Resistive Wall Mode Research Articles

Gray-box modeling of resistive wall modes with vacuum-plasma separation and optimal control design for EXTRAP T2R

This paper presents a graybox methodology to model the resistive wall mode instability by combining first principle approach and system identification technique. In particular we propose a separate vacuum and plasma modeling with cascade interconnection. The shell is modeled using CARIDDI code which solves the 3D integral formulation of eddy current problem, whereas the plasma response is obtained empirically by system identification. Furthermore the resulting model is used to design an optimal feedback control. The model and feedback control is validated experimentally in EXTRAP T2R reversed-field pinch, where RWMs stabilization and non-axisymmetric mode sustainment is considered.

Investigation of intrinsic toroidal rotation scaling in KSTAR

The behaviors of an intrinsic toroidal rotation without any external momentum sources are investigated in KSTAR. In these experiments, pure ohmic discharges with a wide range of plasma parameters are carefully selected and analyzed to speculate an unrevealed origin of toroidal rotation excluding any unnecessary heating sources, magnetic perturbations, and strong magneto-hydrodynamic activities. The measured core toroidal rotation in KSTAR is mostly in the counter-current direction and its magnitude strongly depends on the ion temperature divided by plasma current (Ti/IP). Especially the core toroidal rotation in the steady-state is well fitted by Ti/IP scaling with a slope of ∼−23, and the possible explanation of the scaling is compared with various candidates. As a result, the calculated offset rotation could not explain the measured core toroidal rotation since KSTAR has an extremely low intrinsic error field. For the stability conditions for ion and electron turbulences, it is hard to determine a dominant turbulence mode in this study. In addition, the intrinsic toroidal rotation level in ITER is estimated based on the KSTAR scaling since the intrinsic rotation plays an important role in stabilizing resistive wall modes for future reference.

An analytic scaling relation for the maximum tokamak elongation against n = 0 MHD resistive wall modes

A highly elongated plasma is desirable in order to increase plasma pressure and energy confinement to maximize fusion power output. However, there is a limit to the maximum achievable elongation which is set by vertical instabilities driven by the n = 0 MHD mode. This limit can be increased by optimizing several parameters characterizing the plasma and the wall. The purpose of our study is to explore how and to what extent this can be done. Specifically, we extend many earlier calculations of the n = 0 mode and numerically determine scaling relations for the maximum elongation as a function of dimensionless parameters describing (1) the plasma profile ( and li), (2) the plasma shape (ϵ and δ), (3) the wall radius (b/a) and (4) most importantly the feedback system capability parameter . These numerical calculations rely on a new formulation of n = 0 MHD theory we recently developed (Freidberg et al 2015 J. Plasma Phys. 81 515810607, Lee et al 2015 J. Plasma Phys. 81 515810608) that reduces the 2D stability problem into a 1D problem. This method includes all the physics of the ideal MHD axisymmetric instability while reducing the computation time significantly, so that many parameters can be explored during the optimization process. The scaling relations we present include the effects of the optimal triangularity and the finite aspect ratio on the maximum elongation, and can be useful for determining optimized plasma shapes in current experiments and future tokamak designs.

Investigation of the n = 1 resistive wall modes in the ITER high-mode confinement

The n = 1 resistive wall mode (RWM) stability of ITER high-mode confinement is investigated with bootstrap current included for equilibrium, together with the rotation and diamagnetic drift effects for stability. Here, n is the toroidal mode number. We use the CORSICA code for computing the free boundary equilibrium and AEGIS code for stability. We find that the inclusion of bootstrap current for equilibrium is critical. It can reduce the local magnetic shear in the pedestal, so that the infernal mode branches can develop. Consequently, the n = 1 modes become unstable without a stabilizing wall at a considerably lower beta limit, driven by the steep pressure gradient in the pedestal. Typical values of the wall position stabilize the ideal mode, but give rise to the ‘pedestal’ resistive wall modes. We find that the rotation can contribute a stabilizing effect on RWMs and the diamagnetic drift effects can further improve the stability in the co-current rotation case. But, generally speaking, the rotation stabilization effects are not as effective as the case without including the bootstrap current effects on equilibrium. We also find that the diamagnetic drift effects are actually destabilizing when there is a counter-current rotation.

Stability of DIII-D high-performance, negative central shear discharges

Tokamak plasma experiments on the DIII-D device (Luxon et al 2005 Fusion Sci. Tech. 48 807) demonstrate high-performance, negative central shear (NCS) equilibria with enhanced stability when the minimum safety factor exceeds 2, qualitatively confirming theoretical predictions of favorable stability in the NCS regime. The discharges exhibit good confinement with an L-mode enhancement factor H89 = 2.5, and are ultimately limited by the ideal-wall external kink stability boundary as predicted by ideal MHD theory, as long as tearing mode (TM) locking events, resistive wall modes (RWMs), and internal kink modes are properly avoided or controlled. Although the discharges exhibit rotating TMs, locking events are avoided as long as a threshold minimum safety factor value is maintained. Fast timescale magnetic feedback control ameliorates RWM activity, expanding the stable operating space and allowing access to values approaching the ideal-wall limit. Quickly growing and rotating instabilities consistent with internal kink mode dynamics are encountered when the ideal-wall limit is reached. The RWM events largely occur between the no- and ideal-wall pressure limits predicted by ideal MHD. However, evaluating kinetic contributions to the RWM dispersion relation results in a prediction of passive stability in this regime due to high plasma rotation. In addition, the ideal MHD stability analysis predicts that the ideal-wall limit can be further increased to by broadening the current profile. This path toward improved stability has the potential advantage of being compatible with the bootstrap-dominated equilibria envisioned for advanced tokamak (AT) fusion reactors.

A reduced resistive wall mode kinetic stability model for disruption forecasting

Kinetic modification of ideal stability theory from stabilizing resonances of mode-particle interaction has had success in explaining resistive wall mode (RWM) stability limits in tokamaks. With the goal of real-time stability forecasting, a reduced kinetic stability model has been implemented in the new Disruption Event Characterization and Forecasting (DECAF) code, which has been written to analyze disruptions in tokamaks. The reduced model incorporates parameterized models for ideal limits on β, a ratio of plasma pressure to magnetic pressure, which are shown to be in good agreement with DCON code calculations. Increased β between these ideal limits causes a shift in the unstable region of δWK space, where δWK is the change in potential energy due to kinetic effects that is solved for by the reduced model, such that it is possible for plasmas to be unstable at intermediate β but stable at higher β, which is sometimes observed experimentally. Gaussian functions for δWK are defined as functions of E × B frequency and collisionality, with parameters reflecting the experience of the National Spherical Torus Experiment (NSTX). The reduced model was tested on a database of discharges from NSTX and experimentally stable and unstable discharges were separated noticeably on a stability map in E × B frequency, collisionality space. The reduced model failed to predict an unstable RWM in only 15.6% of cases with an experimentally unstable RWM and performed well on predicting stability for experimentally stable discharges as well.

Implementation of Neoclassical Effects in Momentum Transport Analysis at LHD

Plasma rotation plays an important role in the suppression of turbulence, leading to an increase in energy and particle confinement. Significant rotation also leads to a stabilisation of the resistive wall mode. The external momentum input from Neutral Beam Injection (NBI) in current generation fusion plasmas may not be available for future self-heated fusion reactors. Therefore it is important to analyse the phenomenon of spontaneous rotation. At NIFS plasma rotation and momentum transport of the Large Helical Device (LHD) plasma is analysed using a code suite called TASK3D-a. In this work neoclassical effects, which can be especially significant in non-axisymmetric plasmas, were implemented in TASK3D-a. Initial analysis of neoclassical radial momentum flux profiles shows that in NBI-driven momentum input neoclassical effects, especially neoclassical damping, become dominant in the non-center plasma region. It was also found that during and after pellet-injection the neoclassical damping force becomes strong. With the implementation of neoclassical effects new features can be examined in the momentum flux-gradient relations; in the damping-dominated situation following pellet injection a large excursion in momentum flux is found. This work can aid in the search for neoclassical transport-optimised configurations for enhanced (spontaneous) plasma rotation.

CASTOR3D: linear stability studies for 2D and 3D tokamak equilibria

The CASTOR3D code, which is currently under development, is able to perform linear stability studies for 2D and 3D, ideal and resistive tokamak equilibria in the presence of ideal and resistive wall structures and coils. For these computations ideal equilibria represented by concentric nested flux surfaces serve as input (e.g. computed with the NEMEC code). Solving an extended eigenvalue problem, the CASTOR3D code takes simultaneously plasma inertia and wall resistivity into account. The code is a hybrid of the CASTOR_3DW stability code and the STARWALL code. The former is an extended version of the CASTOR and CASTOR_FLOW code, respectively. The latter is a linear 3D code computing the growth rates of resistive wall modes in the presence of multiply-connected wall structures. The CASTOR_3DW code, and some parts of the STARWALL code have been reformulated in a general 3D flux coordinate representation that allows to choose between various types of flux coordinates. Furthermore, the implemented many-valued current potentials in the STARWALL part allow a correct treatment of the m = 0, n = 0 perturbation. In this paper, we outline the theoretical concept, and present some numerical results which illustrate the present status of the code and demonstrate its numerous application possibilities.

Multimachine Data–Based Prediction of High-Frequency Sensor Signal Noise for Resistive Wall Mode Control in ITER

The high-frequency noise measured by magnetic sensors, at levels above the typical frequency of resistive wall modes, is analyzed across a range of present tokamak devices including DIII-D, JET, MAST, ASDEX Upgrade, JT-60U, and NSTX. A high-pass filter enables identification of the noise component with Gaussian-like statistics that shares certain common characteristics in all devices considered. A conservative prediction is made for ITER plasma operation of the high-frequency noise component of the sensor signals, to be used for resistive wall mode feedback stabilization, based on the multimachine database. The predicted root-mean-square n = 1 (n is the toroidal mode number) noise level is 104 to 105 G/s for the voltage signal, and 0.1 to 1 G for the perturbed magnetic field signal. The lower cutoff frequency of the Gaussian pickup noise scales linearly with the sampling frequency, with a scaling coefficient of about 0.1. These basic noise characteristics should be useful for the modeling-based design of the feedback control system for the resistive wall mode in ITER.

The eddy current probe array for Keda Torus eXperiment.

In a reversed field pinch device, the conductive shell is placed as close as possible to the plasma so as to balance the plasma during discharge. Plasma instabilities such as the resistive wall mode and certain tearing modes, which restrain the plasma high parameter operation, respond closely with conditions in the wall, in essence the eddy current present. Also, the effect of eddy currents induced by the external coils cannot be ignored when active control is applied to control instabilities. One diagnostic tool, an eddy current probe array, detects the eddy current in the composite shell. Magnetic probes measuring differences between the inner and outer magnetic fields enable estimates of the amplitude and angle of these eddy currents. Along with measurements of currents through the copper bolts connecting the poloidal shield copper shells, we can obtain the eddy currents over the entire shell. Magnetic field and eddy current resolutions approach 2 G and 6 A, respectively. Additionally, the vortex electric field can be obtained by eddy current probes. As the conductivity of the composite shell is high, the eddy current probe array is very sensitive to the electric field and has a resolution of 0.2 mV/cm. In a bench test experiment using a 1/4 vacuum vessel, measurements of the induced eddy currents are compared with simulation results based on a 3D electromagnetic model. The preliminary data of the eddy currents have been detected during discharges in a Keda Torus eXperiment device. The typical value of toroidal and poloidal eddy currents across the magnetic probe coverage rectangular area could reach 3.0 kA and 1.3 kA, respectively.

Plasma-resistivity-induced strong destabilization of the kinetic resistive wall mode

On the example of the plasma and mode parameters relevant, e.g., to the experiments on the rotational stabilization of resistive wall modes (RWMs) reported in Reimerdes et al. [Phys. Rev. Lett. 98, 055001 (2007)], it is shown that the plasma resistivity can have a strong impact on the stability of RWMs. The stability criteria for kinetic RWMs derived in Hu and Betti [Phys. Rev. Lett. 93, 105002 (2004)] has been complemented by the restriction on the maximum mode damping rate.

A comparative study of ideal kink stability in two reactor-relevant tokamak plasma configurations with negative and positive triangularity

The effects of an ideal/resistive conducting wall, the drift kinetic resonances, as well as the toroidal plasma flow, on the stability of the ideal external kink mode are numerically investigated for a reactor-relevant tokamak plasma with strongly negative triangularity (NTR) shaping. Comparison is made for a similar plasma equilibrium, but with positive triangularity (PTR). It is found that the ideal wall stabilization is less efficient for the kink stabilization in the NTR plasma due to a less ‘external’ eigenmode structure compared to the PTR plasma. The associated plasma displacement in the NTR plasma does not ‘balloon’ near the outboard mid-plane, as is normally the case for the pressure-driven kink-ballooning instability in PTR plasmas, but being more pronounced near the X-points. The toroidal flow plays a similar role for the kink stability for both NTR and PTR plasmas. The drift kinetic damping is less efficient for the ideal external kink mode in the NTR plasma, despite a somewhat larger fraction of the particle trapping near the plasma edge compared to the PTR equilibrium. However, the drift kinetic damping of the resistive wall mode (RWM) in the NTR plasma is generally as efficient as that of the PTR plasma, although the RWM window, in terms of the normalized pressure, is narrower for the NTR plasma.

Improved model predictive control of resistive wall modes by error field estimator in EXTRAP T2R

Many implementations of a model-based approach for toroidal plasma have shown better control performance compared to the conventional type of feedback controller. One prerequisite of model-based control is the availability of a control oriented model. This model can be obtained empirically through a systematic procedure called system identification. Such a model is used in this work to design a model predictive controller to stabilize multiple resistive wall modes in EXTRAP T2R reversed-field pinch. Model predictive control is an advanced control method that can optimize the future behaviour of a system. Furthermore, this paper will discuss an additional use of the empirical model which is to estimate the error field in EXTRAP T2R. Two potential methods are discussed that can estimate the error field. The error field estimator is then combined with the model predictive control and yields better radial magnetic field suppression.

Power Amplifiers Based on SiC Technology for MHD Mode Control in Fusion Experiments

The effective control of magnetohydrodynamic instabilities in fusion experiments by means of actively controlled coils calls for challenging dynamic requirements to the relevant power supply (PS) systems. The first part of this paper gives an overview on typical requirements and technological solutions adopted in some present fusion experiments. Then, the advantages of SiC devices with respect to the Si ones are described in this paper, including lower switching losses, higher switching speed, higher operating junction temperature, and higher voltage capability. As an application example, this paper reports on the case of the PS system for resistive wall mode control in the JT-60SA satellite tokamak, where analyses showed the difficulty of satisfying the demanding requirements in terms of high current bandwidth (3 kHz) and short latency ( $ ) with a simple H-bridge topology adopting standard IGBT. On the contrary, the use of Si–SiC IGBT with the same topology can allow meeting the specifications, as demonstrated by the development and test of such a power amplifier, rated for $300~A_{{\mathrm{pk}}}$ and 240 V.

Growth of Axisymmetric Instabilities in ASDEX Upgrade

Modern tokamaks, operating with elongated plasmas, are unstable against vertical displacement. In the presence of conducting walls, this instability assumes the characteristics of a resistive wall mode (RWM), amenable to feedback stabilization. On ASDEX Upgrade (AUG), estimates of the growth rates, entering into the definition of safe operating regimes and into feedback parameter settings, were so far based on a rigid displacement (RD) model. However, for highly triangular plasmas this model was found to predict growth rates that were too small. A code package originally developed for general ideal magnetohydrodynamic (MHD) RWMs (NEMEC/CAS3D/STARWALL) and also capable of handling three-dimensional passive stabilizing loops (PSLs) was therefore applied to the n = 0 case for a large data sample of possible AUG equilibria. The comparison with the previously used rigid vertical displacement model showed that the latter gives a consistently lower limit to the growth rates for typical AUG parameters. A statistical analysis of the RD results brings out the stabilizing effect of triangularity. This stabilizing effect disappears, however, if generalized displacements are taken into account, like in the full MHD resistive wall model. The mode acquires a strong m = 2 component, which allows it also to elude partly the stabilization by the PSL. At low elongation, large triangularity produces even significant additional destabilization, with the mode predominantly m = 2, confined to the outer plasma layers, like predicted in references [see Rosen et al., Phys. Fluids, 18, 482 (1975) and Becker and Lackner, Proc. 6th Int. Conf. Plasma Physics and Controlled Nuclear Fusion Research, Vol. II, p. 401 (1977)]. These results explain the tendencies observed in AUG and will be taken into account in future analyses.

Cancellation of drift kinetic effects between thermal and energetic particles on the resistive wall mode stabilization

Drift kinetic stabilization of the resistive wall mode (RWM) is computationally investigated using MHD-kinetic hybrid code MARS-K following the non-perturbative approach (Liu et al 2008 Phys. Plasmas 15 112503), for both reversed field pinch (RFP) and tokamak plasmas. Toroidal precessional drift resonance effects from trapped energetic ions (EIs) and various kinetic resonances between the mode and the guiding center drift motions of thermal particles are included into the self-consistent toroidal computations. The results show cancellation effects of the drift kinetic damping on the RWM between the thermal particles and EIs contributions, in both RFP and tokamak plasmas, even though each species alone can provide damping and stabilize RWM instability by respective kinetic resonances. The degree of cancellation generally depends on the EIs equilibrium distribution, the particle birth energy, as well as the toroidal flow speed of the plasma.

Kinetic calculation of the resistive wall mode and fishbone-like mode instability in tokamak

Kinetic effects of both trapped thermal and energetic particles on the resistive wall mode (RWM) and on the fishbone-like mode (FLM) are investigated in theory. Here, the trapped thermal particles include both ions and electrons. The FLM is driven by trapped energetic particles. The results demonstrate that thermal particle collisions can either stabilize or destabilize the RWM, depending on the energetic particle pressure βh. Furthermore, the critical value of βh for triggering the FLM is increased when the thermal particle contribution is taken into account. The critical value sensitively depends on the plasma collision frequency. In addition, the plasma inertia is found to have a negligible influence on the FLM.

Multi-region approach to free-boundary three-dimensional tokamak equilibria and resistive wall instabilities

Free-boundary 3D tokamak equilibria and resistive wall instabilities are calculated using a new resistive wall model in the two-fluid M3D-C1 code. In this model, the resistive wall and surrounding vacuum region are included within the computational domain. This implementation contrasts with the method typically used in fluid codes in which the resistive wall is treated as a boundary condition on the computational domain boundary and has the advantage of maintaining purely local coupling of mesh elements. This new capability is used to simulate perturbed, free-boundary non-axisymmetric equilibria; the linear evolution of resistive wall modes; and the linear and nonlinear evolution of axisymmetric vertical displacement events (VDEs). Calculated growth rates for a resistive wall mode with arbitrary wall thickness are shown to agree well with the analytic theory. Equilibrium and VDE calculations are performed in diverted tokamak geometry, at physically realistic values of dissipation, and with resistive walls of finite width. Simulations of a VDE disruption extend into the current-quench phase, in which the plasma becomes limited by the first wall, and strong currents are observed to flow in the wall, in the SOL, and from the plasma to the wall.

Excitation of external kink mode by trapped energetic particles

An unstable fishbone-like non-resonant external kink mode (FLEM) is numerically found to be driven by the precessional drift motion of trapped energetic particles (EPs) in both reversed-field pinch (RFP) and tokamak plasmas, even under the ideal wall boundary condition. In the presence of a sufficiently large fraction of trapped energetic ions in high beta plasmas, the FLEM instability may occur. The excitation condition is discussed in detail. The frequency of the FLEM is linked to the precessional drift frequency of EPs, and varies with the plasma flow speed. Therefore, it is usually much higher than that of the typical resistive wall mode (RWM). In general, the growth rate of FLEM does not depend on the wall resistivity. However, the wall position can significantly affect the mode’s property. The drift kinetic effects from thermal particles (mainly due to the transit resonance of passing particles) play a stabilizing role on FLEMs. In the presence of EPs, the FLEM and the RWM can co-exist or even couple to each other, depending on the plasma parameters. The FLEM instabilities in RFP and tokamaks have rather similar physics nature, although certain sub-dominant characters appear differently in the two configurations.

The Harmonic Analysis of RMP Coil Power Supply in EAST TOKAMAK

In EAST TOKAMAK, resonant magnetic perturbation (RMP) coils, which are powered by RMP coil power supply (PS), are set to research edge localized mode, resistive wall mode and error field correction. In order to meet the requirements of fast response and output voltage ripple proposed by physical experimenters, the structure of multiple modules using cascaded carrier phase-shifted PWM H-bridges is adopted, which enhances the equal switch-frequency of the PS and improves the quality of output voltage. In the paper, theoretical calculation applying Fourier analysis is given to analyze its harmonics. Both simulation results on Matlab and test results of output voltage ripple show that the RMP coil PS can meet the requirements.