Compact Torus Research Articles

Development of a compact torus injector system for central fuelling technology of EAST

In this paper, a compact torus (CT) injector system is designed for experimental advanced superconducting tokamak (EAST). The injector system consists of a coaxial electrode and four pulsed power systems and generates self-organized CT with high density and velocity. In addition, the performance of EAST-CTI is evaluated using a series of diagnostics systems. The experimental results show that the velocity, electron density, and number of particles of the CT are 56.0 km/s, 8.73 × 1020 m-3, and 2.4 × 1018, respectively. The detailed description and experimental results are presented in this paper.

Development of a compact torus injection system for the Keda Torus eXperiment

A compact torus injection system, KTX-CTI, has been developed for the planned injection experiments on the Keda Torus eXperiment (KTX) reversed field pinch (RFP) device to investigate the physics and engineering issues associated with interaction between a compact torus (CT) and RFP. The key interests include fueling directly into the reactor center, confinement improvement, and the injection of momentum and helicity into the RFP discharges. The CT velocity and mass have been measured using a multichannel optical fiber interferometer, and for the first time the time evolution of the CT density profile during CT propagation is obtained. The effects of discharge parameters on the number of injected particles, CT velocity and CT density have been characterized: the maximum hydrogen CT plasma mass, is 50 μg, corresponding to 30% of the mass in a typical KTX plasma; the CT velocity exceeds 120 km s−1. It is observed for the first time that multiple CTs can be produced and emitted during a very short period (<100 μs) in one discharge, which is significant for the future study of repetitive CT injection, even with an ultra-high frequency.

Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions and remarks on equivariant asymptotic expansions

Suppose given a holomorphic and Hamiltonian action of a compact torus on a polarized Hodge manifold. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of the torus on the associated Hardy space which decomposes into isotypes. The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for the torus action. The main ingredient in the proof is the micro-local analysis of the equivariant Szegő kernels. As a particular case we obtain a simple approach for asymptotics of the Lefschetz fixed point formula and traces of Toeplitz operators in the setting of ladder representations. We also consider equivariant asymptotic when the decomposition given by the standard circle action is taken into account, in this case one can recall previous results of X. Ma and W. Zhang or of R. Paoletti. We address some explicit computations for the action of the special unitary group of dimension two.

Investigation of the compact torus plasma motion in the KTX-CTI device based on circuit analyses

Compact torus (CT) injection is one of the most promising methods for the central fuelling of next-generation reactor-grade fusion devices due to its high density, high velocity, and self-contained magnetised structure. A newly compact torus injector (CTI) device in Keda Torus eXperiment (KTX), named KTX-CTI, was successfully developed and tested at the University of Science and Technology in China. In this study, first, we briefly introduce the basic principles and structure of KTX-CTI, and then, present an accurate circuit model that relies on nonlinear regression analysis (NRA) for studying the current waveform of the formation region. The current waveform, displacement, and velocity of CT plasma in the acceleration region are calculated using this NRA-based one-dimensional point model. The model results were in good agreement with the experiments. The next-step upgrading reference scheme of the KTX-CTI device is preliminarily investigated using this NRA-based point model. This research can provide insights for the development of experiments and future upgrades of the device.

Orbit spaces for torus actions on Hessenberg varieties

In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.

Neutrino absorption and other physics dependencies in neutrino-cooled black hole accretion discs

ABSTRACT Black hole (BH) accretion discs formed in compact-object mergers or collapsars may be major sites of the rapid-neutron-capture (r-)process, but the conditions determining the electron fraction (Ye) remain uncertain given the complexity of neutrino transfer and angular-momentum transport. After discussing relevant weak-interaction regimes, we study the role of neutrino absorption for shaping Ye using an extensive set of simulations performed with two-moment neutrino transport and again without neutrino absorption. We vary the torus mass, BH mass and spin, and examine the impact of rest-mass and weak-magnetism corrections in the neutrino rates. We also test the dependence on the angular-momentum transport treatment by comparing axisymmetric models using the standard α-viscosity with viscous models assuming constant viscous length-scales (lt) and 3D magnetohydrodynamic (MHD) simulations. Finally, we discuss the nucleosynthesis yields and basic kilonova properties. We find that absorption pushes Ye towards ∼0.5 outside the torus, while inside increasing the equilibrium value $Y_\mathrm{ e}^{\mathrm{eq}}$ by ∼0.05–0.2. Correspondingly, a substantial ejecta fraction is pushed above Ye = 0.25, leading to a reduced lanthanide fraction and a brighter, earlier, and bluer kilonova than without absorption. More compact tori with higher neutrino optical depth, τ, tend to have lower $Y_\mathrm{ e}^{\mathrm{eq}}$ up to τ ∼ 1–10, above which absorption becomes strong enough to reverse this trend. Disc ejecta are less (more) neutron rich when employing an lt = const. viscosity (MHD treatment). The solar-like abundance pattern found for our MHD model marginally supports collapsar discs as major r-process sites, although a strong r-process may be limited to phases of high mass-infall rates, $\dot{M}\, \, \raise0.14em\rm{\gt }\lower0.28em\rm{\sim }\, \, 2\times 10^{-2}$ M⊙ s−1.

Long-time measurements of line-integrated plasma electron density using a two-color homodyne optical fiber interferometer

A two-color homodyne Mach-Zehnder (M-Z) optical fiber interferometer with wavelengths of 1.55 and 1.31 µm was developed for long-time measurement of line-integrated plasma electron density. A novel phase difference demodulation algorithm based on a single 3 × 3 optical coupler was implemented in a two-color optical fiber interferometer scheme for the first time. Our laboratory tests showed that this new optical fiber interferometer could determine the phase shift due to the low-frequency ambient vibration and could maintain high phase resolution measurement. The resolution of the new interferometer was less than 0.04rad in 1000s, corresponding to a line-averaged electron density of less than 1.0 × 1019m-2. Actual plasma discharge experiments performed on KTX-CTI, which is a new compact torus injector (CTI) constructed at the Keda Torus eXperiment (KTX), showed that this interferometer has excellent several-second stability.

Polarized orbifolds associated to quantized Hamiltonian torus actions

Suppose given a holomorphic and Hamiltonian action of a compact torus T on a polarized Hodge manifold M. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of T on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight ν the ν-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through ν, we give a geometric interpretation of the isotypical components associated to the weights kν, k→+∞, in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of M in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map.

C*-blocks and crossed products for classical p-adic groups

Abstract Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of induced representations. For $p$-adic groups, each connected component of the tempered dual comes with a compact torus equipped with a finite group action, and we prove that a version of Wassermann’s theorem holds true under a certain geometric assumption on the structure of stabilizers for that action. We then focus on the case where $G$ is a quasi-split symplectic, orthogonal or unitary group, and explicitly determine the connected components for which the geometric assumption is satisfied.

Kähler–Einstein metrics on symmetric general arrangement varieties

We calculate Chow quotients of some families of symmetric T-varieties. In complexity two we obtain new examples of Kähler–Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorphism class of the orbit space of the compact torus action.

Performance improvement with plasma edge biasing on SPECTOR

A description of an edge-biasing experiment conducted on the spherical compact toroid plasma injector is presented with results. The insertion of a disk-shaped molybdenum electrode (probe), biased at up to +100 V, into the edge of the Compact Torus (CT), resulted in up to 1.2 kA radial current being drawn from the probe to the wall. Core electron temperature, as measured with a Thomson-scattering diagnostic, and CT lifetime, more than doubled in the optimal configuration tested. Temperature and density profiles become more centrally peaked with biasing. In contrast to previous biasing experiments, a significant reduction in electron density was observed; this may be due to the effect of a transport barrier impeding CT fueling and associated cooling, where, as verified by MHD simulation, a principal fueling source is neutral gas that remains concentrated around the gas valves after CT formation.

A Physical Model for a Radiative, Convective Dusty Disk in AGN

Abstract An accretion disk in an Active Galactic Nucleus harbors and shields dust from external illumination. Our model shows that, at the midplane of the disk around an M BH = 107 M ⊙ black hole, dust can exist at 0.1 pc from the black hole, compared to 0.5 pc outside of the disk where such self-shielding is constrained. We construct a physical model of a disk region approximately located between the radius of dust sublimation at the disk midplane and the radius at which dust sublimes at the disk surface. Our main conclusion is that, for a wide range of model parameters such as local accretion rate and/or opacity, the accretion disk’s own radiation pressure on dust significantly influences its vertical structure. In this region, convection plays important role in the vertical transport of energy. When the local accretion rate exceeds 2.5M ⊙ yr, the 10−2 pc scale disk is supercritical with respect to dust opacity. Such a disk puffs up and transforms from geometrically thin to slim. Our model fits into the narrative of a “failed wind” scenario of Czerny &amp; Hryniewicz and the “compact torus” model of Baskin &amp; Laor, incorporating them as variations of the radiative dusty disk model.

Torus Action on Quaternionic Projective Plane and Related Spaces

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number $$\frac{1}{2}\dim X-\dim T$$ is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $${\mathbb {H}}P^2/T^3\cong S^5$$ and $$S^6/T^2\cong S^4$$ , for the homogeneous spaces $${\mathbb {H}}P^2={{\,\mathrm{Sp}\,}}(3)/({{\,\mathrm{Sp}\,}}(2)\times {{\,\mathrm{Sp}\,}}(1))$$ and $$S^6=G_2/{{\,\mathrm{SU}\,}}(3)$$ . Here, the maximal tori of the corresponding Lie groups $${{\,\mathrm{Sp}\,}}(3)$$ and $$G_2$$ act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of $$T^3$$ . This class generalizes $${\mathbb {H}}P^2$$ . We prove that their orbit spaces are homeomorphic to $$S^5$$ as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold.

Twistor lines in the period domain of complex tori

As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a generic chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in Compl where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of Compl, of degree 2n in the Plucker embedding of Compl.

Algebraic flows on commutative complex Lie groups

We recover results by Ullmo–Yafaev and Peterzil–Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax–Lindemann–Weierstrass theorem for commutative complex Lie groups is also given.

Magnetic levitation and compression of compact tori

The magnetic compression experiment at General Fusion was a repetitive non-destructive test to study plasma physics to magnetic target fusion compression. A compact torus (CT) is formed with a co-axial gun into a containment region with an hour-glass shaped inner flux conserver and an insulating outer wall. External coil currents maintain the CT off the outer wall (radial levitation) and then rapidly compress it inward. The optimal external coil configuration greatly improved both the levitated CT lifetime and the rate of shots with good flux conservation during compression. As confirmed by spectrometer data, the improved levitation field profile reduced plasma impurity levels by suppressing the interaction between plasma and the insulating outer wall during the formation process. Significant increases in the magnetic field, density, and ion temperature were routinely observed at magnetic compression despite the prevalence of an instability, thought be an external kink, at compression. Matching the decay rate of the levitation coil currents to that of the internal CT currents resulted in a reduced level of MHD activity associated with unintentional compression by the levitation field and a higher probability of long-lived CTs. An axisymmetric finite element MHD code that conserves system energy, particle count, angular momentum, and toroidal flux was developed to study CT formation into a levitation field and magnetic compression. An overview of the principal experimental observations and comparisons between simulated and experimental diagnostics are presented.

A model for plasma–neutral fluid interaction and its application to a study of CT formation in a magnetized Marshall gun

A model for plasma/neutral fluid interaction was developed and included in the implementation of non-linear MHD equations to a new code framework. The source rates of ion, electron, and neutral fluid momentum and energy due to ionization and recombination are derived using a simple method that enables the determination of the volumetric rate of thermal energy transfer from electrons to photons and neutral particles in the radiative recombination reaction. This quantity cannot be evaluated with the standard formal procedure of taking moments of the relevant collision operator and has been neglected in other studies. The plasma/neutral fluid interaction model was applied to study compact torus (CT) formation in the Super-Magnetized Ring Test (SMRT) and Spherical Compact Toroid (SPECTOR) magnetized Marshall guns, enabling clarification of the mechanisms behind the significant increases in CT electron density that are routinely observed well after formation on the SPECTOR experiment. Neutral gas, which remains concentrated below the gas valves after CT formation, diffuses up the gun barrel to the CT containment region where it is ionized, leading to the observed electron density increases. This understanding helps account for the exceptionally significant increase in temperature and the markedly reduced density, observed during the electrode edge biasing experiment conducted on SPECTOR. It is thought that edge fueling impediment, a consequence of a biasing-induced transport barrier, is largely responsible for the observed temperature increase and density decrease.

Optical fiber interferometer application for high electron density measurements on compact torus plasmas.

An optical fiber Mach-Zehnder interferometer at a wavelength of 1.55 µm has been developed for measurements of high electron density on compact torus (CT) plasmas with a high time resolution of 0.1 µs and high phase resolution of 6.4 × 10-4 rad. To improve density measurement accuracy, the phase noise of the interferometer has been investigated in detail and optimized. In the bench test, the interferometer was calibrated using a piezoelectric ceramic actuator with known stroke. Initial results on CT plasma show that the optical fiber interferometer provides reliable density measurements at two spatial locations and the bulk velocity of plasma can be determined by the method of time of flight.

Scaling arguments for a plasma experiment relevant to an interplanetary coronal mass ejection

We present the design of a plasma experiment scaled to be relevant to an interplanetary coronal mass ejection (ICME) traveling from the Sun to the Earth. The experiment would launch a compact torus of plasma (the scaled analog of the coronal mass ejected by the Sun) into a magnetized background plasma (the scaled analog of the interplanetary medium, the diffuse magnetized plasma that fills the Solar System). We present the complete process of scaling the ICME system to a laboratory experiment: beginning with the ICME features that we aim to replicate with a laboratory experiment, explaining how these desired features are translated into specific design constraints, and presenting a set of experimental parameters that meet those constraints. We find that the experiment would be possible with plasma conditions typically achieved in large-scale facilities such as the Big Red Ball at the University of Wisconsin or the LArge Plasma Device at the University of California, Los Angeles.

Manifolds of Isospectral Matrices and Hessenberg Varieties

Abstract We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.