Ideal Plasma Response Research Articles

Modeling of resistive plasma response in toroidal geometry using an asymptotic matching approach

The method of solving the linear resistive plasma response, based on the asymptotic matching approach, is developed for full toroidal tokamaks by upgrading the resistive DCON code [A. H. Glasser, Z. R. Wang, and J.-K. Park, Phys. Plasmas 23, 112506 (2016)]. The derived matching matrix, asymptotically matching the outer and inner regions, indicates that the applied three dimension (3-D) magnetic perturbations contribute additional small solutions at each resonant surface due to the toroidal coupling of poloidal modes. In contrast, the resonant harmonic only affects the corresponding resonant surface in the cylindrical plasma. The solution of the ideal outer region is critical to the asymptotic matching and is challenging in toroidal geometry due to the singular power series solution at the resonant surfaces. Thus, a systematic verification of the outer region Δ′ matrix is made by reproducing the well-known analytical Δ′ result in Furth et al. [Phys. Fluids 16, 1054–1063 (1073)] and by making a quantitative benchmark with the PEST3 code [A. Pletzer and R. L. Dewar, J. Plasma Phys. 45, 427–451 (1991)]. Finally, the reconstructed numerical solution of the resistive plasma response from the toroidal matching matrix is presented. Compared with the ideal plasma response, the global structure of the response can be affected by the small finite island at the resonant surfaces.

Robustness of the tokamak error field correction tolerance scaling

This paper presents the subtleties of obtaining robust experimental scaling laws for the core resonant error field threshold that leads to field penetration, locked modes, and disruptions. Recent progress in attempts to project this threshold to new machines has focused on advances in the metric used to quantify the dangerous error fields, incorporating the ideal MHD plasma response in a metric referred to as the ‘dominant mode overlap’. However, the scaling of this or any quantity with experimental parameters known to be important for the complicated tearing layer physics requires regressions performed for databases that, for historical reasons, unevenly sample the available parametric space. This paper presents the distribution of the existing international n = 1 database and details biases in the available sampling and details the sensitivity of ITER projections to simple least-squares regressions. Downsampling and a simple kernel density estimation weighted regression are used here to demonstrate the difference in projections that acknowledging the machine sampling bias can make. This results in more robust projection to parameters far from the ‘usual’ devices built thus far. Two multi-device and multi-parameter scalings of the EF threshold in Ohmic and powered plasmas are presented, projecting the threshold to ITER and investigating the impact of sampling biases.

ELM control based on modeling of plasma response to n = 2 and n = 3 resonant magnetic perturbation fields in DIII-D

Toroidal modeling of plasma response to the resonant magnetic perturbation (RMP) fields, with the n = 2 and 3 configurations (n is the toroidal mode number), is carried out for DIII-D discharge utilizing the MARS-F code. In particular, the relative toroidal phase (coil phase) between the lower and upper rows of the I-coil currents is scanned from −180° to 180°, while computing the plasma response. Both the resistive rotating and ideal static plasma response models have been employed to predict the optimum coil phase with typical safety factor (q95) values that are within the windows for suppressing the edge localized modes (ELMs). Six certain criteria, which are constructed from the MARS-F modeling, are proposed and tested to determine the optimum coil phase. For both plasma response models, it is found that the three criteria, namely, the resonant harmonic field including plasma response at the last rational surface, the maximum amplitude of the plasma displacement due to the peeling mode response, and the normal plasma surface displacement amplitude near the X point, can serve as appropriate indicators for determining the optimum coil phase. It is demonstrated that the even parity RMP coil configuration can better control the ELMs within the typical ELM suppression window in DIII-D tokamak.

Resistive versus ideal plasma response to RMP fields in DIII-D: roles of q95 and X-point geometry

Toroidal computation of the plasma response to the n = 2 (n is the toroidal mode number) resonant magnetic perturbation field, based on an H-mode plasma in DIII-D, is carried out for the purpose of investigating the role of the ideal versus resistive plasma response models while scanning the plasma safety factor (q) at the edge. Both response models, implemented in the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681), show significant amplification of the kink-peeling response in certain q-windows. A longstanding issue addressed in this work is the sensitivity of the q-window versus smoothing of the X-point geometry of the plasma separatrix. For this purpose, scan of the safety factor in 2D space (q95, qa) is carried out, where the q-value at 95% of the equilibrium poloidal flux, q95, is scanned by varying the total plasma current, whilst the edge safety factor qa is varied by gradually smoothing the X-point geometry at fixed total plasma current. Transition to the edge-peeling amplification domain is well described by simple analytic curves relating q95 and qa in the (q95, qa) space. These analytic curves not only help to quantify the sensitivity of the q95-window versus qa variation, but also establish the q95 window in the asymptotic limit of infinite qa (i.e. in the presence of true X-points). The ‘resonant’ q95 values (3.6, 4.05, 4.5) are found to roughly exhibit 1/n periodicity at infinite qa. Both ideal and resistive plasma response models, as well as the computed field at different pick-up locations, yield the same analytic curves describing transition to strong edge-peeling response. Detailed analysis of the computed plasma response and comparison with experiments are also performed.

Magnetic field topology modeling under resonant magnetic perturbations on EAST

In order to understand the mechanism by which the resonant magnetic perturbation (RMP) mitigates or suppresses the edge-localized mode (ELM), the topological study of the edge magnetic field in ELM mitigation or suppression phase is a critical issue. To model the three-dimensional magnetic field topology superposed RMP on Experimental Advanced Superconducting Tokamak, a numerical model using the field line tracing method for both vacuum and ideal plasma response approximations is proposed. Using the numerical model, the topological change and the penetration depth of the stochastic field lines in the edge magnetic field are studied in an RMP experiment. Comparing profiles of minimum ρ on edge stochastic field lines and the particle flux pattern, the ideal plasma response changes the field line penetration depth while remaining similar profile relative to vacuum approximation. To mitigate and suppress ELM strongly, the deep penetration of RMP fields and topological changes of the edge magnetic field is a key from our modeling.

Enhanced fast ion prompt loss due to resonant magnetic perturbations in KSTAR

We report a numerical study for the impact of resonant magnetic perturbation (RMP) on fast ion prompt loss in KSTAR using full orbit following simulation. Experimental observation in KSTAR indicates a sudden increase in RMP-induced fast ion prompt loss when the applied RMP field exceeds a threshold amplitude. Full orbit simulation with 3D perturbed equilibrium computed by the ideal plasma response reproduces the experimentally observed feature of RMP-induced prompt loss of fast ions and the existence of threshold RMP amplitude. Simulation with vacuum fields only shows a small increase in the fast ion loss that is insufficient to explain the observation. We show that the conservation of canonical angular momentum is broken due to the RMP during the fast ion transit, which is consistent with the enhanced fast ion loss above the threshold RMP amplitude. Phase space analysis suggests that trapped and intermediate pitch passing particles modified from initial high pitch passing ones are responsible for the threshold behavior of the fast ion loss in the simulation, while the phase space distribution of lost particles depends on the RMP field configuration.

A Riccati solution for the ideal MHD plasma response with applications to real-time stability control

Active feedback control of ideal MHD stability in a tokamak requires rapid plasma stability analysis. Toward this end, we reformulate the δW stability method with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the generic tokamak ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD matrix Riccati differential equation. Since Riccati equations are prevalent in the control theory literature, such a shift in perspective brings to bear a range of numerical methods that are well-suited to the robust, fast solution of control problems. We discuss the usefulness of Riccati techniques in solving the stiff ordinary differential equations often encountered in ideal MHD stability analyses—for example, in tokamak edge and stellarator physics. We demonstrate the applicability of such methods to an existing 2D ideal MHD stability code—DCON [A. H. Glasser, Phys. Plasmas 23, 072505 (2016)]—enabling its parallel operation in near real-time, with wall-clock time ≪1s. Such speed may help enable active feedback ideal MHD stability control, especially in tokamak plasmas whose ideal MHD equilibria evolve with inductive timescale τ≳ 1s—as in ITER.

Observation of resonant and non-resonant magnetic braking in the n = 1 non-axisymmetric configurations on KSTAR

Toroidal rotation braking by neoclassical toroidal viscosity driven by non-axisymmetric (3D) magnetic fields, called magnetic braking, has great potential to control rotation profile, and thereby modify tokamak stability and performance. In order to characterize magnetic braking in the various 3D field configurations, dedicated experiments have been carried out in KSTAR, applying a variety of static , 3D fields of different phasing of , 0, and . Resonant-type magnetic braking was achieved by phasing fields, accompanied by strong density pump-out and confinement degradation, and explained by excitation of kink response captured by ideal plasma response calculation. Strong resonant plasma response was also observed under phasing at , leading to severe confinement degradation and eventual disruption by locked modes. Such a strong resonant transport was substantially modified to non-resonant-type transport at higher , as the resonant particle transport was significantly reduced and the rotation braking was pushed to plasma edge. This is well explained by ideal perturbed equilibrium calculations indicating the strong kink coupling at lower is reduced at higher discharge. The 0 phasing fields achieved quiescent magnetic braking without density pump-out and confinement degradation, which is consistent with vacuum and ideal plasma response analysis predicting deeply penetrating 3D fields without an excitation of strong kink response.

Enhanced understanding of non-axisymmetric intrinsic and controlled field impacts in tokamaks

An extensive study of intrinsic and controlled non-axisymmetric field (δB) impacts in KSTAR has enhanced the understanding about non-axisymmetric field physics and its implications, in particular, on resonant magnetic perturbation (RMP) physics and power threshold (Pth) for L–H transition. The n = 1 intrinsic non-axisymmetric field in KSTAR was measured to remain as low as δB/B0 ~ 4 × 10−5 even at high-beta plasmas (βN ~ 2), which corresponds to approximately 20% below the targeted ITER tolerance level. As for the RMP edge-localized-modes (ELM) control, robust n = 1 RMP ELM-crash-suppression has been not only sustained for more than ~90 τE, but also confirmed to be compatible with rotating RMP. An optimal window of radial position of lower X-point (i.e. Rx = m) proved to be quite critical to reach full n = 1 RMP-driven ELM-crash-suppression, while a constraint of the safety factor could be relaxed (q95 = 5 0.25). A more encouraging finding was that even when Rx cannot be positioned in the optimal window, another systematic scan in the vicinity of the previously optimal Rx allows for a new optimal window with relatively small variations of plasma parameters. Also, we have addressed the importance of optimal phasing (i.e. toroidal phase difference between adjacent rows) for n = 1 RMP-driven ELM control, consistent with an ideal plasma response modeling which could predict phasing-dependent ELM suppression windows. In support of ITER RMP study, intentionally misaligned RMPs have been found to be quite effective during ELM-mitigation stage in lowering the peaks of divertor heat flux, as well as in broadening the ‘wet’ areas. Besides, a systematic survey of Pth dependence on non-axisymmetric field has revealed the potential limit of the merit of low intrinsic non-axisymmetry. Considering that the ITER RMP coils are composed of 3-rows, just like in KSTAR, further 3D physics study in KSTAR is expected to help us minimize the uncertainties of the ITER RMP coils, as well as establish an optimal 3D configuration for ITER and future reactors.

Comparison of divertor heat flux splitting by 3D fields with field line tracing simulation in KSTAR

We present the experimental and numerical analysis of divertor heat flux spitting by 3D magnetic fields in KSTAR. A dedicated experiment to measure the modification of the divertor heat flux profile in the n = 2 resonant and non-resonant field configurations has been conducted. The measured heat flux profiles were directly compared to the magnetic field line tracing (FLT) simulation using the POCA-FLT code, which fully considers vacuum and ideal plasma response models. Measurements indicate that the 3D fields induce strong splitting of heat flux on the divertor target due to the formation of stochastic field layers. Splitting characteristics largely depend on the applied 3D field structure and its alignment to the axisymmetric magnetic pitch. The modeling of the field line connection length shows good agreement with measurements, reproducing well the pattern of heat flux splitting. Simulations with the ideal plasma response better reproduce measured heat flux profiles by excitation of non-resonant field components in the non-resonant 0° phase and shielding of resonant field components in the resonant 90° phase.

Unified nonlinear theory of spontaneous and forced helical resonant MHD states

It is shown that the theory of the nonlinearly saturated ideal internal kink mode by Rosenbluth et al. [Phys Fluids 16(11), 1894 (1973)] can be used to find the fully nonlinear ideal plasma response to an externally applied resonant magnetic perturbation. It is also demonstrated that the solution leads to a jump in the rotational transform across the resonant surface caused by a zonal current sheet. Its amplitude scales linearly with the plasma perturbation despite the nonlinearity of the solution. This confirms a recent conjecture that three-dimensional MHD equilibria with nested magnetic surfaces generally contain discontinuities in the rotational transform [J. Loizu et al., Phys Plasmas 22(9), 090704 (2015)]. It also lends support to Parker's long-standing suggestion that “almost all” MHD equilibria possess current sheets.

Ideal plasma response to vacuum magnetic fields with resonant magnetic perturbations in non-axisymmetric tokamaks

Ideal plasma shielding and amplification of resonant magnetic perturbations in non-axisymmetric tokamak is presented by field line tracing simulation with full ideal plasma response, compared to measurements of divertor lobe structures. Magnetic field line tracing simulations in NSTX with toroidal non-axisymmetry indicate the ideal plasma response can significantly shield/amplify and phase shift the vacuum resonant magnetic perturbations. Ideal plasma shielding for n = 3 mode is found to prevent magnetic islands from opening as consistently shown in the field line connection length profile and magnetic footprints on the divertor target. It is also found that the ideal plasma shielding modifies the degree of stochasticity but does not change the overall helical lobe structures of the vacuum field for n = 3. Amplification of vacuum fields by the ideal plasma response is predicted for low toroidal mode n = 1, better reproducing measurements of strong striation of the field lines on the divertor plate in NSTX.

Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets

We consider the linear and nonlinear ideal plasma response to a boundary perturbation in a screw pinch. We demonstrate that three-dimensional, ideal-MHD equilibria with continuously nested flux-surfaces and with discontinuous rotational-transform across the resonant rational-surfaces are well defined and can be computed both perturbatively and using fully nonlinear equilibrium calculations. This rescues the possibility of constructing MHD equilibria with current sheets and continuous, smooth pressure profiles. The results predict that, even if the plasma acts as a perfectly conducting fluid, a resonant magnetic perturbation can penetrate all the way into the center of a tokamak without being shielded at the resonant surface.

Decoupled recovery of energy and momentum with correction of n = 2 error fields

Experiments applying known n = 2 ‘proxy’ error fields (EFs) find that the rotation braking introduced by the proxy EF cannot be completely alleviated through optimal n = 2 correction with poorly matched poloidal spectra. This imperfect performance recovery demonstrates the importance of correcting multiple components of the n = 2 field spectrum and is in contrast to previous results with n = 1 EFs despite a similar execution. Measured optimal n = 2 proxy EF correction currents are consistent with those required to null dominant mode coupling to the resonant surfaces and minimize the neoclassical toroidal viscosity (NTV) torque, calculated using ideal MHD plasma response computation. Unlike rotation braking, density pumpout can be fully corrected despite poorly matched spectra, indicating density pumpout is driven only by a single component proportional to the resonant coupling. Through precise n = 2 spectral control density pumpout and rotation braking can thus be decoupled. Rotation braking with n = 2 fields is also found to be proportional to the level of co-current toroidal rotation, consistent with NTV theory. Plasmas with modest counter-current rotation are insensitive to the n = 2 field with neither rotation braking nor density pumpout observed.

The spectral basis of optimal error field correction on DIII-D

Experimental optimum error field correction (EFC) currents found in a wide breadth of dedicated experiments on DIII-D are shown to be consistent with the currents required to null the poloidal harmonics of the vacuum field which drive the kink mode near the plasma edge. This allows the identification of empirical metrics which predict optimal EFC currents with accuracy comparable to that of first-principles modelling which includes the ideal plasma response. While further metric refinements are desirable, this work suggests optimal EFC currents can be effectively fed-forward based purely on knowledge of the vacuum error field and basic equilibrium properties which are routinely calculated in real-time.

Error field detection in DIII-D by magnetic steering of locked modes

Optimal correction coil currents for the n = 1 intrinsic error field of the DIII-D tokamak are inferred by applying a rotating external magnetic perturbation to steer the phase of a saturated locked mode with poloidal/toroidal mode number m/n = 2/1. The error field is detected non-disruptively in a single discharge, based on the toroidal torque balance of the resonant surface, which is assumed to be dominated by the balance of resonant electromagnetic torques. This is equivalent to the island being locked at all times to the resonant 2/1 component of the total of the applied and intrinsic error fields, such that the deviation of the locked mode phase from the applied field phase depends on the existing error field. The optimal set of correction coil currents is determined to be those currents which best cancels the torque from the error field, based on fitting of the torque balance model. The toroidal electromagnetic torques are calculated from experimental data using a simplified approach incorporating realistic DIII-D geometry, and including the effect of the plasma response on island torque balance based on the ideal plasma response to external fields. This method of error field detection is demonstrated in DIII-D discharges, and the results are compared with those based on the onset of low-density locked modes in ohmic plasmas. This magnetic steering technique presents an efficient approach to error field detection and is a promising method for ITER, particularly during initial operation when the lack of auxiliary heating systems makes established techniques based on rotation or plasma amplification unsuitable.

Sustained suppression of type-I edge-localized modes with dominantly n = 2 magnetic fields in DIII-D

Type-I edge-localized modes (ELMs) have been suppressed in DIII-D (Luxon et al 2003 Nucl. Fusion 43 1813) H-mode discharges with a H98Y2 confinement factor near 1.0 using magnetic perturbations (MPs) with dominant toroidal mode number n = 2. This expands access to the ELM-suppressed regime, which was previously attainable in DIII-D only with n = 3 fields. ELM suppression is obtained with two rows of internal coils for 1.8 s with normalized beta of 1.9 and average triangularity of 0.53, corresponding to a scaled version of ITER scenario 2 at an ITER relevant electron collisionality of 0.2. The applied field reduces the pedestal pressure and edge current via the density without degrading the edge thermal transport barrier. ELITE calculations find that the resulting profiles are stable to intermediate-n peeling–ballooning modes. ELM suppression is found within different ranges of q95 depending on the coil configuration used to generate the MP. The edge safety factors associated with suppression do not correspond to those that maximize the pitch-resonant components of the applied vacuum field. Instead, ELM suppression is correlated with an increase in the amplification of kink-resonant components of the calculated ideal MHD plasma response field.

Sensitivity to error fields in NSTX high β plasmas

It was found that error field threshold decreases for high β in NSTX, although the density correlation in conventional threshold scaling implies the threshold would increase since higher β plasmas in our study have higher plasma density. This greater sensitivity to error field in higher β plasmas is due to error field amplification by plasmas. When the effect of amplification is included with ideal plasma response calculations, the conventional density correlation can be restored and threshold scaling becomes more consistent with low β plasmas. However, it was also found that the threshold can be significantly changed depending on plasma rotation. When plasma rotation was reduced by non-resonant magnetic braking, the further increase in sensitivity to error field was observed.

Response of a resistive and rotating tokamak to external magnetic perturbations below the Alfvén frequency

Motivated by the recent experimental observation that plasma stability can be improved by external magnetic perturbations, the general problem of plasma response to external magnetic perturbations is investigated. Different (vacuum, ideal and resistive) plasma response models are considered and compared. Plasma response, in experiments where stabilization was achieved, is obtained through computation using the MARS-F code, with a plasma model that includes both plasma resistivity and rotation. The resultant magnetic field line stochasticity is much reduced from that obtained formerly using the vacuum plasma model. This reduced stochasticity is more consistent with the favourable experimental observation of enhanced stability. Examples are given for the response of an ITER plasma to perturbations generated by the correction coils; and the response of a plasma to external coils (antenna) up to the Alfvén frequency.

Importance of plasma response to nonaxisymmetric perturbations in tokamaks

Tokamaks are sensitive to deviations from axisymmetry as small as δB/B0∼10−4. These nonaxisymmetric perturbations greatly modify plasma confinement and performance by either destroying magnetic surfaces with subsequent locking or deforming magnetic surfaces with associated nonambipolar transport. The Ideal Perturbed Equilibrium Code (IPEC) calculates ideal perturbed equilibria and provides important basis for understanding the sensitivity of tokamak plasmas to perturbations. IPEC calculations indicate that the ideal plasma response, or equivalently the effect by ideally perturbed plasma currents, is essential to explain locking experiments on National Spherical Torus eXperiment (NSTX) and DIII-D. The ideal plasma response is also important for neoclassical toroidal viscosity (NTV) in nonambipolar transport. The consistency between NTV theory and magnetic braking experiments on NSTX and DIII-D can be improved when the variation in the field strength in IPEC is coupled with generalized NTV theory. These plasma response effects will be compared with the previous vacuum superpositions to illustrate the importance. However, plasma response based on ideal perturbed equilibria is still not sufficiently accurate to predict the details of NTV transport and can be inconsistent when currents associated with a toroidal torque become comparable to ideal perturbed currents.