This study presents observations of coherent modes (CMs) in a spherical tokamak using a microwave interferometer near the midplane. The CMs within the 30–60 kHz frequency range were observed during electron cyclotron resonance heating only, and the frequency of the CMs increased proportionally with the square root of the electron temperature near $R = 0.7m$. Generally, these modes displayed bursting and chirping signatures with strong density rise and fall. Their appearance indicated an increase in the intensity of hard x rays, suggesting a deterioration in energetic electron confinement. Furthermore, the effect of CMs on the intensity of energetic electron-driven whistler waves was observed. They decreased when CMs were present and gradually increased with the decrease in CM intensity. The CMs may influence the intensity of whistler waves by affecting the energetic electron confinement.

We describe the interaction of parallel-propagating Alfvén waves with ion-acoustic waves and other Alfvén waves, in magnetized, high-$\beta$ collisionless plasmas. This is accomplished through a combination of analytical theory and numerical fluid simulations of the Chew–Goldberger–Low (CGL) magnetohydrodynamic (MHD) equations closed by Landau-fluid heat fluxes. An asymptotic ordering is employed to simplify the CGL-MHD equations and derive solutions for the deformation of an Alfvén wave that results from its interaction with the pressure anisotropy generated either by an ion-acoustic wave or another, larger-amplitude Alfvén wave. The difference in time scales of acoustic and Alfvénic fluctuations at high-$\beta$ means that interactions that are local in wavenumber space yield little modification to either mode within the time it takes the acoustic wave to Landau damp away. Instead, order-unity changes in the amplitude of Alfvénic fluctuations can result after interacting with frequency-matched acoustic waves. Additionally, we show that the propagation speed of an Alfvén-wave packet in an otherwise homogeneous background is a function of its self-generated pressure anisotropy. This allows for the eventual interaction of separate co-propagating Alfvén-wave packets of differing amplitudes. The results of the CGL-MHD simulations agree well with these predictions, suggesting that theoretical models relying on the interaction of these modes should be reconsidered in certain astrophysical environments. Applications of these results to weak Alfvénic turbulence and to the interaction between the compressive and Alfvénic cascades in strong, collisionless turbulence are also discussed.

We present a framework for analysing plasma flow in a rotating mirror. By making a series of physical assumptions, we reduce the magnetohydrodynamic (MHD) equations in a three-dimensional cylindrical system to a one-dimensional system in a shallow, cuboidal channel within a transverse magnetic field, similar to the Hartmann flow in ducts. We then solve the system both numerically and analytically for a range of values of the Hartmann number and calculate the dependence of the plasma flow speed on the thickness of the insulating end cap. We observe that the mean flow overshoots and decelerates before achieving a steady-state value, a phenomenon that the analytical model cannot capture. This overshoot is directly proportional to the thickness of the insulating end cap and the external electric field, with a weak dependence on the external magnetic field. Our simplified model can act as a benchmark for future simulations of the supersonic mirror device CMFX (centrifugal magnetic fusion experiment), which will employ more sophisticated physics and realistic magnetic field geometries.

A specific set of dimensionless plasma and turbulence parameters is introduced to characterize the nature of turbulence and its dissipation in weakly collisional space and astrophysical plasmas. Key considerations are discussed for the development of predictive models of the turbulent plasma heating that characterize the partitioning of dissipated turbulent energy between the ion and electron species and between the perpendicular and parallel degrees of freedom for each species. Identifying the kinetic physical mechanisms that govern the damping of the turbulent fluctuations is a critical first step in constructing such turbulent heating models. A set of ten general plasma and turbulence parameters are defined, and reasonable approximations along with the exploitation of existing scaling theories for magnetohydrodynamic turbulence are used to reduce this general set of ten parameters to just three parameters in the isotropic temperature case: the ion plasma beta, the ion-to-electron temperature ratio and the isotropic driving wavenumber. A critical step forward in this study is to identify the dependence of all of the proposed kinetic mechanisms for turbulent damping in terms of the same set of fundamental plasma and turbulence parameters. Analytical estimations of the scaling of each damping mechanism on these fundamental parameters are presented. The power of this approach is illustrated in the development of the first phase diagram for the turbulent damping mechanisms as a function of the ion plasma beta and isotropic driving wavenumber for unity ion-to-electron temperature ratio, showing the regions of this two-dimensional parameter space in which ion Landau and transit-time damping, electron Landau and transit-time damping, ion cyclotron damping, ion stochastic heating, collisionless magnetic reconnection and kinetic ‘viscous’ heating play a role in the damping of the turbulent fluctuations.

Many stellarator coil design problems are plagued by multiple minima, where the locally optimal coil sets can sometimes vary substantially in performance. As a result, solving a coil design problem a single time with a local optimization algorithm is usually insufficient and better optima likely do exist. To address this problem, we propose a global optimization algorithm for the design of stellarator coils and outline how to apply box constraints to the physical positions of the coils. The algorithm has a global exploration phase that searches for interesting regions of design space and is followed by three local optimization algorithms that search in these interesting regions (a ‘global-to-local’ approach). The first local algorithm (phase I), following the globalization phase, is based on near-axis expansions and finds stellarator coils that optimize for quasisymmetry in the neighbourhood of a magnetic axis. The second local algorithm (phase II) takes these coil sets and optimizes them for nested flux surfaces and quasisymmetry on a toroidal volume. The final local algorithm (phase III) polishes these configurations for an accurate approximation of quasisymmetry. Using our global algorithm, we study the trade-off between coil length, aspect ratio, rotational transform and quality of quasi-axisymmetry. The database of stellarators, which comprises approximately 200 000 coil sets, is available online and is called QUASR, for ‘quasi-symmetric stellarator repository’.

GX is a code designed to solve the nonlinear gyrokinetic system for low-frequency turbulence in magnetized plasmas, particularly tokamaks and stellarators. In GX, our primary motivation and target is a fast gyrokinetic solver that can be used for fusion reactor design and optimization along with wide-ranging physics exploration. This has led to several code and algorithm design decisions, specifically chosen to prioritize time to solution. First, we have used a discretization algorithm that is pseudospectral in the entire phase space, including a Laguerre–Hermite pseudospectral formulation of velocity space, which allows for smooth interpolation between coarse gyrofluid-like resolutions and finer conventional gyrokinetic resolutions and efficient evaluation of a model collision operator. Additionally, we have built GX to natively target graphics processors (GPUs), which are among the fastest computational platforms available today. Finally, we have taken advantage of the reactor-relevant limit of small $\rho _*$ by using the radially local flux-tube approach. In this paper we present details about the gyrokinetic system and the numerical algorithms used in GX to solve the system. We then present several numerical benchmarks against established gyrokinetic codes in both tokamak and stellarator magnetic geometries to verify that GX correctly simulates gyrokinetic turbulence in the small $\rho _*$ limit. Moreover, we show that the convergence properties of the Laguerre–Hermite spectral velocity formulation are quite favourable for nonlinear problems of interest. Coupled with GPU acceleration, which we also investigate with scaling studies, this enables GX to be able to produce useful turbulence simulations in minutes on one (or a few) GPUs and higher fidelity results in a few hours using several GPUs. GX is open-source software that is ready for fusion reactor design studies.

The femtosecond laser filament-induced air laser plays a significant role for the remote sensing of air pollutants. The spatial distributions of air laser intensity were investigated experimentally in previous studies. However, the mechanism of the air laser propagation properties inside the filament plasma has not been quite clear yet. Moreover, few studies have been dedicated to the reproduction of the air laser profile from nitrogen molecules propagating in the filament plasma based on the numerical simulation method. In this study, the lasing action of the air laser from the transition of the first negative (0,0) band of nitrogen ions at 391 nm was simulated during the femtosecond laser filamentation. The beam profile of the air laser changes from a Gaussian or super-Gaussian shape to an outer ring structure by increasing the filament length or nitrogen ion density, which is in accord with the previous experimental result. A multiple-diffraction effect has been proposed to clarify the mechanism of the outer rings beam pattern formation, which is induced by the dynamical interaction between the lasing effect and diffraction effect of the air laser propagating inside the filament plasma. In addition, the amplified air laser power as a function of both the filament length and nitrogen ion density was investigated. Our study would pave the way to improve the energy conversion efficiency and directivity of remote air lasers, which would be significant for remote sensing applications.

Observations, mainly of outbursts in dwarf novae, imply that the anomalous viscosity in highly ionized accretion discs is magnetic in origin and requires that the plasma ${\beta \sim 1}$. Until now, most simulations of the magnetic dynamo in accretion discs have used a local approximation (known as the shearing box). While these simulations demonstrate the possibility of a self-sustaining dynamo, the magnetic activity generated in these models saturates at $\beta \gg 1$. This long-standing discrepancy has previously been attributed to the local approximation itself. There have been recent attempts at simulating magnetic activity in global accretion discs with parameters relevant to the dwarf novae. These too find values of $\beta \gg 1$. We speculate that the tension between these models and the observations may be caused by numerical magnetic diffusivity. As a pedagogical example, we present exact time-dependent solutions for the evolution of weak magnetic fields in an incompressible fluid subject to linear shear and magnetic diffusivity. We find that the maximum factor by which the initial magnetic energy can be increased depends on the magnetic Reynolds number as ${\mathcal {R}}_{m}^{2/3}$. We estimate that current global numerical simulations of dwarf nova discs have numerical magnetic Reynolds numbers around six orders of magnitude less than the physical value found in dwarf nova discs of ${\mathcal {R}}_{m} \sim 10^{10}$. We suggest that, given the current limitations on computing power, expecting to be able to compute realistic dynamo action in observable accretion discs using numerical MHD is, for the time being, a step too far.

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 radiation hydrodynamic simulations used to design inertial confinement fusion (ICF) and pulsed power experiments, nonlinear radiation diffusion tends to dominate CPU time. This raises the interesting question of whether a quantum algorithm can be found for nonlinear radiation diffusion which provides a quantum speedup. Recently, such a quantum algorithm was introduced based on a quantum algorithm for solving systems of nonlinear partial differential equations (PDEs) which provides a quadratic quantum speedup. Here, we apply this quantum PDE (QPDE) algorithm to the problem of a non-equilibrium Marshak wave propagating through a cold, semi-infinite, optically thick target, where the radiation and matter fields are not assumed to be in local thermodynamic equilibrium. The dynamics is governed by a coupled pair of nonlinear PDEs which are solved using the QPDE algorithm, as well as two standard PDE solvers: (i) Python's py-pde solver; and (ii) the KULL ICF simulation code developed at Lawrence-Livermore National Laboratory. We compare the simulation results obtained using the QPDE algorithm and the standard PDE solvers and find excellent agreement.

Simulating plasma physics on quantum computers is difficult because most problems of interest are nonlinear, but quantum computers are not naturally suitable for nonlinear operations. In weakly nonlinear regimes, plasma problems can be modelled as wave–wave interactions. In this paper, we develop a quantization approach to convert nonlinear wave–wave interaction problems to Hamiltonian simulation problems. We demonstrate our approach using two qubits on a superconducting device. Unlike a photonic device, a superconducting device does not naturally have the desired interactions in its native Hamiltonian. Nevertheless, Hamiltonian simulations can still be performed by decomposing required unitary operations into native gates. To improve experimental results, we employ a range of error-mitigation techniques. Apart from readout error mitigation, we use randomized compilation to transform undiagnosed coherent errors into well-behaved stochastic Pauli channels. Moreover, to compensate for stochastic noise, we rescale exponentially decaying probability amplitudes using rates measured from cycle benchmarking. We carefully consider how different choices of product-formula algorithms affect the overall error and show how a trade-off can be made to best utilize limited quantum resources. This study provides an example of how plasma problems may be solved on near-term quantum computing platforms.

Beam–target reactions in fusion plasmas play an important role in both magnetic confinement fusion and inertial confinement fusion in the condition of low-density plasmas with high-velocity interactions. The traditional method for calculating beam–target reaction rate neglects the transport process of incident particles in inhomogeneous plasmas, leading to errors providing that the temperature and density in the transport path of incident particles vary obviously. An improved method considering the transport process is proposed in this paper to eliminate the deficiencies. Then the method is employed in high-speed plasma collision studies. When the initial plasma density and temperature are set to $0.5\,{\rm g}\,{\rm cm}^{-3}$ and 100 eV, it is found that the beam–target reaction rate calculated by the traditional method is almost identical to that by our method if the collision velocity is less than 600 km s$^{-1}$. However, the traditional method is not suitable for study as the collision velocity gets higher, inducing obvious differences, which can reach 70 % at 1000 km s$^{-1}$. The improved method will make large corrections to evaluate the importance of the non-negligible beam–target reaction for inertial confinement fusion schemes with large implosion velocity such as double-cone ignition and impact ignition, in which the high-speed plasmas collide with each other to realize plasma ignition.

The propagation of electromagnetic waves in a linearly varying index of refraction is a fundamental problem in wave physics, being relevant in fusion science for describing certain wave-based heating and diagnostic schemes. Here, an exact solution is obtained for a given incoming wavefield specified on the boundary transverse to the direction of inhomogeneity by performing a spectral, rather than asymptotic, matching. Two case studies are then presented: a Gaussian beam at oblique incidence and a speckled wavefield at normal incidence. For the Gaussian beam, it is shown that when the waist $W$ is sufficiently large, oblique incidence manifests simply as rigid translation and focal shift of the corresponding diffraction pattern at normal incidence. The destruction of the hyperbolic umbilic caustic (corresponding to a critically focused beam) as $W$ is reduced is then demonstrated. The caustic disappears once $W \lesssim \delta _a \sqrt {L}$ ($L$ being the medium length scale normalized by the Airy skin depth $\delta _a$), at which point the wave behaviour is increasingly described by Airy functions, but experiences less focusing as a result. To maximize the intensity of a launched Gaussian beam at a turning point, one should therefore minimize the imaginary part of the launched complex beam parameter while having the real part satisfy critical focusing. For the speckled wavefield, it is shown that the transverse speckle pattern only couples to the Airy longitudinal pattern when the coupling parameter $\eta = \sqrt {L}/f_{\#}$ is large, with $f_{\#}$ being the f-number of the launching aperture. When $\eta \ll 1$, a reduced description of the total wavefield can be obtained by simply multiplying the incoming speckle pattern with the Airy swelling.

The paper presents experimental results from the SMOLA device, which was built in the Budker Institute of Nuclear Physics for the verification of the helical mirror confinement idea. This concept involves active control of axial losses from the confinement zone in an open magnetic trap through the use of multiple mirrors that move in the plasma frame of reference. The discussed experiments focused on determining the cumulative effect of a helical mirror system in combination with a short segment of a stronger magnetic field. Combination of these two methods of axial flow suppression results in higher efficiency compared with each method individually. Different combinations of the mirrors were tested. The most effective flow suppression was observed if the short mirror was placed between the confinement region and the helical mirror. In this configuration, an effective mirror ratio of $R_{{\rm eff}} = 32.6\pm 7.8$ was achieved, along with a more than three-fold increase in plasma density within the confinement region. The possibility of a cumulative effect of different types of magnetic mirrors offers a way to improve the confinement performance of the reactor-grade mirror confinement devices.

The paper considers the electron transport in toroidal systems taking into account relativistic effects for electrons. The treatment is based on the relativistic drift-kinetic equation with the thermodynamic equilibrium given by the relativistic Maxwell–Jüttner distribution function. The definition of relativistic fluxes is given in a classic-like form using the same set of thermodynamic forces as in the classical (non-relativistic) approach. Such a formulation allows us to apply the currently used non-relativistic solvers for calculation of relativistic mono-energetic transport coefficients. As an example, the procedure for calculating electron fluxes is proposed, in which relativistic effects are taken into account using the DKES code. The model can be easily implemented in various transport codes, developed for the non-relativistic limit, making them accurate also for hot plasmas with non-negligible relativistic effects.

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.

The problem of the assimilation of a cryogenic fuel pellet injected into a hot plasma is considered. Due to the transparency to ambient particles of the plasmoid, the localised region of high-density plasma created by ionisation of the ablated pellet material, electrons reach a ‘quasiequilibrium’ (QE) state which is characterised by a steady-state on the fastest collisional time scale. The simplified electron kinetic equation of the QE state is solved. Taking a velocity moment of the higher-order electron kinetic equation, which is valid on the expansion time scale, permits a fluid closure, yielding an evolution equation for the macroscopic parameters describing the QE distribution function. In contrast to the Braginskii equations, the closure does not require that electrons have a short mean free path compared with the size of density perturbations, and permits an anisotropic and highly non-Maxwellian distribution function. As the QE distribution function accounts for both trapped and passing electrons, the self-consistent electric potential that causes the expansion can be properly described, in contrast to earlier models of pellet plasmoid expansion with an unbounded potential. The plasmoid expansion is simulated using both a Vlasov model and a cold-fluid model for the ions. During the expansion plasmoid ions and electrons obtain nearly equal amounts of energy; as hot ambient electrons provide this energy in the form of collisional heating of plasmoid electrons, the expansion of a pellet plasmoid is expected to be a potent mechanism for the transfer of energy from electrons to ions on a time scale shorter than that of ion–electron thermalisation.

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.

This study investigates the dynamics of various particles within the plasma sheath, focusing on the influence of secondary emissions from charged dust particles. The research concentrates on backscattered electron emission (BEE), inelastic reflection emission (IRE) and true-secondary electron emission (TEE) as key contributors to the behaviour of dust particles within the plasma sheath. Employing the semi-empirical model of Furman and Pivi (F-P model), the study defines the total emission of secondary electrons (EES), comprising these three types. The analysis aims to enhance our understanding of the complex interplay between secondary emission phenomena and the dynamics of charged particles within the plasma sheath, contributing valuable insights to the field. Furthermore, a comparative study has been conducted between the results obtained from the emission of secondary electrons according to the Sternglass theory and the emission of secondary electrons obtained using the F-P model. It is observed that the secondary electron emission (SEE) from the dust, based on the F-P model, demonstrates more pronounced effects on the sheath characteristics, particularly when considering lower values of the superextensive electron parameter ‘$q$’.

Overtaking collisions of large-amplitude solitons are investigated via fluid simulations for a plasma consisting of cold ions and Boltzmann-distributed electrons. To achieve this, a new fluid simulation code is presented. In addition, a novel approach to construct soliton initial conditions is developed. Using these ideas, initial conditions are combined that allows the simulation of overtaking collisions. It is shown that, in the small-amplitude regime, simulation results agree well with the two-soliton solution obtained from reductive perturbation theory. Interestingly, in the large amplitude regime, both the slow and fast solitons re-emerge after the collision with no significant change, showing that the collisions remain elastic. A comparison between reductive perturbation analysis and the simulations show that the only significant effect of higher order nonlinearities on overtaking collisions is a reduction in the magnitude of the phase shifts of both solitons.