H-mode Tokamak Research Articles

Using convolutional neural networks to detect edge localized modes in DIII-D from Doppler backscattering measurements.

In H-mode tokamak plasmas, the plasma is sometimes ejected beyond the edge transport barrier. These events are known as edge localized modes (ELMs). ELMs cause a loss of energy and damage the vessel walls. Understanding the physics of ELMs, and by extension, how to detect and mitigate them, is an important challenge. In this paper, we focus on two diagnostic methods-deuterium-alpha (Dα) spectroscopy and Doppler backscattering (DBS). The former detects ELMs by measuring Balmer alpha emission, while the latter uses microwave radiation to probe the plasma. DBS has the advantages of having a higher temporal resolution and robustness to damage. These advantages of DBS diagnostic may be beneficial for future operational tokamaks, and thus, data processing techniques for DBS should be developed in preparation. In sight of this, we explore the training of neural networks to detect ELMs from DBS data, using Dα data as the ground truth. With shots found in the DIII-D database, the model is trained to classify each time step based on the occurrence of an ELM event. The results are promising. When tested on shots similar to those used for training, the model is capable of consistently achieving a high f1-score of 0.93. This score is a performance metric for imbalanced datasets that ranges between 0 and 1. We evaluate the performance of our neural network on a variety of ELMs in different high confinement regimes (grassy ELM, RMP mitigated, and wide-pedestal), finding broad applicability. Beyond ELMs, our work demonstrates the wider feasibility of applying neural networks to data from DBS diagnostic.

Observation of a new pedestal stability regime in MAST Upgrade H-mode plasmas

The first pedestal stability and structure analysis on the new MAST Upgrade (MAST-U) spherical tokamak H-mode plasmas is presented. Our results indicate that MAST-U pedestals are close to the low toroidal mode number (n) peeling branch of the peeling-ballooning instability, in contrast with MAST H-mode pedestals which were deeply in the high-n ballooning branch. This offers the possibility of reaching the ELM-free quiescent H-mode (Burrell et al 2005 Plasma Phys. Control. Fusion 47 B37–B52) or high-performance super H-mode (Snyder et al 2015 Nucl. Fusion 55 083026; Snyder et al 2019 Nucl. Fusion 59 086017) regimes. In addition, the coupling between the peeling and ballooning branches is weak in MAST-U, suggesting that a path to very high pedestal pressure gradient at high density may exist with sufficient heating power. A possible explanation for the differences between MAST and MAST-U pedestal stability is given in terms of plasma shaping parameters, in particular squareness and elongation, as well as the pedestal top temperature and collisionality.

Theoretical and global simulation analysis of collisional microtearing modes

Microtearing modes (MTMs) are suggested as a candidate for anomalous thermal transport in tokamak H-mode discharges. This study investigates MTMs in tokamak plasmas, employing simulations in the BOUT++ framework. It simplifies and linearizes the governing equations in detailed linear simulations. The study meticulously evaluates various conductivity models under diverse plasma conditions and collision regimes. The research thoroughly assesses different conductivity models across a range of plasma conditions and collision regimes. A unified dispersion relation that includes both MTM and Drift-Alfvén Wave (DAW) instabilities is derived, showing that DAW and MTM instabilities occur at varying distances from the rational surface. Specifically, MTMs become unstable near the rational surface but stabilize farther away, while drift-Alfvén instability appears farther from the rational surface. The study also re-derives MTM dispersion relations using Ohm's law and the vorticity equation, providing a thorough analysis of electromagnetic and electrostatic interactions in tokamaks. Global simulations demonstrate an inverse correlation between MTM growth rates and collisionality, and a direct correlation with temperature gradients. The nonalignment of the rational surface with the peak ω*e stabilizes the MTMs. Nonlinear simulations highlight electron temperature relaxation as the primary saturation mechanism for MTMs, with magnetic flutter identified as the dominant mode of electron thermal transport.

A new type of resonant magnetic perturbation for controlling edge localized modes

A new type of resonant magnetic perturbation (RMP), generated by helical coils, is proposed for controlling the edge localized mode (ELM) in H-mode tokamak plasmas. The helical coil optimization utilizes the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681) computed linear resistive fluid response of the plasma to the applied RMP field. The optimal helical coils are found to be located near the outboard mid-plane of the torus, with relatively simple shape but tilted towards the equilibrium magnetic field line pitch. Compared to the window-frame ELM control coils, the optimal helical coils require 2–4 times less current, in order to achieve the same ELM control performance specified by various figures of merit adopted in this work. The results from the present study show a promising path forward in achieving ELM control with RMP fields in tokamak plasmas.

Helical mode localization and mode locking of ideal MHD instabilities in magnetically perturbed tokamak plasmas

In H-mode tokamak plasmas, the achievable pressure-gradient is limited by type-I Edge Localized Modes (ELMs), which are projected to cause severe damage to future fusion devices. There are several approaches aiming to mitigate/suppress the occurrence of ELMs, such as the application of an external non-axisymmetric magnetic perturbation field, which breaks the axisymmetry of the tokamak plasma. In this work we use the CASTOR3D code to investigate helical localization and mode locking of edge-localized ideal MHD instabilities in rotating and flow-free magnetically perturbed tokamak plasmas with periodicity. Helically localized instabilities are separated into two classes: quasi-locked and strictly locked. In a non-rotating plasma, the localization of quasi-locked modes is determined by an envelope while their precise location under the envelope is arbitrary, whereas strictly locked modes can only occur at a single helical position. Strictly locked modes only rotate if the toroidal plasma rotation exceeds a critical threshold; above the threshold the forced rotation of the strictly locked modes is non-uniform. For quasi-locked modes, no such critical threshold exists; they rotate uniformly beneath their envelope in the case of finite plasma rotation. The helical localization of both quasi-locked and strictly locked instabilities is determined by the energetic decomposition of the instabilities close to the most unstable flux-surface; for example, strongly current-density driven instabilities are aligned with regions of augmented parallel equilibrium current-density. Finally, we compare the computationally determined localization of MHD instabilities to experimental observations. The determined MHD instability is located at the same position as the experimentally measured modes with respect to the equilibrium corrugation, verifying that ideal MHD can describe the experimentally observed instabilities.

Spatial heterodyne spectroscopy for fast local magnetic field measurements of magnetized fusion plasmas.

A novel spectroscopy diagnostic for measuring internal magnetic fields in high temperature magnetized plasmas has been developed. It involves spectrally resolving the Balmer-α (656nm) neutral beam radiation split by the motional Stark effect with a spatial heterodyne spectrometer (SHS). The unique combination of high optical throughput (3.7 mm2sr) and spectral resolution (δλ ∼ 0.1 nm) allows these measurements to be made with time resolution ≪1ms. The high throughput is effectively utilized by incorporating a novel geometric Doppler broadening compensation technique in the spectrometer. The technique significantly reduces the spectral resolution penalty inherent to using large area, high-throughput optics while still collecting the large photon flux provided by such optics. In this work, fluxes of order 1010s-1 support the measurement of deviations of <5 mT (ΔλStark ∼ 10-4 nm) in the local magnetic field with 50 µs time resolution. Example high time resolution measurements of the pedestal magnetic field throughout the ELM cycle of a DIII-D tokamak plasma are presented. Local magnetic field measurements give access to the dynamics of the edge current density, which is essential to understanding stability limits, edge localized mode generation and suppression, and predicting performance of H-mode tokamaks.

Non-linear free boundary simulations of the plasma response to resonant magnetic perturbations in ASDEX Upgrade plasmas

A promising method for the control of Edge Localized Modes (ELMs) in H-Mode tokamak plasmas is the application of Resonant Magnetic Perturbations (RMPs), where small helical field perturbations are introduced into the plasma via a set of external coils. While RMPs are used for suppression of ELMs in many present-day tokamaks, the mechanisms that lead to RMP-ELM control are still subject of debate.Here, we use the non-linear MHD code JOREK to investigate the penetration of the magnetic perturbation fields into ASDEX Upgrade (AUG) plasmas. We present an extension of the coupled JOREK-STARWALL code, that replaces the commonly used fixed boundary treatment with a free boundary treatment. Instead of prescribing the magnetic field at the boundary according to the vacuum field using Dirichlet boundary conditions, natural boundary conditions are applied, so that the magnetic field and plasma current density are evolving freely at the boundary. This allows a fully self-consistent development of the plasma response and the magnetic perturbation in the whole computational domain. The direct comparison of both approaches demonstrates that the artificial suppression of the plasma response with the fixed boundary treatment reduces the excitation of marginally stable modes. An overall larger perturbation is observed using the free boundary approach.The presented simulations are performed in realistic geometry with fully realistic plasma parameters and plasma flows based on reconstructions of experimental AUG equilibria. While the use of realistic plasma parameters makes the simulations particularly challenging, it also allows for quantitative comparisons to the experiment. When the RMP induced corrugation of the boundary is compared to electron density measurements from the lithium beam emission spectroscopy, only the free boundary approach shows excellent agreement with the experiment.

Improved Particle Confinement with Resonant Magnetic Perturbations in DIII-D Tokamak H-Mode Plasmas.

Experiments on the DIII-D tokamak have identified a novel regime in which applied resonant magnetic perturbations (RMPs) increase the particle confinement and overall performance. This Letter details a robust range of counter-current rotation over which RMPs cause this density pump-in effect for high confinement (H mode) plasmas. The pump in is shown to be caused by a reduction of the turbulent transport and to be correlated with a change in the sign of the induced neoclassical transport. This novel reversal of the RMP induced transport has the potential to significantly improve reactor relevant, three-dimensional magnetic confinement scenarios.

Design of a multi-configurations divertor for the DTT facility

The standard positive triangularity H-mode tokamak operation with Single Null Divertor (SND) configuration and tungsten monoblocks targets can face some difficulties in providing a solution scalable towards the realization of the fusion reactor. To provide a safer solution, the adoption of alternative divertor magnetic configurations (ADCs) has been considered to provide on the targets a stationary heat load of less than 10 MW/m2 presently considered an upper technological limit for a nuclear fusion reactor and to reduce enough plasma temperature to avoid tungsten sputtering and its possible plasma contamination. Additionally, different plasma scenarios have been considered to avoid the huge transient energy released due to the type-I ELMs of the high confinement H-mode operation. The new high field superconducting divertor tokamak test facility (DTT) [1] is presently under construction to specifically study power exhaust solutions in reactor relevant regimes. The first DTT divertor will use the ITER-like technology based on full tungsten monoblocks bonded on CuCrZr cooling tubes. To test and compare a wide set of different power exhaust solutions, the divertor is being designed to be compatible with the standard single null (SND) configuration but also with some ADCs, like the X divertor (XD) and the hybrid Super-X/long leg SN. Additionally, the l-mode negative triangularity (NT) operation is considered important to explore as a solution to avoid ELMs. Two different closed divertors with the same grazing angle (α = 2°) for the reference SND configuration have been considered: a so called “narrow” one with normalized divertor parameters similar to those presently foreseen for DEMO divertor and the “wide” one with a 50 % wider target aperture which allows testing long legs configurations as well as snowflake divertor configurations (SFD). For the “wide” divertor different dome sizes have been also considered. The comparison between different shapes and their optimization have been done with the 2D edge fluid-kinetic code SOLEDGE2D-EIRENE. The code easily manages all magnetic configurations and can resolve the heat load on all components thanks the mesh filling the whole divertor and vessel volume. In a representative sub-set of all the possible magnetic equilibrium configurations that can be realized in DTT, the edge modelling shows a better performance of the “wide” divertor compared to the “narrow” one. In fact, the “wide” divertor allows to have lower plasma temperature at the targets in pure deuterium at low PSOL power and also lower impurity concentrations is required to achieve plasma detachment by impurity seeding at the reference full power DTT scenario. The results seem to indicate that the improvement due to the longer legs possible with the “wide” divertor can overcome its lower apparent closure compared to the “narrow” divertor solution.

Effective current drive in the pedestal region of high-confinement tokamak plasma using electron cyclotron waves

Large-amplitude edge-localized modes (ELMs) are a major concern in the development of tokamak fusion energy. According to the peeling–ballooning theoretical model, effective current drive in the pedestal region of high-confinement (H-mode) tokamak plasmas can alter the current profile in this region and hence affect ELM instabilities. Using ray-tracing and Fokker–Planck quasilinear codes, effective and localized current drive by electron cyclotron (EC) waves in the H-mode pedestal region is studied in this paper. Numerical investigations are performed under present-day medium-sized tokamaks with an aspect ratio (A = R/a) covering a typical range of [2.7, 4.0]. Localized non-inductive current can be generated effectively by the Ohkawa mechanism of EC waves in the low-electron collisionality H-mode pedestal region. The Ohkawa mechanism-dominated current drive (OKCD) is much more effective than traditional electron cyclotron current drive (ECCD), which is the Fisch–Boozer mechanism-dominated one. Under 1–2 MW EC power, the current density of both co-OKCD and counter-OKCD has the same magnitude as edge bootstrap current density, which is calculated using the Sauter model. However, edge ECCD is much smaller than the bootstrap current. Consequently, the use of edge ECCD to control ELM failed in experiments, and only the EC heating can affect ELM behavior. Simulations on OKCD are performed in the H-mode pedestal region of realistic DIII-D tokamak plasma, and the results also support the above conclusions. The dependence of effective edge OKCD on the ratio of magnetic field and EC frequency is also found in tokamaks with a typical aspect ratio range. This work opens up a new possible field for active control of ELMs (triggering or suppressing an ELM) by edge current drive in present-day medium-sized tokamaks.

Numerical study on predicting MHD stability of HL-2A tokamak pedestal structure

HL-2A tokamak achieved the first ELMy H-mode discharge operation in 2009 under divertor configuration, and many experimental and simulation researches have been carried out to investigate the pedestal magnetohydrodynamic (MHD) instability. However, there are still few studies on the effect of pedestal structure on MHD stability. Therefore, based on HL-2A experimental parameters, equilibria with different pedestal structures are generated by using TOQ code, and the MHD stability of the equilibria is simulated by using the BOUT++ three-field module. The linear simulations show that reducing the pedestal height, increasing the pedestal width, reducing the pedestal current density and reducing the ion density in the pedestal can improve the MHD stability of pedestal. Using the theory of dispersion relation, the simulation results are explained. Under the premise of MHD stability, the maximum pedestal heights corresponding to different pedestal widths are found, and the data are fitted to obtain an empirical formula that can predict pedestal height, and on this basis, considering the kinetic ballooning mode theory, pedestal height and width are predicted simultaneously. The effect of the pedestal structure on the MHD mode structure is investigated, it is found that the pedestal height can affect the radial width of the mode. Nonlinear simulations show that the pre-growth of instability is affected mainly by a single dominant mode, and the growth of the dominant mode to a certain size will cause the collapse of the pedestal and the eruption of the edge localized mode (ELM). The variation of ELM size after ELM eruption is synchronized with the evolution of the dominant mode. Generally, equilibria with larger linear growth rates have larger ELM sizes and a wider range of pedestal collapse in nonlinear simulations. In this work, the scanning of the pedestal data focuses mainly on the width and height of the pedestal, and other parameters such as small radius, toroidal magnetic field, plasma current, and the pedestal safety factor values will be changed in the future based on the work in this paper, with the aim of enriching the HL-2A pedestal database and predicting the pedestal structure more accurately. Finally these results will be integrated under the HL-2A integrated platform, which in turn will provide a reference for HL-2A tokamak H-mode experiments and integrated simulations.

Ion temperature anisotropy in the tokamak scrape-off layer

Understanding tokamak exhaust-power heat loads on divertor plates depends critically on having a realistic model of the scrape-off layer (SOL) plasma. The Braginskii fluid model is often solved to understand the SOL plasma behavior. This model is based on the collisional limit for transport along the magnetic field . The ions and electron gyrofrequencies are assumed to be much larger than the Coulomb collision frequencies, which are nonetheless, sufficiently large to yield common parallel and perpendicular temperatures for each species, i.e. the temperatures are assumed to be isotropic. In certain circumstances such as encountered for the tokamak H-mode, the ion temperature can be quite anisotropic. In this work, the anisotropy effects are implemented in the two-dimensional (2D) transport code UEDGE. Various geometries (1D slab, 2D slab and a toroidal tokamak geometry) are used to study the 2D structure of ion temperature anisotropy and its effects on plasma transport in detail. Results show that the effects of ion temperature anisotropy on the plasma parallel transport are substantial near the magnetic X-point, which leads to different steady state density profiles in the divertor regions. The extra mirror force introduced by ion temperature anisotropy can be one of the main forces contributing to the plasma flow in the SOL.

Global Theory of Microtearing Modes in the Tokamak Pedestal.

The pedestal of H-mode tokamaks displays strong magnetic fluctuations correlated with the evolution of the electron temperature. The microtearing mode (MTM), a temperature-gradient-driven instability that alters magnetic topology, is compatible with these observations. Here we extend the conventional theory of the MTM to include the global variation of the temperature and density profiles. The offset between the rational surface and the location of the pressure gradient maximum (μ) emerges as a crucial parameter for MTM stability. The extended theory matches observations on the Joint European Torus tokamak.

Guiding center and gyrokinetic orbit theory for large electric field gradients and strong shear flows

The guiding center and gyrokinetic theory of magnetized particle motion is extended to the regime of large electric field gradients perpendicular to the magnetic field. A gradient in the electric field directly modifies the oscillation frequency and causes the Larmor orbits to deform from circular to elliptical trajectories. In order to retain a good adiabatic invariant, there can only be strong dependence on a single coordinate at lowest order, so that resonances do not generate chaotic motion that destroys the invariant. When the gradient across magnetic flux surfaces is dominant, the guiding center drift velocity becomes anisotropic in response to external forces and additional curvature drifts must be included. The electric polarization density remains gyrotropic, but both the polarization and magnetization are modified by the change in gyrofrequency. The theory can be applied to shear flows that are even stronger than those observed in the edge transport barrier of a high-performance tokamak (H-mode) pedestal, even if the toroidal field is as small as or even smaller than the poloidal field. Yet, the theory retains a mathematical form that is similar to the standard case and can readily be implemented within existing simulation tools.

Further modeling of q95 windows for the suppression of edge localized modes by resonant magnetic perturbations in the DIII-D tokamak

An improved resonant plasma response model that more accurately captures the physics of the interaction between a tokamak plasma and a resonant magnetic perturbation (RMP) is developed. The model interpolates between the linear and the nonlinear response regimes and takes into account the fact that the slip-frequency is non-zero in the nonlinear regime. The improved model is incorporated into the extended perturbed equilibrium code (EPEC) toroidal asymptotic matching code. The modified EPEC code is used to investigate RMP-induced edge-localized-mode (ELM) suppression in DIII-D H-mode discharge #145380. Somewhat surprisingly, allowing for a finite slip-frequency (i.e., relaxing the so-called no-slip constraint) is found to only slightly facilitate the locking of driven magnetic island chains to the RMP, and, hence, to only slightly facilitate RMP-induced ELM suppression. This is true despite the fact that the nature of non-locked island solutions is radically different when the no-slip constraint is imposed compared to when it is relaxed (in the first case, the widths of the island chains driven at the rational surfaces pulsate, and in the second case, they remain steady). The previously obtained conclusion that the response of a typical H-mode tokamak plasma to an RMP cannot be accurately modeled by linear theory is confirmed. The previously obtained conclusion that the best agreement between theory and observations is achieved by assuming that the natural frequencies of tearing modes, in the absence of the RMP, are determined by the local equilibrium E×B velocity is also confirmed.

Internal measurement of magnetic turbulence in ELMy H-mode tokamak plasmas

Magnetic turbulence is directly observed internally in the pedestal of ELMy H-mode tokamak plasmas using a newly developed Faraday-effect polarimetry measurement. Fluctuation amplitude is δbr≥15 G (150–500 kHz), with a ratio of magnetic to density fluctuation |δbr/B|/|δn/n|≥0.15. Magnetic turbulence is identified as resulting from micro-tearing-instability and mode growth accompanied by degraded plasma confinement is observed.

Stellarator-tokamak energy confinement comparison based on ASDEX Upgrade and Wendelstein 7-X hydrogen plasmas

A confinement database with mainly electron-heated hydrogen plasmas from ASDEX Upgrade and Wendelstein 7-X was assembled. Stellarator confinement scaling expressions describe both standard discharges in the stellarator and L-mode plasmas in the tokamak similarly well and indicate a similar quality of energy confinement in both devices. While the energy confinement time in ASDEX Upgrade benefits from the smaller aspect ratio of the device, the transport coefficients in Wendelstein 7-X appear to be smaller possibly due to reduced average magnetic field curvature. A physics based confinement scaling is derived from a model that successfully describes transport in tokamaks. The dimensionally correct scaling has very similar parameter dependencies as the stellarator scalings and reproduces also the trends in the data from ITER L- and H-mode databases reasonably well. On the basis of this scaling, which represents the confinement times of the present data base, average tokamak L-mode and H-mode confinement is 7% lower and 76% higher, respectively.

Lithium splashing from flowing liquid lithium limiter and its effect on high confinement plasma performance in EAST tokamak

Abstract To study the potential applications of liquid lithium (Li) as plasma facing material, the third generation of flowing liquid Li (FLiLi) limiter has been designed and successfully tested in EAST tokamak H-mode discharges in the 2018 run campaign. The stability of liquid Li on FLiLi surface, and Li splashing and its effect on the plasma performance are investigated in detail. Li splashing was observed in L-mode and H-mode discharges due to Rayleigh-Taylor (R-T) instability, resulting from electromagnetic forces J → × B → . Li splashing in low-parameter plasma decreased both core and edge electron temperatures, resulting in the degradation of plasma energy confinement and even H-L back transition, however, it almost did not influence plasma confinement performance in high-parameter plasma. These findings indicate that high-parameter plasma has higher tolerance to Li impurities than low-parameter plasma. By contrast, tungsten splashing from the melted layer on the upper divertor was also induced by the R-T instability but generally resulted in plasma disruption, which suggests lower tolerance to tungsten impurities in EAST tokamak.

Effect of non-axisymmetric perturbations on the ambipolar Er and neoclassical particle flux inside the ITER pedestal region

The transport dynamics of impurities in the pedestal region of ITER plasmas is of crucial interest since this regulates the penetration of impurities from the edge into the core plasma, where an excessive accumulation of impurities can degrade their fusion performance. In the pedestal region of H-mode tokamak plasmas, anomalous transport is highly reduced and impurity transport is found to be well described by the neoclassical theory. Under these conditions, perturbations to the axisymmetric tokamak geometry can strongly affect both the radial electric field and particle transport. In this work, we describe the results of numerical studies performed to quantify the effects on the pedestal ambipolar electric field and radial particle fluxes of the non-axisymmetric fields, associated with both the intrinsic toroidal field ripple and extrinsic fields applied for ELM control, for ITER Q = 10 plasma conditions with emphasis on high Z impurity transport. It is found that the effect of the ITER toroidal field ripple on high Z impurity transport is negligible. On the contrary, extrinsic three-dimensional fields applied for ELM control cause a strong modification of the pedestal ambipolar electric (to less negative values) and the appearance of multi-valued solutions for the pedestal electric field, analogue to core stellarator transport significantly increasing the outward character of neoclassical pedestal transport for both the main plasma ions (D and T in ITER) and high Z impurities, suggesting a strong modification of the background plasma profiles. Finally, it is found that for the Z impurity, its quantitative evaluation has uncertainties (with important implications for the radial flow direction) associated with the high poloidal Mach number ~ 1, due to the high pedestal electric field and large ion mass, which renders the first order neoclassical theory questionable. Detailed ITER MHD 3D equilibria and a benchmark of SFINCS against NEO and NCLASS codes has also been obtained.

Theory of edge localized mode suppression by static resonant magnetic perturbations in the DIII-D tokamak

According to a recent paper [Hu et al., Phys. Plasmas 26, 120702 (2019)], mode penetration at the top of the pedestal is a necessary and sufficient condition for the suppression of edge localized modes (ELMs) by a resonant magnetic perturbation (RMPs) in an H-mode tokamak discharge. This paper employs asymptotic matching theory to model a particular DIII-D discharge in which ELMs were suppressed by an externally generated, static, n = 2, RMP whose amplitude was modulated at a frequency of 1 Hz. It is demonstrated that the response of the plasma to the applied RMP, in the immediate vicinities of the rational (i.e., resonant) surfaces, is governed by nonlinear, rather than by linear, physics. This is the case because the magnetic island widths associated with driven reconnection exceed the linear layer widths, even in cases where driven reconnection is strongly suppressed by plasma rotation. The natural frequency at a given rational surface (i.e., the helical frequency at which the locally resonant component of the RMP would need to propagate in order to maximize driven reconnection) is found to be offset from the local E×B frame in the ion diamagnetic direction. The size of the offset is mostly determined by neoclassical poloidal rotation. Finally, the predictions of a fully nonlinear plasma response model are found to be broadly consistent with the DIII-D experimental data.