A new approach to high pressure magnetically-confined plasmas is necessary to design efficient fusion devices. This paper presents an equilibrium combining two solutions of the Grad-Shafranov equation, which describes the magnetohydrodynamic equilibrium in toroidal geometry. The outer equilibrium is paramagnetic and confines the inner equilibrium, whose strong diamagnetism permits to balance large pressure gradients. The existence of both equilibria in the same volume yields a dual equilibrium structure. Their combination also improves free-boundary mode stability.
A free-boundary Tokamak Equilibrium Solver (TES), developed for advanced study of tokamak equilibra, is described with two distinctive features. One is a generalized method to resolve the intrinsic axisymmetric instability, which is encountered after all in equilibrium calculation with a free-boundary condition. The other is an extension to deal with a new divertor geometry such as snowflake or X divertors. For validations, the uniqueness of a solution is confirmed by the independence on variations of computational domain, the mathematical correctness and accuracy of equilibrium profiles are c…
An analytic solution to a generalized Grad-Shafranov equation with flow of arbitrary direction is obtained upon adopting the generic linearizing ansatz for the free functions related to the poloidal current, the static pressure and the electric field. Subsequently, a D-shaped tokamak pertinent equilibrium with sheared flow is constructed using the aforementioned solution.
A local magnetic equilibrium model is presented, with finite aspect ratio and up-down asymmetrically shaped cross section, that depends on eight free parameters. In contrast with other local equilibria, which provide simple magnetic-surface parametrizations at the cost of complex poloidal-field flux descriptions, the proposed model is intentionally built to afford analytically tractable magnetic-field components. Therefore, it is particularly suitable for analytical assessments of equilibrium-shaping effects on a variety of tokamak-plasma phenomena.
Uncertainties and errors in magnetic equilibrium reconstructions are a wide-spread problem in interpreting experimental data measured in the tokamak edge. This study demonstrates errors in EFIT++ reconstructions performed on the COMPASS tokamak by comparing the outer midplane separatrix position to the Velocity Shear Layer (VSL) position. The VSL is detected as the plasma potential peak measured by a reciprocating ball-pen probe. A subsequent statistical analysis of nearly 400 discharges shows a strong systematic trend in the reconstructed separatrix position relative to the VSL, where the pri…
From a common expression for the poloidal electrostatic field of a tokamak, in the limit of large aspect ratio and concentric circular flux surfaces, one may determine the associated potential. This potential satisfies Poisson's equation, which reduces to Laplace's equation when the medium has vanishing charge density, in axial geometry but not toroidal geometry. A simple transformation takes the potential over to the correct harmonic form for tokamak coordinates, and the resulting electrostatic field is calculated. From the radial field one may estimate the supporting charge density on the bo…
Various MHD (magnetohydrodynamic) equilibrium tools, some of which being recently developed or considerably updated, are used on the COMPASS tokamak at IPP Prague. MHD equilibrium is a fundamental property of the tokamak plasma, whose knowledge is required for many diagnostics and modelling tools. Proper benchmarking and validation of equilibrium tools is thus key for interpreting and planning tokamak experiments. We present here benchmarks and comparisons to experimental data of the EFIT++ reconstruction code [L.C. Appel et al., EPS 2006, P2.184], the free-boundary equilibrium code FREEBIE [J…
The Grad-Shafranov equation is solved using spectral elements for tokamak equilibrium with toroidal rotation. The Grad-Shafranov solver builds upon and extends the NIMEQ code [Howell and Sovinec, Comput. Phys. Commun. 185 (2014) 1415] previously developed for static tokamak equilibria. Both geometric and algebraic convergence are achieved as the polynomial degree of the spectral-element basis increases. A new analytical solution to the Grad-Shafranov equation is obtained for Solov'ev equilibrium in presence of rigid toroidal rotation, in addition to a previously obtained analytical solution fo…
Toroidal rotation is well known to play significant roles in the edge transport and L-H transition dynamics of tokamaks. Our recent calculation finds that a sufficiently strong localized toroidal rotation can directly bring out the formation of edge pressure pedestal with reversed magnetic shear that is reminiscent of an H-mode plasma, purely through the effects of toroidal rotation on the tokamak MHD equilibrium itself. In particular, the enhanced edge toroidal rotation enables a substantial peaking of the parallel current profile near edge in higher $β$ regimes, which leads to the flattening…
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…
In this work, we have revisited the Solov'ev analytical solution for a tokamak equilibrium. We point out that the vacuum solution in the Solov'ev formulation is inapplicable, since a distributed current density is assumed to fill the vacuum. The realistic vacuum should be current-free. To amend this vacuum solution problem, we use Green's function method to compute the plasma current contribution, together with a homogeneous solution to the Grad-Shafranov equation, to construct the full solution. Matching with the Solov'ev solution on the last closed flux surface is performed to determine the …
Parallel permittivity elements are derived for radio frequency waves in an axisymmetric D-shaped tokamak with Soloviev type equilibrium under arbitrary aspect ratio, arbitrary elongation and moderate triangularity. The drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value problem, accounting for the transit-time and bounce resonances, respectively. It is shown that the new two groups of trapped particles can appear only in tokamaks with large elongation and large central beta. Our dielectric characteristics …
Axisymmetric free-boundary equilibrium reconstructions of tokamak plasmas in the Lithium Tokamak eXperiment (LTX) are performed using the PSI-Tri equilibrium code. Reconstructions in LTX are complicated by the presence of long-lived non-axisymmetric eddy currents generated by vacuum vessel and first wall structures. To account for this effect, reconstructions are performed with additional toroidal current sources in these conducting regions. The source distributions are fixed poloidally, but their scale is adjusted as part of the full reconstruction. Eddy distributions are computed by toroidal…
The Stacey-Sigmar model for tokamak equilibrium as presented in the literature relies heavily on the neoclassical theory of the electrostatic field. Its neglect of Gauss's law is inconsistent with the potential formulation of electrodynamics. Its treatment of the dynamic electric field generated by the central heating and poloidal field coils also is suspect. Its derivation of the viscosity term remains incomplete and does not account for the varying pitch angle of the magnetic field. A derivation of the viscosity term which respects the pitch angle can account for the radial force commonly as…
The neoclassical prescription to use an equation of motion to determine the electrostatic field within a tokamak plasma is fraught with difficulties. Herein we examine two popular expressions for the equilibrium electrostatic field so determined and show that one fails to withstand a formal scrutiny thereof while the other fails to respect the vector nature of the diamagnetic current. Reconsideration of the justification for the presence of the equilibrium electrostatic field indicates that no field is needed for a neutral plasma when considering the net bound current defined as the curl of th…
Recent development of real-time equilibrium code Equinox [1] using a fixed-point algorithm [2] allow major plasma magnetic parameters to be identified in real-time, using rigorous analytical method. The code relies on the boundary flux code providing flux values on the first wall of vacuum vessel. By means of least-square minimization of differences between magnetic field obtained from previous solution and the next measurements the code identifies the source term of the non-linear Grad-Shafranov equation [3]. The strict use of analytical equations together with a flexible algorithm offers an …
The results of the analysis of the deviation of the force equilibrium for ions from the neoclassical theory prediction, calculated using the direct measurements of the radial electric field, in the view of its possible local and nonlocal correlation with the profiles of electron, Te, and ion, Ti, temperatures in the T-10 tokamak are presented. Local correlations are analyzed by means of the Pearson's correlation. Nonlocal correlations are treated with an inverse problem under the assumption of an integral equation relationship between the deviation and Te and Ti profiles. The discharges with z…
Methods are described that extend fields from reconstructed equilibria to include scrape-off-layer current through extrapolated parametrized and experimental fits. The extrapolation includes both the effects of the toroidal-field and pressure gradients which produce scrape-off-layer current after recomputation of the Grad-Shafranov solution. To quantify the degree that inclusion of scrape-off-layer current modifies the equilibrium, the $χ$-squared goodness-of-fit parameter is calculated for cases with and without scrape-off-layer current. The change in $χ$-squared is found to be minor when scr…
With the resonant magnetic perturbations (RMPs) consolidating as an important tool to control the transport barrier relaxation, the mechanism on how they work is still a subject to be clearly understood. In this work we investigate the equilibrium states in the presence of RMPs for a reduced MHD model using 3D electromagnetic fluid numerical code (EMEDGE3D) with a single harmonic RMP (single magnetic island chain) and multiple harmonics RMPs in cylindrical and toroidal geometry. Two different equilibrium states were found in the presence of the RMPs with different characteristics for each of t…
We analyze an axisymmetric equilibrium of a plasma endowed with toroidal and poloidal velocity fields, with the aim to characterize the influence of the global motion on the morphology of the magnetic confinement. We construct our configuration assuming that the poloidal velocity field is aligned with the poloidal magnetic field lines and, furthermore, we require that the plasma mass density depend on the magnetic flux function (or equivalently, that the plasma fluid be incompressible). We then derive a sort of Grad-Shafranov equation for such an equilibrium and implement it to tokamak relevan…
Analytic solutions of the magnetohydrodynamic equilibrium equations for a cylindrically symmetric magnetically confined plasma with reversed magnetic shear, s < 0, and sheared flow are constructed by prescribing the safety factor-, poloidal velocity- and axial velocity- profiles consistently with experimental ones.
VEQDB: A Compact and Reconstructible Multi-Device Tokamak Equilibrium Database
Beijing VeloAlpha Technology Co., Ltd.（北京维洛阿尔法科技有限公司）；Key Laboratory of Materials Modification by Beams of the Ministry of Education, School of Physics, Dalian University of Technology（大连理工大学物理学院光束材料改性教育部重点实验室）
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.
A Rapid Integrated Tokamak Modelling Approach for Coupled Profile and Equilibrium Evolution
Key Laboratory of Materials Modification by Beams of the Ministry of Education, School of Physics, Dalian University of Technology（大连理工大学物理学院材料束改性教育部重点实验室）；Beijing VeloAlpha Technology Co., Ltd.（北京维洛阿尔法科技有限公司）
A new parallel equilibrium reconstruction code for tokamak plasmas is presented. GPEC allows to compute equilibrium flux distributions sufficiently accurate to derive parameters for plasma control within 1 ms of runtime which enables real-time applications at the ASDEX Upgrade experiment (AUG) and other machines with a control cycle of at least this size. The underlying algorithms are based on the well-established offline-analysis code CLISTE, following the classical concept of iteratively solving the Grad-Shafranov equation and feeding in diagnostic signals from the experiment. The new code a…
Characterizing Flux-Surface Shapes in Tokamaks and Quasi-Symmetric Stellarators
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Max Planck Institute for Plasma Physics（马普等离子体物理研究所）；Dutch Institute for Fundamental Research（荷兰基础研究所）；Eindhoven University of Technology（埃因霍温理工大学）；Ruhr-Universität Bochum（波鸿鲁尔大学）
A local magnetic equilibrium solution is sought around the magnetic axis in order to identify the key parameters defining the magnetic-surface's up-down asymmetry in the core of tokamak plasmas. The asymmetry is found to be determined essentially by the ratio of the toroidal current density flowing on axis to the fraction of the external field's odd perturbation that manages to propagate from the plasma boundary into the core. The predictions are tested and illustrated first with an analytical Solovev equilibrium and then using experimentally relevant numerical equilibria. Hollow current-densi…
A class of parametric distribution functions has been proposed in [C.DiTroia, Plasma Physics and Controlled Fusion,54,2012] as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection, Ion Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly isotropic equilibria for Slowing-Down $alpha$ particles and core thermal plasma populations. These EDFs depend on constants of motion (COMs). Assuming an axisymmetric system with no equilibrium electric field, the EDF dep…
Gyrokinetic simulations of ion temperature gradient mode and trapped electron mode driven impurity transport in a realistic tokamak geometry are presented and compared with results using simplified geometries. The gyrokinetic results, obtained with the GENE code in both linear and non-linear modes are compared with data and analysis for a dedicated impurity injection discharge at JET. The impact of several factors on heat and particle transport is discussed, lending special focus to tokamak geometry and rotational shear. To this end, results using s-alpha and concentric circular equilibria are…
The linear dynamics of Alfven modes in tokamaks is investigated here by means of the global gyrokinetic particle-in-cell code NEMORB. The model equations are shown and the local shear Alfven wave dispersion relation is derived, recovering the continuous spectrum in the incompressible ideal MHD limit. A verification and benchmark analysis is performed for continuum modes in a cylinder and for toroidicity-induced Alfven Eigenmodes. Modes in a reversed-shear equilibrium are also investigated, and the dependence of the spatial structure in the poloidal plane on the equilibrium parameters is descri…
Tokamak stability to, potentially explosive, `ballooning' displacements of elliptical magnetic flux tubes is examined in large aspect ratio equilibrium. Above a critical pressure gradient the energy stored in the plasma may be lowered by finite (but not infinitesimal) displacements of such tubes (metastability). Above a higher pressure gradient, the linear stability boundary, such tubes are linearly and nonlinearly unstable. The flux tube displacement can be of the order of the pressure gradient scale length. Plasma transport from displaced flux tubes may result in rapid loss of confinement.
Rotation of tokamak-plasmas, not at the mechanical equilibrium, is investigated utilizing a theorem of thermodynamics, established by Prigogine. This theorem, suitably applied to toroidally confined plasmas, suggests that the global barycentric rotations of the plasma, in the toroidal and poloidal directions, are pure reversible processes. In case of negligible viscosity and by supposing the validity of the balance equation for the internal forces, we show that the plasma, even not in the mechanical equilibrium, may freely rotate as a rigid body in the toroidal direction with an angular freque…
A new global two-fluid electromagnetic turbulence code, CENTORI, has been developed for the purpose of studying magnetically-confined fusion plasmas on energy confinement timescales. This code is used to evolve the combined system of electron and ion fluid equations and Maxwell equations in toroidal configurations with axisymmetric equilibria. Uniquely, the equilibrium is co-evolved with the turbulence, and is thus modified by it. CENTORI is applicable to tokamaks of arbitrary aspect ratio and high plasma beta. A predictor-corrector, semi-implicit finite difference scheme is used to compute th…
We present several numerical solutions to a generalized Grad-Shafranov equation (GGSE), which governs axisymmetric plasma equilibria with incompressible flows of arbitrary direction, using fully connected, feed-forward, deep neural networks, also known as multi-layer perceptrons. Such artificial neural network (ANNs) are trained to approximate tokamak-relevant equilibria upon minimizing the GGSE mean squared residual in the plasma volume and the poloidal flux function on the plasma boundary. Solutions for the Solovev and the general linearizing ansatz for the free functions involved in the GGS…
Magnetohydrodynamic equilibrium and neutronics study on MAST-U using Jenga framework
Pranos Fusion Private Limited（普拉诺斯融合私人有限公司）；Indian Institute of Science（印度科学学院）；Jawaharlal Nehru Centre for Advanced Scientific Research（贾瓦哈拉尔·尼赫鲁高级科学研究中心）
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.
Spherical tokamaks (STs) have been shown to possess properties desirable for a fusion power plant such as achieving high plasma ? and having increased vertical stability. To understand the confinement properties that might be expected in the conceptual design for a high $β$ ST fusion reactor, a 1GW ST plasma equilibrium was analysed using local linear gyrokinetics to determine the type of micro-instabilities that arise. Kinetic ballooning modes (KBMs) and micro-tearing modes (MTMs) are found to be the dominant instabilities. The parametric dependence of these linear modes was determined and fr…
We present recent results and technical breakthroughs for the Bayesian inference of tokamak equilibria using force-balance as a prior constraint. Issues surrounding model parameter representation and posterior analysis are discussed and addressed. These points motivate the recent advancements embodied in the Bayesian Equilibrium Analysis and Simulation Tool (BEAST) software being presently utilised to study equilibria on the Mega-Ampere Spherical Tokamak (MAST) experiment in the UK (von Nessi \emph{et. al.} 2012 \emph{J. Phys. A} \textbf{46} 185501). State-of-the-art results of using BEAST to …
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…
In this paper the global mode structures of linear ion-temperature-gradient (ITG) modes in tokamak plasmas are obtained by combining results from the local gyrokinetic code GS2 with analytical theory. Local gyrokinetic calculations, using GS2, are performed for a range of radial flux surfaces, ${x}$, and ballooning phase angles, ${p}$, to map out the local complex mode frequency, ${Ω_{0}(x,p)=ω_{0}(x,p)+iγ_{0}(x,p)}$ for a single toroidal mode number, ${n}$. Taylor expanding ${Ω_{0}}$ about ${x=0}$, and employing the Fourier-ballooning representation leads to a second order ODE for the amplitu…
Verification and 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 suppresed. The simulated radial mode structures agree quantitatively with the electron cyclotron emission measurement after adjusting, within the experimental uncer…
