Use of reconstructed 3D VMEC equilibria to match effects of toroidally rotating discharges in DIII-D

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1 PAPER Use of reconstructed D VMEC equilibria to match effects of toroidally rotating discharges in DIII-D To cite this article: A. Wingen et al 0 Nucl. Fusion 0 Manuscript version: Accepted Manuscript Accepted Manuscript is the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an Accepted Manuscript watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors This Accepted Manuscript is 0 IAEA, Vienna. During the embargo period (the month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND.0 licence after the month embargo period. After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address... on /0/0 at :

2 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria to match e ects of toroidally rotating discharges in DIII-D A. Wingen, R.S. Wilcox, M.R. Cianciosa, S.K. Seal,E.A. Unterberg, J.M. Hanson, S.P. Hirshman,L.L.Lao,N.C. Logan, C. Paz-Soldan, M.W. Shafer Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA Columbia University, New York, New York, USA General Atomics, PO Box, San Diego, California, USA Princeton Plasma Physics Laboratory, Princeton, New Jersey, USA wingen@fusion.gat.com Abstract. A technique for tokamak equilibrium reconstructions is used for multiple DIII-D discharges, including L-mode and H-mode cases when weakly -D fields ( B/B ) are applied. The technique couples diagnostics to the non-linear, ideal MHD equilibrium solver VMEC, using the VFIT code, to find the most likely -D equilibrium based on a suite of measurements. It is demonstrated that VFIT can be used to find non-linear -D equilibria that are consistent with experimental measurements of the plasma response to very weak -D perturbations, as well as with -D profile measurements. Observations at DIII-D show that plasma rotation larger than 0 krad/s changes the relative phase between the applied -D fields and the measured plasma response. Discharges with low averaged rotation ( krad/s) and peaked rotation profiles ( krad/s) are reconstructed. Similarities and di erences to forward modeled VMEC equilibria, which do not include rotational e ects, are shown. Toroidal phase shifts of up to are found between the measured and forward modeled plasma responses at the highest values of rotation. The plasma response phases of reconstructed equilibra on the other hand match the measured ones. This is the first time VFIT has been used to reconstruct weakly -D tokamak equilibria. PACS numbers:..fa,..-s,..cv,..kj

3 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D. Introduction -D resonant magnetic perturbations (RMP) have been routinely applied in tokamaks to a ect the discharge in various ways, including the mitigation or suppression of edgelocalized modes (ELMs) [,, ]; substantial changes in particle and energy transport []; stability of -D MHD tokamak equilibria []; the imposition of torques on the plasma that may act either to brake or accelerate the plasma []. Therefore tokamak physics can no longer be considered purely axisymmetric; very often -D equilibria are formed. Examples include the formation of edge corrugation and lobes [] in the plasma edge or so called snakes in the plasma core []. To address the need for -D equilibria in tokamaks, a tool set is currently developed, that allows for fast turn-around -D equilibrium reconstruction for tokamaks. The tool set is based on the VFIT code [], which uses the VMEC code [] as equilibrium solver. A parallel version of VMEC, called PARVMEC [], has recently been released and is currently being coupled to VFIT. The final version of parallel VFIT is designed to run on high performance computer systems. The tool set also includes pre- and postprocessing of experimental data in order to generate the input deck for VFIT. The data is then transferred to the computing facility, processed there and the reconstructed -D equilibrium is returned for further analysis. VFIT minimizes the di erence, given by a total, between experimental data, measured by various diagnostics, and data calculated numerically via synthetic diagnostics from a VMEC equilibrium. The synthetic diagnostics currently available in VFIT include magnetic pick up coils, flux-loops and Rogowski-coils, Motional Stark E ect polarimeter (MSE), soft-x-ray (SXR) and more. Here we focus on small -D corrugations that deform the plasma edge. Diagnostics that are the most sensitive to such edge perturbations are the various types of magnetic pick up coils; we also include MSE to constrain plasma current densities. VFIT has successfully been applied to various machines [], stellarators like HSX and CTH, RFPs like RFX-Mod and MST, and is currently being installed at W-X. Here we apply it to the DIII-D tokamak [] for the first time. Using VMEC for a weakly -D ( B/B ), high performance tokamak configuration creates various challenges. In diverted geometry the plasma includes an X-point, which cannot be represented by the finite poloidal and toroidal Fourier mode spectrum, used in VMEC to represent nested flux surfaces; truncation of the total enclosed flux just inside the separatrix is required. H-mode operation is characterized by steep edge pressure gradients and large current drive in the plasma, especially a significant bootstrap current fraction close to the separatrix. VMEC was originally designed for stellarators, which usually do not show either feature. But, it can handle such a configuration, provided special measures are taken, like non-linear flux surface distribution. A procedure to obtain well converged and reliable VMEC equilibria for weakly -D high performance discharges in DIII-D is outlined in []. In this paper we will show that VFIT can be used to find non-linear -D

4 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D equilibria that are consistent with experimental measurements of the plasma response to very weak -D perturbations, as well as with -D profile measurements. Such a reconstructed -D equilibrium is a better choice for representing experimental reality than a forward modeled one and can even capture physical e ects that are not included in the formulation of the equilibrium model. Plasma rotation for example is not included in the VMEC equilibrium model, but has been shown to a ect perturbed H-mode equilibria in DIII-D. E.g. plasma rotation can help stabilizing resistive-wall-modes (RWM) []. In such cases it was found that -D equilibrium predictions from the MARS-K code recover magnetic plasma response measurements at DIII-D only at low to moderate rotation and if fast-ion e ects are neglected []. Experimentally it has been observed that plasma rotation can shift the relative phase between applied RMP and measured plasma response in H-mode discharges by scanning rotation from low, counter rotation of about krad/s up to krad/s co-rotation []. Here we will show that reconstructed equilibria match the experimental observations, while forward modeled VMEC equilibria do not show any change in the relative phase between applied RMP and measured plasma response for the considered discharges. The latter is clear, since VMEC solves the ideal MHD equations and thus does not include rotation. The paper is organized as follows. In the next section, the VFIT reconstruction is compared against EFIT in a purely axisymmetric case. In Sec. we describe the postprocessing applied to the data taken by the -D magnetics sets in DIII-D [] in order to use it in VFIT. In Sec., the fully -D reconstruction is applied to inner-wall limited L-mode discharges and we show that the reconstructed equilibria are quite similar to the forward modeled ones. Then we move on to H-mode discharges in Sec. which show the above mentioned e ect of plasma rotation. The paper finishes with the conclusion section.. Reconstruction of an axisymmetric equilibrium To run a -D equilibrium reconstruction using the VFIT code, an initial guess for the equilibrium based on VMEC is required. To obtain such a guess, we follow the procedure described in []. Starting point is a kinetic EFIT [, 0], which is an axisymmetric, unperturbed -D equilibrium reconstruction that includes fast-ion pressure and bootstrap current. Based on the kinetic EFIT, VMEC is run in fixed boundary mode, using the current density profile, in order to relax the q-profile. Then, VMEC is run in free boundary mode, using the q-profile from the previous step, in order to relax the current density profile and the boundary. The result is a -D, perturbed equilibrium, which is similar to the kinetic EFIT. Such a -D VMEC equilibrium is the initial guess, used in the VFIT reconstructions presented in this paper. Note that this first -D VMEC equilibrium matches the kinetic EFIT well (as demonstrated for a DIII-D discharge in Appendix A of []), but is not guaranteed to be consistent with experimental measurements. In order to compare modeled equilibria with experimental data, we need to compute

5 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D signals from the equilibrium via sets of synthetic diagnostics. A synthetic diagnostic is a numerical implementation of an existing diagnostic, which, based on a given equilibrium, returns a signal that is directly comparable with a measured signal. VFIT provides awidevarietyofsyntheticdiagnostics,includingmagneticfieldprobes,fluxloopsand MSE; the latter was recently added to VFIT. Those synthetic diagnostic sets have been configured to match the experimental measurements on DIII-D. VFIT then iterates on a number of M selected input parameters of the underlying VMEC equilibrium, the so-called reconstruction parameters P k (k =,...,M), in order to minimize the total [], defined by = NX [S i,model (P,...,P M ) S i,observed ], () i= i with N the total number of individual diagnostics and constraints, S the synthetic ( model ) and measured ( observed ) diagnostic signals and the measurement uncertainty. The final reconstructed equilibrium then matches the experimental measurements best within the M-dimensional parameter space. flux loops B r probes MSE B p probes Figure. (Color online) Illustration of the various diagnostics available for a VFIT reconstruction in DIII-D. The mesh shows the plasma boundary of discharge. Figure is an illustration of the diagnostic sets used for the VFIT reconstructions shown here. Four distinct sets of diagnostics are shown: (i) Poloidal magnetic field probes, or B p probes. These probes are small pick-up coils embedded in the vacuum vessel wall, which are aligned with the wall and perpendicular to the toroidal direction. Therefore, such coils measure the poloidal magnetic field component at the coil location. (ii) Radial magnetic field probes, or B r probes. These probes are essentially flux loops that cover a small area of the vacuum vessel wall. The normal vector of the loop is parallel to the normal vector of the vessel wall at the probe center. They measure

6 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D the radial magnetic flux going through the loop area and, since the area itself is constant, in the end the radial magnetic field component. Note that it is an area average of the radial field. (iii) Flux loops. These are large, circular loops and their normal vector is parallel to the vertical direction Z. Theymeasurethenetpoloidalmagneticfluxgoingthroughthe enclosed area. Since they are full toroidal loops, the measured flux is axisymmetric. These diagnostics are also essential in any EFIT reconstruction. (iv) MSE. Each channel measures the polarization angle of a laser, interacting with a neutral beam at a specific location inside the plasma. The diagnostic measures the ratio of the horizontal to the vertical magnetic field, or the so-called pitch angle of the magnetic field lines, at the measurement location. The sets include a total number of magnetics, B p -, B r -probes and flux loops, as well as MSE channels. Due to operational constraints, we use only MSE channels here. (d) (a) (b) (c) Figure. (Color online) VFIT reconstruction of discharge at ms compared to the initial kinetic EFIT: (a) poloidal cross-section at toroidal angle = 0; (b) q-profile; (c) parallel current density; (d) di erence of reconstructed poloidal field coil (F-coil) currents (black with uncertainty bars) from EFIT values (red). In order to test the VFIT code for DIII-D, it is applied to an axisymmetric case and compared with EFIT. Thereby we use the EFIT equilibrium as the initial guess

7 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D and starting point for the reconstruction. Note that EFIT solves the Grad-Shafranov equation in a toroidally axisymmetric system, while VMEC solves for force balance (rp = J ~ B) ~ on fully -D flux surfaces []. Although an axisymmetric case is reconstructed by VFIT here, VMEC is run in full -D, free boundary mode and would allow for -D flux surfaces. A radial grid resolution of surfaces is used. Convergence with respect to radial resolution has been checked. The result is shown in Fig., which shows the comparison between the VFIT reconstruction in black and EFIT in red for (a) the flux surfaces in a poloidal cross-section, (b) the q-profile, (c) the parallel current density profile and (d) the di erence I F = I F,V FIT I F,EFIT between the reconstructed F-coil currents and the original EFIT ones; the latter are within the uncertainty of the reconstructed currents. All four figures show good agreement relative to measurement uncertainties, as will be discussed in Fig.(a). We reconstruct the total enclosed toroidal flux, the poloidal field coil currents (F-coils), which shape the plasma, and the current density profile. For the latter, we use splines with fixed knots in the radial direction ( = normalized poloidal flux) while reconstructing the current density profile at each knot. In this particular case, the plasma shape is a so called double-null shape, which is close to up-down symmetry. Nevertheless, up-down symmetry is not enforced or exploited in VMEC or VFIT for this run. Both codes allow for up-down asymmetry, but the symmetry in the reconstructed plasma shape occurs naturally. We assume that the pressure profile is well reconstructed by EFIT and therefore keep it unchanged. Σ [%] χ V F-coils current density B p probes flux loops MSE constraints Figure. (Color online) VFIT reconstruction of : (a) of each diagnostic; (b) total e ectiveness of each diagnostic; (c) signal e ectiveness matrix. The matrix shows the correlation between the reconstructed parameters (vertical axis: plasma volume (V), poloidal field coils (F-coils), current density profile) and the diagnostic signals (horizontal axis) To verify the quality of the reconstruction, the signal e ectiveness matrix is (a) (b) (c)

8 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D computed and cross-referenced with the individual i, i =,...,N, of each individual diagnostic. We define the average h i = P i /N and its standard deviation = ( P ( i h i) /N ) /. Figure (a) shows i and highlights individual diagnostics that are either borderline with h i < i apple h i + (yellow marker) or poorly reconstructed with i > h i + (red marker). As can be seen, the vast majority of the diagnostics is well reconstructed with i appleh i (green marker). In the present case we find h i =0. for the VFIT reconstruction compared to h i =0. for EFIT. Figure (b) shows the total e ectiveness of each diagnostic, which gives how much each diagnostic contributes to the reconstruction. It basically represents the sensitivity of an individual diagnostic on the reconstruction result. It is computed by summing the columns of the e ectiveness matrix, shown in (c). The matrix is the cross-correlation between the diagnostics and the reconstruction parameters, and it shows how much influence each diagnostic has on each reconstructed parameter. Note that each row is normalized to. We find that MSE and several B p probes influence the current density profile, while the flux loops as well as another subset of the B p probes determine the F-coil currents, the shaping of the plasma. The additional constraints, imposed on the reconstruction, include the vessel wall as a hard limiter, to ensure that the plasma does not extend outside the wall. In this case the plasma does not touch the wall, so the limiter does not have any e ect on the reconstruction. Furthermore, we constrain the total enclosed plasma current within a small uncertainty of a few percent, based on experimental values, and therefore the resulting poloidal magnetic field. The toroidal magnetic field is kept fixed at the experimental value. By keeping the fields constrained, the reconstructed total enclosed flux essentially represents the plasma volume. So, the plasma current constraint is strongly correlated to the plasma volume reconstruction. To summarize, we can successfully reconstruct the axisymmetric shape and current density of a DIII-D diverted H-mode discharge, using the VFIT code, in very good agreement with EFIT. This is achieved by using the configured synthetic diagnostics, magnetics and MSE, adopting the pressure profile from the kinetic EFIT and constraining major 0-D parameters, like total plasma current and toroidal magnetic field, to the experimental values.. Plasma response measured by -D magnetic diagnostics In the next step, further magnetic diagnostics, the so called -D magnetics [], are added to the suite of measurements used in the reconstruction. The new set of magnetic sensors is used to reconstruct the currents in DIII-D s RMP coils, the C-coils and/or the I-coils, in order to obtain a fully -D equilibrium. Each sensor in this set is a pair of two di erenced magnetic probes. The latter are part of the either the B p -orb r -probe sets. The di erencing removes the majority of the n =0contributiontothemagnetic field, such that non-axisymmetric field strengths relative to the main field on the order of B/B can be measured. Both radial, B r,andpoloidal,b p,componentsof the field are measured using total probes.

9 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D Figure. (Color online) (bottom) Time trace of the raw (black) and processed (red) measured data of the di erenced poloidal magnetic field sensor MPIDA0, aligned with the rotating RMP coil current, here represented by the I-coil current IU (top). Data taken from H-mode discharge 0. Starting with the raw data from the -D magnetics, first the vacuum coupling from each of the applied field coils is subtracted to isolate the plasma response, based on regular calibrations of the vacuum fields. Next, the mean signal over one period of -D coil flips is subtracted, in order to eliminate residual coupling to the magnetic sensors from any fields that are not included in the applied spectrum, especially residual n =0 fields. This also eliminates the e ect of the slow drift on the digitizer channels [], which can be significant over longer time scales (on the order of seconds). Figure shows the initial and final stage of the data processing in a time trace of a representative poloidal field sensor. The same processing is done for all sensors. Once the measured data has been processed, we invert the hardware di erencing (remember that only the di erence between two probes is measured), so that the measurements may be compared directly against the synthetic diagnostic in VMEC. This is done by a Pseudo-Inverse method using the singular value decomposition of the corresponding linear di erencing operator. Finally a mode structure, consisting of multiple toroidal and poloidal modes is fitted to the data using least squares minimization. Here again a singular value decomposition of the final linear combination operator for the least squares fit is used to select the most prominent modes. By evaluating the linear combination of the most prominent modes, the magnetic plasma response fields can be extrapolated in the approximate plane of the sensors (the outboard vessel wall). For the actual reconstruction, only the fields at the point of each probe are used to compare against the synthetic diagnostic in VMEC. The synthetic diagnostic is apointdiagnostic. Duetothelargecross-sectionalareaoftheB r saddle loops, their signals are not presently used in the reconstruction. Therefore, we exclude the B r probes from the -D magnetics set in the following and focus on the B p probes only.. Reconstruction of -D inner-wall-limited L-mode equilibria First we apply the -D reconstruction to inner-wall-limited (IWL) L-mode discharges. Two discharges with significantly di erent plasma rotation, but otherwise rather similar

10 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D equilibria are chosen. Specifically, both discharges have the same shape and nonaxisymmetric magnetic perturbation, a. ka n = perturbationusingthec-coils. The injected power by ECH + NBI is di erent though (. MW for and. MW for ), especially since the neutral beams are used to provide torque. In, the plasma rotation profile, here given by the Carbon rotation c,isratherflat across the minor radius, represented by normalized poloidal flux n,andhardlyexceeds krad/s, as shown in Fig. by the blue curve. Discharge shows a peaked rotation profile with C > krad/s in the core, red curve. Ω C [krad/s] ψ n Figure. (Color online) Rotation profiles for inner-wall-limited discharges at 00 ms (red) and at ms (blue). As mentioned in the introduction, we want to show that we can include certain e ects of plasma rotation on the magnetic plasma response to applied RMP fields in VMEC equilibria, by using VFIT reconstruction, although rotation is not a feature currently included in VMEC. The e ect we focus on here is that a shift in the plasma response s relative phase to applied RMPs is observed at large rotation in H-mode discharges. Such an e ect is not seen in the considered L-mode discharges. For an axisymmetric equilibrium, toroidal rotation mainly a ects the pressure profile, because the mass is shifted by the inertial force of the plasma rotation. If the toroidal Mach number is small enough, the pressure profile with the rotation is almost identical to the profile without the rotation. The Mach numbers are small for both cases here. In the red case the Mach number varies radially between 0. and 0., while in the blue case it varies between 0.0 and 0.. We can represent the applied RMP coil current by a single Fourier mode I coil = A cos(n c), () with A =. kaandn =. Similarly,weFourierdecomposethepoloidalmagnetic plasma response field, as measured by the MPIDM toroidal array, a set of di erenced -D magnetics probes, which are distributed toroidally along the low-fieldside (LFS) midplane. In the decomposition, we focus on the n =modeonlyand obtain the Fourier phase p of the probe signals. We then define B = c p () as the relative phase di erence between the applied field and the measured plasma response. The C-coil currents are periodically flipped every 0 ms in and every

11 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D 0 ms in. Figure shows that the relative plasma response phase is identical for both discharges within the time periods of the same C-coil current phase. Therefore, we can conclude that plasma rotation does not a ect the plasma response phase in these L-mode discharges. B [deg] t [s] Figure. (Color online) Time trace of the relative phase between plasma response and applied RMPs for inner-wall-limited discharges at 00 ms (red) and at ms (blue). Note that the applied RMP flips periodically every 0 ms in and every 0 ms in. Nevertheless, we reconstruct the C-coil currents in both discharges with VFIT using the -D magnetics set and compare the reconstructed equilibria with forward modeled VMEC equilibria. The latter are obtained using the experimental coil currents in the C-coils, as well as the initial kinetic EFITs. Note that neither the pressure nor the current profiles are reconstructed in contrast to the previous -D case. Figure (a) shows a poloidal cross-section of the EFIT in red, the forward modeled VMEC (labeled VMEC ) in black and the reconstructed VMEC (labeled VFIT ) in blue for the high rotation case. VMEC and EFIT match perfectly, except in the very core region. Note that a nonlinear flux surface distribution is used in VMEC that packs % of surfaces in the edge region of > 0.. Therefore surfaces are sparse close to the magnetic axis, which reduces resolution and therefore accuracy in the core. VFIT di ers from the forward modeled case only in the applied RMP field, so we expect the flux surface corrugations in the edge to be the primary di erence. Figure (b) shows the corrugation of the LCFS for EFIT, VMEC and VFIT. EFIT is axisymmetric, so it shows no corrugation at all. The corrugation amplitude of VMEC is about twice as large as VFIT, but the shape and phase is the same. The figure shows the poloidal variation, but since the flux surface corrugations are helical, with the same toroidal and poloidal mode numbers in both discharges, also the toroidal variation is in phase. The simulations use surfaces and the nonlinear surface distribution, introduced in [], which ensures convergence of the corrugations with respect to radial resolution. The figure confirms that, as expected, the relative plasma response phase in the reconstructed equilibrium is identical to the one in the forward modeled equilibrium; the latter does not know anything about plasma rotation and therefore always represents a case of no rotation. The same result is found for the low rotation case.

12 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D (a) (b) Top HFS Bottom corrugation of s = surface Figure. (Color online) VMEC only (black), EFIT (red) and VFIT reconstruction (blue) for inner-wall-limited discharge : (a) poloidal cross-section at = 0; (b) LCFS corrugation.. Reconstruction of D diverted H-mode equilibria From IWL L-mode discharges we now move on to diverted H-mode discharges. Experimental observations show that the relative phase of the plasma response with respect to the applied filed changes with increasing plasma rotation. In this section, we analyze and reconstruct three discharges with di erent rotation in order to show that the reconstructed equilibria account for the proper relative phase shift between the applied perturbation and the plasma response, while forward modeled VMEC equilibria do not... VMEC forward modeling As a reference point, all three discharges are modeled with VMEC only, applying the experimental RMP coil currents, a ka n =evenparityrmpgeneratedbythei-coils. Note that the C-coils are used for error field correction during the experiment. So we neglect the intrinsic error field as well as the applied C-coil field for these discharges, because the residual field of error field + C-coil field is small compared to the applied I-coil field. Figure (a) shows the poloidal cross-section of the discharge, the case with the highest rotation of about krad/s. It is a lower single null diverted plasma. The edge shows small corrugation due to the applied RMP. The black dot at R =. m and Z = 0 m in the figure gives the location of the MPIDM toroidal array. Note that the signals measured by this array include only the n>0componentsofthepoloidal magnetic field. In addition, the vacuum coupling is removed, so that the measured field is only the plasma response field. Using synthetic diagnostics, the signals in the array can be modeled based on the VMEC equilibrium. Figure (b) shows the synthetic signals for all three discharges as dots at each toroidal sensor location. Using Fourier analysis, the n = modeis LFS

13 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D (a) (b) Figure. (Color online) (a) VMEC poloidal cross-section at = 0 for H-mode discharge. The black circle shows the location of the MPIDM toroidal array. (b) Synthetic poloidal plasma response field from VMEC at the location of the MPIDM array for multiple discharges. The markers give the modeled field at each toroidal location, while the solid lines give the toroidal Fourier mode n = only. extracted, which is then given by the solid lines respectively. As expected, the synthetic signals are dominated by the n =componentandareinphaseforallthreedischarges, because VMEC does not include plasma rotation... Reconstruction Now, we reconstruct the coil currents in the I-coils based on the mode fitted plasma response, obtained from Sec.. Note that all other VMEC parameter and profiles are kept fixed. The purpose is to alter the I-coil currents in the model and allow them to deviate from the applied values so that the resulting equilibrium matches the measured plasma response the best. This means, the measured and synthetic signals of the D magnetic diagnostics, as discussed in Sec., are used to search for the minimum in.thei-coilcurrentsinthemodelthenmaynolongeragreewiththeexperimentally applied coil currents. There are individual I-coils, in the upper row and in the lower row. During the experiment, the upper and lower rows are kept in a fixed relation, called parity. For the discharges discussed here, the parity is even, which means a coil in the lower row has the same current as its counterpart (same toroidal location) in the upper row. This results in predominantly pitch-aligned perturbations. The I-coils in the VFIT reconstruction are constrained to have the same parity, which reduces the number of reconstruction parameters to. In addition, we constrain the I-coil currents within a row to have zero average current. This ensures that the reconstructed I-coil currents do not add any n =0componenttothemagneticfieldandthereforealterthe axisymmetric part of the equilibrium; the latter is intentionally kept fixed based on the good agreement found in Sec.. Any changes to the axisymmetric part should be done by manipulating the poloidal field coil currents (F-coils), -D profiles or 0-D parameters,

14 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D like in Sec.. The parameter space for the reconstruction with VFIT is therefore dimensional. I δ/ Figure. (Color online) Variation of I-coil current. The red dashed line gives the n = referencesinewavetodefine for the current distribution used in the simulation (black dots). To demonstrate the operation of VFIT and its e ectiveness, we apply a simplified brute force model in parallel. For this brute force approach the I-coil currents are parameterized within the upper row as an n =sinewavewithanamplitudei and a phase I coil ( )=I sin( + ), () as shown in Fig.. Note that the I-coil current are set in the same way during the experiment. The red dashed line is a =0,I =kareferencecase. Theblackmarker give the experimental coil currents and therefore the configuration used in the VMEC forward modeling cases. I and represent a -D parameter space of possible I-coil configurations with the additional constraint of n =. Note that such a constraint is not used in the VFIT reconstruction later. For each combination of I and a VMEC equilibrium is computed, and the synthetic diagnostic for the -D magnetics is evaluated. A total is obtained, using Eq. (). By scanning I and through hundreds of individual VMEC runs, a -D -map is constructed that shows a minimum at the best possible match between the model and the measured data. Figure shows the -map for discharge 0, the low rotation case. We can identify a clear minimum at about I =kaand =.. The minimum point is marked by a yellow circle in the figure. For comparison, the experimental configuration, used in the VMEC forward model, is marked as an orange star and labeled VMEC only. The obtained phase is a close match to the experimental setup. This is expected, because the discharge has a low rotation of about - krad/s. The amplitude I is reduced by / though. Such a reduction in perturbation amplitude was already observed in the L-mode cases. If we now run VFIT, first we find that the reconstructed individual

15 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D χ Map VFIT VMEC only Figure. (Color online) -map for all Bp probes for discharge 0 at ms. The yellow circle marks the minimum of the map. The red triangle marks the VFIT reconstruction result and the orange star, labeled VMEC only, marks the initial VMEC configuration using the experimental coil currents. The map is constructed out of x individual VMEC equilibria. I-coil currents form a clear n =dominatedconfigurationandthereforenativelymatch the constraint of our reduced model. This is not trivial, since all reconstructed I-coil currents could potentially form any structure with toroidal mode numbers n =,, or or any linear combination of these. Secondly, we find that VFIT reconstructs the I-coil currents to almost the same result as found by the reduced model s -map. The reconstructed configuration is marked in Fig by a red triangle and labeled VFIT. This confirms that VFIT finds the minimum value within its -D parameter space in the same way as the -D brute force model. Δ B (deg) Data ms VMEC VFIT Ω (krad/s) φ,ρ=0. Figure. (Color online) Relative phase of the measured poloidal plasma response field for multiple discharges each at several di erent times (black squares). The blue squares are the measured data points at the time used in the simulations. The blue circles are the VFIT reconstruction results, while the red circles show the respective VMEC only result. The dashed lines indicate the trend for VMEC (red) and VFIT + Data (blue). As a final step, we apply VFIT in the same -D reconstruction parameter

16 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D configuration to the other H-mode discharges. The results are shown in Fig.. The figure shows the relative phase of the plasma response poloidal magnetic field B in dependence of the plasma rotation, with B = p, asineq.()butwith from Eq. (). The black squares mark the experimental observations across the di erent discharges, each at various time slices. Note that the time slices at ms, which are used in the modeling, are highlighted as blue squares. The experimental data clearly shows a change of the the plasma response phase with increasing rotation. The red circles show the forward modeled VMEC equilibria; their phase is independent of rotation. The blue circles show the VFIT reconstructed equilibria. The latter agree well with the experimental observations. This shows that the reconstructed equilibria now reflect the change of plasma response with increasing plasma rotation, although VMEC, the equilibrium solver itself, does not include toroidal rotation in the simulation.. Conclusions VFIT was successfully set up, tested and applied for -D equilibrium reconstruction at DIII-D for the first time. We benchmarked an axisymmetric case against EFIT and found very good agreement. Both codes find the same equilibrium and converge to a quite similar.toreconstructtheaxisymmetric,n =0partofa-Dequilibrium,the extensive set of -D magnetics as well as MSE are used. To reconstruct the -D, n>0 part, the -D magnetics are used, which only measure the di erence of two probes. A method of inverting the subtraction in the -D magnetics and fitting the signals to a predefined mode spectrum is presented. This results in the measured plasma response field at the vessel wall. VFIT then compares the processed data to a synthetic diagnostic that returns the magnetic plasma response field at any single point in space. We found that the -D B r probes show significant di erences between experiment and synthetic modeling and require further studies. Using the -D magnetics allowed us to reconstruct the n = perturbationsin IWL L-mode discharges, which represent the most simple DIII-D case possible to test the -D reconstruction on. It was found that the reconstructed surface corrugations agree in shape and phase with forward modeling, but show a reduction in amplitude. The reduced amplitude is most likely due to a systematic mismatch of HFS probes. It is observed that the synthetic VMEC equilibrium signals for the -D magnetics systematically overestimate measured HFS B p -probe signals by a factor of about. A detailed comparison between modeled and measured -D magnetics signals on the HFS can be found in []. Keep in mind that these probes measure a signal of only a few Gauss within a - Tesla field, which means that requirements to precision are quite high for the diagnostic, but also numerically. Furthermore magnetic diagnostics cannot measure directly the corrugation of weakly perturbed flux surfaces. Detailed measurements of the edge displacements was performed on MAST and other machines [], which can then be compared with VMEC D equilibria. Thomson scattering measurements could be used in DIII-D to improve on the D reconstruction, by constraining the surface

17 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D corrugations better. In a final step, we extended the -D reconstruction to generic DIII-D cases and reconstructed the n = perturbation for several H-mode discharges with di erent plasma rotation. The significant finding was that VFIT finds non-linear -D equilibria that are consistent with experimental measurements of the plasma response to applied weak -D perturbations, as well as with -D profile measurements. The e ect, that rotation shifts the relative phase between applied RMP and plasma response, can be included in VMEC equilibria by using reconstruction. This shows that reconstructed VMEC equilibria are a better choice for representing experimental reality than VMEC forward modeled ones and can even account for a physics e ect, that is not native part of the equilibrium model itself. A similar approach is used in well established -D EFIT equilibrium reconstructions, where the reconstructed poloidal field coil (F-coil) currents often don t quite match the measured coil currents for the benefit that other diagnostics are matched better. Here in these cases of -D reconstruction, the reconstructed I- coil currents mismatch the experimental ones for the benefit of matching the plasma response diagnostics. Acknowledgments This material is based upon work supported by the U.S. Department of Energy, O ce of Science, O ce of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE O ce of Science user facility under awards, DE-AC0-00OR, DE-FG0-0ER, DE-FC0-0ER and DE-AC0-0CH. DIII-D data shown in this paper can be obtained in digital format by following the links at DMP. This work used resources of the Oak Ridge Leadership Computing Facility. References [] T E Evans, R A Moyer, P R Thomas, J G Watkins, T H Osborne, J A Boedo, E J Doyle, M E Fenstermacher, K H Finken, R J Groebner, M Groth, J H Harris, R J La Haye, C J Lasnier, S Masuzaki, N Ohyabu, D G Pretty, T L Rhodes, H Reimerdes, D L Rudakov, M J Scha er, G Wang, and L Zeng. Suppression of Large Edge-Localized Modes in High-Confinement DIII-D Plasmas with a Stochastic Magnetic Boundary. Physical Review Letters, ():, 00. [] Y Liang, H R Koslowski, P R Thomas, E Nardon, B Alper, P Andrew, Y Andrew, G Arnoux, Y Baranov, M Becoulet, M Beurskens, T Biewer, M Bigi, K Crombe, E De La Luna, P de Vries, W Fundamenski, S Gerasimov, C Giroud, M P Gryaznevich, N Hawkes, S Hotchin, D Howell, S Jachmich, V Kiptily, L Moreira, V Parail, S D Pinches, E Rachlew, and O Zimmermann. Active Control of Type-I Edge-Localized Modes with n = Perturbation Fields in the JET Tokamak. Physical Review Letters, ():, 00. [] W Suttrop, T Eich, J C Fuchs, S Günter, A Janzer, A Herrmann, A Kallenbach, P T Lang, T Lunt, M Maraschek, R M McDermott, A Mlynek, T Pütterich, M Rott, T Vierle, E Wolfrum, Q Yu, I Zammuto, and H Zohm. First Observation of Edge Localized Modes Mitigation with Resonant and Nonresonant Magnetic Perturbations in ASDEX Upgrade. Physical Review Letters, ():0, 0.

18 Page of AUTHOR SUBMITTED MANUSCRIPT - NF-.R 0 Use of reconstructed -D VMEC equilibria in DIII-D [] O Schmitz, T E Evans, M E Fenstermacher, H Frerichs, M W Jakubowski, M J Scha er, A Wingen, W P West, N H Brooks, K H Burrell, J S degrassie, Y Feng, K H Finken, P Gohil, M Groth, I Joseph, C J Lasnier, M Lehnen, A W Leonard, S Mordijck, R A Moyer, A Nicolai, T H Osborne, D Reiter, U Samm, K H Spatschek, H Stoschus, B Unterberg, E A Unterberg, J G Watkins, R Wolf, the DIII-D, and TEXTOR Teams. Aspects of three dimensional transport for ELM control experiments in ITER-similar shape plasmas at low collisionality in DIII-D. Plasma Phys. Control. Fusion, ():, December 00. [] M J Lanctot, H Reimerdes, A M Garofalo, M S Chu, Y Q Liu, E J Strait, G L Jackson, R J La Haye, M Okabayashi, T H Osborne, and M J Scha er. Validation of the linear ideal magnetohydrodynamic model of three-dimensional tokamak equilibria. Physics of Plasmas, ():00, 0. [] A M Garofalo, K H Burrell, J C DeBoo, J S degrassie, G L Jackson, M Lanctot, H Reimerdes, M J Scha er, W M Solomon, and E J Strait. Observation of Plasma Rotation Driven by Static Nonaxisymmetric Magnetic Fields in a Tokamak. Physical Review Letters, ():0, 00. [] A Kirk, J Harrison, Yueqiang Liu, E Nardon, I T Chapman, P Denner, and the MAST team. Observation of Lobes near the X Point in Resonant Magnetic Perturbation Experiments on MAST. Phys. Rev. Lett., :0, 0. [] W A Cooper, I T Chapman, O Schmitz, A D Turnbull, B J Tobias, E A Lazarus, F Turco, M J Lanctot, T E Evans, J P Graves, D Brunetti, D Pfe erlé, H Reimerdes, O Sauter, F D Halpern, T M Tran, S Coda, B P Duval, B Labit, A Pochelon, M R Turnyanskiy, L Lao, T C Luce, R Buttery, J R Ferron, E M Hollmann, C C Petty, M van Zeeland, M E Fenstermacher, J M Hanson, and H Lütjens. Bifurcated helical core equilibrium states in tokamaks. Nuclear Fusion, ():0, 0. [] James D Hanson, Steven P Hirshman, Stephen F Knowlton, Lang L Lao, Edward A Lazarus, and John M Shields. VFIT: a code for three-dimensional equilibrium reconstruction. Nuclear Fusion, ():0, 00. [] S P Hirshman. Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Physics of Fluids, ():,. [] S K Seal, Steven P Hirshman, M Cianciosa, Andreas Wingen, E A Unterberg, and R S Wilcox. Development of the PARVMEC Code for Rapid Analysis of D MHD Equilibrium. In th Annual Meeting of the APS Division of Plasma Physics, page JP.00, Savannah, GA, 0. [] J D Hanson, D T Anderson, M Cianciosa, P Franz, J H Harris, G H Hartwell, S P Hirshman, S F Knowlton, L L Lao, E A Lazarus, L Marrelli, D A Maurer, J C Schmitt, A C Sontag, B A Stevenson, and D Terranova. Non-axisymmetric equilibrium reconstruction for stellarators, reversed field pinches and tokamaks. Nuclear Fusion, ():0, 0. [] J L Luxon. A design retrospective of the DIII-D tokamak. Nuclear Fusion, ():, May 00. [] A Wingen, N M Ferraro, M W Shafer, E A Unterberg, J M Canik, T E Evans, D L Hillis, S P Hirshman, S K Seal, P B Snyder, and A C Sontag. Connection between plasma response and resonant magnetic perturbation (RMP) edge localized mode (ELM) suppression in DIII-D. Plasma Phys. Control. Fusion, ():0, October 0. [] H Reimerdes, J W Berkery, M J Lanctot, A M Garofalo, J M Hanson, Y In, M Okabayashi, S A Sabbagh, and E J Strait. Evidence for the Importance of Trapped Particle Resonances for Resistive Wall Mode Stability in High Beta Tokamak Plasmas. Phys. Rev. Lett., :, 0. [] F Turco, A D Turnbull, J M Hanson, and G A Navratil. Modeling of fast neutral-beam-generated ions and rotation e ects on RWM stability in DIII-D plasmas. Nuclear Fusion, ():, September 0. [] C Paz-Soldan, N C Logan, M J Lanctot, J M Hanson, J D King, R J La Haye, R Nazikian, J K Park, and E J Strait. Decoupled recovery of energy and momentum with correction of n = error fields. Nuclear Fusion, ():0, August 0.

19 AUTHOR SUBMITTED MANUSCRIPT - NF-.R Page of 0 Use of reconstructed -D VMEC equilibria in DIII-D [] J D King, E J Strait, R L Boivin, D Taussig, M G Watkins, J M Hanson, N C Logan, C Paz-Soldan, D C Pace, D Shiraki, M J Lanctot, R J La Haye, L L Lao, D J Battaglia, A C Sontag, S R Haskey, and J G Bak. An upgrade of the magnetic diagnostic system of the DIII-D tokamak for non-axisymmetric measurements. Review of Scientific Instruments, ():0, 0. [] L L Lao, H St John, R D Stambaugh, A G Kellman, and W Pfei er. Reconstruction of current profile parameters and plasma shapes in tokamaks. Nuclear Fusion, ():, November. [0] L L Lao, J R Ferron, R J Groebner, W Howl, H St John, E J Strait, and T S Taylor. Equilibrium analysis of current profiles in tokamaks. Nuclear Fusion, ():, June 0. [] C Paz-Soldan, N C Logan, S R Haskey, R Nazikian, E J Strait, X Chen, N M Ferraro, J D King, B C Lyons, and J K Park. Equilibrium drives of the low and high field side n = plasma response and impact on global confinement. Nuclear Fusion, ():00, May 0. [] I T Chapman, M Becoulet, T Bird, J Canik, M Cianciosa, W A Cooper, T Evans, N Ferraro, C Fuchs, M Gryaznevich, Y Gribov, C Ham, J Hanson, G Huijsmans, A Kirk, S Lazerson, Y Liang, I Lupelli, R A Moyer, C Nührenberg, F Orain, D Orlov, W Suttrop, D Yadykin, the ASDEX Upgrade, DIII-D, MAST and NSTX Teams, and EFDA-JET Contributors. Three-dimensional distortions of the tokamak plasma boundary: boundary displacements in the presence of resonant magnetic perturbations. Nuclear Fusion, ():00, 0.