Full Torus Research Articles

Toroidal modified Miller-Turner CME model in EUHFORIA: Validation and comparison with flux rope and spheromak

Rising concerns about the impact of space-weather-related disruptions demand modelling and reliable forecasting of coronal mass ejection (CME) impacts. In this study, we demonstrate the application of the modified Miller-Turner (mMT) model implemented within EUropean Heliospheric FORecasting Information Asset (EUHFORIA) in forecasting the geo-effectiveness of observed coronal mass ejection (CME) events in the heliosphere. Our goal is to develop a model that not only has a global geometry, in order to improve overall forecasting, but is also fast enough for operational space-weather forecasting. We test the original full torus implementation and introduce a new three-fourths Torus version called the Horseshoe CME model. This new model has a more realistic CME geometry, and overcomes the inaccuracies of the full torus geometry. We constrain the torus geometrical and magnetic field parameters using observed signatures of the CMEs before, during, and after the eruption. We perform EUHFORIA simulations for two validation cases -- the isolated CME event of 12 July 2012 and the CME--CME interaction event of 8-10 September 2014. We performed an assessment of the model's capability to predict the most important $B_z$ component using the advanced dynamic time-warping (DTW) technique. The Horseshoe model predictions of CME arrival time and geo-effectiveness for both validation events compare well with the observations and are weighed against the results obtained with the spheromak and FRi3D models, which were already available in EUHFORIA. The runtime of the Horseshoe model simulations is close to that of the spheromak model, which is suitable for operational space weather forecasting. However, the capability of the magnetic field prediction at 1 AU of the Horseshoe model is close to that of the FRi3D model. In addition, we demonstrate that the Horseshoe CME model can be used for simulating successive CMEs in EUHFORIA, overcoming a limitation of the FRi3D model.

Two-stage crash process in resistive drift ballooning mode driven ELM crash

We report a two-stage crash process in edge localized mode (ELM) driven by resistive drift-ballooning modes (RDBMs) numerically simulated in a full annular torus domain with a scale-separated four-field reduced MHD (RMHD) model using the BOUT++ framework. In the early nonlinear phase, the small first crash is triggered by linearly unstable RDBMs, and m/n=2/1 magnetic islands are nonlinearly excited by nonlinear coupling of RDBMs as well as their higher harmonics. Here, m is the poloidal mode number, n is the toroidal mode number, the q = 2 rational surface exists near the pressure gradient peak, and q is the safety factor. Simultaneously, middle-n RDBM turbulence develops but is poloidally localized around X-points of the magnetic islands, leading to the small energy loss. The second large crash occurs in the late nonlinear phase. Higher harmonics of m/n=2/1 magnetic islands well develop around the q = 2 surface via nonlinear coupling and make the magnetic field stochastic by magnetic island overlapping. Turbulence heat transport develops at X-points of higher harmonics of m/n=2/1 magnetic islands, resulting in the turbulence spreading in the poloidal direction. The large second crash is triggered when the turbulence covers the whole poloidal region so that the magnetic island generation and magnetic field stochastization before the large crash can be interpreted as ELM precursors. It is concluded that the ELM trigger is attributed to the turbulent spreading in the poloidal direction in synchronization with the magnetic field stochastization and the crash is driven by E × B convection rather than the conventional Rechester–Rosenbluth anomalous electron heat transport.

Group-valued invariant of knots in the full torus

Knot concordance plays a crucial role in the low-dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group theoretical techniques; it is completely combinatorial, and the groups are very easy to deal with.

Glasner property for unipotently generated group actions on tori

A theorem of Glasner from 1979 shows that if \(A \subset \mathbb{T}= \mathbb{R}/\mathbb{Z}\) is infinite, then for each ϵ > 0 there exists an integer n such that nA is ϵ-dense and Berend—Peres later showed that in fact one can take n to be of the form f(m) for any non-constant f(x) ∈ ℤ[x]. Alon and Peres provided a general framework for this problem that has been used by Kelly—Lê and Dong to show that the same property holds for various linear actions on \({\mathbb{T}^d}\). We complement the result of Kelly—Lê on the ϵ-dense images of integer polynomial matrices in some subtorus of \({\mathbb{T}^d}\) by classifying those integer polynomial matrices that have the Glasner property in the full torus \({\mathbb{T}^d}\). We also extend a recent result of Dong by showing that if Γ ≤ SLd(ℤ) is generated by finitely many unipotents and acts irreducibly on ℝd, then the action \(\Gamma \curvearrowright {\mathbb{T}^d}\) has a uniform Glasner property.

3D full wave fast wave modeling with realistic HHFW antenna geometry and SOL plasma in NSTX-U

This paper reports the significant advancement of our ability to model and to understand how RF waves interact with the SOL plasma, by developing for the full torus 3D scrape-off layer (SOL) plasma simulation together with the antenna and core plasma in the NSTX-U device in HHFW frequency regime for a cold plasma model. The present paper extends the previous modeling (Bertelli et al 2020 AIP Conf. Proc. 2254 030001) to a full 3D device geometry including realistic antenna geometry in order to capture a more realistic experimental situation including the fully 3D wave field effects and the antenna plasma interaction in the SOL plasma, and, at the same time, resulting core wave propagation. The central tool of the present work is the Petra-M code, which is a newly developed state-of-the-art generic electromagnetic simulation tool for modeling RF wave propagation based on MFEM (http://mfem.org), an open source scalable C++ finite element method library.

A BOUT++ extension for full annular tokamak edge MHD and turbulence simulations

For tokamak edge plasma simulation, a plasma simulation framework BOUT++ employs a dual coordinate system to simulate moderate-n and high-n plasma instability with reasonable computational cost, where n is the toroidal mode number. This coordinate system however limits the computational domain to the toroidal wedge (full torus divided into N parts in the toroidal direction) for computational efficiency and the use of flute-ordering approximation in the field solver calculating the flow potential from the vorticity which may not be valid for low-n modes. Improving numerical treatment of low-n modes is however indispensable to address simulations of low-n current-driven edge localized mode (ELM), ELM control by resonant magnetic perturbations (RMPs), edge turbulence with RMPs and so on. In this work, BOUT++ is extended to simulate the interplay between n=0, low-n and high-n plasma components in a full annular tokamak edge domain through hybrid modeling of the flow potential and the vorticity. Low-n modes of flow potential are calculated in an orthogonal flux surface coordinate and high-n modes in the dual coordinate system separately in Fourier space. The proposed scheme can capture an interplay between n=1 global modes and high-n turbulence during pedestal collapse in a full annular torus domain with a circular cross section.

On long knots in the full torus

The aim of this paper is to realize the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of particular interest because of their relation to Legendrian knots, knotoids, [Formula: see text]-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [V. O. Manturov, A free-group valued invariant of free knots, preprint (2020), arXiv:2012.15571v2]. Long knots naturally appear in the study of classical knots [T. Fiedler, More [Formula: see text]-cocycles for classical knots, preprint (2020), arXiv:2004.04624; A. Mortier, A Kontsevich integral of order [Formula: see text], preprint (2018), arXiv:1810.05747].

Solar energy toroidal concentrators

This article deals with toroidal concentrators of solar radiation. The developed design configurations have been described. The base model is a semi-torus obtained by cutting full torus along the symmetry plane, the inner surface of semi-toroidal surface is of specular type. The inner surface of semi-toroidal surface is of specular type. The semi-circles of the concentrator cross-section are the generatrices of the surfaces. The radiation receiver is mounted in the midsection, and the radiation receiver symmetry point coincides with that of the concentrator. Results of experimental and theoretical studies of their characteristics have been discussed. Various models employing the advantages of toroidal concentrators have been considered. Analysis results of the most preferable radiation receivers have been presented. Layout options of integration of photoelectric plants equipped with toroidal concentrators of solar radiation into buildings and landscape design objects have been described. Physical limitations of application of installations based on concentrators of this type have been defined and substantiated. Methods for evaluating coast-related characteristics of toroidal concentrators depending on values of their output parameters and on those of design have been presented.

Minimal lamellar structures in a periodic FitzHugh–Nagumo system

A singular limit of a FitzHugh–Nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions. We study the stationary lamellar set and give a criterion to select out the one with the lowest energy. Such an optimal structure is called a minimal lamella. While the empty set or the full torus is a global minimizer for appropriate parameter regimes, the minimal lamellae beat both in other circumstances. The concept of minimal lamella points out that a preferred 1D mesh size is universal.

What happens to full-f gyrokinetic transport and turbulence in a toroidal wedge simulation?

In order to save the computing time or to fit the simulation size into a limited computing hardware in a gyrokinetic turbulence simulation of a tokamak plasma, a toroidal wedge simulation may be utilized in which only a partial toroidal section is modeled with a periodic boundary condition in the toroidal direction. The most severe restriction in the wedge simulation is expected to be in the longest wavelength turbulence, i.e., ion temperature gradient (ITG) driven turbulence. The global full-f gyrokinetic code XGC1 is used to compare the transport and turbulence properties from a toroidal wedge simulation against the full torus simulation in an ITG unstable plasma in a model toroidal geometry. It is found that (1) the convergence study in the wedge number needs to be conducted all the way down to the full torus in order to avoid a false convergence, (2) a reasonably accurate simulation can be performed if the correct wedge number N can be identified, (3) the validity of a wedge simulation may be checked by performing a wave-number spectral analysis of the turbulence amplitude |δΦ| and assuring that the variation of δΦ between the discrete kθ values is less than 25% compared to the peak |δΦ|, and (4) a frequency spectrum may not be used for the validity check of a wedge simulation.

The TOKAM3X code for edge turbulence fluid simulations of tokamak plasmas in versatile magnetic geometries

The new code TOKAM3X simulates plasma turbulence in full torus geometry including the open field lines of the Scrape-off Layer (SOL) and the edge closed field lines region in the vicinity of the separatrix. Based on drift-reduced Braginskii equations, TOKAM3X is able to simulate both limited and diverted plasmas. Turbulence is flux driven by incoming particles from the core plasma and no scale separation between the equilibrium and the fluctuations is assumed so that interactions between large scale flows and turbulence are consistently treated. Based on a domain decomposition, specific numerical schemes are proposed using conservative finite-differences associated to a semi-implicit time advancement. The process computation is multi-threaded and based on MPI and OpenMP libraries. In this paper, fluid model equations are presented together with the proposed numerical methods. The code is verified using the manufactured solution technique and validated through documented simple experiments. Finally, first simulations of edge plasma turbulence in X-point geometry are also introduced in a JET geometry.

Laminar-Turbulent Bifurcation Scenario in 3D Rayleigh-Benard Convection Problem

We are considering two initial-boundary value problems for Rayleigh-Benard convection in Oberbeck-Boussinesq approximation for incompressible fluid in 3D-rectangular domain with 4:4:1 geometric ratio with periodicity in two directions and cubic domain with 1:1:1 ratio and zero velocity and temperature gradient boundary conditions. For this purpose, we use two numerical method: one is a Pseudo-Spectral-Galerkin method with trigonometric-Chebyshev polynomials and the other is finite element/volume method with WENO interpolation for advection term. Numerical methods are presented shortly and are benchmarked against known DNS data and against one another (for quasi-periodic domain problem). Then we perform stability analysis using analytical expression for main stationary solutions and eigenvalue numerical analysis by applying Implicitly Restarted Arnoldi (IRA) method. The IRA is used to perform linear stability analysis, find bifurcations of stationary points and analyze eigenvalues of monodromy matrices. Thus characteristic exponents of the system for time periodic solutions (limited cycles of various periods and resonance invariant tori) are computed. We show, numerically, the existence of multistable rotes to chaos through chaotic fractal attractors, full Feigenbaum-Sharkovski cascades and multidimensional torus attractors (Landau-Hopf scenario). The existence of these attractors is shown through analysis of phase subspaces projections, Poincare sections and eigenvalue analysis of numerically computed DNS data. These attractors burst into chaos with the increase of Rayleigh number either through resonance and phase-locking or through emergence of singular chaotic attractors.

On the range of a random walk in a torus and random interlacements

Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a $k$-iterated log factor from those on the full torus for arbitrary $k$. The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a random walk on random interlacements.

3D thermal-hydraulic analysis of an ITER vacuum vessel regular Field Joint

The ITER vacuum vessel (VV), located inside the cryostat and housing the in-vessel components, is made of 9 40° sectors, connected through splice plates to form the full torus. The regions at the interface between adjacent sectors are the so-called Field Joints (FJs). While each sector has its own cooling loop to remove the heat deposition due to nuclear heating, each FJ is separately cooled. Individual inlet/outlet pipes for the water flow are thus provided for each FJ, located in the outboard bottom segment and on the upper port frame, respectively. The coolant flow splits in two streams, inboard and outboard, passing through the borated In-Wall Shielding (IWS). In this paper we present the 3D steady state thermal-hydraulic analysis of one so-called regular FJ (RFJ), at the interface between two VV regular sectors, using the commercial CFD software ANSYS-FLUENT®. The water flow field, the pressure drop, the temperature maps and the heat transfer coefficients are computed, and the effects of considering different levels of simplification of the IWS model, as well as the influence of buoyancy (natural convection), are discussed.

Heat equation approach to geometric changes of the torus Laughlin state

We study the second quantized or guiding center description of the torus Laughlin state. Our main focus is the change of the guiding center degrees of freedom with the torus geometry, which we show to be generated by a two-body operator. We demonstrate that this operator can be used to evolve the full torus Laughlin state at given modular parameter $\ensuremath{\tau}$ from its simple (Slater-determinant) thin torus limit, thus giving rise to a new presentation of the torus Laughlin state in terms of its ``root partition'' and an exponential of a two-body operator. This operator therefore generates, in particular, the adiabatic evolution between Laughlin states on regular tori and the quasi-one-dimensional thin torus limit. We make contact with the recently introduced notion of a ``Hall viscosity'' for fractional quantum Hall states to which our two-body operator is naturally related and which serves as a demonstration of our method to generate the Laughlin state on the torus.

Archimedean zeta integrals on GL n × GL m and SO2n+1 × GL m

In this paper, we evaluate archimedean zeta integrals for automorphic L-functions on GL n × GL n-1+l and on SO2n+1 × GL n+l , for l = −1, 0, and 1. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals. When l = 0 or l = 1, the archimedean zeta integrals amount to integrals over the full torus. We show that, as has been predicted by Bump for such domains of integration, these zeta integrals are equal to the corresponding local L-factors—which are simple rational combinations of Gamma functions. (In the cases of GL n × GL n-1 and GL n × GL n this has, in large part, been shown previously by the second author of the present work, though the results here are more general in that they do not require the assumption of trivial central characters. Our techniques here are also quite different. New formulas for GL(n, R) Whittaker functions, obtained recently by the authors of this work, allow for substantially simplified computations). In the case l = −1, we express our archimedean zeta integrals explicitly in terms of Gamma functions and certain Barnes-type integrals. These evaluations rely on new recursive formulas, derived herein, for GL(n, R) Whittaker functions. Finally, we indicate an approach to certain unramified calculations, on SO2n+1 × GL n and SO2n+1 × GL n+1, that parallels our method herein for the corresponding archimedean situation. While the unramified theory has already been treated using more direct methods, we hope that the connections evoked herein might facilitate future archimedean computations.

Topological self-joinings of Cartan actions by toral automorphisms

We show that if r≥3 and α is a faithful Zr-Cartan action on a torus Td by automorphisms, then any closed subset of (Td)2 which is invariant and topologically transitive under the diagonal Zr-action by α is homogeneous, in the sense that it is either the full torus (Td)2, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.

Gyrokinetic simulations with external resonant magnetic perturbations: Island torque and nonambipolar transport with plasma rotation

Static external resonant magnetic field perturbations (RMPs) have been added to the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J. Comp. Phys. 186, 545 (2003)]. This allows nonlinear gyrokinetic simulations of the nonambipolar radial current flow jr, and the corresponding j→×B→ plasma torque (density) R[jrBp/c], induced by magnetic islands that break the toroidal symmetry of a tokamak. This extends the previous GYRO formulation for the transport of toroidal angular momentum (TAM) [R. E. Waltz, G. M. Staebler, J. Candy, and F. L. Hinton, Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)]. The focus is on electrostatic full torus radial slice simulations of externally induced q=m/n=6/3 islands with widths 5% of the minor radius or about 20 ion gyroradii. Up to moderately strong E×B rotation, the island torque scales with the radial electric field at the resonant surface Er, the island width w, and the intensity I of the high-n micro-turbulence, as ErwI. The radial current inside the island is carried (entirely in the n=3 component) and almost entirely by the ion E×B flux, since the electron E×B and magnetic flutter particle fluxes are cancelled. The net island torque is null at zero Er rather than at zero toroidal rotation. This means that while the expected magnetic braking of the toroidal plasma rotation occurs at strong co- and counter-current rotation, at null toroidal rotation, there is a small co-directed magnetic acceleration up to the small diamagnetic (ion pressure gradient driven) co-rotation corresponding to the zero Er and null torque. This could be called the residual stress from an externally induced island. At zero Er, the only effect is the expected partial flattening of the electron temperature gradient within the island. Finite-beta GYRO simulations demonstrate almost complete RMP field screening and n=3 mode unlocking at strong Er.

Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulence

The Eulerian gyrokinetic turbulence code gene has recently been extended to a full torus code. Moreover, it now provides Krook-type sources for gradient-driven simulations where the profiles are maintained on average as well as localized heat sources for a flux-driven type of operation. Careful verification studies and benchmarks are performed successfully. This setup is applied to address three related transport issues concerning nonlocal effects. First, it is confirmed that in gradient-driven simulations, the local limit can be reproduced—provided that finite aspect ratio effects in the geometry are treated carefully. In this context, it also becomes clear that the profile widths (not the device width) may constitute a more appropriate measure for finite-size effects. Second, the nature and role of heat flux avalanches are discussed in the framework of both local and global, flux- and gradient-driven simulations. Third, simulations dedicated to discharges with electron internal barriers are addressed.

Fabrication of the KSTAR vacuum vessel and ports

The construction of the steady-state-capable superconducting KSTAR tokamak is in close proximity to the finalization. As one of the main components of the KSTAR device, the vacuum vessel is designed and manufactured during the construction period. The KSTAR vacuum vessel is composed of two large sectors forming the 337.5° of a full torus, and the remaining 22.5° section consisting of 24 small pieces. The large two sectors were welded at the site, and the 22.5° space was used for toroidal field coil assembly. The remaining 22.5° section of the vacuum vessel was assembled after 16 toroidal field coils assembly. The total 72 penetration ports were used to connect the vacuum vessel body and the cryostat. The major fabrication activity started in January 2003 after the finalization of fabrication design. The final components and structures were warehoused in June 2004 and site assembly is finished in 2007. Details of analysis, shop fabrication, and inspection results of the vacuum vessel including ports are summarized in the present work.