Toroidal Volume Research Articles

Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices

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’.

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.

Observation of Long-Lived Plasma Formations in a Toroidal air Vortex under Normal Conditions

Purpose. Experimentally investigate the difference between the characteristic lifetime of plasma formations in a stationary and rotating toroidal air volume under normal ambient conditions.Methods. Experimental study of plasmoids was carried out by videotaping with a high-speed camera with subsequent frame-by-frame processing of the recording. Generation of plasmoids was performed by mechanical rupture of the plasma cord of the welding arc in the field of permanent magnets. The creation of toroidal rotation of the plasma formation in air occurs when it passes through a hole in a solid screen. Numerical estimation of mechanical forces acting on ions in the air vortex is performed by order of magnitude using experimental data.Results. Two different in design laboratory installations for observation of plasma formations in air under normal external conditions have been created. The lifetime of such "long-lived" plasmoids in air is equal to about two milliseconds, which significantly exceeds the lifetime of a plasma-forming welding arc in the free state under the same conditions. The increase in plasmoid lifetime is considered and explained within the framework of mechanical ideas about spatial separation of heavy and light ions inside the plasma under the action of inertia forces as a result of rotational motion of air. The proposed mechanical model of the phenomenon is obviously simplified and does not take into account diffusion, recombination and other related phenomena, since the main and only difference between the observed "longlived" and "ordinary" plasmoids lies in the nature of the mechanical motion of the medium.Conclusion: Two experimental setups allowing to generate and observe the behavior of plasma formations with a characteristic lifetime of the order of two milliseconds are created and described. The results of observations indirectly confirm a simple mechanical hypothesis about the connection between the lifetime of a free plasma formation and the rotational motion of the gas inside it.

The Proton and Occam’s Razor

Otto Stern’s 1933 measurement of the unexpectedly large proton magnetic moment indicated to most physicists that the proton is not a point particle. At that time, many physicists modeled elementary particles as point particles, and therefore Stern’s discovery initiated the speculation that the proton might be a composite particle. In this work, we show that despite being an elementary particle, the proton is an extended particle. Our work is motivated by the experimental data, which we review in section 1.By applying Occam’s Razor principle, we identify a simple proton structure that explains the origin of its principal parameters. Our model uses only relativistic and electromagnetic concepts, highlighting the primary role of the electromagnetic potentials and of the magnetic flux quantum Φ M = h/e. Unlike prior proton models, our methodology does not violate Maxwell’s equation, Noether’s theorem, or the Pauli exclusion principle.Considering that the proton has an anapole (toroidal) magnetic moment, we propose that the proton is a spherical shaped charge that moves at the speed of light along a path that encloses a toroidal volume. A magnetic flux quantum Φ M = h/e stabilizes the proton’s charge trajectory. The two curvatures of the toroidal and poloidal current loops are determined by the magnetic forces associated with Φ M . We compare our calculations against experimental data.

Structural optimisation of the DEMO alternative divertor configurations based on FE and RBF mesh morphing

The DEMO tokamak exhibits extraordinary complexity due to the constraints and requirements pertaining to different fields of physics and engineering. The multidisciplinary nature of the DEMO system makes its design phase extremely challenging since different and often opposite requirements need to be accounted for. Toroidal field (TF) coils generate the toroidal magnetic field required to magnetically confine the plasma particles and support at the same time the poloidal field coils. They must bear tremendous loads deriving from electromagnetic interactions between the coil currents and the generated magnetic field. An efficient tokamak design aims at minimizing the energy stored in its magnetic field and hence at reducing the toroidal volume within the TF coils whose shape would hence ideally mimic co-centrically the shape of the plasma. In order to bear the enormous forces a D-shape is most suitable for the TF coils as it allows them to resist the very large compression on the inner side and to carry the electro-magnetic (EM) pressure mainly by membrane stresses preventing large bending to occur on the outer side. At the same time the divertor structures must fit within the TF coils and this requires adaptations of the TF coil shape in the case of so-called advanced divertor configurations (ADCs), which require larger divertor structures. This article shows the TF coils adapted to ADCs using a structural optimisation procedure applied to the reference shape. The introduced strategy takes as structural optimum the iso-stress profile associated to each coil. A continuous transformation, based on radial basis functions mesh morphing, turns the baseline finite element (FE) model into its iso-stress counterpart, with a series of intermediate configurations available for electromagnetic and structural investigations as output. The adopted strategy allowed to determine, for each of the ADC cases, a candidate shape. Static membrane stress levels during magnetization could be reduced significantly from more than 700 MPa to below 450 MPa.

Electron gases in toroidal shells: Mode coupling and state functions

Eigenvalues and wave functions describing free electron gases in toroidal shells are determined using a basis set expansion natural to the system geometry. Couplings between azimuthal and poloidal modes are found to be appreciable at lower values of ${m_\phi}$, modifying single particle density distributions within the shells. Thermodynamic state functions are calculated and contrasted with those of a solid toroidal volume.

Existence of weakly quasisymmetric magnetic fields without rotational transform in asymmetric toroidal domains

A quasisymmetry is a special symmetry that enhances the ability of a magnetic field to trap charged particles. Quasisymmetric magnetic fields may allow the realization of next generation fusion reactors (stellarators) with superior performance when compared with tokamak designs. Nevertheless, the existence of such magnetic configurations lacks mathematical proof due to the complexity of the governing equations. Here, we prove the existence of weakly quasisymmetric magnetic fields by constructing explicit examples. This result is achieved by a tailored parametrization of both magnetic field and hosting toroidal domain, which are optimized to fulfill quasisymmetry. The obtained solutions hold in a toroidal volume, are smooth, possess nested flux surfaces, are not invariant under continuous Euclidean isometries, have a non-vanishing current, exhibit a vanishing rotational transform, and fit within the framework of anisotropic magnetohydrodynamics. Due to the vanishing rotational transform, these solutions are however not suitable for particle confinement.

Neutral flow interaction with a magnetic dipole plasma II. Global model

A simulation study is presented for a collisional, low-temperature dipole plasma encountered by rarefied, hypersonic neutral gas flow. A global model is developed and averaged over a toroidal control volume to describe the mass and energy exchange between the neutral stream and plasma. Simulations over a large range of magnet and freestream parameters reveal three distinct physical regimes that have significant bearing on the magnitude of plasma/flow interaction. The transitions between these regimes exhibit characteristics of resistive critical ionization, whereby the relative kinetic energy between plasma and neutral gas collisionally heats electrons, driving rapid and complete ionization of the gas. Two regime transitions are observed here with sudden exponential increases in plasma density occurring at velocity thresholds that depend on several energy loss mechanisms. The higher-velocity transition is a classical presentation of critical ionization where flow neutrals are ionized directly by plasma electrons. The other is a unique case in which charge exchange between ions and flow neutrals supplies both the particles and energy required to initiate critical ionization. This transition is distinct from any critical ionization effect reported in literature and indicates the existence of a lower critical velocity governed by collisional and diffusive effects as opposed to ionization energy losses only. Drag force on the magnetic field is considered by examining absorption and deflection of ionized flow by the dipole. The critical ionization thresholds increase the force on the magnet by up to two orders of magnitude compared to aerodynamic drag on an equivalently sized flow impediment.

Simulations of Fragment Correlations in the Disintegration of Non-Compact Nuclear Configurations

Non-spherical and hollow nuclear configurations have been predicted for intermediate-energy heavy-ion collisions. We have developed a simulation of the emitted heavy fragments in order to establish observables indicative of non-spherical break-up geometries. Starting with a spherical, bubble, spheroidal or a toroidal freeze-out volume configuration, a primary fragment distribution is defined with a given radial collective energy, thermal momentum distribution and intrinsic excitation energy. The outgoing fragments are allowed to propagate along their Coulomb trajectories in a self-consistent manner. Account is taken of the de-excitation of the primary fragments by nucléon and cluster evaporation. Considering 5-fragment partitions, azimuthal distributions and two-fragment correlation functions are constructed and discussed in connection with their associated break-up geometries.

Modeling the consequences of fuel assembly bowing on PWR core neutronics using a Monte-Carlo code

The effect of assemblies bowing in PWR nuclear reactors onto core neutronics is an observed phenomenon still poorly understood which can lead to Power Ratio Tilt. Studies of the consequences of rod/assembly bowing involve many different fields addressed by nuclear power plant, such as neutronics, thermohydraulics, mechanics… in a complex combination of multi-physical interactions. For the neutronic part, the modeling of bowed assemblies in Monte Carlo codes must allow to correctly describe the shape of fuel rods. In this article, two discrete ways to model bowed geometries are tested: the first one consists in a stacking of vertical small cylinders following the shape of the fuel rod by small shifts between neighboring cylinders; the second one, newly introduced in the present research, consists in a sequence of rotated cylindrical segments arranged to as to follow the shape of the fuel rod more closely. Both models are used to reproduce two specific bowing patterns, namely C-shape and S-shape, for which a reference modeling involving an analytical toroidal volume cut by planes is available for use with CEA’s Monte Carlo code Tripoli-4®. Results of comparisons between both models and analytical reference show that, even if the segment modeling requires a specific effort to handle implementation constraints, it appears preferable compared to stacking modeling. It provides accuracy with fewer discrete entities and is therefore computationally affordable and it is far more robust when increasing bowing deflection. This approach is thus only considered ready for its application to any kind of bowing patterns.

Wave functions and spectra of charge carriers in a toroidal nanostructure: The transition from ring to bulk states

A basis set expansion is employed to calculate spectra and eigenstates of charge carriers within a toroidal volume characterized by major radius R and minor radius a immersed in an azimuthally symmetric magnetic field. The angular variables appearing in the Schrödinger equation are eliminated by contour methods, yielding effective potentials that reduce computational time by a large factor. An approximation formula for the single particle spectrum is presented that allows efficient construction of the partition function necessary for rapid calculation of thermodynamic quantities. The heat capacity is calculated as a function of torus aspect ratio and temperature, showing the transition from ring-type to bulk-type behavior.

Empirical Modeling of the Quiet and Storm Time Geosynchronous Magnetic Field

AbstractA dynamical empirical model of the near‐geosynchronous magnetic field has been constructed, based on a recently developed radial basis functions approach and a multiyear set of spacecraft data taken by Time History of Events and Macroscale Interactions during Substorms (THEMIS), Polar, Cluster, and Van Allen Probes missions including 133 geomagnetic storms in the time interval between 1996 and 2016. The model describes the field as a function of Cartesian solar magnetic coordinates, dipole tilt angle, and solar wind ram pressure and of a set of dynamic variables representing the response of the magnetosphere to the external driving/loading during the active phase of a space weather event, followed by the internal relaxation/dissipation during the storm recovery. In terms of the disturbance level, the model's validity range extends to intense storms with peak SYM‐H values down to −150 nT. The spatial validity domain is a toroidal volume bounded by the inner (L ∼ 5) and outer (L ∼ 9) dipolar L shells, which allows the model to be used for tracing field lines to magnetically map geosynchronous spacecraft locations down to low altitudes. The model has been validated on independent out‐of‐sample magnetic field data and compared with an earlier empirical model and GOES 15 data taken in 2012 and 2015.

Diffusion Tensor Imaging in Patients with Glioblastoma Multiforme Using the Supertoroidal Model.

PurposeDiffusion Tensor Imaging (DTI) is a powerful imaging technique that has led to improvements in the diagnosis and prognosis of cerebral lesions and neurosurgical guidance for tumor resection. Traditional tensor modeling, however, has difficulties in differentiating tumor-infiltrated regions and peritumoral edema. Here, we describe the supertoroidal model, which incorporates an increase in surface genus and a continuum of toroidal shapes to improve upon the characterization of Glioblastoma multiforme (GBM).Materials and MethodsDTI brain datasets of 18 individuals with GBM and 18 normal subjects were acquired using a 3T scanner. A supertoroidal model of the diffusion tensor and two new diffusion tensor invariants, one to evaluate diffusivity, the toroidal volume (TV), and one to evaluate anisotropy, the toroidal curvature (TC), were applied and evaluated in the characterization of GBM brain tumors. TV and TC were compared with the mean diffusivity (MD) and fractional anisotropy (FA) indices inside the tumor, surrounding edema, as well as contralateral to the lesions, in the white matter (WM) and gray matter (GM).ResultsThe supertoroidal model enhanced the borders between tumors and surrounding structures, refined the boundaries between WM and GM, and revealed the heterogeneity inherent to tumor-infiltrated tissue. Both MD and TV demonstrated high intensities in the tumor, with lower values in the surrounding edema, which in turn were higher than those of unaffected brain parenchyma. Both TC and FA were effective in revealing the structural degradation of WM tracts.ConclusionsOur findings indicate that the supertoroidal model enables effective tensor visualization as well as quantitative scalar maps that improve the understanding of the underlying tissue structure properties. Hence, this approach has the potential to enhance diagnosis, preoperative planning, and intraoperative image guidance during surgical management of brain lesions.

Ion Beam Transport Simulations for the 1.7 MV Tandem Accelerator at the Michigan Ion Beam Laboratory

The Michigan Ion Beam Laboratory houses a 1.7 MV tandem accelerator. For many years this accelerator was configured to run with three ion sources: a TORoidal Volume Ion Source (TORVIS), a Duoplasmatron source and a Sputter source. In this article we describe an application we have created using the SIMION® code to simulate the trajectories of ion beams produced with these sources through the accelerator. The goal of this work is to have an analytical tool to understand the effect of each electromagnetic component on the ion trajectories. This effect is shown in detailed drawings. Each ion trajectory simulation starts at the aperture of the ion source and ends at the position of the target. Using these simulations, new accelerator operators or users quickly understand how the accelerator system works. Furthermore, these simulations allow analysis of modifications in the ion beam optics of the accelerator by adding, removing or replacing components or changing their relative positions.

Dynamics of multiple double layers in high pressure glow discharge in a simple torus

Parametric characterization of multiple double layers is done during high pressure glow discharge in a toroidal vessel of small aspect ratio. Although glow discharge (without magnetic field) is known to be independent of device geometry, but the toroidal boundary conditions are conducive to plasma growth and eventually the plasma occupy the toroidal volume partially. At higher anode potential, the visibly glowing spots on the body of spatially extended anode transform into multiple intensely luminous spherical plasma blob structures attached to the tip of the positive electrode. Dynamics of multiple double layers are observed in argon glow discharge plasma in presence of toroidal magnetic field. The radial profiles of plasma parameters measured at various toroidal locations show signatures of double layer formation in our system. Parametric dependence of double layer dynamics in presence of toroidal magnetic field is presented here.

A synthesis of data from steel spot welded joints of reduced thickness by means of local SED

In some previous works two fatigue scatterbands were given in terms of the local strain energy density (SED) for welded joints made of structural steel and aluminium alloy. The material-dependent radius R0 of the control volume was identified with reference to conventional arc welding processes. It resulted equal to 0.28mm for welded joints made of structural steel tested under prevailing mode I, with fatigue failures both at the weld root and the weld toe. The scatterband did not include data from thin welded joints, the main plate thickness being greater than 6mm. The SED approach is used here to summarise more than 400 fatigue data from steel spot welds in lap–shear joints, coach–peel specimens and cross–tension specimens of small thickness. Due to the typical crack initiation and propagation phases, an appropriate choice to summarise the data in a narrow scatterband seems to be the use of a toroidal control volume. The volume circumferentially encapsulates the slit tip along the weld spot nugget. The SED value at 2 million cycles is found to match that of the previous scatterband for welded joints of higher thicknesses.

ATLAS: Accelerated torus-like angular source for fusion neutronics applications

There is increasing demand for fast and accurate neutronics analysis to be conducted on specific ITER components. The ATLAS (accelerated torus-like angular source) algorithm has been developed to accelerate such neutronics calculations by approximating the 3-D plasma volume source as a surface source at the first-wall region of interest. The ATLAS routine rapidly calculates the angular distribution of the unscattered neutron flux at the chosen first wall location, on a discrete angular grid, and writes a surface source for both MCNP and Attila. This discretised source can then be used to accurately reproduce the full 3-D toroidal volume source in angle and energy, and to quickly simulate the irradiation of individual components or regions in the ITER model. While designed for ITER, the method is general in implementation and can be applied to any tokamak.

Evaluation of water-hammer reduction in a pipe containing a deformable body

The introduction of a coaxial deformable body, for example, a cylindrical rod, over the entire length of a pipeline, or a portion thereof, is one of the means recommended for lowering the spread velocity of excitations in a pipeline filled with a liquid, and thereby lowering the magnitude of a water hammer [1]. The bulk-elastic modulus Er of the rod should be lower than that of the liquid El, whereupon its deformation should reduce the magnitude of the water hammer. This rod may be homogeneous; sponge rubber, into the pores of which no water flows, may then serve as the rod material. A gas-filled pipe with walls fashioned from an elastically deformable material, for example, a thin-wall rubber tube) may serve as the rod. To assess the effectiveness of these recommendations, let us derive the differential equations of a water hammer for a pipeline containing a rod. For convenience of equation derivation, the geometric axes of the rod and pipeline are common. Figure 1 shows the toroidal reference volume V, which is bounded by reference surface A indicated by the hatchures. Reference surface A consists of free cross-sectional areas u1 and u2, and solid boundaries Ap and Ar .I n the section through free cross-sectional area u1, a liquid with a density n flows at an average velocity v = v1, while a liquid with a density x x flows at a velocity v = v2 through free cross-sectional area u2 (which has the same area as u1, since the reference surface is not deformed). When the pressure is increased by Ap, additional tensile stresses, which, according to the boiler formula, are equal to pD 2 ,

Detecting bifurcations in numerical simulation of fluid flow

We present a general method for detecting bifurcations in numerical simulations of dissipative fluid flow whose solutions possess attracting sets for certain values of the parameters of the system. We describe the method as 'fast-slow' steering. The parameters governing system behaviour are slowly steered within a chosen range of interest, as the system evolves in time. The parameter changes are continuous and are 'slow' relative to the 'fast' rate of the time stepping of the system. This allows the simulated system to evolve continuously through a series of states sufficiently close to a trajectory of attractors of the system held fixed at each successive value on the trajectory. We present the method in its general form which can be applied to any suitable dynamical system. We then present a detailed unfolding of a bifurcation sequence in the numerical solutions of magnetohydrodynamic equations modelling dynamo acting in a toroidal volume. Torus-shaped discs have been observed in astrophysical contexts around collapsed objects (e.g. black holes). Our results conform to those obtained using the traditional computational approach of 'quasi-static' steering, and are also compatible with the mathematical analysis.

String dynamics and ejection along the axis of a spinning black hole

Relativistic current carrying strings moving axisymmetrically on the background of a Kerr black hole are studied. The boundaries and possible types of motion of a string with a given energy and current are found. Regions of parameters for which the string falls into the black hole, or is trapped in a toroidal volume, or can escape to infinity, are identified, and representative trajectories are examined by numerical integration, illustrating various interesting behaviors. In particular, we find that a string can start out at rest near the equatorial plane and, after bouncing around, be ejected out along the axis, some of its internal (elastic or rotational kinetic) energy having been transformed into translational kinetic energy. The resulting velocity can be an order unity fraction of the speed of light. This process results from the presence of an outer tension barrier and an inner angular momentum barrier that are deformed by the gravitational field. We speculatively discuss the possible astrophysical significance of this mechanism as a means of launching a collimated jet of MHD plasma flux tubes along the spin axis of a gravitating system fed by an accretion disk.