The most massive baryonic component of galaxy clusters is the ‘intracluster medium’ (ICM), a diffuse, hot, weakly magnetized plasma that is most easily observed in the X-ray band. Despite being observed for decades, the macroscopic transport properties of the ICM are still not well constrained. A path to determine macroscopic ICM properties opened up with the discovery of ‘cold fronts’. These were observed as sharp discontinuities in surface brightness and temperature in the ICM, with the property that the denser side of the discontinuity is the colder one. The high spatial resolution of the Chandra X-ray Observatory revealed two puzzles about cold fronts. First, they should be subject to Kelvin–Helmholtz instabilities, yet in many cases they appear relatively smooth and undisturbed. Second, the width of the interface between the two gas phases is typically narrower than the mean free path of the particles in the plasma, indicating negligible thermal conduction. It was thus realized that these special characteristics of cold fronts may be used to probe the properties of the cluster plasma. In this review, we will discuss the recent simulations of cold fronts in galaxy clusters, focusing on those which have attempted to use these features to constrain ICM physics. In particular, we will examine the effects of magnetic fields, viscosity, and thermal conductivity on the stability properties and long-term evolution of cold fronts. We conclude with a discussion on what important questions remain unanswered, and the future role of simulations and the next generation of X-ray observatories.

The large number of ${\it\gamma}$ -ray pulsars discovered by the Fermi Gamma-Ray Space Telescope since its launch in 2008 dwarfs the handful that were previously known. The variety of observed light curves makes possible a tomography of both the ensemble-averaged field structure and the high-energy emission regions of a pulsar magnetosphere. Fitting the ${\it\gamma}$ -ray pulsar light curves with model magnetospheres and emission models has revealed that most of the high-energy emission, and the particles acceleration, takes place near or beyond the light cylinder, near the current sheet. As pulsar magnetosphere models become more sophisticated, it is possible to probe magnetic field structure and emission that are self-consistently determined. Light curve modelling will continue to be a powerful tool for constraining the pulsar magnetosphere physics.

These lecture notes and example problems are based on a course given at the University of Cambridge in Part III of the Mathematical Tripos. Fluid dynamics is involved in a very wide range of astrophysical phenomena, such as the formation and internal dynamics of stars and giant planets, the workings of jets and accretion discs around stars and black holes and the dynamics of the expanding Universe. Effects that can be important in astrophysical fluids include compressibility, self-gravitation and the dynamical influence of the magnetic field that is ‘frozen in’ to a highly conducting plasma. The basic models introduced and applied in this course are Newtonian gas dynamics and magnetohydrodynamics (MHD) for an ideal compressible fluid. The mathematical structure of the governing equations and the associated conservation laws are explored in some detail because of their importance for both analytical and numerical methods of solution, as well as for physical interpretation. Linear and nonlinear waves, including shocks and other discontinuities, are discussed. The spherical blast wave resulting from a supernova, and involving a strong shock, is a classic problem that can be solved analytically. Steady solutions with spherical or axial symmetry reveal the physics of winds and jets from stars and discs. The linearized equations determine the oscillation modes of astrophysical bodies, as well as their stability and their response to tidal forcing.

In the present work there is presented a model describing transfer of the Crab pulsar’s spin-down energy into the powerful synchrotron emission of the nebula. The process of the energy transfer consists of several consecutive stages. The physical processes underlying the theoretical model provide us with the synchrotron emission spectrum, which fits well with the observed one.

Motivated by recent advances in direct statistical simulation (DSS) of astrophysical phenomena such as out-of-equilibrium jets, we perform a direct numerical simulation (DNS) of the helical magnetorotational instability (HMRI) under the generalised quasilinear approximation (GQL). This approximation generalises the quasilinear approximation (QL) to include the self-consistent interaction of large-scale modes, interpolating between fully nonlinear DNS and QL DNS whilst still remaining formally linear in the small scales. In this paper we address whether GQL can more accurately describe low-order statistics of axisymmetric HMRI when compared with QL by performing DNS under various degrees of GQL approximation. We utilise various diagnostics, such as energy spectra in addition to first and second cumulants, for calculations performed for a range of Reynolds and Hartmann numbers (describing rotation and imposed magnetic field strength respectively). We find that GQL performs significantly better than QL in describing the statistics of the HMRI even when relatively few large-scale modes are kept in the formalism. We conclude that DSS based on GQL (GCE2) will be significantly more accurate than that based on QL (CE2).

A critical revision of the essential principles of the physics of relativistic flows of cold plasma is given. We prove that the approximation of ideal magnetic hydrodynamics of the cold plasma is equivalent to the drift approximation of motion of charged particles in an electromagnetic field. The equations of magnetohydrodynamics are obtained from equations for the drift motion of the charged particles. The conditions of application of the equations of ideal magnetohydrodynamics are obtained. In the case of the Crab pulsar the violation of the frozen-in condition can happen at a distance that well exceeds the distance to the termination shock. One fluid MHD can be incorrect at the light cylinder provided that the Lorentz factor of the plasma exceeds $10^{4}$ and the curvature radius of the flow line is comparable with the light cylinder. It is shown that the electric currents in the cold plasma are the result of the inertial drift motion of the charged particles in the crossed electric and magnetic fields.

We want to understand what drives magnetospheric dissipation in the equatorial current sheet. Numerical simulations have limitations and, unless we have a clear a priori understanding of the physical processes involved, their results can be misleading. We argue that the canonical pulsar magnetosphere is strongly dissipative and that a large fraction (up to 30–40 % in an aligned rotator) of the spindown luminosity is redirected towards the equator where it is dissipated into particle acceleration and emission of radiation. We show that this is due to the failure of the equatorial electric current to cross the Y-point at the tip of the corotating zone.

We review our high-time-resolution radio observations of the Crab pulsar and compare our data to a variety of models for the emission physics. The Main Pulse and the Low Frequency Interpulse come from regions somewhere in the high-altitude emission zones (caustics) that also produce pulsed X-ray and ${\it\gamma}$-ray emission. Although no emission model can fully explain these two components, the most likely models suggest they arise from a combination of beam-driven instabilities, coherent charge bunching and strong electromagnetic turbulence. Because the radio power fluctuates on a wide range of time scales, we know the emission zones are patchy and dynamic. It is tempting to invoke unsteady pair creation in high-altitude gaps as the source of the variability, but current pair cascade models cannot explain the densities required by any of the likely models. It is harder to account for the mysterious High Frequency Interpulse. We understand neither its origin within the magnetosphere nor the striking emission bands in its dynamic spectrum. The most promising models are based on analogies with solar zebra bands, but they require unusual plasma structures which are not part of our standard picture of the magnetosphere. We argue that radio observations can reveal much about the upper magnetosphere, but work is required before the models can address all of the data.

Pulsar electrodynamics is reviewed emphasizing the role of the inductive electric field in an oblique rotator and the incomplete screening of its parallel component by charges, leaving ‘gaps’ with $E_{\Vert }\neq 0$. The response of the plasma leads to a self-consistent electric field that complements the inductive electric field with a potential field leading to an electric drift and a polarization current associated with the total field. The electrodynamic models determine the charge density, ${\it\rho}$, and the current density, $\boldsymbol{J}$; charge starvation refers to situations where the plasma cannot supply ${\it\rho}$, resulting in a gap and associated particle acceleration and pair creation. It is pointed out that a form of current starvation also occurs implying a new class of gaps. The properties of gaps are discussed, emphasizing that static models are unstable, the role of large-amplitude longitudinal waves and the azimuthal dependence that arises across a gap in an oblique rotator. Wave dispersion in a pulsar plasma is reviewed briefly, emphasizing its role in radio emission. Pulsar radio emission mechanisms are reviewed, and it is suggested that the most plausible is a form of plasma emission.

Chaotic electromagnetic fields are common in many relativistic plasma environments, where they can be excited by instabilities on kinetic spatial scales. When strong electric fluctuations exist on sub-electron scales, they may lead to small angle, stochastic deflections of the electrons’ pitch angles. Under certain conditions, this closely resembles the effect of Coulomb collisions in collisional plasmas. The electric pitch-angle diffusion coefficient acts as an effective collision – or ‘quasi-collision’ – frequency. We show that quasi-collisions may radically alter the expected radiative transport properties of candidate plasmas. In particular, we consider the quasi-collisional generalization of the classical Faraday effect.

Propagation of radio emission in a pulsar magnetosphere is reviewed. The effects of polarization transfer, induced scattering and reprocessing to high energies are analysed with a special emphasis on the implications for the gamma-ray pulsars. The possibilities of the pulsar plasma diagnostics based on the observed radio pulse characteristics are also outlined. As an example, the plasma number density profiles obtained from the polarization data for the Vela and the gamma-ray millisecond pulsars J1446-4701, J1939+2134 and J1744-1134 are presented. The number densities derived tend to be the highest/lowest when the radio pulse leads/lags the gamma-ray peak. In the PSR J1939+2134, the plasma density profiles for the main pulse and interpulse appear to fit exactly the same curve, testifying to the origin of both radio components above the same magnetic pole and their propagation through the same plasma flow in opposite directions. The millisecond radio pulse components exhibiting flat position angle curves are suggested to result from the induced scattering of the main pulse by the same particles that generate gamma rays. This is believed to underlie the wide-sense radio/gamma-ray correlation in the millisecond pulsars. The radio quietness of young gamma-ray pulsars is attributed to resonant absorption, whereas the radio loudness to the radio beam escape through the periphery of the open field line tube.

Magnetized high-energy-density plasmas can often have strong electromagnetic fluctuations whose correlation scale is smaller than the electron Larmor radius. Radiation from the electrons in such plasmas – which markedly differs from both synchrotron and cyclotron radiation – is tightly related to their energy and pitch-angle diffusion. In this paper, we present a comprehensive theoretical and numerical study of particle transport in cold, ‘small-scale’ Whistler-mode turbulence and its relation to the spectra of radiation simultaneously produced by these particles. We emphasize that this relation is a superb diagnostic tool of laboratory, astrophysical, interplanetary and solar plasmas with a mean magnetic field and strong small-scale turbulence.

A novel large-scale dynamo mechanism, the magnetic shear-current effect, is discussed and explored. The effect relies on the interaction of magnetic fluctuations with a mean shear flow, meaning the saturated state of the small-scale dynamo can drive a large-scale dynamo – in some sense the inverse of dynamo quenching. The dynamo is non-helical, with the mean field ${\it\alpha}$ coefficient zero, and is caused by the interaction between an off-diagonal component of the turbulent resistivity and the stretching of the large-scale field by shear flow. Following up on previous numerical and analytic work, this paper presents further details of the numerical evidence for the effect, as well as an heuristic description of how magnetic fluctuations can interact with shear flow to produce the required electromotive force. The pressure response of the fluid is fundamental to this mechanism, which helps explain why the magnetic effect is stronger than its kinematic cousin, and the basic idea is related to the well-known lack of turbulent resistivity quenching by magnetic fluctuations. As well as being interesting for its applications to general high Reynolds number astrophysical turbulence, where strong small-scale magnetic fluctuations are expected to be prevalent, the magnetic shear-current effect is a likely candidate for large-scale dynamo in the unstratified regions of ionized accretion disks. Evidence for this is discussed, as well as future research directions and the challenges involved with understanding details of the effect in astrophysically relevant regimes.

In previous galactic dynamo theories of the origin of the magnetic field in our galaxy, the subject of flux-freezing has been omitted. As a consequence, the equation of mass flow has generally also been omitted, particularly in the halo where the galactic gravitational field will operate on the mass flow. In this paper, it has been shown that this neglect could have serious consequences for the results obtained from those galactic dynamo simulations that include the halo. A modification of these dynamo theories is proposed which involves the expulsion of very small pieces of the magnetic field lines, rather than the wholesale expulsion of the complete magnetic lines encapsulated in the previous theories. This expulsion is accomplished by a spike instability that arises from superbubbles when they break out of the galactic disc and their shells fragment. This leads to a cut in the lines of force that still remain in the disc. Subsequently, normal disc turbulence rotates the cut lines and thus dissipates their mean flux, removing them from a role in the dynamo theory. This new process takes a length of time comparable to, but slightly longer than, the previous growth time of the disc dynamo, but avoids the previous difficulties associated with flux freezing and flux expulsion.

Accretion disc theory is less developed than stellar evolution theory although a similarly mature phenomenological picture is ultimately desired. While the interplay of theory and numerical simulations has amplified community awareness of the role of magnetic fields in angular momentum transport, there remains a long term challenge to incorporate the insights gained from simulations into improving practical models for comparison with observations. What has been learned from simulations that can lead to improvements beyond SS73 in practical models? Here, we emphasize the need to incorporate the role of non-local transport more precisely. To show where large-scale transport would fit into the theoretical framework and how it is currently missing, we review why the wonderfully practical approach of Shakura & Sunyaev (Astron. Astrophys., vol. 24, 1973, pp. 337–355, SS73) is necessarily a mean field theory, and one which does not include large-scale transport. Observations of coronae and jets, combined with the interpretation of results from shearing box simulations, of the magnetorotational instability (MRI) suggest that a significant fraction of disc transport is indeed non-local. We show that the Maxwell stresses in saturation are dominated by large-scale contributions and that the physics of MRI transport is not fully captured by a viscosity. We also clarify the standard physical interpretation of the MRI as it applies to shearing boxes. Computational limitations have so far focused most attention toward local simulations, but the next generation of global simulations should help to inform improved mean field theories. Mean field accretion theory and mean field dynamo theory should in fact be unified into a single theory that predicts the time evolution of spectra and luminosity from separate disc, corona and outflow contributions. Finally, we note that any mean field theory, including that of SS73, has a finite predictive precision that needs to be quantified when comparing the predictions to observations.

Clusters of galaxies are embedded in halos of optically thin, gravitationally stratified, weakly magnetized plasma at the system’s virial temperature. Owing to radiative cooling and anisotropic heat conduction, such intracluster medium (ICM) is subject to local instabilities, which are combinations of the thermal, magnetothermal and heat-flux-driven buoyancy instabilities. If the ICM rotates significantly, its stability properties are substantially modified and, in particular, also the magnetorotational instability (MRI) can play an important role. We study simple models of rotating cool-core clusters and we demonstrate that the MRI can be the dominant instability over significant portions of the clusters, with possible implications for the dynamics and evolution of the cool cores. Our results give further motivation for measuring the rotation of the ICM with future X-ray missions such as ASTRO-H and ATHENA.

A rotating and magnetized three-dimensional axisymmetric equilibrium for hot plasma confined by a gravitational field is found. The plasma density and current can exhibit strong equatorial plane localization, resulting in disk equilibria with open magnetic field lines. The associated equatorial plane pinching results in magnetic field flaring, implying a strong gravitational squeezing of the plasma carrying ambient magnetic field lines toward the gravitational source. At high plasma pressure, the magnetic field becomes strongly radial outside the disk. The model predicts the rotation frequency bound, the condition for a plasma disk, and the requirement for strong magnetic field flaring.

In collisional fluids, a number of key processes rely on the frequency of binary collisions. Collisions seem necessary to generate a shock wave when two fluids collide fast enough, to fulfill the Rankine–Hugoniot (RH) relations, to establish an equation of state or a Maxwellian distribution. Yet, these seemingly collisional features are routinely either observed or assumed, in relation with collisionless astrophysical plasmas. This article will review our current answers to the following questions: How do colliding collisionless plasmas end-up generating a shock as if they were fluids? To which extent are the RH relations fulfilled in this case? Do collisionless shocks propagate like fluid ones? Can we use an equation of state to describe collisionless plasmas, like MHD codes for astrophysics do? Why are Maxwellian distributions ubiquitous in particle-in-cell simulations of collisionless shocks? Time and length scales defining the border between the collisional and the collisionless behavior will be given when relevant. In general, when the time and length scales involved in the collisionless processes responsible for the fluid-like behavior may be neglected, the system may be treated like a fluid.

Magnetic reconnection and associated heating of ions and electrons in strongly magnetized, weakly collisional plasmas are studied by means of gyrokinetic simulations. It is shown that an appreciable amount of the released magnetic energy is dissipated to yield (irreversible) electron and ion heating via phase mixing. Electron heating is mostly localized to the magnetic island, not the current sheet, and occurs after the dynamical reconnection stage. Ion heating is comparable to electron heating only in high-β plasmas, and results from both parallel and perpendicular phase mixing due to finite Larmor radius (FLR) effects; in space, ion heating is mostly localized to the interior of a secondary island (plasmoid) that arises from the instability of the current sheet.

The Weibel/filamentation instability is known to play a key role in the physics of weakly magnetized collisionless shock waves. From the point of view of high energy astrophysics, this instability also plays a crucial role because its development in the shock precursor populates the downstream with a small-scale magneto-static turbulence which shapes the acceleration and radiative processes of suprathermal particles. The present work discusses the physics of the dissipation of this Weibel-generated turbulence downstream of relativistic collisionless shock waves. It calculates explicitly the first-order nonlinear terms associated to the diffusive nature of the particle trajectories. These corrections are found to systematically increase the damping rate, assuming that the scattering length remains larger than the coherence length of the magnetic fluctuations. The relevance of such corrections is discussed in a broader astrophysical perspective, in particular regarding the physics of the external relativistic shock wave of a gamma-ray burst.