We have performed two-dimensional hybrid simulations of non-relativistic collisionless shocks in the presence of pre-existing energetic particles (‘seeds’); such a study applies, for instance, to the re-acceleration of galactic cosmic rays (CRs) in supernova remnant (SNR) shocks and solar wind energetic particles in heliospheric shocks. Energetic particles can be effectively reflected and accelerated regardless of shock inclination via a process that we call diffusive shock re-acceleration. We find that re-accelerated seeds can drive the streaming instability in the shock upstream and produce effective magnetic field amplification. This can eventually trigger the injection of thermal protons even at oblique shocks that ordinarily cannot inject thermal particles. We characterize the current in reflected seeds, finding that it tends to a universal value $J\simeq en_{\text{CR}}v_{\text{sh}}$, where $en_{\text{CR}}$ is the seed charge density and $v_{\text{sh}}$ is the shock velocity. When applying our results to SNRs, we find that the re-acceleration of galactic CRs can excite the Bell instability to nonlinear levels in less than ${\sim}10~\text{yr}$, thereby providing a minimum level of magnetic field amplification for any SNR shock. Finally, we discuss the relevance of diffusive shock re-acceleration also for other environments, such as heliospheric shocks, galactic superbubbles and clusters of galaxies.

Mean field electrodynamics (MFE) facilitates practical modelling of secular, large scale properties of astrophysical or laboratory systems with fluctuations. Practitioners commonly assume wide scale separation between mean and fluctuating quantities, to justify equality of ensemble and spatial or temporal averages. Often however, real systems do not exhibit such scale separation. This raises two questions: (I) What are the appropriate generalized equations of MFE in the presence of mesoscale fluctuations? (II) How precise are theoretical predictions from MFE? We address both by first deriving the equations of MFE for different types of averaging, along with mesoscale correction terms that depend on the ratio of averaging scale to variation scale of the mean. We then show that even if these terms are small, predictions of MFE can still have a significant precision error. This error has an intrinsic contribution from the dynamo input parameters and a filtering contribution from differences in the way observations and theory are projected through the measurement kernel. Minimizing the sum of these contributions can produce an optimal scale of averaging that makes the theory maximally precise. The precision error is important to quantify when comparing to observations because it quantifies the resolution of predictive power. We exemplify these principles for galactic dynamos, comment on broader implications, and identify possibilities for further work.

We develop a model of gamma-ray flares of the Crab Nebula resulting from the magnetic reconnection events in a highly magnetised relativistic plasma. We first discuss physical parameters of the Crab Nebula and review the theory of pulsar winds and termination shocks. We also review the principle points of particle acceleration in explosive reconnection events [Lyutikov et al., J. Plasma Phys., vol. 83(6), p. 635830601 (2017a); J. Plasma Phys., vol. 83(6), p. 635830602 (2017b)]. It is required that particles producing flares are accelerated in highly magnetised regions of the nebula. Flares originate from the poleward regions at the base of the Crab’s polar outflow, where both the magnetisation and the magnetic field strength are sufficiently high. The post-termination shock flow develops macroscopic (not related to the plasma properties on the skin-depth scale) kink-type instabilities. The resulting large-scale magnetic stresses drive explosive reconnection events on the light-crossing time of the reconnection region. Flares are produced at the initial stage of the current sheet development, during the X-point collapse. The model has all the ingredients needed for Crab flares: natural formation of highly magnetised regions, explosive dynamics on the light travel time, development of high electric fields on macroscopic scales and acceleration of particles to energies well exceeding the average magnetic energy per particle.

Pedestal modelling is crucial to predict the performance of future fusion devices. Current modelling efforts suffer either from a lack of kinetic physics, or an excess of computational complexity. To ameliorate these problems, we take a first-principles multiscale approach to the pedestal. We will present three separate sets of equations, covering the dynamics of edge localised modes (ELMs), the inter-ELM pedestal and pedestal turbulence, respectively. Precisely how these equations should be coupled to each other is covered in detail. This framework is completely self-consistent; it is derived from first principles by means of an asymptotic expansion of the fundamental Vlasov–Landau–Maxwell system in appropriate small parameters. The derivation exploits the narrowness of the pedestal region, the smallness of the thermal gyroradius and the low plasma $\unicode[STIX]{x1D6FD}$ (the ratio of thermal to magnetic pressures) typical of current pedestal operation to achieve its simplifications. The relationship between this framework and gyrokinetics is analysed, and possibilities to directly match our systems of equations onto multiscale gyrokinetics are explored. A detailed comparison between our model and other models in the literature is performed. Finally, the potential for matching this framework onto an open-field-line region is briefly discussed.

Symmetries of a fluid-gyrokinetic model are investigated using Lie group techniques. Specifically, the nonlinear system constructed by Zocco & Schekochihin (Phys. Plasmas, vol. 18, 2011, 102309), which combines nonlinear fluid equations with a drift-kinetic description of parallel electron dynamics, is studied. Significantly, this model is fully gyrokinetic, allowing for arbitrary $k_{\bot }\unicode[STIX]{x1D70C}_{i}$, where $k_{\bot }$ is the perpendicular wave vector of the fluctuations and $\unicode[STIX]{x1D70C}_{i}$ the ion gyroradius. The model includes integral operators corresponding to gyroaveraging as well as the moment equations relating fluid variables to the kinetic distribution function. A large variety of exact symmetries is uncovered, some of which have unexpected form. Using these results, new nonlinear solutions are constructed, including a helical generalization of the Chapman–Kendall solution for a collapsing current sheet.

The problem of pressure driven infernal type perturbations near the plasma edge is addressed analytically for a circular limited tokamak configuration which presents an edge flattened safety factor. The plasma is separated from a metallic wall, either ideally conducting or resistive, by a vacuum region. The dispersion relation for such types of instabilities is derived and discussed for two classes of equilibrium profiles for pressure and mass density.

We have explored the thermodynamics of compressed magnetized plasmas in laboratory experiments and we call these studies ‘magnetothermodynamics’. The experiments are carried out in the Swarthmore Spheromak eXperiment device. In this device, a magnetized plasma source is located at one end and at the other end, a closed conducting can is installed. We generate parcels of magnetized plasma and observe their compression against the end wall of the conducting cylinder. The plasma parameters such as plasma density, temperature and magnetic field are measured during compression using HeNe laser interferometry, ion Doppler spectroscopy and a linear ${\dot{B}}$ probe array, respectively. To identify the instances of ion heating during compression, a PV diagram is constructed using measured density, temperature and a proxy for the volume of the magnetized plasma. Different equations of state are analysed to evaluate the adiabatic nature of the compressed plasma. A three-dimensional resistive magnetohydrodynamic code (NIMROD) is employed to simulate the twisted Taylor states and shows stagnation against the end wall of the closed conducting can. The simulation results are consistent to what we observe in our experiments.

A drift-kinetic calculation in an axisymmetric tokamak, with super-diamagnetic flows, is presented to elucidate the relation between the radial electric field, $E_{r}$, zonal flows and the rapid precession of the trapped particle (TP) population. It has been shown earlier (Rosenbluth & Hinton, Phys. Rev. Lett., vol. 80(4), 1998, p. 724, hereafter RH) that an initial radial electric field results in geodesic acoustic mode oscillations which subsequently Landau damp, resulting in a much smaller final residual electric field, and accompanying parallel zonal flows. We observe an apparent paradox: the final angular momentum in the RH parallel zonal flow is much smaller than the angular momentum expected from the well-known rapid precession of the trapped particle population in the RH residual electric field. We reconcile this paradox by illuminating the presence of a population of reverse circulating particle flows that, dominantly, are equal and opposite to the rapid TP precession. Mathematically, the calculation is facilitated by transforming to an energy coordinate shifted from conventional by an amount proportional to $E_{r}$. We also discuss the well-known RH coefficient in the context of effective mass and show how the TP precession and the opposite circulating flows contribute to this mass. Finally, we show that in the long wavelength limit, the RH flows and RH coefficient arise as a natural consequence of conservation of toroidal angular momentum and the second adiabatic invariant.

In this paper, weak-turbulence theory is used to investigate the nonlinear evolution of the parametric instability in three-dimensional low-$\unicode[STIX]{x1D6FD}$ plasmas at wavelengths much greater than the ion inertial length under the assumption that slow magnetosonic waves are strongly damped. It is shown analytically that the parametric instability leads to an inverse cascade of Alfvén wave quanta, and several exact solutions to the wave kinetic equations are presented. The main results of the paper concern the parametric decay of Alfvén waves that initially satisfy $e^{+}\gg e^{-}$, where $e^{+}$ and $e^{-}$ are the frequency ($f$) spectra of Alfvén waves propagating in opposite directions along the magnetic field lines. If $e^{+}$ initially has a peak frequency $f_{0}$ (at which $fe^{+}$ is maximized) and an ‘infrared’ scaling $f^{p}$ at smaller $f$ with $-1<p<1$, then $e^{+}$ acquires an $f^{-1}$ scaling throughout a range of frequencies that spreads out in both directions from $f_{0}$. At the same time, $e^{-}$ acquires an $f^{-2}$ scaling within this same frequency range. If the plasma parameters and infrared $e^{+}$ spectrum are chosen to match conditions in the fast solar wind at a heliocentric distance of 0.3 astronomical units (AU), then the nonlinear evolution of the parametric instability leads to an $e^{+}$ spectrum that matches fast-wind measurements from the Helios spacecraft at 0.3 AU, including the observed $f^{-1}$ scaling at $f\gtrsim 3\times 10^{-4}~\text{Hz}$. The results of this paper suggest that the $f^{-1}$ spectrum seen by Helios in the fast solar wind at $f\gtrsim 3\times 10^{-4}~\text{Hz}$ is produced in situ by parametric decay and that the $f^{-1}$ range of $e^{+}$ extends over an increasingly narrow range of frequencies as $r$ decreases below 0.3 AU. This prediction will be tested by measurements from the Parker Solar Probe.

Evolving magnetic fields are shown to generically reach a state of fast magnetic reconnection in which magnetic field line connections change and magnetic energy is released at an Alfvénic rate. This occurs even in plasmas with zero resistivity; only the finiteness of the mass of the lightest charged particle, an electron, is required. The speed and prevalence of Alfvénic or fast magnetic reconnection imply that its cause must be contained within the ideal evolution equation for magnetic fields, $\unicode[STIX]{x2202}\boldsymbol{B}/\unicode[STIX]{x2202}t=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B})$, where $\boldsymbol{u}(\boldsymbol{x},t)$ is the velocity of the magnetic field lines. For a generic $\boldsymbol{u}(\boldsymbol{x},t)$, neighbouring magnetic field lines develop a separation that increases exponentially, as $e^{\unicode[STIX]{x1D70E}(\ell ,t)}$ with $\ell$ the distance along a line. This exponentially enhances the sensitivity of the evolution to non-ideal effects. An analogous effect, the importance of stirring to produce a large-scale flow and enhance mixing, has been recognized by cooks through many millennia, but the importance of the large-scale flow $\boldsymbol{u}$ to reconnection is customarily ignored. In part this is due to the sixty-year focus of recognition theory on two-coordinate models, which eliminate the exponential enhancement that is generic with three coordinates. A simple three-coordinate model is developed, which could be used to address many unanswered questions.

Large-angle Coulomb collisions lead to an avalanching generation of runaway electrons in a plasma. We present the first fully conservative large-angle collision operator, derived from the relativistic Boltzmann operator. The relation to previous models for large-angle collisions is investigated, and their validity assessed. We present a form of the generalized collision operator which is suitable for implementation in a numerical kinetic equation solver, and demonstrate the effect on the runaway-electron growth rate. Finally we consider the reverse avalanche effect, where runaways are slowed down by large-angle collisions, and show that the choice of operator is important if the electric field is close to the avalanche threshold.

In collisionless and weakly collisional plasmas, such as hot accretion flows onto compact objects, the magnetorotational instability (MRI) can differ significantly from the standard (collisional) MRI. In particular, pressure anisotropy with respect to the local magnetic-field direction can both change the linear MRI dispersion relation and cause nonlinear modifications to the mode structure and growth rate, even when the field and flow perturbations are very small. This work studies these pressure-anisotropy-induced nonlinearities in the weakly nonlinear, high-ion-beta regime, before the MRI saturates into strong turbulence. Our goal is to better understand how the saturation of the MRI in a low-collisionality plasma might differ from that in the collisional regime. We focus on two key effects: (i) the direct impact of self-induced pressure-anisotropy nonlinearities on the evolution of an MRI mode, and (ii) the influence of pressure anisotropy on the ‘parasitic instabilities’ that are suspected to cause the mode to break up into turbulence. Our main conclusions are: (i) The mirror instability regulates the pressure anisotropy in such a way that the linear MRI in a collisionless plasma is an approximate nonlinear solution once the mode amplitude becomes larger than the background field (just as in magnetohyrodynamics). This implies that differences between the collisionless and collisional MRI become unimportant at large amplitudes. (ii) The break up of large-amplitude MRI modes into turbulence via parasitic instabilities is similar in collisionless and collisional plasmas. Together, these conclusions suggest that the route to magnetorotational turbulence in a collisionless plasma may well be similar to that in a collisional plasma, as suggested by recent kinetic simulations. As a supplement to these findings, we offer guidance for the design of future kinetic simulations of magnetorotational turbulence.

The extreme properties of the gamma-ray flares in the Crab nebula present a clear challenge to our ideas on the nature of particle acceleration in relativistic astrophysical plasma. It seems highly unlikely that standard mechanisms of stochastic type are at work here and hence the attention of theorists has switched to linear acceleration in magnetic reconnection events. In this series of papers, we attempt to develop a theory of explosive magnetic reconnection in highly magnetized relativistic plasma which can explain the extreme parameters of the Crab flares. In the first paper, we focus on the properties of the X-point collapse. Using analytical and numerical methods (fluid and particle-in-cell simulations) we extend Syrovatsky’s classical model of such collapse to the relativistic regime. We find that the collapse can lead to the reconnection rate approaching the speed of light on macroscopic scales. During the collapse, the plasma particles are accelerated by charge-starved electric fields, which can reach (and even exceed) values of the local magnetic field. The explosive stage of reconnection produces non-thermal power-law tails with slopes that depend on the average magnetization $\unicode[STIX]{x1D70E}$. For sufficiently high magnetizations and vanishing guide field, the non-thermal particle spectrum consists of two components: a low-energy population with soft spectrum that dominates the number census; and a high-energy population with hard spectrum that possesses all the properties needed to explain the Crab flares.

The properties of a non-relativistic magnetised low-beta electron–positron plasma in slab geometry are investigated. The two species are taken to be drift kinetic while we retain Larmor radius effects in quasi-neutrality, and inertia in Ohm’s law. A linear analysis shows that, for small magnetic perturbations, Alfvénic perturbations travel at the electron Alfvén speed, which is based on the electron mass. We discuss the role of the displacement current when Larmor-scale and Debye-scale effects are both retained. We predict the existence of a kinetic electron Alfvén wave which connects to the K-modes of Mishchenko et al. (J. Plasma Phys., 2017 (submitted)) in the electrostatic limit. It is found that linear drift waves are not supported by the system if the two species have the same temperature. Tearing modes can be driven unstable by equilibrium current density gradients. Also in this case, the characteristic time is based on the electron Alfvén speed. Nonlinear hybrid fluid-kinetic equations are also derived. It is shown that each species is described, to leading order, by the kinetic reduced electron heating model (KREHM) kinetic equation of Zocco & Schekochihin (Phys. Plasmas, vol. 18, 2011, 102309). The model is extended to retain first-order Larmor radius effects. It supports collisionless dispersive waves, which can greatly impact nonlinear magnetic reconnection. Diamagnetic effects enter the nonlinear equations via the first-order magnetic compressibility. A minimal nonlinear model for two-dimensional low-frequency isothermal pair plasmas is derived.

The basic physics of the processes playing the most important role in divertor plasma detachment is reviewed. The models used in the two-dimensional edge plasma transport codes that are widely used to address different issues of the edge plasma physics and to simulate the experimental data, as well as the numerical schemes and convergence issues, are described. The processes leading to ultimate divertor plasma detachment, the transition to and the stability of the detached regime, as well as the impact of the magnetic configuration and divertor geometry on detachment, are considered. A consistent, integral physical picture of ultimate detachment of a tokamak divertor plasma is developed.

In this paper, we report a new type of instability of electron holes (EHs) interacting with passing ions. The nonlinear interaction of EHs and ions is investigated using a new theory of hole kinematics. It is shown that the oscillation in the velocity of the EH parallel to the magnetic field direction becomes unstable when the hole velocity in the ion frame is slower than a few times the cold ion sound speed. This instability leads to the emission of ion-acoustic waves from the solitary hole and decay in its magnitude. The instability mechanism can drive significant perturbations in the ion density. The instability threshold, oscillation frequency and instability growth rate derived from the theory yield quantitative agreement with the observations from a novel high-fidelity hole-tracking particle-in-cell code.

Two signatures of quantum effects on radiation reaction in the collision of a ${\sim}$GeV electron beam with a high intensity (${>}3\times 10^{20}~\text{W}~\text{cm}^{-2}$) laser pulse have been considered. We show that the decrease in the average energy of the electron beam may be used to measure the Gaunt factor $g$ for synchrotron emission. We derive an equation for the evolution of the variance in the energy of the electron beam in the quantum regime, i.e. quantum efficiency parameter $\unicode[STIX]{x1D702}\not \ll 1$. We show that the evolution of the variance may be used as a direct measure of the quantum stochasticity of the radiation reaction and determine the parameter regime where this is observable. For example, stochastic emission results in a 25 % increase in the standard deviation of the energy spectrum of a GeV electron beam, 1 fs after it collides with a laser pulse of intensity $10^{21}~\text{W}~\text{cm}^{-2}$. This effect should therefore be measurable using current high-intensity laser systems.

A brief critique is presented of some different classes of magnetohydrodynamic equilibrium solutions based on their continuity properties and whether the magnetic field is integrable or not. A generalized energy functional is introduced that is comprised of alternating ideal regions, with nested flux surfaces with an irrational rotational transform, and Taylor-relaxed regions, possibly with magnetic islands and chaos. The equilibrium states have globally continuous magnetic fields, and may be constructed for arbitrary three-dimensional plasma boundaries and appropriately prescribed pressure and rotational-transform profiles.

A one-dimensional electrostatic filtered Vlasov–Poisson simulation study is discussed. The transition from persisting to arrested Landau damping that is produced by increasing the strength of a sinusoidal perturbation on a background Vlasov–Poisson equilibrium is explored. Emphasis is placed on observed features of the electron phase-space distribution when the perturbation strength is near the transition value. A single ubiquitous waveform is found perturbing the space-averaged phase-space distribution at almost any time in all of the simulations; the sole exception is the saturation stage that can occur at the end of the arrested damping scenario. This waveform contains relatively strong, very narrow structures in velocity bracketing $\pm v_{\text{res}}$ – the velocities at which electrons must move to traverse the dominant field mode wavelength in one of its oscillation periods – and propagating with $\pm v_{\text{res}}$ respectively. Local streams of electrons are found in these structures crossing the resonant velocities from low speed to high speed during Landau damping and from high speed to low speed during Landau growth. At the arrest time, when the field strength is briefly constant, these streams vanish. It is conjectured that the expected transfer of energy between electrons and field during Landau growth or damping has been visualized for the first time. No evidence is found in the phase-space distribution to support recent well-established discoveries of a second-order phase transition in the electric field evolution. While trapping is known to play a role for larger perturbation strengths, it is shown that trapping plays no role at any time in any of the simulations near the transition perturbation strength.

Determining the physical mechanisms that extract energy from turbulent fluctuations in weakly collisional magnetized plasmas is necessary for a more complete characterization of the behaviour of a variety of space and astrophysical plasmas. Such a determination is complicated by the complex nature of the turbulence as well as observational constraints, chiefly that in situ measurements of such plasmas are typically only available at a single point in space. Recent work has shown that correlations between electric fields and particle velocity distributions constructed from single-point measurements produce a velocity-dependent signature of the collisionless damping mechanism. We extend this work by constructing field–particle correlations using data sets drawn from single points in strongly driven, turbulent, electromagnetic gyrokinetic simulations to demonstrate that this technique can identify the collisionless mechanisms operating in such systems. The velocity-space structure of the correlation between proton distributions and parallel electric fields agrees with expectations of resonant mechanisms transferring energy collisionlessly in turbulent systems. This work motivates the eventual application of field–particle correlations to spacecraft measurements in the solar wind, with the ultimate goal to determine the physical mechanisms that dissipate magnetized plasma turbulence.