Monte Carlo methods are often employed to numerically integrate kinetic equations, such as the particle-in-cell method for the plasma kinetic equation, but these methods suffer from the introduction of counting noise to the solution. We report on a cautionary tale of counting noise modifying the nonlinear saturation of kinetic instabilities driven by unstable beams of plasma. We find a saturated magnetic field in under-resolved particle-in-cell simulations due to the sampling error in the current density. The noise-induced magnetic field is anomalous, as the magnetic field damps away in continuum kinetic and increased particle count particle-in-cell simulations. This modification of the saturated state has implications for a broad array of astrophysical phenomena beyond the simple plasma system considered here, and it stresses the care that must be taken when using particle methods for kinetic equations.

A fluid system is derived to describe electrostatic magnetized plasma turbulence at scales somewhat larger than the Larmor radius of a given species. It is related to the Hasegawa–Mima equation, but does not conserve enstrophy, and, as a result, exhibits a forward cascade of energy, to small scales. The inertial-range energy spectrum is argued to be shallower than a $-11/3$ power law, as compared to the $-5$ law of the Hasegawa–Mima enstrophy cascade. This property, confirmed here by direct numerical simulations of the fluid system, may help explain the fluctuation spectrum observed in gyrokinetic simulations of streamer-dominated electron-temperature-gradient driven turbulence (Plunk et al., Phys. Rev. Lett., vol. 122, 2019, 035002), and also possibly some cases of ion-temperature-gradient driven turbulence where zonal flows are suppressed (Plunk et al., Phys. Rev. Lett., vol. 118, 2017, 105002).

A runaway avalanche can result in a conversion of the initial plasma current into a relativistic electron beam in high-current tokamak disruptions. We investigate the effect of massive material injection of deuterium–noble gas mixtures on the coupled dynamics of runaway generation, resistive diffusion of the electric field and temperature evolution during disruptions in the deuterium–tritium phase of ITER operations. We explore the dynamics over a wide range of injected concentrations and find substantial runaway currents, unless the current quench time is intolerably long. The reason is that the cooling associated with the injected material leads to high induced electric fields that, in combination with a significant recombination of hydrogen isotopes, leads to a large avalanche generation. Balancing Ohmic heating and radiation losses provides qualitative insights into the dynamics; however, an accurate modelling of the temperature evolution based on energy balance appears crucial for quantitative predictions.

We report on an analytical and numerical study of the dynamics of a three-dimensional array of identical magnetic flux tubes in the reduced-magnetohydrodynamic description of the plasma. We propose that the long-time evolution of this system is dictated by flux-tube mergers, and that such mergers are dynamically constrained by the conservation of the pertinent (ideal) invariants, viz. the magnetic potential and axial fluxes of each tube. We also propose that in the direction perpendicular to the merging plane, flux tubes evolve in a critically balanced fashion. These notions allow us to construct an analytical model for how quantities such as the magnetic energy and the energy-containing scale evolve as functions of time. Of particular importance is the conclusion that, like its two-dimensional counterpart, this system exhibits an inverse transfer of magnetic energy that terminates only at the system scale. We perform direct numerical simulations that confirm these predictions and reveal other interesting aspects of the evolution of the system. We find, for example, that the early time evolution is characterized by a sharp decay of the initial magnetic energy, which we attribute to the ubiquitous formation of current sheets. We also show that a quantitatively similar inverse transfer of magnetic energy is observed when the initial condition is a random, small-scale magnetic seed field.

This work presents updates in the diagnostics systems, magnetohydrodynamics (MHD) calculations and simulations of microwave heating scenarios of the small modular Stellarator of Costa Rica 1 (SCR-1). Similarly, the design of a flexible bolometer and magnetic diagnostics (a set of Mirnov coils, Rogowski coils and two diamagnetic loops) are introduced. Furthermore, new MHD equilibrium calculations for the plasma of the SCR-1 device were performed using the VMEC code including the poloidal cross-section of the magnetic flux surfaces at different toroidal positions, profiles of the rotational transform, magnetic well, magnetic shear and total magnetic field norm. Charged particle orbits in vacuum magnetic field were computed by the magnetic field solver BS-SOLCTRA (Vargas et al. In 27th IAEA Fusion Energy Conference (FEC 2018), 2018. IAEA). A visualization framework was implemented using Paraview (Solano-Piedra et al. In 23rd IAEA Technical Meeting on the Research Using Small Fusion Devices (23rd TM RUSFD), 2017) and compared with magnetic mapping results (Coto-Vílchez et al. In 16th Latin American Workshop on Plasma Physics (LAWPP), 2017, pp. 43–46). Additionally, simulations of microwave heating scenarios were performed by the IPF-FDMC full-wave code. These simulations calculate the conversion of the ordinary waves to extraordinary waves and allow us to identify the location where the conversion takes place. Finally, the microwave heating scenarios for the $330^{\circ }$ toroidal position are presented. The microwave heating scenarios showed that the O–X–B mode conversion is around 12–14 %. It was possible to identify the spatial zone where the conversion takes place (upper hybrid frequency).

A magnetic field configuration of an island divertor for a quasi-axisymmetric stellarator (CFQS) is proposed. The configuration incorporates large islands surrounding the core confinement region. The interface between the core region and the peripheral region of the island divertor is a clear magnetic separatrix similar to a tokamak divertor. The structure of divertor magnetic field lines is very regular without stochasticity and the connection length is sufficiently long for good divertor performance. Such a divertor configuration is produced in the magnetic field configuration for the CFQS device, which is now under construction in China.

In this paper, we discuss the results of a new particle pusher in realistic ultra-strong electromagnetic fields such as those encountered around rotating neutron stars. After presenting the results of this algorithm in simple fields and comparing them to expected exact analytical solutions, we present new simulations for a rotating magnetic dipole in vacuum for a millisecond pulsar by using the Deutsch solution. Particles are injected within the magnetosphere, neglecting radiation reaction, interaction among them and their feedback on the fields. Our simulations are therefore not yet fully self-consistent because the Maxwell equations are not solved according to the current produced by these particles. The code highlights the symmetrical behaviour of particles of opposite charge to mass ratio, $q/m$, with respect to the north and south hemispheres. The relativistic Lorentz factor $\gamma$ of the accelerated particles is proportional to this ratio $q/m$: protons reach up to $\gamma _p \simeq 10^{10.7}$, whereas electrons reach up to $\gamma _e \simeq 10^{14}$. Our simulations show that particles could either be captured by the neutron star, trapped around it or ejected far from it, well outside the light cylinder. Actually, for a given charge to mass ratio, particles follow similar trajectories. These particle orbits show some depleted directions, especially at high magnetic inclination with respect to the rotation axis for positive charges and at low inclination for negative charges because of symmetry. Other directions are preferred and loaded with a high density of particles, some directions concentrating the highest or lowest acceleration efficiencies.

Kinetic plasma simulations are nowadays commonly used to study a wealth of nonlinear behaviours and properties in laboratory and space plasmas. In particular, in high-energy physics and astrophysics, the plasma usually evolves in ultra-strong electromagnetic fields produced by intense laser beams for the former or by rotating compact objects such as neutron stars and black holes for the latter. In these ultra-strong electromagnetic fields, the gyro-period is several orders of magnitude smaller than the time scale on which we desire to investigate the plasma evolution. Some approximations are required such as, for instance, artificially decreasing the electromagnetic field strength, which is certainly not satisfactory. The main flaw of this downscaling is that it cannot reproduce particle acceleration to ultra-relativistic speeds with a Lorentz factor above $\gamma \approx 10^3$–$10^4$. In this paper, we design a new algorithm able to catch particle motion and acceleration to a Lorentz factor of up to $10^{15}$ or even higher by using Lorentz boosts to special frames where the electric and magnetic field are parallel. Assuming that these fields are locally uniform in space and constant in time, we solve analytically the equation of motion in a tiny region smaller than the length scale of the spatial and temporal gradient of the field. This analytical integration of the orbit severely reduces the constraint on the time step, allowing us to use large time steps, avoiding resolving the ultra-high gyro-frequency. We performed simulations in ultra-strong spatially and time-dependent electromagnetic fields, showing that our particle pusher is able to follow accurately the exact analytical solution for very long times. This property is crucial to properly capture for instance lepton electrodynamics in electromagnetic waves produced by fast rotating neutron stars. We conclude with a simple implementation of our new pusher into a one-dimensional relativistic electromagnetic particle-in-cell code, testing it against plasma oscillations, two-stream instabilities and strongly magnetized relativistic shocks.

In this semi-analytic study we develop a mathematical model for determining the electromagnetic field due to a current-driven antenna immersed in a cold, magnetized plasma, valid for frequencies below the electron plasma frequency. At each point in the plasma, it is shown that the vacuum electric field of the antenna couples to the plasma conductivity tensor and acts as an infinitesimal source term to drive plasma currents – the total field is then found from the aggregate sum of these point sources, expressed as an integral across the vacuum field. A general solution is provided for both azimuthally symmetric cylindrical coordinates as well as a fully generalized Cartesian solution. As an example of how this general solution may be applied, we solve for the field due to an electric dipole antenna of length $\ell$, aligned along the background field, at frequencies below the ion cyclotron frequency. It is found that the near field decays exponentially with increasing $k_{\bot }z$, whereas the far field exhibits wave-like behaviour. The radiation zone exhibits propagation cones emanating from either end of the dipole, with a propagation angle that is consistent with past analytic studies of inertial Alfvén waves. The mathematical model presented here may be advantageous over other numerical methods, as it allows the user to solve parts of the problems analytically, thereby cutting down significantly on computation time, as well as offering physical insight into the system that may not be evident with other numerical solvers.

A Hamiltonian six-field gyrofluid model is constructed, based on closure relations derived from the so-called ‘quasi-static’ gyrokinetic linear theory where the fields are assumed to propagate with a parallel phase velocity much smaller than the parallel particle thermal velocities. The main properties captured by this model, primarily aimed at exploring fundamental problems of interest for space plasmas such as the solar wind, are its ability to provide a reasonable agreement with kinetic theory for linear low-frequency modes, and at the same time to ensure a Hamiltonian structure in the absence of explicit dissipation. The model accounts for equilibrium temperature anisotropy, ion and electron finite Larmor radius corrections, electron inertia, magnetic fluctuations along the direction of a strong guide field and parallel Landau damping, introduced through a Landau-fluid modelling of the parallel heat transfers for both gyrocentre species. Remarkably, the quasi-static closure leads to exact and simple expressions for the nonlinear terms involving gyroaveraged electromagnetic fields and potentials. One of the consequences is that a rather natural identification of the Hamiltonian structure of the model becomes possible when Landau damping is neglected. A slight variant of the model consists of a four-field Hamiltonian reduction of the original six-field model, which is also used for the subsequent linear analysis. In the latter, the dispersion relations of kinetic Alfvén waves and the firehose instability are shown to be correctly reproduced, relatively far in the sub-ion range (depending on the plasma parameters), while the spectral range where the slow-wave dispersion relation and the field-swelling instabilities are precisely described is less extended. This loss of accuracy originates from the breaking of the condition of small phase velocity, relative to the parallel thermal velocity of the electrons (for kinetic Alfvén waves and firehose instability) or of the ions (in the case of the field-swelling instabilities).

The beam–plasma instability can be addressed as a reduced model in several contexts of plasma physics, from space to fusion plasma. In this paper, we review and refine some nonlinear features of this model. Specifically, by analysing the dependence of the nonlinear velocity spread as a function of the linear growth rate, we discuss the effective size of the resonance in view of its role in the spectral overlap at saturation. The relevance of this characterization relies on the necessity of a quantitative determination of the overlap degree to discriminate among different transport regimes of the self-consistent dynamics. The analysis is enriched with a study of the phase-space dynamics by means of the Lagrangian coherent structure technique, in order to define the transport barriers of the system describing the relevant features of the overlap process. Finally, we discuss relevant features related to the mode saturation levels.

Massive material injection has been proposed as a way to mitigate the formation of a beam of relativistic runaway electrons that may result from a disruption in tokamak plasmas. In this paper we analyse runaway generation observed in eleven ASDEX Upgrade discharges where disruption was triggered using massive gas injection. We present numerical simulations in scenarios characteristic of on-axis plasma conditions, constrained by experimental observations, using a description of the runaway dynamics with a self-consistent electric field and temperature evolution in two-dimensional momentum space and zero-dimensional real space. We describe the evolution of the electron distribution function during the disruption, and show that the runaway seed generation is dominated by hot-tail in all of the simulated discharges. We reproduce the observed dependence of the current dissipation rate on the amount of injected argon during the runaway plateau phase. Our simulations also indicate that above a threshold amount of injected argon, the current density after the current quench depends strongly on the argon densities. This trend is not observed in the experiments, which suggests that effects not captured by zero-dimensional kinetic modelling – such as runaway seed transport – are also important.

We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

We apply field–particle correlations – a technique that tracks the time-averaged velocity-space structure of the energy density transfer rate between electromagnetic fields and plasma particles – to data drawn from a hybrid Vlasov–Maxwell simulation of Alfvén-ion cyclotron turbulence. Energy transfer in this system is expected to include both Landau and cyclotron wave–particle resonances, unlike previous systems to which the field–particle correlation technique has been applied. In this simulation, the energy transfer rate mediated by the parallel electric field $E_{\Vert }$ comprises approximately 60 % of the total rate, with the remainder mediated by the perpendicular electric field $E_{\bot }$. The parallel electric field resonantly couples to protons, with the canonical bipolar velocity-space signature of Landau damping identified at many points throughout the simulation. The energy transfer mediated by $E_{\bot }$ preferentially couples to particles with $v_{tp}\lesssim v_{\bot }\lesssim 3v_{tp}$, where $v_{tp}$ is the proton thermal speed, in agreement with the expected formation of a cyclotron diffusion plateau. Our results demonstrate clearly that the field–particle correlation technique can distinguish distinct channels of energy transfer using single-point measurements, even at points in which multiple channels act simultaneously, and can be used to determine quantitatively the rates of particle energization in each channel.

We present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $v^{2}$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.

Finding coil sets with desirable physics and engineering properties is a crucial step in the design of modern stellarator devices. Existing stellarator coil optimization codes ultimately produce zero-thickness filament coils. However, stellarator coils have finite depth and thickness, which can make the single-filament model a poor approximation, particularly when coil build dimensions are relatively large compared to the coil–plasma distance. In this paper, we present a new method for designing coils with finite builds and present a mechanism to optimize the orientation of the winding pack. We approximate finite-build coils with a multi-filament model. A numerical implementation has been developed, and applications to the Helically Symmetric eXperiment stellarator and a new UW-Madison quasihelically symmetric configuration are shown.

The Dimits shift is the shift between the threshold of the drift-wave primary instability and the actual onset of turbulent transport in a magnetized plasma. It is generally attributed to the suppression of turbulence by zonal flows, but developing a more detailed understanding calls for consideration of specific reduced models. The modified Terry–Horton system has been proposed by St-Onge (J. Plasma Phys., vol. 83, 2017, 905830504) as a minimal model capturing the Dimits shift. Here, we use this model to develop an analytic theory of the Dimits shift and a related theory of the tertiary instability of zonal flows. We show that tertiary modes are localized near extrema of the zonal velocity $U(x)$, where $x$ is the radial coordinate. By approximating $U(x)$ with a parabola, we derive the tertiary-instability growth rate using two different methods and show that the tertiary instability is essentially the primary drift-wave instability modified by the local $U'' \doteq {\rm d}^2 U/{\rm d} x^2 $. Then, depending on $U''$, the tertiary instability can be suppressed or unleashed. The former corresponds to the case when zonal flows are strong enough to suppress turbulence (Dimits regime), while the latter corresponds to the case when zonal flows are unstable and turbulence develops. This understanding is different from the traditional paradigm that turbulence is controlled by the flow shear $| {\rm d} U / {\rm d} x |$. Our analytic predictions are in agreement with direct numerical simulations of the modified Terry–Horton system.

Radiative-magnetohydrodynamic (RMHD) equations along with a full set of Maxwell's equations are followed to formulate the charged surface waves at the interface of an incompressible, radiative, magnetized dusty plasma and vacuum, while assuming that the characteristic wave frequency is much smaller than the ion gyrofrequency, having an equilibrium background state. It is found that the separation of charges on the surface is followed by thermal motion, which further leads to a negative pressure gradient normal to the surface, hence the plasma–vacuum interface is under tension due to two different types of oppositely directed pressures. The dusty plasma RMHD set of equations admits a linear dispersion relation of surface Jeans instability of an incompressible dusty plasma, which exhibits a strong coupling between the electron surface charge and dust surface mass densities and we conclude that the surface densities of both electrons and dust as well as the dust inertia play major roles in the gravitational collapse of the surface of astrophysical objects such as stars, galaxies etc. Further, the growth rate of radiative surface waves is found to be function of both the temperature inhomogeneity, appearing due to thermal radiation heat flux, as well as the thermal radiation pressure. The present findings of charged surface waves may seek application at the astroscales.

We consider the tilting instability of a magnetically confined spheromak using three-dimensional magnetohydrodynamic and relativistic particle-in-cell calculations with an application to astrophysical plasmas, specifically those occurring in magnetar magnetospheres. The instability is driven by the counter-alignment of the spheromak's intrinsic magnetic dipole with the external magnetic field. Initially, the spheromak rotates – tilts – trying to lower its magnetic potential energy. As a result, a current sheet forms between the internal magnetic field of a spheromak and the confining field. Magnetic reconnection sets in; this leads to the annihilation of the newly counter-aligned magnetic flux of the spheromak. This occurs on a few Alfvén time scales. In the case of a higher-order (second-order) spheromak, the internal core is first pushed out of the envelope, resulting in formation of two nearly independent tilting spheromaks. Thus, the magnetically twisted outer shell cannot stabilize the inner core. During dissipation, helicity of the initial spheromak is carried away by torsional Alfvén waves, violating the assumptions of the Taylor relaxation theorem. In applications to magnetar giant flares, fast development of tilting instabilities and no stabilization of the higher-order spheromaks make it unlikely that trapped spheromaks are responsible for the tail emission lasting hundreds of seconds.