The magnetohydrodynamic (MHD) equilibrium and stability properties of the Infinity Two fusion pilot plant baseline plasma physics design are presented. The configuration is a four-field period, aspect ratio $A = 10$ quasi-isodynamic stellarator optimised for excellent confinement at elevated density and high magnetic field $B = 9\,T$. Magnetic surfaces exist in the plasma core in vacuum and retain good equilibrium surface integrity from vacuum to an operational $\beta = 1.6 \,\%$, the ratio of the volume average of the plasma and magnetic pressures, corresponding to $800\ \textrm{MW}$ deuterium–tritium fusion operation. Neoclassical calculations show that a self-consistent bootstrap current of the order of ${\sim} 1\ \textrm{kA}$ slightly increases the rotational transform profile by less than 0.001. The configuration has a magnetic well across its entire radius. From vacuum to the operating point, the configuration exhibits good ballooning stability characteristics, exhibits good Mercier stability across most of its minor radius and it is stable against global low-n MHD instabilities up to $\beta = 3.2\,\%$.

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

The selection, design and optimization of a suitable blanket configuration for an advanced high-field stellarator concept is seen as a key feasibility issue and has been incorporated as a vital and necessary part of the Infinity Two fusion pilot plant physics basis. The focus of this work was to identify a baseline blanket which can be rapidly deployed for Infinity Two while also maintaining flexibility and opportunities for higher performing concepts later in development. Results from this analysis indicate that gas-cooled solid breeder designs such as the helium-cooled pebble bed (HCPB) are the most promising concepts, primarily motivated by the neutronics performance at applicable blanket build depths, and the relatively mature technology basis. The lithium lead (PbLi) family of concepts, particularly the dual-cooled lithium lead, offer a compelling alternative to solid blanket concepts as they have synergistic developmental pathways while simultaneously mitigating much of the technical risk of those designs. Homogenized three-dimensional neutronics analysis of the Infinity Two configuration indicates that the HCPB achieves an adequate tritium breeding ratio (TBR) (1.30 which enables sufficient margin at low engineering fidelity), and near appropriate shielding of the magnets (average fast fluence of 1.3 ${\times}$ 10$^{18}$ n cm$^{-2}$ per full-power year). The thermal analysis indicates that reasonably high thermal efficiencies (greater than 30 %) are readily achievable with the HCPB paired with a simple Rankine cycle using reheat. Finally, the tritium fuel cycle analysis for Infinity Two shows viability, with anticipated operational inventories of less than one kilogram (approximately 675 g) and a required TBR (TBR$_{\textrm {req}}$) of less than 1.05 to maintain fuel self-sufficiency (approximately 1.023 for a driver blanket with no inventory doubling). Although further optimization and engineering design are still required, at the physics basis stage all initial targets have been met for the Infinity Two configuration.

The kinetic stability of collisionless, sloshing beam-ion ($45^\circ$ pitch angle) plasma is studied in a three-dimensional (3-D) simple magnetic mirror, mimicking the Wisconsin high-temperature superconductor axisymmetric mirror experiment. The collisional Fokker–Planck code CQL3D-m provides a slowing-down beam-ion distribution to initialize the kinetic-ion/fluid-electron code Hybrid-VPIC, which then simulates free plasma decay without external heating or fuelling. Over $1$–$10\;\mathrm{\unicode{x03BC} s}$, drift-cyclotron loss-cone (DCLC) modes grow and saturate in amplitude. The DCLC scatters ions to a marginally stable distribution with gas-dynamic rather than classical-mirror confinement. Sloshing ions can trap cool (low-energy) ions in an electrostatic potential well to stabilize DCLC, but DCLC itself does not scatter sloshing beam-ions into the said well. Instead, cool ions must come from external sources such as charge-exchange collisions with a low-density neutral population. Manually adding cool $\mathord {\sim } 1\;\mathrm{keV}$ ions improves beam-ion confinement several-fold in Hybrid-VPIC simulations, which qualitatively corroborates prior measurements from real mirror devices with sloshing ions.

In this study, we conducted an electrical analysis of the effects of cold plasma on the properties of distilled water, using a corona discharge in a tip–plane configuration. The discharge was initiated by applying a voltage of 7.17 kV with a 2 mm gap between the tip and the water surface. We investigated the impact of plasma treatment on the total dissolved solids (TDS) and conductivity of 20 mL of distilled water, with exposure times ranging from 2 to 12 min. The results show that plasma treatment leads to a significant increase in conductivity and TDS, with a proportional increase relative to the exposure time. In addition to these measurements, we performed a detailed electrical analysis to evaluate the energy efficiency of the plasma treatment. This analysis involved calculating the useful power and energy efficiency using an equivalent electrical model of the corona discharge reactor, as direct measurement of these parameters is challenging in this context. The model allowed us to calculate energy consumption and analyse the electrical behaviour of the system throughout the treatment process. This study also enables us to monitor, control and optimize the energy during plasma treatment, providing insights into the energy dynamics involved. The findings have potential applications in improving energy efficiency in industrial and environmental processes.

Transport characteristics and predicted confinement are shown for the Infinity Two fusion pilot plant baseline plasma physics design, a high field stellarator concept developed using modern optimization techniques. Transport predictions are made using high-fidelity nonlinear gyrokinetic turbulence simulations along with drift kinetic neoclassical simulations. A pellet-fuelled scenario is proposed that enables supporting an edge density gradient to substantially reduce ion temperature gradient turbulence. Trapped electron mode turbulence is minimized through the quasi-isodynamic configuration that has been optimized with maximum-J. A baseline operating point with deuterium–tritium fusion power of $P_{{fus,DT}}=800$ MW with high fusion gain $Q_{{fus}}=40$ is demonstrated, respecting the Sudo density limit and magnetohydrodynamic stability limits. Additional higher power operating points are also predicted, including a fully ignited ($Q_{{fus}}=\infty$) case with $P_{{fus,DT}}=1.5$ GW. Pellet ablation calculations indicate it is plausible to fuel and sustain the desired density profile. Impurity transport calculations indicate that turbulent fluxes dominate neoclassical fluxes deep into the core, and it is predicted that impurity peaking will be smaller than assumed in the transport simulations. A path to access the large radiation fraction needed to satisfy exhaust requirements while sustaining core performance is also discussed.

One of the critical challenges in future high-current tokamaks is the avoidance of runaway electrons during disruptions. Here, we investigate disruptions mitigated with combined deuterium and noble gas injection in SPARC. We use multi-objective Bayesian optimisation of the densities of the injected material, taking into account limits on the maximum runaway current, the transported fraction of the heat loss and the current quench time. The simulations are conducted using the numerical framework Dream (disruption runaway electron analysis model). We show that during deuterium operation, runaway generation can be avoided with material injection, even when we account for runaway electron generation from deuterium–deuterium induced Compton scattering. However, when including the latter, the region in the injected-material-density space corresponding to successful mitigation is reduced. During deuterium–tritium operation, acceptable levels of runaway current and transported heat losses are only obtainable at the highest levels of achievable injected deuterium densities. Furthermore, disruption mitigation is found to be more favourable when combining deuterium with neon, compared with deuterium combined with helium or argon.

The propagation and absorption of the slow waves in the plasma of the Joint European Torus (JET) tokamak have been investigated by ray tracing. The study aims to obtain a qualitative notion of the penetration into the plasma and absorption of the slow wave excited by the A2 ITER-like antenna. The slow waves are radiated by antennas in the ion cyclotron resonance frequency inverted minority heating or mode conversion heating regimes. It has been discovered that the rays propagate in the toroidal direction over a significant distance, up to 6 × 103 cm, from the antenna. Spreading in the peripheral plasma, mainly between the separatrix and the wall, they slowly shift in the poloidal direction and can reach the divertor region. The change in equilibrium of the JET tokamak has a strong influence on both the propagation and absorption of slow waves. Absorption of the slow waves is caused by ion–electron collisions and Landau damping. In the minority heating regimes, the slow waves are strongly damped in the cyclotron resonance of minority ions even at very low minority density.

Landau’s collisionless result for a weakly damped plasma wave is precisely recovered in a weakly collisional, steady state plasma by treating the physics of the narrow collisional boundary layer associated with the resonant electrons. To recover Landau’s results, the collision frequency must be large enough that islands are unable to form and/or the wave amplitude must be small enough to allow linearization. However, the Landau treatment fails once the collision frequency becomes too weak and/or the wave amplitude too large. Remarkably, Landau’s weakly damped plasma wave results require collisions and are shown to be inappropriate in the collisionless limit for a nonlinear, finite amplitude, steady state wave!

Optimal-mode theory (Landreman et al. 2015 J. Plasma Phys. 81, 905810501) can be used to derive upper bounds on growth rates of local gyrokinetic instabilities (Helander & Plunk 2021 Phys. Rev. Lett. 127, 155001). These bounds follow from thermodynamic principles (specifically on the Helmholtz free energy) (Helander & Plunk Phys. Rev. Lett. 127, 2021, p. 155001), and thus apply to any instability and geometry, independently of many plasma parameters. In this work, we compare these upper bounds with the growth rates of linear gyrokinetic eigenmodes. Experimentally relevant scenarios of density-gradient- and ion-temperature-gradient-driven instabilities are considered. The difference between the upper bounds and the numerically computed growth rates is always positive, as it must be, but depends strongly on the instability in question and on the geometry of the magnetic field. The nature of this difference can be analysed by examining the contributions of optimal modes to gyrokinetic eigenmodes. This approach exploits the completeness and orthogonality properties of optimal modes.

Accurate modelling of runaway electron generation and losses during tokamak disruptions is crucial for the development of reactor-scale tokamak devices. In this paper, we present a reduced model for runaway electron losses due to flux surface scrape-off caused by the vertical motion of the plasma. The model is made compatible with computationally inexpensive one-dimensional models averaging over a fixed flux-surface geometry, by formulating it as a loss term outside an estimated time-varying minor radius of the last closed flux surface. We then implement this model in the disruption modelling tool DREAM and demonstrate its impact on selected scenarios relevant for ITER. Our results indicate that scrape-off losses may be crucial for making complete runaway avoidance possible even in a $15\,\rm MA$ DT H-mode ITER scenario. The results are however sensitive to the details of the runaway electron generation and phenomena affecting the current density profile, such as the current profile relaxation at the beginning of the disruption.

We provide an assessment of the Infinity Two fusion pilot plant (FPP) baseline plasma physics design. Infinity Two is a four-field period, aspect ratio $A = 10$, quasi-isodynamic stellarator with improved confinement appealing to a max-$J$ approach, elevated plasma density and high magnetic fields ($ \langle B\rangle = 9$ T). Here $J$ denotes the second adiabatic invariant. At the envisioned operating point ($800$ MW deuterium-tritium (DT) fusion), the configuration has robust magnetic surfaces based on magnetohydrodynamic (MHD) equilibrium calculations and is stable to both local and global MHD instabilities. The configuration has excellent confinement properties with small neoclassical transport and low bootstrap current ($|I_{bootstrap}| \sim 2$ kA). Calculations of collisional alpha-particle confinement in a DT FPP scenario show small energy losses to the first wall (${\lt}1.5 \,\%$) and stable energetic particle/Alfvén eigenmodes at high ion density. Low turbulent transport is produced using a combination of density profile control consistent with pellet fueling and reduced stiffness to turbulent transport via three-dimensional shaping. Transport simulations with the T3D-GX-SFINCS code suite with self-consistent turbulent and neoclassical transport predict that the DT fusion power$P_{{fus}}=800$ MW operating point is attainable with high fusion gain ($Q=40$) at volume-averaged electron densities $n_e\approx 2 \times 10^{20}$ m$^{-3}$, below the Sudo density limit. Additional transport calculations show that an ignited ($Q=\infty$) solution is available at slightly higher density ($2.2 \times 10^{20}$ m$^{-3}$) with $P_{{fus}}=1.5$ GW. The magnetic configuration is defined by a magnetic coil set with sufficient room for an island divertor, shielding and blanket solutions with tritium breeding ratios (TBR) above unity. An optimistic estimate for the gas-cooled solid breeder designed helium-cooled pebble bed is TBR $\sim 1.3$. Infinity Two satisfies the physics requirements of a stellarator fusion pilot plant.

The bootstrap current in stellarators can be presented as a sum of a collisionless value given by the Shaing–Callen asymptotic formula and an off-set current, which non-trivially depends on plasma collisionality and radial electric field. Using NEO-2 modeling, analytical estimates and semi-analytical studies with the help of a propagator method, it is shown that the off-set current in the $1/\nu$ regime does not converge with decreasing collisionality $\nu _\ast$ but rather shows oscillations over $\log \nu _\ast$ with an amplitude of the order of the bootstrap current in an equivalent tokamak. The convergence to the Shaing–Callen limit appears in regimes with significant orbit precession, in particular, due to a finite radial electric field, where the off-set current decreases as $\nu _\ast ^{3/5}$. The off-set current strongly increases in case of nearly aligned magnetic field maxima on the field line where it diverges as $\nu _\ast ^{-1/2}$ in the $1/\nu$ regime and saturates due to the precession at a level exceeding the equivalent tokamak value by ${v_E^\ast }^{-1/2}$, where $v_E^\ast$ is the perpendicular Mach number. The latter off-set, however, can be minimized by further aligning the local magnetic field maxima and by fulfilling an extra integral condition of “equivalent ripples” for the magnetic field. A criterion for the accuracy of this alignment and of ripple equivalence is derived. In addition, the possibility of the bootstrap effect at the magnetic axis caused by the above off-set is also discussed.

The intense applied lower hybrid electric fields used to drive current in tokamaks can result in the formation of velocity space island structure. When this happens the lower hybrid current drive efficiency can be calculated for a monochromatic wave and is shown to be below the quasilinear level.

Even if the magnetic field in a stellarator is integrable, phase-space integrability for energetic particle guiding-center trajectories is not guaranteed. Both trapped and passing particle trajectories can experience convective losses, caused by wide phase-space island formation, and diffusive losses, caused by phase-space island overlap. By locating trajectories that are closed in the angle coordinate but not necessarily closed in the radial coordinate, we can quantify the magnitude of the perturbation that results in island formation. We characterize island width and island overlap in quasihelical (QH) and quasiaxisymmetric (QA) equilibria with finite plasma pressure $\beta$ for both trapped and passing energetic particles. For trapped particles in QH, low-shear toroidal precession frequency profiles near zero result in wide island formation. While QA transit frequencies do not cross through the zero resonance, we observe that island overlap is more likely since higher shear results in the crossing of more low-order resonances.

When analysing stellarator configurations, it is common to perform an asymptotic expansion about the magnetic axis. This so-called near-axis expansion is convenient for the same reason asymptotic expansions often are, namely, it reduces the dimension of the problem. This leads to convenient and quickly computed expressions of physical quantities, such as quasisymmetry and stability criteria, which can be used to gain further insight. However, it has been repeatedly found that the expansion diverges at high orders in the distance from axis, limiting the physics the expansion can describe. In this paper, we show that the near-axis expansion diverges in vacuum due to ill-posedness and that it can be regularised to improve its convergence. Then, using realistic stellarator coil sets, we demonstrate numerical convergence of the vacuum magnetic field and flux surfaces to the true values as the order increases. We numerically find that the regularisation improves the solutions of the near-axis expansion under perturbation, and we demonstrate that the radius of convergence of the vacuum near-axis expansion is correlated with the distance from the axis to the coils.

Steady, helical perturbations known as ‘density snakes’ with poloidal and toroidal mode numbers $m=1$, $n=1$ have been studied in several tokamak experiments. These three-dimensional, helical states are interesting due to their stability and persistence, including their coexistence with the sawtooth cycle. Presented here are studies of density snakes in tokamak plasmas in the Madison Symmetric Torus (MST) device. They are diagnosed using an 11-chord interferometer, internal and edge magnetic coils and impurity ion spectroscopy. Compared with observations in other tokamak plasmas, snakes in MST form with relatively high resistivity and low edge safety factor, $ q(a) \geqslant 2.2$, which moves the $q=1$ resonant surface outward in radius and probably forms a large magnetic island. As a result, the density perturbation associated with the snake is larger, the structure occupies a broader span of minor radius and the snakes are somewhat less stable. The helical structure and distribution of snake events are characterized, including whether they are best described as ideal or resistive kink modes. Finally, an analysis of their perturbation or destruction during sawtooth crashes is given.