Finding less complicated coils that have adequately low field errors is a crucial step in stellarator development. One coil metric that is of high importance is the maximum curvature of the coil centreline, or coil single filament. Conductors cannot be bent below some threshold minimum radius of curvature. High coil curvatures can cause strains to exceed acceptable levels, especially in superconducting coils. We investigate three ways to optimize coil curvature and find that applying penalty functions to the coil curvature solves for coils that have a constrained maximum curvature and low field error. Penalty functions are implemented in FOCUS and coil solutions optimized for an HSX-like ‘plasma boundary’ are presented.

In an early study of the properties and capabilities of the multiregion, relaxed magnetohydrodynamic model, Hole, Hudson & Dewar claim that they are able to construct a multiregion stepped pressure cylindrical equilibrium which does not require the existence of surface currents. We present a brief argument showing that this claim is incorrect, and clarify the meaning of their statement. Furthermore, even with the statement clarified, we demonstrate that it is not possible to find solutions to reproduce the equilibrium corresponding to the parameters given in the article. We invite the authors to provide a corrigendum with the correct values of the equilibrium they constructed.

In strong electromagnetic regimes, gyrokinetic simulations have linked a substantial ion-scale turbulence stabilization to the presence of supra-thermal particles, capturing qualitatively well the experimental observations in different devices worldwide. An explanation for the underlying physical mechanism responsible for the fast-ion-induced turbulent transport reduction observed in the numerical simulations has been proposed only recently by Di Siena et al. (Nucl. Fusion, vol. 59, 2019, p. 124001; Nucl. Fusion, vol. 60, 2020, p. 089501). It involves a nonlinear cross-scale coupling (nonlinear interaction involving different modes at different wavenumbers) between ion-temperature-gradient and marginally stable Alfvén eigenmodes, which in turn increases zonal flow activity. In view of an optimization of this turbulence-stabilizing effect, the key parameters controlling the nonlinear cross-scale coupling are here identified. At the same time, these findings provide useful insights for reduced-turbulence models and integrative approaches, which might be trained on the results presented in this paper to grasp the underlying physics and the parameter scaling of the beneficial effects of fast particles on plasma turbulence.

In this study we consider the Hamiltonian approach for the construction of a map for a system with nonlinear resonant interaction, including phase trapping and phase bunching effects. We derive basic equations for a single resonant trajectory analysis and then generalize them into a map in the energy/pitch-angle space. The main advances of this approach are the possibility of considering effects of many resonances and to simulate the evolution of the resonant particle ensemble on long time ranges. For illustrative purposes we consider the system with resonant relativistic electrons and field-aligned whistler-mode waves. The simulation results show that the electron phase space density within the resonant region is flattened with reduction of gradients. This evolution is much faster than the predictions of quasi-linear theory. We discuss further applications of the proposed approach and possible ways for its generalization.

Recent discoveries have demonstrated that matter can be distinguished on the basis of topological considerations, giving rise to the concept of topological phase. Introduced originally in condensed matter physics, the physics of topological phase can also be fruitfully applied to plasmas. Here, the theory of topological phase is introduced, including a discussion of Berry phase, Berry connection, Berry curvature and Chern number. One of the clear physical manifestations of topological phase is the bulk-boundary correspondence, the existence of localized unidirectional modes at the interface between topologically distinct phases. These concepts are illustrated through examples, including the simple magnetized cold plasma. An outlook is provided for future theoretical developments and possible applications.

A plasma gun for forming a plasma stream in the open magnetic mirror trap with additional helicoidal field SMOLA is described. The plasma gun is an axisymmetric system with a planar circular hot cathode based on lanthanum hexaboride and a hollow copper anode. The two planar coils are located around the plasma source and create a magnetic field of up to 200 mT. The magnetic field forms the magnetron configuration of the discharge and provides a radial electric insulation. The source typically operates with a discharge current of up to 350 A in hydrogen. Plasma parameters in the SMOLA device are Ti ~ 5 eV, Te ~ 5–40 eV and ni ~ (0.1–1)  × 1019 m−3. Helium plasma can also be created. The plasma properties depend on the whole group of initial technical parameters: the cathode temperature, the feeding gas flow, the anode-cathode supply voltage and the magnitude of the cathode magnetic insulation.

Upcoming tokamak experiments fuelled with deuterium and tritium are expected to have large alpha particle populations. Such experiments motivate new attention to the theory of alpha particle confinement and transport. A key topic is the interaction of alpha particles with perturbations to the tokamak fields, including those from ripple and magnetohydrodynamic modes like Alfvén eigenmodes. These perturbations can transport alphas, leading to changed localization of alpha heating, loss of alpha power and damage to device walls. Alpha interaction with these perturbations is often studied with single-particle theory. In contrast, we derive a drift kinetic theory to calculate the alpha heat flux resulting from arbitrary perturbation frequency and periodicity (provided these can be studied drift kinetically). Novel features of the theory include the retention of a large effective collision frequency resulting from the resonant alpha collisional boundary layer, correlated interactions over many poloidal transits and finite orbit effects. Heat fluxes are considered for the example cases of ripple and the toroidal Alfvén eigenmode (TAE). The ripple heat flux is small. The TAE heat flux is significant and scales with the square of the perturbation amplitude, allowing the derivation of constraints on mode amplitude for avoidance of significant alpha depletion. A simple saturation condition suggests that TAEs in one upcoming experiment will not cause significant alpha transport via the mechanisms in this theory. However, saturation above the level suggested by the simple condition, but within numerical and experimental experience, which could be accompanied by the onset of stochasticity, could cause significant transport.

Most if not all tokamak heating scenarios involve multiple ion populations being heated simultaneously. To allow the simulation of various aspects of physics dynamics determining the characteristics of operational scenarios in a flexible way, speedy yet sufficiently accurate models are needed, and they should be connected to each other via a ‘backbone’. Under the umbrella of EUROfusion's Integrated Modelling efforts, such a structure is provided. The present paper focuses on one physics aspect: auxiliary heating. After solving the wave equation or beam source equation, this requires solving a set of coupled Fokker–Planck equations for the various populations involved. The adopted modules – enabling accounting for the Coulomb collisional interaction of several non-Maxwellian (minority, majority and beam) populations – are discussed and a practical example of their use is provided: the JET ‘baseline’ scenario heating a minority of {}^3\textrm {He}${}^3\textrm {He}$ ions in a balanced D+$+$T mix heated by D and T neutral beams.

A Hamiltonian two-field gyrofluid model for kinetic Alfvén waves (KAWs) in a magnetized electron–proton plasma, retaining ion finite-Larmor-radius corrections and parallel magnetic field fluctuations, is used to study the inverse cascades that develop when turbulence is randomly driven at sub-ion scales. In the directions perpendicular to the ambient field, the dynamics of the cascade turns out to be non-local and the ratio \chi _f$\chi _f$ of the wave period to the characteristic nonlinear time at the driving scale affects some of its properties. For example, at small values of \chi _f$\chi _f$, parametric decay instability of the modes driven by the forcing can develop, enhancing for a while inverse transfers. The balanced state, obtained at early time when the two counter-propagating waves are equally driven, also becomes unstable at small \chi _f$\chi _f$, leading to an inverse cascade. For \beta _e$\beta _e$ smaller than a few units, the cascade slows down when reaching the low-dispersion spectral range. For higher \beta _e$\beta _e$, the ratio of the KAW to the Alfvén frequencies displays a local minimum. At the corresponding transverse wavenumber, a condensate is formed, and the cascade towards larger scales is then inhibited. Depending on the parameters, a parallel inverse cascade can develop, enhancing the elongation of the ion-scale magnetic vortices that generically form.

Lane formation dynamics in externally driven pair-ion plasma (PIP) particles is studied in the presence of external magnetic field using Langevin dynamics (LD) simulation. The phase diagram obtained distinguishing the no-lane and lane states is systematically determined from a study of various Coulomb coupling parameter values. A peculiar lane formation-disintegration parameter space is identified; lane formation area extended to a wide range of Coulomb coupling parameter values is observed before disappearing to a mixed phase. The different phases are identified by calculating the order parameter. This and the critical parameters are calculated directly from LD simulation. The critical electric field strength value above which the lanes are formed distinctly is obtained, and it is observed that in the presence of the external magnetic field, the PIP system requires a higher value of the electric field strength to enter into the lane formation state than that in the absence of the magnetic field. We further find out the critical value of electric field frequency beyond which the system exhibits a transition back to the disordered state and this critical frequency is found as an increasing function of the electric field strength in the presence of an external magnetic field. The movement of the lanes is also observed in a direction perpendicular to that of the applied electric and magnetic field directions, which reveals the existence of the electric field drift in the system under study. We also use an oblique force field as the external driving force, both in the presence and absence of the external magnetic field. The application of this oblique force changes the orientation of the lane structures for different applied oblique angle values.

Type III radio bursts are radio emissions associated with solar flares. They are considered to be caused by electron beams travelling from the solar corona to the solar wind. Magnetic reconnection is a possible accelerator of electron beams in the course of solar flares since it causes unstable distribution functions and density inhomogeneities (cavities). The properties of radio emission by electron beams in an inhomogeneous environment are still poorly understood. We capture the nonlinear kinetic plasma processes of the generation of beam-related radio emissions in inhomogeneous plasmas by utilizing fully kinetic particle-in-cell code numerical simulations. Our model takes into account initial electron velocity distribution functions (EVDFs) as they are supposed to be created by magnetic reconnection. We focus our analysis on low-density regions with strong magnetic fields. The assumed EVDFs allow two distinct mechanisms of radio wave emissions: plasma emission due to wave–wave interactions and so-called electron cyclotron maser emission (ECME) due to direct wave–particle interactions. We investigate the effects of density inhomogeneities on the conversion of free energy from the electron beams into the energy of electrostatic and electromagnetic waves via plasma emission and ECME, as well as the frequency shift of electron resonances caused by perpendicular gradients in the beam EVDFs. Our most important finding is that the number of harmonics of Langmuir waves increases due to the presence of density inhomogeneities. The additional harmonics of Langmuir waves are generated by a coalescence of beam-generated Langmuir waves and their harmonics.

While the front of a fluid shock is a few mean-free-paths thick, the front of a collisionless shock can be orders of magnitude thinner. By bridging between a collisional and a collisionless formalism, we assess the transition between these two regimes. We consider non-relativistic, non-magnetized, planar shocks in electron–ion plasmas. In addition, our treatment of the collisionless regime is restricted to high-Mach-number electrostatic shocks. We find that the transition can be parameterized by the upstream plasma parameter \varLambda$\varLambda$ which measures the coupling of the upstream medium. For \varLambda \lesssim 1.12$\varLambda \lesssim 1.12$, the upstream is collisional, i.e. strongly coupled, and the strong shock front is about \mathcal {M}_1 \lambda _{\mathrm {mfp},1}$\mathcal {M}_1 \lambda _{\mathrm {mfp},1}$ thick, where \lambda _{\mathrm {mfp},1}$\lambda _{\mathrm {mfp},1}$ and \mathcal {M}_1$\mathcal {M}_1$ are the upstream mean free path and Mach number, respectively. A transition occurs for \varLambda \sim 1.12$\varLambda \sim 1.12$ beyond which the front is \sim \mathcal {M}_1\lambda _{\mathrm {mfp},1}\ln \varLambda /\varLambda$\sim \mathcal {M}_1\lambda _{\mathrm {mfp},1}\ln \varLambda /\varLambda$ thick for \varLambda \gtrsim 1.12$\varLambda \gtrsim 1.12$. Considering that \varLambda$\varLambda$ can reach billions in astrophysical settings, this allows an understanding of how the front of a collisionless shock can be orders of magnitude smaller than the mean free path, and how physics transitions continuously between these two extremes.

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer, Phys. Fluids B, vol. 3, 1991, pp. 2805–2821; 2822–2834; Plunk & Helander, J. Plasma Phys., vol. 84, 2018, 905840205), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic partial differential equation for a function \eta$\eta$ such the field strength satisfies B = B(\eta )$B = B(\eta )$. Closed-form solutions are obtained in cylindrical, slab and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasihelical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side. The local solutions thus obtained can be continued globally only for special initial surfaces.

The propagation characteristics of linear waves in high-energy-density magnetoplasmas are investigated using a relativistic magnetohydrodynamic model based on the framework of relativistic quantum theory. Based on the covariant Wigner function approach, a relativistic quantum magnetohydrodynamic model is established. Starting from the relativistic quantum magnetohydrodynamic equations and the Maxwell equations, the dispersion equation for relativistic quantum magnetoplasmas is derived. The contributions of both quantum effects and relativistic effects are shown in the dispersion relations for perpendicular, parallel propagation with respect to a background magnetic field. Results show that the corrections of both quantum effects and relativistic effects are significant when choosing the plasma parameters of laser-based plasma compression schemes.

The electron runaway phenomenon in plasmas depends sensitively on the momentum- space dynamics. However, efficient simulation of the global evolution of systems involving runaway electrons typically requires a reduced fluid description. This is needed, for example, in the design of essential runaway mitigation methods for tokamaks. In this paper, we present a method to include the effect of momentum-dependent spatial transport in the runaway avalanche growth rate. We quantify the reduction of the growth rate in the presence of electron diffusion in stochastic magnetic fields and show that the spatial transport can raise the effective critical electric field. Using a perturbative approach, we derive a set of equations that allows treatment of the effect of spatial transport on runaway dynamics in the presence of radial variation in plasma parameters. This is then used to demonstrate the effect of spatial transport in current quench simulations for ITER-like plasmas with massive material injection. We find that in scenarios with sufficiently slow current quench, owing to moderate impurity and deuterium injection, the presence of magnetic perturbations reduces the final runaway current considerably. Perturbations localised at the edge are not effective in suppressing the runaways, unless the runaway generation is off-axis, in which case they may lead to formation of strong current sheets at the interface of the confined and perturbed regions.

This work aims to analyse the possibility of directly driven imploding spherical targets in order to create a source of energetic particles (neutrons, protons, alphas, tritium and 3He ions) for the Laser MégaJoule facility. D3He gas-filled spherical SiO2 glass pellets, irradiated by an absorbed laser intensity of 1014 W cm−2 or 1015 W cm−2 have been considered. Depending on the absorbed laser intensity and the amount of the ablated glass layer two distinct regimes have been identified: a massive pusher and an exploding pusher. Both regimes are analysed in terms of hydrodynamics and fast particle spectra. Energetic particle time-resolved spectra are calculated and used to infer ionic temperatures and total areal densities. A parametric study has been performed by varying the shell thickness and target inner radius for both laser absorbed intensities.

In the present paper, we use a coarse-graining approach to investigate the nonlinear redistribution of free energy in both position and scale space for weakly collisional magnetised plasma turbulence. For this purpose, we use high-resolution numerical simulations of gyrokinetic (GK) turbulence that span the proton–electron range of scales, in a straight magnetic guide field geometry. Accounting for the averaged effect of the particles’ fast gyro-motion on the slow plasma fluctuations, the GK approximation captures the dominant energy redistribution mechanisms in strongly magnetised plasma turbulence. Here, the GK system is coarse grained with respect to a cut-off scale, separating in real space the contributions to the nonlinear interactions from the coarse-grid scales and the sub-grid scales (SGS). We concentrate on the analysis of nonlinear SGS effects. Not only does this allow us to investigate the flux of free energy across the scales, but also to now analyse its spatial density. We find that the net value of scale flux is an order of magnitude smaller than both the positive and negative flux density contributions. The dependence of the results on the filter type is also analysed. Moreover, we investigate the advection of energy in position space. This rather novel approach for GK turbulence can help in the development of SGS models that account for advective unstable structures for space and fusion plasmas, and with the analysis of the turbulent transport saturation.

A method of random forcing with a constant power input for two-dimensional gyrokinetic turbulence simulations is developed for the study of stationary plasma turbulence. The property that the forcing term injects the energy at a constant rate enables turbulence to be set up in the desired range and energy dissipation channels to be assessed quantitatively in a statistically steady state. Using the developed method, turbulence is demonstrated in the large-scale fluid and small-scale kinetic regimes, where the theoretically predicted scaling laws are reproduced successfully.

An encoder–decoder neural network has been used to examine the possibility for acceleration of a partial integro-differential equation, the Fokker–Planck–Landau collision operator. This is part of the governing equation in the massively parallel particle-in-cell code XGC, which is used to study turbulence in fusion energy devices. The neural network emphasizes physics-inspired learning, where it is taught to respect physical conservation constraints of the collision operator by including them in the training loss, along with the \ell _2$\ell _2$ loss. In particular, network architectures used for the computer vision task of semantic segmentation have been used for training. A penalization method is used to enforce the ‘soft’ constraints of the system and integrate error in the conservation properties into the loss function. During training, quantities representing the particle density, momentum and energy for all species of the system are calculated at each configuration vertex, mirroring the procedure in XGC. This simple training has produced a median relative loss, across configuration space, of the order of 10^{-4}$10^{-4}$, which is low enough if the error is of random nature, but not if it is of drift nature in time steps. The run time for the current Picard iterative solver of the operator is O(n^2)$O(n^2)$, where n$n$ is the number of plasma species. As the XGC1 code begins to attack problems including a larger number of species, the collision operator will become expensive computationally, making the neural network solver even more important, especially since its training only scales as O(n)$O(n)$. A wide enough range of collisionality has been considered in the training data to ensure the full domain of collision physics is captured. An advanced technique to decrease the losses further will be subject of a subsequent report. Eventual work will include expansion of the network to include multiple plasma species.