Dipole and stellarator geometries are capable of confining plasmas of arbitrary neutrality, ranging from pure electron plasmas through to quasineutral. The diocotron mode is known to be important in non-neutral plasmas and has been widely studied. However, drift mode dynamics, dominating quasineutral plasmas, has received very little by way of attention in the non-neutral context. Here, we show that non-neutral plasmas can be unstable respect to both density-gradient- and temperature-gradient-driven instabilities. A local shearless slab limit is considered for simplicity. A key feature of non-neutral plasmas is the development of strong electric fields, in this local limit of straight field line geometry, the effect of the corresponding $\boldsymbol{E}\times \boldsymbol{B}$ drift is limited to the Doppler shift of the complex frequency $\unicode[STIX]{x1D714}\rightarrow \unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{E}$. However, the breaking of the quasineutrality condition still leads to interesting dynamics in non-neutral plasmas. In this paper we address the behaviour of a number of gyrokinetic modes in electron–ion and electron–positron plasmas with arbitrary degree of neutrality.

These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental contexts disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) small- and large-scale magnetohydrodynamic (MHD) dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small- and large-scale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.

We present three-dimensional direct numerical simulations and an analytic model of reflection-driven magnetohydrodynamic (MHD) turbulence in the solar wind. Our simulations describe transverse, non-compressive MHD fluctuations within a narrow magnetic flux tube that extends from the photosphere, through the chromosphere and corona and out to a heliocentric distance $r$ of 21 solar radii $(R_{\odot })$. We launch outward-propagating ‘$\boldsymbol{z}^{+}$ fluctuations’ into the simulation domain by imposing a randomly evolving photospheric velocity field. As these fluctuations propagate away from the Sun, they undergo partial reflection, producing inward-propagating ‘$\boldsymbol{z}^{-}$ fluctuations’. Counter-propagating fluctuations subsequently interact, causing fluctuation energy to cascade to small scales and dissipate. Our analytic model incorporates dynamic alignment, allows for strongly or weakly turbulent nonlinear interactions and divides the $\boldsymbol{z}^{+}$ fluctuations into two populations with different characteristic radial correlation lengths. The inertial-range power spectra of $\boldsymbol{z}^{+}$ and $\boldsymbol{z}^{-}$ fluctuations in our simulations evolve toward a $k_{\bot }^{-3/2}$ scaling at $r>10R_{\odot }$, where $k_{\bot }$ is the wave-vector component perpendicular to the background magnetic field. In two of our simulations, the $\boldsymbol{z}^{+}$ power spectra are much flatter between the coronal base and $r\simeq 4R_{\odot }$. We argue that these spectral scalings are caused by: (i) high-pass filtering in the upper chromosphere; (ii) the anomalous coherence of inertial-range $\boldsymbol{z}^{-}$ fluctuations in a reference frame propagating outwards with the $\boldsymbol{z}^{+}$ fluctuations; and (iii) the change in the sign of the radial derivative of the Alfvén speed at $r=r_{\text{m}}\simeq 1.7R_{\odot }$, which disrupts this anomalous coherence between $r=r_{\text{m}}$ and $r\simeq 2r_{\text{m}}$. At $r>1.3R_{\odot }$, the turbulent heating rate in our simulations is comparable to the turbulent heating rate in a previously developed solar-wind model that agreed with a number of observational constraints, consistent with the hypothesis that MHD turbulence accounts for much of the heating of the fast solar wind.

Turbulence is commonly observed in nearly collisionless heliospheric plasmas, including the solar wind and corona and the Earth’s magnetosphere. Understanding the collisionless mechanisms responsible for the energy transfer from the turbulent fluctuations to the particles is a frontier in kinetic turbulence research. Collisionless energy transfer from the turbulence to the particles can take place reversibly, resulting in non-thermal energy in the particle velocity distribution functions (VDFs) before eventual collisional thermalization is realized. Exploiting the information contained in the fluctuations in the VDFs is valuable. Here we apply a recently developed method based on VDFs, the field–particle correlation technique, to a $\unicode[STIX]{x1D6FD}=1$, solar-wind-like, low-frequency Alfvénic turbulence simulation with well-resolved phase space to identify the field–particle energy transfer in velocity space. The field–particle correlations reveal that the energy transfer, mediated by the parallel electric field, results in significant structuring of the VDF in the direction parallel to the magnetic field. Fourier modes representing the length scales between the ion and electron gyroradii show that energy transfer is resonant in nature, localized in velocity space to the Landau resonances for each Fourier mode. The energy transfer closely follows the Landau resonant velocities with varying perpendicular wavenumber $k_{\bot }$ and plasma $\unicode[STIX]{x1D6FD}$. This resonant signature, consistent with Landau damping, is observed in all diagnosed Fourier modes that cover the dissipation range of the simulation.

In toroidal magnetic confinement devices, such as tokamaks and stellarators, neoclassical transport is usually an order of magnitude larger than its classical counterpart. However, when a high-collisionality species is present in a stellarator optimized for low Pfirsch–Schlüter current, its classical transport can be comparable to the neoclassical transport. In this letter, we compare neoclassical and classical fluxes and transport coefficients calculated for Wendelstein 7-X (W7-X) and Large Helical Device (LHD) cases. In W7-X, we find that the classical transport of a collisional impurity is comparable to the neoclassical transport for all radii, while it is negligible in the LHD cases, except in the vicinity of radii where the neoclassical transport changes sign. In the LHD case, electrostatic potential variations on the flux surface significantly enhance the neoclassical impurity transport, while the classical transport is largely insensitive to this effect in the cases studied.

The work described in this article was carried out to investigate how permanent magnets (PM) affect the plasma confinement and ion beam properties in an inductively coupled plasma which expands from a helicon source. The cylindrical plasma device Njord has a 13 cm long and 20 cm wide stainless steel port connecting the source chamber and the diffusion chamber. The source chamber has an axial magnetic field produced by two coils, with magnetic field lines expanding into the diffusion chamber. Simulations have shown that the field lines leaving the edge of the source hit the port wall, causing a loss of electrons in this section. In the experiments performed in this work, PMs were added around the port walls near the exit of a plasma source and the effect was investigated experimentally by means of a retarding field energy analyser probe. The plasma potential, ion density and ion beam parameters were estimated, and the results with and without the PMs were compared. The results showed that the plasma density in the centre can in some cases be doubled, and the density at the edges of the plasma increased significantly with PMs in place. Although the plasma potential was slightly affected, and the beam velocity dropped by ${\sim}$10 %, the ion beam flux increased by a factor of 1.5.

The use of radio frequency (RF) waves in fusion plasmas for heating, for non-inductive current generation, for profile control and for diagnostics has been well established. The RF waves, excited by antenna structures placed near the wall of a fusion device, have to propagate through density fluctuations at the plasma edge. These fluctuations can modify the properties of the RF waves that propagate towards the core of the plasma. A full-wave electromagnetic computational code ScaRF based on the finite difference frequency domain (FDFD) method has been developed to study the effect of density turbulence on RF waves. The anisotropic plasma permittivity used in the scattering studies is that for a magnetized, cold plasma. The code is used to study the propagation of an RF plane wave through a modulated, spatially periodic density interface. Such an interface could arise in the edge region due to magnetohydrodynamic instability or drift waves. The frequency of the plane wave is taken to be in the range of the electron cyclotron frequency. The scattering analysis is applicable to ITER-like plasmas, as well as to plasmas in medium sized tokamaks such as TCV, ASDEX-U and DIII-D. The effect of different density contrasts across the interface and of different spatial modulations are discussed. While ScaRF is used to study a periodic density fluctuation, the code is general enough to include different varieties of density fluctuations in the edge region – such as blobs and filaments, and spatially random fluctuations.

We study the damping of collisionless Alfvénic turbulence in a strongly magnetised plasma by two mechanisms: stochastic heating (whose efficiency depends on the local turbulence amplitude $\unicode[STIX]{x1D6FF}z_{\unicode[STIX]{x1D706}}$) and linear Landau damping (whose efficiency is independent of $\unicode[STIX]{x1D6FF}z_{\unicode[STIX]{x1D706}}$), describing in detail how they affect and are affected by intermittency. The overall efficiency of linear Landau damping is not affected by intermittency in critically balanced turbulence, while stochastic heating is much more efficient in the presence of intermittent turbulence. Moreover, stochastic heating leads to a drop in the scale-dependent kurtosis over a narrow range of scales around the ion gyroscale.

Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle $\unicode[STIX]{x1D703}=0$) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation $|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$, where $z=\unicode[STIX]{x1D714}/k_{\Vert }c$ is the phase speed and $\unicode[STIX]{x1D6FD}_{\text{A}}c$ is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function, $z^{2}W(z)$, which is negative for $|z|<z_{0}$ and has a sharp maximum at $|z|=z_{\text{m}}$, with $1-z_{\text{m}}<1-z_{0}\ll 1$. We give numerical estimates for the maximum of $z^{2}W(z)$ and for $z_{\text{m}}$ and $z_{0}$ for a one-dimensional Jüttner distribution. The L and A modes reconnect, for $z_{\text{A}}>z_{0}$, to form the O and Alfvén modes for oblique propagation ($\unicode[STIX]{x1D703}\neq 0$). For $z_{\text{A}}<z_{0}$ the Alfvén and O mode curves reconnect forming a new mode that exists only for $\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$. The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around $\unicode[STIX]{x1D703}\approx 0$. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.

The shape gradient quantifies the change in some figure of merit resulting from differential perturbations to a shape. Shape gradients can be applied to gradient-based optimization, sensitivity analysis and tolerance calculation. An efficient method for computing the shape gradient for toroidal three-dimensional magnetohydrodynamic (MHD) equilibria is presented. The method is based on the self-adjoint property of the equations for driven perturbations of MHD equilibria and is similar to the Onsager symmetry of transport coefficients. Two versions of the shape gradient are considered. One describes the change in a figure of merit due to an arbitrary displacement of the outer flux surface; the other describes the change in the figure of merit due to the displacement of a coil. The method is implemented for several example figures of merit and compared with direct calculation of the shape gradient. In these examples the adjoint method reduces the number of equilibrium computations by factors of $O(N)$, where $N$ is the number of parameters used to describe the outer flux surface or coil shapes.

Kinetic treatments of drift tearing modes that match an inner, resonant layer solution to an external magnetohydrodynamic (MHD) solution, characterised by $\unicode[STIX]{x1D6E5}^{\prime }$, can fail to match the ideal MHD boundary condition on the parallel electric field, $E_{\Vert }=0$. In this paper we demonstrate how consideration of ion sound and ion Landau damping effects achieves this, placing the theory on a firm footing. These effects are found to modify the effective critical $\unicode[STIX]{x1D6E5}^{\prime }$ for instability of drift tearing modes, in particular for weak electron temperature gradients. The implications for a realistic hot plasma resonant layer model – involving large ion Larmor radius and semi-collisional electron physics (Connor et al., Plasma Phys. Control. Fusion, vol. 54, 2012, 035003) – are determined.

Magnetic helicity flux gives information about the topology of a magnetic field passing through a boundary. In solar physics applications, this boundary is the photosphere and magnetic helicity flux has become an important quantity in analysing magnetic fields emerging into the solar atmosphere. In this work we investigate the evolution of magnetic helicity flux in magnetohydrodynamic (MHD) simulations of solar flux emergence. We consider emerging magnetic fields with different topologies and investigate how the magnetic helicity flux patterns correspond to the dynamics of emergence. To investigate how the helicity input is connected to the emergence process, we consider two forms of the helicity flux. The first is the standard form giving topological information weighted by magnetic flux. The second form represents the net winding and can be interpreted as the standard helicity flux less the magnetic flux. Both quantities provide important and distinct information about the structure of the emerging field and these quantities differ significantly for mixed sign helicity fields. A novel aspect of this study is that we account for the varying morphology of the photosphere due to the motion of the dense plasma lifted into the chromosphere. Our results will prove useful for the interpretation of magnetic helicity flux maps in solar observations.

In a magnetized, collisionless plasma, the magnetic moment of the constituent particles is an adiabatic invariant. An increase in the magnetic-field strength in such a plasma thus leads to an increase in the thermal pressure perpendicular to the field lines. Above a $\unicode[STIX]{x1D6FD}$-dependent threshold (where $\unicode[STIX]{x1D6FD}$ is the ratio of thermal to magnetic pressure), this pressure anisotropy drives the mirror instability, producing strong distortions in the field lines on ion-Larmor scales. The impact of this instability on magnetic reconnection is investigated using a simple analytical model for the formation of a current sheet (CS) and the associated production of pressure anisotropy. The difficulty in maintaining an isotropic, Maxwellian particle distribution during the formation and subsequent thinning of a CS in a collisionless plasma, coupled with the low threshold for the mirror instability in a high-$\unicode[STIX]{x1D6FD}$ plasma, imply that the geometry of reconnecting magnetic fields can differ radically from the standard Harris-sheet profile often used in simulations of collisionless reconnection. As a result, depending on the rate of CS formation and the initial CS thickness, tearing modes whose growth rates and wavenumbers are boosted by this difference may disrupt the mirror-infested CS before standard tearing modes can develop. A quantitative theory is developed to illustrate this process, which may find application in the tearing-mediated disruption of kinetic magnetorotational ‘channel’ modes.

We present a complete analysis of all wave modes in a cold pair plasma, significantly extending standard textbook treatments. Instead of identifying the maximal number of two propagating waves at fixed frequency $\unicode[STIX]{x1D714}$, we introduce a unique labelling of all 5 mode pairs described by the general dispersion relation $\unicode[STIX]{x1D714}(k)$, starting from their natural ordering at small wavenumber $k$. There, the 5 pairs start off as Alfvén (A), fast magnetosonic (F), modified electrostatic (M) and electromagnetic O and X branches, and each $\unicode[STIX]{x1D714}(k)$ branch smoothly connects to large wavenumber resonances or limits. For cold pair plasmas, these 5 branches show avoided crossings, which become true crossings at exactly parallel or perpendicular orientation. Only for those orientations, we find a changed connectivity between small and large wavenumber behaviour. Analysing phase and group diagrams for all 5 wave modes, distinctly different from the Clemmow–Mullaly–Allis representation, reveals the true anisotropy of the A, M and O branches.

This tutorial covers validation of gyrokinetic turbulent-transport models via comparison of measured turbulence with realistic simulations of fusion plasmas. It presents a brief history of validation of gyrokinetic simulations, the principal challenges encountered, a limited survey of common turbulence diagnostics used on tokamaks and stellarators, an overview of the fundamentals of synthetic diagnostic models and a discussion of frontiers in turbulent-transport model validation.

We study the resistive evolution of a localized self-organizing magnetohydrodynamic equilibrium. In this configuration the magnetic forces are balanced by a pressure force caused by a toroidal depression in the pressure. Equilibrium is attained when this low-pressure region prevents further expansion into the higher-pressure external plasma. We find that, for the parameters investigated, the resistive evolution of the structures follows a universal pattern when rescaled to resistive time. The finite resistivity causes both a decrease in the magnetic field strength and a finite slip of the plasma fluid against the static equilibrium. This slip is caused by a Pfirsch–Schlüter-type diffusion, similar to what is seen in tokamak equilibria. The net effect is that the configuration remains in magnetostatic equilibrium whilst it slowly grows in size. The rotational transform of the structure becomes nearly constant throughout the entire structure, and decreases according to a power law. In simulations this equilibrium is observed when highly tangled field lines relax in a high-pressure (relative to the magnetic field strength) environment, a situation that occurs when the twisted field of a coronal loop is ejected into the interplanetary solar wind. In this paper we relate this localized magnetohydrodynamic equilibrium to magnetic clouds in the solar wind.

The formation of zonal flows from inhomogeneous drift-wave (DW) turbulence is often described using statistical theories derived within the quasilinear approximation. However, this approximation neglects wave–wave collisions. Hence, some important effects such as the Batchelor–Kraichnan inverse-energy cascade are not captured within this approach. Here we derive a wave kinetic equation that includes a DW collision operator in the presence of zonal flows. Our derivation makes use of the Weyl calculus, the quasinormal statistical closure and the geometrical-optics approximation. The obtained model conserves both the total enstrophy and energy of the system. The derived DW collision operator breaks down at the Rayleigh–Kuo threshold. This threshold is missed by homogeneous-turbulence theory but expected from a full-wave quasilinear analysis. In the future, this theory might help better understand the interactions between drift waves and zonal flows, including the validity domain of the quasilinear approximation that is commonly used in the literature.

The confinement of the guiding-centre trajectories in a stellarator is determined by the variation of the magnetic field strength $B$ in Boozer coordinates $(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$, but $B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ depends on the flux surface shape in a complicated way. Here we derive equations relating $B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ in Boozer coordinates and the rotational transform to the shape of flux surfaces in cylindrical coordinates, using an expansion in distance from the magnetic axis. A related expansion was done by Garren and Boozer (Phys. Fluids B, vol. 3, 1991a, 2805) based on the Frenet–Serret frame, which can be discontinuous anywhere the magnetic axis is straight, a situation that occurs in the interesting case of omnigenity with poloidally closed $B$ contours. Our calculation in contrast does not use the Frenet–Serret frame. The transformation between the Garren–Boozer approach and cylindrical coordinates is derived, and the two approaches are shown to be equivalent if the axis curvature does not vanish. The expressions derived here help enable optimized plasma shapes to be constructed that can be provided as input to VMEC and other stellarator codes, or to generate initial configurations for conventional stellarator optimization.

We report the observation of a nonlinear wave packet propagating through a relaxed Taylor state in the Swarthmore Spheromak eXperiment (SSX) device. The wave packet is launched by a fast, pulsed, high current (${\approx}21~\text{kA}$) single-turn theta-pinch coil mounted outside the plasma vessel. The theta-pinch coil is energized by discharging a 40 kV, 2 kJ capacitor circuit. The wave packet velocity is super-thermal and super-Alfvénic; its group velocity is more consistent with a whistler pulse than other characteristic velocities. We also observe a fast density pulse which indicates that it is not Alfvénic in nature.

In tokamak plasmas, sheared flows perpendicular to the driving temperature gradients can strongly stabilise linear modes. While the system is linearly stable, regimes with persistent nonlinear turbulence may develop, i.e. the system is subcritical. A perturbation with small but finite amplitude may be sufficient to push the plasma into a regime where nonlinear effects are dominant and thus allow sustained turbulence. The minimum threshold for nonlinear instability to be triggered provides a criterion for assessing whether a tokamak is likely to stay in the quiescent (laminar) regime. At the critical amplitude, instead of transitioning to the turbulent regime or decaying to a laminar state, the trajectory will map out the edge of chaos. Surprisingly, a quasi-travelling-wave solution is found as an attractor on this edge manifold. This simple advecting solution is qualitatively similar to, but simpler than, the avalanche-like bursts seen in earlier turbulent simulations and provides an insight into how turbulence is sustained in subcritical plasma systems. For large flow shearing rate, the system is only convectively unstable, and given a localised initial perturbation, will eventually return to a laminar state at a fixed spatial location.