The rapidly emerging technology of high-temperature superconductors (HTS) opens new opportunities for the development of non-planar non-insulated HTS magnets. This type of HTS magnet offers attractive features via its simplicity and robustness, and is well suited for modest size steady-state applications such as a mid-scale stellarator. In non-planar coil applications the HTS tape may be subject to severe hard-way bending strain ($\epsilon _{\textrm {bend}}$), torsional strains ($\epsilon _{\textrm {tor}}$) and magnetic field components transverse to the HTS tape plane ($B_{\perp }$), all of which can limit the magnet operating space. A novel method of winding angle optimization is here presented to overcome these limitations for fixed input non-planar coil filamentary geometry. Essentially, this method: (i) calculates the peak $\epsilon _{\textrm {bend}}$ and $B_{\perp }$ for arbitrary winding angle along an input coil filamentary trajectory, (ii) defines a cost function including both and then (iii) uses tensioned splines to define a winding angle that reduces $\epsilon _{\textrm {tor}}$ and optimizes the $\epsilon _{\textrm {bend}}$ and $B_{\perp }$ cost function. As strain limits are present even without $B_{\perp }$, this optimization is able to provide an assessment of the minimum buildable size of an arbitrary non-planar non-insulating HTS coil. This optimization finds that for standard 4 mm wide HTS tapes the minimum size coils of the existing HSX, NCSX and W7-X stellarator geometries are around 0.3–0.5 m in mean coil radius. Identifying the minimum size provides a path to specify a mid-scale stellarator capable of achieving high-field or high-temperature operation with minimal HTS tape length. For coils larger than this size, strain optimization allows use of wider (higher current capacity) HTS tapes or alternatively permitting a finite (yet tolerable) strain allows reduction of $B_{\perp }$. Reduced $B_{\perp }$ enables a reduction of the HTS tape length required to achieve a given design magnetic field or equivalently an increase in the achievable magnetic field for fixed HTS tape length. The distinct considerations for optimizing a stellarator coilset to further ease compatibility with non-insulated HTS magnets are also discussed, highlighting relaxed curvature limits and the introduction of limits to the allowable torsion.

The purpose of this paper is to bridge kinetic plasma descriptions and low-frequency single-fluid models. More specifically, the asymptotics leading to magnetohydrodynamic regimes starting from the Vlasov–Maxwell system are investigated. The analogy with the derivation, from the Vlasov–Poisson system, of a fluid representation for ions coupled to the Boltzmann relation for electrons is also outlined. The aim is to identify asymptotic parameters explaining the transitions from one microscopic description to a macroscopic low-frequency model. These investigations provide groundwork for the derivation of multi-scale numerical methods, model coupling or physics-based preconditioning.

We investigate kinetic entropy-based measures of the non-Maxwellianity of distribution functions in plasmas, i.e. entropy-based measures of the departure of a local distribution function from an associated Maxwellian distribution function with the same density, bulk flow and temperature as the local distribution. First, we consider a form previously employed by Kaufmann & Paterson (J. Geophys. Res., vol. 114, 2009, A00D04), assessing its properties and deriving equivalent forms. To provide a quantitative understanding of it, we derive analytical expressions for three common non-Maxwellian plasma distribution functions. We show that there are undesirable features of this non-Maxwellianity measure including that it can diverge in various physical limits and elucidate the reason for the divergence. We then introduce a new kinetic entropy-based non-Maxwellianity measure based on the velocity-space kinetic entropy density, which has a meaningful physical interpretation and does not diverge. We use collisionless particle-in-cell simulations of two-dimensional anti-parallel magnetic reconnection to assess the kinetic entropy-based non-Maxwellianity measures. We show that regions of non-zero non-Maxwellianity are linked to kinetic processes occurring during magnetic reconnection. We also show the simulated non-Maxwellianity agrees reasonably well with predictions for distributions resembling those calculated analytically. These results can be important for applications, as non-Maxwellianity can be used to identify regions of kinetic-scale physics or increased dissipation in plasmas.

A Hamiltonian and action principle formalism for deriving three-dimensional gyroviscous magnetohydrodynamic models is presented. The uniqueness of the approach in constructing the gyroviscous tensor from first principles and its ability to explain the origin of the gyromap and the gyroviscous terms are highlighted. The procedure allows for the specification of free functions, which can be used to generate a wide range of gyroviscous models. Through the process of reduction, the noncanonical Hamiltonian bracket is obtained and briefly analysed.

In this work, the FALCON code is adopted for illustrating the features of shear Alfvén and sound continuous spectra in toroidal fusion plasmas. The FALCON codes employ the local Floquet analysis discussed in (Phys. Plasmas, vol. 26, issue 8, 2019, 082502) for computing global structures of continuous spectra in general toroidal geometry. As particular applications, reference equilibria for the divertor tokamak test and ASDEX Upgrade plasmas are considered. In particular, we illustrate the importance of mode polarization for recognizing the physical relevance of the various branches of the continuous spectra in the ideal magnetohydrodynamics limit. We also analyse the effect of plasma compression and the validity of the slow sound approximation.

High-resolution spectra emitted by laboratory plasmas provide invaluable diagnostic tools for the measurement of plasma properties. To be implemented, they require a large amount of atomic data and transition rates, which are available in several spectral codes. In this paper we present a new feature added to the CHIANTI code, which allows us to calculate the Zeeman splitting of spectral lines in the presence of a magnetic field with known intensity and orientation. When combined with the CHIANTI database and software to calculate level populations and line emissivities, this new feature returns the emissivities in all four Stokes parameters, that can be utilized for the measurement of the magnetic field inside laboratory plasma chambers, along with other plasma parameters. This new feature can be applied to the analysis of the emission of laboratory plasmas created in different devices.

We focus our attention on some relevant aspects of the beam–plasma instability in order to refine some features of the linear and nonlinear dynamics. After a re-analysis of the Poisson equation and of the assumption dealing with the background plasma in the form of a linear dielectric, we study the non-perturbative properties of the linear dispersion relation, showing the necessity for a better characterization of the mode growth rate in those flat regions of the distribution function where the Landau formula is no longer predictive. We then upgrade the original $N$-body approach in O'Neil et al. (Phys. Fluids, vol. 14, 1971, pp. 1204–1212), in order to include a return current in the background plasma. This correction term is responsible for smaller saturation levels and growth rates of the Langmuir modes, as result of the energy density transferred to the plasma via the return current. Finally, we include friction effects, as those due to the collective influence of all the plasma charges on the motion of the beam particles. The resulting force induces a progressive resonance detuning, because particles are losing energy and decreasing their velocity. This friction phenomenon gives rise to a deformation of the distribution function, associated with a significant growth of the less energetic particle population. The merit of this work is to show how a fine analysis of the beam–plasma instability outlines a number of subtleties about the linear, intermediate and late dynamics which can be of relevance when such a system is addressed as a paradigm to describe relevant nonlinear wave–particle phenomena (Chen & Zonca, Rev. Mod. Phys., vol. 88, 2016, 015008).

In this work, the first nonlinear particle-in-cell simulations carried out in a stellarator with the global gyrokinetic code EUTERPE using adiabatic electrons and realistic plasma parameters are reported. Several studies are conducted with the aim of enabling reliable nonlinear simulations in stellarators with this code. First, EUTERPE is benchmarked against ORB5 in both linear and nonlinear settings in a tokamak configuration. Next, the use of noise control and stabilization tools, a Krook-type collision operator, markers’ weight smoothing and heating sources is investigated. It is studied in detail how these tools influence the linear growth rate of instabilities in both tokamak and stellarator geometries, and their influence on the linear zonal flow evolution in a stellarator. Then, it is studied how these tools allow improvement of the quality of the results in a set of nonlinear simulations of electrostatic turbulence in a stellarator configuration. Finally, these tools are applied to a W7-X magnetic configuration using experimental plasma parameters.

The saturated state of turbulence driven by the ion-temperature-gradient instability is investigated using a two-dimensional long-wavelength fluid model that describes the perturbed electrostatic potential and perturbed ion temperature in a magnetic field with constant curvature (a $Z$-pinch) and an equilibrium temperature gradient. Numerical simulations reveal a well-defined transition between a finite-amplitude saturated state dominated by strong zonal-flow and zonal temperature perturbations, and a blow-up state that fails to saturate on a box-independent scale. We argue that this transition is equivalent to the Dimits transition from a low-transport to a high-transport state seen in gyrokinetic numerical simulations (Dimits et al., Phys. Plasmas, vol. 7, 2000, 969). A quasi-static staircase-like structure of the temperature gradient intertwined with zonal flows, which have patch-wise constant shear, emerges near the Dimits threshold. The turbulent heat flux in the low-collisionality near-marginal state is dominated by turbulent bursts, triggered by coherent long-lived structures closely resembling those found in gyrokinetic simulations with imposed equilibrium flow shear (van Wyk et al., J. Plasma Phys., vol. 82, 2016, 905820609). The breakup of the low-transport Dimits regime is linked to a competition between the two different sources of poloidal momentum in the system – the Reynolds stress and the advection of the diamagnetic flow by the $\boldsymbol {E}\times \boldsymbol {B}$ flow. By analysing the linear ion-temperature-gradient modes, we obtain a semi-analytic model for the Dimits threshold at large collisionality.

The SPARC tokamak is a critical next step towards commercial fusion energy. SPARC is designed as a high-field ($B_0 = 12.2$ T), compact ($R_0 = 1.85$ m, $a = 0.57$ m), superconducting, D-T tokamak with the goal of producing fusion gain $Q>2$ from a magnetically confined fusion plasma for the first time. Currently under design, SPARC will continue the high-field path of the Alcator series of tokamaks, utilizing new magnets based on rare earth barium copper oxide high-temperature superconductors to achieve high performance in a compact device. The goal of $Q>2$ is achievable with conservative physics assumptions ($H_{98,y2} = 0.7$) and, with the nominal assumption of $H_{98,y2} = 1$, SPARC is projected to attain $Q \approx 11$ and $P_{\textrm {fusion}} \approx 140$ MW. SPARC will therefore constitute a unique platform for burning plasma physics research with high density ($\langle n_{e} \rangle \approx 3 \times 10^{20}\ \textrm {m}^{-3}$), high temperature ($\langle T_e \rangle \approx 7$ keV) and high power density ($P_{\textrm {fusion}}/V_{\textrm {plasma}} \approx 7\ \textrm {MW}\,\textrm {m}^{-3}$) relevant to fusion power plants. SPARC's place in the path to commercial fusion energy, its parameters and the current status of SPARC design work are presented. This work also describes the basis for global performance projections and summarizes some of the physics analysis that is presented in greater detail in the companion articles of this collection.

SPARC is designed to be a high-field, medium-size tokamak aimed at achieving net energy gain with ion cyclotron range-of-frequencies (ICRF) as its primary auxiliary heating mechanism. Empirical predictions with conservative physics indicate that SPARC baseline plasmas would reach $Q\approx 11$, which is well above its mission objective of $Q>2$. To build confidence that SPARC will be successful, physics-based integrated modelling has also been performed. The TRANSP code coupled with the theory-based trapped gyro-Landau fluid (TGLF) turbulence model and EPED predictions for pedestal stability find that $Q\approx 9$ is attainable in standard H-mode operation and confirms $Q > 2$ operation is feasible even with adverse assumptions. In this analysis, ion cyclotron waves are simulated with the full wave TORIC code and alpha heating is modelled with the Monte–Carlo fast ion NUBEAM module. Detailed analysis of expected turbulence regimes with linear and nonlinear CGYRO simulations is also presented, demonstrating that profile predictions with the TGLF reduced model are in reasonable agreement.

In order to inform core performance projections and divertor design, the baseline SPARC tokamak plasma discharge is evaluated for its expected H-mode access, pedestal pressure and edge-localized mode (ELM) characteristics. A clear window for H-mode access is predicted for full field DT plasmas, with the available 25 MW of design auxiliary power. Additional alpha heating is likely needed for H-mode sustainment. Pressure pedestal predictions in the developed H-mode are surveyed using the EPED model. The projected SPARC pedestal would be limited dominantly by peeling modes and may achieve pressures in excess of 0.3 MPa at a density of approximately 3 × 1020 m−3. High pedestal pressure is partially enabled by strong equilibrium shaping, which has been increased as part of recent design iterations. Edge-localized modes (ELMs) with >1 MJ of energy are projected, and approaches for reducing the ELM size, and thus the peak energy fluence to divertor surfaces, are under consideration. The high pedestal predicted for SPARC provides ample margin to satisfy its high fusion gain (Q) mission, so that even if ELM mitigation techniques result in a 2× reduction of the pedestal pressure, Q > 2 is still predicted.

Owing to its high magnetic field, high power, and compact size, the SPARC experiment will operate with divertor conditions at or above those expected in reactor-class tokamaks. Power exhaust at this scale remains one of the key challenges for practical fusion energy. Based on empirical scalings, the peak unmitigated divertor parallel heat flux is projected to be greater than 10 GW m−2. This is nearly an order of magnitude higher than has been demonstrated to date. Furthermore, the divertor parallel Edge-Localized Mode (ELM) energy fluence projections (~11–34 MJ m−2) are comparable with those for ITER. However, the relatively short pulse length (~25 s pulse, with a ~10 s flat top) provides the opportunity to consider mitigation schemes unsuited to long-pulse devices including ITER and reactors. The baseline scenario for SPARC employs a ~1 Hz strike point sweep to spread the heat flux over a large divertor target surface area to keep tile surface temperatures within tolerable levels without the use of active divertor cooling systems. In addition, SPARC operation presents a unique opportunity to study divertor heat exhaust mitigation at reactor-level plasma densities and power fluxes. Not only will SPARC test the limits of current experimental scalings and serve for benchmarking theoretical models in reactor regimes, it is also being designed to enable the assessment of long-legged and X-point target advanced divertor magnetic configurations. Experimental results from SPARC will be crucial to reducing risk for a fusion pilot plant divertor design.

Ion cyclotron range of frequencies (ICRF) heating will be the sole auxiliary heating method on SPARC for both full-field (Bt0 ~ 12 T) D–T operation and reduced field (Bt0 ~ 8 T) D–D operation. Using the fast wave at ~120 MHz, good wave penetration and strong single-pass absorption is expected for D–T(3He), D(3He), D(H) and 4He(H) heating scenarios. The dependences of wave absorption on ${k_\parallel }$, 3He concentration, resonance location, antenna poloidal location and losses on alpha particles and ash have been studied. The antenna loading has been assessed by comparison with the Alcator C-Mod antennae. An antenna spectrum of ${k_\parallel }\sim 15\text{--}18\,{\textrm{m}^{ - 1}}$ is shown to be good for both core absorption and edge coupling. For the control of impurity sources, the antenna straps are rotated ~10° to be perpendicular to the B field and the straps can run with different power levels in order to optimize the antenna spectrum and to minimize the image current on the antenna frame. Combining the physics constraints with the SPARC port design, maintenance requirement and contingency against antenna failure during D–T operation, we plan to mount on the inner wall of the vacuum vessel a total of 12 4-strap antennae in 6 ports while keeping 3-strap antennae that are insertable and removable on port plugs as the backup option.

SPARC is being designed to operate with a normalized beta of $\beta _N=1.0$, a normalized density of $n_G=0.37$ and a safety factor of $q_{95}\approx 3.4$, providing a comfortable margin to their respective disruption limits. Further, a low beta poloidal $\beta _p=0.19$ at the safety factor $q=2$ surface reduces the drive for neoclassical tearing modes, which together with a frozen-in classically stable current profile might allow access to a robustly tearing-free operating space. Although the inherent stability is expected to reduce the frequency of disruptions, the disruption loading is comparable to and in some cases higher than that of ITER. The machine is being designed to withstand the predicted unmitigated axisymmetric halo current forces up to 50 MN and similarly large loads from eddy currents forced to flow poloidally in the vacuum vessel. Runaway electron (RE) simulations using GO+CODE show high flattop-to-RE current conversions in the absence of seed losses, although NIMROD modelling predicts losses of ${\sim }80$ %; self-consistent modelling is ongoing. A passive RE mitigation coil designed to drive stochastic RE losses is being considered and COMSOL modelling predicts peak normalized fields at the plasma of order $10^{-2}$ that rises linearly with a change in the plasma current. Massive material injection is planned to reduce the disruption loading. A data-driven approach to predict an oncoming disruption and trigger mitigation is discussed.

Potential loss of energetic ions including alphas and radio-frequency tail ions due to classical orbit effects and magnetohydrodynamic instabilities (MHD) are central physics issues in the design and experimental physics programme of the SPARC tokamak. The expected loss of fusion alpha power due to ripple-induced transport is computed for the SPARC tokamak design by the ASCOT and SPIRAL orbit-simulation codes, to assess the expected surface heating of plasma-facing components. We find good agreement between the ASCOT and SPIRAL simulation results not only in integrated quantities (fraction of alpha power loss) but also in the spatial, temporal and pitch-angle dependence of the losses. If the toroidal field (TF) coils are well-aligned, the SPARC edge ripple is small (0.15–0.30 %), the computed ripple-induced alpha power loss is small (${\sim } 0.25\,\%$) and the corresponding peak surface power density is acceptable ($244\ \textrm{kW}\ \textrm {m}^{-2}$). However, the ripple and ripple-induced losses increase strongly if the TF coils are assumed to suffer increasing magnitudes of misalignment. Surface heat loads may become problematic if the TF coil misalignment approaches the centimetre level. Ripple-induced losses of the energetic ion tail driven by ion cyclotron range of frequency (ICRF) heating are not expected to generate significant wall or limiter heating in the nominal SPARC plasma scenario. Because the expected classical fast-ion losses are small, SPARC will be able to observe and study fast-ion redistribution due to MHD including sawteeth and Alfvén eigenmodes (AEs). SPARC's parameter space for AE physics even at moderate $Q$ is shown to reasonably overlap that of the demonstration power plant ARC (Sorbom et al., Fusion Engng Des., vol. 100, 2015, p. 378), and thus measurements of AE mode amplitude, spectrum and associated fast-ion transport in SPARC would provide relevant guidance about AE behaviour expected in ARC.

Turbulent space and astrophysical plasmas exhibit a complex dynamics, which involves nonlinear coupling across different temporal and spatial scales. There is growing evidence that impulsive events, such as magnetic reconnection instabilities, lead to a spatially localized enhancement of energy dissipation, thus speeding up the energy transfer at small scales. Capturing such a diverse dynamics is challenging. Here, we employ the Multidimensional Iterative Filtering (MIF) method, a novel technique for the analysis of non-stationary multidimensional signals. Unlike other traditional methods (e.g. based on Fourier or wavelet decomposition), MIF does not require any previous assumption on the functional form of the signal to be identified. Using MIF, we carry out a multiscale analysis of Hall-magnetohydrodynamic (HMHD) and hybrid particle-in-cell (HPIC) numerical simulations of decaying plasma turbulence. The results assess the ability of MIF to spatially identify and separate the different scales (the MHD inertial range, the sub-ion kinetic and the dissipation scales) of the plasma dynamics. Furthermore, MIF decomposition allows localized current structures to be detected and their contribution to the statistical and spectral properties of turbulence to be characterized. Overall, MIF arises as a very promising technique for the study of turbulent plasma environments.

The isotope dependence of plasma transport will have a significant impact on the performance of future D-T experiments in JET and ITER and eventually on the fusion gain and economics of future reactors. In preparation for future D-T operation on JET, dedicated experiments and comprehensive transport analyses were performed in H, D and H-D mixed plasmas. The analysis of the data has demonstrated an unexpectedly strong and favourable dependence of the global confinement of energy, momentum and particles in ELMy H-mode plasmas on the atomic mass of the main ion species, the energy confinement time scaling as ${\tau _E}\sim {A^{0.5}}$ (Maggi et al., Plasma Phys. Control. Fusion, vol. 60, 2018, 014045; JET Team, Nucl. Fusion, vol. 39, 1999, pp. 1227–1244), i.e. opposite to the expectations based only on local gyro-Bohm (GB) scaling, ${\tau _E}\sim {A^{ - 0.5}}$, and stronger than in the commonly used H-mode scaling for the energy confinement (Saibene et al., Nucl. Fusion, vol. 39, 1999, 1133; ITER Physics Basis, Nucl. Fusion, vol. 39, 1999, 2175). The scaling of momentum transport and particle confinement with isotope mass is very similar to that of energy transport. Nonlinear local GENE gyrokinetic analysis shows that the observed anti-GB heat flux is accounted for if collisions, E × B shear and plasma dilution with low-Z impurities (9Be) are included in the analysis (E and B are, respectively the electric and magnetic fields). For L-mode plasmas a weaker positive isotope scaling ${\tau _E}\sim {A^{0.14}}$ has been found in JET (Maggi et al., Plasma Phys. Control. Fusion, vol. 60, 2018, 014045), similar to ITER97-L scaling (Kaye et al., Nucl. Fusion, vol. 37, 1997, 1303). Flux-driven quasi-linear gyrofluid calculations using JETTO-TGLF in L-mode show that local GB scaling is not followed when stiff transport (as is generally the case for ion temperature gradient modes) is combined with an imposed boundary condition taken from the experiment, in this case predicting no isotope dependence. A dimensionless identity plasma pair in hydrogen and deuterium L-mode plasmas has demonstrated scale invariance, confirming that core transport physics is governed, as expected, by the 4 dimensionless parameters ρ*, ν*, β, q (normalised ion Larmor radius, collisionality, plasma pressure and safety factor) consistently with global quasi-linear gyrokinetic TGLF calculations (Maggi et al., Nucl. Fusion, vol. 59, 2019, 076028). We compare findings in JET with those in different devices and discuss the possible reasons for the different isotope scalings reported from different devices. The diversity of observations suggests that the differences may result not only from differences affecting the core, e.g. heating schemes, but are to a large part due to differences in device-specific edge and wall conditions, pointing to the importance of better understanding and controlling pedestal and edge processes.

The results of flux-driven, two-fluid simulations in single-null configurations are used to investigate the processes determining the turbulent transport in the tokamak edge. Three turbulent transport regimes are identified: (i) a developed transport regime with turbulence driven by an interchange instability, which shares a number of features with the standard L-mode of tokamak operation; (ii) a suppressed transport regime, characterized by a higher value of the energy confinement time, low-amplitude relative fluctuations driven by a Kelvin–Helmholtz instability, a strong $\boldsymbol {E}\times \boldsymbol {B}$ sheared flow and the formation of a transport barrier, which recalls the H-mode; and (iii) a degraded confinement regime, characterized by a catastrophically large interchange-driven turbulent transport, which recalls the crossing of the Greenwald density limit. We derive an analytical expression of the pressure gradient length in the three regimes. The transition from the developed transport regime to the suppressed transport regime is obtained by increasing the heat source or decreasing the collisionality and vice versa for the transition from the developed transport regime to the degraded confinement regime. An analytical expression of the power threshold to access the suppressed transport regime, linked to the power threshold for H-mode access, as well as the maximum density achievable before entering the degraded confinement regime, related to the Greenwald density, are also derived. The experimental dependencies of the power threshold for H-mode access on density, tokamak major radius and isotope mass are retrieved. The analytical estimate of the density limit contains the correct dependence on the plasma current and on the tokamak minor radius.