The properties of a collisional magnetized plasma sheath containing non-thermal electrons, multi-component ions (${\text{He}}^{+}$ and $ {\text{Ar}}^{+}$), neutral atoms and negatively charged dust particles are analysed. Using a one-dimensional fluid model, the parametric changes in sheath dynamics are investigated in the presence of nanometre-sized charged dust particles and an oblique magnetic field. The influence of charged dust, ionization, ion–neutral collisions, ion loss and non-thermal electrons on sheath parameters such as ion densities, velocities, electron density and potential is explored through theoretical modelling and numerical analysis. The results indicate that the ion density (${\text{He}}^{+}$ and $ {\text{Ar}}^{+}$) increases throughout the sheath region with rising ionization frequency in the absence of charged dust. However, when charged dust is present, the density of ${\text{He}}^{+}$ ions decreases while the density of $\text{Ar}^{+}$ ions increases, exhibiting a sharp peak near the sheath edge. It is also noted that the increase in ion–neutral collision frequency enhances the density, particularly near the sheath edge. Additionally, the presence of non-thermal electrons initially leads to an increase in ion density near the sheath edge, followed by a decrease within the sheath region. A qualitative explanation of the above phenomena, which occur due to different physical parameters, is provided.

Curvature-driven instabilities are ubiquitous in magnetised fusion plasmas. By analysing the conservation laws of the gyrokinetic system of equations, we demonstrate that the well-known spatial localisation of these instabilities to regions of ‘bad magnetic curvature’ can be explained using the conservation law for a sign-indefinite quadratic quantity that we call the gyrokinetic field invariant. Its evolution equation allows us to define the local effective magnetic curvature whose sign demarcates the regions of ‘good’ and ‘bad’ curvature, which, under some additional simplifying assumptions, can be shown to correspond to the inboard (high-field) and outboard (low-field) sides of a tokamak plasma, respectively. We find that, given some reasonable assumptions, electrostatic curvature-driven modes are always localised to the regions of bad magnetic curvature, regardless of the specific character of the instability. More importantly, we also deduce that any mode that is unstable in the region of good magnetic curvature must be electromagnetic in nature. As a concrete example, we present the magnetic-drift mode, a novel good-curvature electromagnetic instability, and compare its properties with the well-known electron-temperature-gradient instability. Finally, we discuss the relevance of the magnetic drift mode for high-$\beta$ fusion plasmas, and in particular its relationship with microtearing modes.

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.

This paper presents a novel approach for simulating plasma instabilities in tokamak plasmas using the piecewise field-aligned finite element method in combination with the particle-in-cell method. Our method traditionally aligns the computational grid, but defines the basis functions in piecewise field-aligned coordinates to avoid grid deformation while naturally representing the field-aligned mode structures. This scheme is formulated and implemented numerically. It also applies to the unstructured triangular meshes in principle. We have conducted linear benchmark tests, which agree well with previous results and traditional schemes. Furthermore, multiple-$n$ simulations are also carried out as a proof of principle, demonstrating the efficiency of this scheme in nonlinear turbulence simulations within the framework of the finite element method.

Steady-state distribution functions can be used to calculate stability conditions for modes, radiation energy losses and particle loss rates. Heuristic analytic approximations to these distributions can capture key behaviors of the true distributions such as the relative speeds of different transport processes while possessing computational advantages over their numerical counterparts. In this paper, we motivate and present a closed-form analytic model for a distribution of particles in a centrifugal or tandem mirror. We find that our model outperforms other known models in approximating numerical steady-state simulations outside of a narrow range of low confining potentials. We demonstrate the model’s suitability in the high confining potential regime for applications such as loss-cone stability thresholds, fusion yields and available energy.

The decay of a turbulent magnetic field is slower with helicity than without. Furthermore, the magnetic correlation length grows faster for a helical than a non-helical field. Both helical and non-helical decay laws involve conserved quantities: the mean magnetic helicity density and the Hosking integral. Using direct numerical simulations in a triply periodic domain, we show quantitatively that in the fractionally helical case the mean magnetic energy density and correlation length are approximately given by the maximum of the values for the purely helical and purely non-helical cases. The time of switchover from one to the other decay law can be obtained on dimensional grounds and is approximately given by $I_{H}^{1/2}I_{M}^{-3/2}$, where $I_{H}$ is the Hosking integral and $I_{M}$ is the mean magnetic helicity density. An earlier approach based on the decay time is found to agree with our new result and suggests that the Hosking integral exceeds naive estimates by the square of the same resistivity-dependent factor by which also the turbulent decay time exceeds the Alfvén time. In the presence of an applied magnetic field, the mean magnetic helicity density is known to be not conserved, and we show that then also the Hosking integral is not conserved.

We investigate a model of turbulent magnetic reconnection introduced by (Higashimori, Yokoi and Hoshino 2013 Phys. Rev. Lett. 110, 255001) and show that the classic two-dimensional, steady-state Sweet–Parker and Petschek reconnection solutions are supported. We present evidence that these are the only two steady-state reconnection solutions, and we determine the criterion for their selection. Sweet–Parker reconnection occurs when there is no growth in turbulent energy, whereas Petschek reconnection occurs when the current density in the reconnecting current sheet is able to surpass a critical value, allowing for the growth of turbulent energy that creates the diffusion region. Further, we show that the Petschek solutions are self-similar, depending on the value of the turbulent time scale, and produce a universal steady reconnection rate. The self-consistent development of Petschek reconnection through turbulence, within the model, is an example of fast and steady magnetic reconnection without an explicit need for the collisionless terms in an extended Ohm’s law.

We present an extended investigation of a recently introduced model of gravitationally confined, collisionless plasma (Barbieri et al. 2024a A&A vol. 681, p. L5), which showed that rapid temperature fluctuations at the base of the plasma, occurring on time scales much shorter than the electron crossing time, can drive the system into a non-thermal state characterised by anti-correlated temperature and density profiles, commonly referred to as temperature inversion. To describe this phenomenon, a temporal coarse-graining formalism was developed (Barbieri et al., 2024b J. Plasma Phys. vol. 90, p. 905900511). In this work, we generalise that approach to cover regimes where the time scales of temperature fluctuations are comparable to or exceed the electron crossing time. We derive a set of kinetic equations that incorporate an additional term arising from the coarse-graining procedure, which was not present in the earlier formulation. Through numerical simulations, we analyse the plasma dynamics under these broader conditions, showing that the electric field influences the system when fluctuation time scales approach the electron crossing time. However, for time scales much larger than the proton crossing time, the electric field becomes negligible. The observed behaviours are interpreted within the framework of the extended temporal coarse-graining theory, and we identify the regimes and conditions in which temperature inversion persists.

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.

Standard approaches to neoclassical theory do not extend into regions of strong gradients in tokamaks such as the pedestal and internal transport barriers. Here, we calculate the modifications to neoclassical electron physics inside strong gradient regions of large aspect ratio tokamaks in the banana regime. We show that these modifications are due to the different ion flow and the strong poloidal variation of the potential. We also provide a physical interpretation of the mechanisms that drive poloidal asymmetries and hence a poloidal electric field. We apply our model to two specific example cases of pedestal profiles, calculating the neoclassical electron flux and the bootstrap current. We find that, depending on the ion flow, weak gradient neoclassical theory overestimates or underestimates the neoclassical electron transport and the bootstrap current in regions with strong gradients. We show that the determination of the mean parallel flow is more complex than in weak gradient neoclassical theory. For vanishing turbulence, we can determine the radial electric field for a given flow profile in the pedestal.

It is common wisdom that collisionless shocks become non-planar and non-stationary at sufficiently high Mach numbers. Whatever the shock structure, the upstream and downstream fluxes of the mass, momentum and energy should be equal. At low Mach numbers, these conservation laws are satisfied when the shock front is planar and stationary. When this becomes impossible, inhomogeneity and time dependence, presumably in the form of rippling, develop. In this study, we show that the shock structure changes as a kind of ‘phase transition’ when the Mach number is increased while other shock parameters are kept constant.

This study reports on a set of experiments designed to clarify the impact of the rotational transform on confinement quality at the TJ-II stellarator. For this purpose, the net plasma current is controlled using external coils, resulting in the modification of the rotational transform profile. Significant and systematic variations of the edge electron density gradients (up to $50\,\%{-}60\,\%$) and the plasma energy content ($20\,\%{-}30\,\%$) are achieved. The explanation of this behaviour relies on the placement of low-order rational surfaces in relation to the edge gradient region, which affect local turbulence fluctuation levels, facilitating the formation of zonal flows and concomitant transport barriers. This hypothesis is confirmed experimentally on the basis of a broad array of diagnostic measurements. Calculations based on a resistive magnetohydrodynamic turbulence model provide qualitative support for this hypothesis, clarifying the impact on confinement of specific rational surfaces and highlighting the complex nature of magnetically confined fusion plasmas.

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.

In this work, we investigate the generation of the ambipolar electric field in a gravitationally stratified, collisionless plasma atmosphere. In such environments, gravity tends to separate charged species. To prevent separation an electric field, classically described by the Pannekoek–Rosseland expression, is usually imposed externally. Here, we propose a self-consistent method to recover this field based on a multi-mode Fourier expansion of the electrostatic interaction. We show that, under suitable conditions, this approach naturally leads to the ambipolar electric field and restores charge neutrality. The method is tested in both isothermal and multi-temperature plasma configurations. This framework provides a foundation for future developments that may include collisions, ionisation and asymmetric boundary conditions to model more realistic stellar atmospheres.

We analyse distributions of the spatial scales of coherent intermittent structures – current sheets – obtained from fully kinetic, two-dimensional simulations of relativistic turbulence in a collisionless pair plasma using unsupervised machine-learning data dissection. We find that the distribution functions of sheet length $\ell$ (longest scale of the analysed structure in the direction perpendicular to the dominant guide field) and curvature $r_c$ (radius of a circle fitted to the structures) can be well-approximated by power-law distributions, indicating self-similarity of the structures. The distribution for the sheet width $w$ (shortest scale of the structure) peaks at the kinetic scales and decays exponentially at larger values. The data shows little or no correlation between $w$ and $\ell$, as expected from theoretical considerations. The typical $r_c$ depends linearly on $\ell$, which indicates that the sheets all have a similar curvature relative to their sizes. We find a weak correlation between $r_c$ and $w$. Our results can be used to inform realistic magnetohydrodynamic subgrid models for plasma turbulence in high-energy astrophysics.

An ionized gas medium (plasma state) turns to a complex state of plasma or dusty plasma if micrometre- to submicrometre-sized solid dust particles are introduced in it. The dusty plasma medium exhibits fluid- as well as solid-like characteristics at different background plasma conditions. It supports various linear and nonlinear dynamical structures because of external perturbation and internal instabilities. The vortical or coherent structure in the dusty plasma medium is a kind of self-sustained dynamical structure that is formed either by instabilities or by external forcing. In this review article, the author discusses the past theoretical, experimental and computational investigations on vortical and coherent structures in unmagnetized and magnetized dusty plasmas. The possible mechanisms of the formation of vortices in a dust-grain medium are discussed in detail. The studies on the evolution of vortices and their correlation with turbulence are also reviewed.

Magnetic reconnection, a fundamental plasma process, is pivotal in understanding energy conversion and particle acceleration in astrophysical systems. While extensively studied in two-dimensional (2-D) configurations, the dynamics of reconnection in three-dimensional (3-D) systems remains under-explored. In this work, we extend the classical tearing mode instability to three dimensions by introducing a modulation along the otherwise uniform direction in a 2-D equilibrium, given by $g(y)$, mimicking a flux-tube-like configuration. We perform linear stability analysis (both analytically and numerically) and direct numerical simulations to investigate the effects of three-dimensionality. Remarkably, we find that a tearing-like instability arises in three dimensions as well, even without the presence of guide fields. Further, our findings reveal that the 3-D tearing instability exhibits reduced growth rates compared with two dimensions by a factor of $\int g(y)^{1/2} {\rm d}y\,/\int {\rm d}y$, with the dispersion relation maintaining similar scaling characteristics. We show that the modulation introduces spatially varying resistive layer properties, which influence the reconnection dynamics.

Fully relativistic particle-in-cell (PIC) simulations are crucial for advancing our knowledge of plasma physics. Modern supercomputers based on graphics processing units (GPUs) offer the potential to perform PIC simulations of unprecedented scale, but require robust and feature-rich codes that can fully leverage their computational resources. In this work, this demand is addressed by adding GPU acceleration to the PIC code Osiris. An overview of the algorithm, which features a CUDA extension to the underlying Fortran architecture, is given. Detailed performance benchmarks for thermal plasmas are presented, which demonstrate excellent weak scaling on NERSC's Perlmutter supercomputer and high levels of absolute performance. The robustness of the code to model a variety of physical systems is demonstrated via simulations of Weibel filamentation and laser-wakefield acceleration run with dynamic load balancing. Finally, measurements and analysis of energy consumption are provided that indicate that the GPU algorithm is up to ${\sim }$14 times faster and $\sim$7 times more energy efficient than the optimized CPU algorithm on a node-to-node basis. The described development addresses the PIC simulation community's computational demands both by contributing a robust and performant GPU-accelerated PIC code and by providing insight into efficient use of GPU hardware.

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.

We provide a fundamentally new perspective on subcritical turbulence in plasmas, based on coherent structures, which are obtained and characterised via direct numerical solution. The domains where these coherent states exist appear to be closely connected to the those where related turbulent states can exist, so there may be a deep connection between the stability of these coherent structures and the domain where sustained turbulence is possible. In contrast to previous descriptions of turbulence in terms of a stochastic collection of linear waves, we present a fundamentally nonlinear representation based on more general classes of translating oscillatory nonlinear solutions. In turbulent tokamak plasmas, the transport can often be completely suppressed by introducing a background shear flow, whose amplitude is an important control parameter. As this parameter is decreased below a critical value, radially localised structures appear, becoming larger and more complex, in both gyrokinetic simulations and a simpler fluid model of the plasma. For the fluid model, we directly solve for a particular class of nonlinear solutions, relative periodic orbits, and determine their stability, thus explaining why these isolated structures appear in initial-value simulations. The increase of complexity as the flow shear is reduced is explained by a series of Hopf bifurcations of these nonlinear solutions, which we quantify via stability analysis. In gyrokinetic simulations, we are able to indirectly determine the underlying relative periodic orbits by imposing symmetry conditions on the simulations.