Starting from the Braginskii equations, relevant for the tokamak edge region, a complete set of nonlinear equations for the geodesic acoustic modes (GAM) has been derived which includes collisionality, plasma beta and external sources of particle, momentum and heat. Local linear analysis shows that the GAM frequency increases with collisionality at low radial wave number $k_{r}$ and decreases at high $k_{r}$. GAM frequency also decreases with plasma beta. Radial profiles of GAM frequency for two Tore Supra shots, which were part of a collisionality scan, are compared with these calculations. Discrepency between experiment and theory is observed, which seems to be explained by a finite $k_{r}$ for the GAM when flux surface averaged density $\langle n \rangle$ and temperature $\langle T \rangle$ are assumed to vanish. It is shown that this agreement is incidental and self-consistent inclusion of $\langle n \rangle$ and $\langle T \rangle$ responses enhances the disagreement more with $k_r$ at high $k_{r}$ . So the discrepancy between the linear GAM calculation, (which persist also for more "complete" linear models such as gyrokinetics) can probably not be resolved by simply adding a finite $k_{r}$.
The complete refuelling of the plasma density loss (pump-out) caused by mitigation of Edge Localised Modes (ELMs) is demonstrated on the ASDEX Upgrade tokamak. The plasma is refuelled by injection of frozen deuterium pellets and ELMs are mitigated by external resonant magnetic perturbations (RMPs). In this experiment relevant dimensionless parameters, such as relative pellet size, relative RMP amplitude and pedestal collisionality are kept at the ITER like values. Refuelling of density pump out requires a factor of two increase of nominal fuelling rate. Energy confinement and pedestal temperat…
The effect of small deviations from a Maxwellian equilibrium on turbulent momentum transport in tokamak plasmas is considered. These non-Maxwellian features, arising from diamagnetic effects, introduce a strong dependence of the radial flux of co-current toroidal angular momentum on collisionality: As the plasma goes from nearly collisionless to weakly collisional, the flux reverses direction from radially inward to outward. This indicates a collisionality-dependent transition from peaked to hollow rotation profiles, consistent with experimental observations of intrinsic rotation.
Tokamak plasmas rotate even without external injection of momentum. A Doppler backscattering system installed at MAST has allowed this intrinsic rotation to be studied in Ohmic L-mode and H-mode plasmas, including the first observation of intrinsic rotation reversals in a spherical tokamak. Experimental results are compared to a novel 1D model, which captures the collisionality dependence of the radial transport of toroidal angular momentum due to the effect of neoclassical flows on turbulent fluctuations. The model is able to accurately reproduce the change in sign of core toroidal rotation, …
Two-fluid, three-dimensional, flux-driven, global, electromagnetic turbulence simulations carried out by using the GBS code are used to identify the main parameters controlling turbulent transport in the tokamak boundary and to delineate an electromagnetic phase space of edge turbulence. Four turbulent transport regimes are identified: (i) a regime of fully developed turbulence appearing at intermediate values of collisionality and $β$, with turbulence driven by resistive ballooning modes, related to the L-mode operation of tokamaks, (ii) a regime of reduced turbulent transport at low collisio…
A drift-kinetic model to describe the plasma dynamics in the scrape-off layer region of tokamak devices at arbitrary collisionality is derived. Our formulation is based on a gyroaveraged Lagrangian description of the charged particle motion, and the corresponding drift-kinetic Boltzmann equation that includes a full Coulomb collision operator. Using a Hermite-Laguerre velocity space decomposition of the gyroaveraged distribution function, a set of equations to evolve the coefficients of the expansion is presented. By evaluating explicitly the moments of the Coulomb collision operator, distribu…
The extended-MHD NIMROD code [C.R. Sovinec and J.R. King, J. Comput. Phys. 229, 5803 (2010)] is verified against the ideal-MHD ELITE code [H.R. Wilson et al. Phys. Plasmas 9, 1277 (2002)] on a diverted tokamak discharge. When the NIMROD model complexity is increased incrementally, resistive and first-order finite-Larmour radius effects are destabilizing and stabilizing, respectively. The full result is compared to local analytic calculations which are found to overpredict both the resistive destabilization and drift stabilization in comparison to the NIMROD computations.
The accumulation of impurities in the core of magnetically confined plasmas, resulting from standard collisional transport mechanisms, is a known threat to their performance as fusion energy sources. Whilst the axisymmetric tokamak systems have been shown to benefit from the effect of temperature screening, that is an outward flux of impurities driven by the temperature gradient, impurity accumulation in stellarators was thought to be inevitable, driven robustly by the inward pointing electric field characteristic of hot fusion plasmas. We have shown in Helander et. al. (2017b) that such scree…
The plasma beta, ratio of kinetic to magnetic pressure, inside a tokamak should stay below the Troyon limit to avoid major plasma instabilities. However, this paper argues that Troyon limit occurs only when current profiles cannot sustain high beta equilibria. This paper shows that plasmas with peak beta on the order of unity are stable to ideal MHD modes. While the theoretical (ideal-MHD) stability of unity beta equilibrium has been resolved, the experimental existence of high beta equilibria in tokamaks remains an unsolved problem of plasma physics.
The density dependence of edge-localized-mode (ELM) suppression and density pump-out (density reduction) by n = 2 resonant magnetic perturbations (RMPs) is consistent with the effects of narrow well-separated magnetic islands at the top and bottom of the H-mode pedestal in DIII-D low-collisionality plasmas. Nonlinear two-fluid MHD simulations for DIII-D ITER Similar Shape (ISS) discharges show that, at low collisionality, low pedestal density is required for resonant field penetration at the pedestal top, consistent with the ubiquitous low density requirement for ELM suppression in these DIII-…
Neutral atoms can strongly influence the intrinsic rotation and radial electric field at the tokamak edge. Here, we present a framework to investigate these effects when the neutrals dominate the momentum transport. We explore the parameter space numerically, using highly flexible model geometries and a state of the art kinetic solver. We find that the most important parameters controlling the toroidal rotation and electric field are the major radius where the neutrals are localized and the plasma collisionality. This offers a means to influence the rotation and electric field by, for example,…
A potential threat to the performance of magnetically confined fusion plasmas is the problem of impurity accumulation, which causes the concentration of highly charged impurity ions to rise uncontrollably in the center of the plasma and spoil the energy confinement by excessive radiation. It has long been thought that the collisional transport of impurities in stellarators always leads to such accumulation (if the electric field points inwards, which is usually the case), whereas tokamaks, being axisymmetric, can benefit from "temperature screening", i.e., an outward flux of impurities driven …
Access conditions for full suppression of Edge Localised Modes (ELMs) by Magnetic Perturbations (MP) in low density high confinement mode (H-mode) plasmas are studied in the ASDEX Upgrade tokamak. The main empirical requirements for full ELM suppression in our experiments are: 1. The poloidal spectrum of the MP must be aligned for best plasma response from weakly stable kink-modes, which amplify the perturbation, 2. The plasma edge density must be below a critical value, $3.3 \times 10^{19}$~m$^{-3}$. The edge collisionality is in the range $ν^*_i = 0.15-0.42$ (ions) and $ν^*_e = 0.15-0.25$ (e…
The onset and characteristics of Micro-Tearing Modes (MTM) in the core of spherical (NSTX) and conventional tokamaks (ASDEX-UG and JET) are studied through local linear gyrokinetic simulations with gyro [J. Candy and E. Belli, General Atomics Report GA-A26818 (2011)]. For experimentally relevant core plasma parameters in the NSTX and ASDEX-UG tokamaks, in agreement with previous works, we find MTMs as the dominant linear instability. Also, for JET-like core parameters considered in our study an MTM is found as the most unstable mode. In all these plasmas, finite collisionality is needed for MT…
This work, which extends Squire et al. [ApJL, 830 L25 (2016)], explores the effect of self-generated pressure anisotropy on linearly polarized shear-Alfvén fluctuations in low-collisionality plasmas. Such anisotropies lead to stringent limits on the amplitude of magnetic perturbations in high-beta plasmas, above which a fluctuation can destabilize itself through the parallel firehose instability. This causes the wave frequency to approach zero, "interrupting" the wave and stopping its oscillation. These effects are explored in detail in the collisionless and weakly collisional "Braginskii" reg…
A simulated annealing (SA) relaxation method is used for calculation of high-beta reduced magnetohydrodynamics (MHD) equilibria in toroidal geometry. The SA method, based on artificial dynamics derived from the MHD Hamiltonian structure, is used to calculate equilibria of large-aspect-ratio and circular-cross-section tokamaks as well as toroidally averaged stellarators. Tokamak equilibria including incompressible poloidal rotations are obtained and the Shafranov shift is seen to increase nearly quadratically in the rotation speed. A mapping procedure between non-rotating and poloidally rotatin…
This work reports on the development and numerical implementation of the linear electromagnetic gyrokinetic (GK) model in a tokamak flux-tube geometry using a moment approach based on the expansion of the perturbed distribution function on a velocity-space Hermite-Laguerre polynomials basis. A hierarchy of equations of the expansion coefficients, referred to as the gyro-moments (GM), is derived. We verify the numerical implementation of the GM hierarchy in the collisionless limit by performing a comparison with the continuum GK code GENE, recovering the linear properties of the ion-temperature…
It is shown that low-collisionality plasmas cannot support linearly polarized shear-Alfvén fluctuations above a critical amplitude $δB_{\perp}/B_{0} \sim β^{\,-1/2}$, where $β$ is the ratio of thermal to magnetic pressure. Above this cutoff, a developing fluctuation will generate a pressure anisotropy that is sufficient to destabilize itself through the parallel firehose instability. This causes the wave frequency to approach zero, interrupting the fluctuation before any oscillation. The magnetic field lines rapidly relax into a sequence of angular zig-zag structures. Such a restrictive bound …
The characteristic profile and magnitude are predicted in theory for the neoclassical toroidal viscosity (NTV) torque induced by plasma response to the resonant magnetic perturbation (RMP) in a tokamak with an edge pedestal. For a low-$β$ equilibrium, the NTV torque is dominated by the toroidal component with the same dominant toroidal mode number of RMP. The NTV torque profile is found to be localized, whose peak location is determined by profiles of both the equilibrium temperature (pressure) and the plasma response. In general, the peak of NTV torque profile is found to track the pedestal p…
We propose an asymptotic-preserving (AP), uniformly convergent numerical scheme for the relativistic collisional Drift-Kinetic Equation (rDKE) to simulate runaway electrons in axisymmetric toroidal magnetic field geometries typical of tokamak devices. The approach is derived from an exact Green's function solution with numerical approximations of quantifiable impact, and results in a simple, two-step operator-split algorithm, consisting of a collisional Eulerian step, and a Lagrangian orbit-integration step with analytically prescribed kernels. The AP character of the approach is demonstrated …
Non-dimensional confinement scaling in similar negative triangularity plasmas on the DIII-D and TCV tokamaks
University of California San Diego（加州大学圣地亚哥分校）；General Atomics（通用原子能公司）；Ecole Polytechnique Federale de Lausanne（洛桑联邦理工学院）；Consorzio RFX（RFX联合体）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Columbia University（哥伦比亚大学）；College of William and Mary（威廉玛丽学院）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Results from a database analysis of H-mode electron density profiles on the Tokamak à Configuration Variable (TCV) in stationary conditions show that the logarithmic electron density gradient increases with collisionality. By contrast, usual observations of H-modes showed that the electron density profiles tend to flatten with increasing collisionality. In this work it is reinforced that the role of collisionality alone, depending on the parameter regime, can be rather weak and in these, dominantly electron heated TCV cases, the electron density gradient is tailored by the underlying turbulenc…
A four-field system of equations which includes the neoclassical flow damping effects and the lowest-order finite-Larmor-radius (FLR) corrections is deduced from a system of gyrofluid equations. The FLR corrections to the poloidal flow damping are calculated by solving a simplified version of the gyrokinetic equation. This system of equations is applied to the study of a chain of freely rotating magnetic islands in a tokamak, resulting from the nonlinear evolution of a resistive tearing mode, to determine the islands rotation velocity consistently with the fields radial profiles close to the r…
The results of flux-driven, two-fluid simulations in single-null configurations are used to investigate the processes determining the turbulent transport in the tokamak edge. Three turbulent transport regimes are identified: (i) a developed transport regime with turbulence driven by an interchange instability, which shares a number of features with the standard L-mode of tokamak operation, (ii) a suppressed transport regime, characterized by a higher value of the energy confinement time, low-amplitude relative fluctuations driven by a Kelvin-Helmholtz instability, a strong E x B sheared flow, …
Microtearing modes are an important drive of turbulent heat transport in present-day fusion plasmas. We investigate their linear stability under very-low collisionality regimes, expected for the next generations of devices, using gyrokinetic and drift-kinetic approaches. At odds with current opinion, we show that collisionless microtearing instabilities may occur in certain experimental conditions, particularly relevant for such devices as reversed field pinches and spherical tokamaks.
Most tokamak devices including ITER exploit the D-T reaction due to its high reactivity, but the wall loading caused by the associated 14MeV neutrons will limit the further development of fusion performance at high beta. To explore p-11B fusion cycle, a tokamak system code is extended to incorporate the relativistic bremsstrahlung since the temperature of electrons is approaching the rest energy of electron. By choosing an optimum p-11B mix and ion temperature, some representative sets of parameters of p-11B tokamak reactor, whose fusion gain exceeds 1, have been found under the thermal wall l…
In some conditions, I-mode plasmas can feature pedestal relaxation events (PREs) that transiently enhance the energy reaching the divertor target plates. To shed light into their appearance, characteristics and energy reaching the divertor targets, a comparative study between two tokamaks $-$ Alcator C-Mod and ASDEX Upgrade $-$ is carried out. It is found that PREs appear only in a subset of I-mode discharges, mainly when the plasma is close to the H-mode transition. Also, the nature of the triggering instability is discussed by comparing measurements close to the separatrix in both devices. T…
In large hot tokamaks like JET, the width of the reconnecting layer for resistive modes is determined by semi-collisional electron dynamics and is much less than the ion Larmor radius. Firstly a dispersion relation valid in this regime is derived which provides a unified description of drift-tearing modes, kinetic Alfven waves and the internal kink mode at low beta. Tearing mode stability is investigated analytically recovering the stabilising ion orbit effect, obtained previously by Cowley et al. [Phys. Fluids (29) 3230 1986], which implies large values of the tearing mode stability parameter…
In this work, a purely data-driven discharge prediction model was developed and tested without integrating any data or results from simulations. The model was developed based on the experimental data from the Experimental Advanced Superconducting Tokamak (EAST) campaign 2010-2020 discharges and can predict the actual plasma current $I_{p}$, normalized beta $β_{n}$, toroidal beta $β_{t}$, beta poloidal $β_{p}$, electron density $n_{e}$, store energy $W_{mhd}$, loop voltage $V_{loop}$, elongation at plasma boundary $κ$, internal inductance $l_{i}$, q at magnetic axis $q_{0}$, and q at 95% flux s…
A model system of equations has been derived to describe a toroidally rotating tokamak plasma, unstable to Resistive Wall Modes (RWMs) and metastable to Neoclassical Tearing Modes (NTMs), using a linear RWM model and a nonlinear NTM model. If no wall is present, the NTM growth shows the typical threshold/saturation island widths, whereas a linearly unstable kink mode grows exponentially in this model plasma system. When a resistive wall is present, the growth of the linearly unstable RWM is accelerated by an unstable island: a form of coupled RWM-NTM mode. Crucially, this coupled system has no…
A four-dimensional plasma model able to describe the scrape-off layer region of tokamak devices at arbitrary collisionality is derived in the drift-reduced limit. The basis of the model is provided by a drift-kinetic equation that retains the full non-linear Coulomb collision operator and describes arbitrarily far from equilibrium distribution functions. By expanding the dependence of distribution function over the perpendicular velocity in a Laguerre polynomial basis and integrating over the perpendicular velocity, a set of four-dimensional moment equations for the expansion coefficients of t…
Boundary plasma physics plays an important role in tokamak confinement, but is difficult to simulate in a gyrokinetic code due to the scale-inseparable nonlocal multi-physics in magnetic separatrix and open magnetic field geometry. Neutral particles are also an important part of the boundary plasma physics. In the present paper, noble electrostatic gyrokinetic techniques to simulate the flux-driven, low-beta electrostatic boundary plasma is reported. Gyrokinetic ions and drift-kinetic electrons are utilized without scale-separation between the neoclassical and turbulence dynamics. It is found …
Recently-proposed tokamak concepts use magnetic fields up to 12 T, far higher than in conventional devices, to reduce size and cost. Theoretical and computational study of trends in plasma behavior with increasing field strength is critical to such proposed devices. This paper considers trends in Alfvén eigenmode (AE) stability. Energetic particles, including alphas from D-T fusion, can destabilize AEs, possibly causing loss of alpha heat and damage to the device. AEs are sensitive to device magnetic field via the field dependence of resonances, alpha particle beta, and alpha orbit width. We d…
Geodesic acoustic modes (GAMs) are studied by means of the gyrokinetic global particle-in-cell code ORB5. Linear electromagnetic simulations in the low electron beta limit have been performed, in order to separate acoustic and Alfvénic time scales and obtain more accurate measurements. The dependence of the frequency and damping rate on several parameters such as the safety factor, the GAM radial wavenumber and the plasma elongation is studied. All simulations have been performed with kinetic electrons with realistic electron/ion mass ratio. Interpolating formulae for the GAM frequency and dam…
The resonant and non-resonant effects of core plasma on the excitation of low frequency modes with $ω< ω_{BAE}$, such as Beta-induced Alfvén Acoustic Eigenmodes (BAAEs) and Kinetic Ballooning Modes (KBMs) are examined in the framework of the generalized fishbone-like dispersion relation. The formalism of the fishbone-like equation contains all the necessary ingredients to describe the features of these low frequency fluctuations, and explain experimental findings. Core plasma properties (diamagnetic frequency and precession resonance with trapped ions) strongly affect the excitation of the mod…
When electron cyclotron (EC) driven current is first applied to the inside of a magnetic island, the current spreads throughout the island and after a short period achieves a steady level. Using a two equation fluid model for the EC current that allows us to examine this early evolution in detail, we analyze high-resolution simulations of a 2/1 classical tearing mode in a low-beta large aspect-ratio circular tokamak. These simulations use a nonlinear 3D reduced-MHD fluid model and the JOREK code. During the initial period where the EC driven current grows and spreads throughout the magnetic is…
A general methodology is proposed to differentiate the likelihood of energetic-particle-driven instabilities to produce frequency chirping or fixed-frequency oscillations. The method employs numerically calculated eigenstructures and multiple resonance surfaces of a given mode in the presence of energetic ion drag and stochasticity (due to collisions and micro-turbulence). Toroidicity-induced, reversed-shear and beta-induced Alfven-acoustic eigenmodes are used as examples. Waves measured in experiments are characterized and compatibility is found between the proposed criterion predictions and …
Toroidal flow alone is generally thought to have important influence on tokamak edge pedestal stability, even though theory analysis often predicts merely a weak stabilizing effect of toroidal flow on the edge localized modes (ELMs) in experimental parameter regimes. For the first time, we find from two-fluid MHD calculations that such a stabilization, however, can be significantly enhanced by increasing the edge plasma density. Our finding resolves a long-standing mystery whether or how toroidal rotation can indeed have effective influence on ELMs, and explains why the ELM mitigation and supp…
Linear gyrokinetic simulations covering the collisional -- collisionless transitional regime of the tearing instability are performed. It is shown that the growth rate scaling with collisionality agrees well with that predicted by a two-fluid theory for a low plasma beta case in which ion kinetic dynamics are negligible. Electron wave-particle interactions (Landau damping), finite Larmor radius, and other kinetic effects invalidate the fluid theory in the collisionless regime, in which a general non-polytropic equation of state for pressure (temperature) perturbations should be considered. We …
The effect of changes in plasma parameters, that are characteristic near or at an L-H transition in fusion edge plasmas, on fluctuation correlation lengths are analysed by means of drift-Alfven turbulence computations. Scalings by density gradient length, collisionality, plasma beta, and by an imposed shear flow are considered. It is found that strongly sheared flows lead to the appearence of long-range correlations in electrostatic potential fluctuations parallel and perpendicular to the magnetic field.
The electron Bernstein wave (EBW) is typically the only wave in the electron cyclotron (EC) range that can be applied in spherical tokamaks for heating and current drive (H&CD). Spherical tokamaks (STs) operate generally in high-beta regimes, in which the usual EC O- and X- modes are cut-off. In this case, EBWs seem to be the only option that can provide features similar to the EC waves---controllable localized H&CD that can be utilized for core plasma heating as well as for accurate plasma stabilization. The EBW is a quasi-electrostatic wave that can be excited by mode conversion from a suita…
A detailed understanding of island seeding is crucial to avoid (N)TMs and their negative consequences like confinement degradation and disruptions. In the present work, we investigate the growth of 2/1 islands in response to magnetic perturbations. Although we use externally applied perturbations produced by resonant magnetic perturbation (RMP) coils for this study, results are directly transferable to island seeding by other MHD instabilities creating a resonant magnetic field component at the rational surface. Experimental results for 2/1 island penetration from ASDEX Upgrade are presented e…
