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.

Particle image velocimetry is a fluid measurement technique that has been used for more than 20 years to characterize the particle transport and thermal state of dusty plasma systems. This manuscript provides an overview of this diagnostic technique, highlighting the strengths and limitations that are associated with its use. Additionally, the variations of this technique that have been applied in the study of dusty plasma systems will be discussed, along with a small selection of measurements that can be made with the technique. Potential future directions for this diagnostic tool within the dusty plasma community will also be discussed.

Counter-propagating and suitably polarized light (laser) beams can provide conditions for pair production. Here, we consider in more detail the following two situations: (i) in the homogeneity regions of anti-nodes of linearly polarized ultra-high intensity laser beams, the Schwinger process is dynamically assisted by a second high-frequency field, e.g. by an XFEL beam; and (ii) a high-energy probe photon beam colliding with a superposition of co-propagating intense laser and XFEL beams gives rise to the laser-assisted Breit–Wheeler process. The prospects of such bi-frequent field constellations with respect to the feasibility of conversion of light into matter are discussed.

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.

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.

Sheared flows perpendicular to the magnetic field can be driven by the Reynolds stress or ion pressure gradient effects and can potentially influence the stability and turbulent saturation level of edge plasma modes. On the other hand, such flows are subject to the transverse Kelvin–Helmholtz (KH) instability. Here, the linear theory of KH instabilities is first addressed with an analytic model in the asymptotic limit of long wavelengths compared with the flow scale length. The analytic model treats sheared $\boldsymbol{E}\times \boldsymbol{B}$ flows, ion diamagnetism (including gyro-viscous terms), density gradients and parallel currents in a slab geometry, enabling a unified summary that encompasses and extends previous results. In particular, while ion diamagnetism, density gradients and parallel currents each individually reduce KH growth rates, the combined effect of density and ion pressure gradients is more complicated and partially counteracting. Secondly, the important role of realistic toroidal geometry is explored numerically using an invariant scaling analysis together with the 2DX eigenvalue code to examine KH modes in both closed and open field line regions. For a typical spherical torus magnetic geometry, it is found that KH modes are more unstable at, and just outside of, the separatrix as a result of the distribution of magnetic shear. Finally implications for reduced edge turbulence modelling codes are discussed.

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.

The SIESTA magnetohydrodynamic (MHD) equilibrium code has been used to compute a sequence of ideally stable equilibria resulting from numerical variation of the helical resonant magnetic perturbation (RMP) applied to an axisymmetric DIII-D plasma equilibrium. Increasing the perturbation strength at the dominant $m=2$, $n=-1$ resonant surface leads to lower MHD energies and increases in the equilibrium island widths at the $m=2$ (and sidebands) surfaces, in agreement with theoretical expectations. Island overlap at large perturbation strengths leads to stochastic magnetic fields which correlate well with the experimentally inferred field structure. The magnitude and spatial phase (around the dominant rational surfaces) of the resonant (shielding) component of the parallel current are shown to change qualitatively with the magnetic island topology.

The measurement of vacuum polarisation effects, in particular vacuum birefringence, using combined optical and X-ray laser pulses are now actively pursued. Here we briefly examine the feasibility of two alternative set-ups. The first utilises an alternative target, namely a converging dipole pulse, and the second uses an alternative probe, namely the synchrotron-like emission from highly energetic particles, themselves interacting with a laser pulse. The latter set-up has been proposed for experiments at ELI-NP.

Particle-based simulations, such as in particle-in-cell (PIC) codes, are widely used in plasma physics research. The analysis of particle energy transfers, as described by the second moment of the Boltzmann equation, is often necessary within these simulations. We present computationally efficient, analytically derived equations for evaluating collisional energy transfer terms from simulations using discrete particles. The equations are expressed as a sum over the properties of the discrete particles.

In this paper we revisit the paradigm of space science turbulent dissipation traditionally considered as myth (Coroniti, Space Sci. Rev., vol. 42, 1985, pp. 399–410). We demonstrate that due to approach introduced by Pitaevskii (Sov. J. Expl Theor. Phys., vol. 44, 1963, pp. 969–979; (in Russian)) (the effect of a finite Larmor radius on a classical collision integral) dissipation induced by effective interaction with microturbulence produces a significant effect on plasma dynamics, especially in the vicinity of the reconnection region. We estimate the multiplication factor of collision frequency in the collision integral for short wavelength perturbations. For waves propagating transverse to the background magnetic field, this factor is approximately $({\it\rho}_{e}k_{x})^{2}$ with ${\it\rho}_{e}$ an electron gyroradius and where $k_{x}$ is a transverse wavenumber. We consider recent spacecraft observations in the Earth’s magnetotail reconnection region to the estimate possible impact of this multiplication factor. For small-scale reconnection regions this factor can significantly increase the effective collision frequency produced both by lower-hybrid drift turbulence and by kinetic Alfvén waves. We discuss the possibility that the Pitaevskii’s effect may be responsible for the excitation of a resistive electron tearing mode in thin current sheets formed in the outflow region of the primary X-line.

We report on the experimental demonstration of a technique to generate steep density gradients in gas-jet targets of interest to laser–plasma ion acceleration. By using an intentional low-energy prepulse, we generated a hydrodynamic blast wave in the gas to shape the target prior to the arrival of an intense CO$_{2}$ (${\it\lambda}\approx 10~{\rm\mu}\text{m}$) drive pulse. This technique has been recently shown to facilitate the generation of ion beams by shockwave acceleration (Tresca et al., Phys. Rev. Lett., vol. 115 (9), 2015, 094802). Here, we discuss and introduce a model to understand the generation of these blast waves and discuss in depth the experimental realisation of the technique, supported by hydrodynamics simulations. With appropriate prepulse energy and timing, this blast wave can generate steepened density gradients as short as $l\approx 20~{\rm\mu}\text{m}$ ($1/e$), opening up new possibilities for laser–plasma studies with near-critical gaseous targets.

The fundamentals of stereoscopy and their application to dusty plasmas are described. It is shown that stereoscopic methods allow us to measure the three-dimensional particle positions and trajectories with high spatial and temporal resolution. The underlying technical implications are presented and requirements and limitations are discussed. The stereoscopic method is demonstrated for dust particles in dust-density waves under microgravity conditions.

V. D. Shafranov was a key person in the fusion program. The paper presents the recollections of one of his close colleagues about Shafranov’s impact on the early days of tokamak research.

In honour of the 50th anniversary of the influential review/monograph on plasma turbulence by B. B. Kadomtsev as well as the seminal works of T. H. Dupree and J. Weinstock on resonance-broadening theory, an introductory tutorial is given about some highlights of the statistical–dynamical description of turbulent plasmas and fluids, including the ideas of nonlinear incoherent noise, coherent damping, and self-consistent dielectric response. The statistical closure problem is introduced. Incoherent noise and coherent damping are illustrated with a solvable model of passive advection. Self-consistency introduces turbulent polarization effects that are described by the dielectric function ${\mathcal{D}}$. Dupree’s method of using ${\mathcal{D}}$ to estimate the saturation level of turbulence is described; then it is explained why a more complete theory that includes nonlinear noise is required. The general theory is best formulated in terms of Dyson equations for the covariance $C$ and an infinitesimal response function $R$, which subsumes ${\mathcal{D}}$. An important example is the direct-interaction approximation (DIA). It is shown how to use Novikov’s theorem to develop an $\boldsymbol{x}$-space approach to the DIA that is complementary to the original $\boldsymbol{k}$-space approach of Kraichnan. A dielectric function is defined for arbitrary quadratically nonlinear systems, including the Navier–Stokes equation, and an algorithm for determining the form of ${\mathcal{D}}$ in the DIA is sketched. The independent insights of Kadomtsev and Kraichnan about the problem of the DIA with random Galilean invariance are described. The mixing-length formula for drift-wave saturation is discussed in the context of closures that include nonlinear noise (shielded by ${\mathcal{D}}$). The role of $R$ in the calculation of the symmetry-breaking (zonostrophic) instability of homogeneous turbulence to the generation of inhomogeneous mean flows is addressed. The second-order cumulant expansion and the stochastic structural stability theory are also discussed in that context. Various historical research threads are mentioned and representative entry points to the literature are given. Some outstanding conceptual issues are enumerated.