Free-boundary Equilibrium Research Articles

Current ramps in tokamaks: from present experiments to ITER scenarios

In order to prepare adequate current ramp-up and ramp-down scenarios for ITER, present experiments from various tokamaks have been analysed by means of integrated modelling in view of determining relevant heat transport models for these operation phases. A set of empirical heat transport models for L-mode (namely, the Bohm–gyroBohm model and scaling based models with a specific fixed radial shape and energy confinement time factors of H96−L = 0.6 or HIPB98 = 0.4) has been validated on a multi-machine experimental dataset for predicting the li dynamics within ±0.15 accuracy during current ramp-up and ramp-down phases. Simulations using the Coppi–Tang or GLF23 models (applied up to the LCFS) overestimate or underestimate the internal inductance beyond this accuracy (more than ±0.2 discrepancy in some cases). The most accurate heat transport models are then applied to projections to ITER current ramp-up, focusing on the baseline inductive scenario (main heating plateau current of Ip = 15 MA). These projections include a sensitivity study to various assumptions of the simulation. While the heat transport model is at the heart of such simulations (because of the intrinsic dependence of the plasma resistivity on electron temperature, among other parameters), more comprehensive simulations are required to test all operational aspects of the current ramp-up and ramp-down phases of ITER scenarios. Recent examples of such simulations, involving coupled core transport codes, free-boundary equilibrium solvers and a poloidal field (PF) systems controller are also described, focusing on ITER current ramp-down.

The CEDRES++ equilibrium code and its application to ITER, JT-60SA and Tore Supra

Free-boundary equilibrium codes are an essential tool for tokamak design studies and scenario development. In this contribution, the CEDRES++ code is presented and applications to ITER, JT-60SA and Tore Supra are reported on. A satisfactory benchmark of CEDRES++ with the DINA code was performed on an ITER Scenario 2 case at 32 different times across the discharge, both in direct and inverse mode. Ongoing design studies for the poloidal field coils system of JT-60SA, performed in the frame of the Broader Approach, were cross-checked with CEDRES++. Finally, a dimensioning study was carried out for the coils and power supplies for plasma vertical control in the frame of a feasibility study for the implementation of a divertor in Tore Supra.

Free-boundary magnetohydrodynamic equilibria with flow

The finite-element M3D code [W. Park et al., Phys. Plasmas 6, 1796 (1999)] has been modified to include a free-boundary equilibrium solver with arbitrary toroidal and poloidal flows. With this modification, the M3D code now has the capability to self-consistently model two essential ingredients necessary for equilibrium calculations in the edge region, namely, free-boundary and arbitrary flow. As a free-boundary code, M3D includes the separatrix and scrape-off layer regions in the equilibrium calculation. Poloidal flows in the subsonic, supersonic, and transonic regimes can be calculated with the new version of the M3D code. Calculation results show that the presence of equilibrium flows, in particular those next to the plasma boundary, can considerably influence the position of the X-point and magnetic separatrix shape/location and hence the position of the strike point on the divertor plates. Moreover, it is shown that poloidal flow is not a rigid-body rotation, with the fastest flows occurring on the inboard side of the plasma. A numerical confirmation of the “de Laval nozzle” model of Betti and Freidberg [R. Betti and J. P. Freidberg, Phys. Plasmas 7, 2439 (2000)] for free-boundary equilibrium calculations is obtained, with the formation of the predicted discontinuities between regions of subsonic and supersonic flows (with respect to the poloidal sound speed). Finally, a detailed comparison between isentropic and isothermal equilibria is presented, showing qualitative analogies and quantitative differences.

Validation of the thermal transport model used for ITER startup scenario predictions with DIII-D experimental data

Control simulations of ITER startup using 2D free-boundary equilibrium and 1D transport codes rely on accurate predictions of the electron and ion temperature profiles that determine the electrical conductivity and pressure profiles during the current rise. We present results of validation studies that apply the transport model used by the ITER team to DIII-D discharge evolution and compare predictions with data from similarity experiments. Results presented here detail difficulties and sensitivities associated with the modelling of time-dependent current profile evolution required to asses performance of the poloidal-field coil system and controllers on ITER.

Stellarator coil design and plasma sensitivity

The rich information contained in the plasma response to external magnetic perturbations can be used to help design stellarator coils more effectively. We demonstrate the feasibility by first developing a simple direct method to study perturbations in stellarators that do not break stellarator symmetry and periodicity. The method applies a small perturbation to the plasma boundary and evaluates the resulting perturbed free-boundary equilibrium to build up a sensitivity matrix for the important physics attributes of the underlying configuration. Using this sensitivity information, design methods for better stellarator coils are then developed. The procedure and a proof-of-principle application are given that (1) determine the spatial distributions of external normal magnetic field at the location of the unperturbed plasma boundary to which the plasma properties are most sensitive, (2) determine the distributions of external normal magnetic field that can be produced most efficiently by distant coils, and (3) choose the ratios of the magnitudes of the efficiently produced magnetic distributions so the sensitive plasma properties can be controlled. Using these methods, sets of modular coils are found for the National Compact Stellarator Experiment (NCSX) which are either smoother or can be located much farther from the plasma boundary than those of the present design.

Analysis of high β regimes for DEMO

Stability limit calculations are presented for a range of tokamak power plant equilibria. The current drive requirements to sustain the optimised equilibrium profiles are confirmed by a transport code and the plasma shape is obtained from free-boundary equilibrium calculations. A pressure pedestal is included according to empirical scaling and ballooning mode stability limits. A terative optimisation of the profiles is undertaken to improve the baseline profiles in order to achieve the highest possible plasma performance and most favourable magnetohydrodynamic stability within conservative assumptions in order to increase confidence in the availability and control of the plasma. This results in a fully noninductive baseline operating scenario for a tokamak power plant design which has a broad low-shear q-profile which is meant to complement previous advanced tokamak design studies.

Simulation technique of free-boundary equilibrium evolution in plasma ramp-up phase

We describe a new scheme for a consistent linkage between the transport and equilibrium solvers in the integrated transport code TOPICS for a transport simulation in tokamaks during a plasma volume ramp-up phase. Although the flux conserving tokamak (FCT) equilibrium solver implemented in TOPICS is well consistent with the transport solver, it cannot evaluate a toroidal magnetic flux associated with an evolving equilibrium by itself. The evolution of the equilibrium is then modeled with a combination of the determination of the toroidal magnetic flux by the iterative solution of the Grad–Shafranov equation and the FCT scheme. The new scheme developed is applied to a predictive simulation with the volume expansion in a tokamak comparable to JT-60U for both the limiter and the divertor cases.

Simulating ITER plasma startup and rampdown scenarios in the DIII-D tokamak

DIII-D experiments have investigated ITER startup scenarios, including an initial phase where the plasma was limited on low field side poloidal bumper limiters. In addition, li feedback control has been tested with the goal of producing discharges in ITER within the capabilities of the poloidal field coil set and favourable to the intended mode of operations in the subsequent constant current (flattop) phase. These discharges have been modelled using the Corsica free boundary equilibrium code. High performance hybrid scenario discharges (βN = 2.8, H98,y2 = 1.4) and ITER H-mode baseline discharges (βN > 1.6, H98,y2 = 1–1.2) have been obtained experimentally in an ITER similar shape after the ITER-relevant startup. Studies have been initiated to develop a reliable scenario for exiting the burn phase and ramping down the plasma current in ITER without disruptions.

Simulations of KSTAR high performance steady state operation scenarios

We report the results of predictive modelling of high performance steady state operation scenarios in KSTAR. Firstly, the capabilities of steady state operation are investigated with time-dependent simulations using a free-boundary plasma equilibrium evolution code coupled with transport calculations. Secondly, the reproducibility of high performance steady state operation scenarios developed in the DIII-D tokamak, of similar size to that of KSTAR, is investigated using the experimental data taken from DIII-D. Finally, the capability of ITER-relevant steady state operation is investigated in KSTAR. It is found that KSTAR is able to establish high performance steady state operation scenarios; βN above 3, H98(y, 2) up to 2.0, fBS up to 0.76 and fNI equals 1.0. In this work, a realistic density profile is newly introduced for predictive simulations by employing the scaling law of a density peaking factor. The influence of the current ramp-up scenario and the transport model is discussed with respect to the fusion performance and non-inductive current drive fraction in the transport simulations. As observed in the experiments, both the heating and the plasma current waveforms in the current ramp-up phase produce a strong effect on the q-profile, the fusion performance and also on the non-inductive current drive fraction in the current flattop phase. A criterion in terms of qmin is found to establish ITER-relevant steady state operation scenarios. This will provide a guideline for designing the current ramp-up phase in KSTAR. It is observed that the transport model also affects the predictive values of fusion performance as well as the non-inductive current drive fraction. The Weiland transport model predicts the highest fusion performance as well as non-inductive current drive fraction in KSTAR. In contrast, the GLF23 model exhibits the lowest ones. ITER-relevant advanced scenarios cannot be obtained with the GLF23 model in the conditions given in this work. Finally, ideal MHD stability is investigated for the ITER-relevant advanced scenarios in KSTAR. The methods and results presented in this paper are expected to contribute to improving the ITER and beyond ITER predictive simulations.

Full tokamak discharge simulation of ITER by combining DINA-CH and CRONOS

A full tokamak discharge simulator has been developed by combining a free-boundary equilibrium evolution code, DINA-CH, and an advanced transport modelling code, CRONOS. The combined tokamak discharge simulator provides a full simulation of a whole tokamak discharge, including non-linear coupling effects between the evolution of the free-boundary plasma equilibrium and transport. The free-boundary plasma equilibrium evolution is self-consistently calculated with the plasma current diffusion, in response to currents flowing in the PF coils and the surrounding conducting system. The heat and current source profiles calculated taking the free-boundary plasma equilibrium are used for the plasma transport. The constraints in operating a tokamak, such as the PF coil current and voltage limits, are taken into account. The potential of the combined tokamak discharge simulator is demonstrated by simulating whole operation phases of the inductive 15 MA ELMy H-mode ITER scenario 2. Issues related to ITER operation, such as respecting the coil current limit, vertical instability and poloidal flux consumption, are investigated. ITER hybrid mode operation is studied focusing on the capability of operating the plasma with a stationary flat safety factor profile.

Derivation of free-boundary equilibrium linear models by a flux perturbation method

A new method to derive linear models of the axisymmetric plasma dynamic is detailed. The method follows the modified inductance approach but adopts a different set of independent variables to perturb the equilibrium. This different basis permits one to reduce the numerical computation of the model and has revealed an interesting property of the inverse inductance matrix whose elements are influenced by the plasma only if they refer to circuits facing directly the plasma column. In terms of approximations, the method proposed is equivalent to the standard modified inductance method but has the additional advantage of ensuring numerical convergence of the perturbed equilibria and therefore it allows using Picard based numerical Grad–Shafranov solvers. A linear model of the RFX-mod machine was derived with this approach and it was validated against the experiment.

Linear and Nonlinear Development of the m = 0 Instability in Z-Pinch Equilibria with Axial Sheared Flows

The effect of sheared flows on the m = 0 instability development in a z-pinch is numerically investigated using a 2D magnetohydrodynamic (MHD) code. The behavior of both internal and free-boundary modes is studied by using two types of initial configurations: a diffuse Bennett equilibrium and a free-boundary parabolic equilibrium. It was found that sheared flows change the m = 0 development by reducing the linear growth rates, decreasing the saturation amplitude, and modifying the instability spectrum. Full stability can be obtained for supersonic plasma flows, but a larger shear is needed to obtain stabilization of free-boundary modes.

Fast particle losses due to NTMs and magnetic field ripple

We present a detailed numerical study of the interaction between fast particles and large-scale magnetic perturbations and toroidal field ripple. In particular we focus our study on the losses of fast ions created by neutral beam injection (NBI) for an ASDEX Upgrade discharge with neoclassical tearing mode (NTM) activity. For these investigations, we use as input an equilibrium carefully reconstructed from experimental data. The magnetic field ripple is self-consistently included by a three-dimensional, free-boundary equilibrium computation. The magnetic islands caused by a (2,1)-NTM are introduced by a field perturbation superimposed on the equilibrium magnetic field. The experimental data are used to reproduce size and location of those islands numerically. Starting from a realistic seed distribution, the guiding centres of about 100 000 fast ions are traced up to a given time limit, or until they hit plasma-facing structures. A detailed analysis of the particle trajectories provides important information on the underlying loss mechanisms such as: (i) losses of passing particles caused by drift island formation, and (ii) losses of trapped particles due to stochastic diffusion.

Calculations of current ramp-up scenarios for the Component Test Facility tokamak using the DINA code

The free-boundary equilibrium time evolution code DINA is used for calculations of the current ramp-up scenarios in the Component Test Facility (CTF) machine based on the concept of low-aspect-ratio tokamaks. The current ramp-up and subsequent support of the steady phase operation in the CTF is realized by the current driven by the neutral beam (NB), the bootstrap current and the current produced owing to the variation in the poloidal field from the control coils. The problems of seeded current formation are discussed. The self-consistent scenario for current ramp-up for the CTF project is worked out. The steady-state regime parameters are analysed. NB power absorption and current drive calculations are carried out by the DINA code tested on different devices.

NCSX Magnetic Configuration Flexibility and Robustness

The National Compact Stellarator Experiment (NCSX) will study the physics of low-aspect ratio, high-β, quasi-axisymmetric stellarators. To achieve the scientific goals of the NCSX mission, the device must be capable of supporting a wide range of variations in plasma configuration about a reference baseline equilibrium. We demonstrate the flexibility of NCSX coils to support such configuration variations and demonstrate the robustness of performance of NCSX plasmas about reference design values of the plasma current Ip, β, and profile shapes. The robustness and flexibility calculations make use of free-boundary plasma equilibrium constructions using a combination of nonaxisymmetric modular coils and axisymmetric toroidal and poloidal field coils. The primary computational tool for the studies is STELLOPT, a free-boundary optimization code that varies coil currents to target configurations with specific physics properties.

Plasma profile and shape optimization for the advanced tokamak power plant, ARIES-AT

An advanced tokamak plasma configuration is developed based on equilibrium, ideal MHD stability, bootstrap current analysis, vertical stability and control, and poloidal field coil analysis. The plasma boundaries used in the analysis are forced to coincide with the 99% flux surface from the free-boundary equilibrium. Using an accurate bootstrap current model and external current drive profiles from ray tracing calculations in combination with optimized pressure profiles, β N values above 7.0 have been obtained. The minimum current drive requirement is found to lie at a lower β N of 6.0. The external kink mode is stabilized by a tungsten shell located at 0.33 times the minor radius and a feedback system. Plasma shape optimization has led to an elongation of 2.2 and triangularity of 0.9 at the separatrix. Vertical stability could be achieved by a combination of tungsten shells located at 0.33 times the minor radius and feedback control coils located behind the shield. The poloidal field coils were optimized in location and current, providing a maximum coil current of 8.6 MA. These developments have led to a simultaneous reduction in the power plant major radius and toroidal field from those found in a previous study [S.C. Jardin, C.E. Kessel, C.G. Bathke, D.A. Ehst, T.K. Mau, F. Najmabadi, T.W. Petrie, the ARIES Team, Physics basis for a reversed shear tokamak power plant, Fusion Eng. Design 38 (1997) 27].

PIES free boundary stellarator equilibria with improved initial conditions

The MFBE procedure developed by Strumberger (1997 Nucl. Fusion 37 19) is used to provide an improved starting point for free boundary equilibrium computations in the case of W7-X (Nührenberg and Zille 1986 Phys. Lett. A 114 129) using the Princeton iterative equilibrium solver (PIES) code (Reiman and Greenside 1986 Comput. Phys. Commun. 43 157). Transferring the consistent field found by the variational moments equilibrium code (VMEC) (Hirshmann and Whitson 1983 Phys. Fluids 26 3553) to an extended coordinate system using the VMORPH code, a safe margin between plasma boundary and PIES domain is established. The new EXTENDER_P code implements a generalization of the virtual casing principle, which allows field extension both for VMEC and PIES equilibria. This facilitates analysis of the 5/5 islands of the W7-X standard case without including them in the original PIES computation.

Measurement of edge currents in DIII-D and their implication for pedestal stability

The present performance limits of tokamak discharges are strongly coupled to the stability and transport properties of the edge plasma. Both experimental and modeling efforts have shown a clear connection between the edge pressure pedestal height and core plasma confinement. The key to understanding the stability and performance limits of the pedestal revolves around an accurate knowledge of the plasma current in this region. Using the Zeeman effect in an injected 30keV lithium beam, we have measured the currents in the edge of the DIII-D [J. L. Luxon, Nucl. Fusion 42, 6114 (2002)] tokamak for various confinement modes. This method of determining j(r) is insensitive to the large electric fields which coexist in the pedestal region and which complicate motional Stark effect measurements. For the high confinement cases, where substantial pedestal pressures exist, we find large (~MA∕m2), localized (ΔR∼1–2cm) currents in the pedestal region, located near the maximum in the pressure gradient. These values are consistent with calculations of edge bootstrap current using the neoclassical NCLASS [W. A. Houlberg, K. C. Shaing, S. P. Hirshman, and M. C. Zarnstorff, Phys. Plasmas 4, 3230 (1997)] and Sauter [O. Sauter, C. Angioni, and Y. R. Lin-Lin, Phys. Plasmas 6, 2834 (1999)] models and the measured pedestal density and temperature profiles. The apparent consistency of the measured jEDGE with neoclassical predictions occurs despite the violation of one of the fundamental tenets of the theory, namely, ε=ρi∕LP⪡1, where ρi is the ion poloidal gyroradius and Lp is the pressure gradient scale length. The measured jEDGE has also been used to generate self-consistent reconstructions using the free boundary equilibrium solvers CORSICA [T. A. Casper, T. B. Kaiser, R. A. Jong, L. L. LoDestro, J. Moller, and L. D. Pearlstein, Plasma Phys. Controlled Fusion 45, 1193 (2003)] and EFIT [L. L. Lao, H. E. St. John, R. D. Stambough, A. G. Kellman, and W. Pfeiffer, Nucl. Fusion 25, 1611 (1985)]. These equilibria allow us, in conjunction with the edge localized instabilities in tokamak experiments [P. B. Snyder, H. R. Wilson, J. R. Ferron, L. L. Lao, A. W. Leonard, T. H. Osborne, A. D. Turnbull, D. Mossessian, M. Murakami, and X. Q. Xu, Phys. Plasmas 9, 2037 (2002); H. R. Wilson, P. B. Snyder, G. T. A. Huysmans, and R. L. Miller, Phys. Plasmas 9, 1277 (2002)] magnetohydrodynamic stability code, to assess the linear stability of the edge to peeling∕ballooning modes. These results are then compared to the measured edge localized mode onset conditions and again good agreement is found between the experimental and model limits on the maximum permissible jEDGE.

Equilibrium and Stability of High-β Plasmas in Wendelstein 7-AS

One of the major goals for Wendelstein 7-AS (W7-AS) was the testing of the theoretical basis for the optimized configuration of Wendelstein 7-X (W7-X), currently under construction in Greifswald, Germany. In the last experimental campaign of W7-AS, volume-averaged β values >3% have been achieved. The underlying experimental changes leading to these results are briefly reviewed. The equilibrium characteristics expected from magnetohydrodynamic (MHD) theory are modeled in a simplified picture and compared with three-dimensional equilibrium calculations. A wide range of parameters has been covered in the experiments with and without net toroidal currents. Experimental data are compared with free-boundary equilibrium calculations and exhibit good agreement. The high-β equilibria usually showed only small MHD activity. The most prominent activities are low-frequency pressure-driven modes connected with low-order rationals also expected from numerical calculations using the CAS3D code, and Alfvén modes driven by energetic particles from the tangential neutral beam injection. Comparison of experimentally measured frequencies and mode structures from soft-X-ray tomography with theoretical predictions also shows the improving understanding of these modes in stellarators. The agreement of experiment and theory gives confidence in the predictions for W7-X.

Second ballooning stability in high-β, compact stellarators

Second ballooning stability is examined in quasipoloidally symmetric, compact stellarator configurations. These high-β (volume-average β>4%) free-boundary equilibria are calculated using a reference Quasi-Poloidal Stellarator (QPS) configuration. QPS plasmas have low-shear, stellarator-like rotational transform profile with |B| that is approximately poloidally symmetric. The high-β QPS equilibria are similar in their magnetic configuration to previously studied tokamak-stellarator hybrids which have a high-shear, tokamak-like rotational transform profile. Both types of configurations have strong magnetic wells and consequently high interchange stability β limits. Free-boundary QPS equilibria have regions of second stability at high β. For infinite-n ballooning modes in QPS plasmas, the boundary for first instability is 〈β〉∼2% and the boundary for second stability is 〈β〉∼6%. Finite-n ballooning mode calculations show higher β limits, 〈β〉>5%. Increasing plasma current (for fixed plasma pressure) can lower the finite-n ballooning mode β limit to 〈β〉=3% by reducing magnetic shear. QPS plasmas with Ohmic current profiles (peaked on-axis) have both a lower infinite-n ballooning β-limit for the onset of first instability and a higher β-limit for the onset of second stability relative to QPS plasma with bootstrap current profiles (peaked off-axis). QPS plasmas are stable to low-n ideal magnetohydrodynamic kink modes and vertical modes for values of β in this range (〈β〉∼6%) due to the low level of plasma current in QPS relative to an equivalent tokamak.