We demonstrate a fast adjoint-based method to optimise tokamak and stellarator equilibria against a pressure-driven instability known as the infinite-$n$ ideal ballooning mode. We present three finite-$\beta$ (the ratio of thermal to magnetic pressure) equilibria: one tokamak equilibrium and two stellarator equilibria that are unstable against the ballooning mode. Using the self-adjoint property of ideal magnetohydrodynamics, we construct a technique to rapidly calculate the change in the eigenvalue, a measure of ideal ballooning instability. Using the SIMSOPT optimisation framework, we then implement our fast adjoint gradient-based optimiser to minimise the eigenvalue and find stable equilibria for each of the three originally unstable equilibria.

A novel curved surface nanowire target is proposed to improve the cutoff energy of accelerated protons via target normal sheath acceleration. The interaction of a laser of intensity $1.37\times 10^{20}\ {\rm W}\ {\rm cm}^{-2}$ with a curved surface nanowire target is studied by two-dimensional particle-in-cell simulations. The numerical results indicate that the sheath electric field at the target rear side is significantly enhanced by this simple target design, compared with using the planar nanowire target. The transverse motion of hot electrons is effectively confined and the energy density of electrons is naturally increased. A series of simulations with various target parameters is carried out to investigate the performance of this novel target. This tailored target may provide implications for generating high-quality proton beams in experiments.

Free energy, widely used as a measure of turbulence intensity in weakly collisional plasmas, has been recently found to be a suitable basis to describe both linear and nonlinear growth in a wide class gyrokinetic systems. The simplicity afforded by this approach is accompanied by some drawbacks, notably the lack of any explicit treatment of wave–particle effects, which makes the theory unable to describe things like stability thresholds or dependence on the geometry of the background magnetic field. As a step toward overcoming these limitations, we propose an extension of the theory based on a generalization of the free energy. With this, it is demonstrated that resonance effects are recovered, and the bounds on growth are significantly reduced. The simplicity and efficient computation of the associated ‘optimal’ growth rates makes the theory potentially applicable to stellarator optimization.

In this research, the process of electron acceleration and wakefield generation by Gaussian-like (GL), super-Gaussian (SG) and Bessel–Gaussian (BG) laser pulses through cold collisionless plasma in the presence of a planar magnetostatic wiggler are studied. Three different types of laser spatial profiles, GL, SG and BG, are considered. Additionally, using the hydrodynamics fluid equations, Maxwell's equations as well as the perturbation technique for GL, SG and BG laser pulses in the weakly nonlinear regime and in the presence of a planar magnetostatic wiggler, governing equations for analysing the laser wakefield and electron acceleration have been derived and compared correspondingly. In addition, the effect of some important factors, including the wiggler field strengths, laser intensity, pulse length, plasma electron density and laser frequency on the wakefield and the electron energy gain, have been investigated. Numerical results show that enhancing the wiggler magnetic field results in an increase in the amplitude of the wakefield. Furthermore, it is observed that in comparison with the wakefield amplitude excited by SG and GL laser pulses, the amplitude of the wakefield excited by BG laser pulse is larger when the wiggler field is enhanced. Moreover, it is realized that the type of the laser profile, selected laser parameters and wiggler magnetic field are the most decisive and effective factors in the wakefield amplitude and shape of wakefield generation through cold collisionless plasma. Also, it is seen that as the pulse length declines, the amplitude of the wakefield increases, and correspondingly the resonance positions shift to higher ${({\varOmega _w}/{\varOmega _p})_{max}}$ values.

The magnetohydrodynamic stability of plasmas with an anisotropic pressure component is analysed for a low magnetic field configuration of the large helical device. Magnetic equilibria are calculated by the anisotropic Neumann inverse moments equilibrium code, an extension of the three-dimensional variational moments equilibrium code. A modified version of the bi-Maxwellian is used to model the anisotropic particle velocity distribution. Magnetohydrodynamic stability calculations for the $n=1$ mode family are carried out by TERPSICHORE, which has been expanded by the Kruskal–Oberman energy principle. For on-axis particle deposition, the growth rate and plasma displacement show that the parallel dominant plasmas are significantly more stable than isotropic or perpendicular dominant plasmas. For off-axis particle deposition, the growth rate and the Mercier criterion in the peripheral region $\rho =0.9$, show that low field (LF) deposition perpendicular dominant plasmas are most unstable. For the most realistic off-axis deposition profile, it is found that parallel dominant plasmas are most stable for LF deposition, while perpendicular dominant plasmas are most stable for high field deposition. We conclude that, under low magnetic field conditions in the large helical device, tangential neutral beam injection heating has a stabilising influence on the plasma.

The DESC stellarator optimization code takes advantage of advanced numerical methods to search the full parameter space much faster than conventional tools. Only a single equilibrium solution is needed at each optimization step thanks to automatic differentiation, which efficiently provides exact derivative information. A Gauss–Newton trust-region optimization method uses second-order derivative information to take large steps in parameter space and converges rapidly. With just-in-time compilation and GPU portability, high-dimensional stellarator optimization runs take orders of magnitude less computation time with DESC compared to other approaches. This paper presents the theory of the DESC fixed-boundary local optimization algorithm along with demonstrations of how to easily implement it in the code. Example quasi-symmetry optimizations are shown and compared to results from conventional tools. Three different forms of quasi-symmetry objectives are available in DESC, and their relative advantages are discussed in detail. In the examples presented, the triple product formulation yields the best optimization results in terms of minimized computation time and particle transport. This paper concludes with an explanation of how the modular code suite can be extended to accommodate other types of optimization problems.

Good fast ion confinement in quasi-isodynamic stellarators still has to be demonstrated experimentally. In the absence of fusion alphas, auxiliary heating systems can be used to generate fast ions. For this purpose, W7-X has neutral beam injection and its newly installed ion cyclotron heating system. In practice, both systems can run simultaneously, the synergetic running of both is the focus of the current modelling development. With the upgrades to the SCENIC code described here, we find that, compared with pure neutral beam injection (NBI), the fast ion tail is significantly enhanced when radio frequency (RF) heating is enabled. But the effectiveness of heating and fast ion tail generation strongly depends on the plasma parameters and antenna frequency, which calls for further study, both numerical and experimental.

This paper investigates energetic electron transport in magnetized toroidal plasmas with magnetic fields characterized by island chains and regions of stochastic field lines produced by coil perturbations. We report on experiments performed in the DIII-D tokamak, which utilize electron cyclotron heating and current drive pulses to ‘tag’ electron populations within different locations across the discharge. The cross-field transport of these populations is then inferred from electron cyclotron emission measurements and gamma emission signals from scintillator detectors. Two types of energetic particles are distinguished and discussed: non-relativistic suprathermal electrons and relativistic runaway electrons. The magnetic field topology in each discharge is reconstructed with field-line tracing codes, which are also used to determine the location and scale of magnetic islands and stochastic regions. Comparison of simulations and experiments suggests that suprathermal transport is suppressed when the tagging is performed at a smaller radial location than the location of the $q = 1$ island chain and enhanced otherwise. Here q is the safety factor. We further demonstrate that increasing the width of the stochastic region within the edge plasma yields enhancement of the suprathermal electron transport.

It is shown that the extensive literature on leaky modes in coronal magnetic flux tubes should be revised since those modes have no physical meaning.

An approximate model for pure electron plasma compression is developed for the case where the rotating wall (RW) electric field couples to the $E\times B$ rotation and axial bounce motion of the electrons. The key assumption in the model is that, throughout the compression, the plasma remains in a slowly evolving thermal equilibrium defined by the plasma temperature and angular momentum. Linearized drift kinetic theory is employed to derive an expression for torque exerted by the RW field on the plasma through coupling to the resonant plasma particles, and averaging is used to find the torque that both compresses and heats the plasma. The evolution equations for the angular velocity and temperature of the plasma include the compression and heating from the torque and cooling from cyclotron radiation.

Gyrokinetic theories, even ‘global’ models, typically rely on a separation in scale (in perpendicular wavelength) between the fluctuations and the system scale. In such models direct simulation of system-scale dynamics like magnetohydrodynamic (MHD) motion is formally not consistent. Drift-kinetic theory, on the other hand, may be directly applied to model system-scale MHD-ordered behaviour. I review the long-wavelength limit of standard gyrokinetics and drift kinetics to present the relationships between these theories in an elementary fashion. This provides a pathway to global gyrokinetic modelling, resulting in an approach that is structurally similar to kinetic MHD, and I present dynamical equations for solving global field evolution in this framework. Departures from certain earlier global gyrokinetic theories include the appearance of magnetosonic (fast) modes, and the cross-coupling of the parallel and perpendicular currents with perpendicular and parallel magnetic field components. A periodic two-dimensional testcase is outlined as a benchmarking and implementation target, to help clarify practical aspects of these theories, with minimal complexity in terms of boundary conditions, and a proof-of-principle implementation of a field-solver is exhibited. To motivate this work, I first illustrate certain limitations of existing global gyrokinetic frameworks and directly identify how scale separation approximations lead to certain ‘missing’ system-scale field terms in global gyrokinetics, largely as a result of simplifications associated with the field representation in terms of $A_{\|}$ and $B_{\|}$. As a result, the currents in the gyrokinetic Ampère's law resulting from a gyrokinetic equilibrium distribution do not match the currents implied by Ampère's law in a force-balance MHD equilibrium. I present a simple choice of equilibrium distribution function whose drift-kinetic currents are consistent with MHD currents; a specific Grad–Shafranov equilibrium is used to illustrate the size of the components of the parallel currents.

This paper presents the calculation of the bounce-averaged drift of trapped particles in a near-axis framework for axisymmetric and quasisymmetric magnetic fields that possess up-down and stellarator symmetry, respectively. This analytic consideration provides important insight on the dependence of the bounce-averaged drift on the geometry and stability properties of the field. In particular, we show that although the maximum-$\mathcal {J}$ property is unattainable in quasisymmetric stellarators, one may approach it through increased plasma $\beta$ and triangular shaping, albeit going through a reduced precession scenario with potentially higher particle losses. The description of trapped particles allows us to calculate the available energy of trapped electrons analytically in two asymptotic regimes, providing insight into the behaviour of this measure of turbulence. It is shown that the available energy is intimately related to magnetohydrodynamics (MHD) stability, providing a potential synergy between this measure of gyrokinetic turbulence and MHD stability.

In this paper, we summarize our recent work on gyrokinetic applications in electron–positron and non-neutral plasma. The electrostatic stability of electron–positron plasmas was investigated in dipole and slab geometries, with and without ion admixture. The gyrokinetic dispersion relation was derived and, for the slab case, extended to non-neutral plasmas. Here, we further extend the gyrokinetic formulation to the relativistic regime. Electron–positron plasmas are found to be remarkably stable as long as perfect symmetry between the two species prevails, but instabilities appear if this symmetry is broken, for instance by the introduction of impurities or magnetic curvature.

Colliding collisionless shocks appear in a great variety of astrophysical phenomena and are thought to be possible sources of particle acceleration in the Universe. We have previously investigated particle acceleration induced by single super-critical shocks (whose magnetosonic Mach number is higher than the critical value of 2.7) (Yao et al., Nat. Phys., vol. 17, issue 10, 2021, pp. 1177–1182; Yao et al., Matter Radiat. Extrem., vol. 7, issue 1, 2022, 014402), as well as the collision of two sub-critical shocks (Fazzini et al., Astron. Astrophys., vol. 665, 2022, A87). Here, we propose to make measurements of accelerated particles from interpenetrating super-critical shocks to observe the ‘phase-locking effect’ (Fazzini et al., Astron. Astrophys., vol. 665, 2022, A87) from such an event. This effect is predicted to significantly boost the energy spectrum of the energized ions compared with a single super-critical collisionless shock. We thus anticipate that the results obtained in the proposed experiment could have a significant impact on our understanding of one type of primary source (acceleration of thermal ions as opposed to secondary acceleration mechanisms of already energetic ions) of ion energization of particles in the Universe.

This paper discusses thermal equilibrium states of single-species plasmas, such as pure electron plasmas and pure positron plasmas, that are confined in a dipole trap. Thermal equilibrium states for such plasmas are routinely realized in the homogeneous magnetic field of Penning–Malmberg traps. We generalize the theory of these states to include inhomogeneous magnetic dipole fields. The approach to thermal equilibrium takes place in two stages with well separated time scales. On the collision time scale, thermal equilibrium is established along each magnetic field line. On the much longer transport time scale, heat conduction and viscosity bring the plasmas on different flux contours into thermal equilibrium, we call this a state of global thermal equilibrium. We present numerical results for local and global thermal equilibria. These results agree with the analytic predictions for charge collections that are large compared with the Debye length. There is, in principle, no limit to the confinement time of a single-species plasma in a global thermal equilibrium state. Experiments with hot electron–ion plasmas performed in the LDX and RT1 devices give us confidence that, in contrast to a Penning–Malmberg trap, a magnetic dipole field can also confine cold quasi-neutral electron–positron pair plasmas on the time scale of the phenomena of interest. Such pair plasmas are assumed to form in the magnetosphere of neutron stars but have so far not been realized in a laboratory. The creation of an electron–positron pair plasma is the main goal of the APEX collaboration.

Available energy (Æ), which quantifies the maximum amount of thermal energy that may be liberated and converted into instabilities and turbulence, has shown to be a useful metric for predicting saturated energy fluxes in trapped-electron-mode-driven turbulence. Here, we calculate and investigate the Æ in the analytical tokamak equilibria introduced by Miller et al. (Phys. Plasmas, vol. 5, issue, 4, 1998, pp. 973–978). The Æ of trapped electrons reproduces various trends also observed in experiments; negative shear, increasing Shafranov shift, vertical elongation and negative triangularity can all be stabilising, as indicated by a reduction in Æ, although it is strongly dependent on the chosen equilibrium. Comparing Æ with saturated energy flux estimates from the TGLF (trapped gyro-Landau fluid) model, we find fairly good correspondence, showcasing that Æ can be useful to predict trends. We go on to investigate Æ and find that negative triangularity is especially beneficial in vertically elongated configurations with positive shear or low gradients. Furthermore, we extract a gradient-threshold-like quantity from Æ and find that it behaves similarly to gyrokinetic gradient thresholds: it tends to increase linearly with magnetic shear, and negative triangularity leads to an especially high threshold. We next optimise the device geometry for minimal Æ and find that the optimum is strongly dependent on equilibrium parameters, for example, magnetic shear or pressure gradient. Investigating the competing effects of increasing the density gradient, the pressure gradient, and decreasing the shear, we find regimes that have steep gradients yet low Æ, and that such a regime is inaccessible in negative-triangularity tokamaks.

The understanding of cross-field transport is crucial for optimizing the properties of magnetic confinement in fusion devices. In this work, a two-dimensional, simplified model is used to study the turbulent dynamics in the region of the scrape-off layer. The numerical model, based on the reduced Braginskii equations, is able to describe the formation and the evolution of blob-like structures. The dynamics has been investigated by using both classical Eulerian analysis and the Lagrangian approach, by varying the ambient conditions of the plasma. The major goals are (i) a detailed and systematic study of turbulence by varying the plasma jump conditions in the edge tokamaks plasmas and (ii) a Lagrangian diffusion study of the edge turbulence by using an appropriate Braginskii model. It has been found that both the magnetic shear and the plasma mean profiles of density and temperature are crucial for setting the properties of the transport. By following fluid tracers, diffusive transients for the radial transport are observed, at length scales larger than the typical blob size. This work is relevant for the comprehension of the turbulent transport at tokamaks edges.

The ability to quickly refresh gas-jet targets without cycling the vacuum chamber makes them a promising candidate for laser-accelerated ion experiments at high repetition rate. Here we present results from the first high repetition rate ion acceleration experiment on the VEGA-3 PW-class laser at CLPU. A near-critical density gas-jet target was produced by forcing a 1000 bar H$_2$ and He gas mix through bespoke supersonic shock nozzles. Proton energies up to 2 MeV were measured in the laser forward direction and 2.2 MeV transversally. He$^{2+}$ ions up to 5.8 MeV were also measured in the transverse direction. To help maintain a consistent gas density profile over many shots, nozzles were designed to produce a high-density shock at distances larger than 1 mm from the nozzle exit. We outline a procedure for optimizing the laser–gas interaction by translating the nozzle along the laser axis and using different nozzle materials. Several tens of laser interactions were performed with the same nozzle which demonstrates the potential usefulness of gas-jet targets as high repetition rate particle source.

This paper documents the design and construction of MUSE, the world's first permanent magnet (PM) stellarator and the first quasi-axisymmetric experiment. The purpose of MUSE is to develop and assess a new way of building optimised stellarators that uses simple planar coils PMs. Our PM optimisation algorithm consists of initialising a geometry to pack dipoles densely, running the FAMUS code to minimise surface field error subject to PM constraints and applying discrete jumps to reach a physically realisable solution. FAMUS treats the PM system as a set of ideal point dipoles. From there we construct finite-volume magnet towers to be housed in 3D-printed PM holders. We describe the design of the PM holders, which were validated by laser metrology. We analyse the effects of finite permeability, sensitivity to perturbations and magnetostatic forces. An exact analytic formula for the magnetic field from a finite-volume PM tower is presented to compute PM–PM forces and stress on the PM holder. Stellarator construction is complete and experiments are underway.