Blade: A package for block-triangular form improved Feynman integrals decomposition
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Guan, Xin↗
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Competing higher order topological superconducting phases in triangular lattice magnet-superconductor hybrid systems
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Wong, Ka Ho (ORCID:0000000176026686)↗
Zonal flow screening in negative triangularity tokamaks
Here, this paper presents a comparative study of zonal flow screening in positive and negative triangularity tokamaks. Neoclassical screening sets the strength of zonal flow shear for fixed drive and damping. Orbit calculations show that the radial excursions of trapped particles are reduced in negative triangularity tokamaks, as compared to positive triangularity. Yet surprisingly, the neoclassical dielectric susceptibility actually increases with decreasing triangularity, such that the susceptibility is higher for negative triangularity than for positive triangularity. This is because the reduction in neoclassical polarization by shrinking the banana width is offset by the increase in neoclassical polarization by the enhancement of trapped fraction for negative triangularity. As a result, the zonal flow screening length is actually enhanced for negative triangularity, as compared to positive triangularity. Hence, the zonal flow residual is smaller for negative triangularity than for positive triangularity. Results from gyrokinetic simulations support the analytic calculations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Divertor characterization and access to dissipative divertor conditions in negative triangularity discharges in DIII-D
Reduced radial transport, short midplane-to-target parallel connection lengths, and a strong effect of cross-field drifts were responsible for the high densities required for detachment in strong negative triangularity (NT) configurations in DIII-D. Dissipative divertor conditions were achieved in NT discharges at different triangularity, injected power, plasma current, and toroidal field direction. Differences between negative and positive triangularity (PT) discharges are analyzed in this paper to understand the requirements for access to detached divertor conditions: power balance, geometry, radial transport and effect of cross-field particle drifts. Parametric dependencies of access to detachment on plasma current and power flowing into the scrape-off layer remained similar in negative and PT and impurity seeding was observed to reduce the density needed to detach by up to 30% at the expense of core impurity dilution. The impact of triangularity on core-edge integration was tested varying bottom triangularity at fixed top triangularity. The high density needed to detach was not intrinsic to the NT edge as shapes with positive lower triangularity and negative upper triangularity were able to detach at lower upstream densities while maintaining an ELM-free NT edge. Confinement degradation at deeper detachment levels was however observed in all NT shapes, often associated with radiation instabilities.
detachment↗
Effect of anisotropic thermal transport on tearing mode stability in negative versus positive triangularity plasmas
The combined effects of anisotropic thermal transport and the plasma shaping, including negative triangularity, on the n = 1 (n is the toroidal mode number) tearing mode (TM) stability are numerically investigated utilizing the MARS-F code [Liu et al., Phys. Plasmas 7, 3681–3690 (2000)]. While varying the plasma boundary triangularity, the TM stability is found to be dictated by the competing effects of the Shafranov shift induced stabilization and the bad-curvature induced destabilization. The negative triangularity shape increases the Shafranov shift (stabilizing) in the plasma core but also enlarges bad-curvature regions (destabilizing) near the plasma edge, with the net effect being largely destabilizing for the TM as compared to the positive triangularity counter-part. Large negative triangularity however can also lead to more stabilization for the plasma core-localized TM. Anisotropic thermal transport reduces the stabilizing effect on the TM associated with the favorable averaged curvature, resulting in more unstable core-localized TMs in both negative and positive triangularity plasmas. But the opposite effect can also take place for the edge-localized TM in finite-pressure plasmas with negative triangularity.
Physics↗
Difference in neoclassical edge flows between strongly negative and positive triangularities in the XGC gyrokinetic simulation
The neoclassical baseline study of a strongly negative triangularity (NT) plasma and the corresponding positive triangularity (PT) plasma is performed using the edge-specialized, total-f gyrokinetic code (X-point Gyrokinetic Code, XGC). A DIII-D-like plasma is used, based on the negative triangularity discharge of DIII-D #193 793. An artificial positive triangularity (PT) equilibrium has been constructed to compare the edge rotation physics at the same triangularity strength but with opposite sign, while keeping the same elongation and other geometric parameters. Carbon(6+) ions are added to the deuterium plasma at an experimentally relevant level. By using the experimental profile of carbon toroidal rotation profile as an input, XGC finds that the deuteron rotation is significantly different from the carbon rotation at the inboard and outboard midplanes, mostly caused by the difference in the Pfirsch–Schlüter rotation. More importantly, a significant difference in the X-point orbit loss physics, thus the rotation source, is found between the positive and negative triangularity equilibrium models. However, it is also found that the agreement between the present neoclassical simulation and the experimental NT data is validated only within the middle of the pedestal slope, indicating the importance of edge turbulence. Finally, this study could establish a baseline for the multiphysics, multiscale studies that include turbulence of negative triangularity plasmas.
Deuterium↗
Geometric dependencies of the mean E × B shearing rate in negative triangularity tokamaks
Abstract This paper presents a comparative study of the poloidal distribution of the mean E × B shearing rate for positive triangularity (PT) and negative triangularity (NT) tokamaks. The effects of flux surface up–down asymmetry due to asymmetric upper and lower triangularities are also considered. Both direct eddy straining and the effects on Shafranov shift feedback loops are examined. Shafranov shift increases the shearing rate at all poloidal angles for all triangularities, due to flux surface compression. The maximum shearing rate bifurcates at a critical triangularity δ crit ( ≲ 0 ) . Thus, the shearing rate is maximal off the outboard midplane for NT, while it is maximal on the outboard midplane for PT. For up–down asymmetric triangularity, the usual up–down symmetry of the shearing rate is broken. The shearing rate at the outboard midplane is lower for NT than for PT, suggesting that the shearing efficiency in NT is reduced. Implications for turbulence stabilisation and confinement improvement in high- β p NT and internal transport barrier discharges are discussed.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Effect of negative triangularity on tearing mode stability in tokamak plasmas
Abstract The influence of negative triangularity (NT) of the plasma shape on the n = 1 (n is the toroidal mode number) tearing mode (TM) stability has been numerically investigated, with results compared to that of the positive triangularity (PT) counterpart. By matching the safety factor profile for a series of toroidal equilibria, several important plasma parameters, including the triangularity, the plasma equilibrium pressure, the plasma resistivity as well as the toroidal rotation, have been varied. The TM localized near the plasma edge is found to be more unstable in the NT plasmas as compared to the PT counterpart. The fundamental reason for this difference is the lack of favorable average curvature stabilization in NT configurations. Direct comparison of the Mercier index corroborates this conclusion. For the core-localized mode, where the difference in the local triangularity between NT and PT becomes small and the curvature stabilization is significantly reduced, larger Shafranov shift in the plasma core associated with the NT configuration results in more stable TM. The plasma toroidal flow generally stabilizes the TM in plasmas with both NTs and PTs. The flow stabilization is however weaker in the case of negative triangularity with finite plasma pressure.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Understanding the negative triangularity ELM trigger and ELM free state on DIII-D with ECE-imaging
Here, the Electron Cyclotron Emission Imaging (ECEI) diagnostic was used to observe a finite-n interchange mode structure in the edge of negative triangularity shaped plasmas on DIII-D. At a small negative triangularity (δ u = -0.2), the plasma is in the H-mode with ELMs that are triggered by a low-n interchange mode. At a larger negative triangularity (δ u = -0.4) and low NBI power (2 MW), a dithering oscillation is observed that is triggered by a low-n interchange mode, whereas at higher NBI power (>2 MW), the edge reverts to L-mode and the low-n interchange mode is present continuously. In all cases, the edge pressure gradient is clamped when the interchange mode is present. It is concluded that the low-n interchange mode prevents the plasma from transitioning to H-mode at a large negative triangularity. This agrees with linear BOUT++ simulations which suggest that the interchange-type MHD can be a resistive ballooning mode whereby resistivity can significantly increase the finite-n ballooning mode growth rate. The absence of H-mode at large negative triangularity can, thus, be explained by the excitation of low-n pressure driven resistive ballooning modes in the plasma edge.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Comparison of detachment in Ohmic plasmas with positive and negative triangularity
Abstract In recent years, negative triangularity (NT) has emerged as a potential high-confinement L-mode reactor solution. In this work, detachment is investigated using core density ramps in lower single null Ohmic L-mode plasmas across a wide range of upper, lower, and average triangularity (the mean of upper and lower triangularity: δ ) in the TCV tokamak. It is universally found that detachment is more difficult to access for NT shaping. The outer divertor leg of discharges with δ ≈ − 0.3 could not be cooled to below 5 eV through core density ramps alone. The behavior of the upstream plasma and geometrical divertor effects (e.g. a reduced connection length with negative lower triangularity) do not fully explain the challenges in detaching NT plasmas. Langmuir probe measurements of the target heat flux widths ( λ q ) were constant to within 30% across an upper triangularity scan, while the spreading factor S was lower by up to 50% for NT, indicating a generally lower integral scrape-off layer width, λ int . The line-averaged core density was typically higher for NT discharges for a given fuelling rate, possibly linked to higher particle confinement in NT. Conversely, the divertor neutral pressure and integrated particle fluxes to the targets were typically lower for the same line-averaged density, indicating that NT configurations may be closer to the sheath-limited regime than their PT counterparts, which may explain why NT is more challenging to detach.
Physics↗
On the effect of negative triangularity on ion temperature gradient turbulence in tokamaks
Considering the same magnetic equilibrium and plasma conditions as in Duff et al., we perform linear and nonlinear simulations of electrostatic ion temperature gradient turbulence investigating the role of triangularity δ. Differently from what was previously reported, we find that triangularity increases the transport level regardless of its sign, but more strongly when δ is positive. Here, for the case analyzed, we identify the shear of triangularity as the critical parameter determining the transport level, indicating that even in the local limit negative triangularity can reduce the transport efficiently, suggesting that confinement improvement can also be expected for larger devices.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Verification of the kinetic electron role in the microinstabilities in a negative triangularity model equilibrium
Effect of kinetic electrons on negative triangularity plasmas has been investigated and compared against the corresponding positive triangularity plasmas, using the global gyrokinetic code X-point Gyrokinetic Code with scale-separated delta-f option without Coulomb collisions. Our model magnetic equilibria have strong positive and negative triangularities and weak magnetic shear. However, unusually large ρ i > a and low density plasmas are chosen to maximize the nonlocal effect to investigate the finite ρ i effect and to be clearly away from kinetic ballooning modes. Similar conclusions to previous flux tube and global simulations have been obtained in this highly nonlocal model plasma: it is essential to include kinetic electrons in the micro-instability study of negative triangularity plasmas. Most physics findings agree with existing reports, with some disagreement. We offer a new “effective trapping fraction” concept that can add to the explanation of the growth rate difference between NT and PT plasmas, pointing to the significant variation in trapped particle fractions that have turning points in the mode growth regions.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Alfvén mode spectrum in the advanced steady-state tokamak scenario with negative triangularity
In the earlier studies [Zheng et al. Fundam. Plasma Phys. 10, 100051 (2024)], the negative triangularity tokamak was shown to have a unique and outstanding feature for the steady-state confinement. In addition to relieving the divertor heat load concern, it can reach a much higher beta limit in the nearly steady-state confinement, about twice the Troyon limit in the positive triangularity case. In this work, we extend the investigation to the Alfvén spectrum and toroidal Alfvén eigenmodes (TAEs). This helps to evaluate the property of energetic particle confinement in this advanced scenario of a negative triangularity tokamak. We found that, because of the large safety factor and reversed magnetic shear in the core plasma region near the magnetic axis, the TAE gap tends to be closed or have only a short radial extension between the continua. in the on-axis core region and open primarily in the off-axis region, where TAEs are shown to be present. This indicates that energetic particle confinement in the core region near the magnetic axis, where nuclear fusion reactions prevail, is not seriously affected by the excitation of TAEs in this scenario. The configuration with off-axis TAEs helps the confinement of energetic particles as compared to the configuration with near-axis TAEs. The results show that the advanced steady-state tokamak scenario with negative triangularity, because of the rapid variation of the safety factor, also has a favorable feature for energetic particle confinement.
Fourier analysis↗
Plasma edge and scrape-off layer turbulence in gyrokinetic simulations of negative triangularity plasmas
Gyrokinetic simulations in the long-wavelength or drift-kinetic limit are carried out of DIII-D inner-wall-limited (IWL) plasmas to investigate the effect of triangularity on edge and scrape-off layer (SOL) turbulence. The effect of neutral interactions and triangularity on plasma blobs is explored due to the impact blobs can have in setting the SOL width or introducing impurities through interactions with plasma-facing components. Seeded blob simulations with neutrals in shaped SOL scenarios demonstrate that increasing elongation, triangularity, or Shafranov shift decreases radial blob velocities, but neutral interactions have a minor effect. Fully turbulent simulations of DIII-D IWL plasmas include both open- and closed-field-line regions. The negative triangularity (NT) simulation has lower average core T e , lower normalized T e fluctuations, and lower fluxes, but a greater number of coherent structures (blobs) identified with increased size and velocity, on average. Density and electron temperature profiles are within a factor of 2 of experimental values. In conclusion, the increased trapped electron particle fraction in NT simulations is consistent with previous studies.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Manipulating Cu vacancy in Yb triangular lattice Materials: Investigating the quantum disordered compound CuYbSe 2 and spin glass Cu 2.73 YbP 2
Yb-based triangular lattice materials have attracted significant attention due to their potential for hosting exotic magnetic quantum states. In this paper, we report the structural, magnetic, and thermodynamic properties of two Yb triangular lattice compounds with vacancy disorder at nonmagnetic Cu sites: CuYbSe 2 and Cu 2.73 YbP 2 . Single-crystal x-ray diffraction determines notable Cu-site vacancies in both materials. Unlike a single Cu site with 0.5 occupancy in CuYbSe 2 , Cu 2.73 YbP 2 only presents 27% vacancies on one of three Cu sites. Magnetic susceptibility measurements indicate dominant antiferromagnetic interactions between Yb 3+ ions. The effective magnetic moment is consistent with an effective spin-1/2 state of Yb 3+ . CuYbSe 2 exhibits strong magnetic anisotropy between in-plane and out-of-plane directions with the estimated exchange interactions 𝐽 𝑎 /𝑘 B = 6.48 K and 𝐽 𝑐 /𝑘 B = 2.55 K. No magnetic ordering is observed down to 0.4 K, supporting its quantum disordered ground state and candidacy as a quantum spin liquid. In contrast, Cu 2.73 YbP 2 shows clear evidence of spin freezing at 4.5 K in both magnetization and heat capacity data, indicating a spin glass state. The field-induced magnetic orders are found in CuYbSe 2 . These results highlight the critical role of structural vacancies in perturbing Yb-based triangular lattices. In conclusion, we demonstrate that while the intrinsic Yb triangular lattices are comparable, the presence of Cu vacancies can fundamentally alter the magnetic ground state—tuning the system between quantum disordered and spin glass behavior.
Frustrated magnetism↗
Structure and magnetism of the triangular lattice material YbBO 3
Abstract YbBO 3 is a member of the orthoborate family of materials which contains a triangular arrangement of Yb 3+ ions. Here we study the physical properties of YbBO 3 with neutron diffraction, inelastic neutron scattering, specific heat, and ac susceptibility measurements. The neutron diffraction measurements confirm that our samples of YbBO 3 crystallize in the monoclinic space group C 2 / c (#15) which contains two crystallographically distinct Yb 3+ sites decorating a slightly distorted triangular lattice. Heat capacity and ac susceptibility measurements indicate a potential transition to magnetic order at 0.4 K. In agreement with these observations, neutron diffraction measurements at 0.044 K observe magnetic Bragg peaks which can be indexed by a propagation vector of (0 0 1). Although determining a unique spin configuration corresponding to the observed magnetic Bragg peaks is not possible, model refinements indicate that the ordered moments are likely in the range of 0.4–0.9 μ B and, notably, require moments on both Yb sites. In addition to the magnetic Bragg peaks, diffuse scattering at low Q is observed indicating that the transition does not correspond to complete long range magnetic order. The two-site picture for YbBO 3 is further evidenced by the number of crystal field excitations observed by inelastic neutron scattering measurements. Together these results show that YbBO 3 is a two-site triangular lattice material with signatures of long-range order as well as shorter ranged spin correlations.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Characterization of the ELM-free negative triangularity edge on DIII-D
Tokamak plasmas with strong negative triangularity (NT) shaping typically exhibit fundamentally different edge behavior than conventional L-mode or H-mode plasmas. On DIII-D, every plasma with sufficiently negative triangularity (δ < δ crit ≃ -0.12) is found to be inherently free of edge localized modes (ELMs), even at injected powers well above the predicted L-H power threshold. It is also possible to access an ELM free state at weaker average triangularities provided that at least one of the two x-points is still sufficiently negative. Access to the ELM-free NT scenario is found to coincide with the closure of the second stability region for infinite-n ballooning modes, suggesting that ballooning stability may play a role in limiting the accessible pressure gradient in NT plasmas. Despite this, NT plasmas are able to support small pedestals and are typically characterized by an enhancement of edge pressure gradients beyond those found in traditional L mode plasmas. Further, the pressure gradient inside of this small pedestal is unusually steep, allowing access to high core performance that is competitive with other ELM-free regimes previously achieved on DIII-D. Since ELM-free operation in NT is linked directly to the magnetic geometry, NT fusion pilot plants are predicted to maintain advantageous edge conditions even in burning plasma regimes, potentially eliminating reactor core-integration issues caused by ELMs.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Gyrokinetic electromagnetic particle simulations in triangular meshes with C1 finite elements
The triangular mesh-based gyrokinetic scheme enables comprehensive axis-to-edge studies across the entire plasma volume. Our approach employs triangular finite elements with first-derivative continuity (C1), building on previous work to facilitate gyrokinetic simulations. Additionally, we have adopted the mixed variable/pullback scheme for gyrokinetic electromagnetic particle simulations. The filter-free treatment in the poloidal cross-section with triangular meshes introduces unique features and challenges compared to previous treatments using structured meshes. Our implementation has been validated through benchmarks using ITPA-toroidicity-induced Alfvén eigenmode parameters, showing its capability in moderate to small electron skin depth regimes. Additional examinations using experimental parameters confirm its applicability to realistic plasma conditions.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗