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 %.

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.

We present analytic solutions for three-dimensional magnetized axisymmetric equilibria confining rotating hot plasma in a gravitational field. Our up–down symmetric solution to the full Grad–Shafranov equation can exhibit equatorial plane localization of the plasma density and current, resulting in disk equilibria for the plasma density. For very weak magnetic fields and high plasma pressure, we find strongly rotating thin plasma disk gravitational equilibria that satisfy strict Keplerian motion provided the gravitational energy is much larger than the plasma pressure, which must be large compared to the magnetic energy of the poloidal magnetic field. When the rotational energy exceeds the gravitational energy and it is larger than the plasma pressure, diffuse disk equilibrium solutions continue to exist provided the poloidal magnetic energy remains small. For stronger magnetic fields and lower plasma pressure and rotation, we can also find gravitational equilibria with strong localization to the equatorial plane. However, a toroidal magnetic field is almost always necessary to numerically verify these equilibria are valid solutions in the presence of gravity for the cases considered in Catto & Krasheninnikov (J. Plasma Phys., vol. 81, 2015, 105810301). In all cases both analytic and numerical results are presented.

Ideal magnetohydrodynamics (IMHD) is strongly constrained by an infinite number of microscopic constraints expressing mass, entropy and magnetic flux conservation in each infinitesimal fluid element, the latter preventing magnetic reconnection. By contrast, in the Taylor relaxation model for formation of macroscopically self-organized plasma equilibrium states, all these constraints are relaxed save for the global magnetic fluxes and helicity. A Lagrangian variational principle is presented that leads to a new, fully dynamical, relaxed magnetohydrodynamics (RxMHD), such that all static solutions are Taylor states but also allows state with flow. By postulating that some long-lived macroscopic current sheets can act as barriers to relaxation, separating the plasma into multiple relaxation regions, a further generalization, multi-region relaxed magnetohydrodynamics (MRxMHD) is developed.

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.

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.

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 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.

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.

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.

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.

An explanation is provided for the disruptive instability in diverted tokamaks when the safety factor $q$ at the 95 % poloidal flux surface, $q_{95}$ , is driven below 2.0. The instability is a resistive kink counterpart to the current-driven ideal mode that traditionally explained the corresponding disruption in limited cross-sections (Shafranov, Sov. Phys. Tech. Phys., vol. 15, 1970, p. 175) when $q_{edge}$ , the safety factor at the outermost closed flux surface, lies just below a rational value $m/n$ . Experimentally, external kink modes are observed in limiter configurations as the current in a tokamak is ramped up and $q_{edge}$ decreases through successive rational surfaces. For $q_{edge}<2$ , the instability is always encountered and is highly disruptive. However, diverted plasmas, in which $q_{edge}$ is formally infinite in the magnetohydrodynamic (MHD) model, have presented a longstanding difficulty since the theory would predict stability, yet, the disruptive limit occurs in practice when $q_{95}$ , reaches 2. It is shown from numerical calculations that a resistive kink mode is linearly destabilized by the rapidly increasing resistivity at the plasma edge when $q_{95}<2$ , but $q_{edge}\gg 2$ . The resistive kink behaves much like the ideal kink with predominantly kink or interchange parity and no real sign of a tearing component. However, the growth rates scale with a fractional power of the resistivity near the $q=2$ surface. The results have a direct bearing on the conventional edge cutoff procedures used in most ideal MHD codes, as well as implications for ITER and for future reactor options.