In solving the Ion Cyclotron Resonance Heating wave equation, plane wave Fourier mode $\exp[ikx]$ base functions are often exploited to produce expressions for the dielectric response when including the effect of finite temperature, requiring locally integrating the equation of motion of charged particles by hand. Standard Fourier analysis notation is used: $x$ is the position, $k$ is the wave vector component in the $x$-direction and $i$ is the imaginary number. In contrast, finite element techniques adopt base functions – typically low - order polynomials – that individually are only non-zero in a small domain. They are a natural and easy approach to capture inhomogeneity effects and readily allow grid refinement to zoom in on regions where this is required. To enable ample realism of kinetic effects while profiting from the rich pool of numerical tools available for solving differential and integro-differential equations relying on finite elements, it is desirable to have a procedure allowing us to profit from both approaches: the detailed physics brought by finite temperature effects – commonly described in terms of Fourier modes – as well as the simplicity from a local polynomial representation. A novel technique is offered to achieve that. It consists of finding the Fourier representation of the localised base functions exploited in finite elements so that the richness of the dielectric response in $k$-space can be accounted for. The resulting equation is assembled just like the finite element method prescribes but the coefficients of the linear local system are assembled differently. The technique allows us to capture finite temperature corrections in both the parallel and perpendicular directions for a dielectric tensor model of choice. The focus in this paper is on the numerical technique while the expressions for the dielectric response are assumed to be known. A few first examples are briefly discussed: the wave equation solutions for a typical minority heating (H)-D JET plasma are provided for (i) an all-FLR model, (ii) the FLR-0 ‘tepid’ equivalent and (iii) a cold plasma model; FLR refers to the Finite Larmor Radius expansion exploited to account for finite temperature corrections.

The two-plasmon decay instability of high-power microwaves used for electron cyclotron resonance heating has been observed experimentally in multiple fusion devices. This type of instability is a nonlinear three-wave interaction that can transfer energy away from the cyclotron harmonics. Analytical models quantify exponential growth rates and power thresholds, but typically evaluate the growth as a spatially averaged gain over the interaction region of the instability. This description effectively excludes field inhomogeneities and noise, leaving their impact on the instability growth uncertain. We assess this assumption by solving the full nonlinear system on a spatial grid. Across all cases considered, we find that the instabilities drive the wave fields toward the spatially averaged behaviour. After a transient period that scales inversely with the instability growth rate, the simulated growth converges to the averaged prediction, indicating that the established models are asymptotically valid.

This study investigates the Lagrangian properties of ion turbulent transport driven by drift-type turbulence in tokamak plasmas focusing on the Cyclone Base Case. Despite the compressible and inhomogeneous nature of Eulerian gyrocentre drifts, numerical simulations with the T3ST code reveal approximate ergodicity, stationarity and time-symmetry. These characteristics are attributed to broad initial phase-space distributions that support ergodic mixing. Moreover, relatively minor constraints on the initial distributions are found to have negligible effects on transport levels.

Geometrical optics (GO) is widely used for reduced modelling of waves in plasmas, but it fails near reflection points, where it predicts a spurious singularity of the wave amplitude. We show how to avoid this singularity by adopting a different representation of the wave equation. Instead of the physical coordinate $x$ and the wavevector $k$, we use the ray time $\tau$ as the new canonical coordinate and the ray energy $h$ as the associated canonical momentum. To derive the envelope equation in the $\tau$-representation, we construct the Weyl symbol calculus on the $(\tau , h)$ space and show that the corresponding Weyl symbols are related to their $(x, k)$ counterparts by the Airy transform. This allows us to express the coefficients in the envelope equation through the known properties of the original dispersion operator. When necessary, solutions of this equation can be mapped to the $x$-space using a generalised metaplectic transform. However, the field per se might not even be needed in practice. Instead, knowing the corresponding Wigner function usually suffices for linear and quasilinear calculations. As a Weyl symbol itself, the Wigner function can be mapped analytically, using the aforementioned Airy transform. We show that the standard Airy patterns that form in regions where conventional GO fails are successfully reproduced within metaplectic GO (MGO) simply by remapping the field from the $\tau$-space to the $x$-space. An extension to mode-converting waves is also presented. This formulation, which we call generalised MGO, can be particularly useful, for example, for reduced modelling of the O–X conversion in inhomogeneous plasma near the critical density, an effect that is important for fusion applications and also occurs in the ionosphere. Overall, MGO can replace GO for any practical purposes, because it better handles cutoffs and is similar otherwise.

A magnetohydrodynamic (MHD) shock is a discontinuity through which plasma flows, while the mass, momentum and energy are conserved, and the entropy increases. In a collisionless system, the entropy is strictly constant along the phase-space trajectory. The MHD required entropy increase is attributed to dissipative processes causing relaxation to local thermodynamic equilibrium. The search for such dissipative processes in the shock observations usually explores variations of the entropy density or entropy per particle across the shock front. We show that both quantities may increase or decrease across the shock front even when the entropy is strictly conserved. Entropy production should be assessed using the entropy flux density.

In this work we investigate the landscape of quasi-isodynamic (QI) stellarators using the near-axis expansion of the magnetic field. Building on recent theoretical developments, we construct a database of more than 800 000 stable, approximately QI vacuum magnetic configurations. These configurations span a range of field period numbers and other geometric control parameters, including the magnetic axis shape and plasma elongation. To evaluate each configuration, we use a broad set of measures, including effective ripple, sensitivity of the Shafranov shift to changes in plasma $\beta$, the prevalence of maximum-$\mathcal{J}$ trapped particles and the Rosenbluth–Hinton residual, among others. This enables an exhaustive, thorough and quantitative characterisation of the database. Statistical analysis and modern machine learning techniques are then employed to find correlations and identify key descriptors and heuristics to help understand tendencies that govern the behaviour of numerical optimisation. The database provides baseline configurations for further studies and to serve as tailored initial conditions for optimisation. With this work we initiate a long-term program to complete a systematic exploration of QI stellarator design space.

To successfully show that fusion is an attractive energy source, the ARC$^{\scriptstyle \mathrm{TM}}$ fusion power plant will need to operate with a robust, integrated power and particle exhaust solution. To maximise ARC’s fusion power output while avoiding excessive erosion of the plasma-facing components, we will need to radiatively dissipate most of the power crossing the last-closed flux surface, injecting radiating impurities such as argon or neon to access divertor detachment. Divertor detachment will need to be integrated with a high-performance core plasma, and with efficient impurity pumping to prevent the accumulation of helium ash in the core. To access and control detachment in high-performance plasmas, we have designed ARC with up–down-symmetric divertors, with secondary X-points in long, tightly baffled outer legs. Using a core-edge modelling workflow, we predict that with this divertor design, ARC will access detachment with modest argon seeding in the divertor (${c_{Ar,div}}\sim {0.9\,\%}$), which should have minimal impact on the core (${\Delta Z_{\textit{eff},\textit{core}}}\lt {0.5}$) for reasonable argon enrichment (${c_{Ar,div}/c_{Ar,\textit{core}}}={6}$). Due to the high predicted divertor neutral pressure (${p_{\textit{div}}}\sim {20\,\mathrm{Pa}}$), we predict that ARC will sufficiently pump helium to limit ash accumulation in the core (${c_{\textit{He},\textit{core}}}\lt {2\,\%}$) for a helium enrichment of ${c_{\textit{He},\textit{div}}/c_{\textit{He},\textit{core}}}={0.4}$. ARC’s divertor design is expected to increase the stability of a detachment front in the outer divertor leg, helping to prevent divertor reattachment during smaller heat-flux transients such as scrape-off-layer filaments associated with the quasi-continuous exhaust regime. However, this buffering will not be sufficient to prevent divertor reattachment during large type-I edge-localised modes (ELMs), and as such these will need to be avoided on ARC. Experiments on SPARC will be used to select an integrated scenario which avoids or mitigates type-I-ELMs while maintaining access to detachment, good core fusion performance and sufficient impurity exhaust. SPARC experiments will also be used to finalise ARC’s divertor design, by studying the impact of magnetic and first-wall geometry on detachment stability, impurity enrichment and neutral baffling under conditions similar to those expected for ARC. In conclusion, our analysis finds that ARC will have a viable power and particle exhaust solution which is compatible with high-power operations, and this solution will be validated in experiments on SPARC.

The ARC$^{\textrm {TM}}$ tokamak, a high-field ($B_T$ = 11.4 T) fusion power plant, under development by Commonwealth Fusion Systems, is studied using a suite of integrated modelling tools to predict its fusion power generation ($P_{fus}$), transport and confinement properties. Analysis is based off an ARC operational point scoped first with zero-dimensional (0-D) plasma operational contour (POPCON) modelling to produce 1.13 GW of fusion power. A suite of integrated modelling tools (TRANSP, ASTRA and TORAX) were applied to predict the performance and kinetic profiles of the ARC design point, yielding a range of predicted performance spanning from ${\sim} 900$ to 1300 MW in rough quantitative agreement with POPCON predictions. The sensitivity of these results to uncertain modelling inputs was probed using scans of pedestal boundary conditions around EPED-predicted values (total pressure and temperature ratios), tungsten concentration and seperatrix density around their nominal assumptions. Pedestal pressure and pedestal top $(T_i/T_e)$ play a large role in 1.5-dimensional performance predictions, able to modify the predicted $P_{fus}$ by a factor of 2 within reasonable assumptions. High-fidelity core nonlinear gyrokinetic profile predictions, performed using CGYRO (Candy et al. 2016 J. Comput. Phys., vol. 324, pp. 73–93) coupled with the PORTALS (Rodriguez-Fernandez et al. 2024 Nucl. Fusion, vol. 64, 076034; Phys. Plasmas, vol. 31, 2024, 062501) framework, yield substantially lower performance ($P_{fus} = 677$ MW) compared with 0-D and medium-fidelity modelling for nominal assumptions, showing that there is non-negligible uncertainty between models and that future work on SPARC may help resolve discrepancies. Lower overall performance results from significantly reduced volume-averaged densities and temperatures, along with reduced levels of density and temperature peaking. Turbulence and transport are largely dominated by ion temperature gradient across the profile, confirmed by both linear stability and the response of the nonlinear fluxes to changes in gradients, with some impact of kinetic ballooning modes in the deep core. This work represents one of the most complete scoping of potential fusion power plant conditions performed to date. The extensive integrated modelling provides confidence in ARC performance approaching 1 GW, while nonlinear gyrokinetic modelling results in open questions into the physics of density and temperature peaking in fusion-power-plant-relevant operational space. A discussion of results and the role that the SPARC tokamak (Creely et al. 2020 J. Plasma Phys., vol. 86, 865860502) will play in informing ARC design, performance and operation is presented.

Commonwealth Fusion Systems (CFS) plans to operate a tokamak power plant called ARC in the early 2030s. Tokamak plasmas have stability limits that, if crossed, lead to a rapid termination of the plasma, referred to as a disruption. Disruptions pose a melt risk to the first wall resulting from thermal and non-thermal particle heat fluxes, and an electromagnetic loading risk on all metal components within the equilibrium coils. A comprehensive set of models is used herein to provide an assessment of both mitigated and unmitigated ARC disruption loads. A preliminary massive gas injection system is baselined and a runaway electron mitigation coil option is proposed to close possible gaps in the baseline. It is predicted that all ARC disruption loads are within a factor of 2 of the disruption loads in SPARC, a tokamak presently under construction by CFS, and therefore SPARC provides an opportunity to calibrate models, test solutions and inform the design of ARC. The goal for ARC is disruption-free operation, however, the pragmatic design target is to withstand one mitigated disruption per day, and to restart the plasma following mitigation in tens of seconds without interrupting the power output. Unmitigated disruptions must be rare, and experience with unmitigated disruption impacts in SPARC will better define what rare means. The implications of this strategy for plasma disruptivity and disruption prediction are discussed, and operating the ARC scenario on SPARC is expected to refine the ARC final design and operational plan.

Commonwealth Fusion Systems plans to build ARC as the first fusion power plant at a site in Chesterfield County, Virginia, USA by the early 2030s. We present an overview of analysis comprising the physics basis of the ARC V3A design, a high-magnetic-field tokamak with $B_0=11.4 \ \text{T}$, $I_p=12.0 \ \text{MA}$, $R_0=4.62 \, \text{m}$, $a=1.18 \, \text{m}$. ARC V3A is designed to produce $P_{fus} \approx 1.13$ GW DT fusion power and deliver $\geqslant$400 MW net electric power to the grid. This overview includes quantitative analysis of fundamental issues for design of and operational plasma scenarios for a tokamak power plant, and lays out the design targets and strategic choices for ARC, including empirical fusion performance projections, assessment of H-mode access, ion cyclotron resonance heating simulations, alpha particle physics and time-dependent full-pulse simulations. This is complemented by topical papers on fusion performance and transport, disruption physics, boundary physics and magnetohydrodynamic stability. Critically, these studies identify key model uncertainties and physics risks to be retired through SPARC operation. Due to the modular nature of ARC, early results from SPARC can be incorporated into the design of the first ARC as well as subsequent replacements of the ARC vacuum vessel.

The paper presents a fusion plant concept called VOYAGER developed by leading groups in fusion research (BINP SB RAS) and high-temperature superconductivity (SuperOx LLC). This plant has an open trap configuration with high-temperature superconducting (HTS) vortex (helical) magnets for plasma confinement. The plant is designed with deuterium–deuterium fusion in the ignited regime (without external plasma heating) providing an unlimited supply of reactor fuel, using already proven technologies, and thus maximising the plant’s electrical power output and revenue. The costs for construction and operation of the power plant, as well as levelised cost of electricity (LCOE) for the plant, are calculated. The VOYAGER plant has comparable construction and operating costs with existing fusion plant projects; however, VOYAGER, due to the high electrical power output, has the lowest LCOE. The low LCOE value for the proposed plant makes it competitive with conventional electricity sources. Reduced research and development costs due to the use of proven technologies and modular nature of open trap confinement make VOYAGER an even more attractable preposition among fusion plant concepts.

We present a fast, spectrally (exponentially) accurate, automatically differentiable bounce-averaging algorithm that is used to simplify kinetic models. Using this algorithm, implemented in the DESC stellarator optimisation suite, we can perform efficient optimisation of many objectives to improve stellarator performance, such as the effective ripple $\epsilon _{\mathrm{eff}}$ metric for the neoclassical transport coefficient in the low collisionality regime and proxies for energetic particle confinement. For the first time, we optimise a finite-beta stellarator to directly reduce neoclassical ripple transport using reverse-mode differentiation. This ensures the computational cost of differentiation is independent of the number of controllable parameters.

In collisionless astrophysical plasmas, turbulence mediates the partitioning of free energy among cascade channels, and its dissipation into ion and electron heat. The resulting ion heating is often anisotropic, with ions observed to be preferentially heated perpendicular to the local magnetic field; understanding the mechanisms responsible for this heating is a key step in understanding the evolution of such plasmas. In this paper, we use the framework of quasi-linear theory to compute analytically the heating rates of ions interacting with turbulent, large-scale Alfvénic fluctuations. We show how the imbalance of the turbulence (the difference in energies between Alfvénic fluctuations travelling parallel and antiparallel to the magnetic field) modifies the spatiotemporal spectrum of these fluctuations, allowing the heating mechanism to smoothly transition between stochastic heating in balanced turbulence and cyclotron-resonant heating in imbalanced turbulence. The resultant heating rate is found to have a general form regardless of the level of imbalance, exhibiting a suppression related to the conservation of the ions’ magnetic moment at small turbulent amplitudes and recovering previous empirical results in a formal calculation. The results of this work help to consolidate our qualitative understanding of ion heating within astrophysical plasmas, as well as yielding specific quantitative predictions to analyse simulations and observations.

Structures in tokamak plasmas are elongated along the direction of the magnetic field and short in the directions perpendicular to the magnetic field. Many tokamak simulation codes take advantage of this by using a field-aligned coordinate system. However, field-aligned coordinate systems have a coordinate singularity at magnetic X-points where the poloidal magnetic field vanishes, which makes it difficult to use field-aligned coordinate systems when simulating the core and scrape-off layer simultaneously. Here, we present an algorithm for grid generation and computing geometric quantities in a standard field-aligned coordinate system that avoids the singularity and allows one to conduct two-dimensional gyrokinetic axisymmetric simulations in X-point geometries. Convergence tests of advection, boundary value problems and geometric quantities all show greater than first-order convergence even in the vicinity of the X-point. We also demonstrate the geometric consistency of our algorithm with an example simulation of the spherical tokamak for energy production, which shows machine-precision particle conservation.

This work presents an integrated modelling study of fast-proton distributions generated by ion cyclotron range of frequency (ICRF) minority heating in the Experimental Advanced Superconducting Tokamak (EAST). Using a series of high-confinement (H-mode) discharges with increasing ICRF power levels from 0.8 to 2.4 MW, fast protons were produced via minority heating mechanisms and analysed through simulations using the ASCOT code. The results reveal that the fast protons are primarily concentrated near the fundamental cyclotron resonance layer and exhibit strong power-dependent behaviour in both real-space (R–Z) distribution and velocity space, where R is the major radius and Z is the vertical coordinate. As the ICRF power increases, the energetic proton population shows significant spatial broadening and energy enhancement, reaching up to 1 MeV. The fast-ion pitch-angle distribution becomes increasingly anisotropic, with high-energy ions concentrated around $|\textit{v}_{\|}/\textit{v}| \lt 0.5$, where $\nu$ is the magnitude (speed) of the full velocity vector of the particle. Furthermore, the energy density of fast ions aligns well with the ICRF power deposition profile, confirming efficient central-core heating. These findings, which provide insight into fast-ion behaviour and ICRF heating characteristics in EAST plasmas, also support future fast-ion diagnostics and performance control strategies in EAST and similar experimental conditions.

We propose a methodology to infer collision operators from phase space data of plasma dynamics. Our approach combines a differentiable kinetic simulator, whose core component in this work is a differentiable Fokker–Planck solver, with a gradient-based optimisation method to learn the collisional operators that best describe the phase space dynamics. We test our method using data from two-dimensional particle-in-cell simulations of spatially uniform thermal plasmas, and learn the collision operator that captures the self-consistent electromagnetic interaction between finite-size charged particles over a wide variety of simulation parameters. We demonstrate that the learned operators are more accurate than alternative estimates based on particle tracks, while making no prior assumptions about the relevant time scales of the processes and significantly reducing memory requirements. We find that the retrieved operators, obtained in the non-relativistic regime, are in excellent agreement with theoretical predictions derived for electrostatic scenarios. Our results show that differentiable simulators offer a powerful and computational efficient approach to infer novel operators for a wide range of problems, such as electromagnetically dominated collisional dynamics and stochastic wave–particle interactions.

An analytical framework is presented for determining the ground states of a collisionless plasma with a given density profile, and its associated available energy, which bounds the field energy, is calculated. We show that the bound can be tightened by enforcing that the ground state is physically realisable, i.e. that the released energy is consistently stored in the fields supported by the density profile. A simple waterbag model is employed to retrieve nonlinear bounds for particularly simple conditions, further finding a phase-transition-like behaviour at the critical energy where the constraint of non-negative number densities becomes relevant. We verify the derived bounds with one-dimensional particle-in-cell simulations, where it is found that the true energy released is typically ${\sim} 20\,\%$ of the bound derived. Next, we present an asymptotic framework for calculating ground states with given density profiles of distribution functions close to this ground state. This framework is employed to describe a magnetised plasma, where it is shown that the ground state seeks long-wavelength structures for large electron-to-ion temperature ratios, highlighting unfavourable transport properties in this regime. By using the corrected adiabatic response in toroidal geometry (with a flux-surface average subtracted), it is found that the ground state seeks long-wavelength zonal structures. Furthermore, a crude model of a toroidal, multispecies, quasineutral plasma highlights that there is non-trivial dependence on the impurity content with certain optimal mixing ratios. A similar model is constructed to capture the effects of a fast-ion population, showing a non-trivial dependence on the exact shape of the fast-ion distribution. These results provide rigorous, density-aware limits on energy release in collisionless plasmas, and similar methods may be used to account for electromagnetic effects.