Equilibrium Reconstruction Research Articles

Implementation of a Kalman filter-based eddy current estimator for the P-EFIT magnetic equilibrium reconstruction code

This article discusses the integration of a Kalman filter in the P-EFIT equilibrium reconstruction code, with the aim of estimating the currents induced in the passive structures of a tokamak. The filter is based on a vacuum electromagnetic model of the reactor, and takes advantage of an estimate of the effect of the plasma on the magnetics, provided by the equilibrium reconstruction algorithm. On the other hand, the observer is integrated into the equilibrium reconstruction, which exploits the eddy currents estimates provided by the Kalman filter to refine the obtained solution. To analyze the interplay of the reconstruction code and the proposed observer, the ITER tokamak is considered as a case-study, and the algorithm is tested on a variety of plasma conditions, selected in such a way to maximize the relevance of an accurate knowledge of the passive currents. The code performance is evaluated in terms of convergence metrics, eddy currents estimation accuracy and reconstruction of plasma-related quantities such as plasma–wall gaps, plasma current and plasma profile parameters.

Equilibrium reconstruction of DIII-D plasmas using predictive modeling of the pressure profile

New workflows have been developed for predictive modeling of magnetohydrodynamic (MHD) equilibrium in tokamak plasmas. The goal of this work is to predict the MHD equilibrium in tokamak discharges without having measurements of the kinetic profiles. The workflows include a cold start tool, which constructs all the profiles and power flows needed by transport codes; a Grad–Shafranov equilibrium solver; and various codes for the sources and sinks. For validation purposes, a database of DIII-D tokamak discharges has been constructed that is comprised of scans in the plasma current, toroidal magnetic field, and triangularity. Initial efforts focused on developing a workflow utilizing an empirically derived pressure model tuned to DIII-D discharges with monotonic safety factor profiles. This workflow shows good agreement with experimental kinetic equilibrium calculations, but is limited in that it is a single fluid (equal ion and electron temperatures) model and lacks H-mode pedestal predictions. The best agreement with the H-mode database is obtained using a theory-based workflow utilizing pressure profile predictions from a coupled TGLF turbulent transport and EPED pedestal models together with external magnetics and Motional Stark Effect (MSE) data to construct the equilibrium. Here, we obtain an average root mean square error of 5.1% in the safety factor profile when comparing the predicted and experimental kinetic equilibrium. We also find good agreement with the plasma stored energy, internal inductance, and pressure profiles. Including MSE data in the theory-based workflow results in noticeably improved agreement with the q-profiles in high triangularity discharges in comparison with the results obtained with magnetic data only. The predictive equilibrium workflow is expected to have wide applications in experimental planning, between-shot analysis, and reactor studies.

Recent progress on the J-TEXT three-wave polarimeter-interferometer

The J-TEXT three-wave polarimeter-interferometer system (POLARIS), which measures time-space distribution of electron density and current density, has been optimized with both the optical system and the equilibrium reconstruction method. The phase resolution of a Faraday rotation angle has been improved from 0.1 to 0.06 degree in chords from –0.18 to 0.18 m (plasma minor radius), and the sawtooth oscillation behavior has been detected by Faraday rotation angle measurement. By combining the POLARIS measured data and the equilibrium and fitting code (EFIT), an upgraded equilibrium reconstruction method has been developed, which provides a more accurate temporal and spatial distribution of current density and electron density. By means of the optimized POLARIS and improved equilibrium reconstruction, variations of profiles with increasing density have been carried out, under both Ohmic and electron cyclotron resonance heating discharges.

The GBS code for the self-consistent simulation of plasma turbulence and kinetic neutral dynamics in the tokamak boundary

A new version of GBS (Ricci et al. (2012) [27]; Halpern et al. J. Comput. Phys. 315 (2016) 388-408; Paruta et al. (2018) [11]) is described. GBS is a three-dimensional, flux-driven, global, two-fluid turbulence code developed for the self-consistent simulation of plasma turbulence and kinetic neutral dynamics in the tokamak boundary. In the new version presented here, the simulation domain is extended to encompass the whole plasma volume, avoiding an artificial boundary with the core, hence retaining the core-edge-SOL interplay. A toroidal coordinate system is introduced to increase the code flexibility, allowing for the simulation of arbitrary magnetic configurations (e.g. single-null, double-null and snowflake configurations), which can also be the result of the equilibrium reconstruction of an experimental discharge. The implementation of a new iterative solver for the Poisson and Ampère equations is presented, leading to a remarkable speed-up of the code with respect to the use of direct solvers, therefore allowing for efficient electromagnetic simulations that avoid the use of the Boussinesq approximation. The self-consistent kinetic neutral model, initially developed for limited configurations, is ported to the magnetic configurations considered by the present version of GBS and carefully optimized. A new MPI parallelisation is implemented to evolve the plasma and neutral models in parallel, thus improving the code scalability. The numerical implementation of the plasma and neutral models is verified by means of the method of manufactured solutions. As an example of the simulation capabilities of the new version of GBS, a simulation of a TCV tokamak discharge is presented.

Magnetohydrodynamic dynamo effect from electrostatic drift velocity field on sustainment of reversed field pinch plasmas in three-dimensional KTX equilibrium

A dynamo is necessary for reversed field pinch (RFP) configuration sustainment in a continuous way or oscillation cycles by self-organization. A dynamo velocity field from electrostatic drift is studied in three-dimensional (3D) equilibrium for Keda Torus eXperiment RFP plasmas in quasi-multi-helical (QMH) and quasi-single-helical (QSH) states. The 3D equilibrium reconstruction is computed by VMEC code. It is found that the electrostatic potential provides helical modulation of plasma flow in QSH states. The electrostatic potential in QMH states is processed as a superposition of limited helical harmonics, and it shows an axisymmetric pattern. Non-axisymmetric components are negligible in QMH states because of flux surface averaging of the electrostatic field and corresponding electrostatic drift flow. The electrostatic drift velocity is comparable to the total plasma drift in QSH states; this implies that the electrostatic drift may play an important role in sustaining the helical configuration in RFP plasma.

The measurements by diamagnetic loops in EAST

The measurement of the diamagnetic flux for poloidal beta and energy content of EAST based on diamagnetic loops is described in this article. The error sources can be well compensated with the diamagnetic loop and the compensation loop signals. And the maximum error of vacuum flux under experiments without plasmas is about 0.11 mWb, which meet the typical experimental requirements for high βp discharges (βp∼2.5). For ohmic discharges, the stored energy determined from the diamagnetic flux measurement Wdia is in good agreement with that from the equilibrium reconstruction Wmhd. After the auxiliary heating power are injected into plasmas, the increment of the energy content is observed to verify the heating effects of different heating systems. With the upgrade of 2021, the recent campaign has shown a good response between the diamagnetic flux measurements and rapid stored energy drop induced by edge localized modes (ELMs). The preliminary experimental results indicate that plasma anisotropic degree decreases with increasing density under operation conditions with high density and high heating power.

Validation of the strike point position estimation with the local expansion method for MAST upgrade on the DIII-D tokamak

A local expansion method has been proposed for estimating the strike point position for the advanced divertor configuration on the MAST Upgrade tokamak. The paper discusses the application and assesses the performance of the technique on a long-legged divertor plasma configuration on an operating device - the DIII-D tokamak. A comparison of the spatial location of the outer strike point estimated with the local expansion method against the plasma boundary reconstruction and divertor diagnostics on the DIII-D tokamak is reported. A good agreement with the equilibrium reconstruction and diagnostic data is achieved with respect to estimation of the spatial location of the outer strike point for the long-legged divertor plasma discharge.

First equilibrium reconstruction for ITER with the code NICE

In this short paper we present the first application of the IMAS compatible code NICE to equilibrium reconstruction for ITER geometry. The inverse problem is formulated as a least square problem and the numerical methods implemented in NICE in order to solve it are presented. The results of a numerical experiment are shown: a reference equilibrium is computed from which a set of synthetic magnetic measurements are extracted. Then these measurements are used successfully to reconstruct the equilibrium of the plasma.

Advances in the DTT poloidal interferometer/polarimeter design

A multi-channels poloidal interferometer/polarimeter is under development for the Divertor Tokamak Test (DTT) facility, a new tokamak device whose construction is starting in Italy. The aim of the diagnostics is the simultaneous measurement of the line integrated electron density, Faraday rotation and Cotton Mouton effect. The measurement of two polarimetric signals together with the interferometric one would allow for a robust electron density estimate, for the internal magnetic field measurement as well as for the magnetic equilibrium reconstruction. In this work, we present the advances in the diagnostics design whose characteristics are constrained by scientific requirements as well as the DTT structural aspects. In this regards, we have analyzed the possibility of accommodating up to 16 lines of sight in the available space and the contribution of different chords positions in the magnetic equilibrium reconstruction. In particular, we have used the VMEC and V3FIT codes to evaluate the expected interferometric/polarimetric signals and their contribution in the reconstruction of the q profile. These results have been used to optimize the chords positions. Eventually, we present a realistic CAD-driven design of the diagnostic, with details of the most critical components (e.g. the corner cube retroreflectors in the high field side), which would allow for the selected chords configurations implementation.

Verification and validation of linear gyrokinetic and kinetic-MHD simulations for internal kink instability in DIII-D tokamak

Verification and linear validation of the internal kink instability in tokamak have been performed for both gyrokinetic (GTC) and kinetic-MHD codes (GAM-solver, M3D-C1-K, NOVA, XTOR-K). Using realistic magnetic geometry and plasma profiles from the same equilibrium reconstruction of the DIII-D shot #141216, these codes exhibit excellent agreement for the growth rate and mode structure of the internal kink mode when all kinetic effects are suppressed. The simulated radial mode structures, obtained from linear simulations, are in reasonable agreement with the normalised electron cyclotron emission measurement after adjusting, within the experimental uncertainty, the safety factor q = 1 flux-surface location in the equilibrium reconstruction. Compressible magnetic perturbations strongly destabilize the kink, while poloidal variations of the equilibrium current density reduce the growth rate of the kink. Furthermore, kinetic effects of thermal ions are found to decrease the kink growth rate in kinetic-MHD simulations, but increase the kink growth rate in gyrokinetic simulations, due to the additional drive of the ion temperature gradient and parallel electric field. Kinetic thermal electrons are found to have negligible effects on the internal kink instability.

New Identification Approach and Methods for Plasma Equilibrium Reconstruction in D-Shaped Tokamaks

The paper deals with the identification of plasma equilibrium reconstruction in D-shaped tokamaks on the base of plasma external magnetic measurements. The methods of such identification are directed to increase their speed of response when plasma discharges are relatively short, like in the spherical Globus-M2 tokamak (Ioffe Inst., St. Petersburg, Russia). The new approach is first to apply to the plasma discharges data the off-line equilibrium reconstruction algorithm based on the Picard iterations, and obtain the gaps between the plasma boundary and the first wall, and the second is to apply new identification methods to the gap values, producing plasma shape models operating in real time. The inputs for on-line robust identification algorithms are the measurements of magnetic fluxes on magnetic loops, plasma current, and currents in the poloidal field coils measured by the Rogowski loops. The novel on-line high-performance identification algorithms are designed on the base of (i) full-order observer synthesized by linear matrix inequality (LMI) methodology, (ii) static matrix obtained by the least square technique, and (iii) deep neural network. The robust observer is constructed on the base of the LPV plant models which have the novelty that the state vector contains the gaps which are estimated by the observer, using input and output signals. The results of the simulation of the identification systems on the base of experimental data of the Globus-M2 tokamak are presented.

Predictive multi-channel integrated modeling of a reversed magnetic shear H-mode discharge with internal transport barrier in EAST

A reversed magnetic shear high plasma performance H-mode discharge with internal transport barrier (ITB) has been realized and modeled on experimental advanced superconducting tokamak (EAST) in deuterium (D) plasmas with an ITER-like tungsten (W) upper divertor. In this shot (#71326), the dimensionless parameter G () can reach to 0.25 (H 89 ⩽ 2, β N ⩽ 2, q 95 ∼ 4, I p ⩽ 450 kA, B T ⩽ 1.6 T). The ITB has been observed in the channels of particle (electron density n e), electron and ion temperature (T e and T i) and toroidal rotation velocity (V t). However, in the formation process of V t and n e ITBs, the heavy impurities accumulate in the core plasmas, which may be one of the most important reasons for the limitation of the improvement of plasma performance. A time slice (t = 4.65 s) of the high plasma performance phase with n e, T i and V t ITBs has been analyzed. It is found that the position of minimum safety factor (q min) is at about normalized radius ρ = 0.4 where may be the location of ion ITB foot, during equilibrium reconstruction from EFIT using external magnetic and internal polarimeter-INTferometer (POINT) measurement constraints. Based on this equilibrium, the source is self-consistently calculated by ONETWO and NUBEAM. Then the n e, T e, T i and V t have been modeled predictively and simultaneously by TGYRO utilizing the reconstructed equilibrium, the self-consistent source and the experimental radiated power for the first time in EAST. This modeling well reproduces the experimental n e and T i profiles in the reversed magnetic shear plasmas. In addition, it is shown that reversed magnetic shear rather than shear of radial electric field (E r) mainly produced by V t plays a key role to the formation of n e and T i ITBs. The reasons for central accumulation of W impurity are also clarified by computing both neoclassical and turbulent transport components of W. It turns out that the neoclassical transport dominates over turbulent transport for W and neoclassical pinch of W causes its central accumulation.

The magnetic structure of long-wavelength magnetohydrodynamic modes in current-carrying stellarator plasmas

Magnetic field fluctuations are observed in current-carrying stellarator plasmas when the rotational transform is close to a rational value at the edge of the plasma. At low plasma pressure, these fluctuations are associated with perturbed currents parallel to the equilibrium magnetic field lines. A model for these magnetohydrodynamic modes in a low-β, three-dimensional stellarator equilibria has been developed. A set of helical current filaments are constrained to mimic the structure of magnetic field lines on rational surfaces derived from three-dimensional (3D) equilibrium reconstructions. Transformation to straight field line coordinates then allows fitting of the poloidal magnetic sensor data to a single harmonic function, which fixes the modeled toroidal mode structure via the field line flow geometry. The developed procedure accurately captures phase and amplitude variation for m/n = 3/2, 3/1, and 4/1 modes in the 3D equilibria of the compact toroidal hybrid experiment.

Modification of the magneto-hydrodynamic equilibrium by the lower-hybrid wave driven fast electrons on the TST-2 spherical tokamak

Presence of a large population of fast particles may qualitatively modify the tokamak equilibrium from the one that can be described by standard magneto-hydrodynamics (MHD). The kinetic modification of the MHD equilibrium was studied for a lower-hybrid (LH) wave driven plasma on the TST-2 spherical tokamak. The analysis was performed using an equilibrium reconstruction method based on an extended MHD model that considers a two-component plasma of bulk MHD fluid and kinetic fast electrons. Scrape-off-layer current appeared naturally in the extended MHD analysis because of the finite electron orbit excursion from the flux surfaces. This reduced the bulk pressure contribution to the toroidal current substantially. The resulting poloidal flux profile was more consistent with that inferred from the Thomson scattering measurement.

Kinetic equilibrium reconstruction and the impact on stability analysis of KSTAR plasmas

High fidelity kinetic plasma equilibrium reconstructions are an essential requirement for accurate stability and disruption prediction analyses to support continuous operation of high beta tokamak plasmas. The present kinetic equilibrium reconstructions of plasmas in the KSTAR device include plasma density and temperature profiles from Thomson scattering and ion temperature from charge exchange spectroscopy diagnostics, and allowance for fast particle pressure. In addition, up to 25 channels of motional Stark effect diagnostic data are used to constrain the magnetic field pitch angle profile in the plasma to produce a reliable computation of the safety factor, q, profile. H-mode plasmas exhibit clear pedestal characteristics in the reconstructed pressure profile compared to internal transport barrier or L-mode plasmas. The plasma configuration and vertical position of inner strike points are validated by CCD and infrared camera images. Ideal and resistive magnetohydrodynamic (MHD) stability analyses using the DCON and resistive DCON codes utilize these kinetic equilibrium reconstructions to compare to experimental plasma stability. Equilibria with sufficiently low convergence error can provide reliable computation of ideal and resistive MHD stability analysis.

Initial characterization of electron temperature and density profiles in PEGASUS spherical tokamak discharges driven solely by local helicity injection

Local helicity injection (LHI) is a non-solenoidal startup technique that utilizes electron current injectors at the plasma edge to initiate tokamak discharges. Viable non-solenoidal startup techniques require high central Te to combat resistive losses and enhance coupling to auxiliary methods of current drive/heating. Thomson scattering measurements of LHI discharges in Pegasus showed peaked Te profiles at Ip∼0.15 MA and Bt∼0.15 T with Te,0∼100– 150 eV. These results are similar to Te profiles observed with Ohmic induction. At lower levels of Bt, LHI Te profiles were hollow with Te,0∼40 eV and Te,max≤120 eV depending upon the helicity input. Regardless of the Bt level and helicity input, the electron pressure profiles were flat/peaked with hollow JR profiles. Equilibrium reconstructions and measurements of core absolute extreme ultraviolet radiation suggest the hollow Te profiles are a result of very low resistive heating power in the core due to the edge-localized nature of LHI and low-Z line radiation losses. Estimates of Zeff from the plasma conductivity indicate averaged values of ∼1 or ∼3 assuming neoclassical or Spitzer conductivity, respectively. When auxiliary heating power from magnetic reconnection is considered, this observed LHI performance is comparable to expectations from a linear Ohmic confinement scaling estimate and a collisional stochastic confinement scaling estimate of the core plasma region.

Plasma initiation and preliminary magnetic control in the HL-2M tokamak

The first plasmas have been achieved in the HL-2M tokamak by fulfilling the usual criteria for plasma breakdown, i.e. optimal working gas pressure, field null configuration of poloidal magnetic field, reasonable toroidal electrical field and clean wall conditions. The characterization of plasma initiation, such as identification of initiation time and position, is performed by diagnostic data analysis and numerical simulation. A model-based method, which enables dedicated pick-up sensors to estimate plasma radial position in real time, is applied for the limiter plasma shape in the first plasma commissioning phase. For a circular cross-section plasma, feedback control on plasma current and radial position allowed us to extend discharge durations to ∼250 ms associated with a 120 kA plasma current in flat-top. Preliminary plasma equilibrium reconstruction by the NICE code from pick-up sensors and flux loops, as well as interferometry measurements, gives plasma boundaries, plasma profiles of toroidal current density and safety factor, as well as electron density, in an offline way.

A new tool GSEQ-FRC for two-dimensional field-reversed configuration equilibrium

A new tool (GSEQ-FRC) is developed for solving two-dimensional equilibrium of the field-reversed configuration (FRC) based on fixed boundary and free boundary conditions that includes external coils. Benefiting from the two-parameter modified rigid rotor radial equilibrium model and the numerical approaches presented (Ma et al 2021 Nucl. Fusion 61 036046) in this work, the GSEQ-FRC code is used to study the equilibrium properties of FRC quantitatively and will be used for fast FRC equilibrium reconstruction. In GSEQ-FRC, the FRC equilibrium is determined by two parameters, i.e. the ratio between thermal pressure and magnetic pressure at the separatrix β s , and the normalized scrape of layer width δ s . Examples with fixed and free boundary conditions are given to demonstrate the capability of GSEQ-FRC for equilibrium calculations. This new tool is used to quantitatively study the factors affecting the shape of the FRC separatrix, which reveals how the FRC changes from racetrack-like to ellipse-like.

Measurements and modelling of diamagnetic flux in ASDEX Upgrade

The reproducibility of diamagnetic flux measurements in vacuum field shots for the internal and external diamagnetic flux measurements in the ASDEX Upgrade tokamak is investigated. A comparison of diamagnetic flux measurements and predictions from equilibrium reconstruction, including pressure constraints from thermal and fast-ion contributions, is reported. The ideal magnetohydrodynamic model involved in interpreting diamagnetic flux measurements in a tokamak with anisotropic pressure is summarised. The plasma energy calculated from equilibrium reconstruction and inferred from diamagnetic flux measurements is compared with the respective values calculated by modelling with TRANSP and IDE/RABBIT. It is found that at low densities, in discharges with parallel neutral beam sources, the plasma energy inferred from the diamagnetic flux measurements is smaller than the plasma energy calculated from the equilibrium reconstruction. In addition, details of the modelling are discussed to stress the interdependence of the various modelling inputs and the interpretation of the results. A comparison of diamagnetic flux measurements with calculations from equilibrium reconstruction by IDE and fast-ion profiles from RABBIT and TRANSP simulations can identify anisotropic plasma discharges and provide a consistency check of the input data for modelling.

Development of a thin high-frequency and high-precision magnetic probe array in Sino-United Spherical Tokamak

Since the major/minor radius of the Sino-United Spherical Tokamak (SUNIST) is 0.3/0.23m, respectively, the space left for magnetic diagnostics in the high field side (HFS) is quite limited. At the same time, a good signal-to-noise ratio and a high-frequency response (up to 1MHz) are required for equilibrium reconstruction (ER) and Alfven eigenmode studies. Such a magnetic probe array must be extremely thin and tightly close to the central column, not exceeding the inner limiter and leaving the aspect ratio of the spherical tokamak unchanged. Therefore, the turn number and the shape of windings should be highly optimized to enable both a high-frequency response and an enough effective area. A 32-channel magnetic probe array fulfilling these requirements on the central column is designed, calibrated, and installed in the SUNIST. The array consists of 16 probes. Each of them consists of two perpendicular windings, which can measure toroidal and poloidal magnetic fields simultaneously. The effective area and frequency response of each probe are calibrated using a Helmholtz coil and an LCR bridge based on an equivalent probe-and-cable circuit model. After that, an expression of the magnetic diagnostic response to the field coil currents is used to calibrate the installation error. With the full coverage of magnetic probes in the poloidal direction, more reliable ER can be obtained, and the features of magnetohydrodynamic activities in the HFS can be studied.