Landau damping is one of the cornerstones of plasma physics. Based on the initial-value approach adopted by Landau in his original derivation of Landau damping, we examine the solutions of the linear Vlasov–Poisson system for different equilibrium distribution functions f_0(v)$f_0(v)$, going beyond the traditional focus on the root with largest imaginary part and investigating the full set of roots that the dispersion relation of the system generally admits. Specifically, we provide analytical insights into the number and the structure of the roots for entire and meromorphic functions f_0(v)$f_0(v)$, such as Maxwellian and \kappa$\kappa$ distributions, we discuss the potential issues related to the redefinition of \partial{f}_0/\partial{v}$\partial{f}_0/\partial{v}$ as a complex variable function and we show how different sigmoids affect the root structure associated with non-meromorphic cut-off distribution functions. Finally, based on the comparison of the several root structures considered, we wonder if the multiple roots might hint at a deeper understanding of the Landau damping phenomenon.

This study investigates the impact of local thermal non-equilibrium on the stability analysis of partially ionized plasma within a porous medium. The plasma, heated from below, is enclosed by various combinations of bounding surfaces. Both nonlinear (via the energy method) and linear (utilizing the normal mode analysis method) analyses are performed. Eigenvalue problems for both analyses are formulated and solved using the Galerkin method. The study also explores the effects of compressibility, medium permeability and magnetic fields on system stability. The collisional frequency among plasma components and the thermal diffusivity ratio significantly influence energy decay. The results reveal that the Rayleigh–Darcy number is identical for both nonlinear and linear analyses, thus eliminating the possibility of a subcritical region and confirming global stability. The principle of exchange of stabilities is validated, indicating the absence of oscillatory convection modes. Medium permeability, heat-transfer coefficient and compressibility delay the onset of convection, demonstrating stabilizing effects. Conversely, the porosity-modified conductivity ratio hastens the convection process, indicating destabilizing effects. Rigid–rigid bounding surfaces are found to be thermally more stable for confining the partially ionized plasma. Additionally, the magnetic field exerts a stabilizing influence.

High-beta magnetised plasmas often exhibit anomalously structured temperature profiles, as seen from galaxy cluster observations and recent experiments. It is well known that when such plasmas are collisionless, temperature gradients along the magnetic field can excite whistler waves that efficiently scatter electrons to limit their heat transport. Only recently has it been shown that parallel temperature gradients can excite whistler waves also in collisional plasmas. Here, we develop a Wigner–Moyal theory for the collisional whistler instability starting from Braginskii-like fluid equations in a slab geometry. This formalism is necessary because, for a large region in parameter space, the fastest-growing whistler waves have wavelengths comparable to the background temperature gradients. We find additional damping terms in the expression for the instability growth rate involving inhomogeneous Nernst advection and resistivity. They (i) enable whistler waves to re-arrange the electron temperature profile via growth, propagation and subsequent dissipation, and (ii) allow non-constant temperature profiles to exist stably. For high-beta plasmas, the marginally stable solutions take the form of a temperature staircase along the magnetic field lines. The electron heat flux can also be suppressed by the Ettingshausen effect when the whistler intensity profile is sufficiently peaked and oriented opposite the background temperature gradient. This mechanism allows cold fronts without magnetic draping, might reduce parallel heat losses in inertial fusion experiments and generally demonstrates that whistler waves can regulate transport even in the collisional limit.

We investigate the global stability properties of an electron–positron pair plasma in the linear regime. The plasma is confined by the magnetic field of an infinitely long wire. This configuration is the large-aspect-ratio limit of the levitated dipole experiment of the APEX collaboration. The stability is governed by the diocotron mode and the interchange mode. The diocotron mode dominates in the case of a cold, non-neutral plasma. For specific density profiles we find analytic solutions. We derive a necessary condition for instability and find unstable solutions if the plasma forms a thin shell around the wire. Solutions for arbitrary density profiles with finite temperature are obtained numerically. We find that finite-temperature effects stabilise the diocotron mode. The interchange mode, on the other hand, dominates if the plasma is neutral and has a finite temperature. This mode becomes unstable for a steep-enough density gradient, that is aligned with the gradient of the magnetic field strength and is stabilised by the equilibrium E\times B$E\times B$ drift of a non-neutral plasma.

The change in direction of the wavevector and group velocity experienced by a wave refracted at the interface of an anisotropic medium in uniform linear motion are determined analytically. These transmission conditions, which are shown to be consistent with the generalised Snell’s law written in the laboratory frame, are then used to examine the effect of motion on waves incident on a magnetised plasma. For an incident wave in the plane perpendicular to the magnetic field the motion is observed to lead to non-negligible deviation of the low-frequency X-mode, as well as to non-symmetrical total reflection angles. These effects are shown to be further complicated when the magnetic field is in the plane formed by the incident wavevector and the medium’s velocity, as the anisotropy now competes with the motion-induced drag. Although obtained in simplified configurations, these results suggest that accounting for motion when modelling plasma wave trajectories could be important under certain conditions, calling for a more detailed quantification of the effect of motion in actual diagnostics and plasma control schemes.

This paper presents a theoretical model of plasma equilibrium in the diamagnetic confinement mode in an axisymmetric mirror device with neutral beam injection. The hot ionic component is described within the framework of the kinetic theory, since the Larmor radius of the injected ions appears to be comparable to or even larger than the characteristic scale of the magnetic field inhomogeneity. The electron drag of the hot ions is taken into account, while the angular scattering of the hot ions due to Coulomb collisions is neglected. The background warm plasma, on the contrary, is considered to be in local thermal equilibrium, i.e. has a Maxwellian distribution function and is described in terms of magnetohydrodynamics. The density of the hot ions is assumed to be negligible compared with that of the warm plasma. Both the conventional gas-dynamic loss and the non-adiabatic loss specific to the diamagnetic confinement mode are taken into account. In this work, we do not consider the effects of the warm plasma rotation as well as the inhomogeneity of the electrostatic potential. A self-consistent theoretical model of the plasma equilibrium is constructed. In the case of the cylindrical bubble, this model is reduced to a simpler one. The numerical solutions in the limit of a thin transition layer of the diamagnetic bubble are found. Examples of the equilibria corresponding to the gas-dynamic multiple-mirror trap device are considered.

The near-axis description of optimised stellarators, at second order in the expansion, provides important information about the field, both of physical and practical importance for stellarator optimisation. It, however, remains relatively underdeveloped for an important class of such stellarators, called quasi-isodynamic (QI). In this paper we develop the theoretical and numerical framework, applying the second-order omnigeneity conditions derived in Rodríguez & Plunk (2023), to make explicit construction of equilibrium solutions. We find that the case of QI stellarators calls for the careful treatment of continuity, smoothness and periodicity of the various functions involved, especially for so-called half-helicity fields, which feature prominently in existing QI designs. The numerical implementation of necessary elements is described, and several examples are constructed and quantitatively verified in detail. This work establishes a basis for further systematic exploration of the space of QI stellarators, and the development of both theoretical and practical tools to facilitate effective optimisation of QI stellarators.

Kinetic theory of particles near resonances is a current topic of discussion in plasma physics and astrophysics. We extend this discussion to the kinetic theory of the interaction between alpha particles (energetic particles predicted to exist in large quantities in next-generation fusion experiments) and a neoclassical tearing mode (NTM), a resistively-driven perturbation which sometimes exists in a tokamak. We develop a quasilinear treatment of the interaction between alpha particles and an NTM, showing why an NTM can be a source of significant passing alpha particle transport in tokamaks. The limitations on quasilinear theory constrain our theory's applicability to small amplitude NTMs, highlighting the importance of nonlinear studies.

This paper presents a novel approach for simulating plasma instabilities in tokamak plasmas using the piecewise field-aligned finite element method in combination with the particle-in-cell method. Our method traditionally aligns the computational grid, but defines the basis functions in piecewise field-aligned coordinates to avoid grid deformation while naturally representing the field-aligned mode structures. This scheme is formulated and implemented numerically. It also applies to the unstructured triangular meshes in principle. We have conducted linear benchmark tests, which agree well with previous results and traditional schemes. Furthermore, multiple-n$n$ simulations are also carried out as a proof of principle, demonstrating the efficiency of this scheme in nonlinear turbulence simulations within the framework of the finite element method.

The decay of a turbulent magnetic field is slower with helicity than without. Furthermore, the magnetic correlation length grows faster for a helical than a non-helical field. Both helical and non-helical decay laws involve conserved quantities: the mean magnetic helicity density and the Hosking integral. Using direct numerical simulations in a triply periodic domain, we show quantitatively that in the fractionally helical case the mean magnetic energy density and correlation length are approximately given by the maximum of the values for the purely helical and purely non-helical cases. The time of switchover from one to the other decay law can be obtained on dimensional grounds and is approximately given by I_{H}^{1/2}I_{M}^{-3/2}$I_{H}^{1/2}I_{M}^{-3/2}$, where I_{H}$I_{H}$ is the Hosking integral and I_{M}$I_{M}$ is the mean magnetic helicity density. An earlier approach based on the decay time is found to agree with our new result and suggests that the Hosking integral exceeds naive estimates by the square of the same resistivity-dependent factor by which also the turbulent decay time exceeds the Alfvén time. In the presence of an applied magnetic field, the mean magnetic helicity density is known to be not conserved, and we show that then also the Hosking integral is not conserved.

We analyse distributions of the spatial scales of coherent intermittent structures – current sheets – obtained from fully kinetic, two-dimensional simulations of relativistic turbulence in a collisionless pair plasma using unsupervised machine-learning data dissection. We find that the distribution functions of sheet length \ell$\ell$ (longest scale of the analysed structure in the direction perpendicular to the dominant guide field) and curvature r_c$r_c$ (radius of a circle fitted to the structures) can be well-approximated by power-law distributions, indicating self-similarity of the structures. The distribution for the sheet width w$w$ (shortest scale of the structure) peaks at the kinetic scales and decays exponentially at larger values. The data shows little or no correlation between w$w$ and \ell$\ell$, as expected from theoretical considerations. The typical r_c$r_c$ depends linearly on \ell$\ell$, which indicates that the sheets all have a similar curvature relative to their sizes. We find a weak correlation between r_c$r_c$ and w$w$. Our results can be used to inform realistic magnetohydrodynamic subgrid models for plasma turbulence in high-energy astrophysics.

It is common wisdom that collisionless shocks become non-planar and non-stationary at sufficiently high Mach numbers. Whatever the shock structure, the upstream and downstream fluxes of the mass, momentum and energy should be equal. At low Mach numbers, these conservation laws are satisfied when the shock front is planar and stationary. When this becomes impossible, inhomogeneity and time dependence, presumably in the form of rippling, develop. In this study, we show that the shock structure changes as a kind of ‘phase transition’ when the Mach number is increased while other shock parameters are kept constant.

A charged particle in a suitably strong magnetic field spirals along the field lines while slowly drifting transversely. This note provides a brief derivation of an effective Lagrangian formulation for the guiding-centre approximation that captures this dynamics without resolving the gyro motion. It also explains how the effective Lagrangian may, for special magnetic fields, admit a ‘quasi-symmetry’ which can give rise to a conserved quantity helpful for plasma confinement in fields lacking a geometric isometry. The aim of this note is to offer a pedagogical introduction and some perspectives on this well-established subject.

We investigate a model of turbulent magnetic reconnection introduced by (Higashimori, Yokoi and Hoshino 2013 Phys. Rev. Lett. 110, 255001) and show that the classic two-dimensional, steady-state Sweet–Parker and Petschek reconnection solutions are supported. We present evidence that these are the only two steady-state reconnection solutions, and we determine the criterion for their selection. Sweet–Parker reconnection occurs when there is no growth in turbulent energy, whereas Petschek reconnection occurs when the current density in the reconnecting current sheet is able to surpass a critical value, allowing for the growth of turbulent energy that creates the diffusion region. Further, we show that the Petschek solutions are self-similar, depending on the value of the turbulent time scale, and produce a universal steady reconnection rate. The self-consistent development of Petschek reconnection through turbulence, within the model, is an example of fast and steady magnetic reconnection without an explicit need for the collisionless terms in an extended Ohm’s law.

An ionized gas medium (plasma state) turns to a complex state of plasma or dusty plasma if micrometre- to submicrometre-sized solid dust particles are introduced in it. The dusty plasma medium exhibits fluid- as well as solid-like characteristics at different background plasma conditions. It supports various linear and nonlinear dynamical structures because of external perturbation and internal instabilities. The vortical or coherent structure in the dusty plasma medium is a kind of self-sustained dynamical structure that is formed either by instabilities or by external forcing. In this review article, the author discusses the past theoretical, experimental and computational investigations on vortical and coherent structures in unmagnetized and magnetized dusty plasmas. The possible mechanisms of the formation of vortices in a dust-grain medium are discussed in detail. The studies on the evolution of vortices and their correlation with turbulence are also reviewed.

Fully relativistic particle-in-cell (PIC) simulations are crucial for advancing our knowledge of plasma physics. Modern supercomputers based on graphics processing units (GPUs) offer the potential to perform PIC simulations of unprecedented scale, but require robust and feature-rich codes that can fully leverage their computational resources. In this work, this demand is addressed by adding GPU acceleration to the PIC code Osiris. An overview of the algorithm, which features a CUDA extension to the underlying Fortran architecture, is given. Detailed performance benchmarks for thermal plasmas are presented, which demonstrate excellent weak scaling on NERSC's Perlmutter supercomputer and high levels of absolute performance. The robustness of the code to model a variety of physical systems is demonstrated via simulations of Weibel filamentation and laser-wakefield acceleration run with dynamic load balancing. Finally, measurements and analysis of energy consumption are provided that indicate that the GPU algorithm is up to {\sim }${\sim }$14 times faster and \sim$\sim$7 times more energy efficient than the optimized CPU algorithm on a node-to-node basis. The described development addresses the PIC simulation community's computational demands both by contributing a robust and performant GPU-accelerated PIC code and by providing insight into efficient use of GPU hardware.

We provide a fundamentally new perspective on subcritical turbulence in plasmas, based on coherent structures, which are obtained and characterised via direct numerical solution. The domains where these coherent states exist appear to be closely connected to the those where related turbulent states can exist, so there may be a deep connection between the stability of these coherent structures and the domain where sustained turbulence is possible. In contrast to previous descriptions of turbulence in terms of a stochastic collection of linear waves, we present a fundamentally nonlinear representation based on more general classes of translating oscillatory nonlinear solutions. In turbulent tokamak plasmas, the transport can often be completely suppressed by introducing a background shear flow, whose amplitude is an important control parameter. As this parameter is decreased below a critical value, radially localised structures appear, becoming larger and more complex, in both gyrokinetic simulations and a simpler fluid model of the plasma. For the fluid model, we directly solve for a particular class of nonlinear solutions, relative periodic orbits, and determine their stability, thus explaining why these isolated structures appear in initial-value simulations. The increase of complexity as the flow shear is reduced is explained by a series of Hopf bifurcations of these nonlinear solutions, which we quantify via stability analysis. In gyrokinetic simulations, we are able to indirectly determine the underlying relative periodic orbits by imposing symmetry conditions on the simulations.

We construct a mean-field model that describes the nonlinear dynamics of a spin-polarized electron gas interacting with fixed, positively charged ions possessing a magnetic moment that evolves in time. The mobile electrons are modelled by a four-component distribution function in the two-dimensional phase space (x,v)$(x,v)$, obeying a Vlasov–Poisson set of equations. The ions are modelled by a Landau–Lifshitz equation for their spin density, which contains ion–ion and electron–ion magnetic exchange terms. We perform a linear response study of the coupled Vlasov–Poisson–Landau–Lifshitz (VPLL) equations for the case of a Maxwell–Boltzmann equilibrium, focussing in particular on the spin dispersion relation. Conditions of stability or instability for the spin modes are identified, which depend essentially on the electron spin polarization rate \eta$\eta$ and the electron–ion magnetic coupling constant K$K$. We also develop an Eulerian grid-based computational code for the fully nonlinear VPLL equations, based on the geometric Hamiltonian method first developed by Crouseilles et al. (J. Plasma Phys., vol. 89, no. 2, 2023, p. 905890215). This technique allows us to achieve great accuracy for the conserved quantities, such as the modulus of the ion spin vector and the total energy. Numerical tests in the linear regime are in accordance with the estimations of the linear response theory. For two-stream equilibria, we study the interplay of instabilities occurring in both the charge and the spin sectors. The set of parameters used in the simulations, with densities close to those of solids ({\approx }10^{29}\ {\rm m}^{-3}${\approx }10^{29}\ {\rm m}^{-3}$) and temperatures of the order of 10 eV, may be relevant to the warm dense matter regime appearing in some inertial fusion experiments.

Using the ONEDFEL code we perform free electron laser simulations in the astrophysically important guide-field dominated regime. For wigglers’ (Alfvén waves) wavelengths of tens of kilometres and beam Lorentz factor {\sim }10^3${\sim }10^3$, the resulting coherently emitted waves are in the centimetre range. Our simulations show a growth of the wave intensity over fourteen orders of magnitude, over the astrophysically relevant scale of approximately a few kilometres. The signal grows from noise (unseeded). The resulting spectrum shows fine spectral substructures, reminiscent of those observed in fast radio bursts.

In hydrodynamic (HD) turbulence, an exact decomposition of the energy flux across scales has been derived that identifies the contributions associated with vortex stretching and strain self-amplification (Johnson, Phys. Rev. Lett., vol. 124, 2020 104501; J. Fluid Mech., vol. 922, 2021, A3) to the energy flux across scales. Here, we extend this methodology to general coupled advection–diffusion equations and, in particular, to homogeneous magnetohydrodynamic (MHD) turbulence. We show that several MHD subfluxes are related to each other by kinematic constraints akin to the Betchov relation in HD. Applied to data from direct numerical simulations, this decomposition allows for an identification of physical processes and for the quantification of their respective contributions to the energy cascade, as well as a quantitative assessment of their multi-scale nature through a further decomposition into single- and multi-scale terms. We find that vortex stretching is strongly depleted in MHD compared with HD, and the kinetic energy is transferred from large to small scales almost exclusively by the generation of regions of small-scale intense strain induced by the Lorentz force. In regions of large strain, current sheets are stretched by large-scale straining motion into regions of magnetic shear. This magnetic shear in turn drives extensional flows at smaller scales. Magnetic energy is transferred from large to small scales predominantly by the aforementioned current-sheet thinning in regions of high strain. The contributions from current-filament stretching – the analogue to vortex stretching – and from bending of magnetic field-lines into current filaments by vortical motion are both almost negligible, although the latter induces strong backscatter of magnetic energy. Consequences of these results for subgrid-scale turbulence modelling are discussed.