Challenges in plasma physics are wide. Investigation and advances are made in experiments but at the same time, to understand and to reach the experimental limits, accurate numerical simulations are required from systems of nonlinear equations. The numerical challenges of solving the associated fluid equations are discussed in this paper. Using the framework of the finite difference discretization, the most widely used methods for the problems linked to the diffusion or advection operators are presented.

Magnetic reconnection is thought to be the dynamical mechanism underlying many explosive phenomena observed both in space and in the laboratory, although the question of how fast magnetic reconnection is triggered in such high Lundquist ( $S$ ) number plasmas has remained elusive. It has been well established that reconnection can develop over time scales faster than those predicted traditionally once kinetic scales are reached. It has also been shown that, within the framework of resistive magnetohydrodynamics (MHD), fast reconnection is achieved for thin enough sheets via the onset of the so-called plasmoid instability. The latter was discovered in studies specifically devoted to the Sweet–Parker current sheet, either as an initial condition or an apparent transient state developing in nonlinear studies. On the other hand, a fast tearing instability can grow on an ideal, i.e. $S$ -independent, time scale (dubbed ‘ideal’ tearing) within current sheets whose aspect ratio scales with the macroscopic Lundquist number as $L/a\sim S^{1/3}$ – much smaller than the Sweet–Parker one – suggesting a new way to approach to the initiation of fast reconnection in collapsing current configurations. Here we present an overview of what we have called ‘ideal’ tearing in resistive MHD, and discuss how the same reasoning can be extended to other plasma models commonly used that include electron inertia and kinetic effects. We then discuss a scenario for the onset of ‘ideal’ fast reconnection via collapsing current sheets and describe a quantitative model for the interpretation of the nonlinear evolution of ‘ideally’ unstable sheets in two dimensions.

Coaxially nested intense $E\times B$ sheared flow realized an upgraded stable mirror plasma regime. After such an external control of high vorticity formation due to electron cyclotron heating, significantly unstable plasmas appeared. Thereby, the associated cross-field transport caused a crash of plasmas. Its generalized physics and interpretation could prepare or extend to another possibility of stability in a field-reversed configuration (FRC), for instance. Such underlying physics bases of vorticity formation were essentially or partially performed in tokamaks and stellarators (solved problems). Nevertheless, it remains to be seen whether this mirror-based experimental evidence is applicable or not to open ended FRC devices. This open issue may give a solution of one of unsolved important problems, and possibly provide more generalized and externally controllable opportunities for not only FRC but wider plasma confinement improvements.

The particle-in-cell (PIC) method was used to simulate heat flux mitigation experiments with partially ionised argon. The experiments demonstrate the possibility of reducing heat flux towards a target using magnetic fields. Modelling using the PIC method is able to reproduce the heat flux mitigation qualitatively. This is driven by modified electron transport. Electrons are magnetised and react directly to the external magnetic field. In addition, an increase of radial turbulent transport is also needed to explain the experimental observations in the model. Close to the target an increase of electron density is created. Due to quasi-neutrality, ions follow the electrons. Charge exchange collisions couple the dynamics of the neutrals to the ions and reduce the flow velocity of neutrals by radial momentum transport and subsequent losses. By this, the dominant heat-transport channel by neutrals gets reduced and a reduction of the heat deposition, similar to the experiment, is observed. Using the simulation a diagnostic module for optical emission is developed and its results are compared with spectroscopic measurements and photos from the experiment. The results of this study are in good agreement with the experiment. Experimental observations such as a shrank bright emission region close to the nozzle exit, an additional emission in front of the target and an overall change in colour to red are reproduced by the simulation.

In recent papers, we show how accretion disks around massive black holes could act as dynamos producing magnetic jets similar to the jets that create spheromaks in the laboratory. In this paper, we discuss how these magnetic astrophysical jets might naturally produce runaway ion beams accelerated to $10^{20}$  eV or more, finally ejected as ultra high energy cosmic rays (UHECRs) long regarded as one of the mysteries of astrophysics. The acceleration is mainly due to the drift cyclotron loss cone kinetic instability known from plasma research. Experiments and simulations are suggested to verify the acceleration process.

Rotating plasmas have the potential to offer unique capabilities for isotope and mass separation. Among the various electric and magnetic field configurations offering mass separation capabilities, rotating plasmas produced through static or oscillating fields are shown to be a leading candidate for tackling the unsolved problem of large-scale plasma separation. The successful development and deployment of industrial-scale plasma separation technologies could, among many other applications, provide an innovative path towards advanced sustainable nuclear energy. In this context, the potential and versatility of plasma rotation induced by rotating magnetic fields is uncovered and analysed. Analytical stability diagrams are derived from rotating ion orbits as a function of ion mass. Based on these findings, the basic principles of a rotating field plasma separator are then introduced. In light of these results, challenges associated with this original separation process are underlined, and the main directions for a future research program aimed at this important unsolved problem of applied plasma physics are identified.

Charge carrier transport across interfaces of dissimilar materials (including vacuum) is the essence of all electronic devices. Ultrafast charge transport across a nanometre length scale is of fundamental importance in the miniaturization of vacuum and plasma electronics. With the combination of recent advances in electronics, photonics and nanotechnology, these miniature devices may integrate with solid-state platforms, achieving superior performance. This paper reviews recent modelling efforts on quantum tunnelling, ultrafast electron emission and transport, and electrical contact resistance. Unsolved problems and challenges in these areas are addressed.

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.

The dynamics of passage and capture into resonance of a distribution of particles driven by a chirped frequency perturbation is discussed. The resonant capture in this case involves crossing of the separatrix by individual particles and, therefore, the adiabatic theorem cannot be used in studying this problem no matter how slow the variation of the driving frequency is. It is shown that, if instead of analysing complicated single particle dynamics in passage through resonance, one considers the slow evolution of a whole distribution of initial conditions in phase space, the adiabaticity and phase space incompressibility arguments yield a solution to the resonant passage problem. This approach is illustrated in the case of an ensemble of electrons driven by a chirped frequency wave passing through Cherenkov resonances with the velocity distribution of electrons.

The Universe is permeated by hot, turbulent, magnetized plasmas. Turbulent plasma is a major constituent of active galactic nuclei, supernova remnants, the intergalactic and interstellar medium, the solar corona, the solar wind and the Earth’s magnetosphere, just to mention a few examples. Energy dissipation of turbulent fluctuations plays a key role in plasma heating and energization, yet we still do not understand the underlying physical mechanisms involved. THOR is a mission designed to answer the questions of how turbulent plasma is heated and particles accelerated, how the dissipated energy is partitioned and how dissipation operates in different regimes of turbulence. THOR is a single-spacecraft mission with an orbit tuned to maximize data return from regions in near-Earth space – magnetosheath, shock, foreshock and pristine solar wind – featuring different kinds of turbulence. Here we summarize the THOR proposal submitted on 15 January 2015 to the ‘Call for a Medium-size mission opportunity in ESAs Science Programme for a launch in 2025 (M4)’. THOR has been selected by European Space Agency (ESA) for the study phase.

A new class of exact electrostatic solutions of the Vlasov–Maxwell equations based on the Jeans’s theorem is proposed for studying the evolution and properties of two-dimensional anisotropic plasmas that are far from thermodynamic equilibrium. In particular, the free expansion of a slab of electron–ion plasma into vacuum is investigated.

Global, first-principles study of the kinetic ballooning mode (KBM) is crucial to understand tokamak edge physics in high-confinement mode (H-mode). In contrast to the ion temperature gradient mode and trapped electron mode, the KBM is found to be very sensitive to the equilibrium implementations in gyrokinetic codes. In this paper, we show that a second-order difference in Shafranov shift or geometric coordinates, or a difference between local and global profile implementations can bring a factor of two or more discrepancy in real frequency and growth rate. This suggests that an accurate global equilibrium is required for validation of gyrokinetic KBM simulations.

The effects of ionization, ion loss and ion-neutral collisions on ion acoustic wave modulations in a dusty plasma are investigated. It is found that the dynamics of the modulated wave is governed by a nonlinear Schrödinger equation with a linear dissipative (damping or growth) term. This linear dissipative term arises due to the combined influence of ionization, ion loss and ion-neutral collisions. Analytical and numerical investigations predict the formation of envelope (bright) soliton with growing or decaying amplitude with time. The ionization instability causes the amplitude amplification, whereas, ion loss and ion-neutral collisions are responsible for the amplitude decay. Results are discussed in the context of dusty plasma experiments.

The thermal motion of a dust particle levitated in a plasma chamber is similar to that described by Brownian motion in many ways. The primary difference between a dust particle in a plasma system and a free Brownian particle is that in addition to the random collisions between the dust particle and the neutral gas atoms, there are electric field fluctuations, dust charge fluctuations, and correlated motions from the unwanted continuous signals originating within the plasma system itself. This last contribution does not include random motion and is therefore separable from the random motion in a ‘normal’ temperature measurement. In this paper, we discuss how to separate random and coherent motions of a dust particle confined in a glass box in a Gaseous Electronic Conference (GEC) radio-frequency (RF) reference cell employing experimentally determined dust particle fluctuation data analysed using the mean square displacement technique.

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.

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.

In a recent article (J. Plasma Phys., vol. 82, 2009, 905820104), weakly dissipative dust-ion acoustic wave modulation in dusty plasmas was considered. It is shown in this Comment that the analysis therein involved severe fallacies, and is in fact based on an erroneous plasma fluid model, which fails to satisfy an equilibrium condition, among other shortcomings. The subsequent analysis therefore is dubious and of limited scientific value.

In a three-dimensional non-null magnetic reconnection, the process of magnetic reconnection takes place in the absence of a null point where the magnetic field vanishes. By randomly injecting a population of 10 000 protons, the trajectory and energy distribution of accelerated protons are investigated in the presence of magnetic and electric fields of a particular model of non-null magnetic reconnection with the typical parameters for the solar corona. The results show that protons are accelerated along the magnetic field lines away from the non-null point only at azimuthal angles where the magnitude of the electric field is strongest and therefore particles obtain kinetic energies of the order of thousands of MeV and even higher. Moreover, the energy distribution of the population depends strongly on the amplitude of the electric and magnetic fields. Comparison shows that a non-null magnetic reconnection is more efficient in accelerating protons to very high GeV energies than a null-point reconnection.

Plasma free energy is that part of the total energy that feeds the growth of turbulence. The most successful free energy formulation in plasma physics is the MHD Energy Principle – successful because, within magnetohydrodynamics (MHD), the free energy $\unicode[STIX]{x1D6FF}W$ is both exact and self-adjoint (or Hermitian). A corresponding result in Vlasov theory is the free energy of equilibria neighbouring stable Maxwellian states – again giving a free energy of Hermitian form for the linearized equations. Since quantum mechanics is inherently Hermitian, here I speculate that quantum free energy is the ultimate way to understand classical plasma dynamics.