Linearized gravity around a rotating black hole or compact object introduces the concept of a gravitomagnetic field, which originates from the matter–current in the rotating object. Plasma in proximity to this object is subsequently subjected to motion guided by this gravitomagnetic term (where mass serves as the effective charge) in addition to the conventional magnetic field. Such an interplay of fields complicates the accessible plasma waves of the system and thus merits exploration to delineate this interplay, to identify any observable signatures of the gravitomagnetic field in weakly magnetized systems, and to motivate future numerical work in a fully relativistic setting where the effects may be stronger. In this work we analyse the dispersion of the upper and lower hybrid electrostatic waves in a plasma immersed in both magnetic and gravitomagnetic fields. In particular, we discuss the effective augmentation or cancellation of the two fields under the right conditions for the upper hybrid wave. In contrast, the lower hybrid wave experiences a frequency up-shift from the gravitomagnetic field regardless of whether it is parallel or antiparallel to the magnetic field for the studied field strengths.

In a recent publication (Toler et al., J. Plasma Phys., vol. 89, issue 2, 2023, p. 905890210), we demonstrated that for axisymmetric geometries, the Kapur–Rokhlin quadrature rule provided an efficient and high-order accurate method for computing the normal component, on the plasma surface, of the magnetic field due to the toroidal current flowing in the plasma, via the virtual-casing principle. The calculation was indirect, as it required the prior computation of the magnetic vector potential from the virtual-casing principle, followed by the computation of its tangential derivative by Fourier differentiation, to obtain the normal component of the magnetic field. Our approach did not provide the other components of the virtual-casing magnetic field. In this letter, we show that a more direct and more general approach is available for the computation of the virtual-casing magnetic field. The Kapur–Rokhlin quadrature rule accurately calculates the principal value integrals in the expression for all the components of the magnetic field on the plasma boundary, and the numerical error converges at a rate nearly as high as the indirect method we presented previously.

We present a theoretical analysis of electron pitch-angle scattering by ion-acoustic electrostatic fluctuations present in the Earth's bow shock and, presumably, collisionless shocks in general. We numerically simulate electron interaction with a single wave packet to demonstrate the scattering through phase bunching and phase trapping and quantify electron pitch-angle scattering in dependence on the wave amplitude and wave normal angle to the local magnetic field. The iterative mapping technique is used to model pitch-angle scattering of electrons by a large number of wave packets, which have been reported in the Earth's bow shock. Assuming that successive electron scatterings are not correlated, we revealed that the long-term dynamics of electrons is diffusive. The diffusion coefficient depends on the ratio $\varPhi _0/W$ between the wave packet amplitude and electron energy, $D\propto (\varPhi _0/W)^{\nu }$. A quasi-linear scaling ($\nu \approx 2$) is observed for sufficiently small wave amplitudes, $\varPhi _0\lesssim 10^{-3}W$, while the diffusion is nonlinear ($1<\nu <2$) above this threshold. We show that pitch-angle diffusion of ${\lesssim }1$ keV electrons in the Earth's bow shock can be nonlinear. The corresponding diffusion coefficient scales with the intensity $E_{w}^{2}$ of the electrostatic fluctuations in a nonlinear fashion, $D\propto E_{w}^{\nu }$ with $\nu <2$, while its expected values in the Earth's bow shock are $D\sim 0.1\unicode{x2013}100$ $(T_{e}/W)^{\nu -1/2}\,{\rm rad}^{2}\,{\rm s}^{-1}$. We speculate that in the Earth's quasi-perpendicular bow shock the stochastic shock drift acceleration mechanism with pitch-angle scattering provided by the electrostatic fluctuations can contribute to the acceleration of thermal electrons up to approximately 1 keV. The potential effects of a finite perpendicular coherence scale of the wave packets on the efficiency of electron scattering are discussed.

This paper establishes a mean-field equation set and an energy theorem to provide a theoretical basis in view of the development of self-consistent, physics-based turbulent transport models for mean-field transport codes. A rigorous averaging procedure identifies the exact form of the perpendicular turbulent fluxes which are modelled by ad hoc diffusive terms in mean-field transport codes, next to other closure terms which are not commonly considered. Earlier work suggested that the turbulent $E\times B$ particle and heat fluxes, which are thus identified to be important closure terms, can be modelled to reasonable accuracy using the kinetic energy in the $E\times B$ velocity fluctuations ($k_{E}$). The related enstrophy led to further modelling improvements in an initial study, although further analysis is required. To support this modelling approach, transport equations are derived analytically for both quantities. In particular, an energy theorem is established in which the various source and sink terms of $k_{E}$ are shown to couple to mean-field and turbulent parallel kinetic energy, kinetic energy in the other perpendicular velocity components, the thermal energy and the magnetic energy. This provides expressions for the interchange, drift-wave and Reynolds stress terms amongst others. Note that most terms in these energy equations are in turn closure terms. It is suggested to evaluate these terms using reference data from detailed turbulence code simulations in future work.

In the limit of sufficiently fast rotation, rotating mirror traps are known to be stable against the loss-cone modes associated with conventional (non-rotating) mirrors. This paper calculates how quickly a mirror configuration must rotate in order for several of these modes to be stabilized (in particular, the high-frequency convective loss cone, drift cyclotron loss cone and Dory–Guest–Harris modes). Commonalities in the stabilization conditions for these modes then motivate a modified formulation of the Gardner free energy and diffusively accessible free energy to be used for systems in which the important modes have wavevectors that are orthogonal or nearly orthogonal to the magnetic field, as well as a modification to include the effects of a loss region in phase space.

We solve the linearized gyrokinetic equation, quasineutrality condition and Ampere's law to obtain the dispersion relation of microtearing modes (MTMs) in collisionless low-$\beta$ toroidal plasmas. Consistent with past studies, we find that MTMs are driven unstable by the electron temperature gradient and that this instability drive is mediated by magnetic drifts. The dispersion relation that we derive can be evaluated numerically very quickly and may prove useful for devising strategies to mitigate MTM instability in fusion devices.

The gyrokinetic theory of the residual flow, in the electrostatic limit, is revisited, with optimized stellarators in mind. We consider general initial conditions for the problem, and identify cases that lead to a non-zonal residual electrostatic potential, i.e. one having a significant component that varies within a flux surface. We investigate the behaviour of the ‘intermediate residual’ in stellarators, a measure of the flow that remains after geodesic acoustic modes have damped away, but before the action of the slower damping that is caused by unconfined particle orbits. The case of a quasi-isodynamic stellarator is identified as having a particularly large such residual, owing to the small orbit width achieved by optimization.

Large-amplitude current-driven plasma instabilities, which can transition to the Buneman instability, were observed in one-dimensional simulations to generate high-energy back-streaming ions. We investigate the saturation of multi-dimensional plasma instabilities and its effects on energetic ion formation. Such ions directly impact spacecraft thruster lifetimes and are associated with magnetic reconnection and cosmic ray inception. An Eulerian Vlasov–Poisson solver employing the grid-based direct kinetic method is used to study the growth and saturation of 2D2V collisionless, electrostatic current-driven instabilities spanning two dimensions each in the configuration (D) and velocity (V) spaces supporting ion and electron phase-space transport. Four stages characterise the electric potential evolution in such instabilities: linear modal growth, harmonic growth, accelerated growth via quasi-linear mechanisms alongside nonlinear fill-in and saturated turbulence. Its transition and isotropisation process bears considerable similarities to the development of hydrodynamic turbulence. While a tendency to isotropy is observed in the plasma waves, followed by electron and then ion phase spaces after several ion-acoustic periods, the formation of energetic back-streaming ions is more limited in the 2D2V than in the 1D1V simulations. Plasma waves formed by two-dimensional electrostatic kinetic instabilities can propagate in the direction perpendicular to the net electron drift. Thus, large-amplitude multi-dimensional waves generate high-energy transverse-streaming ions and eventually limit energetic backward-streaming ions along the longitudinal direction. The multi-dimensional study sheds light on interactions between longitudinal and transverse electrostatic plasma instabilities, as well as fundamental characteristics of the inception and sustenance of unmagnetised plasma turbulence.

Recent developments in the design of magnetic confinement fusion devices have allowed the construction of exceptionally optimized stellarator configurations. The near-axis expansion in particular has been proven to enable the construction of magnetic configurations with good confinement properties while taking only a fraction of the usual computation time to generate optimized magnetic equilibria. However, not much is known about the overall features of fast-particle orbits computed in such analytical, yet simplified, equilibria when compared with those originating from accurate equilibrium solutions. This work aims to assess and demonstrate the potential of the near-axis expansion to provide accurate information on particle orbits and to compute loss fractions in moderate to high aspect ratios. The configurations used here are all scaled to fusion-relevant parameters and approximate quasi-symmetry to various degrees. This allows us to understand how deviations from quasi-symmetry affect particle orbits and what are their effects on the estimation of the loss fraction. Guiding-centre trajectories of fusion-born alpha particles are traced using gyronimo and SIMPLE codes under the NEAT framework, showing good numerical agreement. Discrepancies between near-axis and magnetohydrodynamic fields have minor effects on passing particles but significant effects on trapped particles, especially in quasi-helically symmetric magnetic fields. Effective expressions were found for estimating orbit widths and passing–trapped separatrix in quasi-symmetric near-axis fields. Loss fractions agree in the prompt losses regime but diverge afterwards.

The pellet ablation mechanism is an interesting subject for plasma fuelling in fusion plasmas. In GAMMA 10/PDX, pellet injection experiments for higher density plasma production are planned to conduct detached plasma experiments in the higher density plasma condition. We measured the pellet ablation cloud by using the two-directional simultaneous photography system in GAMMA 10/PDX. The tomography reconstruction technique was used for considering the pellet trajectory in the plasma and pellet ablation. The three-dimensional pellet trajectory and pellet ablation images in the plasma were clearly obtained for the first time, to the best of our knowledge.

Quasisymmetry (QS), a hidden symmetry of the magnetic field strength, is known to support nested flux surfaces and provide superior particle confinement in stellarators. In this work, we study the ideal magnetohydrodynamic (MHD) equilibrium and stability of high-beta plasma in a large-aspect-ratio stellarator. In particular, we show that the lowest-order description of a near-axisymmetric equilibrium vastly simplifies the problem of three-dimensional quasisymmetric MHD equilibria, which can be reduced to a standard elliptic Grad–Shafranov equation for the flux function. We show that any large-aspect-ratio tokamak, deformed periodically in the vertical direction, is a stellarator with approximate volumetric QS. We discuss exact analytical solutions and numerical benchmarks. Finally, we discuss the ideal ballooning and interchange stability of some of our equilibrium configurations.

We present new stellarator equilibria that have been optimized for reduced turbulent transport using nonlinear gyrokinetic simulations within the optimization loop. The optimization routine involves coupling the pseudo-spectral GPU-native gyrokinetic code GX with the stellarator equilibrium and optimization code DESC. Since using GX allows for fast nonlinear simulations, we directly optimize for reduced nonlinear heat fluxes. To handle the noisy heat flux traces returned by these simulations, we employ the simultaneous perturbation stochastic approximation (SPSA) method that only uses two objective function evaluations for a simple estimate of the gradient. We show several examples that optimize for both reduced heat fluxes and good quasi-symmetry as a proxy for low neoclassical transport. Finally, we run full transport simulations using the T3D stellarator transport code to evaluate the changes in the macroscopic profiles.

We present the results of numerical simulation by two-dimensional hybrid particle-in-cell code of high-beta plasma with hot ions in an axisymmetric mirror machine. Two particular effects are discussed: the self-rotating of plasma with Maxwellian ions in regime of diamagnetic confinement and the excitation of axisymmetric magnetosonic waves in a high-beta plasma with sloshing ions.

Some stellarators tend to benefit from favourable average magnetic curvature for trapped particles when the plasma pressure is sufficiently high. This so-called maximum-$J$ property has several positive implications, such as good fast-particle confinement, magnetohydrodynamic stability and suppression of certain trapped-particle instabilities. This property cannot be attained in quasisymmetric stellarators, in which deeply trapped particles experience average bad curvature and therefore precess in the diamagnetic direction close to the magnetic axis. However, quasi-isodynamic stellarators offer greater flexibility and allow the average curvature to be favourable and the precession to be reversed. We find that it is possible to design such stellarators so that the maximum-$J$ condition is satisfied for the great majority of all particles, even when the plasma pressure vanishes. The qualitative properties of such a stellarator field can be derived analytically by examining the most deeply and the most shallowly trapped particles, although some small fraction of the latter will inevitably not behave as desired. However, through numerical optimisation, we construct a vacuum field in which 99.6 % of all trapped particles satisfy the maximum-$J$ condition.

Collisionless shocks are frequently analysed using the magnetohydrodynamics (MHD) formalism, even though MHD assumes a small mean free path. Yet, isotropy of pressure, the fruit of binary collisions and assumed in MHD, may not apply in collisionless shocks. This is especially true within a magnetized plasma, where the field can stabilize an anisotropy. In a previous article (Bret & Narayan, J. Plasma Phys., vol. 88, no. 6, 2022b, p. 905880615), a model was presented capable of dealing with the anisotropies that may arise at the front crossing. It was solved for any orientation of the field with respect to the shock front. Yet, for some values of the upstream parameters, several downstream solutions were found. Here, we complete the work started in Bret & Narayan (J. Plasma Phys., vol. 88, no. 6, 2022b, p. 905880615) by showing how to pick the physical solution out of the ones offered by the algebra. This is achieved by 2 means: (i) selecting the solution that has the downstream field obliquity closest to the upstream one. This criterion is exemplified on the parallel case and backed up by particle-in-cell simulations. (ii) Filtering out solutions which do not satisfy a criteria already invoked to trim multiple solutions in MHD: the evolutionarity criterion, that we assume valid in the collisionless case. The end result is a model in which a given upstream configuration results in a unique, or no downstream configuration (as in MHD). The largest departure from MHD is found for the case of a parallel shock.

The Okubo–Weiss (Okubo, Deep-Sea Res., vol. 17, issue 3, 1970, pp. 445–454; Weiss, Physica D, vol. 48, issue 2, 1991, pp. 273–294) criterion has been widely used as a diagnostic tool to divide a two-dimensional (2-D) hydrodynamical flow field into hyperbolic and elliptic regions. This paper considers extension of these ideas to 2-D magnetohydrodynamic (MHD) flows, and presents an Okubo–Weiss-type criterion to parameterize the magnetic field topology in 2-D MHD flows. This ensues via its topological connections with the intrinsic metric properties of the underlying magnetic flux manifold, and is illustrated by recasting the Okubo–Weiss-type criterion via the 2-D MHD stationary generalized Alfvénic state condition to approximate the slow-flow-variation ansatz imposed in its derivation. The Okubo–Weiss-type parameter then turns out to be related to the sign definiteness of the Gaussian curvature of the magnetic flux manifold. A similar formulation becomes possible for 2-D electron MHD flows, by using the generalized magnetic flux framework to incorporate the electron-inertia effects. Numerical simulations of quasi-stationary vortices in 2-D MHD flows in the decaying turbulence regime are then given to demonstrate that the Okubo–Weiss-type criterion is able to separate the MHD flow field into elliptic and hyperbolic field configurations very well.

Magnetic reconnection is an important process in astrophysical environments, as it reconfigures magnetic field topology and converts magnetic energy into thermal and kinetic energy. In extreme astrophysical systems, such as black hole coronae and pulsar magnetospheres, radiative cooling modifies the energy partition by radiating away internal energy, which can lead to the radiative collapse of the reconnection layer. In this paper, we perform two- and three-dimensional simulations to model the MARZ (Magnetic Reconnection on Z) experiments, which are designed to access cooling rates in the laboratory necessary to investigate reconnection in a previously unexplored radiatively cooled regime. These simulations are performed in GORGON, an Eulerian two-temperature resistive magnetohydrodynamic code, which models the experimental geometry comprising two exploding wire arrays driven by 20 MA of current on the Z machine (Sandia National Laboratories). Radiative losses are implemented using non-local thermodynamic equilibrium tables computed using the atomic code Spk, and we probe the effects of radiation transport by implementing both a local radiation loss model and $P_{1/3}$ multi-group radiation transport. The load produces highly collisional, super-Alfvénic (Alfvén Mach number $M_A \approx 1.5$), supersonic (Sonic Mach number $M_S \approx 4-5$) strongly driven plasma flows which generate an elongated reconnection layer (Aspect Ratio $L/\delta \approx 100$, Lundquist number $S_L \approx 400$). The reconnection layer undergoes radiative collapse when the radiative losses exceed the rates of ohmic and compressional heating (cooling rate/hydrodynamic transit rate = $\tau _{\text {cool}}^{-1}/\tau _{H}^{-1}\approx 100$); this generates a cold strongly compressed current sheet, leading to an accelerated reconnection rate, consistent with theoretical predictions. Finally, the current sheet is also unstable to the plasmoid instability, but the magnetic islands are extinguished by strong radiative cooling before ejection from the layer.

Isothermal fluid turbulence simulations have been performed in the edge and scrape-off layer (SOL) of an analytic stellarator configuration with an island divertor, thereby providing numerical insight into edge turbulence in regions around islands in a stellarator. The steady-state transport follows the a curvature drive that is inverse to the major radius ($1/R$) toward the outboard side, but large fluctuations are present throughout the island divertor region, with the average wavelength of similar size to the island width. The system exhibits a prominent $m=2$, $n=5$ mode, where m is the poloidal mode number and n is the toroidal mode number, although other modes are present. The amplitude and radial extent of the density fluctuations are similar throughout the edge and SOL, but can decrease near island O-points. The fluctuations exhibit a predominantly positive skewness on the outboard midplane, indicating blob-like perturbations for the transport into the outer SOL. It is determined that a point on the separatrix is generally more correlated with regions outside of the SOL than a nearby reference point which does not lie on the separatrix.

The centrifugal mirror confinement scheme incorporates supersonic rotation of a plasma into a magnetic mirror device. This concept has been shown experimentally to drastically decrease parallel losses and increase plasma stability as compared with prior axisymmetric mirrors. MCTrans++ is a dimensionless (0-D) scoping tool which rapidly models experimental operating points in the Centrifugal Mirror Fusion Experiment (CMFX) at the University of Maryland. In the low-collisionality regime, parallel losses can be modelled analytically. A confining potential is set up that is partially ambipolar and partially centrifugal. Due to the stabilizing effects of flow shear, the perpendicular losses can be modelled as classical. Radiation losses such as bremsstrahlung and cyclotron emission are taken into account. A neutrals model is included, and, in some circumstances, charge-exchange losses are found to exceed all other loss mechanisms. We use the SUNDIALS ARKODE library to solve the underlying equations of this model; the resulting software is suitable for scanning large parameter spaces, and can also be used to model time-dependent phenomena such as a capacitive discharge. MCTrans++ has been used to verify results from prior centrifugal mirrors, create an experimental plan for CMFX and find configurations for future reactor-scale fusion devices.

This paper is based on a report at the 2nd International Fusion Plasma Conference & 13th International Conference on Open Magnetic Systems for Plasma Confinement (iFPC & OS 2023), August 21–25, 2023, Busan, Korea and provides a brief overview of the status of work at the Budker Institute on the study of hot plasma confinement in open-type magnetic traps with a linear axisymmetric configuration. The main attention is paid to key problems: magnetohydrodynamics (MHD) stability in regimes with extremely high relative pressure, longitudinal electronic thermal conductivity, stability with respect to the development of kinetic modes and transverse transport. This paper provides an overview of the methods we are developing to address these problems, the experimental and theoretical results achieved and plans for future development. The last section of the article provides brief information about the preliminary design of the gas-dynamic multiple-mirror trap device, the development of which has been completed.