Ideal MHD Code Research Articles

Nonlinear two-fluid modeling of plasma response to RMPs for the ELM control in the ITER baseline

Numerical modeling, combining the toroidal ideal MHD code GPEC and the nonlinear two-fluid MHD code TM1, was used for comprehensive studies of the plasma response to resonant magnetic perturbations (RMPs) with toroidal mode number n = 1–5 for controlling edge-localized modes (ELMs) in ITER for the standard operation scenario (15 MA Q = 10). Several issues related to RMP ELM control are investigated, including the optimization of the RMP coils configuration, the evaluation of the magnitude of density pump-out and the q 95 windows of ELM suppression. GPEC calculates the magnetic response, which consistently includes the very important edge kink/peeling response to static magnetic perturbations. GPEC two-dimensional scans of the relative coil current phasing among the three rows of internal coils, at fixed coil current amplitude, reveal the optimal phasing for the RMP coil configuration with n = 1–5, respectively. The poloidal half wavelength of resonant mode at the edge of plasma calculated by GPEC indicates that the midplane row coils have the best resonant coupling with the plasma for n = 2, while the upper and lower row coils have the best resonant coupling with the plasma for n = 3. Based on the plasma kinetic equilibrium and the GPEC calculations of the magnetic response, TM1 was used to simulate the conditions for RMP field penetration in the ITER pedestal. TM1 shows magnetic island formation at the foot of ITER pedestal with RMP coil current threshold ranging from 4 kAt to 8 kAt with n = 2 to 4. These magnetic islands at the pedestal-foot lead to density pump-out, the magnitude of which scales as and ranges from 5% to 20% at the pedestal-top when scanning the coil current from 4 to 60 kAt. The density pump-out is found to be weaker for higher n RMP. The nonlinear TM1 simulations also show field penetration at the pedestal-top, where the threshold of RMP coil current depends on the q 95. The alignment of the magnetic island and the location of the pedestal-top decreases the height and width of the pedestal to suppress ELMs. Simulations by two-dimensional scans of RMP coil current and q 95 reveal the accessible q 95 windows of ELM suppression for both n = 3 and 4 RMPs. The predicted q 95 windows of ELM suppression are very similar to the ones in currently operating tokamaks and the required RMP coil current for ELM suppression is less than 40–50 kAt, which is well within the designed capability for ITER. In addition, the simulations indicate that wide q 95 windows of ELM suppression may be accessible in ITER by operating with dominant n = 4 (or n = 5) RMPs.

Investigation of mode activity in NBI-heated experiments of Wendelstein 7-X

The 2018 operation phase (OP 1.2b) of the stellarator Wendelstein 7-X (W7-X) included, for the first time, neutral beam injection (NBI) to heat the plasma. Since the injection geometry at W7-X is not parallel, this generates both passing and trapped fast particles. During longer phases of NBI injection, with the primary purpose to study the heating efficiency of this system, Alfvén eigenmodes (AEs) were observed by a number of diagnostics, including the phase contrast imaging (PCI) system, the magnetic pick-up coils (Mirnov coils), and the soft x-ray multi-camera tomography system (XMCTS).Alfvén eigenmodes are of great interest for future fusion reactors as it has been shown that the resonant interaction of fast ions with self-excited AEs can lead to enhanced transport of fast ions and potentially to energy losses. This is especially true for so-called gap-modes, Alfvén eigenmodes with frequencies in gaps of the continuous spectrum, since they lack continuum damping. These modes are commonly known to be excited by fast ions, but other destabilizing mechanisms, e.g. the electron-pressure gradient are also possible.In this article we present a first analysis of the experimentally observed frequencies from the theoretical side. The calculation of shear Alfvén wave continua for selected cases and the assignment of observed frequencies to the gaps of the continuous spectra are presented. Using the ideal-MHD code CKA (Könies A. 2007 10th IAEA TM on Energetic Particles in Magnetic Confinement System), we find gap modes that match the experimental measurements in terms of the observed frequencies. We emphasize the crucial roles played by the coupling of sound and Alfvén waves as well as of the Doppler shift arising as a consequence of the radial electric field in W7-X.We employ the perturbative gyrokinetic code CKA-EUTERPE (Feh´er 2013 Simulation of the interaction between Alfv´en waves and fast particles), using a slowing-down distribution function for the fast ions as calculated by the Monte-Carlo particle following code ASCOT (Hirvijoki et al 2014 Comput. Phys. Commun. 185 1310–21) to assess the fast-ion drive. We find that the fast-ion drive is insufficient to overcome the background-plasma damping. The fact that unstable modes were observed experimentally may point to problems with the modelling or indicate the existence of other destabilizing mechanisms, e.g. associated with the electron-pressure gradient (Windisch et al 2017 Plasma Phys. Control. Fusion 59 105002) that sensitively depend on the profiles of the background plasma.

Validation of Alfvén eigenmode simulations on DIII-D and projection for CFETR scenario

Alfvén eigenmodes (AEs) are driven unstable by on-axis neutral beam (NB) injection in DIII-D reversed magnetic shear discharges #166496 and #159243, a high-beta steady-state demonstration discharge and an L-mode current ramp discharge, respectively. The experimental results are used to validate the ideal MHD code with kinetic extensions NOVA-K and the gyro-Landau-fluid code TGLFEP. Both codes predict low toroidal mode number (n = 1–2) toroidal Alfvén eigenmode (TAE) instability in the former discharge and low-n (n = 2–6) reverse shear Alfvén eigenmode (RSAE) instability in the latter discharge, consistent with observations. Discrepancies between the two codes are discussed. The same two codes are also employed to predict linear AE instabilities driven by α particles in the China fusion engineering test reactor (CFETR). For the CFETR reversed shear q profile, the growth rates of AEs oscillate with increasing toroidal mode number, depending on the dominant AE type, and the RSAEs near a rational surface always have larger growth rates than TAEs. AEs are found to be further destabilized by the addition of 500 keV off-axis injected NB. The effects of q-profile variation on AE stability in CFETR are also investigated and the results show that raising qmin can further destabilize RSAEs, while increasing q0 with fixed qmin, to create stronger negative magnetic shear, is stabilizing. Increasing plasma density is beneficial for AE stability due to reduced fast ion drive because of shorter slowing down time and higher electron collisional damping.

Parametric study of fast-ion-driven modes in Wendelstein 7-X

In the 2018 experimental campaign, fast ions in the stellarator Wendelstein 7-X will be generated by neutral beam injection. Later operation phases will also include ion cyclotron resonance heating. The fast ions may excite instabilities in the plasma which can lead to enhanced fast-ion transport and can, in severe cases, cause damage to plasma-facing components.We present a numerical study of fast-ion-driven Alfvén eigenmodes in a Wendelstein 7-X high-mirror equilibrium. Realistic fast-ion parameters are obtained using the ASCOT code. To model the instabilities, we use the CKA-EUTERPE code package. This model is perturbative, since a fixed mode structure – computed by the ideal-MHD code CKA – is used throughout the calculation. The non-linear gyro-kinetic code EUTERPE computes the power transfer from the fast particles to the mode which defines the growth rate of the instability.We show that having a fast-ion collision operator present in the simulations is required to accurately predict the non-linear saturation level of the mode. The scaling of the saturated amplitude with respect to fast-ion drag and the pitch-angle collision frequency is investigated and found to vary for different Alfvén eigenmodes.Furthermore, we study the impact of several other actuators that might be of experimental relevance for finding operation windows that show Alfvén-eigenmode activity. Examples are the effects of a radial electric field and the composition of the background plasma (hydrogen versus helium). While growth rates are found to be reduced in helium plasmas, including a radial electric field, typically present in Wendelstein 7-X, seems to have little influence on the modes.

Simulations of peeling-ballooning modes in the Globus-M tokamak

Analysis of stability of the peeling-ballooning modes in the edge plasma of the Globus-M tokamak is presented. Studies were performed using the ideal 3D MHD code written in the BOUT++ framework. The edge localized modes which are frequently observed in the Globus-M plasma could be described using MHD theory. It was found that the parameters of the Globus-M edge plasma are in the range of stability for the peeling-ballooning modes. The instability modes with a toroidal number of n = 12 and typical distances between filaments of approximately ∼12 cm were found to be the most unstable. This statement agrees with the Doppler backscattering measurements. It was also considered whether the plasma in the Globus-M2 tokamak will be stable against the PB instability.

Alfvén eigenmode evolution computed with the VENUS and KINX codes for the ITER baseline scenario

A new application of the VENUS code is described, which computes alpha particle orbits in the perturbed electromagnetic fields and its resonant interaction with the toroidal Alfven eigenmodes (TAEs) for the ITER device. The ITER baseline scenario with Q = 10 and the plasma toroidal current of 15 МА is considered as the most important and relevant for the International Tokamak Physics Activity group on energetic particles (ITPA-EP). For this scenario, typical unstable ТАЕ-modes with the toroidal index n = 20 have been predicted that are localized in the plasma core near the surface with safety factor q = 1. The spatial structure of ballooning and antiballooning modes has been computed with the ideal MHD code KINX. The linear growth rates and the saturation levels taking into account the damping effects and the different mode frequencies have been calculated with the VENUS code for both ballooning and antiballooning TAE-modes.

Experiment-theory comparison for low frequency BAE modes in the strongly shaped H-1NF stellarator

Recent advances in the modeling, analysis, and measurement of fluctuations have significantly improved the diagnosis and understanding of Alfvén eigenmodes in the strongly shaped H-1NF helical axis stellarator. Experimental measurements, including 3D tomographic inversions of high resolution visible light images, are in close agreement with beta-induced Alfvén eigenmodes (BAEs) calculated using the compressible ideal MHD code, CAS3D. This is despite the low β in H-1NF, providing experimental evidence that these modes can exist due to compression that is induced by the strong shaping in stellarators, in addition to high β, as is the case in tokamaks. This is confirmed using the CONTI and CAS3D codes, which show significant gap structures at lower frequencies which contain BAE and beta-acoustic Alfvén eigenmodes (BAAEs). The BAEs are excited in the absence of a well confined energetic particle source, further confirming previous studies that thermal particles, electrons, or even radiation fluctuations can drive these modes. Datamining of magnetic probe data shows the experimentally measured frequency of these modes has a clear dependence on the rotational transform profile, which is consistent with a frequency dependency due to postulated confinement related temperature variations.

Alfvén modes in the Madison Symmetric Torus

This work presents a theoretical and computational analysis of core-localized energetic particle driven modes observed near the magnetic axis in the Madison Symmetric Torus [L. Lin, W. X. Ding, D. L. Brower et al., Phys. Plasmas 20, 030701 (2013)]. Using measured safety factor and plasma pressure profiles as input, the linear ideal MHD code Adaptive EiGenfunction Independent Solution (AEGIS) [L. J. Zheng and M. Kotschenreuther, J. Comput. Phys. 211, 748 (2006)] reveals Alfvénic modes close to the measured frequencies. The AEGIS results together with a reduced analytical model demonstrate that the modes are essentially “cylindrical” and dominated by a single poloidal component (m = 1). The modes are localized at the plasma core where the magnetic shear is weak and continuum damping is minimal. Detailed analysis establishes constraints on the safety factor and plasma pressure, under which two modes can exist simultaneously.

Properties of Alfvén eigenmodes in the Toroidal Alfvén Eigenmode range on the National Spherical Torus Experiment-Upgrade

A second Neutral Beam (NB) injection line is being installed on the NSTX Upgrade device, resulting in six NB sources with different tangency radii that will be available for heating and current drive. This work explores the properties of instabilities in the frequency range of the Toroidal Alfvén Eigenmode (TAE) for NSTX-U scenarios with various NB injection geometries, from more perpendicular to more tangential, and with increased toroidal magnetic field with respect to previous NSTX scenarios. Predictions are based on analysis through the ideal MHD code NOVA-K. For the scenarios considered in this work, modifications of the Alfvén continuum result in a frequency up-shift and a broadening of the radial mode structure. The latter effect may have consequences for fast ion transport and loss. Preliminary stability considerations indicate that TAEs are potentially unstable with ion Landau damping representing the dominant damping mechanism.

Study of chirping toroidicity-induced Alfvén eigenmodes in the National Spherical Torus Experiment

Chirping toroidicity-induced Alfvén eigenmodes (TAEs) are destabilized during neutral beam injection on the National Spherical Torus Experiment (NSTX (Ono M. et al 2000 Nucl. Fusion 40 557)) by super-Alfvénic ions with velocities up to five times larger than the Alfvén velocity. TAEs exhibit repeated bursts in amplitude and down-chirps in frequency. Larger bursts, so-called TAE avalanches, are eventually observed and correlate with a loss of fast ions up to 30% over ∼1 ms. Frequency, amplitude and radial structure of TAEs are characterized via magnetic pickup coils and a multi-channel reflectometer system. The modes have a broad radial structure, which appears to be unaffected by the large frequency and amplitude variations. However, the large mode amplitude does impact the modes' dynamics by favouring the coupling among different modes. In addition, the coupling involves kink-like modes and can therefore degrade the thermal plasma confinement. In spite of the non-linear regime characterizing the TAE dynamics, the measured properties are found to be in reasonable agreement with solutions from the ideal MHD code NOVA.

Alfvén eigenmode stability and fast ion loss in DIII-D and ITER reversed magnetic shear plasmas

Neutral beam injection into reversed-magnetic shear DIII-D plasmas produces a variety of Alfvénic activity including toroidicity-induced Alfvén eigenmodes (TAEs) and reversed shear Alfvén eigenmodes (RSAEs). With measured equilibrium profiles as inputs, the ideal MHD code NOVA is used to calculate eigenmodes of these plasmas. The postprocessor code NOVA-K is then used to perturbatively calculate the actual stability of the modes, including finite orbit width and finite Larmor radius effects, and reasonable agreement with the spectrum of observed modes is found. Using experimentally measured mode amplitudes, fast ion orbit following simulations have been carried out in the presence of the NOVA calculated eigenmodes and are found to reproduce the dominant energy, pitch and temporal evolution of the losses measured using a large bandwidth scintillator diagnostic. The same analysis techniques applied to a DT 8 MA ITER steady-state plasma scenario with reversed-magnetic shear and both beam ion and alpha populations show Alfvén eigenmode instability. Both RSAEs and TAEs are found to be unstable with maximum growth rates occurring for toroidal mode number n = 6 and the majority of the drive coming from fast ions injected by the 1 MeV negative ion beams. AE instability due to beam ion drive is confirmed by the non-perturbative code TAEFL. Initial fast ion orbit following simulations using the unstable modes with a range of amplitudes (δB/B = 10−5–10−3) have been carried out and show negligible fast ion loss. The lack of fast ion loss is a result of loss boundaries being limited to large radii and significantly removed from the actual modes themselves.

Simulation of edge localized modes using BOUT++

The BOUT++ code is used to simulate edge localized modes (ELMs) in a shifted circle equilibrium. Reduced ideal MHD simulations are first benchmarked against the linear ideal MHD code ELITE, showing good agreement. Diamagnetic drift effects are included finding the expected suppression of high toroidal mode-number modes. Nonlinear simulations are performed, making the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron thermal diffusivity. This allows simulations with realistically high Lundquist numbers (S = 108), finding ELM sizes of 5–10% of the pedestal stored thermal energy. Scans show a strong dependence of the ELM size on resistivity at low Lundquist numbers, with higher resistivity leading to more violent eruptions. At high Lundquist numbers relevant to high-performance discharges, ELM size is independent of resistivity as hyper-resistivity becomes the dominant dissipative effect.

Fast Ion Induced Shearing of 2D Alfvén Eigenmodes Measured by Electron Cyclotron Emission Imaging

Two-dimensional images of electron temperature perturbations are obtained with electron cyclotron emission imaging (ECEI) on the DIII-D tokamak and compared to Alfvén eigenmode structures obtained by numerical modeling using both ideal MHD and hybrid MHD-gyrofluid codes. While many features of the observations are found to be in excellent agreement with simulations using an ideal MHD code (NOVA), other characteristics distinctly reveal the influence of fast ions on the mode structures. These features are found to be well described by the nonperturbative hybrid MHD-gyrofluid model TAEFL.

Local simulations of the magnetized Kelvin-Helmholtz instability in neutron-star mergers

Context. Global MHD simulations show Kelvin-Helmholtz (KH) instabilities at the contact surface of two merging neutron stars. That region has been identified as the site of efficient amplification of magnetic fields. However, these global simulations, due to numerical limitations, were unable to determine the saturation level of the field strength, and thus the possible back-reaction of the magnetic field onto the flow. Aims. We investigate the amplification of initially weak fields in KH unstable shear flows, and the back-reaction of the field onto the flow. Methods. We use a high-resolution ideal MHD code to perform 2D and 3D local simulations of shear flows. Results. In 2D, the magnetic field is amplified in less than 0.01ms until it reaches locally equipartition with the kinetic energy. Subsequently, it saturates due to resistive instabilities that disrupt the KH vortex and decelerate the shear flow on a secular time scale. We determine scaling laws of the field amplification with the initial field strength and the grid resolution. In 3D, this hydromagnetic mechanism may be dominated by purely hydrodynamic instabilities limiting the amplification. We find maximum magnetic fields of 10^16 G locally, and r.m.s. maxima within the box of 10^15 G. However, such strong fields exist only for a short period. In the saturated state, the magnetic field is mainly oriented parallel to the shear flow for strong initial fields, while weaker initial fields tend to lead to a more balanced distribution of the field energy. In all models the flow shows small-scale features. The magnetic field is at most in equipartition with the decaying shear flow. (abridged)

Observation of Reversed Shear Alfvén Eigenmodes between Sawtooth Crashes in the Alcator C-Mod Tokamak

Groups of frequency chirping modes observed between sawtooth crashes in the Alcator C-Mod tokamak are interpreted as reversed shear Alfvén eigenmodes near the q=1 surface. These modes indicate that a reversed shear q profile is generated during the relaxation phase of the sawtooth cycle. Two important parameters, q_{min} and its radial position, are deduced from comparisons of measured density fluctuations with calculations from the ideal MHD code NOVA. These studies provide valuable constraints for further modeling of the sawtooth cycle.

LIGKA: A linear gyrokinetic code for the description of background kinetic and fast particle effects on the MHD stability in tokamaks

In a plasma with a population of super-thermal particles generated by heating or fusion processes, kinetic effects can lead to the additional destabilisation of MHD modes or even to additional energetic particle modes. In order to describe these modes, a new linear gyrokinetic MHD code has been developed and tested, LIGKA (linear gyrokinetic shear Alfvén physics) [Ph. Lauber, Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks, Ph.D. Thesis, TU München, 2003; Ph. Lauber, S. Günter, S.D. Pinches, Phys. Plasmas 12 (2005) 122501], based on a gyrokinetic model [H. Qin, Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks, Ph.D. Thesis, Princeton University, 1998]. A finite Larmor radius expansion together with the construction of some fluid moments and specification to the shear Alfvén regime results in a self-consistent, electromagnetic, non-perturbative model, that allows not only for growing or damped eigenvalues but also for a change in mode-structure of the magnetic perturbation due to the energetic particles and background kinetic effects. Compared to previous implementations [H. Qin, mentioned above], this model is coded in a more general and comprehensive way. LIGKA uses a Fourier decomposition in the poloidal coordinate and a finite element discretisation in the radial direction. Both analytical and numerical equilibria can be treated. Integration over the unperturbed particle orbits is performed with the drift-kinetic HAGIS code [S.D. Pinches, Ph.D. Thesis, The University of Nottingham, 1996; S.D. Pinches et al., CPC 111 (1998) 131] which accurately describes the particles’ trajectories. This allows finite-banana-width effects to be implemented in a rigorous way since the linear formulation of the model allows the exchange of the unperturbed orbit integration and the discretisation of the perturbed potentials in the radial direction. Successful benchmarks for toroidal Alfvén eigenmodes (TAEs) and kinetic Alfvén waves (KAWs) with analytical results, ideal MHD codes, drift-kinetic codes and other codes based on kinetic models are reported.

Discrete compressional Alfvén eigenmode spectrum in tokamaks

The spectrum of compressional Alfvén eigenmodes (CAE) is analysed and shown to be discrete in tokamaks with low aspect ratio, such as the National Spherical Torus Experiment (NSTX), as well as in conventional tokamaks, such as DIII-D. The study is focused on recent similarity experiments on NSTX and DIII-D in which sub-cyclotron frequency instabilities of CAEs were observed at similar plasma conditions (W.W. Heidbrink et al 2006 Nucl. Fusion 46 324). The global ideal MHD code NOVA recovers the main properties of these modes predicted by theory and observed in both devices. The discrete spectrum of CAEs is characterized by three quantum mode numbers for each eigenmode, (M, S and n), where M, S and n are poloidal, radial and toroidal mode numbers, respectively. The expected mode frequency splitting corresponding to each of these mode numbers seems to be observed in experiments and is consistent with our numerical analysis. The polarization of the observed magnetic field oscillations in NSTX was measured and is also consistent with the numerical analysis, which helps to identify them as CAE activity. CAE mode structure was obtained and shown to be localized in both radial and poloidal directions with typical radial localization toward the plasma edge and poloidal localization at the low field side of the plasma cross section.

Interpretation of the finite pressure gradient effects in the reversed shear Alfvén eigenmode theory

Ideal MHD equations employed in the NOVA code are analysed analytically and numerically in order to investigate the role of the pressure gradient on global reversed shear Alfvén eigenmodes (RSAEs) or Alfvén cascades. We confirm both numerically and analytically conclusions obtained earlier using the ideal MHD code NOVA [1] and analytically [10] that the plasma pressure gradient plays a key role in the existence condition and in the dispersion relation for the mode. The effect of the plasma pressure gradient is to shift the mode frequency up at the low part of the RSAE frequency chirp and downshift the mode frequency when the frequency approaches the TAE gap. This finding is contrary to predictions in a recent publication [2], where the pressure gradient is found to be always stabilizing by means of downshifting the RSAE frequency and enhancing its interaction with the continuum. We resolve this discrepancy by showing that neglecting the pressure gradient effect on the plasma equilibrium (modification of the Shafranov shift and the averaged curvature) leads to conclusions at variance with the numerical and analytical results presented here. A new variational approximation of the RSAE is introduced which compares remarkably well with NOVA solutions. With this new approximation we clearly demonstrate the diagnostic potential and limitations of the RSAE frequency measurement for MHD spectroscopy.

Edge localized modes control by stochastic magnetic fields

This paper describes a new approach for modelling the pedestal energy transport in the presence of a small radial magnetic perturbation. The cases of a ballooning instability leading to Type I edge localized modes (ELMs) and a magnetic perturbation generated by external coils are treated. The model for Type I ELMs is based on the linear ideal MHD code MISHKA coupled with the non-linear energy transport code TELM in a realistic tokamak geometry. The main mechanism of the increased transport through the external transport barrier in this model of ELMs is due to the appearance of a radial velocity and a radial magnetic field perturbation due to the MHD mode. Both lead to additional transport perpendicular to the magnetic surface and hence to a relaxation of the pressure profile in the unstable zone. The typical Type I ELM time-cycle was reproduced numerically including the destabilization of the ballooning modes leading to the fast (250 μ s) collapse of the pedestal pressure followed by the edge pressure profile re-building on a diffusive time scale. A possible mechanism for the control of Type I ELMs using a stochastic plasma boundary created by external coils is modelled in this paper using data on ELM suppression by I-coils from the DIII-D experiment. In the stochastic layer the transverse transport is effectively increased by diffusion of the magnetic field lines. The modelling results demonstrate the possibility of decreasing the edge pressure gradient to a value that is just below the ideal ballooning limit, leading to a high confinement regime without Type I ELMs.

Effect of the resistive wall on the growth rate of weakly unstable external kink mode in general 3D configurations

The formulation of a method for the systematic computation of the growth rate of the weakly unstable resistive wall mode (RWM) in 3D configurations by using results from ideal stability codes is presented. It is shown that the growth rate of the RWM is given, approximately, by the rate at which the available free energy for the ideal external kink mode can be dissipated by the resistive wall. The eigenfunction is also approximated by that of the external kink mode. This formulation is demonstrated by coupling to the ideal MHD code KSTEP with computation of the dissipation on the resistive wall. Results of the stability of the RWM in LHD plasmas and discussion of the validity and improvement of the computation are also included.