Tokamak Geometry Research Articles

Verification of an improved equation-free projective integration method for neoclassical plasma-profile evolution in tokamak geometry

A brute-force, long-time gyrokinetic simulation of plasma profile evolution in magnetic fusion devices is not desirable due to large computational resource requirements and a possible accumulation of numerical error. The equation-free projective integration method of Keverekidis et al. [Commun. Math. Sci. 1(4), 715–762 (2003)] is one of the outstanding candidates in projecting micro-scale simulations to a longer timescale. However, its application to tokamak plasma has not been fruitful due to the appearance of spurious transient oscillations in the lifting process, which are present when the kinetic simulations are initialized with a simplified model distribution function and which make the kinetic simulations to deviate from the desired paths. In this work, a kinetically informed lifting algorithm is added to the equation-free projective integration method, which is then verified in the electrostatic gyrokinetic particle-in-cell code XGCa [R. Hager and C. S. Chang, Phys. Plasmas 23, 042503 (2016)] for a neoclassical ion heat transport problem with adiabatic electrons. This new lifting operator is demonstrated to control spurious transients, enabling an over four-times reduction in the overall computing time in the time-evolution of the ion temperature profile in an axisymmetric toroidal plasma. Further reduction in the computing time is found to be limited due to the stability properties of the linear least squares projective integrator.

Validity limits of the passive treatment of impurities in gyrokinetic tokamak simulations

In gyrokinetic simulations of turbulent impurity transport, trace impurity species are often treated as passive species, in the sense that they are not included in Maxwell equations. This is consistent with the assumption that impurities with low enough concentrations are impacted by turbulence generated by electrons and main ions, but do not impact it significantly in return. In this work, we relax this assumption, and investigate the active impacts of impurity on impurity transport as a function of its concentration, in the presence of trapped-particle-driven turbulence. We focus on W40+ tungsten, which is relevant for modern tokamaks, and adopt a reduced gyrokinetic bounce-averaged model for trapped particles in a simplified tokamak geometry. The impacts depend on the relationship between equilibrium density gradient and temperature gradient. When these gradients are equal, we observe that tungsten can be treated as a passive species for concentrations below . Above this concentration, the impurity significantly impacts both density and heat transport, essentially quenching them for concentrations above 10−3. This quenching occurs as electric potential fluctuations become in phase with impurity density fluctuations.

Coordinate transformation and construction of finite element mesh in a diverted tokamak geometry

AbstractA coordinate transformation technique between straight magnetic field line coordinate system (Ψ, θ) and Cartesian coordinate system (R, Z) is presented employing a Solov'ev solution of the Grad‐Shafranov equation. Employing the equilibrium solution, the poloidal magnetic flux Ψ(R, Z) of a diverted tokamak, magnetic field line equation is solved computationally to find curves of constant poloidal angle θ, which provides us with explicit relations R = R(Ψ, θ) and Z = Z(Ψ, θ). Correspondingly, conversion from one coordinate to the other along particle trajectories in the vicinity of separatrix is demonstrated. Based on the magnetic structure, a finite element mesh is generated in a diverted tokamak geometry to solve Poisson's equation.

Development and testing of an unstructured mesh method for whole plasma gyrokinetic simulations in realistic tokamak geometry

In this work, we have formulated and implemented a mixed unstructured mesh-based finite element–Fourier decomposition scheme for gyrokinetic simulations in realistic tokamak geometry. An efficient particle positioning (particle-triangle mapping) scheme for the charge deposition and field scattering using an intermediate grid as the search index for triangles has been implemented, and a significant speed-up by a factor of ∼30 is observed as compared with the brute force scheme for a medium-size simulation. The TRIMEG (TRIangular MEsh based Gyrokinetic) code has been developed. As an application, the ion temperature gradient (ITG) mode is simulated using the simplified gyrokinetic Vlasov-Poisson model. Our simulation and that using the ORB5 code for the DIII-D Cyclone case show reasonable agreement. As an additional application, ITG simulations using an ASDEX Upgrade equilibrium have been performed with density and temperature gradient profiles similar to the Cyclone case. Capabilities of the TRIMEG code for simulations with realistic experimental equilibria in the plasma core and in the whole plasma volume with open field lines are demonstrated.

IMPGYRO: The full-orbit impurity transport code for SOL/divertor and its successful application to tungsten impurities

In future fusion reactors, tungsten (W) is currently regarded as one of the most feasible candidates for the plasma-facing components (PFCs). Once the W impurities sputtered from the PFCs penetrate into the core plasma, the large radiation cooling and fuel dilution, which leads to the deterioration of the core plasma performance, take place. To understand and to control W impurity transport are indispensable for future fusion reactors. In order to understand W transport, we are continuing to develop the kinetic impurity transport code IMPGYRO for SOL/divertor in fusion reactors. The IMPGYRO has obtained following unique features compared to other existing kinetic impurity transport codes through the 15-years development activity; (i) the exact Larmor motion of impurity ions are computed so that the effects of drifts are automatically taken into account, (ii) the Coulomb collision between impurities and background plasma ions are modelled by the Binary Collision Method which kinetically calculates more precise friction and thermal forces, and (iii) the background plasma transport and impurity transport are computed self-consistently by the coupled calculation with SOL/divertor plasma transport code such as SOLPS-ITER. Although the IMPGYRO transport model has significantly improved from the initial development phase, the modelling improvement of the IMPGYRO has never been described in detail. In this paper, (a) the current W generation/transport model, and (b) the coupling strategy between the SOLPS-ITER and the IMPGYRO are summarized. The new results (c) the check of the prompt re-deposition model of the IMPGYRO, and (d) the further extension of the Coulomb collision modelling of the IMPGYRO towards the neoclassical (NC) transport handling capability has been performed. The effects of the NC transport processes on a simple circular tokamak geometry are compared with the anomalous diffusion by means of the extended IMPGYRO to obtain better understanding of cross-field transport process of W. The results show that the NC transport process may become non-negligible compared to the anomalous diffusion at least in this configuration. Based on these recent improvements, the present version of the IMPGYRO is very useful tool for understanding and prediction of W transport in SOL/divertor regions of fusion devices.

Runaway electron generation in axisymmetric tokamak geometry

The generation of relativistic electrons in tokamak plasmas has been the subject of extensive research due to their intrinsic interest as well as the threat they pose to reactor-scale tokamak devices. Runaway electron generation becomes particularly robust for the very large inductive electric fields that are often present during tokamak disruptions. In this letter we show that when such large inductive electric fields are present, the physics of how tokamak geometry impacts runaway generation processes is qualitatively modified compared to the more commonly studied limit of weak inductive electric-field strengths. This modification leads to a significant increase in the efficiency of all runaway generation processes.

Toroidal effect on runaway vortex and avalanche growth rate

The momentum-space dynamics of runaway electrons in a slab geometry, in terms of both the geometry and topological transition of the runaway vortex when synchrotron radiative damping is taken into account, has recently been shown to play a crucial role in runaway mitigation and avoidance. In a tokamak geometry, magnetic trapping arises from parallel motion along the magnetic field that scales as 1/R in strength with R the major radius. Since the transit time for a runaway electron moving along the field is of order 10−8 s while the collisional time is of ∼0.01 s in ITER-like plasmas, a bounce-averaged formulation can drastically reduce computational cost. Here, the Los Alamos Plasma Simulation – Relativistic Fokker-Planck Solver code's implementation of a bounce-averaged relativistic Fokker-Planck model, along with the essential physics of synchrotron radiation damping and knock-on collisions, is described. It is found that the magnetic trapping can reduce the volume of the runaway vortex as the momentum-space fluxes are strongly modified inside the trapped-region. As a result, the avalanche growth rate is reduced at off-axis locations. In addition to benchmarking with previous calculations that did not take into account radiation damping, we also clarify how synchrotron radiation damping modifies the avalanche growth rate in a tokamak.

Verification of the global gyrokinetic stellarator code XGC-S for linear ion temperature gradient driven modes

XGC (X-point Gyrokinetic Code) is a whole-volume, total-f gyrokinetic particle-in-cell code developed for modeling tokamaks. In recent work, XGC has been extended to model more general 3D toroidal magnetic configurations, such as stellarators. These improvements have resulted in the XGC-S version. In this paper, XGC-S is benchmarked in the reduced delta-f limit for linear electrostatic ion temperature gradient-driven microinstabilities, which can underlie turbulent transport in stellarators. An initial benchmark of XGC-S in tokamak geometry shows good agreement with the XGC1, ORB5, and global GENE codes. A benchmark between XGC-S and the EUTERPE global gyrokinetic code for stellarators has also been performed, this time in the geometry of the optimized stellarator Wendelstein 7-X. Good agreement has been found for the mode number spectrum, mode structure, and growth rate.

Effects of plasmoid formation on sawtooth process in a tokamak

For realistic values of the Lundquist number in tokamak plasmas, 1/1 magnetic islands lead to the formation of a secondary thin current sheet, which breaks up into a chain of small magnetic islands, called plasmoids. The role of plasmoid dynamics during the sawtooth reconnection process in fusion plasmas remains an unresolved issue. In this study, systematic simulations are performed to investigate the resistive internal kink mode using the full resistive magnetohydrodynamics equations implemented in the NIMROD code in a simplified tokamak geometry. For Lundquist number S≥1.6×107, the secondary current sheet is found to be unstable to plasmoids during the nonlinear resistive kink mode evolution with a critical aspect ratio of the current sheet of ∼70. The merging of small plasmoids leads to the formation of a monster plasmoid that can significantly affect the primary island evolution. This may provide an explanation for the partial reconnection observed in sawtooth experiments.

Helical equilibrium magnetohydrodynamic flow effects on the stability properties of low-n ideal external-infernal modes in weak shear tokamak configurations

The impact of equilibrium helical flows on the stability properties of low shear tokamak plasmas is assessed. The corrections due to such helical flow to the equilibrium profiles (mass density, pressure, Shafranov shift, magnetic fluxes) are computed by minimising order by order the generalised Grad–Shafranov equation. By applying the same minimisation procedure, a set of three coupled equations, suitable for the study of magnetohydrodynamic perturbations localised within core or edge transport barriers is derived in circular tokamak geometry. We apply these equations to modelling the impact of strong poloidal flow shear in the edge region caused by a radial electric field on the stability of edge infernal modes retaining vacuum effects. Due to the poloidal flow shearing, the effect of plasma rotation is not simply a Doppler shift of the eigenfrequency. Stabilisation is found even for weak flow amplitude.

A geometric model of ion orbit loss under the influence of a radial electric field

A geometric approach is used to find the threshold energies for ion orbit loss in experimental tokamak geometry, like that of ASDEX Upgrade. The effects of any pre-existing radial electric field are maintained as they nontrivially affect the calculations. The associated velocity-space loss hole is investigated using equilibria and profiles from ASDEX Upgrade discharges 30628, in the L-mode just before the L-H transition and in the resultant H-mode, and 31533, an H-mode shot with a comparably high ion temperature. We use a novel technique to calculate the shift of the passing-trapped boundary and see that the experimental radial electric field allows for upwardly bounded regions (UBRs) of the loss hole, which are now studied in detail. We further develop a new loss-hole-shape sensitive method to treat the steady-state ion orbit losses by considering relaxation processes along the border of the loss cone. Typical characteristic times of the steady-state losses in the pedestal region of an H-mode are found to be between 1 and 10 ms. The UBRs significantly enhance the steady-state losses in two ways: these velocity-space loss regions reach further into thermally accessible regions and the extensions are shaped such that they have the ability to refill under both diffusive and slowing down relaxation processes.

Avalanche mechanism for runaway electron amplification in a tokamak plasma

The avalanche of runaway electrons is thought to pose a significant obstacle to the success of reactor scale devices such as ITER. As a result, a significant effort has been devoted toward quantifying both the threshold for the initiation of the avalanche of runaway electrons and the efficiency of the avalanche mechanism. In this work, these two quantities are computed utilizing a guiding-center formulation with large-angle collision operators of varying physics fidelity. The use of a guiding-center formulation, while computationally more costly compared to bounce-averaged approaches, provides a conceptually straightforward means of incorporating tokamak geometry. It is found that while the avalanche threshold is only weakly impacted by toroidal geometry for fully ionized low-Z plasmas, it can be significantly impacted if high-Z impurities are present. Furthermore, it is shown that the efficiency of the avalanche mechanism depends sensitively on the impurity content, the charge state of the underlying impurities, and the radial profile of the seed electron population. Finally, the commonly employed Møller secondary source term used to model the generation of secondary electrons is shown to yield avalanche growth rates and thresholds in good agreement with a more complete conservative large-angle collision operator.

On non-linear frequency chirping in connection with collisions

The non-linear behaviour of toroidicity-induced Alfvén eigenmodes, destabilized by fast ions, is investigated in tokamak geometry and for a Wendelstein 7-X high-mirror equilibrium. Both cases show frequency chirping in the non-linear phase. The focus of this paper is on how particle collisions influence the non-linear dynamics and the associated frequency chirping. Pitch-angle scattering and fast-ion drag, which together are described by the fast-ion collision operator, are considered. We study the effect of a Krook operator, relaxing the distribution function to its unperturbed value, on the non-linear dynamics. The Krook operator leads to a periodic re-appearance of the chirping. This is also observed in experiments in which a fast-particle source is usually present.The simulations are carried out using the non-linear and fully three-dimensional CKA-EUTERPE model. The model is perturbative in the sense that a fixed mode structure is used. Since such an investigation is undertaken for the first time for the stellarator Wendelstein 7-X, the tokamak case as well as analytical theory are used for comparison. The parameters of the fast-ion distribution function in Wendelstein 7-X are inspired by the 2018 experimental campaign which, for the first time, includes neutral beam injection to supply fast ions.

Dynamics of ideal modes and subsequent ELM crashes in 3D tokamak geometry from external magnetic perturbations

The impact of three-dimensional (3D) tokamak geometry from external magnetic perturbations on edge instabilities has been examined in high confinement mode plasmas with edge localised modes (ELMs) in ASDEX Upgrade. The 3D geometry has been probed using rigidly rotating MP fields. The measured distortions of the plasma boundary are compared to single-fluid ideal magnetohydrodynamic (MHD) equilibria using VMEC and MARS-F applying ideal and resistive MHD, whereas VMEC uses only ideal MHD. Both codes accurately reproduce the measured radial displacements of the edge density and temperature profiles in amplitude, toroidal phase and their dependence on the applied poloidal mode spectrum.The induced 3D geometry distorts the local magnetic shear, which locally reduces the stabilising effect from field-line bending at certain most unstable field lines. Around these field lines, we observe additional stable ideal MHD modes with clear ballooning structure in-between ELMs. After their immediate appearance, they saturate and then grow on timescales of the pedestal pressure recovery. The subsequent ELMs show strongly localised magnetic perturbations of the initial crash and accompanied energetic electrons around the same most unstable field lines. These are strong signatures that filaments at the ELM onset preferentially erupt on these most unstable (‘bad’) field lines with their unfavourable 3D geometry where preceding ballooning modes are observed.

Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2018

The beautiful site of Villa Monastero welcomed the 2018 edition of the joint Varenna-Lausanne international workshop on the theory of fusion plasmas on August 27-31. The audience attracted a record number of participants, and the discussions were lively and intense. It is hard to imagine a better place for discussing new ideas and concepts in the theory community. We believe that this special issue reflects the high quality of presentations and scientific exchanges.Let us bring forward some highlights that mark this special issue. MHD and turbulent transport were given some special attention. This is no surprise given the importance of these issues in present and future devices. It is quite clear that gyrokinetic codes incorporate more and more ingredients to describe the plasma edge. The route is still long toward a full integrated description of the edge and core, but certainly the first signs of this evolution were visible at the conference. Quite a large number of papers address kinetic MHD in both tokamak and stellarator geometries. Again this was to be expected, as MHD modes driven by energetic particles are foreseen to be triggered in ITER and JT-60SA “advanced” scenarios, but also in the W7X stellarator. Also is should not be forgotten that kinetic effects due to thermal ions also matter, and indeed this physics is dealt with in several contributions. Overall this is the signature of another clear trend in our field, from fluid to kinetic MHD. Other outstanding topics were addressed, like edge physics – from reduced to the most sophisticated models - and propagation of RF waves. Last but not the least, blue sky theory brightly shines, as witnessed by a wealth of advanced analytical methods. This is complemented by innovative methods for code verification, numerical schemes, and data analysis.The present volume of Journal of Physics Conference Series contains 22 outstanding papers – all peer reviewed. Let us mention another set of about 20 papers to appear in a special issue of Plasma Physics and Controlled Fusion. We hope the reader will enjoy it, as we did!Xavier Garbet, Olivier SauterOctober, 26 2018

A Study of the von Neumann Stability Analysis of a Semi-implicit Numerical Method Applied Over the Radial Impurity Transport Equation in Tokamak Plasma

The radial impurity transport equation for tokamak plasma is a form of diffusion–convection–reaction equation. The impurity transport equation is solved to determine the distribution of impurity (non-fuel) ion species with different ionization states perpendicular to magnetic surfaces of tokamak plasma. The equation for each charge (ionization) state Z is a non-linear, second-order in space, first-order in time, parabolic partial differential equation coupled to the previous Z − 1 and the next Z + 1 charge states of the impurity species through its reaction term. The number of differential equations to be solved simultaneously is hence determined by the number of ionization states of the impurity species studied. The solution to the set of these coupled equations can be obtained using a semi-implicit numerical method applied on it. The present study describes the application of von Neumann stability analysis over the semi-implicit numerical method applied over the radial impurity transport equation and determines a generic stability criterion for the method. The stability analysis is further illustrated using the geometry of Aditya tokamak installed at the Institute for Plasma Research Gandhinagar, India as an example. The impurity species considered is oxygen (Atomic number = 8). This leads to a set of eight coupled equations for charge states Z = 1 to 8 over which von Neumann analysis is illustrated in present study.

Free boundary 3D ideal MHD equilibrium calculations for non-linearly saturated current driven external kink modes in tokamaks

It is shown that free boundary 3D equilibrium calculations in tokamak geometry are capable of capturing the physics of non-linearly saturated external kink modes for monotonic current and q profiles typical of standard (baseline) plasma scenarios. The VMEC ideal MHD equilibrium model exhibits strong flux surface corrugations of the plasma vacuum boundary, driven by the core current profile. A method is presented which conveniently extracts the amplitude of the corrugation in terms of Fourier components in straight field line coordinates. The Fourier spectrum, and condition for non-linear corrugation agrees well with linear simulations, and the saturated amplitude agrees well with non-linear analytic calculations.

A power-balance model for local helicity injection startup in a spherical tokamak

A 0D circuit model for predicting in Local Helicity Injection (LHI) discharges is developed. Analytic formulas for estimating the surface flux of finite- plasmas developed by Hirshman and Neilson (1986 Phys. Fluids 29 790) are modified and expanded to treat highly shaped, ultralow- tokamak geometry using a database of representative equilibria. Model predictions are compared to sample LHI discharges in the Pegasus spherical tokamak, and are found to agree within 15% of experimental . High performance LHI discharges are found to follow the Taylor relaxation current limit for approximately the first half of the current ramp, or 75 kA. The second half of the current ramp follows a limit imposed by power-balance as plasmas expand from high-A to ultralow-A. This shape evolution generates a significant drop in external plasma inductance, effectively using the plasma’s initially high inductance to drive the current ramp and provide >70% of the current drive V-s. Projections using this model indicate the relative influences of higher helicity input rate and injector current on the attainable total plasma current.

Multi‐level model of radiation transport in inhomogeneous plasma

Radiation transport in plasma can be important for a number of problems, for example, first wall‐shielding effects or influence of hydrogen isotope radiation on tokamak edge plasma. Existing calculations in the field employ many simplifying assumptions. For example, the model of radiation transport in one of the most used codes, SOLPS, suggests that photoexcitation can be ignored for all transitions, not including the ground level. We develop a collisional radiative model of radiation transport in a multi‐level system, fully accounting for all transitions between levels. The role of the Stark effect in line shape is taken into account. The validity of some of the approximations made in SOLPS is analysed for detached divertor plasma in DIII‐D‐like tokamak geometry. It is shown that total radiation flux is directed away from the divertor target due to nonuniform plasma temperature and density.

The effect of a sheared flow on magnetic islands in plasmas with non-axisymetric geometry

ABSTRACTThe stability of a magnetic island in a toroidal magnetic confinement device depends on various factors besides the usual tearing-mode stability parameter , determined by the local current profile.The presence of a sheared flow in the vicinity of the rational surface that supports the island is one of the factors that affects its stability since it can give rise to a polarization current around the island position. The contribution of the polarization current to the stability has been computed for a tokamak geometry. Here, we consider the case of magnetic islands with a shear flow in a stellarator which has a non-axisymmetric magnetic geometry. The main difference is a contribution to the polarization current from the toroidal electrostatic oscillation. A correction due to the global toroidal magnetic geometry is also present. It is found that the regime where the stability is affected corresponds to the large island width relative to the ion gyroradius. Thus, the contribution is relevant for low-temperature regimes. In that case, the polarization current is destabilizing for frequencies larger than the ion diamagnetic frequency. Our results imply that the sheared flow can produce a growth of the magnetic island in a cold plasma but it can become narrower as the temperature rises.