Sheared flows perpendicular to the magnetic field can be driven by the Reynolds stress or ion pressure gradient effects and can potentially influence the stability and turbulent saturation level of edge plasma modes. On the other hand, such flows are subject to the transverse Kelvin–Helmholtz (KH) instability. Here, the linear theory of KH instabilities is first addressed with an analytic model in the asymptotic limit of long wavelengths compared with the flow scale length. The analytic model treats sheared $\boldsymbol{E}\times \boldsymbol{B}$ flows, ion diamagnetism (including gyro-viscous terms), density gradients and parallel currents in a slab geometry, enabling a unified summary that encompasses and extends previous results. In particular, while ion diamagnetism, density gradients and parallel currents each individually reduce KH growth rates, the combined effect of density and ion pressure gradients is more complicated and partially counteracting. Secondly, the important role of realistic toroidal geometry is explored numerically using an invariant scaling analysis together with the 2DX eigenvalue code to examine KH modes in both closed and open field line regions. For a typical spherical torus magnetic geometry, it is found that KH modes are more unstable at, and just outside of, the separatrix as a result of the distribution of magnetic shear. Finally implications for reduced edge turbulence modelling codes are discussed.

The SIESTA magnetohydrodynamic (MHD) equilibrium code has been used to compute a sequence of ideally stable equilibria resulting from numerical variation of the helical resonant magnetic perturbation (RMP) applied to an axisymmetric DIII-D plasma equilibrium. Increasing the perturbation strength at the dominant $m=2$, $n=-1$ resonant surface leads to lower MHD energies and increases in the equilibrium island widths at the $m=2$ (and sidebands) surfaces, in agreement with theoretical expectations. Island overlap at large perturbation strengths leads to stochastic magnetic fields which correlate well with the experimentally inferred field structure. The magnitude and spatial phase (around the dominant rational surfaces) of the resonant (shielding) component of the parallel current are shown to change qualitatively with the magnetic island topology.

V. D. Shafranov was a key person in the fusion program. The paper presents the recollections of one of his close colleagues about Shafranov’s impact on the early days of tokamak research.

Using a numerical code based on guiding centre drift equations, collisionless high energy particle losses, and in particular ${\it\alpha}$-particle losses, are studied for a number of stellarator configurations in the presence of magnetic islands caused by resonant perturbations of magnetic surfaces. Standard stellarator configurations, as well as an optimized quasi-helically symmetric stellarator, are used in this study. It is found that the role of islands in collisionless ${\it\alpha}$-particle losses is practically negligible for standard stellarators, however, for optimized stellarators, islands can have a negative impact.

The understanding of plasma disruptions in tokamaks and predictions of their effects require realistic simulations of electric current excitation in three-dimensional vessel structures by the plasma touching the walls. As discovered at JET in 1996 (Litunovski JET Internal Report contract no. JQ5/11961, 1995; Noll et al., Proceedings of the 19th Symposium on Fusion Technology, Lisbon (ed. C. Varandas & F. Serra), vol. 1, 1996, p. 751. Elsevier) the wall-touching kink modes are frequently excited during vertical displacement events and cause large sideways forces on the vacuum vessel which are difficult to withstand in large tokamaks. In disruptions, the sharing of electric current between the plasma and the wall plays an important role in plasma dynamics and determines the amplitude and localization of the sideways force (Riccardo et al., Nucl. Fusion, vol. 40, 2000, p. 1805; Riccardo & Walker, Plasma Phys. Control. Fusion, vol. 42, 2000, p. 29; Zakharov, Phys. Plasmas, vol. 15, 2008, 062507; Riccardo et al., Nucl. Fusion, vol. 49, 2009, 055012; Bachmann et al., Fusion Engng Des., vol. 86, 2011, pp. 1915–1919). This paper describes a flat triangle representation of the electric circuits of a thin conducting wall of arbitrary three-dimensional geometry. Implemented into the shell simulation code (SHL) and the source sink current code (SSC), this model is suitable for modelling the electric currents excited in the wall inductively and through current sharing with the plasma.

Plasma confinement is based on the use of nested toroidal magnetic surfaces. In axisymmetric configurations the nestedness is provided by the existence of a flux function describing the magnetic surfaces explicitly. In the case of a three-dimensional magnetic field, the nested surfaces represent an exception. More typically, magnetic islands are formed on the resonant surfaces. The islands could degrade the plasma performance. The rigorous condition for the existence of nested surfaces without islands was formulated by Hamada (Nucl. Fusion, vol. 2, 1962, pp. 23–37) but was not implemented directly into numerical codes used, for example, for designing the stellarator configurations. This paper introduces a method of implementation of the Hamada principle in numerical algorithms. The proposed approach allows for simple linearized equilibrium equations (LEE) and potentially very efficient three-dimensional calculations of nested equilibria.

In this and the accompanying paper, the problem of the maximally achievable elongation ${\it\kappa}$ in a tokamak is investigated. The work represents an extension of many earlier studies, which were often focused on determining ${\it\kappa}$ limits due to (i) natural elongation in a simple applied pure vertical field or (ii) axisymmetric stability in the presence of a perfectly conducting wall. The extension investigated here includes the effect of the vertical stability feedback system which actually sets the maximum practical elongation limit in a real experiment. A basic resistive wall stability parameter, ${\it\gamma}{\it\tau}_{w}$, is introduced to model the feedback system which although simple in appearance actually captures the essence of the feedback system. Elongation limits in the presence of feedback are then determined by calculating the maximum ${\it\kappa}$ against $n=0$ resistive wall modes for fixed ${\it\gamma}{\it\tau}_{w}$. The results are obtained by means of a general formulation culminating in a variational principle which is particularly amenable to numerical analysis. The principle is valid for arbitrary profiles but simplifies significantly for the Solov’ev profiles, effectively reducing the 2-D stability problem into a 1-D problem. The accompanying paper provides the numerical results and leads to a sharp answer of ‘how much elongation is too much’?

The analytic theory presented in Paper I is converted into a form convenient for numerical analysis. A fast and accurate code has been written using this numerical formulation. The results are presented by first defining a reference set of physical parameters based on experimental data from high performance discharges. Scaling relations of maximum achievable elongation (${\it\kappa}_{max}$) versus inverse aspect ratio (${\it\varepsilon}$) are obtained numerically for various values of poloidal beta (${\it\beta}_{p}$), wall radius $(b/a)$ and feedback capability parameter (${\it\gamma}{\it\tau}_{w}$) in ranges near the reference values. It is also shown that each value of ${\it\kappa}_{max}$ occurs at a corresponding value of optimized triangularity (${\it\delta}$), whose scaling is also determined as a function of ${\it\varepsilon}$. The results show that the theoretical predictions of ${\it\kappa}_{max}$ are slightly higher than experimental observations for high performance discharges, as measured by high average pressure. The theoretical ${\it\delta}$ values are noticeably lower. We suggest that the explanation is associated with the observation that high performance involves not only $n=0$ MHD stability, but also $n\geqslant 1$ MHD modes described by ${\it\beta}_{N}$ in the Troyon limit and transport as characterized by ${\it\tau}_{E}$. Operation away from the $n=0$ MHD optimum may still lead to higher performance if there are more than compensatory gains in ${\it\beta}_{N}$ and ${\it\tau}_{E}$. Unfortunately, while the empirical scaling of ${\it\beta}_{N}$ and ${\it\tau}_{E}$ with the elongation (${\it\kappa}$) has been determined, the dependence on ${\it\delta}$ has still not been quantified. This information is needed in order to perform more accurate overall optimizations in future experimental designs.

Plasma stabilization due to a nearby conducting wall can provide access to better performance in some scenarios in tokamaks. This was proved by experiments with an essential gain in ${\it\beta}$ and demonstrated as a long-lasting effect at sufficiently fast plasma rotation in the DIII-D tokamak (see, for example, Strait et al., Nucl. Fusion, vol. 43, 2003, pp. 430–440). The rotational stabilization is the central topic of this review, though eventually the mode rotation gains significance. The analysis is based on the first-principle equations describing the energy balance with dissipation in the resistive wall. The method emphasizes derivation of the dispersion relations for the modes which are faster than the conventional resistive wall modes, but slower than the ideal magnetohydrodynamics modes. Both the standard thin wall and ideal-wall approximations are not valid in this range. Here, these are replaced by an approach incorporating the skin effect in the wall. This new element in the stability theory makes the energy sink a nonlinear function of the complex growth rate. An important consequence is that a mode rotating above a critical level can provide a damping effect sufficient for instability suppression. Estimates are given and applications are discussed.

This paper is dedicated to Vitaly Shafranov, who became increasingly interested in stellarators. Stellarators have a steady-state magnetic configuration, robust positional stability, and consistency with a plasma current below the level at which runaway electrons become a major issue. The development path for stellarators may be faster and cheaper than for tokamaks: stellarators are amenable to computer design validated by moderate scale experiments to circumvent issues that impede fusion development. This is distinct from the empirical explorations required to find an acceptable nonlinear, self-organized state of a tokamak. Fusion plasmas can be designed and controlled in stellarators in ways that are not possible in tokamaks. This paper outlines computational studies that could be carried at low cost during the next few years that would clarify the reactor potential of the stellarator and are needed for rational planning of the fusion program.

Novel free boundary magnetohydrodynamic equilibrium states with spontaneous three-dimensional (3-D) deformations of the plasma–vacuum interface are computed. The structures obtained look like saturated ideal external kink/peeling modes. Large edge pressure gradients yield toroidal mode number $n=1$ distortions when the edge bootstrap current is large and higher $n$ corrugations when this current is small. Linear ideal MHD stability analyses confirm the nonlinear saturated ideal kink equilibrium states produced and we can identify the Pfirsch–Schlüter current as the main linear instability driving mechanism when the edge pressure gradient is large. The dominant non-axisymmetric component of this Pfirsch–Schlüter current drives a near resonant helical parallel current density ribbon that aligns with the near vanishing magnetic shear region caused by the edge bootstrap current. This current ribbon is a manifestation of the outer mode previously found on JET (Solano 2010). We claim that the equilibrium corrugations describe structures that are commonly observed in quiescent H-mode tokamak discharges.

A formalism for consideration of island formation is presented using a model of a cylindrical resistively unstable plasma. Both current and pressure driven island formation at resonant surfaces are considered. The proposed formalism of perturbed equilibria avoids problems typical for linear analysis of resistive magneto-hydrodynamic instabilities related to extraction of the so-called small solution near the resonant surfaces. The matching technique of this paper is not sensitive to configuration parameters near the resonant surfaces. The comparison of the perturbed equilibrium method with the frequently used quasilinear mode analysis based on a perturbed averaged current density profile shows that the latter is limited in its applicability and underestimates the stability. Presented here for a cylindrical case, the perturbed equilibrium technique can be used in toroidal perturbed equilibrium codes with minor modifications.

The behaviour of runaway electrons have been investigated in lower hybrid current drive (LHCD) and electron cyclotron current drive (ECCD) plasmas as well as the LHCD only plasmas in the HL-2A tokamak. The fast electrons generated by lower hybrid waves (LHWs) and electron cyclotron waves (ECWs) can act as a seed population for runaway electrons. In the LHCD only discharges, a large number of runaway electrons are produced after the termination of lower hybrid (LH) power by conversion of fast electrons into runaway electrons due to the fast electron tail which extends above the runaway critical energy. However, in contrast to LHCD only discharges, during the simultaneous application of LHCD and ECCD discharges, runaway electrons cannot be created by the termination of LH power when the ECCD is on duty. The runaway production is observed to be enhanced until the EC power termination. The loop voltage increase due to the termination of EC power gives rise to a decline in the critical runaway energy, which leads to some of the energetic fast electrons converting into runaway electrons via the acceleration from the toroidal electric field. That is, the fast electrons created by waves can be accelerated into the runaway regime due to the Dreicer process.

In experiments with neutral beam injection at the early stage of a Globus-M discharge, instabilities were observed that were excited by fast ions in the frequency range of 50–200 kHz, which were identified as toroidal Alfvén eigenmodes (TAE) (Petrov et al., Plasma Phys. Rep., vol. 37, 2011, pp. 1001–1005). In contradiction with the NSTX and MAST tokamaks, a regime of TAE generation was realized with strongly developed single modes. Magnetic measurements with fast Mirnov probes have shown that most of the modes have toroidal number $n=1$. The influence of the modes on the fast particle confinement was recorded by means of a tangentially directed neutral particle analyser (NPA) and neutron detector. Hydrogen and deuterium were used as target plasma and injected beam for study of the isotopic effect. At deuterium injection into the deuterium plasma, TAE led to the neutron rate dropping by 25 %, whereas NPA fluxes of high energy dropped by 75 %. At hydrogen injection, the drop in the measured NPA fluxes did not exceed 25 %.

JET experiments have compared the efficacy of low- and high-field side ion cyclotron resonance heating (ICRH) as an actuator to deliberately minimise the sawtooth period. It is found that low-field side ICRH with low minority concentration is optimal for sawtooth control for two main reasons. Firstly, low-field side heating means that any toroidal phasing of the ICRH ($-90^{\circ }$, $+90^{\circ }$ or dipole) has a destabilising effect on the sawteeth, meaning that dipole phasing can be employed, since this is preferable due to less plasma wall interaction from Resonant Frequency (RF) sheaths. Secondly, the resonance position of the low-field side ICRH does not have to be very accurately placed to achieve sawtooth control, relaxing the requirement for real-time control of the RF frequency. These empirical observations have been confirmed by hybrid kinetic–magnetohydrodynamic modelling, and suggest that the ICRH antenna design for ITER is well positioned to provide a control actuator capable of having a significant effect on the sawtooth behaviour.

In honour of the 50th anniversary of the influential review/monograph on plasma turbulence by B. B. Kadomtsev as well as the seminal works of T. H. Dupree and J. Weinstock on resonance-broadening theory, an introductory tutorial is given about some highlights of the statistical–dynamical description of turbulent plasmas and fluids, including the ideas of nonlinear incoherent noise, coherent damping, and self-consistent dielectric response. The statistical closure problem is introduced. Incoherent noise and coherent damping are illustrated with a solvable model of passive advection. Self-consistency introduces turbulent polarization effects that are described by the dielectric function ${\mathcal{D}}$. Dupree’s method of using ${\mathcal{D}}$ to estimate the saturation level of turbulence is described; then it is explained why a more complete theory that includes nonlinear noise is required. The general theory is best formulated in terms of Dyson equations for the covariance $C$ and an infinitesimal response function $R$, which subsumes ${\mathcal{D}}$. An important example is the direct-interaction approximation (DIA). It is shown how to use Novikov’s theorem to develop an $\boldsymbol{x}$-space approach to the DIA that is complementary to the original $\boldsymbol{k}$-space approach of Kraichnan. A dielectric function is defined for arbitrary quadratically nonlinear systems, including the Navier–Stokes equation, and an algorithm for determining the form of ${\mathcal{D}}$ in the DIA is sketched. The independent insights of Kadomtsev and Kraichnan about the problem of the DIA with random Galilean invariance are described. The mixing-length formula for drift-wave saturation is discussed in the context of closures that include nonlinear noise (shielded by ${\mathcal{D}}$). The role of $R$ in the calculation of the symmetry-breaking (zonostrophic) instability of homogeneous turbulence to the generation of inhomogeneous mean flows is addressed. The second-order cumulant expansion and the stochastic structural stability theory are also discussed in that context. Various historical research threads are mentioned and representative entry points to the literature are given. Some outstanding conceptual issues are enumerated.

Disruptions with runaway electron generation have been deliberately induced by injection of argon using a disruption mitigation valve. A second disruption mitigation valve has been utilised to inject varying amounts of helium after a short time delay. No generation of runaway electrons has been observed when more than a critical amount of helium has been injected no later than 5 ms after the triggering of the first valve. The required amount of helium for suppression of runaway electron generation is up to one order of magnitude lower than the critical density according to Connor & Hastie (1975) and Rosenbluth & Putvinski (1997).

This study demonstrates a remarkable flexibility of advanced divertor configurations created with the remote poloidal field coils. The emphasis here is on the configurations with three poloidal field nulls in the divertor area. We are seeking the structures where all three nulls lie on the same separatrix, thereby creating two zones of a very strong flux expansion, as envisaged in the concept of Takase’s cusp divertor. It turns out that the set of remote coils can indeed produce a cusp divertor, with additional advantages of: (i) a large stand-off distance between the divertor and the coils and (ii) a thorough control that these coils exert over the fine features of the configuration. In reference to these additional favourable properties acquired by the cusp divertor, the resulting configuration could be called ‘a super-cusp’. General geometrical features of the three-null configurations produced by remote coils are described. Issues on the way to practical applications include the need for a more sophisticated control system and possible constraints related to excessively high currents in the divertor coils.

For ITER-relevant runaway electron studies, such as suppression, mitigation, termination and/or control of a runaway beam, it is important to obtain the runaway electrons after the disruption. In this paper we report on the first discharges achieved with a post-disruptive runaway electron beam, termed a ‘runaway plateau’, in the COMPASS tokamak. The runaway plateau is produced by a massive gas injection of argon. Almost all of the disruptions with runaway electron plateaus occurred during the plasma current ramp-up phase. The Ar injection discharges with and without a runaway plateau were compared for various parameters. Parametrisation of the discharges shows that the COMPASS disruptions fulfil the range of parameters important for runaway plateau occurrence. These parameters include electron density, electric field, disruption speed, effective safety factor, and the maximum current quench electric field. In addition to these typical parameters, the plasma current value just before the massive gas injection proved to be surprisingly important.