Direct measurements of internal structures of born-locked modes and the key role in triggering tokamak disruptions

A crucial issue in realizing fusion in tokamaks is to avoid major plasma disruptions. The onset of the disruption is commonly associated with low-order, nonrotating magnetic islands, i.e., m = 2/n = 1 islands (m and n: poloidal and toroidal mode numbers), called locked modes.^1–4 Although the physics mechanism describing how the disruption is triggered by locked modes remains elusive, suggestive evidence can be found in previous experiments.^5–7 An interesting early study simulated the plasma major disruptions due to locked modes by ramping up external helical coil current. The result shows that the observed disruption limits are correlated with the onset of overlapping magnetic islands that extend to the limiter.⁵ Therefore, the investigation of the existence of small edge island chains, during the mode locking phase, which may sit between the q = 2 rational surface and the last closed flux surface (LCFS), and their roles in triggering plasma disruption is of great interest.

However, the identification of this process in experiments is quite challenging. The main difficulty is that the essential information on internal structures of eigenmodes is impossible to be diagnosed by existing systems due to a nonrotating character of locked modes. Currently, locked modes are inferred from a localized flattening region in the electron temperature T_e profile. This method is applicable only when the island width is much wider than the critical island width.⁸ It is not applicable if edge island widths are narrow or heat flux toward the edge is substantial. Both conditions may be easily satisfied when plasma thermal quench begins.

For realizing the direct measurements of internal structures, a novel dual tangential soft X-ray imaging system (dual-SXI) has recently been developed in DIII-D tokamaks to measure the radiation symmetry breaking in the soft X-ray (SX) band in the presence of static, three-dimensional helical structures. Based on the dual-SXI data obtained in the dedicated experiments of the born-locked modes, we report the first direct measurements of coupling between the low-order locked island and the plasma separatrix mediated by small nonrotating edge islands preceding a major disruption.

The dual-SXI consists of two duplicate tangentially viewing camera systems with identical views but toroidally 120° apart. Each visible camera observes the phosphor, which monitors filtered SX radiation via pinhole optic,^9,10 as shown in Fig. 1(a). The signal from each image pixel is an integral of SX emission along the sightline, expressed as $∑wij·Isx$⁠, where i and j are the indices of image pixels, w^ij is the geometry weight function, and I_sx is the SX emission. The dual-SXI heavily weighs the plasma edge. These features allow differential imaging from two systems, referred to as δΛ imaging, to extract nonaxisymmetric, nonrotating edge perturbations. The linearized equation is given by

where subscript 0 is the equilibrium quantity; Φ₁ and Φ₂ are the phases of nonrotating perturbation, related to the locations of the two soft X-ray imaging systems correspondingly, defined as mθ–nϕ–χ (θ and ϕ: poloidal and toroidal angles in flux coordinates, respectively; χ: phase angle of the mode). The offset of the differential image, which is dominantly contributed by the tiny asymmetry of views, $w1ij$⁠, i.e., $∑w1ij·Isx0$ term, is estimated before the onset of static modes. The δΛ image reflects the unique combination of mode numbers and the structure, and its intensity linearly scales with the SX emission difference between different phases of the mode. The system has a fine spatial resolution and is able to resolve the island width down to ∼1 cm.

The experiment is conducted in an inner wall limited oval cross-section plasma without error field correction, having a plasma current of ∼1.25 MA, a toroidal magnetic field of ∼–1.9 T, and a line averaged electron density of ∼2 × 10¹⁹ m⁻³. The penetration of the intrinsic error field routinely leads to plasma disruption by opening large stationary magnetic islands, known as “born-locked” modes.^11–13 There is no auxiliary heating in order to minimize the damage to the first wall. The safety factor q at the limiter is $≳4$⁠. Figure 2 shows the SX images obtained in two cameras preceding the onset of locked modes, integrating from 5.0 s to 5.075 s of shot 172102. Note that the amplitude is normalized to the maximum of SX emission observed in the plasma core for both systems. It is found that two images are nearly symmetric, resulting in the absence of the perturbation in the δΛ image, as shown in Fig. 3(a).

The δΛ image with the first clear perturbation is found at ∼5.1 s, as shown in Fig. 3(b) with red and blue bands. This is the time when the slow increase in the integrated n = 1 poloidal magentic field perturbation b_p, meausred by magentics,¹⁴ occurs. The contour lines of the minimum q that each sightline covers are overlaid, showing that the large amplitude of fluctuation resides inside and near the q = 2 rational surface. Here, the equilibrium is reconstructed by the EFIT code.¹⁵ The observed images are similar as time evolves, persisting from ∼5.1 s to ∼5.44 s until the onset of plasma disruption, as seen from Figs. 3(b)–3(d). Figure 3(f) shows the δΛ image conditionally averaged over ∼20 frames in 8 shots during the nonlinear saturation phase of locked modes before the thermal quench. The image is mapped back to the camera views at the tangency plane and is further overlaid by the q = 2, 3, and 4 rational surfaces. It is immediately realized that the dominated perturbation is well-aligned with q = 2 rational surfaces in the poloidal direction.

To understand the measured δΛ images, a forward modeling based on Eq. (1) using the real dual-SXI geometry is developed. Constant SX emission along a magnetic field line is assumed, i.e., a flattened SX emission across the island O-point but a peaked one across the X-point. The similarity between the conditionally averaged images and synthetic images is roughly estimated by the mean squared deviation (MSD). In detail, a spatial calibration that is conducted before the experiment produces a mapping to address how each pixel correlates with the real location on the phosphor. On the other hand, the synthetic images are interpolated on the same locations of the image pixels. Then, the median filter is applied to both the synthetic images (⁠$δΛ̂$⁠) and the experimental images (δΛ), using the same sampling window, to suppress the noise but preserve the image edge. Finally, the similarity metric is calculated as follows:

where i_m and j_m are the maximum values of the indices of the image pixels. Figure 4(a) shows that the optimized phase of the 2/1 island is about 265° in DIII-D machine coordinates, which is consistent with the phase of b_p at ∼9°, as shown in Fig. 3(g), and is also consistent with the phase of the intrinsic error field reported previously in Ref. 16. Scanning the widths of the 2/1 island reveals that a broad width of >8% of normalized minor radius is necessary to interpret the observed image.

Figure 5(a) shows the synthetic images using a 2/1 island, which centers at a normalized poloidal flux of Ψ_n ∼ 0.67 with a width of ∼0.09 of minor radius and a phase of 270°. This does not contradict with the observed flattening of the electron temperature T_e profile measured using a Thomson scattering (TS) system at ∼5.3 s which is centered at Ψ_n ∼0.67 with a width of ∼0.095, as shown in Fig. 6. The synthetic image exhibits a reasonable match with the observed image inside q = 2, illustrating a well-known fact that a large 2/1 island is often excited and locked to the error fields before plasma disruptions.

It is also recognized that a weak perturbation, having fine band structures, develops across the q = 3 rational surface and reaches q = 4, which is an apparent mismatch between the modeling of Fig. 5(a) and the observations of Fig. 3(f). This suggests that the presence of the 2/1 island alone cannot explain the observed vast expansion of perturbations toward the LCFS. Characteristics of these fine structures are as follows: (i) the bands are well-aligned with the projection lines of the q = 3 and 4 rational surfaces in the poloidal direction, for example, indicated by arrows in Figs. 3(c) and 3(f); (ii) the observed perturbation amplitude outside q = 3 projection lines is much weaker, i.e., only ∼20% of that inside q = 2. Moreover, since b_r of the n = 2 and 3 components measured by magnetics are small, it is considered that the 3/1 and 4/1 island chains should be taken into account.

The widths of 3/1 and 4/1 islands are investigated using the above-mentioned best-fitted phase and width of the 2/1 island chain. The results are shown in Fig. 4(b). In contrast to the broad width of the 2/1 island, the narrow widths of these edge islands down below ∼3% fit the conditional averaged image well. The optimized phase for the 3/1 island is around 0° but with a large uncertainty of ∼100°. This may be due to the line-integrated feature of the system. The optimized phase for the 4/1 island is around 60°. By integrating these small edge island chains into the dominant 2/1 island chain, the synthetic image in Fig. 5(d) successfully reproduces most of the fine structures outside the q = 2 rational surface in Fig. 3(f) and exhibits remarkable agreement with the measured δΛ image. The best-fitting values used in Fig. 5(d) suggest that the 3/1 and 4/1 islands are centered at ψ_n ∼ 0.822 and ψ_n ∼ 0.91 with the widths of ∼3% and ∼2.5% of the normalized minor radius and the phases of 0° and 60° in DIII-D machine coordinates, respectively. The synthetic images calculated from only the 3/1 and only the 4/1 island chains are also given in Figs. 5(b) and 5(c). It should be noted that the island chains are treated as being stationary in the forward modeling because the measured variation of χ from magnetics is as small as ∼3° for each frame, as shown in Fig. 3(g). Based on the consistency among the measurements and modeling results, it is concluded that small edge island chains are destabilized and locked between the outer 2/1 island separatrix and the LCFS. Importantly, according to the best-fitted locations and radial widths of 2/1, 3/1, and 4/1 island chains, it is implied that the island chains are isolated from each other before the thermal quench.

Figures 6(a) and 6(b) show the time evolution of T_e and dT_e/dΨ_n contours obtained from the TS system. The following features are recognized: (i) it is found that the localized low T_e gradient but not the fully flattened region appears around q = 3 and 4 rational surfaces, having quite narrow widths of ∼0.03 and ∼0.04 of Ψ_n. These regions are considered to be induced by the 3/1 and 4/1 island chains. (ii) The T_e flattening at the q = 2 rational surface slowly expands but remains isolated from the flattening at the q = 3 rational surface by a high T_e gradient region for several energy confinement times, as seen from Fig. 6(b). On the other hand, the radial width of the high T_e gradient region d₂₃ slowly shrinks, roughly depicted in Fig. 6(b). For example, d₂₃ is inferred as ∼3.5 cm at 5.16 s, but ∼1.5 cm at 5.4 s, given in Fig. 6(d). This is accompanied by the increase in perturbations derived from the corresponding regions of the δΛ image, which are dominated by 2/1 and 3/1 island chains, respectively, shown in Fig. 6(c). The observation supports the existence of the intact flux surface between the 2/1 and 3/1 islands; (iii) although the 2/1 island has a considerable width up to ∼13 cm at 5.4 s, a T_e gradient in the core is still sustained, as seen from Fig. 6(d), and the plasma does not disrupt. That is, the isolated island chains lead to a long quasistationary period of ∼300 ms.

The thermal confinement suddenly bifurcates at ∼5.44 s from a quasistationary period into the quench. The major change observed immediately before the bifurcation is the rapid reduction of the high T_e gradient region between the 2/1 and 3/1 islands after ∼5.4 s, accompanied by a sudden increase in b_p having n = 1, as shown in Fig. 3(g) and the sequential increase in the 3/1 and 2/1 perturbation amplitude measured by dual-SXI, as shown in Fig. 6(c). Quantitatively, b_r, dominated by 2/1 surface islands, increases from 41 G to 53 G, roughly suggesting an increase in the 2/1 island width by ∼1.8 cm. The gap of d₂₃ ∼ 1.5 cm estimated at ∼5.4 s can be overcome, shown in Fig. 6(d), demonstrating that the 2/1 island starts to overlap with the outer edge islands. After ∼5.44 s, a rapid T_e drop over all radii is correlated with the sudden evanescence of coherent b_r. This may indicate the generation of a broad stochastic layer across the entire outer region, which connects the core plasma to the wall along open field lines and flattens the T_e profile deep into the core. It should be noted that the fact that plasma disruption triggered by locked modes is quite similar to that by rotating islands, reported previously in Ref. 17, reveals that the overlap of the large 2/1 island with narrow edge island chains is a common mechanism responsible for the thermal quench and plasma disruption in tokamaks.

A simulation of error field penetration at multiple rational surfaces and its impact on electron thermal confinement by a nonlinear single fluid magnetohydrodynamic code (TM1)^18–20 is qualitatively consistent with the observations, using measured plasma profiles, reconstructed equilibrium, perpendicular heat diffusivities of χ_⊥ = 0.5 m²s⁻¹ and $χ∥/χ⊥=109$⁠.²¹ As seen from Fig. 7(b), the error field of n = 1 with the amplitudes of 2 G, 1 G, and 0.5 G at the corresponding q = 2, 3, and 4 rational surfaces is able to fully penetrate. This process generates a series of locked magnetic island chains, which dominantly consists of 2/1, 3/1, and 4/1 locked islands. The edge small locked islands, i.e., 3/1 and 4/1 islands, govern the edge cooling process near the plasma boundary by producing field stochasticity there. Meanwhile, the temperature of the major 2/1 island slowly decreases as its outer separatrix expands outward. A very flattened T_e profile deep into the core at t = 327 ms in Fig. 7(a) is similar to the observed T_e profile at 5.44 s preceding the plasma disruption in Fig. 6(d). This is the time when the 2/1 island nearly overlaps with the 3/1 island boundary, as illustrated in Fig. 7(c).

Other alternative explanations are considered implausible after examining the power balance equation

where W is the total plasma energy density, P_H is the ohmic heating power, nχ_⊥∇T is the loss from neoclassical and anomalous transport, and P_rad, P_cx, and $Q∥$ are the radiation loss, charge exchange loss, and the loss from the thermal transport parallel to magnetic field line, respectively. The ohmic heating inside the 2/1 island is significantly increased by a factor of three from the decrease in the island O-point temperature by half, i.e., $PH∝Te−1.5$⁠. The total radiated power does not start to exponentially rise until 5.43 s, as measured by the bolometer arrays. That is, the radiation increase does not explain the continuous shrinking of the high T_e gradient region between 2/1 and 3/1 islands, but it is a result of thermal quench. The nχ_⊥∇T component is negligible as the T_e gradient is significantly reduced. Although the charge exchange loss is enhanced by the increase in the neutral penetration depth, this possibility is still excluded as long as it cannot account for the bifurcation of thermal transport. Overall, except for the $Q∥$ term, none of these terms can explain the sudden collapse of thermal confinement in spite of the significant increase in ohmic heating.

Once the excitation and locking of a large island take place, the plasma edge is very vulnerable to the growth of small edge island chains. This is because the locked island commonly drags down the plasma rotation and, as a consequence, lowers the thresholds for the error field penetration at neighboring rational surfaces.¹⁶ A locked island could also flatten the local current density profile, which steepens the current density gradient nearby and increases the drive on neighboring small edge islands. On the other hand, the reduction of the global Lundquist number due to the edge cooling regulated by the growth of small edge islands could further destabilize the major locked islands. Thus, the presence of locked island chains with multiple helicity is thought to be a robust mechanism that leads to thermal quench and disruption in tokamaks.

We conclude that observed edge locked island chains play a crucial role in leading up to the plasma major disruption by bridging the gap between the low-order islands and the LCFS. It would be important to understand the island coupling mechanism and explosive growth of stochasticity in the final stage of plasma disruption for efficiently mitigating or controlling the disruption. Suppression of the small locked edge islands outside the large 2/1 locked island would be a rather efficient way to avoid thermal quench and plasma disruption in tokamak-type fusion devices.

The author (X. D. Du) would like to thank W. W. Heidbrink, R. J. La Haye, H. Q. Wang, C. Paz-Soldan, and M. A. Van Zeeland for fruitful discussions and encouragement. This work was supported by the U.S. DOE under DE-AC05-00OR22725, DE-FC02-04ER54698, DE-AC02-09CH11466, and DE-SC0015878, JSPS KAKENHI Grant No. JP26249144, and Japan/U.S. Cooperation in Fusion Research and Development. This report was prepared as an account of the 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 the authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 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.