Understanding magnetised laser–plasma interactions is important for controlling magneto-inertial fusion experiments and developing magnetically assisted radiation and particle sources. For nanosecond pulses at non-relativistic intensities, interactions are dominated by coherent three-wave interactions, whose nonlinear coupling coefficients became known only recently when waves propagate at oblique angles with the magnetic field. In this paper, backscattering coupling coefficients predicted by warm-fluid theory are benchmarked using particle-in-cell simulations in one spatial dimension, and excellent agreements are found for a wide range of plasma temperatures, magnetic field strengths and laser propagation angles, when the interactions are mediated by electron-dominant hybrid waves. Systematic comparisons between theory and simulations are made possible by a rigorous protocol. On the theory side, the initial boundary value problem of linearised three-wave equations is solved, and the transient-time solutions allow the effects of growth and damping to be distinguished. On the simulation side, parameters are carefully chosen and calibration runs are performed to ensure that comparisons are well controlled. Fitting simulation data to analytical solutions yields numerical growth rates that match theory predictions within error bars. Although warm-fluid theory is found to be valid for a wide parameter range, genuine kinetic effects have also been observed.

The theoretical investigation of relevant turbulent transport mechanisms in H-mode pedestals is a great scientific and numerical challenge. In this study, we address this challenge by global, nonlinear gyrokinetic simulations of a full pedestal up to the separatrix, supported by a detailed characterisation of gyrokinetic instabilities from just inside the pedestal top to the pedestal centre and foot. We present ASDEX Upgrade pedestal simulations using an upgraded version of the gyrokinetic, Eulerian, delta-f code GENE (genecode.org) that enables stable global simulations at experimental plasma $\beta$ values. The turbulent transport is found to exhibit a multi-channel, multi-scale character throughout the pedestal with the dominant contribution transitioning from ion-scale trapped electron modes/micro-tearing modes at the pedestal top to electron-scale electron temperature gradient modes in the steep gradient region. Consequently, the turbulent electron heat flux changes from ion to electron scales and the ion heat flux reduces to almost neoclassic values in the pedestal centre. $E\times B$ shear is found to strongly reduce heat flux levels in all channels (electron, ion, electrostatic, electromagnetic) and the interplay of magnetic shear and pressure gradient is found to locally stabilise ion-scale instabilities.

This paper focuses on the wave inverse cascade instability analysis with self-regulating feedback control for a fixed external potential field and a highly localized finite-amplitude initial pulse. The wave inverse cascade instability analysis is carried out by solving the corresponding two-dimensional generalized nonlinear Schrödinger equation. The wave field firstly suffers from the modulation instability, followed by collapse into turbulence containing the shortest-wavelength modes in the system. This is followed by inverse cascade of the shortest wavelength modes back to the longer-wavelength ones, until a statistical stationary turbulent state is reached. It is found that the inverse cascade is limited to the shorter-wavelength modes with the wavenumber $\left |k\right |\geq 100$. This shows that the viscous damping $p_i$ acts like a control switch to the inverse cascade, and the feedback control can also regulate the intensity of the inverse cascade mode.

The structure of a collisionless shock affects ion motion in the shock front and is affected by the formed ion distribution. In high-Mach-number shocks, a significant fraction of incident ions are reflected by the macroscopic electric and magnetic fields in the shock front. Ions are non-specularly reflected by the combined electric deceleration and magnetic deflection. Here, a first analytical description of the non-specular reflection is presented. The contribution of the increasing magnetic field is evaluated and shown to enhance reflection. The distribution of non-specularly reflected ions ahead of the ramp is calculated and their velocities at the re-entry to the shock are found numerically. Dependence on the angle between the shock normal and the upstream magnetic field vector is illustrated.

Pellet injection is used for fuelling and controlling discharges in tokamaks, and it is foreseen in ITER. During pellet injection, a movement of the ablated material towards the low-field side (or outward major radius direction) occurs because of the inhomogeneity of the magnetic field. Due to the complexity of the theoretical models, computer codes developed to simulate the cross-field drift are computationally expensive. Here, we present a one-dimensional semi-analytical model for the radial displacement of ablated material after pellet injection, taking into account both the Alfvén and ohmic currents which shortcircuit the charge separation creating the drift. The model is suitable for rapid calculation of the radial drift displacement, and can be useful for e.g. modelling of disruption mitigation via pellet injection.

If the electrons in a plasma are suddenly heated, the resulting change in Debye shielding causes the ion kinetic energy to quickly increase. For the first time, this correlation heating, which is much faster than collisional energy exchange, is rigorously derived for a moderately coupled, electron–ion plasma. The electron–ion mass ratio is taken to be the smallest parameter in the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, smaller even than the reciprocal of the plasma parameter. This ordering differs from conventional kinetic theory by making the electron collision rates faster than the ion plasma frequency, which allows stronger coupling and makes the ion heating a function only of the total energy supplied to the electrons. The calculation uses known formulae for correlations in a two-temperature plasma, for which a new, elementary derivation is presented. Suprathermal ions may be created more rapidly by this mechanism than by ion–electron Coulomb collisions. This means that the use of a femtosecond laser pulse could potentially help to achieve ignition in certain fast ignition approaches to inertial confinement fusion.

We show that the saturation of resistive tearing modes in a cylindrical tokamak, as well as the corresponding island width, can be directly calculated with a magnetohydrodynamics (MHD) equilibrium code without solving the dynamics and without considering resistivity. The results are compared with initial-value resistive MHD simulations and to an analytical nonlinear theory. For small enough islands, the agreement is remarkable. For sufficiently large islands, the equilibrium calculations, which assume a flat current profile inside the island, overestimate the saturation amplitude. On the other hand, excellent agreement between nonlinear resistive MHD simulations and nonlinear theory is observed for all the considered tearing unstable equilibria.

The inclusion of kinetic effects into fluid models has been a long standing problem in magnetic reconnection and plasma physics. Generally, the pressure tensor is reduced to a scalar which is an approximation used to aid in the modelling of large scale global systems such as the Earth's magnetosphere. This unfortunately omits important kinetic physics which have been shown to play a crucial role in collisionless regimes. The multi-fluid ten-moment model, however, retains the full symmetric pressure tensor. The ten-moment model is constructed by taking moments of the Vlasov equation up to second order, and includes the scalar density, the vector bulk-flow and the symmetric pressure tensor for a total of ten separate components. Use of the multi-fluid ten-moment model requires a closure which truncates the cascading system of equations. Here we look to leverage data-driven methodologies to seek a closure which may improve the physical fidelity of the ten-moment multi-fluid model in collisionless regimes. Specifically, we use the sparse identification of nonlinear dynamics (SINDy) method for symbolic equation discovery to seek the truncating closure from fully kinetic particle-in-cell simulation data, which inherently retains the relevant kinetic physics. We verify our method by reproducing the ten-moment model from the particle-in-cell (PIC) data and use the method to generate a closure truncating the ten-moment model which is analysed through the nonlinear phase of reconnection.

In decaying magnetohydrodynamic (MHD) turbulence with a strong magnetic field, the spectral magnetic energy density is known to increase with time at small wavenumbers $k$, provided the spectrum at low $k$ is sufficiently steep. This process is called inverse cascading and occurs for an initial Batchelor spectrum, where the magnetic energy per linear wavenumber interval increases like $k^4$. For an initial Saffman spectrum that is proportional to $k^2$, however, inverse cascading has not been found in the past. We study here the case of an intermediate $k^3$ spectrum, which may be relevant for magnetogenesis in the early Universe during the electroweak epoch. This case is not well understood in view of the standard Taylor expansion of the magnetic energy spectrum for small $k$. Using high resolution MHD simulations, we show that, also in this case, there is inverse cascading with a strength just as expected from the conservation of the Hosking integral, which governs the decay of an initial Batchelor spectrum. Even for shallower $k^\alpha$ spectra with spectral index $\alpha >3/2$, our simulations suggest a spectral increase at small $k$ with time $t$ proportional to $t^{4\alpha /9-2/3}$. The critical spectral index of $\alpha =3/2$ is related to the slope of the spectral envelope in the Hosking phenomenology. Our simulations with $2048^3$ mesh points now suggest inverse cascading even for an initial Saffman spectrum.

A new method named B-TIS (Bourgeois & Davoine, J. Comput. Phys., vol. 413, 2020, 109426) has recently been proposed for suppressing the influence of numerical Cherenkov radiation that appears in particle-in-cell (PIC) simulation of laser wakefield acceleration (LWFA). However, while this method provides good results when applied to the already accelerated electrons, we show here that it cannot model correctly most of the plasma electron bulk interacting with the laser field. We thus investigate in this paper the origins of this limitation and propose an improved method for which this limitation is removed. This new method, named B-TIS3, can now be applied to a much broader variety of problems and improve the performance in comparison with the standard PIC algorithm. We show that, for an electron interacting directly with a laser pulse, this new technique offers greater accuracy in terms of momentum and motion than the conventional scheme used in many PIC codes. These improvements translate into more faithful energy spectrum and electric charge for the accelerated beam in simulations of vacuum laser acceleration (VLA) or LWFA involving direct laser acceleration (DLA) at low plasma density. This new method, easy to implement and not computationally demanding, should then prove useful to study in depth and help develop novel VLA, DLA and LWFA techniques.

While it is important to design stellarators with high magnetohydrodynamic stability $\beta$-limit, it is also crucial to ensure that good magnetic surfaces exist in a large range of $\beta$ values. As $\beta$ increases, pressure-driven currents perturb the vacuum magnetic field and often lead to the emergence of magnetic field line chaos, which can worsen the confinement and is the cause of another kind of $\beta$-limit, the so-called equilibrium $\beta$-limit. In this paper, we explore numerically the dependence of the equilibrium $\beta$-limit on the bootstrap current strength in a classical stellarator geometry using the stepped pressure equilibrium code. We develop a diagnostic to determine whether or not magnetic islands are expected to participate significantly to radial transport, and we build an analytical model to predict the expected equilibrium $\beta$-limit, which recovers the main features of the numerical results. This research opens the possibility to include additional targets in stellarator optimization functions, provides additional understanding on the existence of magnetic surfaces at large $\beta$, and is a step forward in the understanding of the equilibrium $\beta$-limit.

We propose a theoretical scheme for the study of the nonlinear interaction of drift-wave-like turbulence and energetic particles in stellarators. The approach is based on gyrokinetics, and features a separation of time and scales, for electromagnetic fluctuations, inspired by linear ballooning theory. Two specific moments of the gyrokinetic equation constitute the main equations of the system, which requires a full kinetic nonlinear solution. This is found iteratively, expanding in the smallness of the bounce-average radial drift frequency, and nonlinear $\boldsymbol {E}\times \boldsymbol {B}$ drift frequency, compared with the inverse time scales of the resonantly interacting energetic particles. Our analysis is therefore valid for neoclassically optimised stellators. The resummation of all iterative and perturbative nonlinear kinetic solutions is discussed in terms of Feynman diagrams. Particular emphasis is put on the role of collisionlessly undamped large-scale structures in phase space, the kinetic equivalent of zonal flows, i.e. phase-space zonal structures, and on wave-like fluctuations generated by energetic particles.

In the present scheme of work, the Hermite–Gaussian (HG) laser beam dynamics has been investigated under the influence of an upward density ramp inside magnetized plasma, where both relativistic and ponderomotive nonlinearities are operative. One can achieve self-focusing of laser beam due to the change in the medium's dielectric function, which comes into operation due to the expulsion of plasma electrons from the high intensity to the low-intensity region by ponderomotive force and their motion at relativistic speeds. The dynamics of the laser beam and terahertz generation have been investigated by using the moment theory approach. It has been observed from the present analysis that the dynamics of the laser beam and the production of terahertz radiations strongly depends upon the HG laser beam and plasma parameters. In addition to this, the effect of density ramp and magnetic field has also been investigated on the efficiency of terahertz generation. It has been observed that higher-order modes of the HG laser beam play a dominant role in the production of terahertz radiations.

Both spin and orbital angular momentum can be exchanged between a rotating wave and a rotating magnetized plasma. Through resonances the spin and orbital angular momentum of the wave can be coupled to both the cyclotron rotation and the drift rotation of the particles. It is, however, shown that the Landau and cyclotron resonance conditions which classically describe resonant energy–momentum exchange between waves and particles are no longer valid in a rotating magnetized plasma column. In this case a new resonance condition which involves a resonant matching between the wave frequency, the cyclotron frequency modified by inertial effects and the harmonics of the guiding centre rotation is identified. A new quasilinear equation describing orbital and spin angular momentum exchanges through these new Brillouin resonances is then derived, and used to expose the wave-driven radial current responsible for angular momentum absorption.

Electron resonant interactions with electromagnetic whistler-mode waves play an important role in electron flux dynamics in various space plasma systems: planetary radiation belts, bow shocks, solar wind and magnetic reconnection regions. Two key wave characteristics determining the regime of wave–particle interactions are the wave intensity and the wave coherency. The classical quasi-linear diffusion approach describes well electron diffusion by incoherent and low-amplitude waves, whereas the nonlinear resonant models describe electron phase bunching and trapping by highly coherent intense waves. This study is devoted to the investigation of the regime of electron resonant interactions with incoherent but intense waves. Although this regime is characterized by electron diffusion, we show that diffusion rates scale linearly with the wave amplitude, $D\propto B_w$, in contrast to the quasi-linear diffusion scaling $D_{QL}\propto B_w^2$. Using observed wave amplitude distributions, we demonstrate that the quasi-linear diffusion model significantly overestimates electron scattering by incoherent, but intense whistler-mode waves. We discuss the results obtained in the context of simulations of long-term electron flux dynamics in space plasma systems.

We analyse the generation of kinetic instabilities and their effect on the energization of ions in non-relativistic, oblique collisionless shocks using a 3D-3V (three spatial with three velocity components) simulation by dHybridR, a hybrid particle-in-cell code. At sufficiently high Mach number, quasi-perpendicular and oblique shocks can experience rippling of the shock surface caused by kinetic instabilities arising from free energy in the ion velocity distribution due to the combination of the incoming ion beam and the population of ions reflected at the shock front. To understand the role of the ripple on particle energization, we devise a new instability isolation method to identify the unstable modes underlying the ripple and interpret the results in terms of the governing kinetic instability. We generate velocity-space signatures using the field–particle correlation technique to look at energy transfer in phase space from the isolated instability driving the shock ripple, providing a viewpoint on the different dynamics of distinct populations of ions in phase space. Together, the field–particle correlation technique and our new instability isolation method provide a unique viewpoint on the different dynamics of distinct populations of ions in phase space and allow us to completely characterize the energetics of the collisionless shock under investigation.

We present orbit classification schemes for use as fast metrics for fusion alpha particle losses implemented in the symplectic guiding-centre code SIMPLE. Two variants respectively based on conservation of the parallel adiabatic invariant, and topology of footprints in Poincaré sections are introduced. Like an existing approach based on the Minkowski fractal dimension, those methods estimate whether a guiding-centre orbit is regular and therefore expected to be confined for infinite time in the collisionless case, or chaotic, which might lead to its loss. Compared with the existing approach, the required orbit tracing time for the novel classifiers is shorter by at least an order of magnitude. This enables massive sampling of orbits across the whole phase space to identify regular and chaotic regions for the purpose stellarator optimization. Based on conservation of the perpendicular invariant, we demonstrate how extended regular regions may act as radial barriers for orbits from the chaotic regions on the radially inboard side. We propose to use a quantified version of this property as a new metric for collisionless fusion alpha losses. As pitch-angle scattering becomes only relevant after alphas have already deposited a significant fraction of their energy, such a metric remains useful also for the case with collisions. This is illustrated by comparison with collisional loss computations. Results are presented for applications to two quasi-isodynamic configurations, a quasi-helical configuration and two quasi-axisymmetric configurations. In addition, the Hamiltonian action-angle formalism is used in quasi-axisymmetric configurations to investigate the overlap of drift-orbit resonances leading to chaos. The respective analysis is performed with the NEO-RT code originally developed for investigation of neoclassical toroidal viscous torque in tokamak plasmas.

We present a quasi-linear treatment of the drift-kinetic equation in the presence of a stochastic magnetic field, which provides a self-contained description of particle, parallel momentum and heat transport. Explicit analytical expressions, which satisfy the Onsager reciprocal relations, are obtained by approximating the distribution function by a local shifted Maxwellian. This theory completes previous formulations (Harvey et al., Phys. Rev. Lett., vol. 47, 1981, p. 102) by including the momentum transport and by generalizing the derivation from the cylindrical tokamak configuration to an arbitrary cylindrical pinch. Application to the reversed field pinch provides satisfactory results.

To comprehensively study the physical properties of inductively coupled plasma (ICP), a finite element method (FEM) simulation model of ICP is developed using the well-established COMSOL software. To benchmark the validation of the FEM model, two key physical parameters, the electron density and the electron temperature of the ICP plasma, are precisely measured by the state-of-the-art laser Thomson scattering diagnostic approach. For low-pressure plasma such as ICP, the local pressure in the generator tube is difficult to measure directly. The local gas pressure in the ICP tube has been calibrated by comparing the experimental and simulation results of the maximum electron density. And on this basis, the electron density and electron temperature of ICP under the same gas pressure and absorbed power have been compared by experiments and simulations. The good agreement between the experimental and simulation data of these two key physical parameters fully verifies the validity of the ICP FEM simulation model. The experimental verification of the ICP FEM simulation model lays a foundation for further study of the distribution of various physical quantities and their variation with pressure and absorption power, which is beneficial for improving the level of ICP-related processes.

The formation time of the surrounding sheath of Langmuir probes in an ionospheric plasma has been studied to better understand the constraints this puts on the sampling frequency of a probe. A fully kinetic three-dimensional particle-in-cell model is used to simulate the temporal effects in the electron saturation region as the sheath forms. The stability of the probe current and the stability of the ion and electron density in the vicinity of the probe have been used to evaluate when the sheath was formed. Simulated results were compared with theoretical models and are in good agreement with the theoretical results. This shows that theoretical models can be used as guidance to estimate the formation time and to determine the sampling rate for a swept bias Langmuir system. Our results also show that the formation time is less affected by the plasma temperature and bias voltage as we move into the thick sheath regime, and will instead be determined by the plasma density. The presented results also show that applying a step function to the probe could be used to characterise ions species composition, or to estimate the ion density.