MHD stability of negative triangularity DIII-D plasmas

Negative triangularity (NT) experiments in DIII-D point to an emergent reactor scenario free of sawteeth, endowed with benign, nondisruptive n = 2 tearing modes, which experience q min ⩾ 1 similar to the positive triangularity hybrid scenario. Plasmas exhibiting this behavior attain 3$?> βN>3 , high enough to reconsider long held views of NT stability. Ideal MHD and tearing stability analysis of well-diagnosed equilibrium reconstructions of experimental hybrid-like plasmas predict that among shape parameters, MHD stability limits are only sensitive to average triangularity. Operation is predicted to be possible at βN relevant to upcoming tokamaks and commercial NT reactors.


Introduction
Surging interest in negative triangularity (NT) plasmas is owed to their intrinsic power handling advantages [1,2]. Edge transport barriers are rendered unnecessary to access H-mode similar confinement, eliminating the issue of edge localized mode mitigation, likely reducing undesirable impurity retention, and removing concerns regarding operation near the L-H power threshold [3][4][5]. Core confinement improvement is thought to be due to reduction of trapped electron mode driven turbulent fluctuations [5].
Though NT boasts strong power handling advantages, the question of MHD stability in performing discharges merits Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
investigation. Initial stability analysis of NT showed lower ideal MHD limits compared to positive triangularity (PT) plasmas at similar parameters [6]. However, modeling has shown that NT can be stable to global kink modes without wall stabilization at β N > 3, where β N = βt ( Ip aB ) [7]. Here, I p is the plasma current in MA, a is the minor radius in m, B is the vacuum toroidal field in T, and β t is the ratio of plasma pressure to magnetic field pressure in %. Operation at high β N is necessary for fusion power production and reduces external current drive requirements through raised bootstrap current fraction f bs . DIII-D NT experiments demonstrating high β N [8], taken with ideal MHD modeling of NT TCV similar equilibria and previous DIII-D plasmas, suggest these limits may be high enough to develop reactor scenarios optimized for substantial β N [9]. Recent diverted NT plasmas operating at β N ∼ 3 have observed significant tearing modes (TMs) for the first time in the DIII-D facility's experience with the regime. Benign n = 2 TMs are observed, which are found to minimally impact confinement quality. One plasma suffered major pressure loss and experienced profile flattening with the onset of an n = 1 TM. No TM laden plasma disrupted. Cases with benign modes exhibit a transition from being burdened with sawteeth and fishbones to a state dominated by n = 2 activity. The minimum safety factor q min in this state is near or just above 1, inviting comparison to the PT H-mode hybrid scenario [10].
Optimization of this NT hybrid-like state, termed here NTH-like, would represent a step toward establishing a strong NT reactor scenario candidate. In that vein, we present studies of n = 1 kink limits and their dependence on plasma shape, with the intent to understand and raise said limits. These studies were done with the DCON [11], GATO [12], and PEST3 [13] codes, predicting no ideal MHD limit cost for increasing plasma volume or for varying shape parameters aside from average triangularity. Furthermore, tearing stability modeling of the 2/1 mode, which is most important for high performance scenarios to avoid, was performed with RDCON and PEST3. This study predicts weak 2/1 TM stability variation as a function of plasma shape parameters in this regime.
This paper is organized as follows: section 2 details the MHD phenomenology of hybrid-like NT plasmas; section 3 presents ideal n = 1 and 2/1 tearing stability modeling of these plasmas; section 4 describes studies of ideal kink MHD limits and tearing stability as functions of shape parameters; section 5 summarizes these studies and presents conclusions.

Phenomenology
Saturated TMs in diverted NT plasmas have been observed in DIII-D experiments at high pressure (figure 1). TMs of toroidal mode number n = 2 onset above β N ∼ 2.8, suggesting tearing limits in this vicinity [14]. Despite these modes, NT plasmas attain β N > 3 for over five energy confinement times (τ E ∼ 70 ms), being limited by loss of neutral beam power or the onset of a confinement-reducing 2/1 TM. The period during which these plasmas host n = 2 TMs resembles the PT hybrid scenario (figure 2) in that sawteeth are absent and fishbones vanish when the n = 2 instability sets on. This suggests q min is raised initially by heating to around unity, removing sawteeth, then rises above unity with the TM. Demonstration of anomalous current diffusion, as has been done for hybrid scenario plasmas, is deferred until this state is achieved in stationary conditions. These plasmas are part of a series of diverted NT experiments in DIII-D whose shapes were developed to reduce δ to negative values without risking damage to unarmored outboard surfaces [15]. All diverted plasmas described here operated at B = 2.0 T on axis and I p = 0.89 MA, with top-bottom averaged triangularity δ avg ∼ −0.14, resulting in q 95 = 4.5. Full DIII-D field was necessary to prevent q 95 from dropping low enough to risk disruptions. Earlier inner wall limited plasmas had operated at the same field and current, but due to average triangularity δ avg ∼ −0.36, q 95 drops to 3.4. Too few NT discharges equilibrated over multiple resistive current diffusion times (τ R ∼ 1 s) exist to definitively say whether these q 95 values, low relative to PT discharges' at similar I p and B, materially degrade stability. DIII-D error fields were corrected with the standard method.
The NTH-like plasmas were heated exclusively with neutral beam injection (NBI). Heating trajectories in most NT DIII-D beam-driven experiments consisted of even duration steps in β N until the end of each discharge, often not injecting enough power to reach β N = 2.8. NTH-like plasmas exceed β N = 2.8 earlier on, encountering n = 2 TMs mid-discharge. This pre-TM period can likely be shortened to maximize NBI availability during flattop. NTH-like plasmas had line averaged density of n e = 4-5.5 m −3 . The effect of electron cyclotron heating on the NTH-like is as yet unclear.
These NTH-like discharges encounter n = 2 TMs that minimally impact energy confinement. The sawteeth and fishbones commonly observed in DIII-D NT plasmas are not present during these TMs. This MHD cascade is pictured in figures 3(c) and 4(c), showing 10 kHz sawtooth blips until ∼2 s supplanted by 15-20 kHz fishbone activity (also figure 5) followed by a 25-30 kHz n = 2 TM period. The location of this TM has been estimated at ρ ∼ 0.35 by observing electron temperature fluctuations measured by the electron cyclotron emission (ECE) diagnostic. The NTH-like state is accessed when the injected power is stepped up to raise β N above the threshold for n = 2 onset (yellow vertical line in figure 6). The mode is then robustly maintained for over 1 s in the first of these discharges. As shown in figure 6, the confinement is recovered to levels prior to the existence of this 3/2 mode, despite the drop in power after ∼3.5 s. Electron density continues to rise for the remainder of this discharge due to weak pumping and sources from NBI fueling.
The higher β N of these discharges (gold in figure 1) encounters an n = 2 TM earlier, tracking β N evolution. While constant injected power is maintained, β N increases marginally due to density accumulation until a large n = 1 TM starts. The 2/1 results in a substantial loss of confinement and profile collapse. It is noteworthy that this plasma ramps down without disruption while hosting a saturated rotating 2/1 mode, the kind that regularly leads to early termination in DIII-D. NT plasmas treated here see q 95 ∼ 4.5 at lower I p and higher on axis B t than PT hybrids at similar parameters, whose q 95 ∼ 6. This is attributable to NT shaped plasmas residing in a region of lower average B t . However, NTH-like plasmas did not experience disruptions induced by TMs at these low q 95 values.
No significant Alfvén eigenmode activity is observed, as can be seen in the electron density fluctuations measurements with CO 2 interferometry (panel a of figures 3 and 4). Instead, this diagnostic sees bursts, including chirping, correlated with magnetic fluctuations characteristic of the fishbone instability. The absence of Alfvén eigenmodes is significant, as similar fast ion stored energy fractions ∼30% usually correlate with toroidal Alfvén eigenmodes deleterious for fast particle confinement in PT scenario plasmas.
PT hybrid plasmas typically require some type of n > 1 TM activity to see anomalous current density profile broadening [10], often hosting a 3/2 or 4/3 TM. NTH-like plasmas require the presence of similar TMs, so the benefits of eliminating fishbones and sawteeth must be weighed against loss of confinement from these small TMs. Despite these TMs, double  null PT hybrids exhibit enhanced levels of energy confinement, achieving H 98y2 ∼ 1.5 and β N ∼ 3.6 while being passively stable to 2/1 TMs [16]. Single null hybrid plasmas at lower β N ∼ 3, similar B T and slightly higher I p to the NTHlike plasmas described here, achieve H 98y2 ∼ 1.25-1.35 [17]. The highest performing NTH-like plasma has so far achieved H 98y2 ∼ 1.15 for 400 ms at β N ∼ 3.1 prior to the confinement being reduced by a 2/1 TM. Candidate steady-state scenarios such as the 'high q min ' scenario also achieve enhanced confinement in the presence of small n > 1 TMs. Loss of confinement in these cases is estimated to be similar to NT cases (5%-6%) using a similar method as applied in figures 7 and 8 [18].
Estimated confinement loss using the Chang-Callen model [19] is calculated for the duration of each TM in NTH plasmas. This method entails using experimental magnetic fluctuation amplitude for each toroidal and poloidal mode number to estimate island width, using EFIT reconstructions constrained by motional stark effect (MSE) and magnetics diagnostics to locate the m/n surface. Island width and rational surface location are taken as input to a calculation of stored energy loss from pressure profile flattening inside the island width at the island location relative to an equilibrium without the island. Modeling confinement loss thusly was only sufficient to explain some of the degradation observed in one of these NTH discharges (figure 7). The remainder of confinement degradation in the second discharge (figure 8) may be explained by the diminution of rotation shear caused by the 2/1 coupling with DIII-D's wall, a phenomenon regularly associated with significantly reduced confinement [20].

Equilibrium reconstruction and preparation
As is typical of MHD stability analysis of experimental equilibrium reconstructions, modifications to the safety factor (q) profile must be made to remove the q = 1 surface [21]. Without this alteration, the ideal internal kink mode is found instead of the ideal external kink at marginal β N values. Equilibria have been reconstructed using the EFIT [22] code with magnetics [23] and MSE [24] diagnostics and the suite of DIII-D kinetic diagnostics. These include electron cyclotron emission [25] and Thomson scattering [26] for electron temperature, CO 2 interferometry [27] and Thomson scattering for electron density as well as charge exchange recombination spectroscopy [28] for main and impurity ion density.
During the high β N period, featuring only n = 2 TMs, these reconstructions show q min < 1 despite the absence of sawtooth and fishbone activity that accompanies the q = 1 surface. DIII-D PT hybrid plasma reconstructions including core MSE have q min < 1 in some cases, despite also lacking sawteeth. Resolution of q min in hybrids should be possible with more accurate core MSE, but this has not been demonstrated. As such, q profile modification as described below not only enables ideal MHD analysis of external kink modes, but also produces equilibria more closely resembling the experiment. We qualify this by stating that 2D EFIT reconstructions of PT hybrids and NT hybrids cannot fully capture equilibria made 3D by core TMs, i.e. helical perturbations. The TEQ code within the Corsica framework [29] was used to replace initial equilibrium  current density (J) profiles with profiles that produce q profiles splined to selected values of q min and q 0 . This method was chosen instead of directly recalculating equilibria with target q profiles as replacing J profiles produced more physical final equilibria. Replacement of q directly often produced negative non-monotonic edge J. To preserve information derived from edge diagnostics, splined core profile sections are joined with the original J and q profiles at the lowest ρ location for which  a solution for the desired q min and q 0 exists (e.g. figure 9). These are adaptively selected for individual time slices, as fitting requirements can vary. Values of internal inductance l i derived from equilibrium reconstructions constrained by kinetic profile diagnostics and equilibria whose q profiles have  been modified are well matched (∼1.1). EFIT01 l i values are slightly below these due to erroneously high edge current. Alterations to pressure profiles are negligible. More details on this q modification method and its consequences for equilibria and their stability can be found in appendix.
A sensitivity scan has been conducted across values of q min and q 0 (figures 10 and 11) to determine appropriate input choices that do not have a significant effect on ideal stability  predictions with the DCON code. The series of profiles selected from this set of input choices resemble hybrid q profiles in the core, having low shear and remaining near q = 1 in the core, q min ∼ 1.09. Selected q profile settings were identified to minimize the incidence of spurious internal modes found by ideal DCON if q min of flat profiles becomes too near 1.

Ideal MHD stability
Time series of ideal MHD limits evaluated for n = 1 external kink modes in the presence of an ideal DIII-D wall and no wall limits suggest these NTH-like plasmas are only weakly wall stabilized ( figure 12). Edge bootstrap current is known to be responsible for the coupling of ideal modes to the vessel wall and its absence explains the weakness of this usually substantial effect [30]. It is also the case that NT plasmas in DIII-D exist in a nonconformal wall to their boundary, unlike PT plasmas. Higher average distance from plasma edge to conducting structures reduces wall stabilization. NTH-like plasmas also appear to operate near the no wall limit predicted for their shape and profiles. This offers further resemblance to PT hybrids, which regularly operate at or in excess of their no wall limits [16]. An off-trend, spurious limit calculation has been removed from the time series presented. The calculated Mercier stability criterion indicates stability to the ideal interchange, with D I in the range The Troyon limit on maximally attainable β N without a conducting wall can be empirically described by the expression β No wall N = kl i where k accounts for modifications to stability due to shaping and profile variation [31,32]. Scenarios with peaked current density profiles tend to see k ∼ 2.5 whereas scenarios' whose profiles are broader see k ∼ 4. These limit calculations suggest k ∼ 3 in NTH-like plasmas (figure 13). Decisive determination of k will require further study of NTHlike plasmas equilibrated to stationarity over scans of shaping parameters, as intuition of k from PT plasmas may not prove correct. Increasing k may represent actionable improvement of NT scenarios operated at substantial β N . It bears mentioning that lower pressure peaking p peak = p0 ⟨p⟩ (figure 13) correlates with increasing β N and with lower l i . Minor improvement in wall stabilization appears as pressure peaking and l i diminish, seen in a wider separation between no wall and ideal wall limits. As τ R in these plasmas is longer than the period of flat β N , attaining stationary β N and understanding the current density profile's role in determining ideal limits requires dedicated experiments. The GATO ideal stability code was employed to corroborate stability limits and mode structure found by DCON. Scans of equally spaced equilibria in increasing plasma pressure were produced with Corsica, each equilibrium having been subjected to DCON n = 1 stability calculation. The equilibrium found to be marginally unstable was then input to GATO. GATO finds the n = 1 external kink marginally unstable, with a larger m = 1 component than is typical of PT scenarios at similar β N . Kink mode structure found for equilibria scaled to marginal pressure was similar in both the fishboning period with experimental β N ∼ 2.5 and the NTH-like period (figure 14) whose experimental β N ∼ 3.1. NTH-like period kinks include relatively higher amplitude m > 1 harmonics vs earlier modes, likely due to the broader profiles at higher β N . These broader modes may account for slightly improved wall stabilization later in the discharge.

Tearing stability
Tearing stability has been investigated for NTH-like plasmas by using the RDCON code [33] to calculate evolution of the linear tearing stability index ψ 2µ s ∆ ′ over time and for scans of β N (see comparisons with PEST3 in the appendix) and shape.
Here, ψ s is normalized poloidal flux at a particular surface, µ = √ −D I , and D I is the ideal interchange stability parameter. This tearing index will be used as a proxy for tearing stability, as well-validated models for the critical tearing index (∆ ′ c ) at which TMs onset in a given equilibrium do not exist. It is likely that ∆ ′ c is parametrized by T e local to a given rational surface [34]. Modifications to equilibria in Corsica are only reflected in total pressure and current density, and in cases of q and shape modification, pressure profiles are not altered significantly. This suggests little change in ∆ ′ c , such that trends in the tearing index alone may be useful.
RDCON is used for equilibria at experimental β N because PEST3 erroneously finds most equilibria to have lower ideal limits than those found by Corsica DCON, RDCON, and suggested by experiment. RDCON also supports a DIII-D conducting wall, which is valuable for comparison to Corsica DCON and to investigate possible effects of eddy currents on tearing stability in a realistic structure rather than a conformal wall found to have a roughly equivalent effect. As with many of fusion's machines and codes alike, PEST3 was not built with NT in mind. The vacuum code that supports a conformal wall demonstrates particular sensitivity to edge equilibrium profiles in NT shapes, further described in the appendix. Given these considerations, all tearing calculations were performed with RDCON. The tearing index approach to ideal limits calculated by PEST3 in successful cases is reasonably replicated by RDCON, lending confidence to primarily using this code ( figure 15). The reasoning employed analyzing such calculations falls from variations in β N typical of experiment causing substantial classical tearing instability when operating near an ideal limit [35]. Evolution of the  tearing index has been calculated over the high β N period of shot 186473, showing consistent 2/1 tearing instability drive at β N ∼ 2.5, then somewhat random tearing index at higher pressure ( figure 16). This is consistent with operation very near an ideal limit, where tearing index calculations become inaccurate [36] as this period of the discharge was calculated to have been by ideal DCON.
Equilibria from experiment were modified to raise q min in the same manner as in ideal stability scans detailed above, setting q min ∼ 1.09 and preserving a fairly flat core q profile. This is a necessary step for tearing stability calculations, as tearing codes will not yield valid tearing index predictions in a regime unstable to an ideal mode, i.e. q min < 1 at nonzero β. The tearing index evolution for this set of experimental equilibria does not show an obvious increase in tearing index values along the approach to 2/1 TM onset. The spread in predictions may be due to operation near ideal limits. Figure 17. The base armor campaign shape (gray) was expanded to fill the DIII-D vessel one shaping parameter at a time. Each shape parameter-elongation (pink), upper inner (purple), and lower (blue) outer squareness-is scanned for use in ideal and tearing stability scans.

Shape modeling
Shaping of the plasma boundary is known to have major repercussions for the confinement and MHD stability of fusion plasmas. Given that NT plasmas face lower global MHD stability limits vs comparable PT plasmas, optimization of available low resource parameters like shaping is essential. Stronger shaping in PT yields improved confinement and MHD limits [37], associated with higher pedestals. Edge bootstrap current produced in all other scenarios by pedestal pressure gradients is absent in NTH-like, thus the effects of shaping parameters on profile shapes merits investigation. The shape modeling presented here is intended to guide experiments on DIII-D. A range of shapes, designed to explore consequences for MHD stability of individual shaping parameter scans, is modeled with Corsica, DCON, and RDCON ( figure 17). These efforts culminate in shapes intended to improve scenario performance without detracting from MHD stability.   [38,39]. Squareness reported here accords with that used by the DIII-D plasma control system for shape control, definition in [39], rather than the definition used in [38]. Lower triangularity is held fixed at −0.55, as these shapes are meant to explore shape options with armor installed in DIII-D's lower outer quadrant, onto which lower single null strike points should terminate. Outer squarenesses are not explored as operation near the DIII-D wall is constrained by heat loading of plasma facing components. As higher squareness is known to not necessarily maximize MHD stability or confinement quality [18], two scans of the same values in lower squareness are performed using the two highest upper squareness values. It is also worth reexamining performance trends in each of these shape parameters as enhanced confinement in shaped PT scenarios has to do with changes in pedestal stability. Historical intuition may not applicable to optimization of an L mode NT scenario.
Trends are modeled at two time slices, one in the lower β N fishboning, pre-NTH-like period and one during the higher β N NTH-like period of shot 186473. These scans aim to understand the trends in n = 1 ideal and tearing stability with each shaping parameter, resulting in an optimized shape whose limits are minorly improved vs a base shape intended for use in dedicated DIII-D NT experiments. This shape's larger crosssectional area makes use of more of the DIII-D vessel, increasing volume by nearly 3 m 3 to 18 m 3 , a similar volume to standard PT scenario discharges.
Having found n = 1 MHD limits insensitive to increased squareness and elongation (figure 18), a shape has been constructed ( figure 19, blue) that aims to improve ideal MHD stability while preserving substantial average NT δ avg ∼ −0.4 and increasing volume. Average triangularity has been associated with maintaining L mode operation [40] and as such experiments should be conducted with strong enough NT shaping so as to not enter H mode at high power operation like will be required in a reactor. This requirement must be balanced with potential reduction of MHD stability limits, suggesting moderate δ avg may be required for a high β N path reactor.
Upper triangularity has been scanned to investigate trends in MHD stability for shapes possible in DIII-D experiments, preserving elongation and lower triangularity. This scan predicts weak stability dependence on δ u , suggesting instead that δ avg , which remains negative in all cases, is the operative quantity. Relocation of minimal edge current density in such peaked L mode equilibria may explain this result. The familiar effects δ has in PT on pedestal height and by extension wall stabilization are absent ( figure 18(d)).

Tearing stability
RDCON was used to evaluate trends in tearing index over a scan of β N for a representative NTH-like equilibrium, shot 186473 at 3480 ms, recalculated in each of the shapes addressed above. RDCON ideal limit calculations are somewhat determinative of tearing index calculations at an absolute β N , especially in the ideal limit's vicinity [14]. By extension, tearing calculations are meaningful in respect to those ideal limits, and as such we have elected to assess trends in the tearing index evolution over β N /β NLim , where β NLim is the ideal limit found for each shape by RDCON. In this manner of analysis, trends in tearing stability with shape parameter can be isolated from the effects of shaping on ideal limits treated above. Tearing index trends are sampled in two regimes within each β N scan: far from the ideal limit, where changes to equilibria are relevant for tearing index predictions, and near the ideal limit, where these details do not impact tearing stability strongly compared to small perturbations natural to experimental plasmas. In the near pole region, small perturbations to equilibrium profiles and shape cause changes in the tearing index larger than changes to equilibria by available actuators. Both high and low β N regimes merit examination for the sake of optimizing either path to a high-performance scenario.
Trends with elongation and UISQ suggest more elongated plasmas, while less stable than less elongated plasmas (figure 20), benefit minorly from stronger squareness far from an ideal limit and are unaffected near an ideal limit ( figure 21). Increased lower squareness is predicted to have a relatively small effect on tearing stability, suggesting shapes can be symmetrized without penalty ( figure 22). The triangularity scan conducted reflects a weak dependence on δ u while δ avg remains negative, minorly favoring higher δ avg . Tearing stability changes most strongly as δ avg nears 0. All cases, in both β N /β NLim regimes sampled, reflect parity with or improvement in stability over tearing index calculations made for experimental equilibria.

Stability of optimized shapes
To extend the findings of shape parameter scans performed thus far, three shapes of negative average triangularity [−0.4, −0.5, −0.6] (termed 'bulged', flat bump, and 'flat' respectively) (figure 23) and optimized parameters for stability and performance based on the study described in sections 4.1 and 4.2 are considered in this section. The strongest NT shape from the aforementioned tearing index scan is compared against two shapes similar on the high field side, whose low field side gaps have been increased to reduce triangularity despite geometric constraints imposed by the DIII-D vessel.
An equilibrium representative of the NTH-like phase is predicted to have significantly lower stability limits at δ avg = −0.6 vs higher δ avg and a similar PT hybrid ( figure 24). This is likely explained by the reduced poloidal curvature on the low field side, leading to compression of flux surfaces and as a consequence greater sensitivity to radial displacement. This is distinct from the stabilizing effect of increasing PT such that field nulls present, which are loci for instabilities, are moved into a region of better curvature. Though this effect is absent from the comparison of flat and bulged shapes here, flux surface compression in the bad curvature region will come as a natural consequence of stronger NT shaping. It is possible that the upper triangularity scan performed ( figure 19) has little effect on ideal stability because shapes treated here are by design lower single null, and the average triangularity does not become positive. Comparison of the no wall limits of the NT shapes (figure 24) indicates most of the loss in stability is independent of wall coupling.
Shaping without accounting for the influence of a wall effects a ∼13% decrease from the bulged NT shape's no wall limit to the flat shape's. Diminished coupling between the ideal wall and a centrally peaked current density profile in the flat     shape (figure 24) yields an ideal wall limit very similar to the no wall limit. Lacking the usual pedestal bootstrap current, neither NT shape sees such an improvement in MHD stability limits from no wall to ideal wall as is seen for the PT hybrid scenario equilibrium. No wall MHD stability limits are higher for the PT hybrid equilibrium than either NT shape, due to both aforementioned triangularity effects. However, the no wall limit in the hybrid is only 16.6% larger than the bulged NT limit. It is unlikely that a vessel conformal to the NT shapes would narrow the gap in wall stabilization with the hybrid, as the hybrid's relative wall stabilization outpaces the bulged NT shape's with decreasing conformal wall distance. This disparity in β N limits widens to 44% at a conformal wall distance equivalent to one tenth the plasma radius.
Tearing stability evaluated with RDCON shows a substantial difference between the flat and flat bump NT cases compared to the flat NT case evaluated here (figure 25). Better stability is predicted in the less strongly NT shapes at both 90% and 65% of the ideal limit found by RDCON. These RDCON ideal limits are in fairly good agreement with the ideal limits found by Corsica DCON.

Conclusions
Diverted NT L-mode plasmas produced in DIII-D have exhibited MHD novel in the facility's history exploring this regime. Similar to the PT hybrid scenario, these plasmas host a TM of toroidal mode number n = 2, absent of sawteeth and fishbones, which are present in all other DIII-D NT plasmas. One plasma ramps down successfully having achieved operation at β N > 3 for 400 ms in this NT hybrid state. Another plasma persists at β N ∼ 2.5 with the n = 2 mode, seeing no reemergence of MHD associated with the q = 1 surface for the remaining 1.5 s of the discharge. As these n = 2 modes minimally impact confinement while sawteeth and fishbones are absent, development and optimization of a NTH scenario would represent progress toward a NT reactor. Also similar to the PT hybrid, it remains to be demonstrated that confinement remains reactor relevant at low torque. Further work in experiment and modeling is required to definitively determine whether this hybridlike state in NT is more stable than NT plasmas without n = 2 modes at similar global parameters.
Ideal MHD stability has been modeled for representative NTH-like equilibria, appearing overall worse than PT hybrid stability. Ideal and tearing stability scans as functions of plasma boundary shape in representative NTH-like equilibria predict limits similar to those predicted for experiment or marginal limit improvement at stronger shaping. This work predicts no n = 1 stability is sacrificed for increasing volume with most shaping parameters. Shapes optimized for performance recover volumes similar to reactor candidate PT scenarios. Such shapes represent two points in δ avg , the shaping parameter our modeling suggests is most relevant for global stability. Ideal and no wall limit predictions change substantially with this reduction in outer gap, but little variation in stability comes of varying δ avg by means of a δ u scan. Tearing stability seems to follow trends in ideal limits. Devices planning operation at low β N (β N < 2) e.g. SPARC and ITER can operate below 75% of the no wall limits of strongly shaped NTH-like plasmas and below 60% of the no wall limits of moderately shaped NTH-like plasmas. Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Appendix. Details of equilibrium modifications and impact on code results
To calculate external kink stability, ideal MHD codes require input equilibrium modifications to raise q min > 1. A method to do this utilizing the Corsica framework and implemented in MATLAB is described here.
This method attempts to reduce user input choices, such that modifications to equilibria are determined on a generalized basis. Presently, the method requires only selection of a target q min and q shear proxy k (q 0 = q min * k). The location in ρ, square root of the normalized toroidal flux, (or normalized poloidal flux ψ n , user's preference) of the unaltered input q profile's q = 1 crossing is used as the target for ρ(q min ). As of Figure A1. Adjusted core safety factor profile is plotted in modified and unmodified cases, having been splined to experimental profiles outside ρ ∼ 0.2, the location of experimental q = 1 surface.
writing, this method produces slightly inverted q profiles, as it finds these solutions much more easily than solutions whose q min = q 0 , where q min > 1.
Having chosen values for q min , q 0 , and ρ(q min ), the method will scan the ρ at which splined q is joined with the original q profile, termed here ρ join . Requirements of the fit for q(ρ < ρ join ) include passing through q min within a tolerance smaller than variations in meaningful equilibrium reconstructions of experimental plasmas, here ±0.02. Included also are constraints on the second derivative of q in ρ, to disallow local minima or maxima from accepted solutions. Solutions for values of ρ join are attempted between [ρ(q min ), 0.35], and the solution minimizing ρ join is selected. The upper limit of 0.35 was selected arbitrarily as the limit of reasonably modifying experimental equilibria without excessive fabrication. The final profile should be devoid of artificial features, having not introduced profile details impossible to accurately measure in experimental tokamak plasmas. Profiles should not be excessively inverted, so as to not enter a regime dominated by negative central shear. The artificial, splined core profile's extent should be minimized, such that the final product resembles experiment as closely as possible, preserving the influence of kinetic diagnostics' constraints. Example profiles have been included for reference.
Modifications to the plasma shape of fixed boundary equilibria can clarify equilibrium stability sensitivity to shape parameters, similar to the analysis presented in section 4. Replacement of boundary shape with Corsica results in modification to profiles in accordance with solutions to the Grad-Shafranov equation calculated with TEQ. As such, the workflow developed here replaces the boundary of experimental equilibrium reconstructions, then modifies q profiles extracted from these modified equilibria. The spline routine used to generate q profile solutions meeting the criteria detailed above generates numerical noise in J par ≡ ⇀ J · ⇀ B B 2 , around ρ join . These B factors are adjusted to preserve J par while I p is held fixed, as is done in this workflow. Additionally, replacement of boundary shapes can create numerical noise in J par (ρ > 0.9), leading to nonphysical profile points at the edge. To ensure equilibrium quality, especially for use as inputs to tearing stability codes, both instances of numerical noise are removed by cubic spline extrapolation. These J par profiles are then stored for use in stability analysis. Of those possible within Corsica, the stability calculation workflow found to reduce numerical error begins by replacing plasma boundary (maroon to gold in figure A1) and then J par appropriate for the new boundary shape. While this requires generation of a unique solution for every equilibrium in a given boundary shape, the resulting q profiles are exactly as specified. Following replacement of boundary and J par , pressure scans are then executed to find stability limits with fixed q.
This equilibrium modification workflow has also been employed to produce pressure scans for analysis with PEST3 Figure A2. Tearing stability calculations in PEST3 from a previous workflow that used J profiles from experimental equilibria before boundary modification (left) and the workflow detailed in this appendix (right). Figure A3. (Left) PEST3 tearing index calculations compared to ideal limit found by Corsica DCON (line) and RDCON tearing index calculations, each using equilibria produced using J profiles calculated from experimental equilibria before boundary modification. (Right) Calculations of tearing index by RDCON and PEST3, plotted vs β N normalized to the ideal pressure limit each code finds. and RDCON, both codes finding similarly well-fixed q profile details. If these details (e.g. q min , q 0, q 95 ) are allowed to float, both tearing codes produce calculations whose scatter clouds any discernible trends with shaping ( figure A2).
Both tearing codes can encounter further difficulty when generating equilibria during pressure scans, if care is not taken. PEST3 requires correct selection of mesh scan points for a given equilibrium to properly resolve both the ideal stability and tearing index. Even if this is done correctly, PEST3 can find ideal limits discrepant with those found by ideal DCON, as exemplified in figure A3 (left). This discrepancy likely stems from differences in integration of the F, P, and q profiles to re-form equilibria within each code. TEQ attempts to fix zero current density at ρ = 1, smoothly forcing P ′ and FF ′ to zero at the edge. Some equilibria produced from Corsica feature edge gradients in P ′ and FF ′ too extreme for PEST3's integration to accurately recalculate, hence PEST3 reconstructs non-physical negative edge current density, which then causes it to find erroneous ideal instability.
This initially appeared to be PEST3 not handling NT, but instead this problem seems to occur when negative edge current density is combined with certain NT shapes which feature x-points. This was investigated by recalculating equilibria in simple Miller geometry shapes, which do not feature x-points or other sharp features of any kind. PEST3 did not find these to be ideally unstable. Thus, shape parameters did not appear to be the cause. After investigating exactly what PEST3 reads as input from Corsica-produced files, P ′ and current density were splined to soften gradients near the edge, yielding equilibria nearly identical otherwise to the originals. PEST3 was then able to correctly integrate P and F, enabling not only use of large volume NT shapes, but also a conformal wall with most NT shapes. Doing this seems to lessen disagreement between PEST3 and RDCON on limits, even though RDCON's ideal stability limit calculations do not see noticeable difference with the edge splining, likely due to edge truncation DCON codes usually perform. There remain unexplained cases in which the conformal wall in PEST3 calculates non-physically high ideal limits once this edge alteration is made, of which investigation is ongoing and lies beyond the scope of this paper.
RDCON approximates a similar approach in tearing index calculations to ideal limits as PEST3 calculates in most cases, as in figure A3 (right), but equilibria produced by workflows allowing q to float and some with replaced boundary shapes via a precise workflow prove to reproducibly produce nonphysical secondary poles far from ideal limits in RDCON's tearing index calculation ( figure A4). This phenomenon has not been explained as of writing. The difficulty of extracting meaningful calculations from these classical tearing codes has been due to the sensitivity of this calculation to very minor perturbations in equilibrium details at the edge. This should give pause to efforts attempting real-time calculations of these quantities from equilibrium reconstructions constrained by a much reduced diagnostic set and no options to constrain the edge to the detail required by these codes to obtain meaningful results.