequency. The corresponding growth rates and threshold conditions are obtained.

The principle of profile consistency states that for fixed limiter safety factor q_a there are unique natural equilibrium profile shapes for the current density j(r) (and consequently q(r)) and the electron temperature T_e(r) for any tokamak plasma, independent of the shapes of the heating power deposition profiles. The mathematical statements of the three basic consequences of this principle for sawtoothing discharges (i.e. discharges with q(0) ≤ 1) are: (1) r₁/a = F₁(l/q_a) (≈ 1/q_a, empirically); (2) ⟨T_e⟩/T_e0 = F₂(l/q_a); and (3) a unique scaling law for the central electron temperature , where q(r₁) = 1, ⟨T_e⟩ is the volume average electron temperature, and F₃(q_a) = q_a/q₀. Since Ohm's law relates j(r) to T_e(r), the principle of profile consistency dictates that this unique set of functions F₁, F₂ and F₃ remain the same for all sawtoothing discharges in any tokamak, regardless of its size (i.e. a and R), I_p, V_L, B_T, etc. The paper presents a rather complete and detailed analysis of this self-consistency of the measured values of T_e(r), F₁, F₂ and F₃ for sawtoothing TFTR discharges. In particular, the analytical predictions of the profiles of Coppi's Gaussian, exponential, modified exponential and trapezoidal models, as well as the model profiles of Kadomtsev and Campbell et al. are compared with TFTR and TFR data. Some of the principal results are: (1) The empirical profile consistency relation r₁/a = 1/q_a is an acceptable solution of q(r₁) = 1 for all q_a dependent profiles. (2) A comparison between experiment and the present analytic results yields [⟨T_e⟩/T_e0]_EXP ≈ [⟨T_e⟩/T_e0]_AN + 0.05 for the profiles of Coppi's Gaussian model, and for the profiles of the models of Kadomtsev and Campbell et al. (3) For all q_a independent profiles, F₃(q_a) = q_a/q₀ = constant, and, consequently, for all q_a dependent profiles, F₃(q_a) = q_a/q₀ ∼ q_a when r₁/a ≈ 1/q_a, and, consequently, . (4) Coppi's and Ohkawa's forms of x_e(r) yield , while the INTOR value of x_e(r) yields . (5) For r₁/a ≈ 1/q_a, the profiles of Coppi's Gaussian model and those of the models of Kadomtsev and Campbell et al. all predict that the normalized sawtooth amplitude ΔT_e/T_e ∼ 1/q_a, in agreement with the experimental observations. (6) For q_a dependent models, universality of profiles exists in suitably reduced co-ordinates when r₁/a ≈ 1/q_a.

The burn condition for D-T plasmas is derived in a form which accounts for the stationary helium concentration determined by the coupling of the fusion power and helium production. The ratio of the global alpha particle confinement time to the energy confinement time τ_E is taken as a quantity characterizing the prevailing confinement and exhaust regime. For an ignited and stationary burning D-T plasma this ratio needs to remain sufficiently below 15 or below 10 for typical impurity concentrations. This poses a lower limit on the required helium exhaust efficiency and indicates that operational regimes are needed where the particle confinement time does not diverge too far from the energy confinement time. Applied to the D-³He fusion reaction, the analysis leads to even more stringent limits on the ratio , which must be smaller by a factor of three to four compared to the D-T case. It is found that, for any given temperature (and for otherwise equal exhaust and recycling conditions), in general burning can be achieved at two different values of the fusion parameter n T τ_E and ash concentration.

Good confinement at low values of q_0.95 and at high values of total beta is crucial for the economical development of a fusion reactor. It is shown that an extension of the domain of H-mode operation in JET to values of q_0.95 ≥ 2.0 (i.e. a plasma current of 3 MA and a toroidal field of 1.2 T), with additional heating at a power of up to 12 MW, does not lead to an appreciable confinement deterioration when q_0.95 is decreased, for total beta values of up to 4.2%, corresponding to 90% of the Troyon limit. Pressure profiles with a peaking factor p₀/⟨p⟩ ∼ 3 can be achieved for such low values of q_0.95. Analysis of the MHD equilibrium and stability shows that the conditions of marginal ballooning instability near the plasma edge are reached in a discharge with β ∼ 4.2%.

The results of Monte Carlo transport simulation at very low collisionality in stellarators are presented. The contribution to transport arising from transition particles is evaluated and the dependence of the proper diffusion coefficient on geometrical factors is studied and compared with analytic theory.

For a continuously burning D-T fusion plasma the particle and energy confinement times have to be appropriate since otherwise the produced ⁴He and the energy cannot be exhausted. The upper limits for the ⁴He particle confinement parameter n_eτ_p(He) are discussed. Comparison with the energy confinement parameter n_eτ_E given by the burn condition shows that there is only a limited operational window for a fusion reactor. For a steady state burning D-T plasma, only small concentrations of impurities are tolerable. With increasing impurity concentration, the window becomes smaller and finally closes.

Direct electron damping of low frequency fast magnetosonic waves has been observed in the centre of high beta hydrogenic JET plasmas where transit time magnetic pumping is a significant component in the electron-wave interaction mechanism. The electron heating power profile was peaked on axis, extended across almost half the minor radius and accounted for 22 ± 5% of the total radiofrequency power coupled to the plasma. After passing through the plasma core, the fast wave was absorbed by hydrogen ions at the second harmonic cyclotron resonance which was placed inboard of the magnetic axis and intersected the equatorial plane at one third of the minor radius.

An explicit formulation is presented to determine widths of magnetic island separatrices generated by magnetic field perturbations in a general toroidal stellarator geometry. The method, which is based on a representation of three-dimensional flux surfaces by double Fourier series, allows rapid and accurate calculation of the island widths in real vacuum field configurations, without the need to follow field lines through numerical integration of the field line equations. The procedure does not involve any simplifying approximations and can be used for fields with multiple small perturbations. Numerical results of the island width obtained in the flux co-ordinate representation for the Advanced Toroidal Facility agree closely with those determined from Poincaré puncture points obtained by following field lines. The effects of resonance coupling are analysed.

The seventh Workshop on electron cyclotron emission (ECE) and electron cyclotron resonance heating (ECRH) as well as the IAEA Technical Committee Meeting on the same subjects were held in Hefei, China, from 9 to 11 May 1989. The Workshop was organized by the Institute of Plasma Physics, Academia Sinica (ASIPP), Hefei, under the auspices of the IAEA. About 40 delegates attended, with 30 participants coming from outside China. 50 papers were given, including six invited papers and six posters. The main subjects of the meeting were the theory of electron cyclotron resonance phenomena and the exploitation of these phenomena in plasma diagnostics (ECE) and plasma heating (ECRH). The meeting was lively and informative, and for almost every topic covered there were results which represent significant progress.