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

The KSTAR has three rows (Top/Middle/Bottom) of versatile IVCC (capable of up to n  =  2) with 2 turns in low field side [21], whose configuration is similar to that of the planned ITER RMP coils (capable of up to n  =  4). Unlike other tokamaks equipped with two (Top/Bottom) rows of RMP coils, the KSTAR is the only major tokamak that has in-vessel mid-plane RMP coils. Hence, any uncertainties involved in ITER in-vessel mid-RMP coils can be directly addressed in KSTAR. Incorporating broadband switching power amplifiers (SPA) capable of up to 5 kA/turn in DC to 1 kHz, various 3D configurations with n  =  1 and/or n  =  2 fields can be applied in conjunction with three dozens of assorted patch panels in KSTAR [22]. Besides, along with an upgraded magnetic diagnostic (i.e. magnetic probes installed on passive plates), various plasma responses for 3D field physics study can be directly diagnosed by the state-of-the-art diagnostics, such as electron cyclotron emission imaging (ECEI) [17] and microwave interferometry (MIR) [23].

Figure 1 shows a summary of the n  =  1 intrinsic EF in low/intermediate/high-β plasmas in KSTAR. In particular, it is to be noted that the n  =  1 intrinsic EF in high-β plasmas (β_N ~ 2) was measured to be as low as ~4  ×  10⁻⁵ B₀, consistent with a linear extrapolation of the EF measurements on low-β [10], and intermediate-β plasmas. This corresponds to approximately 20% below the targeted ITER tolerance level (5  ×  10⁻⁵ B₀) [4]. Unlike a typical 'compass scan' that uses either mode-locking or angular momentum variations [24], the n  =  1 intrinsic EF at high-β plasmas was determined based on rotation collapse due to n  =  1 field penetration [12]. Hence, a linear projection suggests that the n  =  1 intrinsic EF would remain low enough for KSTAR to easily reach the no-wall stability limit (nominally β_N ~ 2.6), and possibly up to β_N ~ 3 (due to kinetic effects (e.g [25])) without dedicated EF correction. While the identification of n  >  1 intrinsic EF based on non-axisymmetric plasma response is limited in KSTAR due to the insufficient number of the toroidal coil sectors, no evidence of n  >  1 EF has been found yet.

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As one of the important milestones, KSTAR has demonstrated a record-long (>10 s, more than 90 τ_E) sustainment of RMP-driven ELM-crash-suppressed H-modes, as shown in figure 2. In addition to a typical ELM-suppressed discharge in the past in KSTAR (near q₉₅ ~ 6), a very wide ELM suppression window of the safety factor (q₉₅) has been identified near 5.0  ±  0.25 with a level of RMP field δB_RMP/B_T ~ (5–7)  ×  10⁻⁴ at pedestal top (with magnetic field B_T  =  1.8 T, plasma current I_P  =  0.5 MA, and external NBI heating power P_NBI ~ 2.8 MW) with collisionality ν^* ~ 0.2–0.25 (assuming Z_eff  =  1) [19]. Such superb performance of RMP-driven ELM-crash-suppression has been ascertained to be very reproducible (with more than 100 discharges). More importantly, the methodology we have developed now guides us to a new frontier of RMP physics in terms of both accessibility and robustness. Specifically, we have found a distinctive shape dependence of radial position of lower X-point (R_x  =  1.44 m) within $\pm$0.02 m, which corresponds to the lower triangularity of δ_l  =  0.74 $\pm$ 0.04 [19].

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However, as shown in figure 3, such 'performance-oriented' configuration (in black) cannot be easily diagnosed by a divertor camera overlooking downward from the top [28], in that the outer striking point would reside on the vertical leg of the divertor. Hence, taking into account the spatially limited view of infra-red camera for divertor heat flux measurements, another set of systematic parameter scans was conducted with R_x  =  1.39 m nearly fixed (in red in figure 3), in which the outer strike point would be located at the center of the divertor. Fortunately, another optimal window of R_x was found in the vicinity of the original R_x (at 1.44 m in black) with relatively minor changes of other plasma parameters at the expense of much more challenging operational sensitivities. As a result, the 2nd optimal group with more strict constraints have been found at q₉₅  =  4.95  ±  0.05, and R_x  =  1.39  ±  0.01 m, (i.e. δ_l  =  0.85 $\pm$ 0.02) respectively. In fact, figure 4 shows such a newly established ELM-crash-suppression discharge, whose optimal lower R_x is positioned at 1.39 m. Although the duration (~50 τ_E) of ELM-suppressed stage is shorter than that of the performance-oriented configuration (as shown in figure 2), the availability of divertor heat flux measurement during ELM-controlled periods enabled us to directly diagnose the 3D configuration impact on the divertor. Figure 5 shows the divertor heat flux footprints measured using an infra-red camera during n  =  1 RMP-driven ELM-crash-suppression from 3.2 s to 8.7 s on 16661 shown in figure 4. The striation patterns consist of the axisymmetric peak (R ~ 1.45 m), and n  =  1 non-axisymmetric peak (R ~ 1.47 m), while each peak is below ~1.2 MW m⁻² at P_NBI  =  3.4 MW with ν^* ~ 0.3–0.4 (assuming Z_eff  =  1), as summarized in the right figure of figure 5. Throughout this discharge, a peak of the axisymmetric lobe remains higher than that of non-axisymmetric lobe. Interestingly, even during ELM-suppressed stage, there is a noticeable evolution of the non-axisymmetric peak, which appears to have taken a few seconds to be built up from the early stage (at 3.5 s marked at 'A'), prior to the fully grown stage (at 5.0 s, up to a similar level marked at 'B'). This suggests there is another transport process to establish a heat flux channel from edge plasma to divertor, which could be virtually independent of any onset criteria of ELM suppression.

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Similarly, we have confirmed the validity of the operational approach even on the n  =  2 (near q₉₅ ~ 4) RMP ELM control, but so far observed the marginal suppression or strong mitigation frequently, rather than full suppression. While a further study needs to be done, we are speculating the limited capability of n  =  2 phasing adjustment in KSTAR could be partially responsible for non-optimal n  =  2 RMP coupling to edge plasmas. Here, the phasing refers to the toroidal phase difference between adjacent rows. Since the existing IVCCs are composed of 4 coils in a row, an arbitrary n  =  2 toroidal phase cannot be easily accommodated. As a result, the n  =  2 phasing adjustment, unlike n  =  1 phasing to be discussed in the next section, is very limited to either even or odd parities between adjacent rows.

One of the important goals in RMP physics is not only to achieve the ELM suppression, but also to secure a reliable path to it. Since an optimal phasing was reported to be quite critical in suppressing ELMs using n  =  2 RMP in DIII-D [29], the n  =  1 phasing dependence in KSTAR has been systematically investigated.

Although the n  =  1, +90° phasing is known to be quite effective in KSTAR, that may not warrant it would be the most optimal for various plasma conditions (e.g. q₉₅ or edge collisionalities (ν^*) etc). Thus, assuming that resonant components in an optimal RMP phasing should be configured (i) maximally at the edge to control ELMs [${\sum}_{\text{edge}}\left(\delta B\right)$] but (ii) minimally in the core to avoid mode-locking [${\sum}_{\text{core}}\left(\delta B\right)$], a systematic map of the ratio of these two quantities [${\sum}_{\text{edge}}\left(\delta B\right)$/${\sum}_{\text{core}}\left(\delta B\right)$] has been established in vacuum and ideal plasma response modelling, using IPEC [26], as summarized in the polor plots of (I_MID, ϕ) in figure 6. With top/bottom rows fixed at 5 kA/turn (effective; 10 kA), the phasing (ϕ) between rows is shown counterclockwise, while the radius (I_MID) of the circle represents the current amplitude at the middle row. According to the vacuum and ideal plasma response calculations, depending on the phasing, there are three distinctive bands (locking(brown), non-resonant(green) and suppression(blue)). To validate whether a proper level of RMP currents for ELM suppression and mode-locking would exist at a specific phasing, one of the well diagnosed discharges, similar to the discharge in figure 2, was referenced. That is, the ELM suppression is predicted in the window defined by ${\sum}_{\text{edge}}\left(\delta B\right)$  >  [${\sum}_{\text{edge}}\left(\delta B\right)$]_Ref, while keeping ${\sum}_{\text{core}}\left(\delta B\right)$  <  [${\sum}_{\text{core}}\left(\delta B\right)$]_Ref, as the I_MID varies. Then, a set of experiments was conducted to test this newly predicted ELM suppression window (in blue on I and II in figure 6), by varying I_MID in each of the fixed phasings at (a) 90 (b) 75 (c) 60 (d) 45 and (e) 315 (°). The experiments indeed demonstrated the existence of the ELM suppression windows in the predicted range of currents at all the tested phasings, as summarized in the middle figure of figure 6 with the experimental results on ELM suppression and locked mode disruption. The agreement shown here with ideal 3D response is quite remarkable, given the narrowness of the windows in such a large coil configuration.

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Interestingly, it is to be noted that a non-conventional phasing (e.g. 315° in figure 6(e)), which we rarely pay attention to, tends to follow the modeling prediction, as well. To access the ELM suppression window without mode-locking, the RMP current has been arranged to be reduced from high currents, rather than increased from low currents (compared to other phasings). Without having an ideal plasma response prediction, it was very unlikely that we would have attempted the dominantly non-resonant phasing (equivalent to  −45° phasing), which would be almost orthogonal to a typical phasing (i.e. near  +90° phasing). Surely, this new modelling capability is expected to help us chart a new route even to a seemingly unlikely phasing, including other unexplored 3D configurations. Meanwhile, the vacuum superposition method critically failed in the prediction, as shown in the left figure of figure 6. The importance of plasma response on the RMP ELM control has been recently highlighted in the DIII-D [29], but this investigation consolidates their findings with the greatly improved isolation of plasma response using 3 rows of coils. The details will be reported elsewhere [30].

Since the introduction of non-axisymmetric field (δB) for RMP-driven ELM control affects the power thresholds (P_th), a very systematic study was conducted in DIII-D, showing that unless the RMP strength gets high, there would be minimal impact on P_th for L–H transition [31]. Ever since, many other devices reported a similar trend of P_th, which appears insensitive in low δB, but linearly increasing in high δB. However, considering that KSTAR has a much lower level of n  =  1 intrinsic EF than other tokamaks, any accumulative effects of 'uncorrected' multiple low-n intrinsic EFs are expected to be also minimal in KSTAR. Figure 7 shows the measurement results based on three sets of non-axisymmetric fields. As a result, the n  =  1 δB shows a very weak linear dependence at low δB, while showing a slightly stronger linear dependence at high δB. However, the n  =  2 δB shows a strongly linear dependence even at low δB, an insensitivity at the intermediate δB, but a strongly linear dependence at high δB. To address any merit of low non-axisymmetric field in KSTAR, we added an n  =  1 field that would be equivalent to a typical n  =  1 intrinsic EF in conventional devices (e.g. nominal n  =  1 δB/B₀ ~ 2.7  ×  10⁻⁴ at edge, where pedestal top could be formed) on top of n  =  2 field scan. It turns out that such a mixed δB (magenta square) leads to much higher P_th, once n  =  2 gets higher than δB/B₀ ~ 2.2  ×  10⁻⁴, resulting in P_th ~ 1.5 MW. This suggests that due to multiple 'uncorrected' EFs in tokamaks, the power threshold P_th would get much higher than what a single non-axisymmetric field could dictate.

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As a note, the previous study in DIII-D reported P_th ~ 1.6 MW (n_e  =  0.29  ×  10²⁰ m⁻³, B_T  =  1.7 T, S  =  56 m²) [31], while its standard n  =  1 EF correction was applied without n  =  3 RMP [32]. However, it is ~19% higher than the estimation using Martin's scaling law P_{th, Martin}  =  c $n_{\text{e}20}^{0.72}$ $B_{\text{T}}^{0.80}$ S^0.941, where c  =  0.049 [33] (P_{th, Martin}  =  1.35 MW). Here, S refers to the surface area of plasma boundary. Meanwhile, the power threshold without RMP in KSTAR was P_th  =  0.86 MW (n_e  =  0.2  ×  10²⁰ m⁻³, B_T  =  1.8 T, S  =  49 m²), which is ~9% lower than the estimation of 0.95 MW. Since all the three plasma parameters (n_e, B_T, S) in two devices vary somewhat, it may not be straightforward to make a direct comparison between DIII-D and KSTAR experimental results. Thus, assuming the DIII-D experimental result is the reference that has a higher coefficient than that of Martin's scaling law (c'  =  (1.6 MW/1.35 MW)c), the KSTAR experimental conditions can be projected to 1.13 MW, as drawn in horizontal line in figure 7. It is to be noted that the multi-device empirical scaling [33] has no δB dependence and that KSTAR has approximately 24% lower P_th than the projection that is based on a similar discharge in DIII-D. Overall, this suggests that the merit of low intrinsic EFs on P_th might have been inadvertently overlooked in previous studies. Specifically, owing to the presence of multiple 'uncorrected' intrinsic EFs, it is possible that the experimental results on P_th dependence might not have been sufficiently resolved in the existing devices. Therefore, the lower P_th in KSTAR could be attributable to low level of intrinsic non-axisymmetry, which is likely due to not only low n  =  1 δB, but also possibly other low n  >  1 δB harmonics that have not been measured yet. At the same time, one needs to be reminded that the non-axisymmetric field influence alone cannot explain the large scatters shown in the worldwide database used in the establishment of Martin scaling [33], which requires separate extensive studies in multiple devices.

Figure 8 shows the time evolution of various plasma parameters on discharge (16822) with intentionally misaligned RMP configurations, while figure 9 shows the details of RMP amplitude, phase and phasing in the relevant RMP configurations. Considering that full ELM-crash-suppression was achieved at n  =  1, +90° phasing (compatible with the divertor-camera view, as shown in figures 4 and 5), the misalignment has been imposed on top and bottom rows with respect to mid-row by 5° increment and decrement respectively, as defined at the bottom of figure 9. Specifically, the phasing between top and mid rows increases every 2 s, while the counterpart between mid and bottom rows decreases at the same time. Thus, with the phase of mid-row fixed at 0 (°), the toroidal phases of top and bottom rows are configured at (ϕ_Top, ϕ_Bot)  =  (−95, +85), (−100, +80), (−105, +75) and (−110, +70) (°) successively, rather than at a fixed (ϕ_Top, ϕ_Bot)  =  (−90, +90) (°). (Since the phasing is defined as the phase difference between upper and lower rows, the n  =  1, +90 phasing corresponds to the toroidal phases of (ϕ_Top, ϕ_Mid, ϕ_Bot)  =  (−90,0, +  90) (°) in standard configuration in KSTAR, for example.) Also, at a given misaligned phasing, both static and rotating RMPs have been configured to be sustained for 1 s respectively. The main purpose of a rotating RMP is to diagnose the toroidally asymmetric divertor heat flux footprints that cannot be accessed by the toroidally fixed divertor camera with static RMP.

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Figure 10 shows the time evolution of divertor heat flux footprints and divertor D_α signal, along with the profiles at a few timeslices of interest respectively. Despite no full suppression of ELMs, the striation patterns of ELM mitigation appear quite similar to those of ELM suppression, except the peak of non-axisymmetric lobe. Specifically, in comparison with figure 5 that has full ELM-crash-suppression, the axisymmetric peaks during ELM-crash-mitigation remain at a similar level of below ~1.2 MW m⁻², though the non-axisymmetric peak has been reduced by almost a half (down to ~0.5 MW m⁻²) in figure 10. When comparing various divertor heat flux profiles in static misaligned RMP configurations (right figure of figure 10), it is not difficult to conclude that the divertor heat flux peaks and widths get lower and broadened respectively, as the degree of misalignment (i.e. dephasing) increases.

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Since such static RMP results cannot capture any toroidally asymmetric heat flux footprints properly, a toroidal average of rotating RMP would be of great help to confirm the validity of the static RMP results. In fact, the left figure of figure 11 shows such toroidally-averaged divertor heat flux at each rotating misaligned RMP configuration. As expected, the toroidal average removes the non-axisymmetric lobes but the toroidally averaged axisymmetric lobe appears less peaked with broader width, corroborating the static RMP analysis results shown in figure 10. Meanwhile, it is much more important to compare the normalized shapes of divertor heat fluxes at various RMP configurations, when the RMP applicability to ITER and future devices is concerned. For that reason, the right figure of figure 11 confirms the effectiveness of the intentionally misaligned RMP configurations, where the increase of misalignment leads to a broadened heat flux shape. Technically, the misaligned RMP configurations can be easily realized to reduce the localized heat flux loading without resorting to any special hardware other than independent phase control capability at each row (which is already planned to be equipped in ITER). Once such a favourable outcome is consistently reproduced even during RMP-driven ELM-crash-suppression, various concerns related to time-varying RMP application would be significantly alleviated in ITER and future reactors.

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Nevertheless, we are aware that the RMP alone might not be sufficient to reduce the axisymmetric peak divertor heat flux, but that other means, such as detached plasmas, should be vigorously explored, as well. To be fair, multiple non-axisymmetric lobes induced by higher non-axisymmetric harmonics would be also of help to redistribute the divertor heat flux in a wider area, while the RMP application needs to avoid mode-locking at the plasma core.