We propose the physics-augmented surrogate model for linear instability analysis of high- $n$ toroidal Alfvén eigenmodes (TAEs) in tokamaks. The database for high- $n$ TAE linear instabilities is generated by the global initial-value simulations by Gyrokinetic magnetohydrodynamic Energetic-particle Code (GMEC), and also the local eigenvalue code Energetic-particle TAE Eigenvalue Code (ETEC). The equations solved by ETEC are much simplified from the first-principle equations solved by GMEC, with high- $n$ ballooning representation. The local solutions from ETEC exhibit a strong correlation with the global solutions from GMEC. The multi-layer perceptron based surrogate models are trained, taking local normalized equilibrium parameters as inputs, and the real frequency and growth rate of TAEs obtained from GMEC as outputs. Our results indicate that incorporating ETEC results as prior physical information into the inputs leads to a substantial improvement in the generalization performance of the surrogate model.

An optimized stellarator at finite plasma beta is realized by single-stage optimization by simply modifying the coil currents of the Compact Stellarator with Simple Coils (CSSC) [Yu et al., J. Plasma Phys. 88, 905880306 (2022)]. The CSSC is an optimized stellarator obtained by direct optimization via coil shapes, with its coil topology similar to that of the Columbia Non-neutral Torus [T. S. Pederson and A. H. Boozer, Phys. Rev. Lett. 88, 205002 (2002)]. Due to its vacuum-based optimization, the CSSC exhibits detrimental finite beta effects on neoclassical confinement. The results of optimization show that the finite beta effects can be largely mitigated by reducing the coil currents of CSSC.

A machine learning based surrogate model for fishbone linear instability in tokamaks is constructed. Hybrid simulations with the kinetic-magnetohydrodynamic (MHD) code M3D-K is used to generate the database of fishbone linear instability, through scanning the four key parameters which are thought to determine the fishbone physics. The four key parameters include (1) central total beta of both thermal plasma and fast ions, (2) the fast ion pressure fraction, (3) central value of safety factor q and (4) the radius of q = 1 surface. Four machine learning methods including linear regression, support vector machines (SVM) with linear kernel, SVM with nonlinear kernel and multi-layer perceptron are used to predict the fishbone instability, growth rate and real frequency, mode structure respectively. Among the four methods, SVM with nonlinear kernel performs very well to predict the linear instability with accuracy approximate to 95 % , growth rate and real frequency with R 2 approximate to 98%, mode structure with R 2 approximate to 98%.

An efficient numerical code FP3D has been developed to calculate particle orbits and evaluate particle confinement in 3D magnetic fields including stellarators and tokamaks with 3D fields. The magnetic field is either calculated from coils directly or obtained from equilibrium codes. FP3D has been verified with the 3D equilibrium code VMEC (S. P. Hirshman, Phys. Fluids 26, 3553 (1983)) for magnetic field calculation and with the drift-kinetic code SFINCS (M. Landreman, Physics of Plasmas 21 (4) (2014)) for neoclassical transport. The code has been applied successfully to the NCSX stellarator (B. Nelson, Fusion Engineering and design 66 (2003)) for the calculation of neoclassical transport coefficient with the 3D magnetic field obtained directly from coils. FP3D is also used to calculate ripple losses in the tokamak EAST (W. Yuanxi, Plasma Science Technology 8 (3) (2006)).

We have developed a hybrid code GMEC: Gyro-kinetic Magnetohydrodynamics (MHD) Energetic-particle Code that can numerically simulate energetic particle-driven Alfvén eigenmodes and energetic particle transport in tokamak plasmas. In order to resolve the Alfvén eigenmodes with high toroidal numbers effectively, the field-aligned coordinates and meshes are adopted. The extended MHD equations are solved with the five-point finite difference method and the fourth-order Runge–Kutta method. The gyrokinetic equations are solved by particle-in-cell method for the perturbed energetic particle pressures that are coupled into the MHD equations. Up to now, a simplified version of the hybrid code has been completed with several successful verifications, including linear simulations of toroidal Alfvén eigenmodes and reversed shear Alfvén eigenmodes.

The aim of this work is to analyze the alpha particle transport induced by Alfven eigenmodes (AEs) based on the latest design of China Fusion Engineering Test Reactor (CFETR). Firstly, the stability of AEs is analyzed in order to understand the physical characteristics of AEs in CFETR with latest design parameters. AEs driven by alpha particles are analyzed using the gyrokinetic ion/fluid electron hybrid code GEM. The transport of alpha particles is investigated using a reduced energetic particle transport model based on the resonance broadened quasi-linear model. GEM results show that AEs with many toroidal mode numbers can be destabilized simultaneously by alpha particles in CFETR, and the toroidal mode number of the most unstable mode is n=8 . It is found that the excitation threshold of the alpha particle central beta for the n=8 mode is substantially below the expected alpha particle beta value of CFETR. All these results indicate that AEs can be driven strongly in CFETR plasmas. Furthermore, simulation results using the reduced transport model show that, due to multiple unstable AEs, the alpha particle density decreases in the core and alpha particles are transported from the core region ( r/a<0.35 ) to the outer region ( r/a>0.35 ). In addition, we find that the minimum value of the safety factor qmin and the central beta value of alpha particle have significant effects on the alpha particle transport, and the AE-induced alpha particle transport level becomes lower with smaller linear growth rates. Finally, it is found that the radial range of the redistributed alpha particle density profile depend on the qmin radial location.

A new magnetohydrodynamics (MHD) code based on initial value approach, GMEC_I, has been developed for simulating various MHD physics in tokamak plasmas, as the MHD foundation of the gyrokinetic-MHD energetic particle simulation code (GMEC) family. GMEC_I solves multi-level reduced-MHD models that form a hierarchy of physics complexity, which provide conveniences for the cross-code verification and the identification of key physics effect in tokamak geometry. The field-aligned coordinates are used to represent mode structure efficiently. High-order finite difference methods are used for spatial discretization. The shifted metric methods are used for numerical stability. The discrete expansion forms of physics equations in the code are generated symbolically using the compile-time symbolic solver, which is specifically developed to reduce the complexity of the high-order finite difference form of the MHD equations. Advanced computational techniques have been implemented for optimizing memory access and code parallelization that show a good efficiency using both thread building block and message passing interface. Benchmarks between GMEC_I and the eigenvalue code MAS are presented for ballooning modes without and with diamagnetic drift effects, and tearing modes, which show excellent agreements.

We perform a systematic simulation study of energetic passing particle-driven instabilities in KSTAR using the kinetic-MHD hybrid code M3D-K. Linear simulation results show that the observed n = 1 mode in the early phase of the discharge is the low-frequency fishbone driven by energetic passing beam ions. The mode frequency computed is in a good agreement with the experimental measurement. Nonlinear simulations show that the frequency of the n = 1 mode jumps up to a higher value corresponding to the beta-induced Alfv & eacute;n eigenmode (BAE). In the later phase of the discharge, the simulated n = 5 mode is identified as a BAE in its linear phase. In the nonlinear phase, the n = 5 mode exhibits a similar frequency jump to a higher value of an energetic particle (EP) mode after mode saturation. Analysis of perturbed beam ion distributions in phase space shows that these new modes in nonlinear stages are driven by new resonances due to nonlinearly evolved beam ion distributions. Further simulations of a beam beta scan for the n = 5 mode show that the frequency jump disappears for a sufficiently small beam beta or beam ion drive. This result may explain the non-existence of frequency jump in the experiment. Finally, the impact of toroidal rotation on mode characteristics is investigated, showing that it has a marginal influence on EP driven modes.

Fishbone bursts have been observed to strongly correlate to internal transport barrier (ITB) formation in a number of tokamak devices. A simple model incorporating the fishbone dynamics and ion pressure gradient evolution is proposed in order to investigate the key physics parameters assisting the triggering of ITB. The time evolution of fishbone is described by the well-known predator–prey model. For each burst cycle, the energetic particles (EPs) resonantly interact with fishbone and are radially expelled from inner region leading to a radial current. A compensating bulk plasma return current and, hence, poloidal flow can be induced if the fishbone cycle frequency is greater than the poloidal flow damping rate. When the shear of the poloidal flow exceeds a critical value, the turbulent fluctuations are suppressed and the bulk ion pressure gradient transits to the high-confinement state. It is shown that this process is only sensitive to the deposition rate of the trapped EPs within the $q=1$ surface, but not sensitive to other parameters. A quantitative formula for the shearing rate of poloidal flow induced by fishbone bursts is derived and verified numerically.

A strong nonlinear saturation mechanism of the reverse shear Alfven eigenmode (RSAE)induced by thermal electrons is observed in gyrokinetic delta f particle-in-cell simulation. This nonlinear effect occurs at moderate mode amplitude, delta B-perpendicular to/B < 10(-3), and is associated with the electron parallel streaming along the perturbed magnetic field lines, the term v(parallel to)delta B-perpendicular to//B center dot del delta f, in the electron drift-kinetic equation. In the case of an n=4 RSAE, where n is the toroidal modenumber, this magnetic fluttering nonlinearity leads to strong mode coupling and broadening of the saturation spectrum. The nonlinear components, which are highly damped through the interaction between thermal electrons and the parallel electric field, modifies the RSAE mode structure and suppresses energetic particle drive significantly. The split-weight scheme (Chen and Parker 2007 J. Comput. Phys.220 839-55) is used to simulate kinetic electrons.

A systematic simulation study of the $n/m=1/1$ instability driven by energetic counter-passing particles in tokamak plasmas has been carried out using the kinetic-MHD (Magnetohydrodynamics) hybrid code M3D-K. The safety factor's radial profile is monotonically increasing with central value $q_0$ less than unity. The linear simulation results show that the instability is either a $m/n=1/1$ energetic particle mode or a $m/n=1/1$ global Alfvén eigenmode depending on the value of the central safety factor. The mode frequencies are close to the tip of Alfvén continuum spectrum at the magnetic axis. The excited modes are radially localized near the magnetic axis well within the safety factor $q=1$ surface. The main wave particle resonance is found to be $\omega _\phi +2\omega _\theta =\omega$, where ω is the mode frequency. The nonlinear simulation results show that there is a long period of quasi-steady-state saturation phase with frequency chirping up after initial saturation. Correspondingly, the energetic particle distribution with low energies is flattened in the core of the plasma. After this quasi-steady phase, the mode amplitude grows again and frequency jumps down to a low value corresponding to a new mode similar to the energetic co-passing particle-driven low-frequency fishbone while the energetic particle distribution is flattened for higher energies in the core of plasma.

An experimental study is conducted on the onset and evolution characteristics of energetic particle-driven instabilities in Korea Superconducting Tokamak Advanced Research (KSTAR) with dominant tangential neutral beam injection (NBI). A scan of NBI beam energy shows the evanescence of the sawtooth crash and the concomitant onset of the strong passing particle-driven low-frequency fishbone instability. A quantitative analysis shows that the safety factor (q)-profile in the core region is clamped by a balance between the depletion of energetic passing particles by the fishbone instability and their external replenishment. Two synchronized chirping modes with distinct toroidal mode numbers (n = 1 and n = 5) supersede the fishbone instability after a self-organized q-profile is attained. An analysis shows that the n = 1 mode is likely to be a high-frequency beta-induced Alfven eigenmode fishbone branch, while the n = 5 mode is an energetic particle mode (EPM). A dynamic system analysis of the synchronized EPM (S-EPM) shows that a stable S-EPM cycle can exist when the coupling between the two modes involved is insignificant. The potential impact of such EPMs on the establishment of a burning plasma scenario with a flat core q-profile is briefly discussed.

An optimized compact stellarator with four simple coils is obtained from direct optimization via a coil shape. The new stellarator consists of two interlocking coils and two vertical field coils similar to those of the Columbia Non-neutral Torus (CNT) (Pedersen et al., Phys. Rev. Lett., vol. 88, 2002, pp. 205002). The optimized configuration has a global magnetic well and a low helical ripple level comparable to that of Wendelstein 7-X (W7-X) (Wolf et al., Nucl. Fusion, vol. 57, 2017, pp. 102020). The two interlocking coils have a smooth three-dimensional shape much simpler than those of advanced stellarators such as W7-X. This result opens up possibilities of future stellarator reactors with simplified coils.

A linear simulation study of energetic passing particle-driven low-frequency fishbone instability in tokamak plasmas has been carried out using the global kinetic-MHD (magnetohydrodynamics) hybrid code M3D-K. This work is focused on the interaction of energetic passing beam ions and n = 1 mode with a monotonic safety factor q profile and q 0 < 1 . Specifically, the stability and mode frequency as well as mode structure of the n = 1 mode are calculated for scans of parameter values of beam ion beta, beam ion injection energy, beam ion orbit width, beam ion beta profile, as well as background plasma beta. The excited modes are identified as a low-frequency fishbone with the corresponding resonance of ω ϕ + ω θ = ω , where ω ϕ is the beam ion toroidal transit frequency and ω θ is the beam ion poloidal transit frequency. The simulated mode frequency is approximately proportional to the beam ion injection energy and beam ion orbit width. The mode structure is similar to that of internal kink mode. These simulation results are similar to the analytic theory of Yu et al.

Based on the experimental parameters in the HL-2A tokamak, hybrid simulations have been carried out to investigate the linear stability and nonlinear dynamics of the beta-induced Alfvén eigenmode (BAE). It is found that the ( m / n = 3/2) BAE is excited by co-passing energetic ions with q min = 1.5 in linear simulation, and the mode frequency is consistent with the experimental measurement. The simulation results show that the energetic ions β h , the injection velocity v 0 , and orbit width parameter ρ h of energetic ions are important parameters determining the drive of BAE. Furthermore, the effect of q min (with the fixed shape of the q profile) is studied, and it is found that when q min ⩽ 1.5, the excited modes are BAEs, which are located near q = 1.5 rational surfaces; when q min > 1.5, the excited modes are similar to the reversed-shear Alfvén eigenmodes, which are mainly localized around q = q min surfaces. Nonlinear simulation results show that the nonlinear dynamics of BAE are sensitive to the EP drive. For the strongly driven case, firstly, redistribution and transport of energetic ions are trigged by 3/2 BAE, which raised the radial gradient of the distibution function of energetic ions near the q = 2 rational surface, and then an energetic particle mode (EPM) ( m / n = 4/2) is driven in the nonlinear phase. Finally, these two instabilities triggered a significant redistribution of energetic ions, which results in the twice-repeated and mostlydownward frequency chirping of 3/2 BAE. For the weakly driven case, there are no 4/2 EPM being driven nor twice-repeated chirping in the nonlinear phase, since the radial gradient near q = 2 rational surface is small and almost unchanged.

[This retracts the article DOI: 10.1016/j.omtn.2021.02.009.].

A neoclassically optimized compact stellarator with simple coils has been designed. The magnetic field of the new stellarator is generated by only four planar coils including two interlocking coils of elliptical shape and two circular poloidal field coils. The interlocking coil topology is the same as that of the Columbia Non-neutral Torus (CNT). The new configuration was obtained by minimizing the effective helical ripple directly via the shape of the two interlocking coils. The optimized compact stellarator has very low effective ripple in the plasma core implying excellent neoclassical confinement. This is confirmed by the results of the drift-kinetic code SFINCS showing that the particle diffusion coefficient of the new configuration is one order of magnitude lower than CNT's.

A new linked mirror device for magnetic confinement experiment is proposed.The new linked mirror device consists of two straight magnetic mirrors connected by two half-torus.The structure of the configuration as a whole is three dimensional because the two linear mirror sections are not parallel.The angle between the two mirror sections generates rotational transform which results in good magnetic confinement of toroidally processing passing particles.In this way the usual loss cone of the traditional linear mirror machines is eliminated.The single particle confinement is similar to that of tokamaks with most of particles well confined.The calculated neoclassical confinement is very good and is even better than that of an equivalent tokamak.Potentially the proposed linked mirror configuration can allow MHD stable high beta equilibria with good plasma confinement suitable for neutron sources and magnetic fusion reactors.

Significant variations in MHD activity and fast-ion transport are observed in the DIII-D high-beta, steady-state hybrid discharges with a mixture of electron cyclotron (EC) waves and neutral beam injection (NBI). When electron cyclotron heating (ECH) or current drive (ECCD) is applied, Alfven eigenmodes (AEs) are usually suppressed and replaced by low-frequency bursting modes. The analysis of a recently compiled database of hybrid discharges suggests that the change of the fast-ion pressure especially the perpendicular pressure is the main factor responsible for the instability transition although the transition in some discharges can also be explained by a slight drop of the safety factor q(min). The lower ratio of fast-ion injection speed v(inj) to Alfven speed v(alfven) and slight drop of q(min) during ECCD also facilitate the transition. The database shows that AEs mainly occur when the fast-ion fraction P-f/P-total is less than 0.53 and v(inj)/v(alfven) is greater than 0.50, while low-frequency bursting modes appear in the opposite regime. Here, P-f and P-total are the central fast-ion pressure from classical prediction and total plasma pressure, respectively. The correlation with q(min) is weaker, and qmin is around unity in all the cases. The reason why the instability transition correlates with P-f/P-total and v(inj)/v(alfven) is that they can significantly modify the drive of low-frequency bursting modes and AEs. The explanation is supported by the observation that low-frequency bursting modes are rarely seen in the hybrids with NBI only, with EC waves and counter-NBI, or with high plasma density. A careful check of the low-frequency bursting modes suggests that they are mainly chirping (neoclassical) tearing modes (referred to as chirping (N)TMs), i.e. the mode frequency firstly jumps up from the steady (N)TM frequency, then chirps down, and finally returns to the steady (N)TM frequency. Occasionally, the (N)TMs are fully stabilized and replaced with pure fishbones. The resonance condition calculation and 'Kick' model simulations suggest that (N)TMs and fishbones can interact through modification of the fast ion distribution in phase space, which influences the drive.

Linear stability and non-linear dynamics of the fishbone instabilities with reversed safety factor profile have been investigated by the global kinetic-magnetohydrodynamic (MHD) code M3D-K. For the consideration of the fishbone instability with a reversed q profile, there are two different types of the fishbone instability: dual resonant fishbone (DRF) with double q = 1 surfaces and non-resonant fishbone (NRF) with the minimum value of safety factorqmin<ia little larger than unity. Based on EAST-like parameters, linear simulations show that the DRF is excited by the trapped beam ions when the fast ion pressure exceeds a critical value, and the mode structure of DRF exhibits splitting radial structure due to double q = 1 surfaces. When q(min) increases from below unity to above unity, the fishbone instability transits from the DRF to the NRF, and the mode frequency of the NRF is higher than the DRF as the NRF is resonant with fast ions with larger precessional frequency. Nonlinear simulations show that the saturation of the DRF is due to MHD non-linearity with a large n = 0 component. However, the saturation of the NRF is mainly due to the non-linearity of fast ions, and the frequency of the NRF chirps down nonlinearly. The fast ions are redistributed and become flattened due to the DRF or the NRF, and the transport level of the fast ions due to the NRF is weaker with more centrally radial redistribution region in comparison with that of the DRF.