A Hamiltonian and action principle formalism for deriving three-dimensional gyroviscous magnetohydrodynamic models is presented. The uniqueness of the approach in constructing the gyroviscous tensor from first principles and its ability to explain the origin of the gyromap and the gyroviscous terms are highlighted. The procedure allows for the specification of free functions, which can be used to generate a wide range of gyroviscous models. Through the process of reduction, the noncanonical Hamiltonian bracket is obtained and briefly analysed.

The purpose of this paper is to bridge kinetic plasma descriptions and low-frequency single-fluid models. More specifically, the asymptotics leading to magnetohydrodynamic regimes starting from the Vlasov–Maxwell system are investigated. The analogy with the derivation, from the Vlasov–Poisson system, of a fluid representation for ions coupled to the Boltzmann relation for electrons is also outlined. The aim is to identify asymptotic parameters explaining the transitions from one microscopic description to a macroscopic low-frequency model. These investigations provide groundwork for the derivation of multi-scale numerical methods, model coupling or physics-based preconditioning.

The isotope dependence of plasma transport will have a significant impact on the performance of future D-T experiments in JET and ITER and eventually on the fusion gain and economics of future reactors. In preparation for future D-T operation on JET, dedicated experiments and comprehensive transport analyses were performed in H, D and H-D mixed plasmas. The analysis of the data has demonstrated an unexpectedly strong and favourable dependence of the global confinement of energy, momentum and particles in ELMy H-mode plasmas on the atomic mass of the main ion species, the energy confinement time scaling as ${\tau _E}\sim {A^{0.5}}$ (Maggi et al., Plasma Phys. Control. Fusion, vol. 60, 2018, 014045; JET Team, Nucl. Fusion, vol. 39, 1999, pp. 1227–1244), i.e. opposite to the expectations based only on local gyro-Bohm (GB) scaling, ${\tau _E}\sim {A^{ - 0.5}}$, and stronger than in the commonly used H-mode scaling for the energy confinement (Saibene et al., Nucl. Fusion, vol. 39, 1999, 1133; ITER Physics Basis, Nucl. Fusion, vol. 39, 1999, 2175). The scaling of momentum transport and particle confinement with isotope mass is very similar to that of energy transport. Nonlinear local GENE gyrokinetic analysis shows that the observed anti-GB heat flux is accounted for if collisions, E × B shear and plasma dilution with low-Z impurities (9Be) are included in the analysis (E and B are, respectively the electric and magnetic fields). For L-mode plasmas a weaker positive isotope scaling ${\tau _E}\sim {A^{0.14}}$ has been found in JET (Maggi et al., Plasma Phys. Control. Fusion, vol. 60, 2018, 014045), similar to ITER97-L scaling (Kaye et al., Nucl. Fusion, vol. 37, 1997, 1303). Flux-driven quasi-linear gyrofluid calculations using JETTO-TGLF in L-mode show that local GB scaling is not followed when stiff transport (as is generally the case for ion temperature gradient modes) is combined with an imposed boundary condition taken from the experiment, in this case predicting no isotope dependence. A dimensionless identity plasma pair in hydrogen and deuterium L-mode plasmas has demonstrated scale invariance, confirming that core transport physics is governed, as expected, by the 4 dimensionless parameters ρ*, ν*, β, q (normalised ion Larmor radius, collisionality, plasma pressure and safety factor) consistently with global quasi-linear gyrokinetic TGLF calculations (Maggi et al., Nucl. Fusion, vol. 59, 2019, 076028). We compare findings in JET with those in different devices and discuss the possible reasons for the different isotope scalings reported from different devices. The diversity of observations suggests that the differences may result not only from differences affecting the core, e.g. heating schemes, but are to a large part due to differences in device-specific edge and wall conditions, pointing to the importance of better understanding and controlling pedestal and edge processes.

It is demonstrated that finite-pressure, approximately quasi-axisymmetric stellarator equilibria can be directly constructed (without numerical optimization) via perturbations of given axisymmetric equilibria. The size of such perturbations is measured in two ways, via the fractional external rotation and, alternatively, via the relative magnetic field strength, i.e. the average size of the perturbed magnetic field, divided by the unperturbed field strength. It is found that significant fractional external rotational transform can be generated by quasi-axisymmetric perturbations, with a similar value of the relative field strength, despite the fact that the former scales more weakly with the perturbation size. High mode number perturbations are identified as a candidate for generating such transform with local current distributions. Implications for the development of a general non-perturbative solver for optimal stellarator equilibria are discussed.

We consider the tilting instability of a magnetically confined spheromak using three-dimensional magnetohydrodynamic and relativistic particle-in-cell calculations with an application to astrophysical plasmas, specifically those occurring in magnetar magnetospheres. The instability is driven by the counter-alignment of the spheromak's intrinsic magnetic dipole with the external magnetic field. Initially, the spheromak rotates – tilts – trying to lower its magnetic potential energy. As a result, a current sheet forms between the internal magnetic field of a spheromak and the confining field. Magnetic reconnection sets in; this leads to the annihilation of the newly counter-aligned magnetic flux of the spheromak. This occurs on a few Alfvén time scales. In the case of a higher-order (second-order) spheromak, the internal core is first pushed out of the envelope, resulting in formation of two nearly independent tilting spheromaks. Thus, the magnetically twisted outer shell cannot stabilize the inner core. During dissipation, helicity of the initial spheromak is carried away by torsional Alfvén waves, violating the assumptions of the Taylor relaxation theorem. In applications to magnetar giant flares, fast development of tilting instabilities and no stabilization of the higher-order spheromaks make it unlikely that trapped spheromaks are responsible for the tail emission lasting hundreds of seconds.

We apply field–particle correlations – a technique that tracks the time-averaged velocity-space structure of the energy density transfer rate between electromagnetic fields and plasma particles – to data drawn from a hybrid Vlasov–Maxwell simulation of Alfvén-ion cyclotron turbulence. Energy transfer in this system is expected to include both Landau and cyclotron wave–particle resonances, unlike previous systems to which the field–particle correlation technique has been applied. In this simulation, the energy transfer rate mediated by the parallel electric field $E_{\Vert }$ comprises approximately 60 % of the total rate, with the remainder mediated by the perpendicular electric field $E_{\bot }$. The parallel electric field resonantly couples to protons, with the canonical bipolar velocity-space signature of Landau damping identified at many points throughout the simulation. The energy transfer mediated by $E_{\bot }$ preferentially couples to particles with $v_{tp}\lesssim v_{\bot }\lesssim 3v_{tp}$, where $v_{tp}$ is the proton thermal speed, in agreement with the expected formation of a cyclotron diffusion plateau. Our results demonstrate clearly that the field–particle correlation technique can distinguish distinct channels of energy transfer using single-point measurements, even at points in which multiple channels act simultaneously, and can be used to determine quantitatively the rates of particle energization in each channel.

We present the elements required to construct two devices used in an undergraduate plasma physics laboratory. The materials and construction costs of the sources, the vacuum systems and probe drives and electrical circuits are presented in detail in the text and the first appendix. We also provide the software for probe motion and data acquisition as well as the electrical schematics for key components. Experiments which have been performed are listed and two (resonance cones and whistler waves) are described in greater detail. The machines are flexible and original research is possible.

Numerical simulations of Alfvén modes driven by energetic particles are performed with the gyrokinetic (GK) global particle-in-cell code ORB5. A reversed shear equilibrium magnetic field is adopted. A simplified configuration with circular flux surfaces and large aspect ratio is considered. The nonlinear saturation of beta-induced Alfvén eigenmodes (BAE) is investigated. The roles of the wave–particle nonlinearity of the different species, i.e. thermal ions, electrons and energetic ions are described, in particular for their role in the saturation of the BAE and the generation of zonal flows. The nonlinear redistribution of the electron population is found to be important in increasing the BAE saturation level and the zonal flow amplitude.

The possibility of performing electron density and temperature measurements in a high power helicon plasma is a crucial issue in the framework of the AWAKE (Advanced WAKefield Experiment) project, which demonstrates acceleration of particles using $\text{GeV}~\text{m}^{-1}$ electric fields in plasmas. For AWAKE, a helicon is currently envisaged as a candidate plasma source due to its capability for low electron and ion temperature, high electron density and production of an elongated plasma column. A plasma diagnostic to accurately determine the electron density in AWAKE regimes would be a valuable supporting tool. A demonstration Thomson scattering (TS) diagnostic was installed and successfully tested on the resonant antenna ion device (RAID) at the Swiss Plasma Center of Ecole Polytechnique Fédérale de Lausanne. RAID produces a helicon plasma column with characteristics similar to those of the AWAKE helicon source, and is therefore an optimal testbed for application to the AWAKE device. The spectrometer employed in RAID is based on polychromators which collect the light scattered by plasma electrons in spectrally filtered wavelength regions. Results from TS on RAID demonstrate conditions of electron density and temperature respectively of $n_{e}=1.10\,(\pm 0.19)\times 10^{19}~\text{m}^{-3}$ and $T_{e}=2.3\,(\pm 0.6)~\text{eV}$ in a steady-state discharge in an Ar plasma with 5 kW of RF power. If the same polychromator system is used for AWAKE, where the electron density attained is $2\times 10^{20}~\text{m}^{-3}$, the contribution to measurement error due to coherent scattering is ${\sim}2.5\,\%$. Presented here are details of the TS diagnostic and the first tests in RAID, and the expectations for the system when employed on the AWAKE device.

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

Motivated by recent observations of ‘electron-only’ magnetic reconnection, without an ion-scale sheet or ion outflows, in both the Earth’s magnetosheath and in numerical simulations, we study the formation and reconnection of electron-scale current sheets at low plasma $\unicode[STIX]{x1D6FD}$. We first show that ideal sheets collapse to thicknesses much smaller than the ion scales, by deriving an appropriate analogue of the Chapman–Kendall collapse solution. Second, we show that, in practice, reconnection onset happens in these collapsing sheets once they reach a critical aspect ratio, because the tearing instability then becomes faster than their collapse time scale. We show that this can happen for sheet thicknesses larger than the ion scale or at only a few times the electron scale, depending on plasma parameters and the aspect ratio of the collapsing structure, thereby unifying the usual picture of ion-coupled reconnection and the new regime of electron-only reconnection. We derive relationships between plasma $\unicode[STIX]{x1D6FD}$, ion-to-electron temperature ratio, the aspect ratio, electron outflow velocity and the final thickness of the sheets, and thus determine under what circumstances electron-scale sheets form and reconnect.

The confinement of energetic particles in nuclear fusion devices is studied in the presence of an oscillating radial electric field and an axisymmetric magnetic equilibrium. It is shown that, despite the poloidal and toroidal symmetries, initially integrable orbits turn into chaotic regions that can potentially intercept the wall of the tokamak, leading to particle losses. It is observed that the losses exhibit algebraic time decay different from the expected exponential decay characteristic of radial diffusive transport. A dynamical explanation of this behaviour is presented, within the continuous time random walk theory. The central point of the analysis is based on the fact that, contrary to the radial displacement, the poloidal angle is not bounded and a proper statistical analysis can therefore be made, showing for the first time that energetic particle transport can be super-diffusive in the poloidal direction and characterised by asymmetric poloidal displacement. The connection between poloidal and radial positions ensured by the conservation of the toroidal canonical momentum, implies that energetic particles spend statistically more time in the inner region of the tokamak than in the outer one, which explains the observed algebraic decay. This indicates that energetic particles might be efficiently slowed down by the thermal population before leaving the system. Also, the asymmetric transport reveals a new possible mechanism of self-generation of momentum.

The creation of well-thermalized, hot and dense plasmas is attractive for warm dense matter studies. We investigate collisionally induced energy absorption of an ultraintense and ultrashort laser pulse in a solid copper target using particle-in-cell simulations. We find that, upon irradiation by a $2\times 10^{20}~\text{W}\,\text{cm}^{-2}$ intensity, 60 fs duration, circularly polarized laser pulse, the electrons in the collisional simulation rapidly reach a well-thermalized distribution with ${\sim}3.5~\text{keV}$ temperature, while in the collisionless simulation the absorption is several orders of magnitude weaker. Circular polarization inhibits the generation of suprathermal electrons, while ensuring efficient bulk heating through inverse bremsstrahlung, a mechanism usually overlooked at relativistic laser intensity. An additional simulation, taking account of both collisional and field ionization, yields similar results: the bulk electrons are heated to ${\sim}2.5~\text{keV}$, but with a somewhat lower degree of thermalization than in the pre-set, fixed-ionization case. The collisional absorption mechanism is found to be robust against variations in the laser parameters. At fixed laser pulse energy, increasing the pulse duration rather than the intensity leads to a higher electron temperature.

We study the longitudinal stability of beam–plasma systems in the presence of a density inhomogeneity in the background plasma. Previous works have focused on the non-relativistic regime where hydrodynamical models are used to evolve pre-existing Langmuir waves within inhomogeneous background plasmas. Here, for the first time we study the problem with kinetic equations in a fully relativistic way. We do not assume the existence of Langmuir waves, and we focus on the rate and the mechanism by which waves are excited in such systems from an initial perturbation. We derive the structure of the unstable modes and compute an analytical approximation for their growth rates. Our computation is limited to dilute and cold beams, and shows an excellent agreement with particle-in-cell simulations performed using the SHARP code. We show that, due to such an inhomogeneity, the virulent beam–plasma instabilities in the intergalactic medium are not suppressed but their counterparts in the solar wind can be suppressed as evidenced by propagating type-III solar radio bursts.

Accelerated statistical computation of collisionless fusion alpha particle losses in stellarator configurations is presented based on direct guiding-centre orbit tracing. The approach relies on the combination of recently developed symplectic integrators in canonicalized magnetic flux coordinates and early classification into regular and chaotic orbit types. Only chaotic orbits have to be traced up to the end, as their behaviour is unpredictable. An implementation of this technique is provided in the code SIMPLE (symplectic integration methods for particle loss estimation, Albert et al., 2020b, doi:10.5281/zenodo.3666820). Reliable results were obtained for an ensemble of 1000 orbits in a quasi-isodynamic, a quasi-helical and a quasi-axisymmetric configuration. Overall, a computational speed up of approximately one order of magnitude is achieved compared to direct integration via adaptive Runge–Kutta methods. This reduces run times to the range of typical magnetic equilibrium computations and makes direct alpha particle loss computation adequate for use within a stellarator optimization loop.

We present an energy-conserving discontinuous Galerkin scheme for the full-$f$ electromagnetic gyrokinetic system in the long-wavelength limit. We use the symplectic formulation and solve directly for $\unicode[STIX]{x2202}A_{\Vert }/\unicode[STIX]{x2202}t$, the inductive component of the parallel electric field, using a generalized Ohm’s law derived directly from the gyrokinetic equation. Linear benchmarks are performed to verify the implementation and show that the scheme avoids the Ampère cancellation problem. We perform a nonlinear electromagnetic simulation in a helical open-field-line system as a rough model of the tokamak scrape-off layer using parameters from the National Spherical Torus Experiment (NSTX). This is the first published nonlinear electromagnetic gyrokinetic simulation on open field lines. Comparisons are made to a corresponding electrostatic simulation.

In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behaviour is different from that of an $\unicode[STIX]{x1D6FC}^{2}$ dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time dependent. The growing mean field can be modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. (J. Fluid Mech., vol. 783, 2015, pp. 23–45), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15 % supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.

The deviation from thermodynamic equilibrium of the ion velocity distribution functions (VDFs), as measured by the Magnetospheric Multiscale (MMS) mission in the Earth’s turbulent magnetosheath, is quantitatively investigated. Making use of the unprecedented high-resolution MMS ion data, and together with Vlasov–Maxwell simulations, this analysis aims at investigating the relationship between deviation from Maxwellian equilibrium and typical plasma parameters. Correlations of the non-Maxwellian features with plasma quantities such as electric fields, ion temperature, current density and ion vorticity are found to be similar in magnetosheath data and numerical experiments, with a poor correlation between distortions of ion VDFs and current density, evidence that questions the occurrence of VDF departure from Maxwellian at the current density peaks. Moreover, strong correlation has been observed with the magnitude of the electric field in the turbulent magnetosheath, while a certain degree of correlation has been found in the numerical simulations and during a magnetopause crossing by MMS. This work could help shed light on the influence of electrostatic waves on the distortion of the ion VDFs in space turbulent plasmas.

Differential forms provide a coordinate-free way to express many quantities and relations in mathematical physics. In particular, they are useful in plasma physics. This tutorial gives a guide so that you can read the plasma physics literature that uses them and apply them yourself.

We present a detailed guide to advanced collisionless fluid models that incorporate kinetic effects into the fluid framework, and that are much closer to the collisionless kinetic description than traditional magnetohydrodynamics. Such fluid models are directly applicable to modelling the turbulent evolution of a vast array of astrophysical plasmas, such as the solar corona and the solar wind, the interstellar medium, as well as accretion disks and galaxy clusters. The text can be viewed as a detailed guide to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid models that are obtained by closing the fluid hierarchy with simple (non-Landau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the fluid model of Chew–Goldberger–Low (CGL) in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius corrections to the pressure tensor. We consider dispersive effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the non-gyrotropic heat flux strongly influences the maximum growth rate of these instabilities. Furthermore, we discuss fluid models that contain evolution equations for the gyrotropic heat flux fluctuations and that are closed at the fourth-moment level by prescribing a specific form for the distribution function. For the bi-Maxwellian distribution, such a closure is known as the ‘normal’ closure. We also discuss a fluid closure for the bi-kappa distribution. Finally, by considering one-dimensional Maxwellian fluid closures at higher-order moments, we show that such fluid models are always unstable. The last possible non Landau fluid closure is therefore the ‘normal’ closure, and beyond the fourth-order moment, Landau fluid closures are required.