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.

In space and astrophysical plasmas, turbulence is responsible for transferring energy from large scales driven by violent events or instabilities, to smaller scales where turbulent energy is ultimately converted into plasma heat by dissipative mechanisms. The nonlinear interaction between counterpropagating Alfvén waves, denoted Alfvén wave collisions, drives this turbulent energy cascade, as recognized by early work with incompressible magnetohydrodynamic (MHD) equations. Recent work employing analytical calculations and nonlinear gyrokinetic simulations of Alfvén wave collisions in an idealized periodic initial state have demonstrated the key properties that strong Alfvén wave collisions mediate effectively the transfer of energy to smaller perpendicular scales and self-consistently generate current sheets. For the more realistic case of the collision between two initially separated Alfvén wavepackets, we use a nonlinear gyrokinetic simulation to show here that these key properties persist: strong Alfvén wavepacket collisions indeed facilitate the perpendicular cascade of energy and give rise to current sheets. Furthermore, the evolution shows that nonlinear interactions occur only while the wavepackets overlap, followed by a clean separation of the wavepackets with straight uniform magnetic fields and the cessation of nonlinear evolution in between collisions, even in the gyrokinetic simulation presented here which resolves dispersive and kinetic effects beyond the reach of the MHD theory.

We investigate the onset of the classical magnetohydrodynamic (MHD) tearing instability (TI) and focus on non-modal (transient) growth rather than the tearing mode. With the help of pseudospectral theory, the operators of the linear equations are shown to be highly non-normal, resulting in the possibility of significant transient growth at the onset of the TI. This possibility increases as the Lundquist number $S$ increases. In particular, we find evidence, numerically, that the maximum possible transient growth, measured in the $L_{2}$-norm, for the classical set-up of current sheets unstable to the TI, scales as $O(S^{1/4})$ on time scales of $O(S^{1/4})$ for $S\gg 1$. This behaviour is much faster than the time scale $O(S^{1/2})$ when the solution behaviour is dominated by the tearing mode. The size of transient growth obtained is dependent on the form of the initial perturbation. Optimal initial conditions for the maximum possible transient growth are determined, which take the form of wave packets and can be thought of as noise concentrated at the current sheet. We also examine how the structure of the eigenvalue spectrum relates to physical quantities.

We apply a nonlinear mean-field dynamo model which includes a budget equation for the dynamics of Wolf numbers to predict solar activity. This dynamo model takes into account the algebraic and dynamic nonlinearities of the $\unicode[STIX]{x1D6FC}$ effect, where the equation for the dynamic nonlinearity is derived from the conservation law for the magnetic helicity. The budget equation for the evolution of the Wolf number is based on a formation mechanism of sunspots related to the negative effective magnetic pressure instability. This instability redistributes the magnetic flux produced by the mean-field dynamo. To predict solar activity on the time scale of one month we use a method based on a combination of the numerical solution of the nonlinear mean-field dynamo equations and the artificial neural network. A comparison of the results of the prediction of the solar activity with the observed Wolf numbers demonstrates a good agreement between the forecast and observations.

Spacecraft measurements of propagating interplanetary shocks are often interpreted using the ideal magnetohydrodynamic (MHD) model of a planar shock wave travelling with constant velocity $\boldsymbol{V}_{\text{sh}}$ through a spatially uniform plasma. In particular, measurements of the plasma variables upstream and downstream have long been used in conjunction with the Rankine–Hugoniot conditions, also known as the MHD jump conditions, to determine shock velocities and other physical parameters of interplanetary shocks. This procedure is justified only if the shock velocity determined by the MHD jump conditions is unique. In this study the important property of uniqueness is demonstrated for non-perpendicular shocks in MHD media characterized by an isotropic pressure tensor. The primary conclusion is that the shock velocity is uniquely determined by the jump conditions regardless of the type of shock (slow, intermediate or fast). Several new formulas for the shock speed are also derived including one that is independent of the shock normal $\hat{\boldsymbol{n}}$. In principle, the solution technique developed here can be applied to estimate $\boldsymbol{V}_{\text{sh}}$ using solar wind data provided the measurements obey the MHD shock model with sufficient accuracy. That is not its intended purpose, however, and such applications are beyond the scope of this work.

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.

Wave–particle interaction is a key process in particle diffusion in collisionless plasmas. We look into the interaction of single plasma waves with individual particles and discuss under which circumstances this is a chaotic process, leading to diffusion. We derive the equations of motion for a particle in the fields of a magnetostatic, circularly polarized, monochromatic wave and show that no chaotic particle motion can arise under such circumstances. A novel and exact analytic solution for the equations is presented. Additional plasma waves lead to a breakdown of the analytic solution and chaotic particle trajectories become possible. We demonstrate this effect by considering a linearly polarized, monochromatic wave, which can be seen as the superposition of two circularly polarized waves. Test particle simulations are provided to illustrate and expand our analytical considerations.

The electrostatic stability of electron–positron plasmas is investigated in the point-dipole and Z-pinch limits of dipole geometry. The kinetic dispersion relation for sub-bounce-frequency instabilities is derived and solved. For the zero-Debye-length case, the stability diagram is found to exhibit singular behaviour. However, when the Debye length is non-zero, a fluid mode appears, which resolves the observed singularity, and also demonstrates that both the temperature and density gradients can drive instability. It is concluded that a finite Debye length is necessary to determine the stability boundaries in parameter space. Landau damping is investigated at scales sufficiently smaller than the Debye length, where instability is absent.

Mean-field dynamo equations are addressed with the aid of the path integral method. The evolution of magnetic field is treated as a three-dimensional Wiener random process, and the mean magnetic-field equations are obtained with the Wiener integrals taken over all the trajectories of the fluid particles. The form of the equations is just the same as the conventional mean-field equations, but here the equations are derived with the velocity field realisation affected by the force exerted by the magnetic field. In this sense, we derive nonlinear dynamo equations.

New features of the twisted dusty plasma modes and associated instabilities are investigated in permeating plasmas. Using the Vlasov–Poisson model equations, a generalized dispersion relation is obtained for a Maxwellian distributed plasma to analyse the dust-acoustic and dust-ion-acoustic waves with finite orbital angular momentum (OAM) states. Existence conditions for damping/growth rates are discussed and showed significant modifications in twisted dusty modes as compared to straight propagating dusty modes. Numerically, the instability growth rate, which depends on particle streaming and twist effects in the wave potential, is significantly modified due to the Laguerre–Gaussian profiles. Relevance of the study to wave excitations due to penetration of solar wind into cometary clouds or interstellar dusty plasmas is discussed.

An increase in symmetry is observed for a low density non-collisional plasma, in a simple magnetic mirror machine, due to the application of external oscillating magnetic fields of 1.5 MHz frequency. The increase in symmetry is attributed to an increase in stability of the flute mode and is dependent on the field’s polarization and trap magnetic field strength.

Understanding the removal of energy from turbulent fluctuations in a magnetized plasma and the consequent energization of the constituent plasma particles is a major goal of heliophysics and astrophysics. Previous work has shown that nonlinear interactions among counterpropagating Alfvén waves – or Alfvén wave collisions – are the fundamental building block of astrophysical plasma turbulence and naturally generate current sheets in the strongly nonlinear limit. A nonlinear gyrokinetic simulation of a strong Alfvén wave collision is used to examine the damping of the electromagnetic fluctuations and the associated energization of particles that occurs in self-consistently generated current sheets. A simple model explains the flow of energy due to the collisionless damping and the associated particle energization, as well as the subsequent thermalization of the particle energy by collisions. The net particle energization by the parallel electric field is shown to be spatially localized, and the nonlinear evolution is essential in enabling spatial non-uniformity. Using the recently developed field–particle correlation technique, we show that particles resonant with the Alfvén waves in the simulation dominate the energy transfer, demonstrating conclusively that Landau damping plays a key role in the spatially localized damping of the electromagnetic fluctuations and consequent energization of the particles in this strongly nonlinear simulation.

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. These are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have indeed minimum quadratic helicity.

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.

Gyrokinetic simulations of drift waves in low-magnetic-shear stellarators reveal that simulation domains comprised of multiple turns can be required to properly resolve critical mode structures important in saturation dynamics. Marginally stable eigenmodes important in saturation of ion temperature gradient modes and trapped electron modes in the Helically Symmetric Experiment (HSX) stellarator are observed to have two scales, with the envelope scale determined by the properties of the local magnetic shear and an inner scale determined by the interplay between the local shear and magnetic field-line curvature. Properly resolving these modes removes spurious growth rates that arise for extended modes in zero-magnetic-shear approximations, enabling use of a zero-magnetic-shear technique with smaller simulation domains and attendant cost savings. Analysis of subdominant modes in trapped electron mode (TEM)-driven turbulence reveals that the extended marginally stable modes play an important role in the nonlinear dynamics, and suggests that the properties induced by low magnetic shear may be exploited to provide another route for turbulence saturation.

Stellar radiative zones are typically assumed to be motionless in standard models of stellar structure but there is sound theoretical and observational evidence that this cannot be the case. We investigate by direct numerical simulations a three-dimensional and time-dependent model of stellar radiation zones consisting of an electrically conductive and stably stratified anelastic fluid confined to a rotating spherical shell and driven by a baroclinic torque. As the baroclinic driving is gradually increased a sequence of transitions from an axisymmetric and equatorially symmetric time-independent flow to flows with a strong poloidal component and lesser symmetry are found. It is shown that all flow regimes characterised by significant non-axisymmetric components are capable of generating a self-sustained magnetic field. As the value of the Prandtl number is decreased and the value of the Ekman number is decreased, flows become strongly time dependent with progressively complex spatial structure and dynamos can be generated at lower values of the magnetic Prandtl number.

In this paper, weak-turbulence theory is used to investigate the nonlinear evolution of the parametric instability in three-dimensional low-$\unicode[STIX]{x1D6FD}$ plasmas at wavelengths much greater than the ion inertial length under the assumption that slow magnetosonic waves are strongly damped. It is shown analytically that the parametric instability leads to an inverse cascade of Alfvén wave quanta, and several exact solutions to the wave kinetic equations are presented. The main results of the paper concern the parametric decay of Alfvén waves that initially satisfy $e^{+}\gg e^{-}$, where $e^{+}$ and $e^{-}$ are the frequency ($f$) spectra of Alfvén waves propagating in opposite directions along the magnetic field lines. If $e^{+}$ initially has a peak frequency $f_{0}$ (at which $fe^{+}$ is maximized) and an ‘infrared’ scaling $f^{p}$ at smaller $f$ with $-1<p<1$, then $e^{+}$ acquires an $f^{-1}$ scaling throughout a range of frequencies that spreads out in both directions from $f_{0}$. At the same time, $e^{-}$ acquires an $f^{-2}$ scaling within this same frequency range. If the plasma parameters and infrared $e^{+}$ spectrum are chosen to match conditions in the fast solar wind at a heliocentric distance of 0.3 astronomical units (AU), then the nonlinear evolution of the parametric instability leads to an $e^{+}$ spectrum that matches fast-wind measurements from the Helios spacecraft at 0.3 AU, including the observed $f^{-1}$ scaling at $f\gtrsim 3\times 10^{-4}~\text{Hz}$. The results of this paper suggest that the $f^{-1}$ spectrum seen by Helios in the fast solar wind at $f\gtrsim 3\times 10^{-4}~\text{Hz}$ is produced in situ by parametric decay and that the $f^{-1}$ range of $e^{+}$ extends over an increasingly narrow range of frequencies as $r$ decreases below 0.3 AU. This prediction will be tested by measurements from the Parker Solar Probe.

Stability of a plasma cylinder with a current wrapped by a helical plasma flow is studied. Unstable surface modes of magnetohydrodynamic (MHD) oscillations develop at the boundary of the cylinder enwrapped by the plasma flow. Unstable eigenmodes can also develop for which the plasma cylinder is a waveguide. The growth rate of the surface modes is much higher than that for the eigenmodes. It is shown that the asymmetric MHD modes in the plasma cylinder are stable if the velocity of the plasma flow is below a certain threshold. Such a plasma flow velocity threshold is absent for the symmetric modes. They are unstable in any arbitrarily slow plasma flows. For all surface modes there is an upper threshold for the flow velocity above which they are stable. The helicity index of the flow around the plasma cylinder significantly affects both the Mach number dependence of the surface wave growth rate and the velocity threshold values. The higher the index, the lower the upper threshold of the velocity jump above which the surface waves become stable. Calculations have been carried out for the growth rates of unstable oscillations in an equilibrium plasma cylinder with current serving as a model of the low-latitude boundary layer (LLBL) of the Earth’s magnetic tail. A tangential discontinuity model is used to simulate the geomagnetic tail boundary. It is shown that the magnetopause in the geotail LLBL is unstable to a surface wave (having the highest growth rate) in low- and medium-speed solar wind flows, but becomes stable to this wave in high-speed flows. However, it can remain weakly unstable to the radiative modes of MHD oscillations.

Models for astrophysical plasmas often have magnetic field lines that leave the boundary rather than closing within the computational domain. Thus, the relative magnetic helicity is frequently used in place of the usual magnetic helicity, so as to restore gauge invariance. We show how to decompose the relative helicity into a relative field-line helicity that is an ideal-magnetohydrodynamic invariant for each individual magnetic field line, and vanishes along any field line where the original field matches the reference field. Physically, this relative field-line helicity is a magnetic flux, whose specific definition depends on the gauge of the reference vector potential on the boundary. We propose a particular ‘minimal’ gauge that depends only on the reference field and minimises this boundary contribution, so as to reveal topological information about the original magnetic field. We illustrate the effect of different gauge choices using the Low–Lou and Titov–Démoulin models of solar active regions. Our numerical code to compute appropriate vector potentials and relative field-line helicity in Cartesian domains is open source and freely available.