Tokamak Equilibria Research Articles

An analysis and successful benchmarking of the Chapman-Enskog-like (CEL) continuum kinetic closure approach algorithm in NIMROD

Herein, we formulate, analyze, and apply a numerical method for solving a Chapman-Enskog-like (CEL) continuum kinetic model for plasmas. It is shown that centering the heat flux at the beginning of the time step and the ion temperature at the end of the time step in the kinetic equation allows for a numerically-stable time advance of the coupled fluid-kinetic system. In addition, it is shown that numerical stability is impossible to achieve without explicitly enforcing key tenets of the CEL closure approach, in particular, that the number density (n), flow (u), and temperature (T) moments of the kinetic distortion remain small in time. We show that with a method to constrain these moments, it is possible to remove both the numerical growth and numerical damping from the linear modes. We apply the results from the linear stability analysis to allow for a numerically-stable fully nonlinear axisymmetric evolution of profiles in NIMROD, wherein we observe the asymptotic evolution of the flow in a DIII-D tokamak equilibrium (based on DIII-D ITER Baseline Scenario (IBS) discharge 174446 at 3390 ms). We compare the self-consistently computed results to analytics and to results from a previously benchmarked fixed-background δf implementation in NIMROD. Agreement with prediction is found for both the dynamics and asymptotics of the flow. This work demonstrates the first successful published benchmarking of the full CEL approach in a plasma fluid code.

Off-target gradient-driven flows in 3D simulations of ADITYA-Upgrade tokamak scrape-off layer plasma transport

Coupled plasma-neutral transport simulations are performed on ADITYA-Upgrade tokamak scrape-off layer (SOL) plasma, where flows in the core and SOL were measured to reverse signs with density variation. The simulations performed using the EMC3-Eirene plasma-neutral code combination incorporate the toroidally continuous high-field-side belt limiter placed in a moderate circular tokamak equilibrium. The development of mutually counter-propagating toroidal plasma flows in the top and bottom regions of both the SOL and core is recovered for relatively high upstream density cases with high input power (300 kW and 3 m2 s−1). The origin of the flows is traced to the poloidal density variation introduced by high recycling on the inboard localized belt limiter. The results are compared with similar observations, for example, in Doppler-shifted passive charge exchange line emission on the ADITYA-Upgrade (ADITYA-U) tokamak, highlighting the role played by residual stress in the total Reynolds stress. The external stimuli, such as a localized gas puff, are discussed as potential drivers of flow, via residual stress, based on the existing resonant model of the tokamak plasma rotation.

FLIPEC, an ideal MHD free-boundary axisymmetric equilibrium solver in the presence of macroscopic flows

The most relevant features of FLIPEC (Free fLow Iterative Plasma Equilibrium Code) are presented. This new code iteratively calculates free-boundary, axisymmetric ideal MHD equilibria with arbitrary poloidal and toroidal plasma flows. FLIPEC is a mature code that has emerged from a complete overhaul of a previous version (F-Torija Daza 2022 et al Nucl. Fusion 62 126044). It uses a (inverse) curvilinear coordinate representation for the Grad–Shafranov–Bernoulli equation system, which allows FLIPEC to extend its free-boundary capabilities to arbitrary plasma shapes and removes many limitations with regards to the distance between plasma and external coils. Run-time stabilization of vertical modes has also been implemented by means of artificial feedback coils. Finally, active targeting schemes have also been included. These capabilities are illustrated on two very different cases: the ITER tokamak baseline configuration and a NSTX spherical tokamak equilibrium.

On local geometric properties of a tokamak equilibrium

To separate the complexities in plasma physics and geometric effects, compact formulas for local geometric properties of a tokamak equilibrium are presented in this paper. They are written in a form similar to the Frenet formulas. All of the geometric quantities are expressed in terms of curvature and torsion of the three spatial curves for the moving local frame of reference, i.e., local orthogonal vector basis. In this representation, the local magnetic shear and the normalized parallel current are just the differences between two torsions of the vector basis. All of the geometric properties are coordinate invariants and form a prime set of quantities for describing tokamak plasma equilibrium. This prime set can be evaluated in both flux coordinates with closed flux surfaces and cylindrical coordinates including areas with open field lines, which may allow the extension of some analysis on the open field lines outside the last closed surface. Fundamental differential operators for stability and transport studies can be expressed explicitly in terms of these geometric properties. It can also be used to simplify analytic studies.

Multi-scale gyrokinetic simulations of JT-60U L-mode plasma: reduction of the ion scale energy loss due to the nonlinear coupling with the electron scale turbulence

We investigate the effect of the electron temperature gradient (ETG) driven turbulence on the energy transport in JT-60U L-mode plasma by means of the multi-scale gyrokinetic simulation. In the core region at r/a=0.5 , the instability in the ion scale is driven by the ion temperature gradient (ITG), meanwhile strong unstable ETG modes are found in the electron scale. The nonlinear multi-scale gyrokinetic simulation shows that ETG modes are stabilized in the nonlinear phase and the energy transport in the multi-scale simulation is similar to that obtained in the ion scale ITG simulation. In an outer region at r/a=0.6 , the ion scale instability changes to be the trapped electron mode (TEM). The multi-scale simulation shows that both the ion and the electron energy flux are reduced by ∼30% compared to those obtained in the single scale TEM simulation. Interestingly, the electron energy flux is close to the experimental value after this reduction. From the data analyses, we find that ETG turbulence damps the energy of TEM modes through the ion scale/electron scale coupling and the electron scale/electron scale coupling, and can be modeled as a turbulent diffusion of TEM modes. These results suggest that the single ion scale simulation seems to be still valid in the inner region with r/a<0.5 . However, in the outer region it is necessary to include the ETG modes in the gyrokinetic simulations to explain the energy transport in this L-mode plasma. This is the first result showing that ETG turbulence can reduce the electron energy loss via the cross-scale interaction in a real tokamak equilibrium profile.

A new flux coordinates-based solver for fixed-boundary tokamak equilibrium with toroidal flow

The plasma equilibrium plays a crucial role in nuclear fusion studies, serving as the foundation for various aspects of fusion research, including plasma stability, transport, and current drive. In this paper, a new Grad–Shafranov equation solver is developed for the fixed-boundary plasma equilibria with toroidal flow. This solver utilizes the pressure profile, safety factor profile (not current profile), and any two profiles of the toroidal angular velocity, plasma temperature, and square of the Mach number as inputs. The numerical results obtained by this solver exhibit good agreement with known analytic solution under identical parameters, and the potential applications of the solver are demonstrated through several numerical equilibria with toroidal flow. It is very convenient to apply this code to simulate the tokamak equilibrium with a smooth plasma shape. In addition, the effect of toroidal flow on the plasma equilibria is investigated as a simple application. The results reveal a notable outward shift in the contour profiles of magnetic flux surface, density, pressure, and temperature induced by toroidal flow.

Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

SummaryFinite elements of class are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).

An adjoint-based method for optimising MHD equilibria against the infinite-n, ideal ballooning mode

We demonstrate a fast adjoint-based method to optimise tokamak and stellarator equilibria against a pressure-driven instability known as the infinite- $n$ ideal ballooning mode. We present three finite- $\beta$ (the ratio of thermal to magnetic pressure) equilibria: one tokamak equilibrium and two stellarator equilibria that are unstable against the ballooning mode. Using the self-adjoint property of ideal magnetohydrodynamics, we construct a technique to rapidly calculate the change in the eigenvalue, a measure of ideal ballooning instability. Using the SIMSOPT optimisation framework, we then implement our fast adjoint gradient-based optimiser to minimise the eigenvalue and find stable equilibria for each of the three originally unstable equilibria.

Flexible, integrated modeling of tokamak stability, transport, equilibrium, and pedestal physics

The STEP (Stability, Transport, Equilibrium, and Pedestal) integrated-modeling tool has been developed in OMFIT to predict stable, tokamak equilibria self-consistently with core-transport and pedestal calculations. STEP couples theory-based codes to integrate a variety of physics, including magnetohydrodynamic stability, transport, equilibrium, pedestal formation, and current-drive, heating, and fueling. The input/output of each code is interfaced with a centralized ITER-Integrated Modelling &amp; Analysis Suite data structure, allowing codes to be run in any order and enabling open-loop, feedback, and optimization workflows. This paradigm simplifies the integration of new codes, making STEP highly extensible. STEP has been verified against a published benchmark of six different integrated models. Core-pedestal calculations with STEP have been successfully validated against individual DIII-D H-mode discharges and across more than 500 discharges of the H98,y2 database, with a mean error in confinement time from experiment less than 19%. STEP has also reproduced results in less conventional DIII-D scenarios, including negative-central-shear and negative-triangularity plasmas. Predictive STEP modeling has been used to assess performance in several tokamak reactors. Simulations of a high-field, large-aspect-ratio reactor show significantly lower fusion power than predicted by a zero-dimensional study, demonstrating the limitations of scaling-law extrapolations. STEP predictions have found promising scenarios for an EXhaust and Confinement Integration Tokamak Experiment, including a high-pressure, 80%-bootstrap-fraction plasma. ITER modeling with STEP has shown that pellet fueling enhances fusion gain in both the baseline and advanced-inductive scenarios. Finally, STEP predictions for the SPARC baseline scenario are in good agreement with published results from the physics basis.

Gyro orbit simulations of neutral beam injection in Wendelstein 7-X

Simulations exploring neutral beam operation in Wendelstein 7-X (W7-X) at reduced magnetic field are performed using a newly implemented gyro orbit model in the BEAMS3D code. Operation at field strengths below the nominal 2.5 T are seen as a path to explore both high beta plasmas and as a means to access magnetic configurations not possible at 2.5 T. As the field strength becomes smaller, the gyro radius for 55 keV fast protons grows from at 2.5 T to at 0.75 T in a device with minor radius bringing into question the applicability of the gyro center approximation. To address this a gyro orbit model was implemented in the BEAMS3D code. Agreement is found between the gyro center and gyro orbit models in a circular cross section tokamak equilibrium at high field. A set of W7-X equilibria are assessed with fixed density and temperature profiles but decreasing magnetic field strength (increasing plasma beta). Neutral beam deposition is found to be mostly unaffected with changes in the core of the plasma associated with the Shafranov-shift. In general good agreement is found between gyro orbit and gyro center simulations at 2.5 T. Both models indicate increasing losses with decreasing magnetic field strength with the gyro orbit losses being higher at all field strengths. Gyro orbit simulations to the first wall of W7-X show a change in loss pattern with decreasing magnetic field strength. A preliminary assessment of losses to fast ion loss detectors are made.

Retraction Note to: Prevention of Plasma–Surface Interactions by Control of Plasma Equilibrium in Large Aspect Ratio Tokamaks

Retraction Note to: Prevention of Plasma–Surface Interactions by Control of Plasma Equilibrium in Large Aspect Ratio Tokamaks

Quantifying the role of higher order neoclassical corrections to gyrokinetics in tokamak plasmas

We implement the higher order gyrokinetic theory developed in Dudkovskaia et al (2023 Plasma Phys. Control. Fusion 65 045010), reduced to the limit of , where B 0 is the tokamak equilibrium magnetic field, and B ϑ is its poloidal component, in the local gyrokinetic turbulence code, GS2. The principal motivation for this extension is to quantify the importance of neoclassical flows in electromagnetic gyrokinetics, with a particular interest in sharp pressure gradient regions where the bootstrap current becomes dominant. To incorporate neoclassical equilibrium physics, GS2 is coupled to NEO, a multi-species drift kinetic solver. It is found that the regions where microinstabilities are most likely to be influenced by neoclassical equilibrium effects are in a pedestal plasma and a spherical tokamak core plasma.

Controlled movement of lower strike points by applying a divertor coil on the EAST tokamak

A new in-vessel water-cooled coil has been placed below the lower tungsten-covered divertor in EAST, aiming at moving the strike points in the divertor and spreading the heat flow on the targets. To achieve controlled movement of the strike points by the plasma control system (PCS) when applying the in-vessel coil, the plasma equilibrium configurations are predicted by the tokamak equilibrium (TEQ) code. The divertor coil current is involved in these calculations. The shape control parameters in the PCS are adjusted according to the predesigned equilibria from the TEQ code. The strike point movement over a wide range has been achieved in the experiment while keeping the main plasma stable. Thus, this method for divertor strike point movement with the assistance of the divertor coil is feasible.

Drift kinetic theory of the NTM magnetic islands in a finite beta general geometry tokamak plasma

In (Imada et al 2019 Nucl. Fusion 59 046016 and references therein) a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from , where is the trapped ion banana orbit width, to , making it in closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.

Effect of negative triangularity on the bulk ions co-current rotation caused by the ion orbit loss at the edge of the tokamak plasmas

The ion orbit loss (IOL) can drive the bulk ions co-current rotation at the edge of the tokamak plasmas. The effect of triangularity on the IOL is investigated by using an analytical tokamak equilibrium model for the shaped plasmas. The peaking speed of the bulk ions co-current rotation at the tokamak edge will be increased greatly with the negative triangularity.

Numerical Investigation of the Ion Cyclotron Resonance Heating (ICRH) Physics in DTT

An extensive linear analysis of the ICRH propagation and absorption in DTT full heating scenario has been performed by means of three advanced numerical tools: DISEMAG, FELICE and TORIC. The numerical codes solve respectively the full dispersion relation in the complex domain of the wavenumber (DISEMAG) and the integro-differential equation that accounts for the correct evaluation of the ICRH power absorption in a slab plasma (FELICE) and in the tokamak equilibrium configuration (TORIC). Moreover, by incorporating the antenna conceptual design, as released by the engineering design team, in FELICE and TORIC, the power spectrum, radiated by the antenna (3 straps) and coupled to the plasma, has been evaluated in the case of 60 and 90 MHz (3He and H minority heating respectively). By extensively using the abovementioned suite of codes, and after establishing the plasma target (fixing tokamak dimensions, density, temperatures, plasma current, magnetic field, isotopic composition, etc.), and the antenna characteristics (number, dimension, pitch, and radial position of the straps, antenna size, etc.), the power absorption on the various species (electrons, majority and minority ions) has been calculated as function of the minority concentration, parallel wavenumber, frequency, harmonic resonant layer, etc.. Comparison between the codes DISEMAG and TORIC in evaluating the absorption on electrons and minority ions has also been performed and shows a good agreement of the results. A new ion heating scheme based on three ions mixture has also been proposed in DTT.

Linear simulation of kinetic Peeling-Ballooning mode with bootstrap current under the BOUT++ gyro landau fluid code

The kinetic peeling-ballooning mode (KPBM) plays a crucial role in the edge turbulence and transport in a tokamak plasma. However, the impact of the bootstrap current on KPBM is still unclear. Simulations of KPBM using the BOUT++ gyro-Landau-fluid code are presented in this study. To investigate the KPBM in a real tokamak equilibrium, the global equilibrium solver CORSICA was employed to generate a set of realistic equilibria of shifted circular geometry, including the Shafranov shift, elongation effects and bootstrap current. The linear instability property of the KPBM is observed in a wide range of pressure gradient and parallel current density in the pedestal region. Our results indicate that the bootstrap current has a stabilizing effect on the high-n KPBM over the entire pedestal region. Here, n is the toroidal mode number. In comparison with the ideal magnetohydrodynamic peeling-ballooning mode stability diagram, the unstable KPBM region shrinks and shifts to low ( where denotes the pressure at the peak pressure gradient position) with the increase of the bootstrap current. Moreover, we find that the low-n kink modes are driven unstable by the bootstrap current in the second stable region. With an additional external current drive on top of the bootstrap current, the low-n KPBMs can be stabilized when the total edge current is sufficiently large, although the low-n kink mode is still unstable in the region with .

Surrogate models for plasma displacement and current in 3D perturbed magnetohydrodynamic equilibria in tokamaks

A numerical database of over one thousand perturbed three-dimensional (3D) equilibria has been generated, constructed based on the MARS-F (Liu et al 2000 Phys. Plasmas 7 3681) computed plasma response to the externally applied 3D field sources in multiple tokamak devices. Perturbed 3D equilibria with the n = 1–4 (n is the toroidal mode number) toroidal periodicity are computed. Surrogate models are created for the computed perturbed 3D equilibrium utilizing model order reduction (MOR) techniques. In particular, retaining the first few eigenstates from the singular value decomposition (SVD) of the data is found to produce reasonably accurate MOR-representations for the key perturbed quantities, such as the perturbed parallel plasma current density and the plasma radial displacement. SVD also helps to reveal the core versus edge plasma response to the applied 3D field. For the database covering the conventional aspect ratio devices, about 95% of data can be represented by the truncated SVD-series with inclusion of only the first five eigenstates, achieving a relative error (RE) below 20%. The MOR-data is further utilized to train neural networks (NNs) to enable fast reconstruction of perturbed 3D equilibria, based on the two-dimensional equilibrium input and the 3D source field. The best NN-training is achieved for the MOR-data obtained with a global SVD approach, where the full set of samples used for NN training and testing are stretched and form a large matrix which is then subject to SVD. The fully connected multi-layer perceptron, with one or two hidden layers, can be trained to predict the MOR-data with less than 10% RE. As a key insight, a better strategy is to train separate NNs for the plasma response fields with different toroidal mode numbers. It is also better to apply MOR and to subsequently train NNs separately for conventional and low aspect ratio devices, due to enhanced toroidal coupling of Fourier spectra in the plasma response in the latter case.

Free-plasma-boundary solver for axisymmetric ideal MHD equilibria with flow

An efficient iterative, free-plasma-boundary solver for the Grad–Shafranov–Bernoulli system of equations, that describes the ideal MHD equilibrium of a toroidally axisymmetric plasma with flow, is presented. The code implements a numerical scheme recently developed in the context of free-plasma-boundary solvers for ideal static MHD equilibria with magnetic islands and stochastic regions for stellarators. The shape of the plasma edge is permitted to change as needed until the total net force eventually vanishes en route to the equilibrium. Complex coil configurations can be treated in the toroidally axisymmetric approximation. The code opens the possibility of quantifying the changes that plasma flows may induce on important features of a tokamak equilibrium such as the shape of the plasma edge, the plasma confining volume, the position of the magnetic axis or the position of the X-point, among others. Some examples, selected for illustrative purposes, are shown for the ITER baseline magnetic configuration.

ATEQ: Adaptive toroidal equilibrium code

A radially adaptive numerical scheme is developed to solve the Grad–Shafranov equation for axisymmetric magnetohydrodynamic equilibrium. A decomposition with independent solutions is employed in the radial direction, and Fourier decomposition is used in the poloidal direction. The independent solutions are then obtained using an adaptive shooting scheme together with the multi-region matching technique in the radial direction. Accordingly, the adaptive toroidal equilibrium (ATEQ) code is constructed for axisymmetric equilibrium studies. The adaptive numerical scheme in the radial direction improves considerably the accuracy of the equilibrium solution. The decomposition with independent solutions effectively reduces the matrix size in solving the magnetohydrodynamic equilibrium problem. The reduction of the matrix size is about an order of magnitude as compared with the conventional radially grid-based numerical schemes. Also, in this ATEQ numerical scheme, no matter how accuracy in the radial direction is imposed, the size of matrices basically does not change. The small matrix size scheme gives ATEQ more flexibility to address the requirement of the number of Fourier components in the poloidal direction in tough equilibrium problems. These two unique features, the adaptive shooting and small matrix size, make ATEQ useful to improve tokamak equilibrium solutions.