Neutral atoms recycled from wall interaction interact with confined plasma, thereby refuelling it, most strongly in the region closest to the wall. This occurs near the X-point in diverted configurations, or else near the wall itself in limited configurations. A progression of analytic models is developed for neutral density in the vicinity of a planar or linear source in an ionising domain. First-principles neutral transport simulations with DEGAS2 are used throughout to test the validity and limits of the model when using equivalent sources. The model is further generalised for strong plasma gradients or the inclusion of charge exchange. An important part of the problem of neutral fuelling from recycling is thereby isolated and solved with a closed-form analytic model. A key finding is that charge exchange with the confined plasma can be significantly simplified with a reasonable sacrifice of accuracy by treating it as a loss. The several assumptions inherent to the model (and the simulations with which it is compared) can be adapted according to the particular behaviour of neutrals in the divertor and the manner in which they cross the separatrix.

Machine learning (ML)-driven reduced-order modelling is applied to accelerate steady-state convergence in three-dimensional, nonlinear, flux-driven two-fluid simulations of boundary plasma turbulence. A parametric scan of plasma resistivity, heating and density sources is performed to generate comprehensive datasets across various turbulent regimes for model training and validation. To efficiently manage and interpret these datasets, we apply the proper orthogonal decomposition technique to reduce the dimensionality of key plasma quantities such as plasma density, temperature, electric potential and vorticity. Data-driven models are trained to map physical parameters to low-dimensional representation, enabling the rapid generation of quasi-steady-state plasma profiles. The results demonstrate that density, temperature and electric potential are qualitatively well captured with a relatively low number of bases, whereas vorticity requires a larger number of bases due to its fine spatial structures. A comparison between ML-generated restarts and simulations from scratch demonstrates a significant computational advantage of the ML approach, reducing simulation time by up to a factor of three. This hybrid framework, combining data-driven reduced-order modelling with first-principles simulations, highlights the potential of ML to accelerate plasma turbulence modelling, making high-fidelity simulations more computationally feasible for large-scale fusion devices, such as ITER and DEMO.

The effects of negative triangularity (NT) on boundary plasma turbulence in double-null (DN) configurations are investigated using global, nonlinear, three-dimensional, flux-driven two-fluid simulations. Negative triangularity plasmas exhibit suppressed interchange-driven instabilities, resulting in enhanced confinement and lower fluctuation levels compared with positive triangularity (PT) plasmas. This reduction in interchange instability is associated with the weakening of curvature effects in the unfavourable region, caused by the stretching of magnetic field lines at the outer midplane. The magnetic disconnection between the turbulent low-field side and the quiescent high-field side results in most of the heat flux reaching the DN outer targets. In NT plasmas, the power load on the outer target is reduced, while it increases on the inner target, indicating a reduced in–out power asymmetry compared with PT plasmas. Furthermore, the analysis of power load asymmetry between the upper and lower targets shows that the absolute magnitude of up–down power asymmetry is mitigated in NT plasmas, mainly due to the reduced total power crossing the separatrix. The reduction of interchange instabilities in NT plasmas also affects the blob dynamics. A three-dimensional blob analysis reveals that NT plasmas feature smaller blob sizes and slower propagation velocities. Finally, an analytical scaling law for blob size and velocity that includes plasma shaping effects is derived based on the two-region model and is found to qualitatively capture the trends observed in nonlinear simulations.

Linear stability studies are presented for a quasi-axisymmetric stellarator equilibrium which is unstable with respect to external kink and peeling-ballooning modes. Using the three-dimensional linear stability CASTOR3D code, the effects of parallel viscosity, gyro-viscosity, ion diamagnetic drift velocity, ExB velocity and an externally driven flow in direction of the quasi-symmetry are investigated with respect to their influence on growth rate, oscillation frequency and mode structure.

This work demonstrates that magnetohydrodynamic (MHD) stable, quasi-isodynamic (QI) stellarator equilibria with reduced turbulence can be generated with an optimised coilset. We present one such equilibrium which, when being generated by coils, maintains the benefits of its excellent QI quality (low neoclassical transport at small particle collisionality net toroidal current and good fast-particle confinement) while demonstrating ideal-MHD stability and lower ion-temperature-gradient-driven turbulent heat flux than W7-X. As a consequence of its optimised rotational transform profile, this plasma equilibrium has nested flux surfaces and a chain of large islands at the plasma’s edge, for which we present an island divertor design. It additionally features an electron root – a large region in the plasma core in which the radial electric field points outwards, towards the plasma boundary – which provides a potential solution for preventing impurity accumulation in a fusion device.

Based on the High Magnetic field Helicon eXperiment (HMHX), considering parabolic distribution and Gaussian distribution of radial plasma density, the HELIC code was used to study the parameter optimisation design of a helicon wave plasma (HWP) source. Some parameters (antenna type, radio frequency, discharge gas, plasma radius, magnetic field) were selected. The results show that a half-helix antenna is the excitation antenna and a frequency of 13.56 MHz is the most commonly used power source for HMHX. Argon and nitrogen are selected as discharge gases to achieve the best effect of power deposition. In order to realise hydrogen–HWP discharge, a new antenna with plasma radius of 10.5 mm and antenna radius of 13.5 mm can be designed. For the new antenna, when the magnetic field intensity is 1000 Gs, the best discharge effect can be achieved. The results of this paper can provide guidance for the design of a plasma source for HWP discharge under different conditions in the future.

An efficient novel mechanism of laser pulse focusing with the help of a shaped underdense plasma target immersed in an inhomogeneous magnetic field has been demonstrated. These studies have been carried out with the help of two-dimensional particle-in-cell simulation employing the OSIRIS 4.0 platform. It is shown that the divergent magnetic field profile compresses the electromagnetic wave pulse in the transverse direction. A comparative investigation with plane and lens-shaped plasma geometries has also been conducted to find an optimal configuration for focusing the laser at the desirable location. Furthermore, it is also demonstrated that, when the electron cyclotron resonance layer is placed at a suitable location where the laser is focused, a highly energetic electron beam is generated.

This study makes use of plasma-profile data from the EUROfusion pedestal database (Frassinetti et al. 2020 Nucl. Fusion vol. 61, p. 016001), focusing on the electron-temperature and electron-density profiles in the edge region of H-mode ELMy JET ITER-Like-Wall (ILW) pulses. We make systematic predictions of the electron-temperature pedestal, taking engineering parameters of the plasma pulses and the density profiles as inputs. We first present a machine-learning (ML) algorithm which, given more inputs than theory-based modelling, is able to reconstruct unseen temperature profiles within $20\,\%$ of the experimental values. We find a hierarchy of the most consequential engineering parameters for such predictions. This result confirms the conceptual possibility of accurate data-driven prediction. Next, taking a simple theoretical approach that assumes a definite local relationship between the electron-density ($R/L_{n_e}$) and electron-temperature ($R/L_{T_e}$) gradients, we find that a range of power-law scalings $R/L_{T_e}=A(R/L_{n_e})^\alpha$ with $\alpha\approx 0.4$ correctly capture the behaviour of the electron-temperature in the steep-gradient region. Fitting $A$ and $\alpha$ independently for each pedestal reveals a clear one-to-one correlation, suggesting an underlying constraint in pedestal physics. The measured $\eta_e = L_{n_e}/L_{T_e}$ values across the pedestal exhibit a wide distribution, significantly exceeding the slab-ETG linear stability threshold, implying either a non-linear threshold shift or a measurably supercritical saturated turbulent state. Finally, we fit parameters for scalings that relate the turbulent heat flux to the gradients $R/L_{T_e}$ and $R/L_{n_e}$, similarly to models extracted from gyrokinetic simulations. The inclusion of more experimental parameters is necessary for such models to match the accuracy of our ML results.

We detail here a semi-analytical model for the pellet rocket effect, which describes the acceleration of pellets in a fusion plasma due to asymmetries in the heat flux reaching the pellet surface and the corresponding ablation rate. This effect was shown in experiments to significantly modify the pellet trajectory, and previously projected deceleration values of ${\sim} 10^6\,\textrm {m}\,\textrm{s}^{-2}$ for reactor-scale devices indicated that it may severely limit the effectiveness of pellet injection methods. We account for asymmetries stemming both from plasma parameter gradients and an asymmetric plasmoid shielding caused by the drift of the ionised pellet cloud. For high temperature, reactor relevant scenarios, we find a wide range of initial pellet sizes and speeds – particularly those relevant for large fragments of shattered pellet injection for disruption mitigation – where the rocket effect has a major impact on the penetration depth. In these cases, the plasma parameter profile variations dominate the rocket effect. We find that for small and fast pellets, where the rocket effect is less pronounced, plasmoid shielding-induced asymmetries dominate.

To better understand the dependence of the magnetic field structure in the plasma edge on the plasma boundary shape, in the context of X-point and island divertor designs, we define and develop a class of stellarators called umbilic stellarators. These equilibria are characterised by a single continuous high-curvature edge on the plasma boundary that goes around multiple times toroidally before meeting itself. We develop a technique that allows us to simultaneously optimise the plasma boundary along with a curve lying on the boundary on which we impose a high curvature while imposing omnigenity – a property of the magnetic field that ensures trapped particle confinement throughout the plasma volume. We find that umbilic stellarators naturally tend to favour piecewise omnigenity instead of omnigenity with a specific helicity. After generating omnigenous umbilic stellarators, we design coil sets for some of them and explore the fieldline structure in the edge and its sensitivity to small fluctuations in the plasma. Finally, using single-stage optimisation, we simultaneously modify the plasma and coil shape and propose an experiment to modify an existing tokamak to a finite-$\beta$ stellarator using this technique and explore a potentially simpler way to convert a limited tokamak into a diverted stellarator.

This study investigates the feasibility of reconstructing the last closed flux surface in the DIII-D tokamak using neural network models trained on reduced input feature sets, addressing an ill-posed task. Two models are compared: one trained solely on coil currents and another incorporating coil currents, plasma current and loop voltage. The model trained exclusively on coil currents achieved a mean point displacement of $0.04$ m on a held-out test set, while the inclusion of plasma current and loop voltage reduced the error to $0.03$ m. This comparison highlights the trade-offs between input feature complexity and reconstruction accuracy, demonstrating the potential of machine learning algorithms to perform effectively in data-limited environments, such as those expected in fusion power plants due to diagnostic constraints imposed by the presence of blankets and shielding.

The near-axis description of optimised stellarator fields has proven to be a powerful tool both for the design and understanding of this magnetic confinement concept. The description consists of an asymptotic model of the equilibrium in the distance from its centremost axis, and is thus only approximate. Any practical application therefore requires the eventual construction of a global equilibrium. This paper presents a novel way of constructing global equilibria using the DESC code that guarantees the correct asymptotic behaviour imposed by a given near-axis construction. The theoretical underpinnings of this construction are carefully presented, and benchmarking examples provided. This opens the door to an efficient coupling of the near-axis framework and that of global equilibria for future optimisation efforts.

We present a simple, analytically solvable magnetohydrodynamics model of current sheet formation through X-point collapse under optically thin radiative cooling. Our results show that cooling accelerates the collapse of the X-point along the inflows, but strong cooling can arrest or even reverse the current sheet elongation in the outflow direction. Hence, we detail a modification to the radiatively cooled Sweet–Parker model developed by Uzdensky & McKinney (Phys. Plasmas, 1962, vol. 18, issue 4, p. 042105) to allow for varying current sheet length. The steady-state solution shows that, when radiative cooling dominates compressional heating, the current sheet length is shorter than the system size, with an increased reconnection rate compared with the classical Sweet–Parker rate. The model and subsequent results lay out the groundwork for a more complete theoretical understanding of magnetic reconnection in regimes dominated by optically thin radiative cooling.

Cosmic-ray transport in turbulent astrophysical environments remains a multifaceted problem and, despite decades of study, the impact of complex magnetic field geometry – evident in simulations and observations – has only recently received more focussed attention. To understand how ensemble-averaged transport behaviour emerges from the intricate interactions between cosmic rays and structured magnetic turbulence, we run test-particle experiments in snapshots of a strongly turbulent magnetohydrodynamics simulation. We characterise particle–turbulence interactions via the gyro radii of particles and their experienced field-line curvatures, which reveals two distinct transport modes: magnetised motion, where particles are tightly bound to strong coherent flux tubes and undergo large-scale mirroring; and unmagnetised motion, characterised by chaotic scattering through weak and highly tangled regions of the magnetic field. We formulate an effective stochastic process for each mode: compound subdiffusion with long mean free paths for magnetised motion, and a Langevin process with short mean free paths for unmagnetised motion. A combined stochastic walker that alternates between these two modes accurately reproduces the mean squared displacements observed in the test-particle data. Our results emphasise the critical role of coherent magnetic structures in comprehensively understanding cosmic-ray transport and lay a foundation for developing a theory of geometry-mediated transport.

We develop a new scaling theory for the resistive tearing mode instability of a current sheet with a strong shear flow across the layer. The growth rate decreases with increasing flow shear and is completely stabilized as the shear flow becomes Alfvénic: both in the constant-$\varPsi$ regime, as in previous results, but we also show that the growth rate is in fact suppressed more strongly in the nonconstant-$\varPsi$ regime. As a consequence, for sufficiently large flow shear, the maximum of the growth rate is always affected by the shear suppression, and the wavenumber at which this maximum growth rate is attained is an increasing function of the strength of the flow shear. These results may be important for the onset of reconnection in imbalanced MHD turbulence.

In this paper, we calculate the available energy, an upper bound on the thermal energy released that may in turn drive turbulence, due to an ion-temperature-gradient instability driven by curvature in the presence of adiabatic electrons. This is done by choosing an appropriate set of invariants that neglect parallel dynamics, whilst keeping the density profile fixed. Conditions for vanishing available energy are derived and are found to be qualitatively similar to conditions for stability derived from gyrokinetic theory, including strong stabilisation if the ratio of the temperature and density gradient, $\mathrm{d} \ln T / \mathrm{d} \ln n =\eta$, falls in the range $0 \leqslant \eta \leqslant 2/3$. To assess the utility of the available energy, a database consisting of $6 \times 10^4$ local gyrokinetic simulations in randomly sampled tokamak geometries is constructed. Using this database and a similar one sampling stellarators (Landreman et al. 2025 J. Plasma Phys. vol. 91, E120), the available energy is shown to exhibit correlation with the ion energy flux as long as the parallel dynamics is unimportant. Overall it is found that available energy is good at predicting the energy flux variability due to the gradients in density and temperature, but performs worse when it comes to predicting its variability arising from geometry.

First numerical results from the newly developed pseudo-spectral code P-FLARE (Penalised FLux-driven Algorithm for REduced models) are presented. This flux-driven turbulence/transport code uses a pseudo-spectral formulation with the penalisation method to impose radial boundary conditions. Its concise, flexible structure allows implementing various quasi-two-dimensional reduced fluid models in flux-driven formulation. Here, results from simulations of the modified Hasegawa–Wakatani system are discussed, where particle transport and zonal flow formation, together with profile relaxation, are studied. It is shown that coupled spreading/profile relaxation that one obtains for this system is consistent with a simple one-dimensional model of coupled spreading/transport equations. Then, the effect of a particle source is investigated, which results in the observation of sandpile-like critical behaviour. The model displays profile stiffness for certain parameters, with very different input fluxes resulting in very similar mean density gradients. This is due to different zonal flow levels around the critical value for the control parameter (i.e. the ratio of the adiabaticity parameter to the mean gradient) and the existence for this system of a hysteresis loop for the transition from two-dimensional turbulence to a zonal flow dominated state.

In plasmas and in astrophysical systems, particle diffusion faster than normal, namely superdiffusion, has been detected, calling for a generalisation of Fick’s law and of the transport equation. Formally, superdiffusive transport is often described by fractional diffusion equations, where the second-order spatial derivative is changed into a spatial derivative of fractional order less than two, usually in the form of the so-called Riesz derivative. Fractional operators are non-local, so that this involves the contribution of very distant points (far from the particle source) to the particle flux at a given position in the system. To consider the property of non-locality in the case of anomalous transport, we give a simple analytical derivation of the fractional Fick’s law, where the contribution to the flux of distant points is weighted by an inverse power law, and show that this is consistent with use of the Riesz derivative in the transport equation. A numerical procedure for the computation of the non-local flux is presented and applied to both a simple Gaussian density profile and also to density profiles coming from test particle simulations of one-dimensional collisionless shocks. In these simulations, energetic particles can move diffusively or superdiffusively. The latter case naturally gives rise to the emergence of uphill transport in the downstream region, which means a flux of particles in the same direction of the density gradient. This analysis contributes to the interpretation of energetic particle fluxes accelerated at collisionless shock waves in the interplanetary medium.

Bootstrap current plays a crucial role in the equilibrium of magnetically confined plasmas, particularly in quasi-symmetric stellarators and in tokamaks, where it can represent bulk of the electric current density. Accurate modeling of this current is essential for understanding the magnetohydrodynamic (MHD) equilibrium and stability of these configurations. This study expands the modeling capabilities of M3D-C1, an extended-MHD code, by implementing self-consistent physics models for bootstrap current. It employs two analytical frameworks: a generalized Sauter model (Sauter et al. 1999 Phys. Plasmas vol. 6, no. 7, pp. 2834–2839), and a revised Sauter-like model (Redl et al. 2021 Phys. Plasmas vol. 28, no. 2, pp. 022502). The isomorphism described by Landreman et al. (2022 Phys. Rev. Lett. vol. 128, pp. 035001) is employed to apply these models to quasi-symmetric stellarators. The implementation in M3D-C1 is benchmarked against neoclassical codes, including NEO, XGCa and SFINCS, showing excellent agreement. These improvements allow M3D-C1 to self-consistently calculate the neoclassical contributions to plasma current in axisymmetric and quasi-symmetric configurations, providing a more accurate representation of the plasma behavior in these configurations. A workflow for evaluating the neoclassical transport using SFINCS with arbitrary toroidal equilibria calculated using M3D-C1 is also presented. This workflow enables a quantitative evaluation of the error in the Sauter-like model in cases that deviate from axi- or quasi-symmetry (e.g. through the development of an MHD instability).

Accurate reduced models of turbulence are desirable to facilitate the optimisation of magnetic-confinement fusion reactor designs. As a first step towards higher-dimensional turbulence applications, we use reservoir computing, a machine-learning (ML) architecture, to develop a closure model for a limiting case of electrostatic gyrokinetics. We implement a pseudo-spectral Eulerian code to solve the one-dimensional Vlasov–Poisson system on a basis of Fourier modes in configuration space and Hermite polynomials in velocity space. When cast onto the Hermite basis, the Vlasov equation becomes an infinitely coupled hierarchy of fluid moments, presenting a closure problem. We exploit the locality of interactions in the Hermite representation to introduce an ML closure model of the small-scale dynamics in velocity space. In the linear limit, when the kinetic Fourier–Hermite solver is augmented with the reservoir closure, the closure permits a reduction of the velocity resolution, with a relative error within 2 % for the Hermite moment where the reservoir closes the hierarchy. In the strongly nonlinear regime, the ML closure model more accurately resolves the low-order Fourier and Hermite spectra when compared with a naive closure by truncation and reduces the required velocity resolution by a factor of 16.