We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model has already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view to derive dissipative brackets, a means of derivation that is natural for understanding and creating dissipative dynamics without appealing to underlying kinetic theory orderings. Finally, the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, non-local dissipative brackets naturally emerge.

In this work, we highlight an issue that may reduce the accuracy of many local nonlinear gyrokinetic simulations – turbulent self-interaction through the parallel boundary condition. Given a sufficiently long parallel correlation length, individual turbulent eddies can span the full domain and ‘bite their own tails’, thereby altering their statistical properties. Such self-interaction is only modelled accurately when the simulation domain corresponds to a full flux surface, otherwise it is artificially strong. For Cyclone Base Case parameters and typical domain sizes, we find that this mechanism modifies the heat flux by approximately 40 % and it can be even more important. The effect is largest when using kinetic electrons, low magnetic shear and strong turbulence drive (i.e. steep background gradients). It is found that parallel self-interaction can be eliminated by increasing the parallel length and/or the binormal width of the simulation domain until convergence is achieved.

We study numerically the mechanisms of proton acceleration in gas–foil targets driven by an ultraintense femtosecond laser pulse. The target consists of a near-critical-density hydrogen gas layer of a few tens of microns attached to a $2\ \mathrm {\mu }$m-thick solid carbon foil with a contaminant thin proton layer at its back side. Two-dimensional particle-in-cell simulations show that, at optimal gas density, the maximum energy of the contaminant protons is increased by a factor of $\sim$4 compared with a single foil target. This improvement originates from the near-complete laser absorption into relativistic electrons in the gas. Several energetic electron populations are identified, and their respective effect on the proton acceleration is quantified by computing the electrostatic fields that they generate at the protons’ positions. While each of those electron groups is found to contribute substantially to the overall accelerating field, the dominant one is the relativistic thermal bulk that results from the nonlinear wakefield excited in the gas, as analysed recently by Debayle et al. (New J. Phys., vol. 19, 2017, 123013). Our analysis also reveals the important role of the neighbouring ions in the acceleration of the fastest protons, and the onset of multidimensional effects caused by the time-increasing curvature of the proton layer.

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.

Linear gyrokinetic simulations of magnetically confined electron–positron plasmas are performed for the first time in the geometry and parameter regimes likely to be relevant for upcoming laboratory experiments. In such plasmas, the density will be sufficiently small as to render the plasma effectively collisionless. The magnetic field will be very large, meaning that the Debye length will exceed the gyroradius by a few orders of magnitude. We show the results of linear simulations in flux tubes close to the current carrying ring and also in the bulk of the plasma, demonstrating the existence of entropy modes and interchange modes in pair plasmas. We study linear stability and show that in the relevant configurations, almost complete linear stability is attainable in large swathes of parameter space.

Ion cyclotron range of frequencies (ICRF) heating will be the sole auxiliary heating method on SPARC for both full-field (Bt0 ~ 12 T) D–T operation and reduced field (Bt0 ~ 8 T) D–D operation. Using the fast wave at ~120 MHz, good wave penetration and strong single-pass absorption is expected for D–T(3He), D(3He), D(H) and 4He(H) heating scenarios. The dependences of wave absorption on ${k_\parallel }$, 3He concentration, resonance location, antenna poloidal location and losses on alpha particles and ash have been studied. The antenna loading has been assessed by comparison with the Alcator C-Mod antennae. An antenna spectrum of ${k_\parallel }\sim 15\text{--}18\,{\textrm{m}^{ - 1}}$ is shown to be good for both core absorption and edge coupling. For the control of impurity sources, the antenna straps are rotated ~10° to be perpendicular to the B field and the straps can run with different power levels in order to optimize the antenna spectrum and to minimize the image current on the antenna frame. Combining the physics constraints with the SPARC port design, maintenance requirement and contingency against antenna failure during D–T operation, we plan to mount on the inner wall of the vacuum vessel a total of 12 4-strap antennae in 6 ports while keeping 3-strap antennae that are insertable and removable on port plugs as the backup option.

The paper analyses the penetration of a magnetic field into the plasma bridge of nanosecond and microsecond opening switches. For switches with a conduction time of ~100 ns, simple formulae are derived to estimate the magnetic field velocity in collisionless and collisional plasmas. It is shown that in both cases this velocity is determined by the magnetic field rise rate to plasma density ratio raised to the power of 1/2. As the conduction time is increased to ~1 ${\rm \mu}$s, the field velocity starts to depend on the plasma aggregation by a magnetic piston. At the same time, irrespective of the conduction time, the electron flow velocity is limited by the radial drift velocity in crossed magnetic and polarization electric fields. Such a limitation suppresses the current channel conductivity with respect to the Spitzer value by a factor equal to the electron magnetization parameter raised to one or another power. On completion of the conduction phase, the rate of rise of the switch resistance is proportional to the electron drift velocity. The peak switch voltage obtained in calculations is compared with its values recorded in experiments on mega-ampere current switching. A procedure is also presented for calculating the switch parameters to obtain the maximum possible voltage in the phase of current cutoff.

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

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.

Understanding particle drifts in a non-symmetric magnetic field is of primary interest in designing optimized stellarators in order to minimize the neoclassical radial loss of particles. Quasisymmetry and omnigeneity, two distinct properties proposed to ensure radial localization of collisionless trapped particles in stellarators, have been explored almost exclusively for magnetic fields with nested flux surfaces. In this work, we examine radial particle confinement when all field lines are closed. We then study charged particle dynamics in the special case of a non-symmetric vacuum magnetic field with closed field lines obtained recently by Weitzner & Sengupta (Phys. Plasmas, vol. 27, 2020, 022509). These magnetic fields can be used to construct magnetohydrodynamic equilibria for low pressure. Expanding in the amplitude of the non-symmetric fields, we explicitly evaluate the omnigeneity and quasisymmetry constraints. We show that the magnetic field is omnigeneous in the sense that the drift surfaces coincide with the pressure surfaces. However, it is not quasisymmetric according to the standard definitions.

We present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $v^{2}$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.

Scattering phenomena between charged particles and highly excited Rydberg atoms are of critical importance in many processes in plasma physics and astrophysics. While a Maxwell–Boltzmann (MB) energy distribution for the charged particles is often assumed for calculations of collisional rate coefficients, in this contribution we relax this assumption and use two different energy distributions, a bimodal MB distribution and a $\unicode[STIX]{x1D705}$-distribution. Both variants share a high-energy tails occurring with higher probability than the corresponding MB distribution. The high-energy tail may significantly affect rate coefficients for various processes. We focus the analysis to specific situations by showing the dependence of the rate coefficients on the principal quantum number of hydrogen atoms in $n$-changing collisions with electrons in the excitation and ionization channels and in a temperature range relevant to the divertor region of a tokamak device. We finally discuss the implications for diagnostics of laboratory plasmas.

SPARC is being designed to operate with a normalized beta of $\beta _N=1.0$, a normalized density of $n_G=0.37$ and a safety factor of $q_{95}\approx 3.4$, providing a comfortable margin to their respective disruption limits. Further, a low beta poloidal $\beta _p=0.19$ at the safety factor $q=2$ surface reduces the drive for neoclassical tearing modes, which together with a frozen-in classically stable current profile might allow access to a robustly tearing-free operating space. Although the inherent stability is expected to reduce the frequency of disruptions, the disruption loading is comparable to and in some cases higher than that of ITER. The machine is being designed to withstand the predicted unmitigated axisymmetric halo current forces up to 50 MN and similarly large loads from eddy currents forced to flow poloidally in the vacuum vessel. Runaway electron (RE) simulations using GO+CODE show high flattop-to-RE current conversions in the absence of seed losses, although NIMROD modelling predicts losses of ${\sim }80$ %; self-consistent modelling is ongoing. A passive RE mitigation coil designed to drive stochastic RE losses is being considered and COMSOL modelling predicts peak normalized fields at the plasma of order $10^{-2}$ that rises linearly with a change in the plasma current. Massive material injection is planned to reduce the disruption loading. A data-driven approach to predict an oncoming disruption and trigger mitigation is discussed.

The dependence of fishbone cycle on energetic particle intensity has been investigated in EAST low-magnetic-shear plasmas. It is observed that the fishbone mode growth rate, saturation amplitude as well as fishbone cycle frequency clearly increase with increasing neutral beam injection (NBI) power. Moreover, enhanced electron density and temperature perturbations as well as energetic particle loss were observed with greater injected NBI power. Simulation results using M3D-K code show that as the NBI power increases, the resonant frequency and the energy of the resonant particles become higher, and the saturation amplitude of the mode also changes, due to the non-perturbative energetic particle contribution. The relationship between the calculated energetic particle pressure ratio and fishbone cycle frequency is obtained as ${f_{\textrm{FC}}} = 2.2{(1000{\beta _{\textrm{ep,calc}}} - 0.1)^{5.9 \pm 0.5}}$. Results consistent with the experimental observations have been achieved based on a predator–prey model.

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.

Potential loss of energetic ions including alphas and radio-frequency tail ions due to classical orbit effects and magnetohydrodynamic instabilities (MHD) are central physics issues in the design and experimental physics programme of the SPARC tokamak. The expected loss of fusion alpha power due to ripple-induced transport is computed for the SPARC tokamak design by the ASCOT and SPIRAL orbit-simulation codes, to assess the expected surface heating of plasma-facing components. We find good agreement between the ASCOT and SPIRAL simulation results not only in integrated quantities (fraction of alpha power loss) but also in the spatial, temporal and pitch-angle dependence of the losses. If the toroidal field (TF) coils are well-aligned, the SPARC edge ripple is small (0.15–0.30 %), the computed ripple-induced alpha power loss is small (${\sim } 0.25\,\%$) and the corresponding peak surface power density is acceptable ($244\ \textrm{kW}\ \textrm {m}^{-2}$). However, the ripple and ripple-induced losses increase strongly if the TF coils are assumed to suffer increasing magnitudes of misalignment. Surface heat loads may become problematic if the TF coil misalignment approaches the centimetre level. Ripple-induced losses of the energetic ion tail driven by ion cyclotron range of frequency (ICRF) heating are not expected to generate significant wall or limiter heating in the nominal SPARC plasma scenario. Because the expected classical fast-ion losses are small, SPARC will be able to observe and study fast-ion redistribution due to MHD including sawteeth and Alfvén eigenmodes (AEs). SPARC's parameter space for AE physics even at moderate $Q$ is shown to reasonably overlap that of the demonstration power plant ARC (Sorbom et al., Fusion Engng Des., vol. 100, 2015, p. 378), and thus measurements of AE mode amplitude, spectrum and associated fast-ion transport in SPARC would provide relevant guidance about AE behaviour expected in ARC.

Finding coil sets with desirable physics and engineering properties is a crucial step in the design of modern stellarator devices. Existing stellarator coil optimization codes ultimately produce zero-thickness filament coils. However, stellarator coils have finite depth and thickness, which can make the single-filament model a poor approximation, particularly when coil build dimensions are relatively large compared to the coil–plasma distance. In this paper, we present a new method for designing coils with finite builds and present a mechanism to optimize the orientation of the winding pack. We approximate finite-build coils with a multi-filament model. A numerical implementation has been developed, and applications to the Helically Symmetric eXperiment stellarator and a new UW-Madison quasihelically symmetric configuration are shown.

The dynamical behaviours of electromagnetic (EM) solitons formed due to nonlinear interaction of linearly polarized intense laser light and relativistic degenerate plasmas are studied. In the slow-motion approximation of relativistic dynamics, the evolution of weakly nonlinear EM envelope is described by the generalized nonlinear Schrödinger (GNLS) equation with local and nonlocal nonlinearities. Using the Vakhitov–Kolokolov criterion, the stability of an EM soliton solution of the GNLS equation is studied. Different stable and unstable regions are demonstrated with the effects of soliton velocity, soliton eigenfrequency, as well as the degeneracy parameter $R=p_{Fe}/m_ec$, where $p_{Fe}$ is the Fermi momentum and $m_e$ the electron mass and $c$ is the speed of light in vacuum. It is found that the stability region shifts to an unstable one and is significantly reduced as one enters from the regimes of weakly relativistic $(R\ll 1)$ to ultrarelativistic $(R\gg 1)$ degeneracy of electrons. The analytically predicted results are in good agreement with the simulation results of the GNLS equation. It is shown that the standing EM soliton solutions are stable. However, the moving solitons can be stable or unstable depending on the values of soliton velocity, the eigenfrequency or the degeneracy parameter. The latter with strong degeneracy $(R>1)$ can eventually lead to soliton collapse.

We find a new family of solutions for force-free magnetic structures in cylindrical geometry. These solutions have radial power-law dependence and are periodic but non-harmonic in the azimuthal direction; they generalize the vacuum $z$-independent potential fields to current-carrying configurations.