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.

We present a theoretical framework for describing electromagnetic kinetic turbulence in a multi-species, magnetized, pressure-anisotropic plasma. The turbulent fluctuations are assumed to be small compared to the mean field, to be spatially anisotropic with respect to it and to have frequencies small compared to the ion cyclotron frequency. At scales above the ion-Larmor radius, the theory reduces to the pressure-anisotropic generalization of kinetic reduced magnetohydrodynamics (KRMHD) formulated by Kunz et al. (J. Plasma Phys., vol. 81, 2015, 325810501). At scales at and below the ion-Larmor radius, three main objectives are achieved. First, we analyse the linear response of the pressure-anisotropic gyrokinetic system, and show it to be a generalization of previously explored limits. The effects of pressure anisotropy on the stability and collisionless damping of Alfvénic and compressive fluctuations are highlighted, with attention paid to the spectral location and width of the frequency jump that occurs as Alfvén waves transition into kinetic Alfvén waves. Secondly, we derive and discuss a very general gyrokinetic free-energy conservation law, which captures both the KRMHD free-energy conservation at long wavelengths and dual cascades of kinetic Alfvén waves and ion entropy at sub-ion-Larmor scales. We show that non-Maxwellian features in the distribution function change the amount of phase mixing and the efficiency of magnetic stresses, and thus influence the partitioning of free energy amongst the cascade channels. Thirdly, a simple model is used to show that pressure anisotropy, even within the bounds imposed on it by firehose and mirror instabilities, can cause order-of-magnitude variations in the ion-to-electron heating ratio due to the dissipation of Alfvénic turbulence. Our theory provides a foundation for determining how pressure anisotropy affects turbulent fluctuation spectra, the differential heating of particle species and the ratio of parallel and perpendicular phase mixing in space and astrophysical plasmas.

Reconnection outflows have been under intense recent scrutiny, from in situ observations and from simulations. These regions are host to a variety of instabilities and intense energy exchanges, often even superior to the main reconnection site. We report here a number of results drawn from an investigation of simulations. First, the outflows are observed to become unstable to drift instabilities. Second, these instabilities lead to the formation of secondary reconnection sites. Third, the secondary processes are responsible for large energy exchanges and particle energization. Finally, the particle distribution function are modified to become non-Maxwellian and include multiple interpenetrating populations.

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.

The threshold for the resonant destabilisation of ion-temperature-gradient (ITG) driven instabilities that render the modes ubiquitous in both tokamaks and stellarators is investigated. We discover remarkably similar results for both confinement concepts if care is taken in the analysis of the effect of the global shear ${\hat{s}}$ . We revisit, analytically and by means of gyrokinetic simulations, accepted tokamak results and discover inadequacies of some aspects of their theoretical interpretation. In particular, for standard tokamak configurations, we find that global shear effects on the critical gradient cannot be attributed to the wave–particle resonance destabilising mechanism of Hahm & Tang (Phys. Plasmas, vol. 1, 1989, pp. 1185–1192), but are consistent with a stabilising contribution predicted by Biglari et al. (Phys. Plasmas, vol. 1, 1989, pp. 109–118). Extensive analytical and numerical investigations show that virtually no previous tokamak theoretical predictions capture the temperature dependence of the mode frequency at marginality, thus leading to incorrect instability thresholds. In the asymptotic limit ${\hat{s}}\unicode[STIX]{x1D704}\ll 1$ , where $\unicode[STIX]{x1D704}$ is the rotational transform, and such a threshold should be solely determined by the resonant toroidal branch of the ITG mode, we discover a family of unstable solutions below the previously known threshold of instability. This is true for a tokamak case described by a local ${\hat{s}}-\unicode[STIX]{x1D6FC}$ local equilibrium, and for the stellarator Wendelstein 7-X, where these unstable solutions are present even for configurations with a small trapped-particle population. We conjecture they are of the Floquet type and derive their properties from the Fourier analysis of toroidal drift modes of Connor & Taylor (Phys. Fluids, vol. 30, 1987, pp. 3180–3185), and to Hill’s theory of the motion of the lunar perigee (Acta Math., vol. 8, 1886, pp. 1–36). The temperature dependence of the newly determined threshold is given for both confinement concepts. In the first case, the new temperature-gradient threshold is found to be rather insensitive to the temperature ratio $T_{i}/T_{e}$ , at least for $T_{i}/T_{e}\lesssim 1$ , and to be a growing function of the density gradient scale for $T_{i}/T_{e}\gtrsim 1$ . For Wendelstein 7-X, the new critical temperature gradient is a growing function of the temperature ratio. The importance of these findings for the assessment of turbulence in stellarators and low-shear tokamak configurations is discussed.

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.

A numerical investigation is carried out to understand the equilibrium $\unicode[STIX]{x1D6FD}$ -limit in a classical stellarator. The stepped-pressure equilibrium code (Hudson et al., Phys. Plasmas, vol. 19 (11), 2012) is used in order to assess whether or not magnetic islands and stochastic field-lines can emerge at high $\unicode[STIX]{x1D6FD}$ . Two modes of operation are considered: a zero-net-current stellarator and a fixed-iota stellarator. Despite the fact that relaxation is allowed (Taylor, Rev. Mod. Phys., vol. 58 (3), 1986, pp. 741–763), the former is shown to maintain good flux surfaces up to the equilibrium $\unicode[STIX]{x1D6FD}$ -limit predicted by ideal-magnetohydrodynamics (MHD), above which a separatrix forms. The latter, which has no ideal equilibrium $\unicode[STIX]{x1D6FD}$ -limit, is shown to develop regions of magnetic islands and chaos at sufficiently high $\unicode[STIX]{x1D6FD}$ , thereby providing a ‘non-ideal $\unicode[STIX]{x1D6FD}$ -limit’. Perhaps surprisingly, however, the value of $\unicode[STIX]{x1D6FD}$ at which the Shafranov shift of the axis reaches a fraction of the minor radius follows in all cases the scaling laws predicted by ideal-MHD. We compare our results to the High-Beta-Stellarator theory of Freidberg (Ideal MHD, 2014, Cambridge University Press) and derive a new prediction for the non-ideal equilibrium $\unicode[STIX]{x1D6FD}$ -limit above which chaos emerges.

A long-standing problem in magnetic reconnection is to explain why it tends to proceed at or below a normalized rate of 0.1. This article gives a review of observational and numerical evidence for this rate and discusses recent theoretical work addressing this problem. Some remaining open questions are summarized.

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.

Optimised stellarators and other magnetic-confinement devices having the property that the average magnetic curvature is favourable for all particle orbits are called maximum- $J$ devices. They have recently been shown to be immune to trapped-particle instabilities driven by the density gradient. Gyrokinetic simulations reveal, however, that another instability can arise, which is also associated with particle trapping but causes less transport than typical trapped-electron modes. The nature of this instability is clarified here. It is shown to be similar to the ‘ubiquitous mode’ in tokamaks and is driven by ion free energy, but requires trapped electrons to exist.

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.

Reduced fluid models for collisionless plasmas including electron inertia and finite Larmor radius corrections are derived for scales ranging from the ion to the electron gyroradii. Based either on pressure balance or on the incompressibility of the electron fluid, they respectively capture kinetic Alfvén waves (KAWs) or whistler waves (WWs), and can provide suitable tools for reconnection and turbulence studies. Both isothermal regimes and Landau fluid closures permitting anisotropic pressure fluctuations are considered. For small values of the electron beta parameter $\unicode[STIX]{x1D6FD}_{e}$ , a perturbative computation of the gyroviscous force valid at scales comparable to the electron inertial length is performed at order $O(\unicode[STIX]{x1D6FD}_{e})$ , which requires second-order contributions in a scale expansion. Comparisons with kinetic theory are performed in the linear regime. The spectrum of transverse magnetic fluctuations for strong and weak turbulence energy cascades is also phenomenologically predicted for both types of waves. In the case of moderate ion to electron temperature ratio, a new regime of KAW turbulence at scales smaller than the electron inertial length is obtained, where the magnetic energy spectrum decays like $k_{\bot }^{-13/3}$ , thus faster than the $k_{\bot }^{-11/3}$ spectrum of WW turbulence.

The energy budget of a collisionless plasma subject to electrostatic fluctuations is considered, and the excess of thermal energy over the minimum accessible to it under various constraints that limit the possible forms of plasma motion is calculated. This excess measures how much thermal energy is ‘available’ for conversion into plasma instabilities, and therefore constitutes a nonlinear measure of plasma stability. A distribution function with zero available energy defines a ‘ground state’ in the sense that its energy cannot decrease by any linear or nonlinear plasma motion. In a Vlasov plasma with small density and temperature fluctuations, the available energy is proportional to the mean square of these quantities, and exceeds the corresponding energy in ideal or resistive magnetohydrodynamics. If the first or second adiabatic invariant is conserved, ground states generally have inhomogeneous density and temperature. Magnetically confined plasmas are usually not in any ground state, but certain types of stellarator plasmas are so with respect to fluctuations that conserve both these adiabatic invariants, making the plasma linearly and nonlinearly stable to such fluctuations. Similar stability properties can also be enjoyed by plasmas confined by a dipole magnetic field.

According to magnetohydrodynamics (MHD), the encounter of two collisional magnetized plasmas at high velocity gives rise to shock waves. Investigations conducted so far have found that the same conclusion still holds in the case of collisionless plasmas. For the case of a flow-aligned field, MHD stipulates that the field and the fluid are disconnected, so that the shock produced is independent of the field. We present a violation of this MHD prediction when considering the encounter of two cold pair plasmas along a flow-aligned magnetic field. As the guiding magnetic field grows, isotropization is progressively suppressed, resulting in a strong influence of the field on the resulting structure. A micro-physics analysis allows us to understand the mechanisms at work. Particle-in-cell simulations also support our conclusions and show that the results are not restricted to a strictly parallel field.

In the last decade, the relativistic magnetohydrodynamic (MHD) modelling of pulsar wind nebulae, and of the Crab nebula in particular, has been highly successful, with many of the observed dynamical and emission properties reproduced down to the finest detail. Here, we critically discuss the results of some of the most recent studies: namely the investigation of the origin of the radio emitting particles and the quest for the acceleration sites of particles of different energies along the termination shock, by using wisp motions as a diagnostic tool; the study of the magnetic dissipation process in high magnetization nebulae by means of new long-term three-dimensional simulations of the pulsar wind nebula evolution; the investigation of the relativistic tearing instability in thinning current sheets, leading to fast reconnection events that might be at the origin of the Crab nebula gamma-ray flares.

In the single fluid, non-relativistic, ideal magnetohydrodynamic (MHD) plasma description, magnetic field lines play a fundamental role by defining dynamically preserved ‘magnetic connections’ between plasma elements. Here we show how the concept of magnetic connection needs to be generalized in the case of a relativistic MHD description where we require covariance under arbitrary Lorentz transformations. This is performed by defining 2-D magnetic connection hypersurfaces in the 4-D Minkowski space. This generalization accounts for the loss of simultaneity between spatially separated events in different frames and is expected to provide a powerful insight into the 4-D geometry of electromagnetic fields when $\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{B}=0$ .

This paper is dedicated to Vitaly Shafranov, who became increasingly interested in stellarators. Stellarators have a steady-state magnetic configuration, robust positional stability, and consistency with a plasma current below the level at which runaway electrons become a major issue. The development path for stellarators may be faster and cheaper than for tokamaks: stellarators are amenable to computer design validated by moderate scale experiments to circumvent issues that impede fusion development. This is distinct from the empirical explorations required to find an acceptable nonlinear, self-organized state of a tokamak. Fusion plasmas can be designed and controlled in stellarators in ways that are not possible in tokamaks. This paper outlines computational studies that could be carried at low cost during the next few years that would clarify the reactor potential of the stellarator and are needed for rational planning of the fusion program.

Electron–positron (e–p) plasmas are widely thought to be emitted, in the form of ultra-relativistic winds or collimated jets, by some of the most energetic or powerful objects in the Universe, such as black-holes, pulsars, and quasars. These phenomena represent an unmatched astrophysical laboratory to test physics at its limit and, given their immense distance from Earth (some even farther than several billion light years), they also provide a unique window on the very early stages of our Universe. However, due to such gigantic distances, their properties are only inferred from the indirect interpretation of their radiative signatures and from matching numerical models: their generation mechanism and dynamics still pose complicated enigmas to the scientific community. Small-scale reproductions in the laboratory would represent a fundamental step towards a deeper understanding of this exotic state of matter. Here we present recent experimental results concerning the laser-driven production of ultra-relativistic e–p beams. In particular, we focus on the possibility of generating beams that present charge neutrality and that allow for collective effects in their dynamics, necessary ingredients for the testing pair-plasma physics in the laboratory. A brief discussion of the analytical and numerical modelling of the dynamics of these plasmas is also presented in order to provide a summary of the novel plasma physics that can be accessed with these objects. Finally, general considerations on the scalability of laboratory plasmas up to astrophysical scenarios are given.