Upper bounds on the growth of free energy in gyrokinetics are derived. These bounds apply to all local gyrokinetic instabilities in the geometry of a flux tube, i.e. a slender volume of plasma aligned with the magnetic field, regardless of the geometry of field, the number of particle species or collisions. The results apply both to linear instabilities and to the nonlinear growth of finite-amplitude fluctuations.

In the construction of a stellarator, the manufacturing and assembling of the coil system is a dominant cost. These coils need to satisfy strict engineering tolerances, and if those are not met the project could be cancelled as in the case of the National Compact Stellarator Experiment (NCSX) project (R.L. Orbach, 2008, https://ncsx.pppl.gov/DOE_NCSX_052208.pdf). Therefore, our goal is to find coil configurations that increase construction tolerances without compromising the performance of the magnetic field. In this paper, we develop a gradient-based stochastic optimization model which seeks robust stellarator coil configurations in high dimensions. In particular, we design a two-step method: first, we perform an approximate global search by a sample efficient trust-region Bayesian optimization; second, we refine the minima found in step one with a stochastic local optimizer. To this end, we introduce two stochastic local optimizers: BFGS applied to the sample average approximation; and Adam, equipped with a control variate for variance reduction. Numerical simulations performed on a W7-X-like coil configuration demonstrate that our global optimization approach finds a variety of promising local solutions at less than $0.1\,\%$ of the cost of previous work, which considered solely local stochastic optimization.

Energetic-particle-induced geodesic acoustic modes, EGAMs (Fu, Phys. Rev. Let., vol. 101, 2008, pp. 185002), driven by neutral beam injection (NBI), have been observed in many DIII-D tokamak experiments (Nazikian et al., Phys. Rev. Lett., vol. 101, 2008, pp. 185001). This mechanism has been theoretically investigated in (Qiu et al., Plasma Phys. Control. Fusion, vol. 52, 2010, pp. 095003), using a sharp energetic particle distribution function, and in (Qu et al., Plasma Phys. Control. Fusion, vol. 59, 2017, pp. 055018), where the dispersion relation and eigenmode behaviour were obtained for the situation of early beam scenario, that is, for times smaller than the beam slowing down time. In this work, we extend these studies determining the eigenmode for beyond the slowing down time, in a scenario with reverse safety factor $q$ profile, where a small concentration of energetic ions can produce an off-axis maximum in the GAM dispersion relation. The characteristics of EGAM are analytically studied with the drift kinetic equation together with the MHD code NOVA. The toroidal energetic ion transit frequency, coupled with the GAM frequency, produces the maximum in the dispersion relation where the eigenmode can be found. The quantitative correspondence of experimental results with the predictions of the proposed model is analysed.

The reflection and transmission of an incident solitary wave with an arbitrary propagation direction due to an interface are investigated in the present paper. It is found that the propagation direction of the transmitted solitary wave depends on not only the propagation direction of the incident solitary wave, but also on the system parameters such as the masses, the number densities of dust particles in two different regions. Dependence of the transmission angle on the plasma parameters and incident angle are given analytically. Moreover, the number and amplitude of transmitted solitary waves and reflected solitary waves are also given when there is only one exact incident solitary wave. Our result has potential application, for example, we can devise an appropriate experiment to measure the differences of the masses and number densities of dust particles between two different regions by using our present results. Furthermore, we can also measure the electric charge of a dust particle by devising an appropriate experiment by using our results.

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.

Power-transfer and fixed-point analysis of previous NIMROD simulations (Roberds et al., Phys. Plasmas, vol. 23, issue 9, 2016, 092513) improved the understanding of the effect of 3D (non-axisymmetric equilibrium) magnetic fields on sawtooth oscillations in the Compact Toroidal Hybrid (CTH) experiment. Computing the locations of order-1 fixed points, their Greene's residues, and local values for the rotational transform results in a description of CTH sawteeth consistent with Kadomtsev's model. A power-transfer analysis quantifies the distribution of energy among toroidal Fourier modes and their nonlinear interactions. The Lorentz power transfer drives sawtooth growth, and it is unambiguously interpreted as the flow of energy from toroidal mode $n'$ to mode $n$, catalysed by $\boldsymbol {B}_{n-n'}$. It has been reported previously that the CTH sawtooth frequency increases with the 3D field strength. This is attributed to an increased growth rate of the internal kink that drives sawtooth oscillations. Here, 3D fields remove energy from the kink, eliminating the possibility that these fields are an additional energy source that drives growth. Instead, 3D fields catalyse energy transfer from large-to-small scales, where magnetic reconnection is stronger. It is proposed that this energy transfer increases the reconnection rate at small scales, which is consistent with the increased growth rate observed at higher 3D field strengths.

This paper presents quasilinear theory (QLT) for a classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic and gravitational effects are subsumed. A Fokker–Planck equation for the dressed ‘oscillation-centre’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum and energy, and it also satisfies the $\smash {H}$-theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave–plasma system. The action of non-resonant waves is also conserved, unlike in the standard version of QLT. Dewar's oscillation-centre QLT of electrostatic turbulence (Phys. Fluids, vol. 16, 1973, p. 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed.

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.

A fluid dispersion theory in partially magnetised plasmas is analysed to examine the conditions under which large-wavelength modes develop in Penning-type configurations, that is, where an electric field is imposed perpendicular to a homogeneous magnetic field. The fluid dispersion relation assuming a slab geometry shows that two types of low-frequency, gradient drift instabilities occur in the direction of the $\boldsymbol {E} \times \boldsymbol {B}$ and diamagnetic drifts. One type of instability, observed when the equilibrium electric field and plasma density gradient are in the same direction, is similar to the classic modified Simon–Hoh instability. A second instability is found for conditions in which (i) the diamagnetic drift is in the direction opposite to the $\boldsymbol {E} \times \boldsymbol {B}$ drift and (ii) the magnitude of the diamagnetic drift is sufficiently larger than the electron thermal speed. The present fluid dispersion theory suggests that the rotating spokes driven by such fluid instabilities propagate in the same direction as the diamagnetic drift, which can be in the same direction as or opposite to the $\boldsymbol {E} \times \boldsymbol {B}$ drift, depending on the plasma conditions. This finding may account for the observation, in some plasma devices, of the rotation of large-scale structures in both the $\boldsymbol {E} \times \boldsymbol {B}$ and $-\boldsymbol {E} \times \boldsymbol {B}$ directions.

Recent experiments on the collective interaction of gigawatt electron beams with plasma at the GOL-PET facility have shown that the power of the electromagnetic radiation at the plasma frequency increases tens of times if the starting plasma is created with quasiperiodic radial density gradients. One of the mechanisms capable of providing highly efficient conversion of an unstable beam-driven wave into an electromagnetic one in the presence of periodic density perturbations is the mechanism of plasma antenna. However, earlier this mechanism was considered only for strictly longitudinal modulation of the plasma density, when its efficiency dropped drastically if the transverse size of the plasma significantly exceeded the radiation wavelength. In this work, based on both analytical theory and particle-in-cell modelling, we will show that the presence of oblique modulation of the plasma density makes the antenna mechanism effective even in a thick plasma, if the long-wavelength satellite of the most unstable beam-driven wave falls into resonance with natural plasma oscillations.

We develop the formalism of the first-order near-axis expansion of the magnetohydrodynamic equilibrium equations described by Garren & Boozer (Phys. Fluids B, vol. 3, issue 10, 1991, pp. 2805–2821) and Plunk et al. (J. Plasma Phys., vol. 85, issue 6, 2019; J. Plasma Phys., vol. 87, issue 6, 2021) for the case of a quasi-isodynamic, $N$-field-period, stellarator-symmetric, single-well magnetic field equilibrium. The importance of the magnetic axis shape is investigated, and we conclude that control of the curvature and torsion is crucial to obtain omnigenous configurations with finite aspect ratio and low effective ripple, especially for a higher number of field periods. For this reason a method is derived to construct classes of axis shapes with favourable curvature and torsion. Solutions are presented, including a three-field-period configuration constructed at an aspect ratio of $A=20$, with a maximum elongation of $e=3.2$ and an effective ripple under $1\,\%$, which demonstrates that high elongation is not a necessary feature of quasi-isodynamic stellarators.

This work numerically investigates the role of viscosity and resistivity in Rayleigh–Taylor instabilities in magnetized high-energy-density (HED) plasmas for a high Atwood number and high plasma beta regimes surveying across plasma beta and magnetic Prandtl numbers. The numerical simulations are performed using the visco-resistive magnetohydrodynamic equations. Results presented here show that the inclusion of self-consistent viscosity and resistivity in the system drastically changes the growth of the Rayleigh–Taylor instability (RTI) as well as modifies its internal structure at smaller scales. It is seen here that the viscosity has a stabilizing effect on the RTI. Moreover, the viscosity inhibits the development of small-scale structures and also modifies the morphology of the tip of the RTI spikes. On the other hand, the resistivity reduces the magnetic field stabilization, supporting the development of small-scale structures. The morphology of the RTI spikes is seen to be unaffected by the presence of resistivity in the system. An additional novelty of this work is in the disparate viscosity and resistivity profiles that may exist in HED plasmas and their impact on RTI growth, morphology and the resulting turbulence spectra. Furthermore, this work shows that the dynamics of the magnetic field is independent of viscosity and likewise the resistivity does not affect the dissipation of enstrophy and kinetic energy. In addition, power law scalings of enstrophy, kinetic energy and magnetic field energy are provided in both the injection range and inertial sub-range, which could be useful for understanding RTI induced turbulent mixing in HED laboratory and astrophysical plasmas and could aid in the interpretation of observations of RTI-induced turbulence spectra.

The prevailing paradigm for plasma turbulence associates a unique stationary state with given equilibrium parameters. We report the discovery of bistable turbulence in a strongly magnetised plasma with a sheared mean flow. Two distinct states, obtained with identical equilibrium parameters in first-principle gyrokinetic simulations, have turbulent fluxes of particles, momentum and energy that differ by an order of magnitude – with the low-transport state agreeing with experimental observations. Occurrences of the two states are regulated by the competition between an externally imposed mean flow shear and ‘zonal’ flows generated by the plasma. With small turbulent amplitudes, zonal flows have little impact, and the mean shear causes turbulence to saturate in a low-transport state. With larger amplitudes, the zonal shear can (partially) oppose the effect of the mean shear, allowing the system to sustain a high-transport state. This poses a new challenge for research that has so far assumed a uniquely defined turbulent state.

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 safe operation of tokamak reactors requires a reliable modelling capability of disruptions, and in particular the spatio-temporal dynamics of associated runaway electron currents. In a disruption, instabilities can break up magnetic surfaces into chaotic field line regions, causing current profile relaxation, as well as a rapid radial transport of heat and particles. Using a mean-field helicity transport model implemented in the disruption runaway modelling framework Dream, we calculate the dynamics of runaway electrons in the presence of current relaxation events. In scenarios where flux surfaces remain intact in parts of the plasma, a skin current is induced at the boundary of the intact magnetic field region. This skin current region becomes an important centre concerning the subsequent dynamics: it may turn into a hot ohmic current channel, or a sizeable radially localized runaway beam, depending on the heat transport. If the intact region is in the plasma edge, runaway generation in the countercurrent direction can occur, which may develop into a sizeable reverse runaway beam. Even when the current relaxation extends to the entire plasma, the final runaway current density profile can be significantly affected, as the induced electric field is reduced in the core and increased in the edge, thereby shifting the centre of runaway generation towards the edge.

The first experimental campaigns have proven that, due to the optimization of the magnetic configuration with respect to neoclassical transport, the contribution of turbulence is essential to understand and predict the total particle and energy transport in Wendelstein 7-X (W7-X). This has spurred much work on gyrokinetic modelling for the interpretation of the available experimental results and for the preparation of the next campaigns. At the same time, new stellarator gyrokinetic codes have just been or are being developed. It is therefore desirable to have a sufficiently complete, documented and verified set of gyrokinetic simulations in W7-X geometry against which new codes or upgrades of existing codes can be tested and benchmarked. This paper attempts to provide such a set of simulations in the form of a comprehensive benchmark between the recently developed code stella and the well-established code GENE. The benchmark consists of electrostatic gyrokinetic simulations in the W7-X magnetic geometry and includes different flux tubes, linear ion-temperature-gradient (ITG) and trapped-electron-mode stability analyses, computation of linear zonal-flow responses and calculation of ITG-driven heat fluxes.

The paper presents experimental results from the SMOLA device on the testing of the helical mirror confinement hypothesis. Helical mirror confinement is the technique of an active control of axial plasma losses from a confinement zone by multiple magnetic mirrors that move along the axis in the reference frame of the plasma that experiences $\boldsymbol{E} \times \boldsymbol{B}$ rotation due to an applied radial electric field. Theory predicts that a helical mirror will provide an axial force that modifies the plasma flow and, simultaneously, density pinching to the axis. The force direction depends on the plasma rotation direction. Experimental data on the axial plasma losses at different direction of the magnetic mirror movement are presented. If the trapped ions move in the direction opposite to the direction of the axial losses, then the particle flux reduces in the broad range of the plasma density. The confinement improves with the increase of the fraction of the trapped particles (effective mirror ratio was up to $R_{{\rm eff}}=5.8\pm 1.4$). If the trapped ions move in the same direction as the axial losses, then the flux depends on density. At intermediate densities, the integral flux through the transport section rises compared to the plasma flowing through the straight magnetic field. The effective mirror ratio is lower and does not significantly depend on the fraction of the trapped particles (effective mirror ratio at intermediate density was $R_{{\rm eff}}=3.3\pm 0.8$).

A two-field Hamiltonian gyrofluid model for kinetic Alfvén waves retaining ion finite Larmor radius corrections and parallel magnetic field fluctuations is used to study direct turbulent cascades from the magnetohydrodynamic to the sub-ion scales. For moderate energy imbalance and weak enough magnetic fluctuations, the spectrum of the transverse magnetic field and that of the most energetic wave display a steep transition zone near the ion scale, while the parallel transfer (and thus the parallel dissipation) remains weak. In this regime, the perpendicular flux of generalized cross-helicity displays a significant decay past the ion scale, while the perpendicular energy flux remains almost constant. A phenomenological model suggests that the interactions between co-propagative waves present at the sub-ion scales can play a central role in the development of a transition zone in the presence of a helicity barrier.

We study the time-dependent formation and evolution of a current sheet (CS) in a magnetised, collisionless, high-beta plasma using hybrid-kinetic particle-in-cell simulations. An initially tearing-stable Harris sheet is frozen into a persistently driven incompressible flow so that its characteristic thickness gradually decreases in time. As the CS thins, the strength of the reconnecting field increases, and adiabatic invariance in the inflowing fluid elements produces a field-biased pressure anisotropy with excess perpendicular pressure. At large plasma beta, this anisotropy excites the mirror instability, which deforms the reconnecting field on ion-Larmor scales and dramatically reduces the effective thickness of the CS. Tearing modes whose wavelengths are comparable to that of the mirrors then become unstable, triggering reconnection on smaller scales and at earlier times than would have occurred if the thinning CS were to have retained its Harris profile. A novel method for identifying and tracking X-points is introduced, yielding X-point separations that are initially intermediate between the perpendicular and parallel mirror wavelengths in the upstream plasma. These mirror-stimulated tearing modes ultimately grow and merge to produce island widths comparable to the CS thickness, an outcome we verify across a range of CS formation timescales and initial CS widths. Our results may find their most immediate application in the tearing disruption of magnetic folds generated by turbulent dynamo in weakly collisional, high-beta, astrophysical plasmas.

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 $\beta _e$. The finite $\beta _e$ 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 $\beta _e=0$ and tested against numerical simulations. For $\beta _e \ll 1$ 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 $\beta _e$ effects do not produce qualitative modifications in the structures of the Lagrangian invariants associated with Casimirs of the model.