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.

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.

The Van Kampen normal-mode method is applied in a comprehensive study of the linear wave perturbations of a homogeneous, magnetized and finite-temperature plasma, described by the collisionless Vlasov–Maxwell system in its non-relativistic version. The analysis considers a stable, Maxwellian background, but is otherwise completely general in that it allows for arbitrary wave propagation direction relative to the equilibrium magnetic field, multiple plasma species and general polarization states of the perturbed electromagnetic fields. A convenient formulation is introduced whereby the generator of the time advance is a Hermitian operator, analogous to the Hamiltonian in the Schrödinger equation of quantum mechanics. This guarantees a real frequency spectrum and complete bases of normal modes. Expansions in these normal-mode bases yield immediately the solutions of initial-value problems for general initial conditions. With standard initial conditions and propagation direction parallel to the equilibrium magnetic field, all the familiar results obtained following Landau’s Laplace transform approach are recovered. Considering such parallel propagation, the present work shows also explicitly and provides an example of how to construct special initial conditions that result in different, damped but otherwise arbitrarily prescribed time variations of the macroscopic variables. The known dispersion relations for perpendicular propagation are also recovered.

The significance of dust particles for the electron energy probability function (EEPF) and plasma oscillations is studied under varying magnetic field strength in a filamentary discharge hydrogen plasma. The experimental result shows that with an increase in dust density, the electron density decreases as a result of the charging of dust grains in the plasma background. A bi-Maxwellian EEPF is computed in both a pristine hydrogen plasma and a dust-containing plasma at different magnetic field strengths. We have observed that the increase in magnetic field decreases the lower energy electron population. The electron population of the lower energy range shows nearly identical results at magnetic field, $B\leqslant 3.7$ mT whereas the behaviour of the high-energy electron population becomes identical for a field strength $B\leqslant 5.8$ mT. From the observation, we have seen that the mid energy electron population slightly decreases and the high energy electron population slightly increases due to the presence of dust particles as compared to a pristine plasma. Further, very low energy electron population remains almost unchanged. With increase in dust density, the mid energy electron population further decreases whereas the high energy electron population slightly increases for different magnetic fields. But, no changes were observed for the very low energy electron population in the presence of dust particles. From the study of plasma oscillation, it is observed that the dominant frequency associated with the plasma oscillation is matched with the ion cyclotron frequency. The amplitude of the ion cyclotron frequency reduces with the increase of dust density which might be due to the decrease of plasma density.

We show that when a solid plasma foil with a density gradient on the front surface is irradiated by an intense laser pulse at a grazing angle, ${\sim}80^{\circ }$, a relativistic electron vortex is excited in the near-critical-density layer after the laser pulse depletion. The vortex structure and dynamics are studied using particle-in-cell simulations. Due to the asymmetry introduced by non-uniform background density, the vortex drifts at a constant velocity, typically $0.2{-}0.3$ times the speed of light. The strong magnetic field inside the vortex leads to significant charge separation; in the corresponding electric field initially stationary protons can be captured and accelerated to twice the velocity of the vortex (100–200 MeV). A representative scenario – with laser intensity of $10^{21}~\text{W}~\text{cm}^{-2}$ – is discussed: two-dimensional simulations suggest that a quasi-monoenergetic proton beam can be obtained with a mean energy 140 MeV and an energy spread of ${\sim}10\,\%$. We derive an analytical estimate for the vortex velocity in terms of laser and plasma parameters, demonstrating that the maximum proton energy can be controlled by the incidence angle of the laser and the plasma density gradient.

We have numerically studied how an actual confinement magnetostatic field affects power deposition in a helicon source. We have solved the wave propagation by means of two electromagnetic solvers, namely: (i) plaSma Padova Inhomogeneous Radial Electromagnetic solver (SPIREs), a mono-dimensional finite-difference frequency-domain code, and (ii) Advanced coDe for Anisotropic Media and ANTennas (ADAMANT), a full-wave three-dimensional tool based on the method of moments. We have computed the deposited power spectrum with SPIREs, power deposition profile with ADAMANT and the antenna impedance with both codes. First we have verified the numerical accuracy of both SPIREs and ADAMNT. Then, we have analysed two configurations of magnetostatic field, namely produced by Maxwell coils, and Helmholtz coils. For each configuration we have studied three cases: (i) low density $n=10^{17}~\text{m}^{-3}$ and low magnetic field $B_{0}=250$ G; (ii) medium density $n=10^{18}~\text{m}^{-3}$ and medium magnetic field $B_{0}=500$ G; (iii) high density $n=10^{19}~\text{m}^{-3}$ and high magnetic field $B_{0}=1000$ G. We have found that the Maxwell coil configuration does not produces significant changes in the deposited power phenomenon with respect to a perfectly uniform and axial magnetostatic field. While the Helmholtz coil configuration can lead to a power spectrum peaked near the axis of the discharge.

The study of nonlinear waves in water has a long history beginning with the seminal paper by Korteweg & de Vries (Phil. Mag., vol. 39, 1895, p. 240) and more recently for magnetohydrodynamics Danov & Ruderman (Fluid Dyn., vol. 18, 1983, pp. 751–756). The appearance of a Hilbert transform in the nonlinear equation for magnetohydrodynamics (MHD) distinguishes it from the water wave model description. In this paper, we are interested in examining weakly nonlinear interfacial waves in $2+1$ dimensions. First, we determine the wave solution in the linear case. Next, we derive the corresponding generalisation for the Kadomtsev–Petviashvili (KP) equation with the inclusion of an equilibrium magnetic field. The derived governing equation is a generalisation of the Benjamin–Ono (BO) equation called the Benjamin equation first derived in Benjamin (J. Fluid Mech., vol. 245, 1992, pp. 401–411) and in the higher-dimensional context in Kim & Akylas (J. Fluid Mech., vol. 557, 2006, pp. 237–256).

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.

The effects of coherent structures in plasma edge turbulence are an important topic in the study of plasma cross-field transport in magnetically confined plasmas. To properly characterize these structures, their temperature must be measured, which is often done by using Langmuir probes. In this work, the techniques of Langmuir sweep and triple probe are used, considering the effect of plasma sheath expansion in both methods. It is shown that if the sheath expansion is ignored, the triple probe technique gives overestimated temperature values. In addition, the conditional analysis is applied to both techniques in order to evaluate the mean temperature time trace of the coherent structures. It is shown that these time traces can be distorted in the case of the triple probe technique, due to fact that coherent structures destroy the homogeneity condition on the pins array. Therefore, the sweep technique with a conditional selection of points is more suitable than the triple probe to study coherent structures related to bursts, as it performs a local measurement.

We present an ideal two-fluid wave mode analysis for a pair plasma, extending an earlier study for cold conditions to the warm pair plasma case. Starting from the completely symmetrized means for writing the governing linearized equations in the pair fluid rest frame, we discuss the governing dispersion relation containing all six pairs of forward and backward propagating modes, which are conveniently labelled as S, A, F, M, O and X. These relate to the slow (S), Alfvén (A) and fast (F) magnetohydrodynamic waves, include a modified (M) electrostatic mode, as well as the electromagnetic O and X branches. In the dispersion relation, only two parameters appear, which define the pair plasma magnetization $E^{2}\in [0,\infty ]$ and the squared pair plasma sound speed $v^{2}$, measured in units of the light speed $c$. The description is valid also in the highly relativistic regime, where either a high magnetization and/or a relativistic temperature (hence sound speed) is reached. We recover the exact relativistic single-fluid magnetohydrodynamic expressions for the S, A and F families in the low wavenumber–frequency regime, which can be obtained for any choice of the equation of state. We argue that, as in a cold pair plasma, purely parallel or purely perpendicular propagation with respect to the magnetic field vector $\boldsymbol{B}$ is special, and near-parallel or near-perpendicular orientations demonstrate avoided crossings of branches at computable wavenumbers and frequencies. The complete six-mode phase and group diagram views are provided as well, visually demonstrating the intricate anisotropies in all wave modes, as well as their transformations. Analytic expressions for all six wave group speeds at both small and large wavenumbers complement the analysis.

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.