Recent observations on DIII-D have advanced the understanding of plasma response to applied resonant magnetic perturbations (RMPs) in both H-mode and L-mode plasmas. Three distinct 3D features localized in minor radius are imaged via filtered soft x-ray emission: (i) the formation of lobes extending from the unperturbed separatrix in the X-point region at the plasma boundary, (ii) helical kink-like perturbations in the steep-gradient region inside the separatrix, and (iii) amplified islands in the core of a low-rotation L-mode plasma. These measurements are used to test and to validate plasma response models, which are crucial for providing predictive capability of edge-localized mode control. In particular, vacuum and two-fluid resistive magnetohydrodynamic (MHD) responses are tested in the regions of these measurements. At the plasma boundary in H-mode discharges with *n* = 3 RMPs applied, measurements compare well to vacuum-field calculations that predict lobe structures. Yet in the steep-gradient region, measurements agree better with calculations from the linear resistive two-fluid MHD code, M3D-C1. Relative to the vacuum fields, the resistive two-fluid MHD calculations show a reduction in the pitch-resonant components of the normal magnetic field (screening), and amplification of non-resonant components associated with ideal kink modes. However, the calculations still over-predict the amplitude of the measured perturbation by a factor of 4. In a slowly rotating L-mode plasma with *n* = 1 RMPs, core islands are observed amplified from vacuum predictions. These results indicate that while the vacuum approach describes measurements in the edge region well, it is important to include effects of extended MHD in the pedestal and deeper in the plasma core.

## I. INTRODUCTION

Transient heat loads deposited by edge-localized modes (ELMs) in ITER are predicted to cause rapid erosion of the plasma facing materials, making mechanisms to control or mitigate ELMs essential for ITER.^{1,2} One possibility is to use small amplitude non-axisymmetric magnetic field perturbations, which can alter the ELM stability and dynamics. Experiments show ELMs can be suppressed or mitigated with resonant magnetic perturbations (RMPs) in some cases^{3–5} and triggered in others.^{6,7} In the case of suppression, RMPs can modify the pedestal such that the peeling-ballooning limit is never reached, thus arresting the ELM cycle.^{8} However, the RMP mechanism is not well understood and relies on the role of plasma response. Therefore, validation of response models is critical.

Measurements of the plasma response show how the boundary region is perturbed with various levels of agreement with modeling.^{9–14} Models predict various plasma responses to applied small amplitude non-axisymmetric fields. Vacuum field calculations (a linear combination of the magnetic field perturbation and the plasma equilibrium magnetic field) provide a description for the effect of RMPs in the edge/boundary region.^{15,16} However, typical applied perturbations lead to predictions of high stochasticity with this model, which are in conflict with the observed high-temperature pedestal. In contrast, magnetohydrodynamic (MHD) predicts rotational screening effects such that imaging currents develop at rational surfaces to shield the resonant perturbation.^{17} This screening is perfect in the ideal-MHD model, but may be only partial when dissipative effects are included.^{18–21} Two-fluid resistive MHD simulations show weak screening—and often amplification—of pitch-resonant field components in regions of low perpendicular electron flow, $\omega e,\u22a5=\omega E\xd7B+\omega e,*$, but generally good screening elsewhere.^{22} Ideal and resistive calculations also show significant deformations of the magnetic geometry in the edge, which are associated with the excitation of stable kink/peeling modes by the applied fields. These effects represent departures of the fields and magnetic topology from the vacuum predictions to various extents in different regions across the H-mode pedestal.

In this paper, we present an understanding of the plasma response through measurements of the response to RMPs in different radial regions of the plasma with comparisons to modeling. Tangential soft-x-ray (SXR) imaging is used to measure the radial and poloidal structure of density and temperature perturbations due to the applied RMPs. Through the use of different energy filters and plasma conditions, the imaging results are able to examine: (i) resistive low-rotation edge/scrape-off layer (SOL) region of an H-mode discharge, (ii) H-mode steep-gradient region with high $\omega e,\u22a5$, and (iii) low-rotation L-mode core. Each of these regimes exhibits a different topological response and can be used to inform plasma response models. Synthetic diagnostic modeling is used to forward model predicted responses to compare to data. The experimental setup for three discharge conditions and the diagnostic measurement are described in Sec. II. The imaging data is discussed in Sec. III. The modeled responses are presented in Sec. IV and the comparison to data are presented in Sec. V. Finally, the results are summarized in Sec. VI.

## II. EXPERIMENTAL SETUP

The DIII-D tangential SXR system views the lower X-point region as shown in Fig. 1. The system uses pinhole optics focusing radiation onto a scintillator, which is imaged with visible optics.^{23} The view was chosen to exploit flux expansion in elongated or diverted plasmas.

Figure 1(a) shows an example of a diverted discharge where the flux expansion is largest at the poloidal field null at the X-point. The expansion can be estimated by the spacing between the $\Psi n$ = 0.9 and $\Psi n$ = 1.0 surfaces at the X-point compared to the outboard midplane region ( ∼ 10x). The view is line-integrated, with the view-cone illustrated by Fig. 1(b). However, localized features can still be interpreted in images by projecting the equilibrium and vessel wall along the curved tangency plane onto the images. This will be shown in Sec. III. The camera system uses high-pass energy filters to exclude regions of lower energy emission. Two energy filter ranges are used in this paper: $Te$ ≳ 40 eV which extends in to the extreme ultraviolet (EUV) range; and $Te$ ≳ 600 eV which is traditional SXR. The upper energy is limited by the scintillator thickness, which cuts off near 30 keV. These filters combined with different $Te$ profiles allow measurements in different radial regions.

Three discharges are used to examine plasma response effects: two *n* = 3 RMP ELM-suppressed or nearly ELM-suppressed H-mode discharges, but with different energy filters, and one *n* = 1 RMP L-mode low-$Te$ discharge. Here, *n* is the toroidal mode number of the applied field. In these discharges, *n* = 3 is chosen for the H-mode cases because it is more relevant to ELM suppression scenarios. In the L-mode case, *n* = 1 is chosen because islands would be easier to detect given well-separated rational surfaces if they were present. Time traces of plasma current, $\beta n$, divertor $D\alpha $ and RMP perturbation current are shown in Fig. 2. The plasma shape and applied *n* perturbation waveform are shown in the three separate coil current traces. Only the active RMP coils are also shown. The I-coils, above and below the midplane internal to the vacuum vessel are used in the H-mode discharges (dashed red and solid blue). Conversely, the ex-vessel C-coils centered at the midplane are used in the L-mode discharge (dash-dot green). The C-coils are used for the L-mode discharge because they produce a resonant field that reaches deeper into the core compared to the I-coils. The coil locations are also shown in Fig. 1(a). Despite being ex-vessel, the perturbation from the C-coils is slightly larger than that of the I-coils given 4 turns vs. the single turn I-coils. The shaded region of the coil currents corresponds to the range where imaging data is obtained for each RMP phase.

Corresponding radial profiles are shown in Fig. 3 for $ne$, $Te$, $\omega e,\u22a5$, and modeled filtered radiant power density (discussed below) incident on the scintillator with different energy filters. The density profiles are similar except the solid blue profile is slightly higher. This discharge is not fully ELM-suppressed throughout the application of the RMP given the higher density, as indicated from the $D\alpha $ in Fig. 2. The higher density adds to the SXR signal and will also increase the bootstrap current. The $Te$ profiles for the H-mode cases are similar, while the L-mode $Te$ is much lower. The magnitude of $\omega e,\u22a5$ is typically large (> 50 krad/s) in the steep-gradient region of an H-mode pedestal due to diamagnetic flow. The L-mode case has very low rotation and weak pressure gradients leading to a small $\omega e,\u22a5$.

The modeled radiant power density uses the CHIANTI code^{24} with measured electron and carbon densities and the assumption of trace amounts of metallic impurities <0.04%. These profiles are forward modeled to synthetic images and compared to the data for consistency, with the trace impurities varied radially to improve the line-integrated match. The profile modeled with the lower energy 40 eV filter in H-mode (dashed red) extends the measurement into the edge/SOL region. The profile modeled with a 600 eV filter in H-mode (solid blue) in the tradition SXR range creates a large SXR gradient in the pedestal steep-gradient region. The profile modeled with a 700 eV SXR filter in L-mode (dash-dot green) uses the L-mode low-$Te$ profile. This creates a wider SXR gradient into the core of the plasma. The location of the SXR gradient is important because it can provide localization of perturbations detected in the images. The region of high gradient provides high contrast when using the technique of phase-locked differencing. Here, the phase of the RMP is inverted and images from opposite phases are differenced. The perturbation in the line-integrated image is then isolated to the radial region with the largest SXR gradient. In addition to localization, when the difference technique is locked to the applied RMP toroidal phase, the differenced image is dominated by the applied *n*. All other *n*'s are predominantly removed from the resulting image.

## III. SXR IMAGING DATA

Images for each of the three discharges are shown in Fig. 4. The images are normalized to the integration time to take into account different time averaging (shaded window in Fig. 2) and effective spatial resolution mapped onto the scintillator via the pixel binning. Images acquired in the shaded region from Fig. 2 are averaged together while rejecting times near the sharp phase transition of the applied RMP. Each image has the vessel boundary and separatrix projected onto the image along the curved tangency plane from Fig. 1(b). These projections serve as a guide to interpret localized effects in the line-integrated image, e.g., the vessel boundary aligns with clipping of the inner wall. Imaged emission is strongest at the tangency plane and thus allows partial localization, but line-integral effects that wrap around toroidally are also present.

Each of the images in Fig. 4 shows the effects of different energy filters and plasma conditions. Figure 4(a) shows the image with a strong EUV emission gradient in the edge/SOL region. There is an even stronger emission at the vessel surface from lower energy plasma-wall interactions that is still passed by the filter due to the high intensity, which will not be the focus of the paper. Instead the processed image will focus on the boxed regions along the separatrix in the SOL. Figure 4(b) shows the image focused on the H-mode steep-gradient region. The flux surface in the middle of the SXR gradient, $\Psi n$ = 0.975, is projected onto the image along the tangency plane, which will be used later to orient localized perturbations. Figure 4(c) shows the image focused in the L-mode core. This image would normally have less signal than that of Fig. 4(b), but this L-mode plasma was seeded with neon, providing extra line and continuum radiation. The density of fully stripped neon, measured by charge-exchange recombination spectroscopy, was roughly 0.6% of the electron density. This L-mode plasma is limited on the inner wall. The $q=2/1$ ($\Psi n$ ∼ 0.7) surface is projected onto this image ($n=1$ RMP used), which is in the region of the L-mode SXR gradient.

The phase-locked differential images are shown in Fig. 5. The structure in each of these three images is dominated by applied toroidal mode number of the RMP. Figures 5(a) and 5(b) are dominated by $n=3$ perturbations and Fig. 5(c) is dominated by $n=1$ perturbations. In each case, the perturbation is normalized to a specific region of interest in the line-integrated images, described below.

The edge/SOL perturbation is shown in Fig. 5(a). The differenced image outside of the boxed regions is ignored to focus on the perturbations near the unperturbed separatrix. The perturbation shown is normalized to the maximum value in the line-integrated image in the regions plotted. The predominant structures are perturbations along the unperturbed separatrix with apparent lobes extending from the high-field side. The line-integral effect creates toroidal bands from these lobes.

The perturbation in the H-mode steep-gradient region is shown in Fig. 5(b) where the SXR gradient is largest just inside the separatrix. Helical banded structures wrap toroidally due to line-integral effects. The tangency points of these bands are found to be near the $\Psi n$ = 0.975 ± 0.0024 surface plotted along the tangency plane. A Monte Carlo uncertainty analysis incorporating variations in the magnetics, motional Stark effect (MSE) and pressure data was performed on this equilibrium to assess the location of this flux surface. This surface is between the *q* = 11/3 ($\Psi n=0.96$) and *q* = 12/3 ($\Psi n=0.85$). Given this radial localization and that these structures are $n=3$ from the RMP phase isolation, the perturbations are likely $m=11$ or $m=12$ structures, where *m* is the poloidal mode number. The perturbation shown is normalized to the maximum value in the line-integrated image along the projected $\Psi n$ = 0.975 surface.

In Fig. 6, we use a simplified model perturbation to show how a field-aligned, radially localized structure is expected to appear in the SXR images. The model sinusoidal perturbation uses $m=12$, and is Gaussian in $\Psi n$, centered at $\Psi n$ = 0.975. This perturbation has a generic kink-like structure (in contrast to an island-like perturbation), however it is not centered at a rational surface. The perturbation in RZ coordinates (where R is the major radius and Z is the vertical direction in meters) is shown in Fig. 6(a) and the perturbation in straight field-line coordinates (SFL) is shown in Fig. 6(b). The SFL coordinates are constructed such that the toroidal angle is equal the real-space toroidal angle. The line-integrated model image is shown in Fig. 6(c). The radially localized perturbation creates similar toroidal bands to Fig. 5(b) where the tangency points are located at the projected $\Psi n$ = 0.975 surface.

Figure 5(c) shows the perturbations in a low-rotation L-mode core plasma where islands have formed under the application of an *n* = 1 RMP. The 2/1 island is in the region of the SXR gradient, which is confirmed by flattening in the electron cyclotron emission (ECE) profile. Figure 7 shows several radially separated ECE channels. The coil current represents the same n = 1 RMP from Fig. 2. The island forms after 2700 ms and the phase of the island sampled by ECE is flipped between the X-point and O-point via the RMP. During the O-point phase, the channels measure a similar temperature, albeit with a slight peaking within the island. During the X-point phase, the channels show a gradient that is comparable to the non-RMP time. In the SXR image, the presence of a 2/1 island in the image is understood by the null along the projected 2/1 surface. This is the result of the island flattening centered at the rational surface. When the island phase is inverted and the image is subtracted, the largest perturbation is off the rational surface. The 3/1 surface is found toward the tail of the SXR gradient at $\Psi n$ ∼ 0.84 and may be partially detected in this image. This is further discussed in Sec. V. The perturbation shown in Fig. 5(c) is normalized to the maximum value along the projected 2/1 surface in the line-integrated image.

The line-integrated images can be inverted to the RZ plane with certain geometry assumptions and a regularization scheme. Here we have assumed that the 3D structures are field-aligned and that emission is constant along a field-line. A geometric transformation matrix is constructed that maps pixel sightlines to helical emission. This matrix can be inverted using Tikhonov regularization^{25,26} with a Laplacian smoothing condition. Because there are various sources of noise in the image, the inversion process creates artificial structures. Despite the addition of artifacts, subtracting the out-of-phase inversions largely retains the isolated 3D perturbations at one 2D toroidal angle. Inversions of the data from the core and steep-gradient regions will be shown in Sec. V. Inversions of the edge/SOL data are not shown because many of the sightlines intersect wall emission, breaking the helical symmetry assumption.

## IV. FORWARD MODELED RESPONSES

Plasma conditions are simulated with two different models: the prediction of the vacuum model is calculated by field-line integration of the vacuum fields with the MAFOT-TRIP3D code;^{27} and the prediction of a linear, resistive two-fluid model is calculated with the M3D-C1 code.^{22} The vacuum approach to simulate SXR radiation is to assume emission is constant along field lines and to follow field lines 100 revolutions in both positive and negative toroidal directions. These field lines are launched from the tangency plane of the SXR camera on grid with launching resolution of $(\Delta R,\Delta Z)$ = 0.5 mm. The minimum value in $\Psi n$ is recorded for each field line, $\Psi n,min(R,Z)$, which estimates the field line penetration. This provides a mapping of 1D emission in $\Psi n$ to a 2D RZ grid. Finally, a smoothing filter is applied to the simulated 2D SXR that approximates resolution effects from the finite pinhole diameter at the tangency plane. Here, 1.5 cm is assumed to be the lower limit based on the size of pixels projected to the tangency plane.

The simulated radiant power profiles from Fig. 3 mapped to a 2D $\Psi n,min$ grid are shown in Fig. 8 for the three discharges. The perturbed emission in Fig. 8(a) is predominantly from helical lobes extending from the separatrix, which has been previously imaged.^{9,10} The separatrix is predicted to be easily perturbed with a resonant point (X-point).^{28,29} The separatrix is then found to form stable and unstable manifolds. These manifolds are shown for the two RMP phases in green and pink. Remnant islands and intact islands are found deeper in the plasma for these simulations, but the perturbations are much smaller compared to the lobes.

Figure 7(b) simulates the higher energy filtered H-mode SXR profile from Fig. 3 (solid blue) with high contrast in the steep-gradient region, but it still shows a strong component of the lobe perturbation. This is indicative of the typically stochastic predictions of vacuum field line tracing. Despite filtering out lower energy emission, the field lines starting from within the lobes penetrate deep enough into the pedestal to create this perturbation.

Figure 8(c) shows the vacuum prediction for the $n=1$ perturbations in L-mode. Here, the structure is created from islands. In this simulation, the islands are separated enough such that there is no overlap indicating no stochasticity. The perturbation null aligns with the resonant surfaces, indicating islands flattening the emission profile. This core picture is in contrast to the edge perturbation where stochasticity destroys edge islands and leaves predominantly the lobe perturbation.

The M3D-C1 code is used to calculate the two-fluid MHD response in a linearized time-independent approach with Spitzer resistivity.^{22} The linearized approach allows significantly faster resistive two-fluid simulations, but breaks down where the perturbation gets large relative to the equilibrium scale length. This is the case for the edge/SOL lobes where the deviation of the perturbed field line trajectories from the unperturbed axisymmetric surfaces can be enormous. Therefore, the simulated response is not shown for the edge/SOL case. The linear approach does not treat the SOL temperatures well and the perturbation amplitude negligible.

Figure 9 shows the result of the two-fluid simulation for the steep-gradient region.

Figure 9(a) shows the calculated 2D temperature perturbation. The SXR perturbation is calculated in the same method as the 1D profiles from Fig. 3, but with the perturbed density and temperatures from the fluid calculation. The impurities are assumed to be proportional to electron density. The filtered emission response is shown in Fig. 9(b). The dominant perturbation shown is localized to the 11/3 surface. This is a region of high $\omega e,\u22a5$ and thus flow screening. The calculated resonant screening at this surface is $Bm,n,vac/Bm,n,pr=2.5$. The screening factors for nearby rational surfaces are listed in Table I. Islands are not observed in this edge region, as they would show a null at rational surface. Instead, this perturbation is dominantly a driven kink mode. This kinking structure is destabilized by the edge bootstrap current in the steep-gradient region. Islands are present deeper in the plasma with a much smaller perturbation amplitude than that of the driven kink mode.

Surface . | $Bm,n,vac/Bm,n,pr$ . |
---|---|

12/3 | 4.1 |

11/3 | 2.2 |

10/3 | 2.5 |

9/3 | 1.2 |

8/3 | 0.4 |

Surface . | $Bm,n,vac/Bm,n,pr$ . |
---|---|

12/3 | 4.1 |

11/3 | 2.2 |

10/3 | 2.5 |

9/3 | 1.2 |

8/3 | 0.4 |

The SXR perturbation from the two-fluid simulation for the core L-mode case is modeled using the same method as the vacuum perturbation. Specifically, field-line following of the linearized magnetic field solution from M3D-C1 is used to construct a mapping of 1D emission in $\Psi n$ to a 2D RZ grid at the tangency plane of the SXR camera. The results are similar to Fig. 8(c) and will be shown later Sec. V. This method is needed, as opposed to using the fluid solution, because the linearized fluid solution creates a perturbation around the rational surface, never resulting in a flattening of the temperature profile through the island. This is an inherent limitation to the linear solution. Future studies with nonlinear simulations are needed as the measured island width is larger than the linear layer width^{17} and the process of island formation and saturation is a nonlinear.

The validity of the assumption of linearity in the temperature response can be evaluated by using the perturbed temperatures to define the displacement of the magnetic surfaces. As discussed in Ref. 20, when these displacements imply that neighboring surfaces overlap, this indicates a breakdown of the assumption of linearity in the temperature response. At experimentally relevant I-coil currents (4 kA), it is found that this overlap condition implies a breakdown of linearity in nearly the entire region outside $\Psi n$ = 0.9. This indicates that a nonlinear calculation is required in order to obtain a rigorously valid quantitative estimate of the temperature perturbation in this region. Still, the geometric features of the linear response, including the radial localization and poloidal structure, appear to be adequately captured by the linear response.

## V. DATA/MODEL COMPARISON

The comparison of the modeling to the data for the edge/SOL region is shown in Fig. 10. Here, only vacuum modeling is compared. In both images, the calculated manifolds are plotted along the tangency plane and projected onto the image. This shows good agreement along the high-field side, although the lobe structure appears longer than the forward modeled image. This is possible given the uncertainty in modeling the emission at lower temperatures and where the impurities are not well characterized. On the low-field side, the perturbations do not align as well with the manifolds. They do show a similar oscillation along the unperturbed separatrix. Both images have been normalized to the maximum value inside the regions of interest along the separatrix. This allows for a better comparison to the perturbation strength given uncertainty in the overall sensitivity estimate of the camera system. This perturbation amplitude shows that it is reasonably well modeled by the vacuum approach in this edge region.

The comparison for the data in the steep-gradient region is shown in Fig. 11. Here, the linear two-fluid MHD and vacuum modeling is compared directly to the perturbed data image. The two-fluid MHD response agrees better to the measured perturbed image than the vacuum response. The vacuum response results from larger edge/SOL emission. In the region of $\Psi n$ = 0.975, the vacuum response leads to high stochasticity with smaller remnant islands. Each of the three perturbation images are normalized to the maximum value of the line-integrated image along the $\Psi n$ = 0.975 projected flux surface. This shows that while the structure of the two-fluid MHD matches well, the amplitude is approximately a factor of 4 too large. The perturbations in $ne$ and $Te$ scale linearly because this is a linear calculation. In principle, this is not the case for the modeled SXR. However, the modeled SXR emission is found to scale nearly linearly. This is a result of the competing effects of nonlinear line emission scaling with $Te$, high-pass filter effects and continuum scaling ($\u223cne2/Te$).^{30} Therefore, scaling down the applied perturbations in the two-fluid model by a factor of 4 results in a good SXR amplitude match. The need to reduce the RMP current may indicate that the linear modeling is partially breaking down in this case, where some nonlinear saturation mechanism is needed. It is also possible that some assumptions used in the model are not well-satisfied in the experiment, in particular, the perfectly conducting boundary conditions. However, the gross radial and poloidal structure predicted by the linear model appears to be accurate.

The inverted perturbations are shown in Fig. 12. The comparison here shows the measured perturbation structure is in better agreement with the two-fluid MHD simulation showing a driven kink mode than that of vacuum modeling. This also shows that the measured perturbation is radially outward of the predicted kink structure from the two-fluid MHD response. It is possible that the data shows a stronger 12/3 perturbation than the predicted 11/3 perturbation. Both of these surfaces are in the steep-gradient region and separated by approximately 1.5% of $\Psi n$. The strong emission source on the divertor shelf region is an artifact from noise with regularization process. Furthermore, no floor emission is seen in Fig. 4(b). The perturbations are normalized to the maximum value along the $\Psi n$ = 0.975 flux surface for better comparison. Similar to the line integrated images, the two-fluid prediction is too large compared to the measurement, where a reduction of the two-fluid model by a factor of 4 brings better agreement.

The comparison in the L-mode core region with *n* = 1 RMPs is shown in Fig. 13 for the line-integrated case and Fig. 14 for the RZ inverted case. The measured and modeled data for both line-integrated and inverted data are normalized to the maximum value along the projected 2/1 surface in the line-integrated images and the 2/1 surface in the RZ perturbations, respectively. In addition to the vacuum and two-fluid MHD modeling, an ad-hoc 2/1 island is modeled with approximately double the island width of the vacuum prediction. The ad-hoc island was constructed using a single-helicity model where the island represents a flattening in the poloidal flux.^{31} The 1D SXR profile from Fig. 3 is mapped to the 2D perturbed $\psi n,island(R,Z)$ to forward model the island perturbation. Each line-integrated perturbation image shows a null about the projected 2/1 surface. Additional structure in the data may be the result of a 3/1 island, but it does not show a similar null along the projected 3/1 surface.

The inverted measurement is restricted to a smaller region where signal level is high and little artifacts are present. Here, the null is extended along the 2/1 surface and a partial null is along the 3/1 surface. Radial cuts are taken through the perturbation along a contour of constant $\theta SFL$. These cuts are shown in Fig. 14(e) and better characterize the island widths in the data and models. Here, the 2/1 widths are estimated by the separation of the vertical dotted lines. In general, the vacuum-predicted island perturbations are not as large in width and amplitude as the measured perturbation. The widths estimated in Fig. 14(e) show the measured island width double that of the vacuum model. This indicates amplification of pitch-resonant fields at low rotation leading to larger islands. However, the linear two-fluid MHD perturbations show amplification from vacuum, but do not result in sufficient amplification to match experiment. The ad-hoc island structure was made to replicate the approximate size of the measured island to mimic the effect of resonant amplification.

## VI. DISCUSSION AND CONCLUSIONS

The data presented in this paper show the plasma response in three key regions through use of multiple high pass energy filters on a SXR camera and different plasma conditions. These results show that vacuum-based modeling may work in limited regions, but extended MHD is likely needed to describe measured responses deeper in the core and the pedestal.

First, observations show that the edge/SOL region is perturbed by RMPs to produce lobes extending from the unperturbed separatrix. This is reasonably well described by vacuum modeling. However, the results here do not preclude the presence of screening deeper in the plasma, specifically whether or not field lines deeper in the plasma make it into the lobes. Modeling suggests that screening reduces lobe length.^{21,32} Results from Mega-Ampere Spherical Tokamak (UKAEA-Culham) (MAST) indicate full resonant screening for $\Psi n$ < 0.96, allowing fields to penetrated outside that region, is needed to model the measured lobe lengths.^{33} The lobe lengths are difficult to estimate in images presented here given the line-integral effects and proper modeling of the emission source.

Second, in the steep-gradient region of the pedestal, an *n* = 3 kink-like response is measured with m = 11–12. Two-fluid MHD predicts partial screening and kinking in this region, in better agreement with the data than vacuum. However, linear simulations with M3D-C1 over-predict the perturbation amplitude, suggesting nonlinear effects may likely be important in determining the saturated kink amplitude, but not necessarily needed to predict the mode structure. Recent modeling indicates that the edge pressure gradient and bootstrap current drives the kinking, while the screening is dictated more by $\omega e,\u22a5$ flow.^{13} In principle, the measurements and modeling form a compatible picture with the edge/SOL lobe measurement: the pedestal region will always drive a peaked bootstrap current and $\omega e,\u22a5$ from gradient effects, so this kink structure is likely a strong feature of H-mode plasma response. Partial or full screening likely exists in the steep-gradient region, while vacuum fields are likely penetrating toward the bottom of the pedestal. However, the amount of screening just inside the pedestal is still an open question.

Finally, in low-rotating L-mode plasmas, islands open with the application of RMPs, but are amplified relative to the vacuum prediction. Imaged perturbations show a null about the rational surface, consistent with modeling. The measured width of the 2/1 island is approximately twice the vacuum prediction. The linear two-fluid solution predicts amplification from vacuum, but under predicts the magnitude. Since island penetration and saturation are nonlinear processes, they are not well predicted by the linear modeling. Additionally, higher rotating L-mode plasmas do not show evidence of islands, indicating the importance of the flow. This will be the subject of a future paper.

Overall, the linear resistive two-fluid simulations are in better agreement with experiment than those in vacuum calculations. However, the large H-mode edge perturbations predicted by the linear modeling and observed core L-mode islands in the experiment are almost certainly beyond the range of linear validity, and therefore linear calculations are not expected to yield quantitative agreement. Nonlinear M3D-C1 simulations of these regimes are being explored, but practical limits on the resolution in these simulations currently prevent the use of realistic values of dissipation. Additional considerations, such as transport sources and sinks over the long island-saturation timescales, must also be taken into account in nonlinear calculations. Significant progress is being made in these areas, and quantitative comparison of nonlinear modeling with measurements of non-axisymmetric tokamak equilibria will be a subject of future work.

## ACKNOWLEDGMENTS

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Awards DE-AC05-00OR22725, DE-AC02-09CH11466, and DE-FC02-04ER54698. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/D3D_DMP.