The non-perturbative guiding-centre model provides an exact alternative to full-orbit simulations of charged particle dynamics in situations where traditional guiding-centre theory may fail. We demonstrate that the charged particle motion in a homogeneous, time-varying magnetic field is a solvable example of the non-perturbative guiding-centre model. This entails showing that the exact magnetic moment of Qin and Davidson can be constructed to be asymptotic to the adiabatic invariant series of Kruskal. In contrast to the perturbative invariant, the exact invariant contains information about parametric resonances. These resonances destroy the conservation of the usual magnetic moment over very long times. This refutes some previous claims about the all-time invariance of the magnetic moment.

Ion energy–angle distributions (IEADs) at material surfaces are a critical input for plasma–material interaction (PMI) studies in fusion devices, yet they are computationally expensive to obtain using particle-in-cell (PIC) simulations. In this work, we develop a machine learning surrogate based on a deep deconvolutional neural network (DDeCNN) trained on large databases generated with the hPIC2 code. The surrogate is capable of reconstructing IEADs from sheath parameters for both thermal and radio-frequency (RF) plasmas, including cases with multiple ion species. Across thousands of test cases, the model achieves high accuracy, with over 97 % of predictions classified as good or average based on standard error metrics (MAE, MSE, L2). Even in the more challenging RF and multi-species regimes, the surrogate reliably captures the multi-peak structure of PIC results. Once trained, the surrogate produces IEADs in milliseconds on a common workstation, yielding speedups of six to seven orders of magnitude compared with running a full PIC simulation. This computational gain enables dense parameter scans and direct coupling of IEAD predictions with PMI and erosion models on whole-device scales in fusion-relevant conditions.

We derive the hot-electron-limit (HEL) closure for the moment hierarchy used to solve the gyrokinetic equations, known as the gyromoment (GM) approach. By expanding the gyroaveraging kernels in the small ion-to-electron temperature ratio limit, $\tau \ll 1$, and retaining only the essential $\mathcal{O}(\tau )$ terms, we obtain a closed system for the density, parallel velocity and parallel and perpendicular temperatures. In a Z-pinch geometry, the GM system with the HEL closure is analytically equivalent to the one developed by Ivanov et al. (2022 J. Plasma Phys., vol. 88, no. 5, p. 905880506). Numerical benchmarks confirm the closure’s accuracy, reproducing established linear growth rates, nonlinear heat transport and low collisionality dynamics. An extension to the tokamak-relevant $s{-}\alpha$ geometry and a comparison with gyrokinetic simulations reveal the capabilities and limitations of the HEL-closed GM model: while transport levels and the temporal dynamics are qualitatively preserved even at $\tau =1$, the absence of higher-order kinetic moments prevents an accurate prediction of the Dimits shift and of transport suppression.

Numerical computation of the ideal magnetohydrodynamic (MHD) equilibrium magnetic field is at the base of stellarator optimisation and provides the starting point for solving more sophisticated partial differential equations like transport or turbulence models. Conventional approaches solve for a single stationary point of the ideal MHD equations, which is fully defined by three invariants and the numerical scheme employed by the solver. We present the first numerical approach that can solve for a continuous distribution of equilibria with fixed boundary and rotational transform, varying only the pressure invariant. This approach minimises the force residual by optimising parameters of multilayer perceptrons that map from a scalar pressure multiplier to the Fourier Zernike basis as implemented in the modern stellarator equilibrium solver DESC.

ARC is designed to produce ${400}\,\textrm {MW}$ of net electricity and prove the commercial feasibility of a fusion power plant. In order to achieve this goal ARC has to operate with optimal core performance in a stationary scenario that minimises wear on the first wall and divertor. This requires avoiding or mitigating magnetohydrodynamic (MHD) instabilities which have the potential to not only degrade the plasma core but also lead to deleterious transient heat loads on plasma facing components. Therefore, this work aims at characterising the MHD stability of the high performance ARC scenario and inform the design of error field correction coils. Firstly, simulations of vertical displacement events show that an in-vessel coil is not needed and instead the poloidal shaping coils can be used to control vertical stability. These simulations also inform the demands on the corresponding coil power supplies. Stability analysis of the ideal kink mode with or without a conducting wall and kinetic effects suggests that the ARC baseline scenario operates deeply in the stable region. Using RDCON, tearing modes at the $m/n=2/1$ and $3/2$ surfaces (with poloidal mode number $m$, and toroidal mode number $n$) are shown to be linearly stable, and including thermal transport effects in the rational surfaces lead to further stabilisation. However, other transient plasma instabilities can seed neoclassical tearing modes (NTMs). The marginally stable width of NTMs in ARC strongly depends on the internal inductance and can fall below ${0.1}{\,\,\%}$ of the normalised poloidal flux. Furthermore, an empirical cross-machine model of the $n=1$ error field leading to a disruption predicts a critical error field larger than SPARC but smaller than ITER. Three-dimensional coils can be designed with the Generalised Purturbed Equilbium Code based on a simple model that calculates the maximum correctable error field that is limited by the neoclassical toroidal viscosity torque. Broad scans of different coil geometries identify a set of 2 rows of off-midplane coils to be a suitable solution. It is also determined that such a set of three-dimensional coils is capable of correcting $n=2$ error fields to some degree and creating strong enough $n=2$ or $n=3$ edge resonant perturbation fields for the suppression of edge-localised modes at reasonable coil currents. The final design of the first ARC will be further informed by results from SPARC.

Wave interactions with magnetised particles underlie many plasma heating and current drive technologies. Typically, these interactions are modelled by bounce averaging the quasilinear Kennel–Engelmann diffusion tensor over the particle orbit. However, as an object derived in a two-dimensional space, the Kennel–Engelmann tensor does not fully respect the conservation of four-momentum required by the action conservation theorem, since it neglects the absorption of perpendicular momentum. This defect leads to incorrect predictions for the wave-induced cross-field particle transport. Here, we show how this defect can easily be fixed, by extending the tensor from two to four dimensions and matching the form required by four-momentum conservation. The resulting extended tensor, when bounce averaged, recovers the form of the diffusion paths required by action-angle Hamiltonian theory. Importantly, the extended tensor should be easily implementable in Fokker–Planck codes through a mild modification of the existing Kennel–Engelmann tensor.

Predictions of the pedestal temperature profile calculated using a model for electron-temperature-gradient (ETG) turbulent electron heat transport Field et al. (2023 Philos. Trans. R. Soc. A, vol. 381, p. 20210228) are compared with the pedestal structure of H-mode plasmas in JET-Be/W (with Be wall and W divertor) over scans of the deuterium–tritium (D:T) isotope mix and hydrogenic gas fuelling rate Frassinetti et al. (2023 Nucl. Fusion, vol. 63, p. 112009). Predictions for the electron temperature at the location of the density pedestal top $T_e(\psi _N^{n_{e,top}})$ (where $\psi_N$, is the normalised poloidal flux) are found to agree well with measured values over both scans across the full range of D:T ratio. However, the pedestal top temperature $T_{e,ped}$, typically located somewhat inside the density pedestal top, is under-predicted by as much as a factor ${\sim} 2$. This implies that the ETG heat flux scaling appropriate for the steep-density gradient region, on which the model is based, is not applicable where the density gradient is weak. This difference might be attributed to a difference between the physics of the ETG turbulence in regimes where the density gradient is either strong or weak, which are thought to be dominated by either the ‘slab’ or ‘toroidal’ branches of ETG turbulence. Other branches of turbulence might also play a role in the electron heat transport, particularly in regions of weak-density gradient. As in the experiment, the predicted $T_e$ across the pedestal decreases with the ratio of separatrix to pedestal density $n_{e,sep}/n_{e,ped}$, which increases with the gas fuelling rate. Results from three models combining the ETG heat flux model with the EPED1 pedestal (EPED) model (Snyder et al., Phys. Plasmas, 2009, vol. 16, p. 056118) are also presented, including one which also incorporates the density pedestal prediction mode of Saarelma et al. (Nucl. Fusion, 2023, vol. 63, p. 052002), this model providing a complete prediction of the pedestal profiles.

A disrupting plasma in a high-performance tokamak such as ITER or SPARC may generate large runaway electron currents that, upon impact with the tokamak wall, can cause serious damage to the device. To quickly identify regions of safe operation in parameter space, it is useful to develop reduced models and analytical criteria that predict when a significant fraction of the Ohmic current is converted into a current of runaway electrons. In deuterium–tritium plasmas, the seed runaway current may have a significant contribution from – or may even be dominated by – tritium beta decay and Compton scattering. In this work, a criterion for significant runaway electron generation that includes tritium beta decay and Compton scattering sources is developed. The avalanche gain factor includes the effects of partial screening of injected noble gases. The result is an analytical model that can predict significant runaway electron generation in the next generation of activated tokamak devices. The model is validated by fluid simulations using Dream (Hoppe et al. 2021 Comput. Phys. Commun., vol. 268, p. 108098) and is shown to delineate regions in parameter space where significant runaway electron generation may occur.

We propose a novel reformulation of the Vlasov–Ampère equations for plasmas that reveals discrete symmetries that enables simultaneous conservation of mass, momentum and energy; preservation of Gauss’s law; positivity of the distribution function; and consistency with quasi-neutral asymptotics. The approach employs variable and coordinate transformations to yield a coupled system comprising a modified Vlasov equation and associated moment–field equations. The modified Vlasov equation advances a conditional distribution function that excludes mass, momentum and energy densities, which are instead evolved through moment equations enforcing the relevant symmetries, conservation laws and involution constraints. This reformulation aligns naturally with a recent slow-manifold reduction technique, which separates fast electron time scales and simplifies the treatment of the quasi-neutral limit within the reduced moment–field subsystem. Using this framework, we develop a numerical method for the reduced 1D1V subsystem that, for the first time in the literature, satisfies all key physical constraints while maintaining a quasi-neutral asymptotic behaviour. The advantages of the method are demonstrated on canonical electrostatic test problems, including the multiscale ion acoustic shock wave.

Paschen’s law relates the breakdown voltage ($V_B$) of a gas to the product of the gas pressure ($p$) and the inter-electrode distance ($d$), predicting a characteristic minimum breakdown voltage at a specific $pd$ value. In this study, the roles of electrode configurations (symmetric and asymmetric) and inter-electrode spacing on gas breakdown processes (or Paschen’s law) are examined. Numerous sets of experiments are performed with both symmetric and asymmetric electrode configurations of different sizes to obtain Paschen curves at different inter-electrode distances. The experimentally obtained Paschen’s curves for different electrode configurations are fitted using a proposed modified empirical relation for $V_B$, incorporating variable power-law dependencies and fitting parameters to better capture the observed deviations. Upon closer inspection, we observed that the breakdown voltage ($V_B$) and the corresponding $pd$ value ($pd_{\min}$) are influenced by both electrode configurations and inter-electrode discharge gap. The variation in $V_B$ and $pd_{\min}$ for different electrode configurations is explained by analysing the electric field distributions between the electrodes (cathode and anode) for an applied voltage.

Magnetised liner inertial fusion (MagLIF) has attracted attention in the past decade for its high obtained Lawson triple products and prospects to scale to ignition. In this work, we investigate the effect of viscosity on the sausage instability and magneto-Rayleigh–Taylor instability (MRTI) in conditions relevant for MagLIF implosions. First, we quantify the amount of damping that viscosity has on instability growth by deriving an expression for the ratio between viscous and inviscid growth rates. This expression is parameterised by a single non-dimensional number: the Galilei number $Ga$, which measures the ratio of gravitational and viscous forces. We discuss in detail the physical intuition $Ga$ provides on instability growth. The derived growth rates are then validated against FLASH simulations. We then calculate a critical viscosity threshold $\eta _{c}$ required for viscosity to dampen the instability growth rate by 5 %. From this analysis, we show that, for drive currents relevant to laboratory MagLIF experiments (of the order of tens of MA), this critical viscosity threshold is much greater than realistic liner viscosity values except for the shortest perturbation wavelength regimes. We conclude that viscosity does not play a significant role in the initial linear growth of the sausage instability and MRTI in MagLIF liners, but our results motivate future investigation into effects of viscosity in nonlinear and high temperature regimes.

We present a new variational formulation for viscous and resistive Hall magnetohydrodynamics. We first find a variational principle for ideal Hall magnetohydrodynamics by applying the physical assumptions leading to Hall magnetohydrodynamics at the Lagrangian level, and then we add the viscous and resistive terms by means of constrained variations. We also provide a metriplectic reformulation of our formulation, based on two canonical Lie–Poisson brackets for the ideal part and metric 4-brackets for the dissipative part.

We present two-dimensional particle-in-cell simulations of a magnetised, collisionless, relativistic pair plasma subjected to combined velocity and magnetic field shear, a scenario typical at intermittent structures in plasma turbulence. We create conditions where only the Kelvin–Helmholtz instability (KHI) and drift–kink instability (DKI) can develop, while tearing modes are forbidden. The interaction of DKI and KHI generates qualitatively new structures, marked by a thickened shear layer with very weak electromagnetic field, modulated by KH vortices. Over a range of moderately strong velocity shears explored, the interaction of DKI and KHI results in a significant enhancement of dissipation over cases with only velocity shear or only magnetic shear. Moreover, we observe a new and efficient way of particle acceleration where particles are stochastically accelerated by the motional electric field exterior to the shear layer as they meander in an S-shaped pattern in and out of it. This process takes advantage of the bent geometry of the shear layer caused by the DKI–KHI interaction and is responsible for most of the highest-energy particles produced in our simulations. These results further our understanding of dissipation and particle acceleration at intermittent structures, which are present in plasma turbulence across a wide range of astrophysical contexts such as in active galactic nucleus jet sheaths, potentially relevant to limb-brightened emission, etc., and highlight the sensitivity of dissipation to multiple interacting instabilities, thus providing a strong motivation for further studies of their nonlinear interaction at the kinetic level.

The present paper originates from the need to understand nonlinear wave behaviour in dusty plasmas, aiming to explore stable envelope wave propagation and interactions beyond traditional theoretical models. By using nonlinear Schrödinger equation (NLSE) analysis and molecular dynamics (MD) simulations, the study verifies the existence and stable propagation of non-standard envelope waves. It demonstrates elastic like collisions, introduces tuneable parameters for wave shaping and quantifies error trends with nonlinearity. A key breakthrough is confirming that even analytically invalid waveforms remain stable, challenging NLSE constraints. Present results enhance nonlinear wave theory and support precise, tuneable signal transmission in plasma diagnostics and microgravity experiments.

Within the gyrokinetic formalism, we present and analytically study the equations for an explicit treatment of the trapped-electron-modified kinetic ballooning mode (KBM) and the electromagnetic version of the trapped-electron mode, in general geometry. The gradient of the plasma $\beta =8\pi p /B^2,$ the ratio of kinetic to magnetic pressure, is taken to be small enough to avoid including perturbations of the magnetic field strength. Trapped-electron-modified KBMs are first described close to ideal magnetohydrodynamic marginality, retaining the explicit resonant contribution of both ions and trapped electrons, and then in a strongly driven fluid limit. We show that maximum-$\mathcal J$ devices (where $\mathcal J$ is the second adiabatic invariant) enjoy relatively good stability properties at finite $\beta ,$ but the coupling of trapped electron and KBMs might induce modes rotating in the ion direction, thus eluding good maximum-$\mathcal J$ properties. An eigenvalue equation for the finite$\hbox{-}\beta$ trapped-electron-mode is derived and studied. We highlight the possibility of having an electron-temperature-gradient driven electromagnetic instability in regions of bad magnetic curvature. A mechanism for the destabilisation of the trapped-particle-enabled collisionless microtearing mode also is proposed. Our results are general and provide new theoretical ground for the characterisation of several magnetic confinement concepts, such as tokamaks, quasisymmetric and quasi-isodynamic stellarators.

Hosking & Schekochihin (Phys. Rev. X, 2021, vol. 11, 041005) have proposed that statistically isotropic decaying magnetohydrodynamic turbulence without net magnetic helicity conserves the mean square fluctuation level of magnetic helicity in large volumes – or, equivalently, the integral over space of the two-point correlation function of the magnetic-helicity density, denoted $I_{\!H}$. Formally, the conservation and gauge invariance of $I_{\!H}$ require the vanishing of certain boundary terms related to the strength of long-range spatial correlations. These boundary terms represent the ability (or otherwise) of the turbulence to organise fluxes over arbitrarily large distances to deplete or enhance fluctuations of magnetic helicity. In this work, we present a theory of these boundary terms, employing a methodology analogous to that of Batchelor & Proudman (Philos. Trans. R. Soc. A, 1956, vol. 248, p. 369) to determine the relevant asymptotic forms of correlation functions. We find that long-range correlations of sufficient strength to violate the conservation of $I_{\!H}$ cannot develop dynamically if the evolution equation for the magnetic vector potential is chosen to be local in space. Likewise, we find that such correlations cannot develop for a wide class of gauge choices that make this equation non-local (including the Coulomb gauge). Nonetheless, we also identify a class of non-local gauge choices for which correlations that are sufficiently strong to violate the conservation of $I_{\!H}$ do appear possible. We verify our theoretical predictions for the case of the Coulomb gauge with measurements of correlation functions in a high-resolution numerical simulation.

The influence of Tsallis q-entropy on the electron-impact excitation process is derived for a non-extensive plasma. The ionisation probability is obtained as a function of the impact parameter using a semiclassical trajectory analysis. Results indicate that Tsallis q-entropy suppresses the electron-impact ionisation cross-section in a non-extensive plasma. Additionally, the influence of Tsallis q-entropy diminishes as the ratio of the electron temperature to the ion temperature increases. In addition, the influence of Tsallis q-entropy amplifies with increasing projectile electron energy. Furthermore, it is shown that the position of maximum ionisation probability is recessed from the target centre as the q-entropy increases.

An adjoint formulation of energetic particle confinement in axisymmetric tokamak geometry is derived and evaluated using a physics-informed neural network (PINN). The PINN estimates the mean escape time of energetic ions by solving an inhomogeneous adjoint of the drift kinetic equation with a Lorentz collision operator, yielding predictions of fast ion loss in tokamak geometry due to direct ion orbit loss and collisional transport. To our knowledge, this is the first time a PINN has been used to solve the drift kinetic equation in tokamak geometry, a challenging problem due to the large time scale separation between the rapid transit time of energetic ions and their slow collisional time scale. It is shown that a careful and intentional design of a PINN is able to learn the mean escape time across the majority of the plasma volume, suggesting a path towards constructing a rapid surrogate for use within a broader optimisation framework.

Experimental analysis and simulations with the BOUT++ code show that small edge-localised modes (ELMs) in reactor-relevant high-density regimes originate in a region close to the separatrix and only marginally perturb the pedestal structure. The measured divertor peak parallel energy fluence (ε∥,peak) for a database of small ELM scenarios in DIII-D and ASDEX Upgrade can be reproduced, within 40 % accuracy on average, if an ad hoc modification of the Eich peak parallel ELM energy fluence model is applied to account for the small ELM pedestal birth location. This allows for first-order extrapolation of small-ELM divertor ε∥,peak to ITER and SPARC, resulting in values that satisfy the nominal melting threshold of tungsten monoblocks of 12 MJ m−2. The findings reported in this study, both via modelling and direct measurements, constitute a step forward in assessing small ELMs in high edge-collisionality scenarios as a viable plasma regime for the operation of next-generation fusion machines.

Formation of a high-beta plasma in a mirror magnetic field is studied for the first time using three-dimensional semi-implicit particle-in-cell simulations providing a fully kinetic description of not only ions but also electrons. It is shown that, in addition to the longitudinal jump in electric potential between the centre of the trap and the wall, a radial electric field appears in the plasma. Due to this radial field, almost all of the azimuthal electric current required for equilibrium is created by electrons. It was also found that continuous model injection of plasma into the centre of the trap does not result in reaching the magnetohydrodynamic pressure limit ($\beta =1$) due to the development of the flute instability with an azimuthal number $m=1$. The instability growth rate in such a compact system is found to be comparable to the ion-cyclotron frequency. No stabilising effect is observed either from conducting ends or from the perfectly conducting sidewall. The probable reason for that is fast fluctuations of electric field localised inside the injection region that prevent electrons from being frozen into the field lines.