We carry out expansions of non-symmetric toroidal ideal magnetohydrodynamic (MHD) equilibria with nested flux surfaces about a periodic cylinder, in which physical quantities are periodic of period $2\unicode[STIX]{x03C0}$ in the cylindrical angle $\unicode[STIX]{x1D703}$ and $z$. The cross-section of a flux surface at a constant toroidal angle is assumed to be approximately circular, and data are given on the cylindrical flux surface $r=1$. Furthermore, we assume that the magnetic field lines are closed on the lowest-order flux surface, and the magnetic shear is relatively small. We extend earlier work in a flat torus by Weitzner (Phys. Plasmas, vol. 23, 2016, 062512) and demonstrate that a power series expansion can be carried out to all orders using magnetic flux as an expansion parameter. The cylindrical metric introduces certain new features to the expansions compared to the flat torus. However, the basic methodology of dealing with resonance singularities remains the same. The results, even though lacking convergence proofs, once again support the possibility of smooth, low-shear non-symmetric toroidal MHD equilibria.

The $\unicode[STIX]{x1D6FF}f$-PIC method is widely used for electrostatic particle-in-cell (PIC) simulations. Its basic idea is the ansatz $f=f_{0}+\unicode[STIX]{x1D6FF}f$ ($\unicode[STIX]{x1D6FF}f$-ansatz) where the particle distribution function $f$ is split into a usually time-independent background $f_{0}$ and a time-dependent perturbation $\unicode[STIX]{x1D6FF}f$. In principle, it can also be used for electromagnetic gyrokinetic PIC simulations, but the required number of markers can be so large that PIC simulations become impractical. The reason is a decreasing efficiency of the $\unicode[STIX]{x1D6FF}f$-ansatz for the so-called ‘Hamiltonian formulation’ using $p_{\Vert }$ as a dynamic variable. As a result, the density and current moment of the distribution function develop large statistical errors. To overcome this obstacle we propose to solve the potential equations in the symplectic formulation using $v_{\Vert }$ as a dynamic variable. The distribution function itself is still evolved in the Hamiltonian formulation which is better suited for the numerical integration of the parallel dynamics. The contributions from the full Jacobian of phase space, a finite velocity sphere of the simulation domain and a shifted Maxwellian as a background are considered. Special care has been taken at the discretisation level to make damped magnetohydrodynamics (MHD) mode simulations within a realistic gyrokinetic model feasible. This includes devices like e.g. large tokamaks with a small aspect ratio.

This tutorial covers validation of gyrokinetic turbulent-transport models via comparison of measured turbulence with realistic simulations of fusion plasmas. It presents a brief history of validation of gyrokinetic simulations, the principal challenges encountered, a limited survey of common turbulence diagnostics used on tokamaks and stellarators, an overview of the fundamentals of synthetic diagnostic models and a discussion of frontiers in turbulent-transport model validation.

The injection of cryogenic pellets into a magnetically confined plasma is shown to be accompanied by a considerable transfer of thermal energy from the electrons in the background plasma to the ions. The resulting ion heating can be significant, particularly in plasmas with disparate electron and ion temperatures, and can affect the energy balance of the plasma. In recent Wendelstein 7-X experiments, this mechanism can account for a substantial fraction of the ion heating power during pellet injection.

The formation of zonal flows from inhomogeneous drift-wave (DW) turbulence is often described using statistical theories derived within the quasilinear approximation. However, this approximation neglects wave–wave collisions. Hence, some important effects such as the Batchelor–Kraichnan inverse-energy cascade are not captured within this approach. Here we derive a wave kinetic equation that includes a DW collision operator in the presence of zonal flows. Our derivation makes use of the Weyl calculus, the quasinormal statistical closure and the geometrical-optics approximation. The obtained model conserves both the total enstrophy and energy of the system. The derived DW collision operator breaks down at the Rayleigh–Kuo threshold. This threshold is missed by homogeneous-turbulence theory but expected from a full-wave quasilinear analysis. In the future, this theory might help better understand the interactions between drift waves and zonal flows, including the validity domain of the quasilinear approximation that is commonly used in the literature.

Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding-centre particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength $B$. Here, instead, we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in the distance from the magnetic axis, without using optimization. The method involves solution of a one-dimensional nonlinear ordinary differential equation, originally derived by Garren & Boozer (Phys. Fluids B, vol. 3, 1991, p. 2805). We demonstrate the usefulness and accuracy of this optimization-free approach by providing the results of this construction as input to the codes VMEC and BOOZ_XFORM, confirming the purity and scaling of the magnetic spectrum. The space of magnetic fields that are quasisymmetric to this order is parameterized by the magnetic axis shape along with three other real numbers, one of which reflects the on-axis toroidal current density, and another one of which is zero for stellarator symmetry. The method here could be used to generate good initial conditions for conventional optimization, and its speed enables exhaustive searches of parameter space.

The confinement of the guiding-centre trajectories in a stellarator is determined by the variation of the magnetic field strength $B$ in Boozer coordinates $(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$, but $B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ depends on the flux surface shape in a complicated way. Here we derive equations relating $B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ in Boozer coordinates and the rotational transform to the shape of flux surfaces in cylindrical coordinates, using an expansion in distance from the magnetic axis. A related expansion was done by Garren and Boozer (Phys. Fluids B, vol. 3, 1991a, 2805) based on the Frenet–Serret frame, which can be discontinuous anywhere the magnetic axis is straight, a situation that occurs in the interesting case of omnigenity with poloidally closed $B$ contours. Our calculation in contrast does not use the Frenet–Serret frame. The transformation between the Garren–Boozer approach and cylindrical coordinates is derived, and the two approaches are shown to be equivalent if the axis curvature does not vanish. The expressions derived here help enable optimized plasma shapes to be constructed that can be provided as input to VMEC and other stellarator codes, or to generate initial configurations for conventional stellarator optimization.

High-energy ions, such as fusion alphas and ions from external heating, can be very sensitive to any non-axisymmetric features in the confining magnetic field due to their collisionless nature. Since understanding the confinement properties of these ions is crucial for ITER (the first fusion reactor currently under construction in Cadarache, France) and beyond, it is of ultimate importance that the predictive simulations are accurate and free of numerical distortions. Adding the third dimension comes at substantial computational cost, calling for new kinds of approaches and computational platforms. In this contribution we discuss what new features, even new physics, the non-axisymmetry brings with it and how one could cope with the ever-increasing demands on both memory and CPU resources. In the end, a few simulation examples with a varying level of non-axisymmetry are given.

Radio frequency (RF) waves are routinely used in tokamak fusion plasmas for plasma heating, current control, as well as in diagnostics. These waves are excited by antenna structures placed near the tokamak’s wall and they have to propagate through a turbulent layer known as the scrape-off layer, before reaching the core plasma (which is their target). This layer exhibits coherent density fluctuations in the form of filaments and blobs. The scattering processes of RF plane waves by a single filament is studied with the assumption that the filament has a cylindrical shape and infinite length. Furthermore, besides the major toroidal component of the externally imposed magnetic field, there is also a small poloidal magnetic field component. Considering also that the cylindrical filament’s axis is not necessarily aligned with the toroidal direction, the total magnetic field is in general neither aligned with the axis of the cylinder nor with the toroidal direction. The investigation concerns the case of electron cyclotron (EC) waves (of frequency $f_{0}=170~\text{GHz}$) for tokamak applications. The study covers a variety of density contrasts between the filament and the ambient plasma, different magnetic field inclinations with respect to the cylinder axis (for the same magnitude of magnetic induction $B=4.5T$) and a wide range of filament radii.

Kinetic ballooning modes (KBMs) are investigated by means of linear electromagnetic gyrokinetic (GK) simulations in the stellarator Wendelstein 7-X (W7-X), for high-$\unicode[STIX]{x1D6FD}$ plasmas, where $\unicode[STIX]{x1D6FD}$ is the ratio of thermal to magnetic plasma pressure. The analysis shows suppression of ion-temperature-gradient (ITG) and trapped particle modes (TEM) by finite-$\unicode[STIX]{x1D6FD}$ effects and destabilization of KBMs at high $\unicode[STIX]{x1D6FD}$. The results are compared with a generic tokamak case. We show that, for large pressure gradients, the frequency of KBMs evaluated by the GENE code is in agreement with the analytical prediction of the diamagnetic modification of the ideal magnetohydrodynamic limit in W7-X general geometry. Thresholds for destabilization of the KBM are predicted for different W7-X equilibrium configurations. We discuss the relation of these thresholds to the ideal magnetohydrodynamic (MHD) stability properties of the corresponding equilibria.

In tokamak plasmas, sheared flows perpendicular to the driving temperature gradients can strongly stabilise linear modes. While the system is linearly stable, regimes with persistent nonlinear turbulence may develop, i.e. the system is subcritical. A perturbation with small but finite amplitude may be sufficient to push the plasma into a regime where nonlinear effects are dominant and thus allow sustained turbulence. The minimum threshold for nonlinear instability to be triggered provides a criterion for assessing whether a tokamak is likely to stay in the quiescent (laminar) regime. At the critical amplitude, instead of transitioning to the turbulent regime or decaying to a laminar state, the trajectory will map out the edge of chaos. Surprisingly, a quasi-travelling-wave solution is found as an attractor on this edge manifold. This simple advecting solution is qualitatively similar to, but simpler than, the avalanche-like bursts seen in earlier turbulent simulations and provides an insight into how turbulence is sustained in subcritical plasma systems. For large flow shearing rate, the system is only convectively unstable, and given a localised initial perturbation, will eventually return to a laminar state at a fixed spatial location.

Accurate modelling of the interaction between fast electrons and partially ionized atoms is important for evaluating tokamak disruption mitigation schemes based on material injection. This requires accounting for the effect of screening of the impurity nuclei by the cloud of bound electrons. In this paper, we generalize the Fokker–Planck operator in a fully ionized plasma by accounting for the effect of screening. We detail the derivation of this generalized operator, and calculate the effective ion length scales, needed in the components of the collision operator, for a number of ion species commonly appearing in fusion experiments. We show that for high electric fields, the secondary runaway growth rate can be substantially larger than in a fully ionized plasma with the same effective charge, although the growth rate is significantly reduced at near-critical electric fields. Furthermore, by comparison with the Boltzmann collision operator, we show that the Fokker–Planck formalism is accurate even for large impurity content.

The drift surfaces of minority heated ions differ from flux surfaces due to finite poloidal gyroradius effects. As the minority poloidal gyroradius approaches radial scale lengths in the plasma, the difference between drift and flux surfaces can modify the heating and lead to a symmetric spectrum minority counter-current being driven. In response, a corresponding overall net co-current of comparable size is driven. This beneficial symmetric spectrum current drive in a tokamak is due to the parallel velocity asymmetry in the drift departure from a flux surface. As this new source of driven current is a side effect of minority heating it comes without any additional economic cost to reactor power balance. The symmetric spectrum current driven for near Maxwellian minorities is evaluated by an adjoint method and found to be modest. However, minority heating typically results in strong non-Maxwellian features on minority distributions so it may be possible to drive a significantly larger co-current. A related evaluation is performed for alpha particles in a deuterium minority heated plasma with a tritium majority. The low density of the alphas tends to keep this driven symmetric spectrum current small, but at very high heating levels a significant co-current might be driven. Other mechanisms to drive co-current with a symmetric spectrum are discussed and estimated, including asymmetric electron drag and focusing of the applied minority heating radio frequency fields.

Tokamak start-up is strongly dependent on the state of the initial plasma formed during plasma breakdown. We have investigated through numerical simulations the effects that the pre-filling pressure and induced electric field have on pure ohmic heating during the breakdown process. Three breakdown modes during the discharge are found, as a function of different initial parameters: no breakdown mode, successful breakdown mode and runaway mode. No breakdown mode often occurs with low electric field or high pre-filling pressure, while runaway electrons are usually easy to generate at high electric field or low pre-filling pressure (${<}1.33\times 10^{-4}$  Pa). The plasma behaviours and the physical mechanisms under the three breakdown modes are discussed. We have identified the electric field and pressure values at which the different modes occur. In particular, when the electric field is $0.3~\text{V}~\text{m}^{-1}$ (the value at which ITER operates), the pressure range for possible breakdown becomes narrow, which is consistent with Lloyd’s theoretical prediction. In addition, for $0.3~\text{V}~\text{m}^{-1}$, the optimal pre-filling pressure range obtained from our simulations is $1.33\times 10^{-3}\sim 2.66\times 10^{-3}$  Pa, in good agreement with ITER’s design. Besides, we also find that the Townsend discharge model does not appropriately describe the plasma behaviour during tokamak breakdown due to the presence of a toroidal field. Furthermore, we suggest three possible operation mechanisms for general start-up scenarios which could better control the breakdown phase.

In a reactor plasma like demonstration power station (DEMO), when using the radio frequency (RF) for heating or current drive in the lower hybrid (LH) frequency range (Franke et al., Fusion Engng Des., vol. 96–97, 2015, p. 46; Cardinali et al., Plasma Phys. Control. Fusion, vol. 59, 2017, 074002), a large fraction of the ion population (the continuously born $\unicode[STIX]{x1D6FC}$-particle, and/or the neutral beam injection (NBI) injected ions) is characterized by a non-thermal distribution function. The interaction (propagation and absorption) of the LH wave must be reformulated by considering the quasi-linear approach for each species separately. The collisional slowing down of such an ion population in a background of an electron and ion plasma is balanced by a quasi-linear diffusion in velocity space due to the propagating electromagnetic wave. In this paper, both propagations are considered by including the ion distribution function, solution of the Fokker–Planck equation, which describes the collisional dynamics of the $\unicode[STIX]{x1D6FC}$-particles including the effects of frictional slowing down, energy diffusion and pitch-angle scattering. Analytical solutions of the Fokker–Planck equation for the distribution function of $\unicode[STIX]{x1D6FC}$-particles with a background of ions and electrons at steady state are included in the calculation of the dielectric tensor. In the LH frequency domain, ray tracing (including quasi-linear damping), can be analytically solved by iterating with the Fokker–Planck solution, and the interaction of the LH wave with $\unicode[STIX]{x1D6FC}$-particles, thermal ions and electrons can be accounted self-consistently and the current drive efficiency can be evaluated in this more general scenario.

An increase in symmetry is observed for a low density non-collisional plasma, in a simple magnetic mirror machine, due to the application of external oscillating magnetic fields of 1.5 MHz frequency. The increase in symmetry is attributed to an increase in stability of the flute mode and is dependent on the field’s polarization and trap magnetic field strength.

Gyrokinetic simulations of drift waves in low-magnetic-shear stellarators reveal that simulation domains comprised of multiple turns can be required to properly resolve critical mode structures important in saturation dynamics. Marginally stable eigenmodes important in saturation of ion temperature gradient modes and trapped electron modes in the Helically Symmetric Experiment (HSX) stellarator are observed to have two scales, with the envelope scale determined by the properties of the local magnetic shear and an inner scale determined by the interplay between the local shear and magnetic field-line curvature. Properly resolving these modes removes spurious growth rates that arise for extended modes in zero-magnetic-shear approximations, enabling use of a zero-magnetic-shear technique with smaller simulation domains and attendant cost savings. Analysis of subdominant modes in trapped electron mode (TEM)-driven turbulence reveals that the extended marginally stable modes play an important role in the nonlinear dynamics, and suggests that the properties induced by low magnetic shear may be exploited to provide another route for turbulence saturation.

There has been a growing interest, over the past few years, on understanding the effect on radio frequency waves due to turbulence in the scrape-off layer of tokamak plasmas. While the far scrape-off layer density width is of the order of centimetres in contemporary tokamaks, in ITER (International Thermonuclear Experimental Reactor), and in future fusion reactors, the corresponding width will be of the order of tens of centimetres. As such, this could impact the spectral properties of the waves and, consequently, the transport of wave energy and momentum to the core plasma. The turbulence in the scrape-off layer spans a broad range of spatial scales and includes blobs and filaments that are elongated along the magnetic field lines. The propagation of radio frequency waves through this tenuous plasma is given by Maxwell’s equations. The characteristic properties of the plasma appear as a permittivity tensor in the expression for the current in Ampere’s equation. This paper develops a formalism for expressing the permittivity of a turbulent plasma using the homogenization technique. This technique has been extensively used to express the dielectric properties of composite materials that are spatially inhomogeneous, for example, due to the presence of micro-structures. In a similar vein, the turbulent plasma in the scrape-off layer is spatially inhomogeneous and can be considered as a composite material in which the micro-structures are filaments and blobs. The classical homogenization technique is not appropriate for the magnetized plasma in the scrape-off layer, as the radio frequency waves span a broad range of wavelengths and frequencies – from tens of megahertz to hundreds of gigahertz. The formalism in this paper makes use of the Fourier space components of the electric and magnetic fields of the radio frequency waves for the scattered fields and fields inside the filaments and blobs. These are the eigenvectors of the dispersion matrix which, using the Green’s function approach, lead to a homogenized dielectric tensor.

High-$Z$ impurities in magnetic-confinement devices are prone to develop density variations on the flux surface, which can significantly affect their transport. In this paper, we generalize earlier analytic stellarator calculations of the neoclassical radial impurity flux in the mixed-collisionality regime (collisional impurities and low-collisionality bulk ions) to include the effect of such flux-surface variations. We find that only in the homogeneous density case is the transport of highly collisional impurities (in the Pfirsch–Schlüter regime) independent of the radial electric field. We study these effects for a Wendelstein 7-X (W7-X) vacuum field, with simple analytic models for the potential perturbation, under the assumption that the impurity density is given by a Boltzmann response to a perturbed potential. In the W7-X case studied, we find that larger amplitude potential perturbations cause the radial electric field to dominate the transport of the impurities. In addition, we find that classical impurity transport can be larger than the neoclassical transport in W7-X.

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tilde{\unicode[STIX]{x1D711}}$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tilde{\unicode[STIX]{x1D711}}$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio and the structure of $\tilde{\unicode[STIX]{x1D711}}$ on the magnetic surface are analytically derived in the $1/\unicode[STIX]{x1D708}$, $\sqrt{\unicode[STIX]{x1D708}}$ and superbanana-plateau regimes of stellarators close to omnigeneity; i.e. stellarators that have been optimized for neoclassical transport. It is found that the largest $\tilde{\unicode[STIX]{x1D711}}$ that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigenous configuration, the analytical results are verified and illustrated with calculations by the code KNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry.