Toroidal Configuration Research Articles

Analytical calculation of the kinetic q factor and resonant response of toroidally confined plasmas

Symmetry-breaking perturbations in axisymmetric toroidal plasma configurations have a drastic impact on particle, energy, and momentum transport in fusion devices, thereby affecting their confinement properties. The perturbative modes strongly affect particles with specific kinetic characteristics through resonant mode–particle interactions. In this work, we present an analytical calculation of the kinetic q factor, enabling the identification of particles with kinetic properties that meet the resonant conditions. This allows us to predict the locations and structures of the corresponding resonant island chains, as well as the existence of transport barriers in the particle phase space. The analytical results, derived for the case of a large aspect ratio configuration, are systematically compared to numerical simulations, and their domain of validity is thoroughly investigated and explained. Our findings demonstrate that calculating the kinetic q factor and its dependence on both particle and magnetic field characteristics provides a valuable tool for understanding and predicting the resonant plasma response to non-axisymmetric perturbations. Moreover, this approach can be semi-analytically applied to generic realistic experimental equilibria, offering a low-computational-cost method for scenario investigations under various multi-scale perturbative modes.

Global and local dynamics of X-flare-producing active regions during solar cycle 25 peak phase

The configuration of the longitudinally elongated region that active regions (ARs) cluster around, known as a toroid belt, has been shown to be an indicator of intense activity. In particular, complex ARs at locations in the north and/or south toroids tend to appear "tipped-away" with respect to each other. On the other hand, magnetic helicity has been used as an indicator of flare activity in ARs. As solar cycle (SC) 25 approaches its peak, a number of significant (X-class) flares have been produced. Here, we investigate the circumstances surrounding two of the most flare-prolific ARs of solar cycle 25, namely, ARs 13590 and 13514. Two aspects of the evolution of these ARs are investigated in this work: the global-scale magnetic toroid configuration and small-scale magnetic field morphology and topology -- before, during, and after the onset of major flares. We studied the global morphology of the solar magnetic fields near intense flares in terms of the spatial distribution of ARs on magnetic fields synoptic maps. On AR scales, we analyzed the magnetic helicity accumulation, as well as its current-carrying and potential components. Our results are consistent with major flare-prolific ARs from solar cycles 23 and 24. In particular, we observe a consistent dominance of current-carrying magnetic helicity at the time of major flares. The evolution of global magnetic toroids, indicating the occurrence of flare-prolific ARs in the tipped-away portion of the toroid, together with the local dynamics of complex ARs, could offer a few weeks of lead time to prepare for upcoming space weather hazards.

Impact of fuel dilution on Lawson parameter by considering relativistic correction of radiation loss in toroidal magnetic configuration

In this paper, the ignition boundary and height of the saddle point as the lowest heating power on the way to ignition for D-3He toroidal magnetic configuration are derived. Fusion fuel cycle involving deuterium and helium-3, along with side reactions is considered. The effect of fuel dilution, incorporating synchrotron radiation and bremsstrahlung power loss with relativistic correction on ignition criteria, is investigated. The results were compared by considering the classical state of bremsstrahlung power. It is found that the height of the saddle point is increased with increasing the ratio of the proton particle density to the fuel deuteron density. The ignition regime shrinks and the minimum value of the Lawson parameter for the breakeven condition and ignition boundaries is reduced due to fuel dilution.

Origin of the Twice-90° Rotations of the Polarization Angle in GRB 170114A and GRB 160821A

The observed abrupt twice-90° rotations of the polarization angle (PA) in the prompt phase of gamma-ray bursts (GRBs) are difficult to understand within the current one-emitting-shell models. Here, we apply a model with multiple emitting shells to solve this new challenging problem. Two configurations of large-scale ordered magnetic fields in the shells are considered: toroidal and aligned. Together with the light curves and the spectral peak energy evolutions, the twice-90° PA rotations in GRB 170114A and GRB 160821A could be well interpreted with the multishell aligned magnetic field configuration (MFC). Our numerical calculations also show that the multiple shells with the toroidal MFC could not explain the observed twice-90° PA rotations. An aligned MFC in the GRB outflow usually indicates the preference of a magnetar central engine, while a toroidal field configuration is typically related to a central black hole. Therefore, the magnetar central engines for the two GRBs are favored.

Electro-thermal co-design of a variable pole toroidal induction motor

The United States Department of Energy targets an 88 % reduction in motor and power electronics volume by the year 2025 while minimizing or eliminating magnet usage. Induction machines are a magnet-free alternative with advantages such as low cost, high reliability, and capability of withstanding high short-term overload. To meet these societal goals, we develop an induction machine that uses an electro-thermal co-designed converter-integrated thermal management approach. The machine utilizes toroidal windings in the stator, and an 18-phase power converter to achieve 40 kW/L power density. An optimized jacket design was developed to enable the effective thermal management of the stator and integrated power electronics. The toroidal winding configuration provides advantages for thermal performance over conventional distributed windings due to an increased winding copper surface area without reducing slot-fill factor, thereby allowing for a higher stator density. The improved thermal performance of the toroidal winding design was verified using three-dimensional conjugate computational fluid dynamic simulations. A reduced-order transient model was then developed to generate torque-speed-temperature plots at various load conditions. The transient thermal analysis was extended to verify safe operation of the motor during a typical drive cycle for the desired application. The thermal design models developed in this work are robust and scalable to other motor sizes providing a framework for co-designing and developing thermal management systems for electric motors.

Evidence of a toroidal magnetic field in the core of 3C 84

The spatial scales of relativistic radio jets, probed by relativistic magneto-hydrodynamic (RMHD) jet launching simulations and by most very long baseline interferometry (VLBI) observations differ by an order of magnitude. Bridging the gap between these RMHD simulations and VLBI observations requires selecting nearby active galactic nuclei (AGN), the parsec-scale region of which can be resolved. The radio source 3C 84 is a nearby bright AGN fulfilling the necessary requirements: it is launching a powerful, relativistic jet powered by a central supermassive black hole, while also being very bright. Using 22 GHz globe-spanning VLBI measurements of 3C 84 we studied its sub-parsec region in both total intensity and linear polarisation to explore the properties of this jet, with a linear resolution of ∼0.1 parsec. We tested different simulation set-ups by altering the bulk Lorentz factor Γ of the jet, as well as the magnetic field configuration (toroidal, poloidal, helical). We confirm the persistence of a limb brightened structure, which reaches deep into the sub-parsec region. The corresponding electric vector position angles (EVPAs) follow the bulk jet flow inside but tend to be orthogonal to it near the edges. Our state-of-the-art RMHD simulations show that this geometry is consistent with a spine-sheath model, associated with a mildly relativistic flow and a toroidal magnetic field configuration.

A high-density and high-confinement tokamak plasma regime for fusion energy

The tokamak approach, utilizing a toroidal magnetic field configuration to confine a hot plasma, is one of the most promising designs for developing reactors that can exploit nuclear fusion to generate electrical energy1,2. To reach the goal of an economical reactor, most tokamak reactor designs3–10 simultaneously require reaching a plasma line-averaged density above an empirical limit—the so-called Greenwald density11—and attaining an energy confinement quality better than the standard high-confinement mode12,13. However, such an operating regime has never been verified in experiments. In addition, a long-standing challenge in the high-confinement mode has been the compatibility between a high-performance core and avoiding large, transient edge perturbations that can cause very high heat loads on the plasma-facing-components in tokamaks. Here we report the demonstration of stable tokamak plasmas with a line-averaged density approximately 20% above the Greenwald density and an energy confinement quality of approximately 50% better than the standard high-confinement mode, which was realized by taking advantage of the enhanced suppression of turbulent transport granted by high density-gradients in the high-poloidal-beta scenario14,15. Furthermore, our experimental results show an integration of very low edge transient perturbations with the high normalized density and confinement core. The operating regime we report supports some critical requirements in many fusion reactor designs all over the world and opens a potential avenue to an operating point for producing economically attractive fusion energy.

Effects of magnetic helicity on 3D equilibria and self-organized states in KTX reversed field pinch

The reversed field pinch (RFP) is a toroidal magnetic configuration in which plasmas can spontaneously transform into different self-organized states. Among various states, the ‘quasi-single-helical’ (QSH) state has a dominant component for the magnetic field and significantly improves confinement. Many theoretical and experimental efforts have investigated the transitions among different states. This paper employs the multi-region relaxed magnetohydrodynamic model to study the properties of QSH and other states. The stepped-pressure equilibrium code (SPEC) is used to compute MHD equilibria for the Keda Torus eXperiment (KTX). The toroidal volume of KTX is partitioned into two subvolumes by an internal transport barrier. The geometry of this barrier is adjusted to achieve force balance across the interface, ensuring that the plasma in each subvolume is force-free and that magnetic helicity is conserved. By varying the parameters, we generate distinct self-organized states in KTX. Our findings highlight the crucial role of magnetic helicity in shaping these states. In states with low magnetic helicity in both subvolumes, the plasma exhibits axisymmetric behavior. With increasing core helicity, the plasma gradually transforms from an axisymmetric state to a double-axis helical state and finally to a single-helical-axis state. Elevated core magnetic helicity leads to a more pronounced dominant mode of the boundary magnetic field and a reduced core magnetic shear. This is consistent with previous experimental and numerical results in other RFP devices. We find a linear relationship between the plasma current and helicity in different self-organized states. Our findings suggest that KTX may enter the QSH state when the toroidal current reaches 0.72 MA. This study demonstrates that the stellarator equilibrium code SPEC unveils crucial RFP equilibrium properties, rendering it applicable to a broad range of RFP devices and other toroidal configurations.

3D Magnetization Textures: Toroidal Magnetic Hopfion Stability in Cylindrical Samples.

Topologically non-trivial magnetization configurations in ferromagnetic materials on the nanoscale, such as hopfions, skyrmions, and vortices, have attracted considerable attention of researchers during the last few years. In this article, by applying the theory of micromagnetism, I demonstrate that the toroidal hopfion magnetization configuration is a metastable state of a thick cylindrical ferromagnetic nanodot or a nanowire of a finite radius. The existence of this state is a result of the competition among exchange, magnetostatic, and magnetic anisotropy energies. The Dzyaloshinskii-Moriya exchange interaction and surface magnetic anisotropy are of second importance for the hopfion stabilization. The toroidal hopfion metastable magnetization configuration may be reached in the process of remagnetizing the sample by applying an external magnetic field along the cylindrical axis.

Kinetic vs magnetic chaos in toroidal plasmas: A systematic quantitative comparison

Magnetic field line chaos occurs under the presence of non-axisymmetric perturbations of an axisymmetric equilibrium and is manifested by the destruction of smooth flux surfaces formed by the field lines. These perturbations also render the particle motion, as described by the guiding center dynamics, non-integrable and, therefore, chaotic. However, the chaoticities of the magnetic field lines and the particle orbits significantly differ in both strength and radial location in a toroidal configuration, except for the case of very low-energy particles whose orbits closely follow the magnetic field lines. The chaoticity of more energetic particles, undergoing large drifts with respect to the magnetic field lines, crucially determines the confinement properties of a toroidal device but cannot be inferred from that of the underlying magnetic field. In this work, we implement the smaller alignment index method for detecting and quantifying chaos, allowing for a systematic comparison between magnetic and kinetic chaos. The efficient quantification of chaos enables the assignment of a value characterizing the chaoticity of each orbit in the space of the three constants of the motion, namely, energy, magnetic moment, and toroidal momentum. The respective diagrams provide a unique overview of the different effects of a specific set of perturbations on the entire range of trapped and passing particles, as well as the radial location of the chaotic regions, offering a valuable tool for the study of particle energy and momentum transport and confinement properties of a toroidal fusion device.

Simulation study of three new quadrupole ion funnels to improve low-mass ion transmission.

By applying radio frequency (RF) and direct current (DC) voltages to corresponding ring electrodes, ion funnel (IF) can efficiently focus and transmit ions. However, IF has an inherent mass discrimination problem that will greatly limit low mass-to-charge (m/z) ion focusing and transmission. To improve the transmission efficiency (TE) of the IF, this paper explores three new profile quadrupole ion funnels (QIF). Computer simulations of the potential field distributions of QIFs and conventional IFs were performed to assess their focusing characteristics. To compare the TE, ion optics simulation programs SIMION and AXSIM were used to perform a series of simulations. Three QIF types (toroidal, cylindrical, and hyperbolic configurations) were used to improve ion TE, and their transmission and focus performance were also compared with conventional IF. By simulating the trajectories of ions in the IF, the optimum electrical parameters for three new QIFs were obtained and compared with conventional IFs, with TE improvements recorded for m/z < 100 of 16%, 20%, and 13%. The results indicate that studying these three new IF configurations has great research significance for improving sensitivity to low m/z ions in mass spectrometer instruments.

Theoretically assisted and empirical scalings in the problem of determination of internal inductance in tokamaks

The problem of inferring the internal inductance from external magnetic measurements in tokamaks is considered. It was practically resolved in JET (Barana O et al 2002 Plasma Phys. Control. Fusion 44 2271), but several questions remain to be addressed before extrapolation of that method on other tokamaks. These naturally arise because of the integral nature of determined by unknown distribution of the magnetic field inside the plasma. Without universal solutions, the quality of approximations should be examined along with potential experimental enforcements of the existing procedures. This is the main goal of the study that is fully analytical. Several relevant quantities are calculated for the large-aspect-ratio plasma with elliptical shifted magnetic surfaces. Usually the plasma elongation is mentioned as a key factor in the task. Here it is explicitly shown that not only , but its radial derivative too plays a role. The latter parameter did not appear in the JET scaling and in the accompanying discussions. It is also shown that the -dependent term representing there the first iteration actually brings a systematic error. Therefore, several improvements to that scaling can be proposed. The study is based on the use of virial relations for the toroidal axisymmetric configurations.

Toroidal vs cylindrical analytical description of the magnetic field outside the elongated evolving plasma in tokamaks

In the plasma equilibrium theory, Gajewski's analytical expression [Gajewski, Phys. Fluids 15, 70 (1972)] for the poloidal magnetic flux ψ outside the plasma is known. It was obtained as a solution of the two-dimensional Laplace equation outside an infinite straight cylinder with an elliptical cross section and a uniform current density j ζ. An example of its use for analysis of static configurations is given in the study by Porcelli and Yolbarsop [Phys. Plasmas 26, 054501 (2019)]. Here, we consider the question of its applicability in dynamic problems including, for example, the current quench (CQ) or vertical displacement event (VDE), when the electromagnetic response of the vacuum vessel to the plasma magnetic field evolution has to be accounted for. It is shown that the mentioned cylindrical model does not provide enough information for calculation of the current induced in the wall. Mathematically, this manifests itself in the fact that Gajewski's expression contains an indefinite constant of integration ψ b (hereinafter it is ψ at the plasma boundary), which, in analytical applications, is replaced either by zero or by a value that makes ψ = 0 on the magnetic axis. This does not affect the magnitude of the magnetic field B, but it would incorrectly give the electric field at ∂ B / ∂ t ≠ 0. To eliminate this shortcoming, an additional block of calculations in the toroidal geometry is needed. Here, the problem is solved analytically. The resulting final expression with ψ b well-defined in the toroidal configuration also includes the effects of the Shafranov's shift and inhomogeneity of j ζ. The proposed extensions allow generalization of the earlier results to a wider area and cover such events as CQ or VDE.

Magnetohydrodynamic stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators

How much does the cross-section of a toroidal magnetic field configuration tell us about its magnetohydrodynamic (MHD) stability? It is generally believed that positive triangularity (typically leading to bean-shaped cross-sections with their indentation on the inboard side in stellarators) contributes positively to MHD stability. In this paper, we explore the basis of this statement within a near-axis description for axisymmetric and quasisymmetric magnetic configurations. In agreement with the existing literature, we show that positive triangularity stabilises vertically elongated tokamaks. In quasisymmetric stellarators, the toroidal asymmetry of flux surfaces modifies this relation. The behaviour of stellarator-symmetric, quasisymmetric stellarators can still be described in terms of the shape of one of their up–down symmetric cross-sections. However, we show that for a sample of quasisymmetric configurations, the positive-bean-shaped cross-sections do not contribute positively to stability. Unlike in the axisymmetric case, we also learn that finite $\beta$ can improve stability even without magnetic shear.

Spatial Structure of the Plasma Flows in the Magnetic Fields of Laser Plasma

The results of studying the spatial structure of the plasma flows that appear when a laser pulse of relativistic intensity (above 1018 W/cm2) is incident on the surface of a solid target are presented. The ring structure experimentally observed in the cross section of a plasma flow is shown to correspond to the toroidal equilibrium plasma configuration that appears in the strong magnetic fields of laser plasma. A model is proposed to describe astrophysical current jets consisting of a discrete sequence of toroidal equilibrium plasma structures.

Minimizing separatrix crossings through isoprominence

A simple property of magnetic fields that minimizes bouncing to passing type transitions of guiding center orbits is defined and discussed. This property, called isoprominence, is explored through the framework of a near-axis expansion. It is shown that isoprominent magnetic fields for a toroidal configuration exist to all orders in a formal expansion about a magnetic axis. Some key geometric features of these fields are described.

Global simulations of Tayler instability in stellar interiors: a long-time multistage evolution of the magnetic field

ABSTRACT Magnetic fields are observed in massive Ap/Bp stars and are presumably present in the radiative zone of solar-like stars. To date, there is no clear understanding of the dynamics of the magnetic field in stably stratified layers. A purely toroidal magnetic field configuration is known to be unstable, developing mainly non-axisymmetric modes. Rotation and a poloidal field component may lead to stabilization. Here we perform global MHD simulations with the EULAG-MHD code to explore the evolution of a toroidal magnetic field located in a layer whose Brunt-Väisälä frequency resembles the lower solar tachocline. Our numerical experiments allow us to explore the initial unstable phase as well as the long-term evolution of such field. During the first Alfven cycles, we observe the development of the Tayler instability with the prominent longitudinal wavenumber, m = 1. Rotation decreases the growth rate of the instability and eventually suppresses it. However, after a stable phase, energy surges lead to the development of higher-order modes even for fast rotation. These modes extract energy from the initial toroidal field. Nevertheless, our results show that sufficiently fast rotation leads to a lower saturation energy of the unstable modes, resulting in a magnetic topology with only a small fraction of poloidal field, which remains steady for several hundreds of Alfven traveltimes. The system then becomes turbulent and the field is prone to turbulent diffusion. The final toroidal–poloidal configuration of the magnetic field may represent an important aspect of the field generation and evolution in stably stratified layers.

Impact of magnetic fields on Population III star formation

ABSTRACT The theory of the formation of the first stars in the Universe, the so-called Population III (Pop III), has until now largely neglected the impact of magnetic fields. Complementing a series of recent studies of the magnetohydrodynamic (MHD) aspects of Pop III star formation, we here carry out a suite of idealized numerical experiments where we ascertain how the fragmentation properties of primordial protostellar discs are modified if MHD effects are present. Specifically, starting from cosmological initial conditions, we focus on the central region in a select minihalo at redshift z ∼ 25, inserting a magnetic field at an intermediate evolutionary stage, normalized to a fraction of the equipartition value. To explore parameter space, we consider different field geometries, including uniform, radial, toroidal, and poloidal field configurations, with the toroidal configuration being the most realistic. The collapse of the gas is followed for ∼8 orders of magnitude in density after the field was inserted, until a maximum of $10^{15} {\rm \, cm}^{-3}$ is reached. We find that the magnetic field leads to a delay in the collapse of the gas. Moreover, the toroidal field has the strongest effect on the collapse as it inhibits the fragmentation of the emerging disc surrounding the central core and leads to the formation of a more massive core. The full understanding of the formation of Pop III stars and their mass distribution thus needs to take into account the effect of magnetic fields. We further conclude that ideal MHD is only a first step in this endeavour, to be followed up with a comprehensive treatment of dissipative effects, such as ambipolar diffusion and Ohmic dissipation.

Pellet source density in toroidal plasma configurations based on a 2D Gaussian deposition model

We develop a two-dimensional (2D) Gaussian deposition model to calculate the initial pellet deposition density immediately after pellet ablation, which is valid before the ∇B-drift of the ablated material significantly shifts its location. A 2D Gaussian particle distribution is assumed in the ablation cloud cross-section. Applying this new model to a typical EAST plasma, and comparing it with the conventional point deposition model, it is found that the new model can resolve the tangential singularity problem encountered by the point deposition model. In addition, the model predicts that the initial pellet deposition density depends strongly on the ablation cloud radius as well as the form of the radial particle distribution in the ablation cloud with tangential injection. The ∇B-drift is then introduced with the drift displacement estimated based on a scaling formula derived from HPI2 simulations. The model can provide a fast evaluation of pellet deposition density compared to the predictive HPI2 code at the expense of acceptable accuracy loss. This model could be a useful tool for physical studies relevant to pellet injection, such as pellet ELM triggering and particle and energy transport.

The Aharonov–Bohm effect in a closed flux line

The Aharonov–Bohm (AB) effect was convincingly demonstrated using a micro-sized toroidal magnet but it is almost always explained using an infinitely-long solenoid or an infinitely-long flux line. The main reason for this is that the formal treatment of the AB effect considering a toroidal configuration turns out to be too cumbersome. But if the micro-sized toroidal magnet is modelled by a closed flux line of arbitrary shape and size then the formal treatment of the AB effect is exact, considerably simplified, and well-justified. Here we present such a treatment that covers in detail the electromagnetic, topological, and quantum-mechanical aspects of this effect. We demonstrate that the AB phase in a closed flux line is determined by a linking number and has the same form as the AB phase in an infinitely-long flux line which is determined by a winding number. We explicitly show that the two-slit interference shift associated with the AB effect in a closed flux line is the same as that associated with an infinitely-long flux line. We emphasise the topological nature of the AB phase in a closed flux line by demonstrating that this phase is invariant under deformations of the charge path, deformations of the closed flux line, simultaneous deformations of the charge path and the closed flux line, and the interchange between the charge path and the closed flux line. We also discuss the local and nonlocal interpretations of the AB effect in a closed flux line and introduce a non-singular gauge in which the vector potential vanishes in all space except on the surface surrounded by the closed flux line, implying that this vector potential is zero along the trajectory of the charged particle except on the crossing point where this trajectory intersects the surface bounded by the closed flux line, a result that questions the alleged physical significance of the vector potential and thereby the local interpretation of the AB effect.