Tokamak Plasma Equilibrium Research Articles

Calculation of tearing mode stability in an inverse aspect-ratio expanded tokamak plasma equilibrium

The tearing mode stability of an inverse aspect-ratio expanded tokamak plasma equilibrium of general shape is investigated using asymptotic matching techniques. Particular emphasis is placed on the conservation of toroidal electromagnetic angular momentum. The TJ code, which is a specific implementation of the results of the investigation, is described.

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.

Generalized kinetic equation for tokamak plasma equilibrium distribution function

A generalized kinetic equation for the equilibrium distribution function in a finite beta, arbitrary tokamak plasma is derived. The equation is correct to second order in ρ/L (ρ is the particle Larmor radius and L is the system size). Resolving the finite Larmor radius length scales with no restriction on the ratio of poloidal to total equilibrium magnetic field, Bϑ/B, it generalizes the drift kinetic theory of Hazeltine [Phys. Plasmas 15, 77 (1973)] to the limit of Bϑ/B∼1 (e.g., to ensure validity for spherical tokamaks). Two cases are considered. The first provides the equilibrium distribution function, consistent with the generalized gyrokinetic formalism of Dudkovskaia et al. [Plasma Phys. Controlled Fusion 65, 045010 (2023)], derived specifically to capture neoclassical equilibrium currents in the gyrokinetic stability analyses in strong gradient regions. The second assumes short length scales in the direction perpendicular to the magnetic field, which can occur as a result of small coherent magnetic structures in the plasma, such as neoclassical tearing mode magnetic islands close to threshold. This then extends the drift island equations of Dudkovskaia et al. [Nucl. Fusion 63, 016020 (2023)] for the plasma response to magnetic islands to a spherical tokamak plasma configuration. Resolving ρ∼ρϑ (or Bϑ∼B), where ρϑ is the particle poloidal Larmor radius, is also expected to influence calculations of the magnetic island propagation frequency and the associated contributions to the island onset conditions.

Free-Boundary Plasma Equilibrium Computation in Spherical Globus-M2 Tokamak by Means of the pyGSS Code

The pyGSS code constructed for computation of free-boundary plasma equilibrium in sphericalGlobus-M2 tokamak is described. Currents in the coils of the electromagnetic system, their coordinates,plasma current, positions of the limiter and current-conducting wall, etc., are used as the input parameters.Free parameters determining spatial distribution of equilibrium pressure and current density are selected inthe course of code execution in such a way that the results of reconstruction would agree with the experimentalmeasurements of the poloidal magnetic flux by means of toroidally closed loops. The results of computationof equilibrium are compared with those obtained by means of other codes and experimentally measuredthermal plasma energy, position of the separatrix outer leg, the diamagnetic-loop signal, etc.

Improved FCDI Algorithm for Tokamak Plasma Equilibrium Reconstruction

Improved FCDI Algorithm for Tokamak Plasma Equilibrium Reconstruction

Tokamak plasma equilibrium with relativistic runaway electrons

We consider axisymmetric equilibrium of a tokamak plasma that includes current carried by relativistic runaway electrons (REs). Using a guiding center approach, a qualitative picture of the equilibrium of a pure RE beam is elucidated. In a hot thermal plasma, none of the classical drifts of charged particles contribute to the net field-perpendicular current density, which is purely due to magnetization current. In the case of a runaway beam, however, the curvature drift of REs provides the Lorentz force needed to maintain the centripetal acceleration associated with the relativistic toroidal motion. Two different equilibrium formulations are derived for the general case consisting of a mix of thermal and RE current. At higher RE energies, the shift between flux-surfaces and surfaces of constant generalized angular momentum of REs in such equilibria can exceed the radial extent of a typical magnetohydrodynamic mode such that its stability properties could be altered. Simplified one-dimensional governing equations are derived for the absolute and relative orbit shifts in the case of a circular tokamak, enabling quick estimates of parameter dependencies.

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

Retraction Note to: Magnetic Studies of Tokamak Plasma Equilibrium Based on Magnetic System Materials and Characteristics

Retraction Note to: Magnetic Studies of Tokamak Plasma Equilibrium Based on Magnetic System Materials and Characteristics

Application of integrated simulation environment SIEMNED to the analysis of the MEPHIST-0 tokamak operation

This paper presents the results of numerical simulation of plasma equilibrium and stability in the MEPHIST-0 tokamak with SIEMNED software and comparison of simulation results with experiments. The determined characteristics of the vacuum chamber show that it significantly affects the entire discharge. For various scenarios of the inductor operation, a comparison of experimental data and simulated currents and magnetic fields induced in the chamber was carried out. For steady-state tokamak operation, a numerical study of equilibrium plasma configurations was carried out depending on the currents in the poloidal magnetic field coils and plasma current. The vertical plasma instability was investigated. The limiting values of plasma ellipticity preventing the vertical plasma instability were numerically determined. Numerical simulations show that plasma equilibrium is supported by induced currents. It was shown numerically that magnetic configuration with ‘zero of higher order’ were obtained before the plasma shot, suggesting consistency between the simulation results and observations.

Improved FCDI algorithm for tokamak plasma equilibrium reconstruction

The paper presents the improved FCDI plasma equilibrium reconstruction algorithm implemented at the Globus-M2 tokamak. The algorithm has been improved to be able to operate in real time. The FCDI-FF (Fixed Filaments) operation mode was added, in which the plasma is modeled by current filaments at fixed positions. New user interface for the algorithm displays reconstructed plasma separatrix and contours of the internal magnetic surfaces, the poloidal flux distribution, and graphs of the reconstructed plasma parameters, including profiles of plasma pressure, poloidal current and safety factor q, plasma energy, poloidal beta and internal inductance, which can be exported in *.mat, *.json and G-eqdsk formats. Keywords: tokamak, equilibrium reconstruction, plasma shape.

Py-EFIT: A new Python package for plasma equilibrium reconstruction on EAST tokamak

Plasma equilibrium is the basis for various tokamak research topics, such as plasma performance evaluation, plasma control, and physical analysis of instability, etc. In this study, we have developed a Python-based toolkit with a graphical user interface (GUI) to reconstruct the tokamak plasma equilibrium, which is called the Py-EFIT program. The main implementation steps of Py-EFIT program are introduced. Firstly, various magnetic diagnostic signals and machine structure of Experimental Advanced Superconducting Tokamak (EAST) are utilized as the inputs for computation. Secondly, the plasma magnetic flux distribution is numerically determined based on these inputs according to the plasma equilibrium reconstruction principle. Finally, a GUI of Py-EFIT program is developed for function execution and visualization of results. This program provides a convenient and user-friendly platform for plasma equilibrium calculation. It has been successfully applied for experimental analysis and plasma equilibrium reconstruction on EAST, which also indicates its potential for practical applications in more tokamak devices. Program summaryProgram title: Py-EFITCPC Library link to program files:https://doi.org/10.17632/85f8kj7hvp.1Licensing provisions: GNU General Public License 3Programming language: PythonNature of problem: Plasma equilibrium is the basis for many tokamak research topics, such as plasma performance evaluation, plasma control, physical analysis, etc. But plasma equilibrium cannot be obtained from experimental measurements directly, it has to be calculated through ‘equilibrium reconstruction’ technique. This program (Py-EFIT) provides an efficient, open-source, user-friendly, expandable, and portable computation platform for equilibrium reconstruction calculation.Solution method: 1. Code initialization, including grid mesh division and parameter declaration; 2. Give flux function an initial guess; 3. Construct plasma current matrix and response matrix; 4. Solve for plasma current based on diagnostic signals; 5. Calculate updated flux function based on Green-function method; 6. Refine flux value at magnetic axis and X-points; 7. Evaluate the difference between two iteration steps; 8. Output results until convergence condition is satisfied.

FEQ: a new flux coordinates based equilibrium solver including both magnetic axis and separatrix

Accurate tokamak plasma equilibrium solution in flux coordinates is crucial for many stability and transport studies. Different approaches for dealing with singularities in solving the nonlinear Grad–Shafranov (GS) equation in flux coordinates or also known as straight field line coordinates are proposed in this paper. The GS equation is solved by iterating the position of grids directly in flux coordinates, and hence, no additional errors are introduced due to mapping process for a convergent solution. The singularity at magnetic axis in flux coordinates is removed by using a novel coordinate transform technique. Different from other techniques previously developed, no assumption in boundary condition at magnetic axis is used. This is consistent with the fact that there is no physical boundary at the magnetic axis. A flux coordinate system with poloidal coordinate chosen as the geometric poloidal angle is proposed. It conquers the difficulty in no definition of poloidal coordinate in flux coordinates at separatrix because of the singularity at x-point(s) in a divertor configuration. It also simplifies the process for computing poloidal flux coordinate during the iteration for solving the nonlinear GS equation. Non-uniform grids can be applied in both radial and poloidal coordinates, which allows it to increase the spacial resolution near x-point(s) in a divertor configuration. Based on the model proposed in this paper, a new Flux coordinates based EQuilibrium solver (FEQ) in tokamaks is developed. The numerical solutions from this code agree well with both the analytic Solov’ev solution and the numerical one from the EFIT code for a divertor configuration in the EAST tokamak. This code can be applied for simulating different equilibria with prescribed shape, pressure and current profiles, i.e. including both limiter and divertor configurations, positive triangularity and negative triangularity, different β, arbitrary magnetic shear profile etc. It provides a powerful and convenient fixed-boundary inverse equilibrium solver including both magnetic axis and separatrix in the solution for tokamak researches.

Approximate solutions of polarization state evolution in tokamak plasma polarimetry and their precision

Polarimetry measurement results are used in tokamak plasma equilibrium codes as a constraint, decreasing the measurement uncertainty in determination plasma current, safety factor and density profile. In numerical codes, instead of solving rigorous equation of polarimetric state evolution, approximate analytical expression could be used to increase computing efficiency. In this paper is analyzed the precision of three different models compared with exact solution of polarization state evolution equation. Calculations are made for circular plasma model, with plasma parameters representative for operating domain of tokamak device. The obtained results define the limits of applicability of the separate proposed approximated methods in equilibrium codes for plasma in configuration similar to considered in the paper.

A wall-aligned grid generator for non-linear simulations of MHD instabilities in tokamak plasmas

Block-structured mesh generation techniques have been well addressed in the CFD community for automobile and aerospace studies, and their applicability to magnetic fusion is highly relevant, due to the complexity of the plasma-facing wall structures inside a tokamak device. Typically applied to non-linear simulations of MHD instabilities relevant to magnetically confined fusion, the JOREK code was originally developed with a 2D grid composed of isoparametric bi-cubic Bézier finite elements, that are aligned to the magnetic equilibrium of tokamak plasmas (the third dimension being represented by Fourier harmonics). To improve the applicability of these simulations, the grid-generator has been generalised to provide a robust extension method, using a block-structured mesh approach, which allows the simulations of arbitrary domains of tokamak vacuum vessels. Such boundary-aligned grids require the adaptation of boundary conditions along the edge of the new domain. Demonstrative non-linear simulations of plasma edge instabilities are presented to validate the robustness of the new grid, and future potential physics applications for tokamak plasmas are discussed. The methods presented here may be of interest to the wider community, beyond tokamak physics, wherever imposing arbitrary boundaries to quadrilateral finite elements is required.

Vacuum solution for Solov’ev’s equilibrium configuration in tokamaks

In this work, we construct a tokamak plasma equilibrium generated by a combination of currents flowing within the plasma and in distant external magnetic field coils. The plasma toroidal current density takes the simple form inside the plasma (where r and z are cylindrical coordinates, and a, b, and R are constants), and is zero in the surrounding vacuum. The Grad–Shafranov equation possesses a well-known exact analytic solution within the plasma due to Solov’ev. We use a Green’s function method to compute the poloidal magnetic flux generated by plasma currents, together with a parameterized homogeneous solution to the Grad–Shafranov equation (that is well-behaved at small r and z), to construct the vacuum solution. The vacuum solution is matched to the analytic solution on the last closed magnetic flux surface (LCFS) to determine the parameters in the homogeneous solution. This procedure is performed for both up–down-symmetric double-null and up–down-asymmetric single-null equilibria. We find that any magnetic X-points on the LCFS are distorted due to the fact that one quadrant is filled by a current-carrying plasma, whereas the other three are filled by a vacuum in which no current flows. In particular, the vacuum quadrant opposite the plasma-filled quadrant expands at the expense of the other three quadrants.

Hierarchical robust switching control method with the Improved Moving Filaments equilibrium reconstruction code in the feedback for tokamak plasma shape

The paper describes a new hierarchical tokamak plasma shape control method implemented in the system with switching robust controllers and matching state vectors. A system robustness was achieved via the H∞ loop-shaping design. A novel modeling approach was used for the feedback system simulation: tokamak plasma equilibrium was calculated by the reconstruction code receiving the sum of experimental scenario signals and the output signals of a linear parameter varying (LPV) model, and sending the output to the multivariable plasma shape controller to close the loop. The LPV plant model is based on plasma equilibria reconstructed from the experimental data of the spherical tokamak Globus-M and is used to simulate plasma responses to small deviations from tokamak discharge scenario. The control system components were designed and combined to achieve high-performance operation during short pulses of plasma with total duration of about 200 ms and the divertor phase time length of 20–25 ms. The non-iterative Improved Moving Filaments method was developed for the plasma equilibrium reconstruction with the simultaneous determination of coordinates and currents of filaments. The plasma shape is controlled via the magnetic field at X-point and poloidal fluxes on a plasma separatrix, without calculation of the gaps between the plasma separatrix and the first wall. The hierarchical feedback control method was used in simulations of the entire plasma discharge with the transition from the limiter to the divertor plasma magnetic configuration. The control system was discretized and simulated on a high-speed computer to be prepared for application in Globus-M experiments by a real time test bed.

Plasma equilibrium with fast ion orbit width, pressure anisotropy, and toroidal flow effects

We formulate the problem of tokamak plasma equilibrium including the toroidal flow and fast ion (or energetic particle, EP) pressure anisotropy and the finite drift orbit width (FOW) effects. The problem is formulated via the standard Grad–Shafranov equation (GShE) amended by the solvability condition which imposes physical constraints on allowed spacial dependencies of the anisotropic pressure.The GShE problem employs the pressure coupling scheme and includes the dominant diagonal terms and non-diagonal corrections to the standard pressure tensor. The anisotropic tensor elements are obtained via the distribution function represented in the factorized form via the constants of motion. Considered effects on the plasma equilibrium are estimated analytically, if possible, to understand their importance for GShE tokamak plasma problem. The novelty of the proposed approach is in the way FOW is included into the GShE via the non-diagonal pressure tensor, factorized distribution function of the fast ions representation and in the prescription of the spacial dependence of given the spacial dependence of .

Effects of Resonant Helical Field on Toroidal Field Ripple in IR-T1 Tokamak

The toroidal magnetic field which is created by toroidal coils has the ripple in torus space. This magnetic field ripple has an importance in plasma equilibrium and stability studies in tokamak. In this paper, we present the investigation of the interaction between the toroidal magnetic field ripple and resonant helical field (RHF). We have estimated the amplitude of toroidal field ripples without and with RHF (with different q = m/n) ( m = 2, m = 3, m = 4, m = 5, m = 2 & 3, n = 1) using “Comsol Multiphysics” software. The simulations show that RHF has effects on the toroidal ripples.

Optimal control of a coupled partial and ordinary differential equations system for the assimilation of polarimetry Stokes vector measurements in tokamak free-boundary equilibrium reconstruction with application to ITER

The modelization of polarimetry Faraday rotation measurements commonly used in tokamak plasma equilibrium reconstruction codes is an approximation to the Stokes model. This approximation is not valid for the foreseen ITER scenarios where high current and electron density plasma regimes are expected. In this work a method enabling the consistent resolution of the inverse equilibrium reconstruction problem in the framework of non-linear free-boundary equilibrium coupled to the Stokes model equation for polarimetry is provided. Using optimal control theory we derive the optimality system for this inverse problem. A sequential quadratic programming (SQP) method is proposed for its numerical resolution. Numerical experiments with noisy synthetic measurements in the ITER tokamak configuration for two test cases, the second of which is an H-mode plasma, show that the method is efficient and that the accuracy of the identification of the unknown profile functions is improved compared to the use of classical Faraday measurements.

RETRACTED: A new perspective on remote Saddle Sine and Cosine coils technique for determination of tokamak plasma equilibrium status

In this contribution we have presented a new perspective for determination of tokamak plasma column shift based on remote multipole moments technique. First, we presented analytical details for using this technique. Then principle of different models based on this technique for design and fabrication of a six coils will be presented: four modified Rogowski coil (two Cosine coils and two Sine coils) and two Saddle coils (Saddle Sine coil (SSC) and Saddle Cosine coil (SCC)). Also, in order to comparison of results the flux loops technique is used. Because of continuous measurements of magnetic field distribution around the tokamak plasma using multipole coils, this technique give us more reliable information about the plasma current displacement.