Impact of toroidal and poloidal mode spectra on the control of non-axisymmetric fields in tokamaks Free

The magnetic field in a perfect tokamak is axisymmetric in the toroidal direction (the long way around the torus). The field is independent of the toroidal angle, and the helical magnetic field lines remain on the same magnetic surface as they transit the torus. If a field line has a fixed poloidal position at some toroidal angle $ϕ$⁠, then it will return to that position after q toroidal transits, where q is the safety factor, or winding number.

However, due to design and construction imperfections, non-axisymmetric, or three-dimensional (3D), magnetic fields do exist in tokamak devices. When they exceed a certain critical level, the trajectory of magnetic field lines is altered to such a degree so as to create magnetic islands. Islands always degrade confinement of energy, particles, and toroidal angular momentum. In slowly rotating plasmas, islands typically grow locked to the externally applied field. Via viscous effects, these so-called “locked modes” can completely halt the plasma rotation across the plasma volume. This tends to increase the island size (since finite plasma flow acts to shield the externally applied field), spoiling the confinement further. Since the prevalence of plasma current disruptions also increases when islands are present, maintaining 3D fields below the critical threshold is of paramount importance in the design, construction, and operation of tokamaks.

Since it is impractical to avoid creating any 3D fields during device construction, tokamaks typically have a dedicated set of non-axisymmetric trim coils to minimize “error fields.” One of the most frequently observed magnetic islands has a toroidal wavelength equal to the toroidal circumference and a poloidal wavelength equal to half of the poloidal circumference; this field has a poloidal mode number (m) of two and a toroidal mode number (n) of unity. Operationally, 3D coil systems have been generally successful at avoiding the onset of locked modes during the plasma initiation phase of a discharge and in the flattop using only n = 1 error field control (EFC). This result together with the prevalence of m/n = 2/1 islands (relative to higher order islands such as the 3/2 island) has led some to conclude that it is not essential to correct n > 1 error fields. This thinking has so informed the design of 3D coil systems that at present, the design for error field control coils in the ITER device calls for a set of three rows of six superconducting coils, located outside the vacuum vessel, to be hardwired in n = odd pairs thereby preventing application of n = 2 fields. Furthermore, a second coil set (three rows of nine coils), located inside the vacuum vessel, is intended to be used only to intentionally increase n > 1 fields for the purpose of suppressing edge-localized modes (ELMs); these coils are not presently being considered as error field control coils. As a result, initial ITER research plans provide experimental time only for development of n = 1 error field correction with the ex-vessel coils.

While the critical field level for triggering 2/1 islands has been well documented in the literature (since the early 1990s),^1–4 there is no record in the literature documenting the same thresholds for n > 1 fields. Nevertheless, a substantial amount of operational experience with n > 1 fields exists as a result of efforts to understand the suppression of ELMs by 3D field perturbations^5–8 and non-resonant magnetic braking.^9–15 From these studies, it is known that under certain plasma conditions, n > 1 fields can be as detrimental to plasma performance as n = 1 fields with similar levels of plasma rotation braking and performance degradation observed for n = 1 and n = 2 fields. In this paper, we document the thresholds for driven reconnection for n = 2 fields in low density, low rotation, low q discharges, a regime representative of early ITER operation. Data from various tokamaks including DIII-D, EAST, and RFX-mod (operated as a tokamak) will be presented. Similar results have been observed in other devices (e.g., COMPASS) but will be reported in separate publications.

The paper is organized as follows. Section II gives a basic classification of non-axisymmetric fields and discusses the influence of the plasma response on error field sensitivity. Section III presents a criterion (the critical overlap field) that is used to quantify the degree to which applied fields couple through the plasma response in order to drive magnetic reconnection. Section IV introduces recent experiments focused on documenting the critical n = 2 threshold and shows that the n = 2 island onset is followed by the formation of a second n = 1 island chain, which can lead to termination of the plasma discharge. Section V presents the unexpected result that the measured n = 2 thresholds are in fact similar to previously measured n = 1 thresholds and reports on the density, safety factor, external field, and plasma rotation dependence of the thresholds. Section VI discusses implications for the design and operation of the ITER 3D coils systems and a brief summary is given in Section VII.

Non-axisymmetric magnetic fields are classified as “resonant” magnetic perturbations (RMPs) if they can generate magnetic islands. The mathematics used for identifying these relative sparse fields is described in Ref. 16. Using a straight field line coordinate system $( φ , θ , ϕ )$ where $φ$ is the scalar poloidal flux function and θ and $ϕ$ are the poloidal and toroidal angles, the applied radial field δB_r can be Fourier decomposed into toroidal (n) and poloidal (m) harmonics on the 2D magnetic flux surfaces, $B ̃ r ( φ , θ , ϕ , m , n ) = B c , r ( φ , m , n ) + i B s , r ( φ , m , n ) = B ̃ r , m , n ( φ ) e i ( m θ − n φ )$ where B_c,r and B_s,r are the surface-averaged radial field coefficients (Eq. (A.15) in Appendix A of Ref. 16), and $B ̃ r , m , n ( φ ) = B c , r 2 ( φ , m , n ) + B s , r 2 ( φ , m , n )$⁠. If $B ̃ r , m , n ( φ ′ )$ is finite at $q ( φ ′ ) = m / n$⁠, then an m/n island can form at the $φ ′$ surface. For example, whereas the axisymmetric tokamak has laminar flux surfaces (Figure 1(a)), the structure of the n = 1 and n = 2 vacuum islands is evident in the Poincare plots in Figures 1(b) and 1(c). Here, the applied field is from the Internal coil (I-coil) on DIII-D and the 2D equilibrium is taken from the 15 MA equivalent ITER baseline scenario. The equilibrium selected has a q value at the 95% flux surface q₉₅ = 3.2 and q₀ > 1 (even though this scenario typically has large sawtooth oscillations). Fields with toroidal and poloidal harmonics that do not match the rational q = m/n value and fields at other surfaces are all considered “non-resonant”. These fields bend the field lines but do not drive reconnection. Here, the “resonance” is with respect to the helicity of the magnetic field line. It is also important to note that resonant fields are often identified based on calculations using the applied vacuum field (as done here).¹⁷ 

There is a special subset of non-resonant fields that can modify the total non-axisymmetric field by coupling to normal modes of the plasma.¹⁸ Here, we use the term “plasma response” to refer to any additional fields generated inside the plasma itself. Non-axisymmetric coils can be used to excite stable modes as is done in active MHD spectroscopy^19–21 and/or suppress unstable modes as when active magnetic feedback is used to control resistive wall modes (RWMs).²² The plasma response plays an important role in studies of driven magnetic reconnection because it can modify the total resonant field. Therefore, applied vacuum fields that are considered as non-resonant with respect to the magnetic field line can nevertheless provide a drive for magnetic islands by coupling through the plasma response. Depending on plasma conditions, the normal modes can be global, laminar deformations of the field lines (i.e., “kink” modes), and/or can include field line reconnection (i.e., “tearing” modes). To capture fine details of the plasma response, it is expected that nonlinear, two-fluid, visco-resistive MHD simulations are required. These simulations are expensive in terms of both computational resources and execution time. Fortunately, linear, single-fluid, ideal MHD models have been able to provide a quantitative basis for classifying 3D fields in a more tractable way that still captures the dominant features of the plasma response.²⁹ 

A criterion for quantifying the drive for islands using an ideal MHD model was proposed by Park and is referred to as the “overlap” field.²³ The overlap is defined in Eq. (1) of Ref. 24, and can be expressed mathematically as a dot product of the applied vacuum field and the dominant magnetic field, which is an orthonormal eigenfunction taken from a singular-value decomposition (SVD) of a matrix that linearly maps externally applied fields to the total resonant field. The resonant fields are extracted from the shielding current at each rational surface as described by Boozer and Nührenberg.²⁶ In ideal MHD, the shielding current exists because the safety factor profile is preserved in the perturbed equilibrium calculations completed with IPEC. The metric has been useful for understanding n = 1 error field optimization results in DIII-D^23,25 and for assessing the potential of non-axisymmetric coil systems to correct the expected n = 1 error fields in ITER.²⁷ 

In this paper, we use a dimensional form of the overlap. In the notation of Eq. (3) in Ref. 24, $C = | Φ → · Φ d → | / | Φ d → |$⁠, where $Φ →$ is the applied field at a surface near the plasma boundary (e.g., the 99% flux surface) and $Φ d →$ is the dominant field at the same surface, which gives the overlap units of Gauss. In Section V, both the overlap and the “critical overlap” will be used. The critical overlap is defined as the overlap field evaluated at a time just prior to the onset of an island that has the same toroidal mode number as the applied field.

With a linear model, an overlap field is defined independently for each toroidal harmonic and is further decomposed into the orthonormal basis identified by the SVD. As such, the overlap quantifies the plasma-mediated coupling of the external field to the resonant field (i.e., the drive for island formation) through individual modes. Therefore, a key question is under what plasma conditions is the resonant drive dominated by a single mode or by multiple modes. Or, in mathematical terms, what is the relevant basis set for mapping externally applied fields to the resonant field drive.

The critical overlap field was studied in a series of experiments in DIII-D, EAST, and RFX-mod tokamaks. The experiments are part of an ongoing joint experiment in the ITPA MHD Topical Group. In these initial studies, the goal was to obtain data across a number of different plasma parameters including density, q₉₅, toroidal field, and for various coil configurations in order to ascertain what effect, if any, n = 2 fields may have on the plasma as well as to compare the impact of different poloidal spectra.

Although the plasma shapes studied are similar to the single-null plasmas envisioned for ITER, exact reproduction of the ITER shape was not pursued. In DIII-D, upper single-null Ohmic and L-mode discharges with the dominant X-point in the direction opposite to the ion grad-B direction were generated, as shown in Figure 2(a). This was done to avoid transitions into the H-mode when neutral beam injection (NBI) was used for diagnostic purposes and to vary the plasma rotation. External n = 2 fields were applied using the single-turn Internal coil (“I-coil”), two rows of six coils located above and below the midplane, while n = 1 error field correction was done with the four-turn Correction coil (“C-coil”), a single row of six coils at the midplane. In EAST, lower single-null Ohmic plasmas were studied and n = 2 fields applied with two rows of eight four-turn “RMP” coils located between the vacuum vessel and a set of copper vertical stability plates, Figure 2(b). Since the measured n = 1 intrinsic error field in EAST was found to be small,³³ no n = 1 error field correction was used. In RFX-mod, lower single-null Ohmic discharges were studied. RFX-mod is typically run as a reversed field pinch but can be operated as a diverted tokamak.²⁸ The plasma shape during the flattop is shown in Fig. 1(d), together with the radius of the graphite tiles facing the plasma and the non-axisymmetric coils. Fields were applied with 192 non-axisymmetric coils that fully cover the torus. The coils are distributed over 48 toroidal angles at 4 poloidal angles with each coil independently controlled, allowing fine control over different 3D field spectra. As in EAST, n = 1 error field control was not used.

Despite extensive experiments in RFX-mod, it was not possible to access the n = 2 error field threshold. An example attempt is shown in Figures 3(a)–3(d) where the error field ramp is applied during the time interval of 0.3–0.8 s. In order to maximize the plasma sensitivity to applied error fields, a low value of q₉₅ = 2.7 (B_T = 0.55 T) was chosen and the density was kept as low as possible with line-averaged values $n ¯ e = 0.5 × 10 19 m − 3$⁠. The n = 2 error field is produced using only the outboard and upper coil rows so as to mimic a situation typical of larger tokamaks. In this case, the differential phase between upper-outboard coil rows is 0 deg. The n = 2 error field is ramped linearly up to 45 A, close to the maximum allowed coil current of 50 A. The n = 2 coil current amplitude is shown in Figure 3(d) separately for the two rows, while the actual coil current at t = 0.7 s is shown in Figure 3(a). A very clean n = 2 sinusoid is produced owing to the large number of coils in the toroidal direction. The lack of error field penetration is evident in the absence of any change in the frequency of a 2/1 tearing mode as seen in the magnetic spectrogram of a pick-up coil located just behind the graphite tiles on the outboard midplane, Figure 3(c). The tearing mode is always present in these plasmas and its frequency slightly decreases when the n = 2 error field is applied, recovering when it is switched off, which indicates that the rotation braking is small.

Various quantities were scanned while searching for the conditions where n = 2 error field penetration may occur. In all cases, only small rotation braking was observed; no error field penetration was found. The following studies were pursued:

1. Additional coil rows were powered up to full poloidal coverage to increase the applied n = 2 error field amplitude,

2. Using only outboard and upper coils, the differential phase was varied for values 0, π/2, π, and 3/2π. The level of tearing mode braking has a small variation, with a maximum between π and 3/2π.

3. q₉₅ was varied in the range from 3 to 2.3 by increasing the flattop plasma current,

4. The shape was changed to bring the plasma nearer to the coils.

Note that as q₉₅ decreased, the 2/1 tearing mode increased in amplitude and tended to lock disrupting the discharge. Due to this reason, values of q₉₅ below 2.3 could not be explored.

While the n = 2 threshold could not be accessed, it was possible to exceed the n = 1 limit. This was demonstrated in similar plasmas by ramping an n = 1 error field, using only the outboard coils, as shown in Figures 3(e)–3(h). In this case, the disruption threshold occurs at an n = 1 coil current amplitude of about 17 A. Note that this is strictly not a case of error field penetration since an n = 1 island already exists prior to the application of the external field.

The only conditions where n = 2 error field penetration has been obtained are in circular Ohmic plasmas with edge safety factor q(a) below 2. This special regime can be reached by suppressing the m = 2, n = 1 resistive wall mode with magnetic feedback, as shown in Figure 4. The plasma current waveform is tailored so that q(a) first reaches a value of 2.5, and then, a second current ramp is added and q(a) is brought below 2. Feedback control of the 2/1 RWM enabled at 0.3 s successfully maintains the mode at a very small level. During the phase with q(a) < 2, an m = 1, n = 2 error field ramp is added using the entire set of coils and the plasma disrupts at about 0.65 s. The magnetic spectrogram indicates that a small amplitude 3/2 tearing mode is present before the field is applied. The mode gradually slows down up to the point where the disruption occurs. No sign of n = 1 modes at any frequency is found. This indicates that n = 2 error field penetration occurs in these particular cases; however, there is already a saturated n = 2 island before the field is applied, preventing a measurement of the critical field level for island formation.

In Section V, it is explained that in RFX-mod the overlap field is always too small to achieve error field penetration. As discussed in Section IV C, this is not the case in DIII-D and EAST where the non-axisymmetric coils are much smaller in poloidal extent.

In contrast to the results in RFX-mod, it is possible to access the n = 2 error field threshold in DIII-D. In these experiments, n = 2 fields are applied in Ohmic plasmas with various plasma densities, as shown in Figure 5. As the applied n = 2 field is ramped, non-rotating n = 2 islands are generated as measured with external saddle loops on the outboard midplane, Figures 5(e) and 5(d). Interestingly, the n = 2 mode onset is always followed within a few hundred ms by the growth of an n = 1 island that leads to a disruption under certain conditions. As observed also in n = 1 error field penetration studies, n = 2 island onset suppresses the sawtooth crash. In DIII-D, disruptions occur at higher density and are preceded by the growth of the n = 1 island, which tends to grow at the same time as the n = 2 island.

Even when the locked modes do not terminate the discharge, the onset of the n = 1 and 2 island chains leads to a significant reduction in confinement, as shown in Figure 6. A comparison of electron temperature profiles before the field is applied and after the onset of both islands shows the flattening of the gradient near the q = 2 and 3/2 surfaces, a 30% reduction in the core temperature, and a reduction in the thermal energy confinement time from 50 to 20 ms. In this L-mode discharge, diagnostic NBI pulses are used to obtain charge-exchange recombination spectroscopy (CER) measurements of the main ion (deuterium) toroidal rotation.³⁰ Recent progress has been made in overcoming challenges with interpretation of edge main ion CER measurements,^31,32 allowing direct observation of the impact of magnetic islands on the main ion rotation for the first time. In order to minimize the injected torque, in some cases the NBI pulses for the core measurements are slightly offset in time from those for the edge measurements. Following the island onset, the rotation near the q = 2 is reduced close to zero, the rotation shear inside q = 2 is completely lost, and the electron temperature gradient near the q = 2 and 3/2 surfaces goes to zero. In this case, the islands stabilize after the applied field is turned off and the plasma profiles and confinement recover to the level observed before the field is applied.

In the EAST tokamak, as in DIII-D, the n = 2 error field threshold is also accessed in Ohmic discharges. Figure 7 shows a series of lower single-null discharges at $n ¯ e ∼ 1.1 × 10 19 m − 3$⁠) and q₉₅ = 3.2 where the toroidal phase of the applied n = 2 field was varied. Disruptions occur following the penetration of the applied field. In each case, an n = 2 island either precedes or is coincident with an n = 1 island as shown for single discharge in Figure 8. The position of the 3D coils behind a set of copper passive plates complicates detection of the mode onset, owing to large eddy currents in the plates. However, the penetration of the applied field is evident in sensors near the midplane on the low field side (LFS) and high field side (HFS) where the coil-sensor coupling is reduced. The measurements are reported with respect to an offset taken at 2.9 s just before the n = 2 field is applied and compensated for the applied DC field.

In EAST, the disruptions are preceded by an increase in the poloidal flux errors on the high field side from the axisymmetric shape and plasma current control system. Meanwhile, some of the poloidal field coils that make up the central solenoid reach the lower voltage limit, leading to an oscillation in the plasma current, and the later disruption. In contrast, in DIII-D disruptions occur only when the applied field is turned off. This suggests that discharges with finite size islands may be more difficult to control using large superconducting coils located far from the plasma.

In a series of controlled experiments, the error field threshold was measured in DIII-D and EAST over a range of densities, q₉₅, and the poloidal spectrum of the applied field. In this section, the threshold is reported in terms of the critical overlap field and is compared to an existing database of n = 1 error field thresholds.²³ A main finding of this analysis is that the n = 2 critical overlap is similar to the results in n = 1 error field studies, indicating that the plasma is equally sensitive to both toroidal harmonics and that it is the poloidal mode spectrum that determines if the applied field can exceed the error field threshold.

In a series of controlled experiments, the error field threshold was measured in DIII-D Ohmic plasmas over a range of densities from just above the low density locking threshold obtained with optimal n = 1 error field correction, and extending up to the maximum density where n = 2 field penetration could be accessed within the current limits of the coil power supplies. The critical overlap field was computed using IPEC and is plotted in Figure 9 as a function of the local electron density near the q = 3/2 surface. Although further studies and analysis are needed to resolve the parametric dependence on the density, it is clear from the comparison with an existing database for n = 1 thresholds (also shown) that the critical overlap and density dependence are similar. In other words, the plasma is no less sensitive to n = 2 fields relative to n = 1.

In EAST, the critical overlap field was measured as a function of q₉₅ at $n ¯ e = 1.1 × 10 19 m − 3$ by varying both the toroidal field and the plasma current. At each q₉₅ value, the differential toroidal phase between the upper and lower RMP coils (⁠ $Δ ϕ u l$⁠) was varied in order to test the sensitivity to the poloidal spectrum of the applied field. A detailed scan of $Δ ϕ u l$ was completed at q₉₅ = 3.3 and two point scans were done otherwise. At the lowest value of q₉₅, the applied field generated an island for most values of $Δ ϕ u l$ and the lowest critical overlap of 3 Gauss was found; however, at q₉₅ approaching 5 it was not possible to generate any mode up to overlap fields exceeding 6 G. Figure 10 shows the n = 2 critical overlap field as a function of q₉₅ for the cases where locked modes and disruptions were observed; if the plasma was stable, the maximum overlap field is reported. Taken together, the data suggest that the critical overlap increases with q₉₅ by a factor of two. Similar field levels and safety factor trend are seen in the n = 1 critical overlap in low density Ohmic discharges from DIII-D, KSTAR, and JET, showing again that the plasma is equally sensitive to n = 2 fields.

A detailed scan of the critical overlap as a function of toroidal field has not yet been completed at fixed q₉₅, but existing data suggest little to no toroidal field dependence. This is evidenced by the constant critical overlap from two discharges at q₉₅ = 3.9 and 4.2, which had toroidal fields of 1.55 and 2.25 T. Additional data are needed at a higher field (but the same safety factor) to see over what range of toroidal fields this result holds.

A detailed scan of $Δ ϕ u l$⁠, which varies the poloidal spectrum of the applied radial field, demonstrates that in Ohmic discharges the dominant field determines the coil configurations capable of driving tearing. Figure 11 shows the critical overlap and the overlap field for both n = 2 and n = 1 error field experiments along with the overlap field in units of field relative to the coil current. The n = 1 experiments are described in a recent paper by Wang.³³ For both toroidal harmonics, locked modes are readily observed for coil configurations with large overlap fields and no field penetration is found for fields with weak coupling. The critical overlap is similar for both n = 1 and 2, with some apparent systematic variability with $Δ ϕ u l$ that is nearly within the error bars. More careful study of this trend is outside the scope of this paper, but may be warranted since, in principle, the threshold should be independent of the external field configuration, but in fact the critical overlap decreases somewhat with the overlap field. We speculate that the variation is due either to differences in the evolution of the plasma equilibrium as the fields are ramped, or to the role of secondary plasma modes. Despite these variations, it is clear that the coupling to a single dominant mode can identify which poloidal field harmonics (i.e., coil configurations) preferentially increase the resonant field and drive island formation.

IPEC modeling of the overlap field over a range of poloidal spectra and q₉₅ shows that the strength of the coupling to the resonant field decreases with q₉₅, as shown in Figure 12. The calculations were done using a set of generated equilibria created at different plasma current and toroidal field values. The poloidal spectrum of the applied field can be varied so as to drive nearly zero resonant field (i.e., generate a small overlap field), or so as to exceed the error field threshold of 3 G and cause locking. This increase in the sensitivity to 3D fields at low q₉₅ may help to explain the increased disruptivity of the 15 MA ITER operational scenario in DIII-D, which may occur due to the impact of 3D field sources generated by both external currents and plasma-based MHD instabilities.³⁴ 

Since plasma rotation is known to play an important role in the screening of resonant magnetic fields, it is important to understand how error field thresholds vary with plasma rotation and to account for that information when extrapolating error field thresholds to ITER conditions. In DIII-D, an experiment was conducted to investigate the rotation scaling of the n = 2 threshold near the expected rotation level in ITER, q₉₅ = 3.2, and $n ¯ e ∼ 1.5 × 10 19 m − 3$⁠. At the rotation level closest to the intrinsic rotation level, NBI was injected only before the external field was ramped (i.e., NBI was not enabled at the time of island onset) while modulated NBI was used at the highest rotation level, which had 0.35 N m of (time-averaged) injected torque. At the lowest rotation level, the resulting rotation near the q = 3/2 surface is close to a recent prediction of the edge intrinsic rotation in ITER.³⁵ 

At the highest rotation studied, the rotation profile measured before the n = 2 field is applied (at 1405 ms) is a good proxy for the rotation profile just before island onset and rotation collapse, as shown in Figure 13. In this discharge, the n = 2 island onset occurred at 2585 ms. At 2165 ms, the rotation profile remains similar to the profile at 1405 ms, indicating that the applied 3D field torque is negligible. As discussed in Section IV C, the rotation collapses near the q = 3/2 and q = 2 surfaces following the onset of the island chains, leading to a complete loss of rotation shear in the core. After the applied field is turned off, the islands are suppressed and the rotation profile returns to the original level.

Since the 3D field torque appears negligible during the 3D field ramp, it is meaningful to compare the rotation profiles at different torque levels at the time before the field is applied, as shown in Figure 14. In #164938, NBI pulses are applied only before the 3D field is ramped, leading to a main ion toroidal rotation profile that is near zero in the core, and finite near the plasma boundary. As the NBI torque is increased, a rotation profile becomes peaked in the core. At near zero NBI torque, the rotation near the q = 3/2 surface, which is dominated by the intrinsic rotation, is near the intrinsic edge rotation level recently predicted for ITER, around 10 krad/s.³⁵ Here, “edge” refers to the region 0.8 < ρ < 1.0, where ρ is the normalized toroidal flux. Note that at low q₉₅, the q = 3/2 surface is at larger minor radius near ρ ∼ 0.6. The changes in the rotation profile are reminiscent of the observed rotation profile changes in C-MOD discharges where increased density leads to a transition from peaked to hollow rotation profiles.^36,37

The critical overlap field in these discharges exhibits a strong scaling with the rotation near the q = 3/2 surface, as shown in Figure 15, increasing by 57% as the rotation increases by 73%. Given the sparsity of data, it is not meaningful to compare various functional dependencies. However, a simple linear fit to the existing data can be used to extrapolate to the nearby expected ITER rotation level. Assuming no size scaling and neglecting any shielding effect from finite plasma temperature (i.e., constraining the fit to pass through the origin), the expected n = 2 overlap field in ITER is 1.6 Gauss. Since an inverse size scaling is expected from n = 1 field studies, this can be taken as an upper bound for the ITER threshold at this density. Note that the rotation shear is also changing dramatically near the q = 3/2 surface, which may have an impact on tearing mode stability. A scaling of the threshold (in terms of the overlap field) with rotation has also been observed in NSTX, where the threshold increased by 120% when the rotation increased by 73% (Figure 10 in Ref. 24).

As discussed in Section IV B, it was not possible to access the n = 2 error field threshold in RFX-mod (run as a low q tokamak), but it was possible to use n = 1 fields to slow down fast rotating n = 1 islands and trigger disruptions. A model of the RFX-mod coils was developed in IPEC in order to evaluate the overlap fields and compare against the thresholds found in DIII-D and EAST. For this purpose, a plasma equilibrium with q₉₅ = 2.8 and diverted shape was reconstructed using the EFIT code. The n = 1 overlap field that locked at 17 A was computed for a differential toroidal phase shift between the currents in the two coil rows of zero degrees and found to be 0.99 G, while the maximum n = 2 overlap field for the dominant (first) mode was always well below this level at 0.8 G. Assuming that the critical n = 2 overlap is the same as for n = 1, then a 20% increase in the maximum overlap is needed to drive an n = 2 island. Figure 16 shows the trend of the dominant overlap field (overlap 1) along with the overlap field of the second mode. Interestingly, the overlap field for this mode (overlap 2) exceeds 1 G, but the increased stability of this mode must be associated with a higher threshold since a locked mode did not occur.

The reason why the overlap field is small is due to the large poloidal extent of the coils, which produce poloidal harmonics that are too small to couple to the dominant mode. The external field is dominated by poloidal harmonics with m < nq, whereas the dominant mode is dominated by harmonics m > nq. This spectral separation of the applied field and the dominant mode results in a small overlap field.

Although a complete set of empirical scalings of the n = 2 error field penetration threshold has not yet been obtained, it is clear from the existing experimental data that a field of any toroidal harmonic will drive an island if the poloidal spectrum is such that it couples sufficiently well to the resonant field through the plasma response. This effect is captured well by the overlap field, which provides a quantitative metric for assessing the propensity of a field to drive an island in a plasma with a given safety factor profile. The propensity of a field to generate islands appears to be unrelated to the toroidal spectrum. Instead, it is the poloidal spectrum that determines the effect. These results have important implications for setting the engineering tolerances and control requirements for 3D fields in ITER.

First, future error field optimization and assessment studies should consider n > 1 error field sources. Until a validated theory is available, potentially new error field sources and control coil geometries can be assessed using the overlap field, which provides a suitable ordering parameter to assess the drive for magnetic reconnection. Unless different trends are revealed by further experiments, error field thresholds for low q plasmas should be imposed at the same level for all toroidal harmonics.

Second, metrics other than the overlap field should be considered. For example, the neoclassical toroidal viscous torque (NTV) can provide a useful objective function for 3D field optimizations as the plasma pressure increases³⁸ particularly if the overlap field is weak (as it is when m < nq). Previous work has highlighted the role of these non-resonant fields even when error field control currents have been optimized using conventional methods.³⁹ These second order effects are also seen in the dependence of the critical overlap field with the coil figuration as reported in this paper in Figure 11. The trend may be due to the impact of other normal modes of the system, which points to the possible need of multiple rows of coils to control selected plasma parameters for specific functions, and also to allow for higher order optimizations of applied 3D fields.

Third, previous computations of the expected error fields in ITER⁴⁰ have shown that the worst n = 2 error fields are on the order of 1 G. This is close to experimental error field thresholds at low rotation. Since the actual intrinsic error field in ITER cannot be predicted with sufficient accuracy before the experiment is built, the tolerances for ITER should not be increased unless it can be shown that there is sufficient capability to achieve all ITER 3D field tasks with the available control coils.

Also, since error field thresholds appear to increase with the edge safety factor, assessments and optimizations of n > 1 fields are likely to pose a problem only when q₉₅ is reduced below 5. This provides some guidance on when to schedule n = 2 field optimization studies. Furthermore, if the capability to control n > 1 fields does not exist, it may be necessary to design plasma scenario trajectories such that low safety factors are encountered only when other plasma parameters, such as the density, are such as to increase the error field thresholds.

Finally, these results prompt the question as to whether the existing 3D coil configuration in ITER is capable of controlling of n > 1 error fields should they happen to exist at a significant level. The answer depends on the plasma scenario in question, on the availability of the in-vessel coils, on the spectrum of the intrinsic error field, and on the required in-vessel coil current to suppress ELMs. A detailed study of these issues is beyond the scope of this paper. However, we note that in order to allow for the greatest flexibility, a prudent step may be to avoid hardwiring the side ex-vessel coil for n = odd operation so that it is available to control n = 2 fields. These coils could be used together with the in-vessel coils to achieve 3D field tasks (e.g., error field control, ELM suppression, and rotation control).

In summary, tokamak equilibria in several devices operating at low edge safety factor, low density, and low rotation are found to be sensitive to applied n = 2 fields. The measured error field penetration thresholds for n = 2 and n = 1 magnetic fields are found to be similar in the plasma regimes likely to be encountered during ITER operation when the plasma current is increased leading to q₉₅ < 5. Linear plasma response calculations using the single-fluid ideal MHD code (e.g., IPEC) can be used to quantify the degree to which external fields couple to resonant fields through the plasma response. These calculations can be used to calculate the critical overlap field, which provides a meaningful ordering parameter for documenting empirical scaling laws and for comparing thresholds. These results have important implications for the suitability of 3D coil configurations in ITER and future devices and for the required steps for commissioning new devices. In particular, an error field control mission for the ITER in-vessel coils should be considered in light of these new results with some experimental time dedicated to assessing whether n > 1 error fields exist at a level that impacts plasma performance. If for whatever reason the in-vessel coils are not available (e.g., if all of the available current is required to suppress edge localized modes), modifications to the side ex-vessel coil to allow application of n = 2 fields would be prudent.

Although the overlap field can be used to assess the potential drive for tearing, it does not provide a quantitative prediction of the error field threshold itself. Further work is needed using extended MHD models in order to elucidate the relevant physics governing driven magnetic reconnection in tokamaks and provide a predictive capability using a validated physics model. The data presented in this paper would be suitable for such model validation efforts. Given the interactions between islands with multiple toroidal harmonics observed in these experiments (including possible interactions with m/n = 1/1 islands associated with sawtooth oscillations), nonlinear simulations will be required to fully capture the stability characteristics and dynamics leading to destabilization of n = 1 locked modes and eventual disruption.

The authors would like to thank C. Holcomb and P. Zanca for useful input on the original manuscript. This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Awards DE-FC02-04ER54698, DE-AC02-09CH11466, DE-FG02-05ER54809, and DE-FG02-04ER54761. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/D3D_DMP. This work has been partly carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.