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.

Using analytical and numerical methods (fluid and particle-in-cell simulations) we study a number of model problems involving merger of magnetic flux tubes in relativistic magnetically dominated plasma. Mergers of current-carrying flux tubes (exemplified by the two-dimensional ‘ABC’ structures) and zero-total-current magnetic flux tubes are considered. In all cases regimes of spontaneous and driven evolution are investigated. We identify two stages of particle acceleration during flux mergers: (i) fast explosive prompt X-point collapse and (ii) ensuing island merger. The fastest acceleration occurs during the initial catastrophic X-point collapse, with the reconnection electric field of the order of the magnetic field. During the X-point collapse, 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 plasma magnetization $\unicode[STIX]{x1D70E}\leqslant 10^{2}$ the spectrum power-law index is $p>2$; in this case the maximal energy depends linearly on the size of the reconnecting islands. For higher magnetization, $\unicode[STIX]{x1D70E}\geqslant 10^{2}$, the spectra are hard, $p<2$, yet the maximal energy $\unicode[STIX]{x1D6FE}_{\text{max}}$ can still exceed the average magnetic energy per particle, ${\sim}\unicode[STIX]{x1D70E}$, by orders of magnitude (if $p$ is not too close to unity). The X-point collapse stage is followed by magnetic island merger that dissipates a large fraction of the initial magnetic energy in a regime of forced magnetic reconnection, further accelerating the particles, but proceeds at a slower reconnection rate.

We calculate the disruption scale $\unicode[STIX]{x1D706}_{\text{D}}$ at which sheet-like structures in dynamically aligned Alfvénic turbulence are destroyed by the onset of magnetic reconnection in a low-$\unicode[STIX]{x1D6FD}$ collisionless plasma. The scaling of $\unicode[STIX]{x1D706}_{\text{D}}$ depends on the order of the statistics being considered, with more intense structures being disrupted at larger scales. The disruption scale for the structures that dominate the energy spectrum is $\unicode[STIX]{x1D706}_{\text{D}}\sim L_{\bot }^{1/9}(d_{e}\unicode[STIX]{x1D70C}_{s})^{4/9}$, where $d_{e}$ is the electron inertial scale, $\unicode[STIX]{x1D70C}_{s}$ is the ion sound scale and $L_{\bot }$ is the outer scale of the turbulence. When $\unicode[STIX]{x1D6FD}_{e}$ and $\unicode[STIX]{x1D70C}_{s}/L_{\bot }$ are sufficiently small, the scale $\unicode[STIX]{x1D706}_{\text{D}}$ is larger than $\unicode[STIX]{x1D70C}_{s}$ and there is a break in the energy spectrum at $\unicode[STIX]{x1D706}_{\text{D}}$, rather than at $\unicode[STIX]{x1D70C}_{s}$. We propose that the fluctuations produced by the disruption are circularised flux ropes, which may have already been observed in the solar wind. We predict the relationship between the amplitude and radius of these structures and quantify the importance of the disruption process to the cascade in terms of the filling fraction of undisrupted structures and the fractional reduction of the energy contained in them at the ion sound scale $\unicode[STIX]{x1D70C}_{s}$. Both of these fractions depend strongly on $\unicode[STIX]{x1D6FD}_{e}$, with the disrupted structures becoming more important at lower $\unicode[STIX]{x1D6FD}_{e}$. Finally, we predict that the energy spectrum between $\unicode[STIX]{x1D706}_{\text{D}}$ and $\unicode[STIX]{x1D70C}_{s}$ is steeper than $k_{\bot }^{-3}$, when this range exists. Such a steep ‘transition range’ is sometimes observed in short intervals of solar-wind turbulence. The onset of collisionless magnetic reconnection may therefore significantly affect the nature of plasma turbulence around the ion gyroscale.

The Fokker–Planck transport equation describing the motion of energetic particles through a plasma is explored analytically. The latter equation provides a pitch-angle and position-dependent distribution function of the charged particles. In the current paper the first 14 moments of this equation are computed exactly for an arbitrary initial pitch angle. Such analytical forms are required in nonlinear treatments of perpendicular transport and other scenarios in plasma physics and astrophysics.

We review multiwavelength properties of pulsar wind nebulae created by supersonically moving pulsars and the effects of pulsar motion on the pulsar wind nebulae morphologies and the ambient medium. Supersonic pulsar wind nebulae are characterized by bow-shaped shocks around the pulsar and/or cometary tails filled with the shocked pulsar wind. In the past several years significant advances in supersonic pulsar wind nebula studies have been made in deep observations with the Chandra and XMM-Newton X-ray observatories and the Hubble Space Telescope. In particular, these observations have revealed very diverse supersonic pulsar wind nebula morphologies in the pulsar vicinity, different spectral behaviours of long pulsar tails, the presence of puzzling outflows misaligned with the pulsar velocity and far-UV bow shocks. Here we review the current observational status focusing on recent developments and their implications.

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.

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.

We examine with a particle-in-cell (PIC) simulation the collision of two equally dense clouds of cold pair plasma. The clouds interpenetrate until instabilities set in, which heat up the plasma and trigger the formation of a pair of shocks. The fastest-growing waves at the collision speed $c/5$, where $c$ is the speed of light in vacuum, and low temperature are the electrostatic two-stream mode and the quasi-electrostatic oblique mode. Both waves grow and saturate via the formation of phase space vortices. The strong electric fields of these nonlinear plasma structures provide an efficient means of heating up and compressing the inflowing upstream leptons. The interaction of the hot leptons, which leak back into the upstream region, with the inflowing cool upstream leptons continuously drives electrostatic waves that mediate the shock. These waves heat up the inflowing upstream leptons primarily along the shock normal, which results in an anisotropic velocity distribution in the post-shock region. This distribution gives rise to the Weibel instability. Our simulation shows that even if the shock is mediated by quasi-electrostatic waves, strong magnetowaves will still develop in its downstream region.

This paper summarises some of the recent progress that has been made in understanding astrophysical plasma turbulence in the solar wind, from in situ spacecraft observations. At large scales, where the turbulence is predominantly Alfvénic, measurements of critical balance, residual energy and three-dimensional structure are discussed, along with comparison to recent models of strong Alfvénic turbulence. At these scales, a few per cent of the energy is also in compressive fluctuations, and their nature, anisotropy and relation to the Alfvénic component is described. In the small-scale kinetic range, below the ion gyroscale, the turbulence becomes predominantly kinetic Alfvén in nature, and measurements of the spectra, anisotropy and intermittency of this turbulence are discussed with respect to recent cascade models. One of the major remaining questions is how the turbulent energy is dissipated, and some recent work on this question, in addition to future space missions which will help to answer it, are briefly discussed.

Magnetic fields play an important role in astrophysical accretion discs and in the interstellar and intergalactic medium. They drive jets, suppress fragmentation in star-forming clouds and can have a significant impact on the accretion rate of stars. However, the exact amplification mechanisms of cosmic magnetic fields remain relatively poorly understood. Here, I start by reviewing recent advances in the numerical and theoretical modelling of the turbulent dynamo, which may explain the origin of galactic and intergalactic magnetic fields. While dynamo action was previously investigated in great detail for incompressible plasmas, I here place particular emphasis on highly compressible astrophysical plasmas, which are characterised by strong density fluctuations and shocks, such as the interstellar medium. I find that dynamo action works not only in subsonic plasmas, but also in highly supersonic, compressible plasmas, as well as for low and high magnetic Prandtl numbers. I further present new numerical simulations from which I determine the growth of the turbulent (un-ordered) magnetic field component ( $B_{turb}$ ) in the presence of weak and strong guide fields ( $B_{0}$ ). I vary $B_{0}$ over five orders of magnitude and find that the dependence of $B_{turb}$ on $B_{0}$ is relatively weak, and can be explained with a simple theoretical model in which the turbulence provides the energy to amplify $B_{turb}$ . Finally, I discuss some important implications of magnetic fields for the structure of accretion discs, the launching of jets and the star-formation rate of interstellar clouds.

Numerical simulations have consistently shown that the reconnection rate in certain collisionless regimes can be fast, of the order of $0.1v_{A}B_{u}$ , where $v_{A}$ and $B_{u}$ are the Alfvén speed and the reconnecting magnetic field upstream of the ion diffusion region. This particular value has been reported in myriad numerical simulations under disparate conditions. However, despite decades of research, the reasons underpinning this specific value remain mysterious. Here, we present an overview of this problem and discuss the conditions under which the ‘0.1 value’ is attained. Furthermore, we explain why this problem should be interpreted in terms of the ion diffusion region length.

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.

The stability of a collisionless, magnetised plasma to local convective disturbances is examined, with a focus on kinetic and finite-Larmor-radius effects. Specific application is made to the outskirts of galaxy clusters, which contain hot and tenuous plasma whose temperature increases in the direction of gravity. At long wavelengths (the ‘drift-kinetic’ limit), we obtain the kinetic version of the magnetothermal instability (MTI) and its Alfvénic counterpart (Alfvénic MTI), which were previously discovered and analysed using a magnetofluid (i.e. Braginskii) description. At sub-ion-Larmor scales, we discover an overstability driven by the electron-temperature gradient of kinetic-Alfvén drift waves – the electron MTI (eMTI) – whose growth rate is even larger than the standard MTI. At intermediate scales, we find that ion finite-Larmor-radius effects tend to stabilise the plasma. We discuss the physical interpretation of these instabilities in detail, and compare them both with previous work on magnetised convection in a collisional plasma and with temperature-gradient-driven drift-wave instabilities well known to the magnetic-confinement-fusion community. The implications of having both fluid and kinetic scales simultaneously driven unstable by the same temperature gradient are briefly discussed.

Magnetorotational stability is revisited for self-consistent three-dimensional magnetized hot plasma equilibria in a gravitational field. The eikonal analysis presented finds that magnetorotational stability analysis must be performed with some care to retain compressibility and density gradient effects, and departures from strict Keplerian motion. Indeed, retaining these effects highlights differences between the magnetorotational instability found in the absence of gravity (Velikhov, Sov. Phys. JETP, vol. 36, 1959, pp. 995–998) and that found the presence of gravity (Balbus & Hawley, Astrophys. J., vol. 376, 1991, pp. 214–222). In the non-gravitational case, compressibility and density variation alter the stability condition, while these effects only enter for departures from strict Keplerian motion in a gravitational field. The conditions for instability are made more precise by employing recent magnetized equilibrium results (Catto et al., J. Plasma Phys., vol. 81, 2015, 515810603), rather than employing a hydrodynamic equilibrium. We focus on the stability of the $\unicode[STIX]{x1D6FD}>1$ limit for which equilibria were found in the absence of a toroidal magnetic field, where $\unicode[STIX]{x1D6FD}=$  plasma/magnetic pressure.

Pulsar-wind nebulae emit an extremely broad spectrum of continuum radiation, from low radio frequencies to TeV gamma rays. The part of the spectral energy distribution (SED) from radio through MeV gamma rays is due to synchrotron emission from a distribution of relativistic electrons (or pairs) which can be described by one or more power laws. This spectrum exhibits that particle energy distribution, responsible also for the higher-energy (GeV–TeV) part of the SED, due to inverse-Compton upscattering of one of three photon fields: the synchrotron spectrum, the cosmic microwave background, or ambient optical/infrared photons. However, in a few sources, primary hadrons may produce GeV–TeV gamma rays through the decay of neutral pions produced in inelastic cosmic-ray collisions with thermal gas. The higher-energy end of the particle spectrum, producing synchrotron photons above approximately 10 keV, holds clues to the particle acceleration process. However, its detailed study requires imaging spectroscopy in this energy range, not available until the NuSTAR mission beginning in 2012, which performs true imaging between 3 and 78 keV with ${\sim}1^{\prime }$ angular resolution. I review NuSTAR observations of the first three pulsar-wind nebulae (PWNe) to be examined in this way: the Crab Nebula, G21.5–0.9 and MSH 15–52. All three show spectral structure not previously known: spectral steepening in certain locations and overall source shrinkage with increasing photon energy. The Crab Nebula has different shrinkage rates along the torus and along the northwest counter-jet. The latter rate is similar to that for both the other sources (FWHM $\propto E^{m}$ with $m\sim -0.2$ ). I discuss implications of these results for models of particle transport in PWNe.

Neutron stars are fascinating astrophysical objects immersed in strong gravitational and electromagnetic fields, at the edge of our current theories. These stars manifest themselves mostly as pulsars, emitting a timely very stable and regular electromagnetic signal. Even though discovered almost fifty years ago, they still remain mysterious compact stellar objects. In this review, we summarize the most fundamental theoretical aspects of neutron star magnetospheres and winds. The main competing models explaining their radiative properties like multi-wavelength pulse shapes and spectra and the underlying physical processes such as pair creation and radiation mechanisms are scrutinized. A global but still rather qualitative picture slowly emerges thanks to recent advances in numerical simulations on the largest scales. However considerations about pulsar magnetospheres remain speculative. For instance, the exact composition of the magnetospheric plasma is not yet known. Is it solely filled with a mixture of $e^{\pm }$ leptons or does it contain a non-negligible fraction of protons and/or ions? Is it almost entirely filled or mostly empty except for some small anecdotal plasma filled regions? Answers to these questions will strongly direct the description of the magnetosphere to seemingly contradictory results leading sometimes to inconsistencies. Nevertheless, accounts are given as to the latest developments in the theory of pulsar magnetospheres and winds, the existence of a possible electrosphere and physical insight obtained from related observational signatures of multi-wavelength pulsed emission.

For a Weibel shock to form, two plasma shells have to collide and trigger the Weibel instability. At saturation, this instability generates magnetic filaments in the overlapping region with peak field $B_{f}$. In the absence of an external guiding magnetic field, these filaments can block the incoming flow, initiating the shock formation, if their size is larger than the Larmor radius of the incoming particles in the peak field. Here we show that this result still holds in the presence of an external magnetic field $B_{0}$, provided it is not too high. Yet, for $B_{0}\gtrsim B_{f}/2$, the filaments become unable to stop any particle, regardless of its initial velocity.

Successful phenomenological models of pulsar wind nebulae assume efficient dissipation of the Poynting flux of the magnetized electron–positron wind as well as efficient acceleration of the pairs in the vicinity of the termination shock, but how this is realized is not yet well understood. This paper suggests that the corrugation of the termination shock, at the onset of nonlinearity, may lead towards the desired phenomenology. Nonlinear corrugation of the termination shock would convert a fraction of order unity of the incoming ordered magnetic field into downstream turbulence, slowing down the flow to sub-relativistic velocities. The dissipation of turbulence would further preheat the pair population on short length scales, close to equipartition with the magnetic field, thereby reducing the initial high magnetization to values of order unity. Furthermore, it is speculated that the turbulence generated by the corrugation pattern may sustain a relativistic Fermi process, accelerating particles close to the radiation reaction limit, as observed in the Crab nebula. The required corrugation could be induced by the fast magnetosonic modes of downstream nebular turbulence; but it could also be produced by upstream turbulence, either carried by the wind or seeded in the precursor by the accelerated particles themselves.

Supercritical collisionless perpendicular shocks have an average macrostructure determined primarily by the dynamics of ions specularly reflected at the magnetic ramp. Within the overall macrostructure, instabilities, both linear and nonlinear, generate fluctuations and microstructure. To identify the sources of such microstructure, high-resolution two- and three-dimensional simulations have been carried out using the hybrid method, wherein the ions are treated as particles and the electron response is modelled as a massless fluid. We confirm the results of earlier two-dimensional (2-D) simulations showing both field-parallel aligned propagating fluctuations and fluctuations carried by the reflected-gyrating ions. In addition, it is shown that, for 2-D simulations of the shock coplanarity plane, the presence of short-wavelength fluctuations in all magnetic components is associated with the ion Weibel instability driven at the upstream edge of the foot by the reflected-gyrating ions. In 3-D simulations we show for the first time that the dominant microstructure is due to a coupling between field-parallel propagating fluctuations in the ramp and the motion of the reflected ions. This results in a pattern of fluctuations counter-propagating across the surface of the shock at an angle inclined to the magnetic field direction, due to a combination of field-parallel motion at the Alfvén speed of the ramp and motion in the sense of gyration of the reflected ions.