An electromagnetic reduced gyrofluid model for collisionless plasmas, accounting for electron inertia, finite ion Larmor radius effects and Landau-fluid closures for the electron fluid is derived by means of an asymptotic expansion from a parent gyrofluid model. In the absence of terms accounting for Landau damping, the model is shown to possess a non-canonical Hamiltonian structure. The corresponding Casimir invariants are derived and use is made thereof, in order to obtain a set of normal field variables, in terms of which the Poisson bracket and the model equations take a remarkably simple form. The inclusion of perpendicular temperature fluctuations generalizes previous Hamiltonian reduced fluid models and, in particular, the presence of ion perpendicular gyrofluid temperature fluctuations reflects into the presence of two new Lagrangian invariants governing the ion dynamics. The model is applied, in the cold-ion limit, to investigate numerically a magnetic reconnection problem. The Landau damping terms are shown to reduce, by decreasing the electron temperature fluctuations, the linear reconnection rate and to delay the nonlinear island growth. The saturated island width, on the other hand, is independent of Landau damping. The fraction of magnetic energy converted into perpendicular kinetic energy also appears to be unaffected by the Landau damping terms, which, on the other hand, dissipate parallel kinetic energy as well as free energy due to density and electron temperature fluctuations.

An analytic theory is presented of magnetic structures in collisionless, high-$\unicode[STIX]{x1D6FD}$ plasmas. Using a reduced magnetohydrodynamic model, a stationary nonlinear solution is constructed in the form of a Kelvin–Stuart cat’s eyes chain of magnetic islands, on the spatial scale that exceeds the characteristic ion lengths. The solution is imbedded in a background current sheet and possesses both a significant plasma density perturbation trapped inside the magnetic islands and a compressional magnetic field component that is driven mostly by a current loop located at the separatrix of the islands. This theory may provide an explanation for the magnetic structures observed in the solar wind close to the Earth by the Cluster spacecraft.

A family of Lorentz invariant scalar functions of the magnetic field is defined in an ideal relativistic plasma. These invariants are advected by the plasma fluid motion and play the role of the potential magnetic field introduced by Hide in (Ann. Geophys., vol. 1, 1983, 59) along the lines of Ertel’s theorem. From these invariants we recover the Cauchy conditions for the magnetic field components in the mapping from Eulerian to Lagrangian variables. In addition, the adopted procedure allows us to formulate, in a Lorentz invariant form, the Alfvén theorem for the conservation of the magnetic flux through a surface comoving with the plasma.

We discuss the effect of different types of fluctuations on dynamos generated in the limit of scale separation. We first recall that the magnetic field observed in the VKS (von Karman flow of liquid sodium) experiment is not the one that would be generated by the mean flow alone and that smaller scale turbulent fluctuations therefore play an important role. We then consider how velocity fluctuations affect the dynamo threshold in the framework of mean-field magnetohydrodynamics. We show that the detrimental effect of turbulent fluctuations observed with many flows disappears in the case of helical flows with scale separation. We also find that fluctuations of the electrical conductivity of the fluid, for instance related to temperature fluctuations in convective flows, provide an efficient mechanism for dynamo action. Finally, we conclude by describing an experimental configuration that could be used to test the validity of mean-field magnetohydrodynamics in strongly turbulent flows.

A dynamo driven by motions unaffected by viscous forces is termed ‘magnetostrophic’, but cannot be found through today’s numerical simulations, which require substantial viscosity to stabilize solutions of the full magnetohydrodynamic (MHD) dynamo equations. By using an alternative numerical technique, proposed by Taylor (Proc. R. Soc. Lond. A, vol. 274, 1963, pp. 274–283), we recently obtained the first magnetostrophic dynamo solutions ever derived (Wu & Roberts, Geophys. Astrophys. Fluid Dyn., vol. 109, 2014, pp. 84–110). These were axisymmetric and of mean-field type. In an earlier paper (Roberts & Wu, Geophys. Astrophys. Fluid Dyn., vol. 108, 2014, pp. 696–715), we proposed an extension of Taylor’s method. Here we explore its numerical implications, comparing them to the consequences of Taylor’s original proposal. One of the differences between the two approaches is that our modification retains torsional waves but Taylor’s theory does not. A more important difference is that our extension of Taylor’s method is, for reasons presented here, the most general possible that does not suffer from the limitations imposed by viscosity on today’s numerical simulations.

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical magnetohydrodynamics, it is important to verify that such simulations are in agreement with observations. One of the main challenges in this area is to identify robust quantitative measures to compare structures found in simulations with those inferred from astrophysical observations. A similar challenge is to compare quantitatively results from different simulations. Topological data analysis offers a range of techniques, including the Betti numbers and persistence diagrams, that can be used to facilitate such a comparison. After describing these tools, we first apply them to synthetic random fields and demonstrate that, when the data are standardized in a straightforward manner, some topological measures are insensitive to either large-scale trends or the resolution of the data. Focusing upon one particular astrophysical example, we apply topological data analysis to H i observations of the turbulent interstellar medium (ISM) in the Milky Way and to recent magnetohydrodynamic simulations of the random, strongly compressible ISM. We stress that these topological techniques are generic and could be applied to any complex, multi-dimensional random field.

Recent advances in mean-field theory are reviewed and applications to the Sun, late-type stars, accretion disks, galaxies and the early Universe are discussed. We focus particularly on aspects of spatio-temporal non-locality, which provided some of the main new qualitative and quantitative insights that emerged from applying the test-field method to magnetic fields of different length and time scales. We also review the status of nonlinear quenching and the relation to magnetic helicity, which is an important observational diagnostic of modern solar dynamo theory. Both solar and some stellar dynamos seem to operate in an intermediate regime that has not yet been possible to model successfully. This regime is bracketed by antisolar-like differential rotation on one end and stellar activity cycles belonging to the superactive stars on the other. The difficulty in modelling this regime may be related to shortcomings in simulating solar/stellar convection. On galactic and extragalactic length scales, the observational constraints on dynamo theory are still less stringent and more uncertain, but recent advances both in theory and observations suggest that more conclusive comparisons may soon be possible also here. The possibility of inversely cascading magnetic helicity in the early Universe is particularly exciting in explaining the recently observed lower limits of magnetic fields on cosmological length scales. Such magnetic fields may be helical with the same sign of magnetic helicity throughout the entire Universe. This would be a manifestation of parity breaking.

The aim of this short paper is simply to demonstrate the effect of interactions between distinct waves induced by an oscillatory forcing on generation of the large-scale electromotive force (EMF) in a highly electrically conducting plasma in the dynamical regime, i.e. when the back reaction of the Lorentz force on the flow is included. The mean EMF is induced by waves with distinct but close phase speeds, which are called ‘beating waves’ by analogy with the acoustic effect of ‘beat’, in the presence of a locally uniform seed magnetic field. These waves may have either positive or negative helicity, and it is supposed that waves of a single sign of helicity are preferentially excited by a symmetry-breaking mechanism. The formula for the mean EMF in a highly conducting plasma is derived, conditions relevant to those of the early universe before and during galaxy formation. Within the scope of the weak magnetohydrodynamic turbulence dominated by the linear waves an $\unicode[STIX]{x1D6FC}$-effect, that is generation of the large-scale EMF is studied, which leads to amplification of the seed mean magnetic field. Possible subsequent equilibration due to quenching resulting from the presence of the Lorentz force is briefly discussed.

The quasi-geostrophic dynamo model (QGDM) is a multiscale, fully nonlinear Cartesian dynamo model that is valid in the asymptotic limit of low Rossby number. In the additional limit of small magnetic Prandtl number investigated here, the QGDM is a self-consistent, asymptotically exact form of an $\unicode[STIX]{x1D6FC}^{2}$ large-scale dynamo. This article explores methods for simulating the multiscale QGDM and investigates how convection is altered by the magnetic field in the planetary regime of small Rossby number and small magnetic Prandtl number. At present, this combination is beyond the reach of direct numerical simulations. We use a simplified class of solutions whose horizontal structure is restricted to a periodic hexagonal lattice characterized by a single horizontal wavenumber (single mode). In contrast with previous kinematic investigations of the QGDM, the Lorentz force is included to study saturated, self-consistent dynamos. Two methodologies are used to assess handling of the multiple time scales of the QGDM: a stiff, common-in-time approach where all time scales are converted to a single time variable and a heterogeneous multiscale modelling approach employing fast time averaging on the Reynolds, magnetic and buoyancy eddy fluxes that feed back onto the slow scales. These strategies produce consistent results and each illustrates self-similar dynamics as the time-averaging window is increased. The properties of the convection are significantly altered by the dynamo-generated magnetic field. All solutions show a decrease in the overall heat transfer efficiency as compared to non-magnetic convection, suggesting that a change in length scale or flow planform plays a critical role in the enhanced heat transfer efficiency observed in previous dynamo studies. All dynamo solutions show a trend of increasing ohmic dissipation relative to viscous dissipation as the buoyancy forcing is increased.

The theory of mean field electrodynamics, now celebrating its fiftieth birthday, has had a profound influence on our modelling of cosmical dynamos, greatly enhancing our understanding of how such dynamos may operate. Here I discuss some of its undoubted triumphs, but also some of the problems that can arise in a mean field approach to dynamos in fluids (or plasmas) that are both highly turbulent and also extremely good electrical conductors, as found in all astrophysical settings.

We show results on the high-pressure equation of state of hydrogen–helium mixtures obtained from finite-temperature density functional theory molecular dynamics simulations using the van der Waals density functional. We discuss the calculation of non-ideal entropies based on different methods and show how nuclear quantum corrections influence the free enthalpy of mixing. Furthermore, we calculate a Saturn isentrope based on our new equation of state data.

Energetic particle effects in magnetic confinement fusion devices are commonly studied by hybrid kinetic-fluid simulation codes whose underlying continuum evolution equations often lack the correct energy balance. While two different kinetic-fluid coupling options are available (current coupling and pressure coupling), this paper applies the Euler–Poincaré variational approach to formulate a new conservative hybrid model in the pressure-coupling scheme. In our case the kinetics of the energetic particles are described by guiding center theory. The interplay between the Lagrangian fluid paths and phase space particle trajectories reflects an intricate variational structure which can be approached by letting the four-dimensional guiding center trajectories evolve in the full six-dimensional phase space. Then, the redundant perpendicular velocity is integrated out to recover a four-dimensional description. A second equivalent variational approach is also reported, which involves the use of phase space Lagrangians. Not only do these variational structures confer on the new model a correct energy balance, but also they produce a cross-helicity invariant which is lost in the other pressure-coupling schemes reported in the literature.

Properties of the electron mirror instability and its competition with the usually dominant whistler (electron cyclotron) instability driven by the electron perpendicular temperature anisotropy are investigated at the linear level using a Vlasov linear solver and at the nonlinear level using a two-dimensional full particle code. The simulation results show that the linearly subdominant electron mirror instability may compete at the nonlinear level with the whistler instability and may even eventually become the dominant mode that generates robust non-propagating sub-ion-scale coherent structures in the form of magnetic peaks.

The Arbitrary Linear Plasma Solver (ALPS) is a parallelised numerical code that solves the dispersion relation in a hot (even relativistic) magnetised plasma with an arbitrary number of particle species with arbitrary gyrotropic equilibrium distribution functions for any direction of wave propagation with respect to the background field. ALPS reads the background momentum distributions as tables of values on a $(p_{\bot },p_{\Vert })$ grid, where $p_{\bot }$ and $p_{\Vert }$ are the momentum coordinates in the directions perpendicular and parallel to the background magnetic field, respectively. We present the mathematical and numerical approach used by ALPS and introduce our algorithms for the handling of poles and the analytic continuation for the Landau contour integral. We then show test calculations of dispersion relations for a selection of stable and unstable configurations in Maxwellian, bi-Maxwellian, $\unicode[STIX]{x1D705}$-distributed and Jüttner-distributed plasmas. These tests demonstrate that ALPS derives reliable plasma dispersion relations. ALPS will make it possible to determine the properties of waves and instabilities in the non-equilibrium plasmas that are frequently found in space, laboratory experiments and numerical simulations.

Three-dimensional direct numerical simulations are used to study the energy cascade rate in isothermal compressible magnetohydrodynamic turbulence. Our analysis is guided by a two-point exact law derived recently for this problem in which flux, source, hybrid and mixed terms are present. The relative importance of each term is studied for different initial subsonic Mach numbers $M_{S}$ and different magnetic guide fields $\boldsymbol{B}_{0}$ . The dominant contribution to the energy cascade rate comes from the compressible flux, which depends weakly on the magnetic guide field $\boldsymbol{B}_{0}$ , unlike the other terms whose moduli increase significantly with $M_{S}$ and $\boldsymbol{B}_{0}$ . In particular, for strong $\boldsymbol{B}_{0}$ the source and hybrid terms are dominant at small scales with almost the same amplitude but with a different sign. A statistical analysis undertaken with an isotropic decomposition based on the SO(3) rotation group is shown to generate spurious results in the presence of $\boldsymbol{B}_{0}$ , when compared with an axisymmetric decomposition better suited to the geometry of the problem. Our numerical results are compared with previous analyses made with in situ measurements in the solar wind and the terrestrial magnetosheath.

It is known that the filtered, doubly Fourier transformed, Eulerian Vlasov–Poisson simulation method does not exhibit any effects of recurrence. It has not been understood why this is so. It is shown that there is only one boundary in this simulation method where user input is allowed and that the appearance or not of recurrence depends on the character of the boundary data that are imposed. Examples are given that show that the choice of boundary data that leads to recurrence-free simulations does not degrade the accuracy of Landau damping simulations.

An electron acoustic super solitary wave has been derived using the Sagdeev pseudopotential technique for a four component magnetized plasma consisting of the beam and bulk fluid electrons and two ions with Maxwell Boltzmann distributions. This is the first theoretical report of a super solitary wave in a magnetized plasma which has no direct association with the singularity of the pseudopotential. It shows a narrow and spiky subwell near the low potential which causes the lateral inversion of the wiggle for the bipolar electric field vis-á-vis the unmagnetized plasma. An analytical formalism was developed to identify these novel kinds of super solitary waves and their transition processes have been characterized. It was observed that the super solitary wave is directly influenced by the singularity of the pseudopotential lying in the vicinity of the solution. The first ever prediction of the electron acoustic super solitary wave raises the possibility of its application to the interpretation of the satellite observations of the electrostatic field data.

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tilde{\unicode[STIX]{x1D711}}$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tilde{\unicode[STIX]{x1D711}}$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio and the structure of $\tilde{\unicode[STIX]{x1D711}}$ on the magnetic surface are analytically derived in the $1/\unicode[STIX]{x1D708}$, $\sqrt{\unicode[STIX]{x1D708}}$ and superbanana-plateau regimes of stellarators close to omnigeneity; i.e. stellarators that have been optimized for neoclassical transport. It is found that the largest $\tilde{\unicode[STIX]{x1D711}}$ that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigenous configuration, the analytical results are verified and illustrated with calculations by the code KNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry.

We examine the properties of linear electrostatic waves in unmagnetised quantum and classical plasmas consisting of one or two populations of electrons with analytically tractable distribution functions in the presence of a stationary neutralising ion background. Beginning with the kinetic quantum plasma longitudinal susceptibility, we assess the effects due to increasing complexity of the background distribution function. Firstly, we analyse dispersion and Landau damping in one-component plasmas and consider distribution functions with a variety of analytical properties: the Dirac delta function, the Cauchy profile with two complex first-order poles, the squared Cauchy profile with two second-order poles and the inverse-quartic profile with four first-order poles; we also briefly discuss the non-meromorphic totally and arbitrarily degenerate Fermi–Dirac distribution. In order to study electrostatic instabilities, we then turn to plasmas with two populations of electrons streaming relative to each other in two cases: a symmetric case of two counter-streaming identical populations and a bump-on-tail case with a primary population and a delta-function beam. We obtain the corresponding linear kinetic dispersion relations and evaluate the properties of instabilities when the electron distribution functions are of the delta function, Cauchy, squared Cauchy or inverse-quartic types. In agreement with other studies, we find that in general quantum effects reduce the range of wavelengths for unstable modes at long wavelengths. We also find a second window of instability at shorter wavelengths and elucidate its nature as being due to quantum recoil. We note the possible implications for studies of laboratory and astrophysical quantum plasmas.

High-$Z$ impurities in magnetic-confinement devices are prone to develop density variations on the flux surface, which can significantly affect their transport. In this paper, we generalize earlier analytic stellarator calculations of the neoclassical radial impurity flux in the mixed-collisionality regime (collisional impurities and low-collisionality bulk ions) to include the effect of such flux-surface variations. We find that only in the homogeneous density case is the transport of highly collisional impurities (in the Pfirsch–Schlüter regime) independent of the radial electric field. We study these effects for a Wendelstein 7-X (W7-X) vacuum field, with simple analytic models for the potential perturbation, under the assumption that the impurity density is given by a Boltzmann response to a perturbed potential. In the W7-X case studied, we find that larger amplitude potential perturbations cause the radial electric field to dominate the transport of the impurities. In addition, we find that classical impurity transport can be larger than the neoclassical transport in W7-X.