Rational Surfaces Research Articles

Rational surfaces with a non‐arithmetic automorphism group

AbstractIn [Algebraic surfaces and hyperbolic geometry, Cambridge University Press, Cambridge, 2012], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur in Section 7 of [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14–50]. We give examples of rational surfaces with the same property. Our examples are Looijenga pairs, that is, there is a connected singular nodal curve such that .

Numerical study of turbulent eddy self-interaction in tokamaks with low magnetic shear. Part I: Linear simulations

Abstract This study examines the impact of low and zero magnetic shear, along with rational values of safety factor, on linear mode behavior in tokamak plasmas. We focus on how magnetic field line topology and parallel domain length influence linear mode structures and growth rates as magnetic shear decreases. Our results
show that as magnetic shear is reduced to low values, linear modes can transition between different branches, significantly affecting their characteristics. At zero magnetic shear, accurate modeling requires precisely capturing magnetic topology, as small deviations near rational surfaces can greatly impact growth rates. These findings are extended to nonlinear simulations in a companion paper (Volčokas, 2024), revealing that long turbulent eddies and magnetic field line topology play a crucial role in turbulence self-interaction and plasma transport.

Initial results from neoclassical tearing mode stabilization experiment in KSTAR high normalized beta plasmas

Abstract Active stabilization of neoclassical tearing modes (NTMs) is critical for high beta plasma operation in the KSTAR tokamak. In recent device operation, an experiment was conducted to develop a m/n = 2/1 NTM stabilization in high normalized beta (βN) plasmas having βN > 3 by using electron cyclotron current drive (ECCD). The experiment is designed to first demonstrate the direct mode stabilization effect of the device EC system by varying the ECCD deposition location around the mode rational surface to prepare for future NTM stabilization in KSTAR using feedback schemes. In the experiment, the toroidal magnetic field strength, BT, is reduced to 1.5–1.6 T to create a highly localized ECCD profile on a large island width expected to produce a high stabilization effect. To align the ECCD with the mode, BT is varied between discharges by keeping the EC wave injection angles fixed to move the current deposition location across the q = 2 mode rational surface in ~1 cm steps along the plasma midplane. The result shows a prompt reduction of the mode amplitude by ~60% with a partial recovery of the loss of stored energy when the ECCD with 0.7 MW power is applied promptly after the mode onset at BT = 1.54 T. This abrupt reduction of the mode amplitude is significantly weaker or disappears in other discharges having slightly different BT in which the ECCD is deposited a few centimeters away. This result indicates a direct interaction between the driven EC current and the radially localized island structure first observed in the device. The EC ray-tracing analysis using the TORAY code shows that the applied EC-driven current aligns with the mode rational surface when the mode amplitude is observed to decrease. The stability of the observed 2/1 NTM is examined by constructing the modified Rutherford equation (MRE). The calculated EC power requirement for complete mode stabilization by assuming perfect alignment of ECCD on the mode is 0.8-1.7 MW by considering the uncertainties in the equation. This MRE-computed mode stability agrees with the experiment. The result projects that the present KSTAR EC system can stabilize the 2/1 NTM disrupting high βN plasmas, and provides the requirement of the mode-ECCD alignment for complete mode stabilization in future experiments in which an accurate alignment is planned to be made by feedback schemes being developed.

Rare-Earth Metal-Based Materials for Hydrogen Storage: Progress, Challenges, and Future Perspectives.

Rare-earth-metal-based materials have emerged as frontrunners in the quest for high-performance hydrogen storage solutions, offering a paradigm shift in clean energy technologies. This comprehensive review delves into the cutting-edge advancements, challenges, and future prospects of these materials, providing a roadmap for their development and implementation. By elucidating the fundamental principles, synthesis methods, characterization techniques, and performance enhancement strategies, we unveil the immense potential of rare-earth metals in revolutionizing hydrogen storage. The unique electronic structure and hydrogen affinity of these elements enable diverse storage mechanisms, including chemisorption, physisorption, and hydride formation. Through rational design, nanostructuring, surface modification, and catalytic doping, the hydrogen storage capacity, kinetics, and thermodynamics of rare-earth-metal-based materials can be significantly enhanced. However, challenges such as cost, scalability, and long-term stability need to be addressed for their widespread adoption. This review not only presents a critical analysis of the state-of-the-art but also highlights the opportunities for multidisciplinary research and innovation. By harnessing the synergies between materials science, nanotechnology, and computational modeling, rare-earth-metal-based hydrogen storage materials are poised to accelerate the transition towards a sustainable hydrogen economy, ushering in a new era of clean energy solutions.

Nonlinear dynamics and effects of fast-ion driven instabilities in HL-2A NBI heating high-βN H-mode plasmas

High-performance plasmas had been obtained with hybrid scenarios on HL-2A in past several years, and the quality factor G=H89βN/q952 is around 0.3-0.4. In these high-βN H-mode plasmas with βN>2.5 and H98>1.2 Chen et al. (2022) [24], there are multiple energetic-ion driven instabilities, such as toroidal Alfvén eigenmode (TAE) and beta-induced Alfvén eigenmode (BAE), and there are complex magnetohydrodynamic (MHD) dynamics, e.g. nonlinear mode-mode and mode-particle interactions. The TAE amplitude increases explosively, and the frequency is fast chirping in the range of f=90−150 kHz. The TAE toroidal mode-number is n=3−4, and they localize in outer plasma region. The BAE amplitude also increases explosively, and the BAE frequency is fast chirping in the range of f=50−90 kHz. The toroidal mode-number of dominant BAEs is n=2, and these modes localize at q=2 rational surface where there is a large radius internal transport barrier (ITB). Generally, the energetic-ion fraction is low in outer plasma region. It indicates that thermal ions at large radius ITB location possibly provide additional drive and destabilize the BAE in company with energetic-ions. These TAEs/BAEs occur regularly in high-βN region and before large type-I ELM burst. To further illuminate the characteristics of these Alfvénic modes, the 2D and radial mode structures had been calculated by using a well-benchmarked general Alfvénic mode eigenvalue code (MAS). The experimental results suggest that these Alfvénic modes may have important effects on the pedestal transport and ELM-onset.

Detection of Internal Transport Barrier in the T-10 Tokamak Using Thomson Scattering Diagnostics

At the T-10 tokamak, the Thomson scattering diagnostics was upgraded in 2016. The new system is based on the Nd: YAG laser with the pulse repetition frequency of 100 Hz; the measurements can be performed every 10 ms during the entire plasma discharge at the spatial resolution of up to 5 mm. Using the upgraded diagnostics, the electron temperature measurements were performed during experimental campaigns of the T-10 tokamak in 2016—2018. In the regimes with the electron cyclotron resonance heating, it was demonstrated that the regions with increased temperature gradients form in the plasma, which was interpreted as the formation of internal transport barrier. The ASTRA-code-based simulations of the current profile time evolution made it possible to correlate the positions of the barrier and the rational surface q = 1.

Characteristics of disruptions observed in the initial operation phase of JT-60SA

Abstract Disruption database in the JT-60SA first plasma operation period has been constructed. The disruptions have been divided into three categories based on their characteristics, and the characteristics of each category have been discussed. The most commonly observed disruptions were vertical displacement events (VDEs). As expected by simulations using the MHD equilibrium control simulator (MECS), a highly elongated plasma was likely to disrupt by VDE when the gap between the plasma surface and the wall is large. Despite the quite high edge safety factor q 95 of above 10 in the initial operation, some discharge disrupted because of control failure caused by the n = 1 tearing mode. One possible cause of the n = 1 tearing mode is the edge cooling in high density. The last category is the radiative disruptions, which is characterized by slow current decay with multiple spikes in soft x-ray signals. These spikes are thought to be related to the MHD activity that occurred when the cooling front crossed the rational surface. The radiative disruptions are attributed to a sudden change in edge power balance.

Influence of electron cyclotron current drive on the pressure crashes caused by the 2/1 double tearing modes

Abstract A module with self-consistent evolution of driven current is developed and coupled with the resistive-MHD equations in the three-dimensional, toroidal, and nonlinear simulation code (CLT). The driven current equation is solved with a second-order accuracy symmetric scheme, which exhibits good conservation properties. With the new module, we find that the driven current can self-consistently concentrate inside the magnetic island when the parallel diffusion of the driven current is sufficiently large. The efficiency of the driven current on tearing mode suppression will then be much higher than those with stationary distributions. With the new module, the influence of electron cyclotron current drive (ECCD) on the nonlinear evolution of the 2/1 double tearing modes (DTMs) is investigated. When co-ECCD deposits on the outer resonant surface, the local magnetic shear is reduced, and the growth rates of the DTMs decrease; if ctr-ECCD deposits on the outer resonant surface, the local magnetic shear increases, and the DTMs become more unstable. However, things will be different if ECCD deposits on the inner resonant surface since the local magnetic shear is negative. The co-ECCD deposited on the inner resonant surface increases the negative shear and then promotes the growth of the DTMs; while the ctr-ECCD suppresses the DTMs. It is also found that the off-axis and central pressure crashes associated with the 2/1 DTMs can be converted to each other by properly depositing the driven current. To convert a central crash to an off-axis crash, the co-ECCD should be deposited on the outer resonant surface, or the ctr-ECCD deposited on the inner resonant surface. While, the co-ECCD should be deposited on the inner rational surface, or the ctr-ECCD deposited on the outer rational surface to convert an off-axis crash to a central crash. The co- or ctr-ECCD should be larger than a threshold for such transitions, and the threshold value is mainly determined by the location of the inner resonant surface.

Castelnuovo–Mumford regularity and splitting criteria for logarithmic bundles over rational normal scroll surfaces

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for rational normal scroll surfaces. In this setting we prove analogs of some classical properties. We prove splitting criteria for coherent sheaves and a characterization of Ulrich bundles. Finally we study logarithmic bundles associated to arrangements of lines and rational curves.

In-situ TiO2/reduced graphene oxide double-coated LiNi0.5Co0.2Mn0.3O2 microspheres as high-performance Li-ion cathode materials

LiNi0.5Co0.2Mn0.3O2(NCM) layered materials have high energy density and large power density but also exhibit obvious capacity fading and voltage decay during the cycle. A rational surface modification design is reported in this work to address these problems. Herein, tetrabutyl titanate (TBOT) was decomposed into TiO2 by hydrolysis-condensation reaction and high-temperature calcination, thus a homogeneous coating was generated on the surface of NCM. The coating layer of TiO2 stabilizes the crystal structure. Furthermore, rGO and reduced metal (Ni, Co, Ti) would be wrapped on the surface as a highly conductive second coating, which could provide a rapid electron conduction pathway. The dual surface-modified cathode shows an outstanding discharge capacity of 163.4 mAh g−1 at 0.2 C and 120 mAh g−1 at 10 C. This dual coating strategy provides valuable insights for further study on the modification mechanisms of other layered cathode materials.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

The root cause of disruptive NTMs and paths to stable operation in DIII-D ITER baseline scenario plasmas

Abstract Analyses of the DIII-D ITER Baseline Scenario database support that the disruptive m,n=2,1 magnetic islands are pressure gradient driven, non-linear instabilities seeded in a sequence of stochastic transient magnetic perturbations, and that the current profile relaxation does not affect the m,n=2,1 island onset rate. At low torque, these Neoclassical Tearing Modes are most commonly seeded by non-linear 3-wave coupling when the differential rotation between the q=1 & q=2 rational surfaces approaches zero. Lack of statistically significant difference between the current profiles of stable and unstable states, as well as lack of correlation between the tearing mode onset rate and the current profile relaxation both reject causality between the current profile evolution and the 2,1 magnetic island onsets in these plasmas. These support that preserving the differential rotation between the q=1 and q=2 rational surfaces is key to long pulse stable operation in the plasma scenario planned for ITER, while optimization of the current profile within the explored parameter space may lead to much weaker improvements than sustaining the differential rotation.

Comparative Study on the Interest in Non-Uniform Rational B-Splines Representation versus Polynomial Surface Description in a Freeform Three-Mirror Anastigmat

Novel freeform optical design methods can be classified in two categories, depending on whether they focus on the generation of a starting point or the development of new optimization tools. In this paper, we design a freeform three-mirror anastigmat (TMA) and compare different surface representations using either a differential ray tracer as a new optimization tool or a commercial ray tracer (ANSYS-ZEMAX OpticStudio). For differential ray tracing, we used FORMIDABLE (Freeform Optics Raytracer with Manufacturable Imaging Design cApaBiLitiEs), an optical design library with differential ray tracing and Non-Uniform Rational B-Splines (NURBS) optimization capabilities, available under the European Software Community License (ESCL). NURBS allow a freeform surface to be represented without needing any prior knowledge of the surface, such as the polynomial degree in polynomial descriptions. OpticStudio and other commercial optical design software are designed to optimize polynomial surfaces but are not well-suited to optimize NURBS surfaces, requiring a custom optical design library. In order to demonstrate the interest in using NURBS representation, we designed and independently optimized two freeform telescopes over different iteration cycles; with NURBS using FORMIDABLE or with XY polynomials using OpticStudio. We then compared the resulting systems using their root mean square field maps to assess the optimization quality of each surface representation. We also provided a full-system comparison, including mirror freeform departures. This study shows that NURBS can be a relevant alternative to XY polynomials for the freeform optimization of reflective three-mirror telescopes as it achieves more a uniform imaging quality in the field of view.

Effect of toroidal mode coupling on explosive dynamics of m/n = 3/1 double tearing mode

The CLT code was used to quantitatively study the impact of toroidal mode coupling on the explosive dynamics of the m/n = 3/1 double tearing mode. The focus of this study was on explosive reconnection processes, in which the energy bursts and the main mode no longer dominates when the separation between two rational surfaces is relatively large in the medium range. The development of higher m and n modes is facilitated by a relatively large separation between two rational surfaces, a small q min (the minimum value of the safety factor), or low resistivity. The relationships between the higher m and n mode development, explosive reconnection rate, and position exchange of 3/1 islands are summarized for the first time. Separation plays a more important role than q min in enhancing the development of higher m and n modes. At a relatively large separation, the good development of higher m and n modes greatly reduces the reconnection rate and suppresses the development of the main mode, resulting in the main mode not being able to develop sufficiently large to generate the position changes of 3/1 islands.

Inverse Relations for Blossoms and Parametrisations of Rational Curves and Surfaces

Inverse Relations for Blossoms and Parametrisations of Rational Curves and Surfaces

Steenbrink‐type vanishing for surfaces in positive characteristic

AbstractLet be a pair of a normal surface over a perfect field of characteristic and an effective ‐divisor on . We prove that Steenbrink‐type vanishing holds for if it is log canonical and , or it is ‐pure. We also show that rational surface singularities satisfying the vanishing are ‐injective.

Resistive wall tearing mode disruptions

Abstract This paper deals with resistive wall tearing mode (RWTM) disruptions. RWTMs are closely related to resistive wall modes. RWTMs are tearing modes whose linear and nonlinear behavior is strongly dependent on the resistive wall outside the plasma. The consequence for ITER, is that the thermal quench timescale could be much longer than previously conjectured. Active feedback stabilization is another possible way to mitigate or prevent RWTM disruptions. Simulations of RWTM disruptions are reviewed for DIII-D and MST. MST has a longer resistive wall time than ITER, and disruptions are not observed experimentally when MST is operated as a standard tokamak. Simulations indicate that the RWTM disruption time scale is longer than the experimental shot time. Edge cooling causes contraction of the current profile, which can destabilize RWTMs. The equilibria studied here have the q = 2 rational surface close to the edge of the plasma, and low current density between the q = 2 surface and the wall. A sequence of low edge current model equilibria has major disruptions only for a resistive, not ideal, wall, and edge q ⩽ 3.4 . This is consistent with regimes of tokamak disruptivity, suggesting that tokamak disruptions caused by edge cooling at low edge q could be RWTMs.

Optimization and knowledge discovery of profiled end walls in a turbine stage at a low Reynolds number

To comprehensively explore flow control method of profiled end wall for turbine stage at low Reynolds numbers, a surrogate model optimization platform including non-uniform rational B-spline surface parameterization method, support vector regression, and improved chaos particle swarm optimization algorithm is integrated. Optimization designs have been carried out for stator profiled end walls, rotor profiled end wall, and combined end walls, respectively. The results indicate that under the constraint of the output power, the application of various profiled end wall design cases all can effectively improve the aerodynamic performance of the turbine stage. By organizing the flow field of downstream rotor, the profiled end wall of stator can significantly affect the stage efficiency. The flow control benefits of the profiled end wall of the rotor is from the obstruction of the cross migration of the pressure side leg of the horseshoe vortex. The application of profiled end wall on stator has the most practical engineering value. Self-organizing maps and Shapley methods are used to explore potential correlation information of aerodynamic parameters and summarize design experience. The sensitive design variables of profiled end walls are extracted. Based on the local controllability of NURBS surfaces, the regions that affect the stage efficiency are mainly concentrated in the middle of the stator passage, near the stator trailing edge and near the rotor leading edge. The regions with a significant impact on the output power of the turbine stage are near the trailing edge of the rotor and stator. The corresponding design rules of end walls modeling are summarized.

MHD stability trends and improved performance of LHD inward-shifted configurations: The role of the neutral beam current drive and thermal plasma density

The aim of the present study is to analyze the effect of the neutral beam current drive (NBCD), thermal plasma density, and NBI operational regime on the stability of pressure gradient-driven modes (PGDM) and Alfvén eigenmodes (AE) in LHD inward-shifted configurations. The stabilization of n/m=1/2 PGDM (n toroidal mode and m poloidal mode) is observed in the discharge 167 800 during the co-NBCD phase. The iota profile evolution measured by motional stark effect diagnostic may indicate the iota profile up-shift caused by the co-NBCD can induce a non-resonant transition of the rational surface 1/2 before the mode stabilization. The evolution of the iota profile and continuum gaps in the discharge 167 805 during the ctr-NBCD phase leads to the stabilization of the AE, caused by the narrowing of the continuum gap as the iota profile down-shift. Opposite stability trends are identified for PGDM and AE stability with respect to the thermal plasma density. A larger thermal plasma density (larger thermal β) further enhances PGDM although the continuum gaps are narrower leading to configurations with stable AEs. The linear stability of AEs is analyzed using the gyro-fluid FAR3d code to reproduce the AE stability trends observed in the experiments with respect to the NBCD and thermal plasma density. The analysis of hypothetical scenarios dedicated to study different NBI operational regimes with respect to EP energy, and β and radial density profiles indicate off-axis NBI operation shows a higher EP β threshold to destabilize AEs compared to on-axis configuration. This is explained by the presence of a TAE gap in the inner plasma region, easily destabilized by an on-axis NBI injection. The control of the NBCD and thermal plasma in the discharge 167 800 shows a transitory stabilization of PGDM and AEs, as well as an improved discharge performance identified by an increment of the neutron fluxes.

Shape preserving aspects of a novel class of bi-cubic partially blended rational zipper fractal interpolation surfaces

Shape preserving aspects of a novel class of bi-cubic partially blended rational zipper fractal interpolation surfaces