In the presence of a stationary gravitomagnetic field, a weakly rotating self-gravitating fluid relaxes into equilibrium flow configurations which admit both large- and small-scale structures. Unlike the traditional Beltrami equilibria in fluid, the equilibrium states in a rotating self-gravitating fluid in a gravitomagnetic field have structure similar to that of double curl Beltrami equilibria in plasma where the generalized vortical lines are aligned with the flow field. Different equilibrium flow configurations in the rotating fluid can be distinguished by the ratio between total energy and helicity. However, these fluid equilibria do not exhibit diamagnetic behaviours as observed in multi-species plasma equilibria.

Sufficient conditions are found on the ion velocity distribution f_i$f_i$ and potential amplitude for stability of steady electron holes moving at slow speeds, coinciding with the bulk of f_i$f_i$. Fully establishing stability requires calculation of the ion response to shift potential perturbations having an entire range of oscillatory frequencies, because under some conditions real frequencies intermediate between the ion and electron responses prove to be unstable even when the extremes are not. The mechanism of this overstability is explained and calculated in detail. Electron holes of peak potential \psi$\psi$ less than approximately 0.01 times the background temperature (\psi \lesssim 0.01T_0/e$\psi \lesssim 0.01T_0/e$) avoid the oscillatory instability entirely. For them, the necessary condition that there be a local minimum in f_i$f_i$ in which the hole resides is also sufficient, unless the magnetic field B$B$ is low enough to permit the transverse instability having finite wavenumber k$k$ perpendicular to B$B$.

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity ‘ideal’ Ohm's law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.

Propagation characteristics (propagation regions and cutoffs) of parallel propagating modes (Langmuir, right- and left-handed circularly polarized waves) are studied for relativistic, weakly relativistic and non-relativistic magnetized electron plasma using the kinetic model. The dispersion relation for parallel propagating modes in relativistic electron plasma is investigated by employing the Maxwell–Boltzmann–J üttner distribution function and the final dispersion relation obtained is more general since no approximation is used. As the integrals in the relativistic dispersion relation cannot be done analytically so these integrals have been solved with the numerical quadrature approach. For \eta \leq 1$\eta \leq 1$ (ratio of rest mass energy to thermal energy), the increase in the effective mass of electrons will result in a change in the mass-dependent quantities (plasma frequency, electron cyclotron frequency, electron sound velocity, etc.) which in turn significantly affect the propagation characteristics of parallel propagating modes. It is observed that the propagation region for these parallel propagating modes decreases and cutoff points are shifted to lower values when we consider a relativistic plasma environment. Moreover, a low-density and high-temperature plasma is more transparent as compared with a high-density and low-temperature plasma for these modes.

The paper presents experimental results from the SMOLA device that is the first facility with a helical mirror section of the magnetic system. This device was built in the Budker Institute of Nuclear Physics for the verification of the helical mirror confinement idea that is the technique of an active control of axial losses from a confinement zone. Theory predicts that, with rotating plasma, a helical mirror will provide suppression of the axial plasma flow and, simultaneously, density pinching to the axis. Experiments demonstrated the increase in plasma density in the entrance trap by a factor of 1.6 in the helical configuration. The integral axial flux from the transport section drops severalfold. The effective mirror ratio of the helical section was R_{eff} > 10$R_{eff} > 10$. Particle flux returning by the helical mirror section towards the confinement zone was observed. At high corrugation ratios, the axial flux direction is different at the magnetic axis and in the periphery of the plasma in the helical section. All axial fluxes scale linearly with the plasma density, even if the ion mean free path is comparable to the total length of the helical section. Good agreement of the experimental results with theoretical predictions is found.

Kelvin–Helmholtz (KH) instability plays a significant role in transport and mixing in various media such as hydrodynamic fluids, plasmas, geophysical flows and astrophysical situations. In this paper, we numerically explore this instability for a two-dimensional strongly coupled dusty plasma medium with rotational shear flows. We study this medium using a generalized hydrodynamic fluid model which treats it as a viscoelastic fluid. We consider the specific cases of rotating vorticity with abrupt radial profiles of rotation. In particular, single-circulation and multi-circulation vorticity shell profiles have been chosen. We observe the KH vortices at each circular interface between two relative rotating flows along with a pair of ingoing and outgoing wavefronts of transverse shear waves. Our studies show that due to the interplay between KH vortices and shear waves in the strongly coupled medium, the mixing and transport behaviour are much better than those of standard inviscid hydrodynamic fluids. In the interest of substantiating the mixing and transport behaviour, the generalized hydrodynamic fluid model is extended to include Lagrangian tracer particles. The numerical dispersion of these tracer particles in a flow provides an estimate of the diffusion in such a medium. We present the preliminary observations of tracer distribution (cluster formation) and diffusion (mean square displacement) across the medium.

We employ the \lambda ^{3}$\lambda ^{3}$ regime where a near-single-cycle laser pulse is tightly focused, thus providing the highest possible intensity for the minimal energy at a certain laser power. The quantum electrodynamics processes in the course of the interaction of an ultra-intense laser with a solid target are studied via three-dimensional particle-in-cell simulations, revealing the generation of copious \gamma$\gamma$-photons and electron–positron pairs. A parametric study of the laser polarisation, target thickness and electron number density shows that a radially polarised laser provides the optimal regime for \gamma$\gamma$-photon generation. By varying the laser power in the range of 1 to 300 PW we find the scaling of the laser to \gamma$\gamma$-photon energy conversion efficiency. The laser-generated \gamma$\gamma$-photon interaction with a high-Z$Z$ target is further studied using Monte Carlo simulations revealing further electron–positron pair generation and radioactive nuclide creation.

The long-distance stable transport of relativistic electron beams (REBs) in plasmas is studied by full three-dimensional particle-in-cell simulations. Theoretical analysis shows that the beam transport is mainly influenced by three transverse instabilities, where the excitation of self-modulation instability, and the suppression of the filamentation instability and the hosing instability are important to realize the beam stable transport. By modulating the transport parameters such as the electron density ratio, the relativistic Lorentz factor, the beam envelopes and the density profiles, the relativistic bunches having a smooth density profile and a length of several plasma wave periods can suppress the beam-plasma instabilities and propagate in plasmas for long distances with small energy losses. The results provide a reference for the research of long-distance and stable transport of REBs, and would be helpful for new particle beam diagnosis technology and space active experiments.

Adjoint methods can speed up stellarator optimisation by providing gradient information more efficiently compared with finite-difference evaluations. Adjoint methods are herein applied to vacuum magnetic fields, with objective functions targeting quasi-symmetry and a rotational transform value on a surface. To measure quasi-symmetry, a novel way of evaluating approximate flux coordinates on a single flux surface without the assumption of a neighbourhood of flux surfaces is proposed. The shape gradients obtained from the adjoint formalism are evaluated numerically and verified against finite-difference evaluations.

Given the large anisotropy of transport processes in magnetized plasmas, the magnetic field structure can strongly impact heat diffusion: magnetic surfaces and cantori form barriers to transport while chaotic layers and island structures can degrade confinement. When a small but non-zero amount of perpendicular diffusion is included, the structure of the magnetic field becomes less important, allowing pressure gradients to be supported across chaotic regions and island chains. We introduce a metric for the effective volume over which the local parallel diffusion dominates based on the solution to the anisotropic heat diffusion equation. To validate this metric, we consider model fields with a single island chain and a strongly chaotic layer for which analytic predictions of the relative parallel and perpendicular transport can be made. We also analyse critically chaotic fields produced from different sets of perturbations, highlighting the impact of the mode number spectrum on the heat transport. Our results indicate that this metric coincides with the effective volume of non-integrability in the limit \kappa _{\perp } \rightarrow 0$\kappa _{\perp } \rightarrow 0$, where \kappa_{\perp}$\kappa_{\perp}$ is the perpendicular diffusion coefficient. We propose that this metric be used to assess the impact of non-integrability on the heat transport in stellarator equilibria.

In shockwave theory, the density, velocity and pressure jumps are derived from the conservation equations. Here, we address the physics of a weak shock the other way around. We first show that the density profile of a weak shockwave in a fluid can be expressed as a sum of linear acoustic modes. The shock so built propagates at the speed of sound and matter is exactly conserved at the front crossing. Yet, momentum and energy are only conserved up to order 0 in powers of the shock amplitude. The density, velocity and pressure jumps are similar to those of a fluid shock, and an equivalent Mach number can be defined. A similar process is possible in magnetohydrodynamics. Yet, such a decomposition is found impossible for collisionless shocks due to the dispersive nature of ion acoustic waves. Weakly nonlinear corrections to their frequency do not solve the problem. Weak collisionless shocks could be inherently nonlinear, non-amenable to any linear superposition. Or they could be non-existent, as hinted by recent works.

Quasisymmetric stellarators are an attractive class of optimised magnetic confinement configurations. The property of quasisymmetry (QS) is in practice limited to be approximate, and thus the construction requires measures that quantify the deviation from the exact property. In this paper we study three measure candidates used in the literature, placing the focus on their origin and a comparison of their forms. The analysis shows clearly the lack of universality in these measures. As these metrics do not directly correspond to any physical property (except when exactly quasisymmetric), optimisation should employ additional physical metrics for guidance. It is suggested that close to QS minima, one should treat QS metrics through inequality constraints so that additional physics metrics dominate optimisation. The impact of different quasisymmetric measures on optimisation is presented through an example, for which the standard metric that weights the asymmetric Fourier modes of the field magnitude appears to perform best.

Collisionless shocks efficiently convert the energy of the directed ion flow into their thermal energy. Ion distributions change drastically at the magnetized shock crossing. Even in the absence of collisions, ion dynamics within the shock front is non-integrable and gyrophase dependent. The downstream distributions just behind the shock are not gyrotropic but become so quickly due to the kinematic gyrophase mixing even in laminar shocks. During the gyrotropization all information about gyrophases is lost. Here we develop a mapping of upstream and downstream gyrotropic distributions in terms of scattering probabilities at the shock front. An analytical expression for the probability is derived for directly transmitted ions in the narrow shock approximation. The dependence of the probability on the magnetic compression and the cross-shock potential is demonstrated.

The linear and nonlinear evolutions of the tearing instability in a collisionless plasma with a strong guide field are analysed on the basis of a two-field Hamiltonian gyrofluid model. The model is valid for a low ion temperature and a finite \beta _e$\beta _e$. The finite \beta _e$\beta _e$ effect implies a magnetic perturbation along the guide field direction, and electron finite Larmor radius effects. A Hamiltonian derivation of the model is presented. A new dispersion relation of the tearing instability is derived for the case \beta _e=0$\beta _e=0$ and tested against numerical simulations. For \beta _e \ll 1$\beta _e \ll 1$ the equilibrium electron temperature is seen to enhance the linear growth rate, whereas we observe a stabilizing role when electron finite Larmor radius effects become more relevant. In the nonlinear phase, stall phases and faster than exponential phases are observed, similarly to what occurs in the presence of ion finite Larmor radius effects. Energy transfers are analysed and the conservation laws associated with the Casimir invariants of the model are also discussed. Numerical simulations seem to indicate that finite \beta _e$\beta _e$ effects do not produce qualitative modifications in the structures of the Lagrangian invariants associated with Casimirs of the model.

In fusion devices, the geometry of the confining magnetic field has a significant impact on the instabilities that drive turbulent heat loss. This is especially true of stellarators, where the density-gradient-driven branch of the ‘trapped electron mode’ (TEM) is predicted to be linearly stable if the magnetic field has the maximum-J property, as is very approximately the case in certain magnetic configurations of the Wendelstein 7-X experiment (W7-X). Here we show, using both analytical theory and simulations, that the benefits of the optimisation of W7-X also serve to mitigate ion-temperature-gradient (ITG) modes as long as an electron density gradient is present. We find that the effect indeed carries over to nonlinear numerical simulations, where W7-X has low TEM-driven transport, and reduced ITG turbulence in the presence of a density gradient, giving theoretical support for the existence of enhanced confinement regimes, in the presence of strong density gradients (e.g. hydrogen pellet or neutral beam injection).

Using two-dimensional particle-in-cell simulations of collsionless plasmoid instability, the spatial and temporal variations of electric field components are investigated. The initial magnetic field is a Harris sheet which is added by a non-uniform perpendicular magnetic field (guide field). The effects of weak and medium guide field on the evolution of plasmoid instability and associated electric field are discussed with the input parameters of the solar corona with the condition \omega _{{\rm pe}}/\omega _{{\rm ce}}>1$\omega _{{\rm pe}}/\omega _{{\rm ce}}>1$. As the current sheet elongates, magnetic reconnection in an X-point transits into the plasmoid instability, in which secondary X-points are formed and the consequent magnetic islands coalesce and form larger islands. During the instability, electric field components show two types of disturbances. (1) Transient waves, being excited very early, propagate mainly perpendicular to the current sheet and become damped as the instability proceeds to the nonlinear stage. Fourier analysis reveals that electromagnetic right- and left-hand polarized modes parallel to the initial magnetic field and also the extraordinary mode perpendicular to the current sheet are excited. Additionally, electron Langmuir and slow magnetosonic waves are observed in both directions. (2) Inside the current sheet, an electric field is generated by the plasmoid-dominated reconnection and grows in amplitude as the plasmoid instability proceeds. Moreover, in our study, the effects of amplitude-varying guide field on the spatial structure of electric field components outside and inside the current sheet are discussed. This study is valuable because understanding the spatial and temporal variation of the electric field is an essential prerequisite for the issue of particle acceleration in plasmoid-dominated magnetic reconnection.

Magnetic reconnection, especially in the relativistic regime, provides an efficient mechanism for accelerating relativistic particles and thus offers an attractive physical explanation for non-thermal high-energy emission from various astrophysical sources. I present a simple analytical model that elucidates key physical processes responsible for reconnection-driven relativistic non-thermal particle acceleration in the large-system, plasmoid-dominated regime in two dimensions. The model aims to explain the numerically observed dependencies of the power-law index p$p$ and high-energy cutoff \gamma _c$\gamma _c$ of the resulting non-thermal particle energy spectrum f(\gamma )$f(\gamma )$ on the ambient plasma magnetization \sigma$\sigma$, and (for \gamma _c$\gamma _c$) on the system size L$L$. In this self-similar model, energetic particles are continuously accelerated by the out-of-plane reconnection electric field E_{\rm rec}$E_{\rm rec}$ until they become magnetized by the reconnected magnetic field and eventually trapped in plasmoids large enough to confine them. The model also includes diffusive Fermi acceleration by particle bouncing off rapidly moving plasmoids. I argue that the balance between electric acceleration and magnetization controls the power-law index, while trapping in plasmoids governs the cutoff, thus tying the particle energy spectrum to the plasmoid distribution.

We present a new method for formulating closures that learn from kinetic simulation data. We apply this method to phase mixing in a simple gyrokinetic turbulent system – temperature-gradient-driven turbulence in an unsheared slab. The closure, called the learned multi-mode (LMM) closure, is constructed by, first, extracting an optimal basis from a nonlinear kinetic simulation using singular value decomposition. Subsequent nonlinear fluid simulations are projected onto this basis and the results are used to formulate the closure. We compare the closure with other closures schemes over a broad range of the relevant two-dimensional parameter space (collisionality and gradient drive). We find that the turbulent kinetic system produces phase-mixing rates much lower than the linear expectations, which the LMM closure is capable of capturing. We also compare radial heat fluxes. A Hammett–Perkins closure, generalized to include collisional effects, is quite successful throughout the parameter space, producing {\sim }14\,\%${\sim }14\,\%$ root-mean-square (r.m.s.) error. The LMM closure is also very effective: when trained at three (two) points (in a 35 point parameter grid), the LMM closure produces 8\,\%$8\,\%$ (12\,\%$12\,\%$) r.m.s. errors. The LMM procedure can be readily generalized to other closure problems.

This paper discusses the toroidal localisation of magnetic reconnection during sawtooth crashes. Numerical analysis with realistic heat diffusion coefficients shows that heat distributes itself helically along the torus faster than the temporal resolution of any existing ECE diagnostics. It makes local and global (helically axisymmetric) magnetic reconnection indistinguishable for an observer, while a local crash where the heat stays confined within a finite helical region could be distinguished. Statistical analysis of sawtooth crashes with the ECEI diagnostic is conducted in ASDEX Upgrade. The displacement of the heat within a finite helical region has not been observed. The statistical data supports global magnetic reconnection.

For a magnetised plasma in a straight magnetic guide field, we derive a set of four-dimensional kinetic equations, which can capture electromagnetic turbulence in the drift kinetic limit. To do so, we start from the gyrokinetic equations, employ a Laguerre decomposition in the perpendicular velocity direction, retaining only the dominant gyroaverage contributions and only the first two Laguerre moments that source the electromagnetic fluctuations. The model conserves free energy, and can describe electromagnetic turbulence for a plasma at the transition between fluid and gyrokinetic regimes (k_\perp \rho _i\approx 1$k_\perp \rho _i\approx 1$ range of scales), as dominant finite-Larmor-radius (FLR) effects are considered. In addition to the three dimensions in positions space, we retain the parallel velocity dependence, which we describe via a Hermite representation. Employing this system, but without any other physics-based assumptions for the plasma species that can bias results, will allow us to investigate how fluid effects transition into the kinetic range, and analyse the interplay between spatial and velocity space mixing for electromagnetic plasma turbulence.