Tokamak Equilibria Research Articles

Analytical solution of the tokamak equilibrium. II. The free-boundary case

The free-boundary tokamak equilibrium problem is investigated by a surface current equivalence method applied on the plasma boundary. In addition, use is made of a spectral representation for the internal plasma flux surfaces as presented in Paper I [G. O. Ludwig, Phys. Plasma 24, 092502 (2017)]. The surface current distribution is determined by the Cauchy condition imposed by the external equilibrium coils on the plasma boundary. A self-consistent approximate analytic equilibrium is calculated for an up-down asymmetric configuration of the plasma contained by a simplified set of poloidal field coils representing an ITER-like tokamak.

Analytical solution of the tokamak equilibrium. I. The fixed-boundary case

This article presents an analytical method to evaluate the equilibrium of plasmas in axisymmetric, nested magnetic configurations relevant for tokamak experiments. The emphasis is in the use of spectral representations of the magnetic flux surfaces and variational methods to solve the Grad-Shafranov equation. Approximate solutions are obtained based on power series expansions of the variational moment equations and sectionally continuous representations of the Fourier amplitudes. The method is applied to an up-down asymmetric configuration of an ITER-like tokamak in a fixed-boundary equilibrium. The free-boundary case is treated in the Paper II [G. O. Ludwig, Phys. Plasma 24, 092503 (2017)].

Saturation of Alfvén modes in tokamak plasmas investigated by Hamiltonian mapping techniques

Nonlinear dynamics of single toroidal number Alfvén eigenmodes destabilised by the the resonant interaction with fast ions is investigated, in tokamak equilibria, by means of Hamiltonian mapping techniques. The results obtained by two different simulation codes, XHMGC and HAGIS, are presented for n = 2 Beta induced Alfvén eigenmodes and, respectively n = 6 toroidal Alfvén eigenmodes. Simulations of the bump-on-tail instability performed by a 1-dimensional code, PIC1DP, are also analysed for comparison. As a general feature, modes saturate as the resonant-particle distribution function is flattened over the whole region where mode-particle power transfer can take place in the linear phase. Such region is limited by the narrowest of resonance width and mode width. In the former case, mode amplitude at saturation exhibits a quadratic scaling with the linear growth rate; in the latter case, the scaling is linear. These results are explained in terms of the approximate analytic solution of a nonlinear pendulum model. They are also used to prove that the radial width of the single poloidal harmonic sets an upper limit to the radial displacement of circulating fast ions produced by a single-toroidal-number gap mode in the large n limit, irrespectively of the possible existence of a large global mode structure formed by many harmonics.

High resolution equilibrium calculations of pedestal and SOL plasma in tokamaks

For integrated modeling of equilibrium, stability and dynamics of the divertor tokamak plasma with scrape-off layer (SOL) high resolution equilibrium calculations are needed. A new version of the CAXE equilibrium code computes the tokamak equilibrium on a numerical grid adaptive to magnetic surfaces both in the plasma region with closed flux surfaces and in the SOL region with open magnetic lines. The plasma profiles can be prescribed independently in each region with nested flux surfaces, and realistic SOL profiles with very short pressure drop off length can be accurately treated. The influence of the finite current density in SOL on the connection length is studied. From the point of view of the MHD equilibrium and stability modeling, self-consistent calculations of diverted tokamak configurations with finite current density at the separatrix require taking into account plasma outside the separatrix. Calculated high resolution equilibria provide an input to new versions of the ideal MHD stability codes treating tokamak plasma with SOL. The study of the influence of the pressure gradient profile in the pedestal plasma inside and outside the separatrix on the pedestal height limit set by external kink-ballooning mode stability is presented. Another possible application of the high resolution pedestal and SOL equilibrium code is a coupling to the SOLPS code with a purpose to increase equilibrium accuracy and support self-consistent plasma flow/equilibrium modeling.

Effect of scrape-off-layer current on reconstructed tokamak equilibrium

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 scrape-off-layer current is included; however, flux surfaces are shifted by up to 3 cm. The impact on edge modes of these scrape-off-layer modifications is also found to be small and the importance of these methods to nonlinear computation is discussed.

RETRACTED: A first principles study on resonant perturbation field effects on toroidal field ripples in tokamaks

Abstract The toroidal magnetic field which is created by toroidal coils has ripples in torus space. This magnetic field ripple is important for plasma equilibrium and stability studies in tokamaks. In this paper, we have presented the investigation of interaction between the toroidal magnetic field ripple and Resonant Perturbation Field (RPF) using the COMSOL Multiphysics software in the tokamaks. We have simulated the amplitude of toroidal field ripples without and with RPF (with different q = m/n) (m = 2, m = 3, m = 4, m = 5, m = 2 & 3; n = 1). The simulations show that RPF lead to modulation of toroidal field ripples.

Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016

The joint Varenna-Lausanne international workshop on the theory of fusion plasmas took place in Varenna from August 29 to September 2 2016. Several issues of interest for fusion plasmas were addressed, namely MHD stability, RF heating, collisional and turbulent transport, plasma wall interaction, and physics of burning plasmas. The articles published in this special issue illustrate nicely the well balanced combination of physics, applied mathematics, and computer sciences that characterizes this workshop.Let us mention several attractive topics, which are addressed in this issue. The question of 3D MHD equilibrium in tokamaks has received a great deal of attention, in connection with external resonant magnetic perturbations in tokamaks, and also stochastic edge in stellarators. The reader will also find some recent developments related to the effect of current drive and heating on the stability of tearing modes. As usual, turbulent transport is addressed in much detail. Several papers address specific numerical aspects of fluid and gyrokinetic codes, including code optimisation. Physics issues are abundantly dealt with, such as the impact of fast particles on turbulence, and particle transport. New numerical techniques to model wave propagation are presented, which provide significant advances in the field. Refinements such as the effect of density fluctuation on wave propagation, or the interaction between particles and the electromagnetic field near antennas, have also been studied in depth. Finally, specific issues such as nonlocal transport, decay of zonal flows, and the effect of neutrals on rotation have been investigated.A striking feature of the 2016 edition was the large number of young faces among the participants. This is a great satisfaction for the organizers since a new generation of scientists is certainly needed whilst several devices come to operation, or will do so in a foreseeable future.The diversity and quality of the papers published in the present cluster prove once more that the Varenna-Lausanne meeting has a special place among the conferences on magnetic fusion energy. It is certainly an excellent forum for exchanging ideas on theory and modelling, and for advertising the best results obtained in that field.The present volume of Journal of Physics Conference Series contains 16 outstanding papers - all peer reviewed. Let us mention another set of about 20 papers to appear in Plasma Physics and Controlled Fusion. We hope the reader will appreciate this special issue and that it will be a source of inspiration for him.Xavier Garbet, Olivier Sauter November, 8 2016http://varenna-lausanne.epfl.ch

Use of reconstructed 3D VMEC equilibria to match effects of toroidally rotating discharges in DIII-D

A technique for tokamak equilibrium reconstructions is used for multiple DIII-D discharges, including L-mode and H-mode cases when weakly 3D fields are applied. The technique couples diagnostics to the non-linear, ideal MHD equilibrium solver VMEC, using the V3FIT code, to find the most likely 3D equilibrium based on a suite of measurements. It is demonstrated that V3FIT can be used to find non-linear 3D equilibria that are consistent with experimental measurements of the plasma response to very weak 3D perturbations, as well as with 2D profile measurements. Observations at DIII-D show that plasma rotation larger than 20 krad s−1 changes the relative phase between the applied 3D fields and the measured plasma response. Discharges with low averaged rotation (10 krad s−1) and peaked rotation profiles (40 krad s−1) are reconstructed. Similarities and differences to forward modeled VMEC equilibria, which do not include rotational effects, are shown. Toroidal phase shifts of up to are found between the measured and forward modeled plasma responses at the highest values of rotation. The plasma response phases of reconstructed equilibra on the other hand match the measured ones. This is the first time V3FIT has been used to reconstruct weakly 3D tokamak equilibria.

Global and pedestal confinement and pedestal structure in dimensionless collisionality scans of low-triangularity H-mode plasmas in JET-ILW

A dimensionless collisionality scan in low-triangularity plasmas in the Joint European Torus with the ITER-like wall (JET-ILW) has been performed. The increase of the normalized energy confinement (defined as the ratio between thermal energy confinement and Bohm confinement time) with decreasing collisionality is observed. Moreover, at low collisionality, a confinement factor H98, comparable to JET-C, is achieved. At high collisionality, the low normalized confinement is related to a degraded pedestal stability and a reduction in the density-profile peaking.The increase of normalized energy confinement is due to both an increase in the pedestal and in the core regions. The improvement in the pedestal is related to the increase of the stability. The improvement in the core is driven by (i) the core temperature increase via the temperature-profile stiffness and by (ii) the density-peaking increase driven by the low collisionality.Pedestal stability analysis performed with the ELITE (edge-localized instabilities in tokamak equilibria) code has a reasonable qualitative agreement with the experimental results. An improvement of the pedestal stability with decreasing collisionality is observed. The improvement is ascribed to the reduction of the pedestal width, the increase of the bootstrap current and the reduction of the relative shift between the positions of the pedestal density and pedestal temperature.The EPED1 model predictions for the pedestal pressure height are qualitatively well correlated with the experimental results. Quantitatively, EPED1 overestimates the experimental pressure by 15–35%. In terms of the pedestal width, a correct agreement (within 10–15%) between the EPED1 and the experimental width is found at low collisionality. The experimental pedestal width increases with collisionality. Nonetheless, an extrapolation to low-collisionality values suggests that the width predictions from the KBM constraint are reasonable for ITER.

Global kinetic ballooning mode simulations in BOUT++

We report on simulation results of a 3+1 gyro-Landau-fluid (GLF) model in BOUT++ framework, which contributes to increasing the physics understanding of the edge turbulence. We find that there is no second stability region of kinetic ballooning modes (KBM) in the concentric circular geometry. The first unstable β of KBM decreases below the ideal ballooning mode threshold with increasing . In order to study the KBM in the real tokamak equilibrium, we find that the approximation of shifted circular geometry () is not valid for a high β global equilibrium near the second stability region of KBM. Thus we generate a series of real equilibria from a global equilibrium solver CORSICA, including both Shafranov shift and elongation effects, but not including bootstrap current. In these real equilibria, the second stability region of KBM are observed in our global linear simulations. The most unstable mode for different β are the same while the mode number spectrum near the second stability region is wider than the case near the first stability region. The nonlinear simulations show that the energy loss of an ELM keeps increasing with β, because the linear drive of the turbulence remains strong for the case near the second stability region during profile evolution.

Numerical calculation of free boundary equilibrium via meshfree method and its application in Alborz tokamak

In the present study, the calculation of free boundary axisymmetric tokamak equilibrium has been presented via the meshfree numerical method. This method of calculation has not been used previously for free boundary condition. As a practical application of the proposed numerical method two approaches of Alborz plasma equilibrium calculations were taken into account. First, the design of the poloidal field coil system has been carried out with consideration of the limiter. Second, the calculation has been developed by considering the divertor configuration. Considering the absence of analytical solution for free boundary condition, the validity of the code was examined by three different methods. The convergence of this method was checked by error calculation. Also the mathematical numerical correctness and accuracy were investigated by a comparison with the finite difference method and the uniqueness of a solution was confirmed by the independence on variations of the computational domain. The decrement trend in errors behavior was observed and the error for near circular configuration was reached less than 1e−4 in a few iterations. Moreover, the minimum Root Mean Squared (RMS) error achieved by the MLS method was 5.4826e−6 that correspond to the divertor configuration. The good results obtained from the verification process prove that the methodology applied here is reliable for tokamak equilibrium calculations and can be used as an effective tool in future studies.

Sensitivity of kinetic ballooning mode instability to tokamak equilibrium implementations

Global, first-principles study of the kinetic ballooning mode (KBM) is crucial to understand tokamak edge physics in high-confinement mode (H-mode). In contrast to the ion temperature gradient mode and trapped electron mode, the KBM is found to be very sensitive to the equilibrium implementations in gyrokinetic codes. In this paper, we show that a second-order difference in Shafranov shift or geometric coordinates, or a difference between local and global profile implementations can bring a factor of two or more discrepancy in real frequency and growth rate. This suggests that an accurate global equilibrium is required for validation of gyrokinetic KBM simulations.

Impact of pressure anisotropy on magnetic configuration and stability

A number of tools have recently been developed to study equilibrium and stability in tokamaks with pressure anisotropy. Here we apply these tools to a Mega Ampere Spherical Tokamak (MAST) discharge to calculate equilibrium, Alfvén continua and eigenmodes, through to linear growth and nonlinear saturation of a toroidal Alfvén eigenmode (TAE); this is the first study of wave growth and saturation for anisotropic equilibria. Comparisons with the standard tools which assume an isotropic pressure reveal various differences in equilibrium and modes: the safety factor profile in the isotropic reconstruction is reversed shear while the anisotropic reconstruction gives normal shear; the isotropic TAE gap is much narrower than the anisotropic gap; and the TAE radial mode structure is wider in the anisotropic case. These lead to a modification in the resonant regions of fast-ion phase space, and produce a 35% larger linear growth rate and an 18% smaller saturation amplitude for the TAE in the anisotropic analysis compared to the isotropic analysis.

RETRACTED: Feedback controller input design for ignition of deuterium–tritium in NSTX tokamak

Nuclear fusion is a nuclear reaction in which two or more atomic nuclei (such as a deuterium–tritium) come very close and then collide at a very high speed and join to form a new high energy nucleus (Helium). Determination of accurate plasma horizontal position during plasma discharge is essential to transport it to a control system based on feedback. The solutions of Grad-Shafranov equation (GSE) analytically can be used for theoretical studies of plasma equilibrium, transport and magneto hydrodynamic stability. Here we have presented specific choices for source functions, kinetic pressure and poloidal plasma current, to be quadratic in poloidal magnetic flux and derive an analytical solution for Grad-Shafranov equation. With applying this solution to NSTX tokamak, we calculated poloidal magnetic flux, toroidal current density and normalized pressure profiles for this tokamak. Toroidal and poloidal flows can considerably change the equilibrium parameters of tokamak. These effects on the equilibrium of tokamak plasmas are numerically investigated using the code FLOW. As a comparative approach to equilibrium problem, the code is used to model equilibrium of NSTX tokamak for case pure toroidal flow. Comparison of the results of these two methods for NSTX tokamak shows good agreement between two and that our analytical solution can be served as good benchmark against the equilibrium code FLOW.

Improved analytical flux surface representation and calculation models for poloidal asymmetries

An orthogonalized flux-surface aligned curvilinear coordinate system has been developed from an up-down asymmetric variation of the “Miller” flux-surface equilibrium model. It is found that the new orthogonalized “asymmetric Miller” model representation of equilibrium flux surfaces provides a more accurate match than various other representations of DIII-D [J. L. Luxon, Nucl. Fusion 42, 614–633 (2002)] discharges to flux surfaces calculated using the DIII-D Equilibrium Fitting tokamak equilibrium reconstruction code. The continuity and momentum balance equations were used to develop a system of equations relating asymmetries in plasma velocities, densities, and electrostatic potential in this curvilinear system, and detailed calculations of poloidal asymmetries were performed for a DIII-D discharge.

RETRACTED ARTICLE: Low Dissipation with Normalized Flux Surfaces in 2-Dimensional Coordinate for IR-T1 Tokamak Using Grad–Shafranov Equation Solution

Together with the problem of confinement, plasma–wall interactions present the major constraints toward a magnetic fusion reactor. The solutions of Grad–Shafranov equation (GSE) analytically can be used for theoretical studies of plasma equilibrium, transport and magneto-hydrodynamic stability. Here we introduce specific choices for source functions, kinetic pressure and poloidal plasma current, to be quadratic in poloidal magnetic flux and derive an analytical solution for GS equation. With applying this solution to IR-T1 tokamak, we have calculated the poloidal magnetic flux, toroidal current density and normalized pressure profiles for this tokamak. Toroidal and poloidal flows can considerably change the equilibrium parameters of tokamak. These effects on the equilibrium of tokamak plasmas are numerically investigated using a code FLOW. As a comparative approach to equilibrium problem, the code is used to model equilibrium of IR-T1 tokamak for case pure toroidal flow.

Coupling of the Flux Diffusion Equation with the Equilibrium Reconstruction at ASDEX Upgrade

A tokamak equilibrium reconstruction can benefit much from internal measurements of the current distribution. If lacking robust internal measurements, the reconstruction will be ill posed in the plasma core, not allowing for a sensible estimation of the current distribution. Such deficiencies can be compensated for by modeling the current distribution evolution by employing the current diffusion equation between successive equilibria. A scheme for the coupling of the predictive current diffusion equation with the equilibrium reconstruction from an inverse Grad-Shafranov equilibrium solver minimizing a least-squares criterion on measured and modeled data is proposed. The scheme is intended for routine equilibrium analysis shortly after the discharge where all diagnostic data are available. Results from the implementation at ASDEX Upgrade are shown, applied to a reversed-shear plasma with counter-current electron cyclotron current drive and to the start-up phase of the plasma. Results are compared to TRANSP calculations.

Saturated ideal kink/peeling formations described as three-dimensional magnetohydrodynamic tokamak equilibrium states

Free boundary magnetohydrodynamic equilibrium states with spontaneous three dimensional deformations of the plasma-vacuum interface are computed for the first time. The structures obtained have the appearance of saturated ideal external kink/peeling modes. High edge pressure gradients yield toroidal mode number n = 1 corrugations for a high edge bootstrap current and larger n distortions when this current is small. Deformations in the plasma boundary region induce a nonaxisymmetric Pfirsch-Schlüter current driving a field-aligned current ribbon consistent with reported experimental observations. A variation in the 3D equilibrium confirms that the n = 1 mode is a kink/peeling structure. We surmise that our calculated equilibrium structures constitute a viable model for the edge harmonic oscillations and outer modes associated with a quiescent H-mode operation in shaped tokamak plasmas.

RETRACTED ARTICLE: Prevention of Plasma–Surface Interactions by Control of Plasma Equilibrium in Large Aspect Ratio Tokamaks

In tokamak plasma, impurities present a number of problems. One is the radiative power loss, principally due to line radiation from partially stripped ions. Another is fuel dilution. This arises because impurity atoms produce many electrons and, for a given plasma pressure, these electrons lake the place of fuel particles. At high concentrations impurities prevent the plasma being heated. This is particularly a problem during the plasma start-up phase since impurities radiate most strongly at low temperatures before they become highly ionized. Impurities can also lead to disruptions as a result of edge cooling and consequent current profile modification. On the other hand, these problems can be prevented by control of plasma equilibrium (which is defined by Grad–Shafranov (GS) equation). Numerous methods exist to solve the GS equation, describing the equilibrium of plasma confined by an axisymmetric magnetic field. In this paper, we have proposed a new numerical solution to the GS equation of an axisymmetric, transformed in cylindrical coordinates solved with the Chebyshev collocation method, when the source term (current density function) on the right hand side is linear. The Chebyshev collocation method is a method for computing highly accurate numerical solutions of deferential equations. We describe a circular cross section of tokamak and present numerical result of magnetic surfaces on the IR-T1 tokamak and then compare the results with an analytical solution.

Verification of the ideal magnetohydrodynamic response at rational surfaces in the VMEC code

The VMEC nonlinear ideal MHD equilibrium code [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)] is compared against analytic linear ideal MHD theory in a screw-pinch-like configuration. The focus of such analysis is to verify the ideal MHD response at magnetic surfaces which possess magnetic transform (ι) which is resonant with spectral values of the perturbed boundary harmonics. A large aspect ratio circular cross section zero-beta equilibrium is considered. This equilibrium possess a rational surface with safety factor q = 2 at a normalized flux value of 0.5. A small resonant boundary perturbation is introduced, exciting a response at the resonant rational surface. The code is found to capture the plasma response as predicted by a newly developed analytic theory that ensures the existence of nested flux surfaces by allowing for a jump in rotational transform (ι=1/q). The VMEC code satisfactorily reproduces these theoretical results without the necessity of an explicit transform discontinuity (Δι) at the rational surface. It is found that the response across the rational surfaces depends upon both radial grid resolution and local shear (dι/dΦ, where ι is the rotational transform and Φ the enclosed toroidal flux). Calculations of an implicit Δι suggest that it does not arise due to numerical artifacts (attributed to radial finite differences in VMEC) or existence conditions for flux surfaces as predicted by linear theory (minimum values of Δι). Scans of the rotational transform profile indicate that for experimentally relevant levels of transform shear the response becomes increasing localised. Careful examination of a large experimental tokamak equilibrium, with applied resonant fields, indicates that this shielding response is present, suggesting the phenomena is not limited to this verification exercise.