2.1. Electron ‘tagging’ experiments on DIII-D

Here, we consider ten discharges, shot numbers 172 321–172 330, which were part of the 2017 Frontiers in Science campaign at DIII-D. Each shot was an inner-wall limited L-mode discharge (figure 2a) with no neutral beam injection, using only ohmic heating and modulated ECH and ECCD pulses during the plasma current ${I_p}$ plateau (figure 2b). In each shot, the ECCD is obtained by toroidally aiming the ECH launchers (as opposed to only radially aimed, which provides heating only, and no ECCD). Typical parameters for this campaign were ${I_p} = (0.7\unicode{x2013} 1.1)\textrm{MA}$, toroidal magnetic field ${B_t} \approx 2\; T$ and edge safety factor ${q_{95}} = (4.8\unicode{x2013} 7.4)$. Note that two of the current profiles in figure 2(b), shot 172 325 (orange line) and shot 172 326 (light green line) drop sharply at time $\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }5000\ \textrm{ms}$ due to plasma disruption. Thus, the present analysis only includes data for time $< 5000\ \textrm{ms}$. The ${q_{95}}$ factor provides the ratio between the poloidal winding number m and the toroidal winding number n of the magnetic field lines on the surface corresponding to the $95\,\%$ of the magnetic flux. For these discharges, the ratio $m/n$ increases monotonically from the core to the edge surfaces. The ${q_{95}}$ factor is a useful measure quantifying the number of rational surfaces, and hence the number of magnetic islands, that can be supported in the edge plasma. Higher ${q_{95}}$ corresponds to increased number of closely spaced surfaces and higher ${q_{95}}$ shear in the edge, which in turn yields higher number of islands and increased probability for overlap of neighbouring island chains. However, lower ${q_{95}}$ can support wider islands in the edge plasma, which can result in wider stochastic edge region when RMP is applied. This will be further discussed in § 3.2.

Figure 2.

(a) Inner-wall limited L-mode discharge plasma shape for shot 172 330 at 2600 ms showing an ECH/ECCD pulse consisting of electromagnetic waves from six gyrotrons (coloured lines in the plot). The solid lines on the plot represent magnetic flux surfaces. (b) Plasma current ${I_p}$ and (c) RMP coil current ${I_{\textrm{RMP}}}$ time traces for shots 172 321–172 330. The grey shaded rectangles in (b) and (c) mark the time intervals of the ECH/ECCD pulses.

To minimize plasma response, measures were taken to maintain low plasma densities, ${n_e} = (0.9\unicode{x2013} 1.6) \times {10^{19}}\;{\textrm{m}^{ - 3}}$, without triggering locked modes (standard $n = 1$ error field correction applied via the ex-vessel C-coils), or sawtooth instabilities after $t = 1000\ \textrm{ms}$. For the simple inner-wall limited geometry, low plasma density and lack of toroidal plasma rotation (due to no neutral beam heating), the plasma response is likely weak, which allows to infer the relevant transport mechanisms from simulations of the vacuum magnetic field topology (as discussed in § 3.1). In each shot, a magnetic field perturbation was induced using DIII-D in-vessel I-coils (Evans et al. Reference Evans, Moyer, Watkins, Osborne, Thomas, Becoulet, Boedo, Doyle, Fenstermacher, Finken and Groebner2005a,Reference Evans, Moyer, Watkins, Thomas, Osborne, Boedo, Fenstermacher, Finken, Groebner, Groth and Harrisb), whose current amplitude was varied in the range ${I_{\textrm{RMP}}} ={\pm} (1\unicode{x2013} 6)\ \textrm{kA}$ (see figure 2c and table 1). An important goal for these experiments was to measure changes in electron transport as a function of RMP coil current and link them to the amount of magnetic field stochasticity produced by the perturbation. In § 3.2 we will demonstrate that increasing the width of the stochastic edge region yields an increase in the observed suprathermal electron transport.

Table 1.

ECH/ECCD parameters for shots 172 322–172 330 obtained from TORAY code.

To quantify transport, the experiments utilized a technique called ‘electron tagging’, in which an ECH/ECCD pulse produces a temporally modulated and spatially localized enhanced current at a specific location within the discharge. In cyclotron heating, resonance occurs between the cyclotron frequency ${\omega _c}$ of electrons and the frequency ${\omega _{\textrm{ext}}}$ of external beam of electromagnetic radiation. At DIII-D, the frequency of the gyrotrons is fixed at 110 GHz, which corresponds to a vertical resonant layer extending from the core to the edge plasma. The heating location is adjusted by aiming the gyrotrons so that the ECH/ECCD beam intersects the (~vertical) resonance layer at the desired toroidal flux/radial location, so that electrons at that location receive a ‘kick’ from the electromagnetic beam. Then, the time-dependent radial spreading of the electron perturbation can be observed from TS and ECE measurements. Each experimental shot discussed here included six ECH/ECCD pulses with typical pulse duration of ${\approx} 250\ \textrm{ms}$ and amplitudes in the range $(1.1\unicode{x2013} 3.4)\ \textrm{MW}$. The six pulses occurred at fixed times in each discharge (see grey rectangles in figure 2b,c), which allows for meaningful comparison across discharges. Table 1 lists parameters relevant to the ECH/ECCD tagging and RMP perturbations used in the experiments. Here, ${I_{\textrm{RMP}}}$ is the $n = 3$ RMP coil current amplitude, ${P_{\textrm{ABS}}}$ is the integrated power absorbed in each pulse, ${I_{\textrm{ECCD}}}$ is the integrated current drive in each pulse and ${\rho _{\textrm{ECCD}}}$ is the peak (normalized) radial location of the current drive. In all shots, the ECH pulse frequency was ${\nu _{\textrm{ECH}}} = 110\ \textrm{GHz}$, and the ECH polarization was X-mode second harmonic. The selected time slices feature different combinations of perturbation current, ECH absorbed power and ECCD current. All values in table 1 were obtained using TORAY (Prater et al. Reference Prater, Farina, Gribov, Harvey, Ram, Lin-Liu, Poli, Smirnov, Volpe, Westerhof and Zvonkov2008). TORAY is a ray tracing code for studying ECH and current drive in toroidal geometry. Figure 3 shows typical TORAY plots of the radial distribution of ECH heat flux and ECCD current flux, together with the integrated values used in table 1.

Figure 3.

Shot 172 330 at $2600\ \textrm{ms}$: radial distribution of (a) ECH power absorption; (b) ECCD current drive, obtained using TORAY code. Different colours correspond to the six gyrotrons available at DIII-D. The curve in black in each shot represents the total values.

2.2. Observation of suprathermal electron transport

The presence of suprathermal electrons can be inferred from an unusual increase in electron temperature in the edge plasma at normalized radius $\rho \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }1$, observed in ECE radiometer measurements, as shown in figure 4. What is observed in the present experiments is a milder case of the more extreme example of non-thermal ECE shown by Harvey et al. (Reference Harvey, O'Brien, Rozhdestvensky, Luce, McCoy and Kerbel1993) (in general, $\rho$ is the square root of the normalized toroidal flux surface). The DIII-D ECE radiometer (Austin & Lohr Reference Austin and Lohr2003) is a multichannel heterodyne system that provides electron temperature from measurements of optically thick, second harmonic ECE. The instrument viewing is along a horizontal chord at the tokamak midplane at a toroidal angle of 81 degrees. If the radial electron transport evolves according to classical diffusion, the electron temperature measured by ECE cords is expected to decrease with increasing radial location, following a similar pattern as the TS (Ponce-Marquez et al. Reference Ponce-Marquez, Bray, Deterly, Liu and Eldon2010) data (dashed green lines in figure 4). In all examined shots, the electron temperature measured by ECE cord 01 was consistently higher than the temperature measured by ECE cord 02, even though cord 01 is located at larger radial location $\rho$ than cord 02.

Figure 4.

Radial distribution of electron temperature from ECE for shot 172 330 (a) at $2500\ \textrm{ms}$ prior to an ECH/ECCD pulse (b) at $2560\ \textrm{ms}$ during an ECH/ECCD pulse. Dashed green lines represent electron temperature from a fit to the TS data, which assume thermal electron distribution function.

Vertical green lines in figure 4 indicate the location of several rational surfaces, while the vertical red line in figure 4(b) indicates the location of the ECH/ECCD tagging for that discharge. The safety factor, or q profile, and the locations of the $q = 1$ and $q = 2$ surfaces mentioned throughout the paper, are inferred from the magnetics-only Grad--Shafranov equilibrium solution. As no neutral beams were used in these discharges to keep the plasma rotation and plasma response minimized, the motional Stark effect measurements of the pitch angle in the plasma core were not available for the shots presented here.

Figure 4 shows a typical ECE profile from shot 172 330 before and during the ECH/ECCD pulse. While in both cases, the temperature measured by cord 01 is higher than the one measured by cord 02, the tagging technique clearly enhances the observed suprathermal ECE tail. Comparison between figures 4(a) and 4(b) reveals that, during the tagging, the entire ECE radial profile shifts to higher temperatures (note the difference in the abscissa) but the tail feature and the flat region around the $q = 2$ surface are preserved. The flattening of the ECE profile around the $q = 2$ surface, which is observed for all examined discharges, can be associated with a $2/1$ island located on that surface. A main difference between figures 4(a) and 4(b) is that prior to the pulse, the temperature measurements around the $q = 1$ surface are somewhat flat, while during the pulse, the temperature profile in the same region exhibits a steep slope. Flat regions around the $q = 1$ surface can be associated with the presence of a sawtooth instability or of another magnetic island chain. Since the conditions in the present discharges were carefully selected to minimize plasma response, a sawtooth instability was not observed for these shots, suggesting that the flat region is caused by a $q = 1$ island chain. The plasma equilibrium in these discharges is calculated using magnetics-only equilibrium fitting (EFIT) reconstruction (as described in Heidbrink & White (Reference Heidbrink and White2020). The safety factor profile for both time slices shown in figure 4 is almost identical resulting in almost identical location of the rational surfaces. Therefore, the locations of the q surfaces (vertical green lines on the plots) do not change appreciably. Typical uncertainty in TS data is 10% (as verified using OMFIT), while the uncertainty in ECE measurements is assumed as 5%. The error bar on DIII-D ECE measurements is based on when the primary calibration was performed with the Michelson interferometer as described in Austin et al. (Reference Austin, Ellis, Doane and James1997).

Examination of similar ECE distribution plots for the other discharges leads to several observations: (i) the change around the $q = 1$ surface during the tagging is sensitive to ECH power, (ii) the first ECH/ECCD pulse in each shot leads to the most prominent suprathermal tail on the ECE distribution and (iii) the suprathermal tail is present with and without the RMP coil perturbations. However, in § 4, we discuss how the RMP coil perturbations indirectly affect the observed electron transport through the formation of stochastic edge plasma region.

To verify that the observed increase in ECE measurements at $\rho \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }1$ is not caused by the drop of plasma density outside the separatrix, ECE simulations were conducted (using the ECESIM code (Austin & Pandya Reference Austin and Pandya2010)) to obtain the expected electron temperature as a function of frequency expected for a Maxwellian electron distribution. Figure 5 shows the simulation results for discharge 172 330 at $2580\ \textrm{ms}$ (solid red line), together with the ECE measured values (blue crosses). The ECE simulations were informed by TS data. Deviation from predicted ECE temperature is clearly visible for frequencies $< 84\ \textrm{GHz}$. With the low-density L-mode edge, the ECE is marginally blackbody from the region inside the last closed flux surface, where channels >1 reside, and optically thin for the emission measured by channel 1. Nevertheless, as the simulation shows, one does not expect the edge emission to be elevated if the electron distribution is Maxwellian. Thus, this high temperature measured by ECE is likely caused by suprathermal electrons in the plasma, the radiation escaping due to the low optical thickness of the edge channels.

Figure 5.

Electron temperature as a function of frequency for shot 172 330 at $2580\ \textrm{ms}$. Solid red line shows the predicted values from ECE simulations assuming thermal electron distribution. Blue crosses mark experimental data from ECE measurements. The second axis under the plot shows the location of the ECE chords as a function of major axis ${R_m}$.

2.3. Runaway electrons

Runaway electrons could be distinguished from the general non-thermal electron population as they also induced gamma emission on fast neutron scintillator counters (figure 6). The hard X-ray signals are measured by an uncollimated plastic scintillator placed near the vacuum vessel. Signals from an adjacent ZnS scintillator that is an order of magnitude less sensitive to X-rays (Heidbrink Reference Heidbrink1986) confirms that the plastic scintillator is measuring hard X-rays. In figure 6, shot 172 321 (black line) does not exhibit peaks in the scintillator data. For most other shots, a large peak was observed during the first ECH/ECCD pulse followed by a much smaller peak during the second pulse and no peaks in subsequent pulses. This suggests that the pulses cause depletion of the runaway electrons. Shots 172 325 (light green line) and 172 326 (orange line) exhibit peaks both prior to and during the pulses, indicating an overall larger population of relativistic electrons in these shots.

Figure 6.

Runaway-electron-induced gamma emission on the fast neutron scintillator counter for several shots from the same campaign. Shaded rectangle areas indicate the timing of the ECH pulses.

Interestingly, in each discharge, ECE deviations from the TS fits were observed during all ECH/ECCD pulses, but no peaks in the scintillator data were observed after the second pulse, indicating that electrons detected during subsequent pulses were non-thermal but also not relativistic. The ECE radiometers at DIII-D are preferentially sensitive to non-thermal synchrotron emission from energetic electrons. Due to the radial viewing geometry of the ECE radiometers on DIII-D, these diagnostics probe the high pitch-angle $(60\mathrm{^\circ\unicode{x2013}}90\mathrm{^\circ })$ and low-energy $(1\ \textrm{MeV}\unicode{x2013} 5\ \textrm{MeV)}$ energetic electron population (Paz-Soldan et al. Reference Paz-Soldan, Cooper, Aleynikov, Eidietis, Lvovskiy, Pace, Brennan, Hollmann, Liu, Moyer and Shiraki2018). The HXR detectors used here cover all pitch angles $(0\mathrm{^\circ }\; \unicode{x2013} 90\mathrm{^\circ )}$ and electron energy in the range $(5\ \textrm{MeV}\unicode{x2013} 30\ \textrm{MeV)}$. Thus, we expect that the signals showed in figure 6 correspond to energetic electrons with energy $\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }5\ \textrm{MeV}$, while the non-thermal electron populations detected by ECE after the first ECH/ECCD pulse have energies in the range $(1\ \textrm{MeV}\unicode{x2013} 5\ \textrm{MeV)}$. Accurate determination of energy and pitch angle is not available in the present experiments, but will be explored in an upcoming experimental campaign, which will utilize multiple additional diagnostics available at DIII-D, including the gamma ray imager (Pace et al. Reference Pace, Cooper, Taussig, Eidietis, Hollmann, Riso, Van Zeeland and Watkins2016) and visible synchrotron emission from a tangential camera (Yu et al. Reference Yu, Hollmann, Commaux, Eidietis, Humphreys, James, Jernigan and Moyer2013).

The observation of scintillator peaks only during the first ECH/ECCD pulse in some of the examined discharges suggests that the tagging technique can deplete the relativistic runaway population. While this effect has been observed empirically before, the exact mechanism leading to the release of the runaway electrons from the core to the edge plasma is not well understood. Since runaway electrons are accelerated to relativistic speeds likely during the current ramp-up at the beginning of the discharges, we do not expect that their initial transport is strongly affected by the island chain structure or stochastic regions in the magnetic field. However, their transition from strongly confined to lost orbits during an ECH/ECCD pulse can result from spontaneous or induced changes in the magnetic field topology, including stochastization of magnetic field lines near island X-points or island bifurcation. Interestingly, figure 6 shows smaller but distinct peaks in the period $1900\unicode{x2013} 2300\ \textrm{ms}$ for discharges 172 325 and 172 326 (light green and orange dots, respectively.) Since there is no sawtooth instability, ECH/ECCD pulse, or 3-D coil perturbation in this period, the observed peaks in the scintillator data suggest a spontaneous deconfinement of suprathermal electrons. Spontaneous island bifurcation has been recently observed in DIII-D H-mode experiments for rotating islands in the core plasma (Bardóczi & Evans Reference Heidbrink and White2021a).

2.4. Suppression of the suprathermal feature

In the present experiments, the observed suprathermal features did not appear to change appreciably as the amplitude of the RMP current was varied. For example, at 2600 ms the RMP current for shot 172 330 (figure 4b) was ${I_{\textrm{RMP}}} = 0\ \textrm{kA}$; yet the ECE plot exhibits a pronounced suprathermal tail. We note that for all examined cases, the first ECH deposition was performed with no RMP perturbation, and this first pulse always resulted in the largest electron temperature peak and the largest enhancement in the difference between trace 01 and trace 02 measurements. As mentioned in § 2.3, this effect is likely due to the release of runaway electrons. Thus, in § 4, where we discuss how RMP-induced edge stochasticity affects non-thermal electron transport, we do not consider data from these first ECH/ECCD pulses. In § 4, we show that for fixed properties of the ECH/ECCD pulses and fixed RMP current values, the non-thermal transport depends on the size, location and distribution of magnetic islands across the plasma.

To demonstrate that the suprathermal electron tail is persistent even after the first pulse (which depletes the runaway electrons), figure 7(b) shows the ratio between temperature measured by ECE cords 01 and 02 as a function of time for all examined shots. One expects that ${T_{01}}/{T_{02}}$ would be below one for discharges with electron temperature profiles that decrease monotonically in the plasma edge. In the examined discharges, this ratio is bigger than 1 for all cases, except shots 172 328 and 172 329, where the ECE measurement did not extend beyond $\rho \approx 0.8$ and the outermost ECE channels 02 and 01 were located not in the edge, but in the core of the plasma. Therefore, we omit data from 172 328 and 172 329 in figure 7. While for all other shots, the ratio ${T_{01}}/{T_{02}}$ increases during ECH/ECCD pulses (shaded areas in figure 7b,c), the effect is reversed for shot 172 326 (orange line), as can be seen in figure 7(c). Shot 172 325 (light green line) exhibits transitional behaviour, where the ratio is only slightly enhanced at the onset of an ECH/ECCD pulse and gradually decreases over a time period longer than the pulse.

Figure 7.

Shots 172 322−172 327, 172 330: (a) time trace of calibrated ECH power, (b) time trace of the ratio ${T_{01}}/{T_{02}}$ and (c) zoomed in version of the same plot focused around a single ECH pulse. Shaded areas in (b) and (c) indicate the timing of the ECH pulses from (a). Dashed black lines in (a) and (b) mark the ECH pulse shown in (c).

Since the timing of the ECH/ECCD pulses was performed uniformly for all discharges (figure 7a), the observed differences are not due to offset in the pulse timing. From table 1, we see that the RMP coil current, the ECH power and the location of the ECCD current drive for shots 172 325 and 172 326 have similar values to other cases. The only parameter that differs considerably across the nine examined discharges is the ECCD current, which varies in the interval ${I_{\textrm{ECCD}}} = (45\unicode{x2013} 409)\ \textrm{kA}$ (see table 1). However, the relationship between ${T_{01}}/{T_{02}}$ and ${I_{\textrm{ECCD}}}$ does not seem to follow a simple (linear or power law) trend. Instead, the transitional and reversed behaviour of shots 172 325 and 172 326 seems to occur at intermediate values of the ECCD current ${I_{\textrm{ECCD}}} \approx (70 - 90)\ \textrm{kA}$. Thus, it is not clear if the observed phenomenon is caused by the value of ${I_{\textrm{ECCD}}}$. In the next section we compare data from experiments with vacuum field simulations, which suggest that the observed phenomenon is related to the characteristics of the island structure in these discharges.

3.1. Large islands location with respect to ECH/ECCD pulse

As we saw in § 2, figure 7, in most examined cases, the ECH/ECCD pulses tend to enhance the suprathermal effect, which is visible by the time trace of the ${T_{01}}/{T_{02}}$ ratio. However, in shot 172 325, this effect is diminished, and in shot 172 326, the ECH/ECCD pulse leads to a drop of the ${T_{01}}/{T_{02}}$ ratio. One possibility is that the compared discharges had substantially different plasma conditions, perturbation parameters or features of the ECH/ECCD pulses. All discharges were low density, inner-wall limited, with only ohmic heating and no neutral beam heating. The possible role of changing plasma density in the core during ECH/ECCD pulses will be discussed in § 4. Examination of table 1 reveals that the 3-D perturbation current and properties of the ECH/ECCD pulses used in shots 172 325 and 172 326 are very similar to those used in other shots. Specifically, the perturbation current increases from $0\ \textrm{kA}$ to $5.7\ \textrm{kA}$, the ECH power is approximately $2.2\ \textrm{MW}$ and the ECCD current is in the range $(70\unicode{x2013} 90)\ \textrm{kA}$ is deposited at radial location ${\rho _{\textrm{ECCD}}} \in (0.21,0.24)$. One notable difference is that shots 172 325 and 172 326 have the highest plasma current ${I_p}$ and the lowest ${q_{95}}$ values, which suggests that the radial location of the initial island structure (before RMPs are turned on) in these cases differ. This observation is interesting, as it suggests that, for a fixed location of the ECH/ECCD pulse, the location of the magnetic island chains can be manipulated to either enhance or diminish radial electron transport. Similarly, for a fixed distribution of island chains, we expect that differences in the electron transport will be observed when the tagging is performed at different locations with respect to the large islands.

In the present experiments, the ECH/ECCD deposition locations were similar across all shots, as seen from table 1. While, in principle, DIII-D has the flexibility of independently varying ${q_{95}}$ for more-or-less fixed $q$-values in the core, in the present study, shots with large ${q_{95}}$ values have the $q = 1$ surface closer to the magnetic axis. In addition, these shots have steeper rate of change $q^{\prime}$, which is why they can support larger number of rational surfaces, and therefore, more islands in the edge plasma. In contrast, here, shots with smaller ${q_{95}}$ tend to support fewer larger island chains further away from the magnetic axis. Thus, since the location of the ECH/ECCD pulse is roughly fixed to $\rho \approx 0.1\unicode{x2013} 0.2$ for all cases, it is expected that the electron tagging occurred at different locations with respect to the $q = 1$ island for shots with different ${q_{95}}$ values. In the presence of large magnetic islands, the radial electron diffusion should be slower than diffusion across unperturbed nested surfaces since the electrons need to diffuse across additional surfaces in the island interior. In addition, it has been experimentally demonstrated that electron heat diffusivity is reduced inside an island's O-point (Inagaki et al. Reference Inagaki, Tamura, Ida, Nagayama, Kawahata, Sudo, Morisaki, Tanaka and Tokuzawa2004). Thus, we expect that radial electron transport across islands can lead to electron sub-diffusion due to trapping effects. However, the motion of electrons trapped inside the separatrix of an island becomes ergodic and eventually all particles adopt the constants of motion of the exact resonance creating the island perturbation (Heidbrink & White Reference Heidbrink and White2020). As a result, the energy of electrons trapped in surfaces across the island width does not drop with radial location. This effect should be more pronounced for wider islands.

This is evident from the flat regions in the ECE plots (figure 4), which are assumed to coincide with island locations (e.g. consider the region around the $q = 2$ surface). While the ECE temperature stays constant throughout the width of the island, the temperature calculated from TS (green dashed line in figure 4) drops across the same region. Since the TS and ECE have different poloidal and toroidal locations, it is possible that ECE is probing the island chains’ O-points, while TS is probing X-points. It is also possible that the TS prediction yields dropping electron temperature because it is assuming Maxwellian electron distribution, which smoothens irregularities in the raw data. In either case, we expect that if the flat regions in ECE measurements coincide with island chain locations, when electrons escape the exterior surfaces of these islands, they will have higher energy than the energy they would normally have at the same radial location if the island perturbation were not there. In other words, islands can trap electrons, slowing down their cross-field diffusion, but electrons de-confined from islands will exhibit suprathermal transport in comparison with electrons that were not trapped by islands. The ECH/ECCD pulses used in the present experiments are a possible mechanism for such electron release from island surfaces.

ECCD techniques have been previously used to shrink magnetic islands (Gantenbein et al. Reference Gantenbein, Zohm, Giruzzi, Günter, Leuterer, Maraschek, Meskat and Yu2000; Petty et al. Reference Petty, La Haye, Luce, Humphreys, Hyatt, Lohr, Prater, Strait and Wade2004) if the current drive location is carefully chosen to coincide with the O-point of the island. If the current is driven off axis, but inside an island, it may be expected to still cause some shrinking due to perturbation of the separatrix surface and nearby surfaces. In the present analysis, we neglect this effect for the following reason. The current relaxation time for the ECCD in our experiments $\mathrm{\sim }1\ \textrm{s}$, while the pulse duration used here is $250\ \textrm{ms}$. Thus, we expect that the ECCD pulses would not modify the parallel current profile. To confirm this, we compared the $q$profiles for a time slices right before and during ECCD pulses and verified that there are no changes. Therefore, we expect that the ECCD perturbations do not appreciably change the structure of the magnetic islands in the present experiments. This point is further discussed in the Appendix.

Based on the above logic, if the current drive location ${\rho _{\textrm{ECCD}}} < {\rho _{q = 1}}$, where ${\rho _{q = 1}}$ is the location of the $q = 1$ island axis, the tagged electrons will have to diffuse across the large $q = 1$ island, which should cause trapping and reduced suprathermal feature. In contrast, if ${\rho _{\textrm{ECCD}}}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }{\rho _{q = 1}}$, the electrons can only be trapped in the smaller islands distributed at radius $\rho > {\rho _{\textrm{ECCD}}}$, and the suprathermal feature may not be appreciably affected. While the ECH/ECCD pulses are fairly localized, both the power deposition and the current drive have a finite spatial spread (see figure 3). To determine the spatial spread of electron tagging with respect to the spatial spread of the $q = 1$ island chain, TORAY simulations were used to determine the 3-D location (in the radial, poloidal and toroidal directions) of each pulse, while TRIP3D Poincaré maps were computed to identify the underlying vacuum field structure for each discharge. While it was determined that none of the pulse locations coincided exactly with the O-point of the $q = 1$ island, differences in suprathermal electron transport were observed when the tagging occurred at different locations with respect to ${\rho _{q = 1}}$.

Figure 8 shows the TRIP3D Poincaré plots for several shots where the observed electron transport behaviour differed during ECH/ECCD pulses. In these plots, the vacuum field topology was reconstructed for the toroidal plane corresponding to the toroidal location of the ECE diagnostic. Note that the TS diagnostic is located at a different toroidal angle where the magnetic topology is different from the one presented in the figure. A comparison between ECE line of sight and TS line of sight on TRIP3D plots can be found in Evans, et al. (Reference Evans, Ida, Ohdachi, Tanaka, Suzuki, Shafer, Unterberg, Inagaki and Austin2014).

Figure 8.

TRIP3D Poincare plots of shot numbers (a) 172 322 and (b) 172 330, where enhancement of the suprathermal electron feature was observed during the ECH/ECCD pulse and shot numbers (c) 172 325 and (d) 172 326, where reduced transport was observed. All plots are generated for toroidal angle $\phi = 279\mathrm{^\circ }$ (TRIP3D coordinate system), which coincides with the location of the ECE diagnostic ($81\mathrm{^\circ }$ in DIII-D machine coordinates).

In shots 172 322 and 172 330 (figure 8a,b and red/magenta lines in figure 7), the suprathermal feature was enhanced during each pulse, while in shot 172 326 (figure 9d and orange line in figure 7) the suprathermal feature was reduced during each pulse. For shot 172 325 (figure 8c and light green line in figure 7), the suprathermal feature was initially diminished, and subsequently slightly increased during the pulse, which is what we consider an intermediate behaviour. Red rectangles in each plot in figure 8 indicate the location of the ECH/ECCD pulse with respect to a large magnetic island residing on the $q = 1$ surface (marked by a dashed red line). In the enhanced-transport cases, the location of the electron tagging is at ${\rho _{\textrm{ECCD}}} > {\rho _{q = 1}}$ (figure 8a) or close to the X-point of the $q = 1$ island (figure 8b) and the radial ‘line of sight’ (marked by extended light pink rectangles) passes through the X-points of other islands. In contrast, in the reduced-transport cases (figure 8c,d), the tagging occurs at ${\rho _{\textrm{ECCD}}} < {\rho _{q = 1}}$ and the line of sight for the tagged electrons passes through thicker regions of neighbouring islands. Examination of the other discharges confirms the trends observed in figure 8.

Figure 9.

Shot 172 322: histogram of B-line displacements from the original position (a) at $2600\ \textrm{ms}$ when ${I_{\textrm{RMP}}} = 0\ \textrm{kA}$, (b) at $3100\ \textrm{ms}$ when ${I_{\textrm{RMP}}} = 2\ \textrm{kA}$, (c) at $4100\ \textrm{ms}$ when ${I_{\textrm{RMP}}} = 4\ \textrm{kA}$ and (d) at $5100\ \textrm{ms}$ when ${I_{\textrm{RMP}}} = 5\ \textrm{kA}$. The colour bar is a log scale of the normalized number of field lines, crossing a given location in space.

These observations are compared against findings from Evans et al. (Reference Evans, Ida, Ohdachi, Tanaka, Suzuki, Shafer, Unterberg, Inagaki and Austin2014), where measurements of changes in the time it takes for modulated electron cyclotron heat pulses to propagate across the O-point of a $2/1$ magnetic island in DIII-D were consistent with a model in which the island spontaneously bifurcates from smooth flux surfaces to a partially stochastic region. In other words, heat transport was observed to be slower across the unperturbed island surfaces and faster across partially stochasticized regions. In the present study, we find evidence that suprathermal electron transport is suppressed when the electron cyclotron heat pulse diffuses across the large $q = 1$ island and enhanced when the pulse diffuses across island X-points, where local stochasticity is expected to occur. Although these results are preliminary, they hint towards the possible role of magnetic islands into acceleration and trapping of electrons. This will be further explored by the authors in a next experimental campaign where the ECH/ECCD pulse will be delivered at different locations with respect to the large island chains.

3.2. Transition to stochasticity

It has been proposed that raising RMP coil currents can induce a transition from a regime dominated by nested magnetic flux surfaces through a regime with a variety of magnetic islands to a regime where the islands have ‘overlapped’ to produce an extended region of stochastic field lines. A regime with ideal nested flux surfaces is not accessible in the present experiments even at ${I_{\textrm{RMP}}} = 0\ \textrm{kA}$ as magnetic perturbations from intrinsic error fields and EFC coils always lead to the formation of island structures. In particular, even in the absence of RMPs, all examined experiments feature at least two large island chains, one at the $q = 1$ surface and another one at the $q = 2$ surface. To produce a region of stochastic field lines in the edge plasma, RMPs were used to create and grow islands at the $n = 3$ surfaces until island overlap was achieved. We observe that increasing the RMP coil current leads to both the formation of new smaller islands and stochastization of the large islands’ exteriors. As discussed below, the critical current value for the transition to stochasticity is dependent on the ${q_{95}}$ for each discharge.

Possible uncertainties in the calculation of the magnetic islands result from the uncertainties in the location of the resonant surfaces (uncertainties in safety factor profile in Grad--Shafranov equilibrium reconstruction) and the uncertainties in the estimation of the magnitude of the perturbation field in the plasma (uncertainties in the measurements of coil currents and representation of the coil geometries as simplified thin wires). Given the stochastic nature of the fields near the island chains and in the plasma edge, detailed quantification of uncertainties is quite complicated and lies outside of the scope of this paper.

Figure 9 shows histograms of the B-line displacements from their initial position computed with TRIP3D using initial conditions from shot 172 322. In this discharge, the ${I_{\textrm{RMP}}}$ was varied from $0\ \textrm{kA}$ to $5\ \textrm{kA}$ in three steps (red line in figure 2c). The colour bar in each plot from figure 9 represents a log scale of the normalized number of times a B-line intersected a given location in space. In this representation, narrow red line regions correspond to nested magnetic surfaces, or locations in space where the lines did not deviate significantly from their initial positions as the simulation was advanced. Wide rhombus regions on the plots indicate the locations of magnetic island chains, and irregular blue regions suggest stochastization of the field lines.

In figure 9(a), for ${I_{\textrm{RMP}}} = 0\ \textrm{kA}$, the B-line histogram shows two big islands, located at the $q = 1$ and $q = 2$ surfaces, and $4\unicode{x2013} 5$ smaller, but distinct islands located between the $q = 2$ surface and the edge plasma. These initial islands are created by the intrinsic error fields in DIII-D (shifted and tilted poloidal field coils and B-field buswork) and by the $n = 1$ error field correction currents (in the DIII-D C-coils). The values for the intrinsic fields in DIII-D have been previously measured and discussed in more detail in Orlov et al. (Reference Orlov, Moyer, Evans, Mordijck, Osborne, Fenstermacher, Snyder and Unterberg2010, Reference Orlov, Moyer, Evans, Wingen, Buttery, Ferraro, Grierson, Eldon, Watkins and Nazikian2014, Reference Orlov, Moyer, Evans, Paz-Soldan, Ferraro, Nazikian, deGrassie, Grierson, Eldon, Fenstermacher and King2016). As the current in the RMP coils is increased to ${I_{\textrm{RMP}}} = (2\unicode{x2013} 3)\ \textrm{kA}$ (figure 9b,c), new island chains emerge between the $q = 1$ and $q = 2$ surfaces and the previous island chains develop slightly stochastic exterior (blue colour around rhombus structures). For ${I_{\textrm{RMP}}} = 5\ \textrm{kA}$ (figure 9d), more island chains emerge in the edge region and stochastic exteriors are enhanced. However, even at ${I_{\textrm{RMP}}} = 5\ \textrm{kA}$, the island chains in shot 172 322 remain distinct all the way to the edge and no significant formation of a connected stochastic layer is observed.

The transition to a regime where the edge islands overlap, resulting in a region of stochastic field lines, is visible when higher RMP current is applied. Figure 10 shows that, at ${I_{\textrm{RMP}}} =- 6\ \textrm{kA}$ for shot 172 330 (figure 10a) and at ${I_{\textrm{RMP}}} = 5.7\ \textrm{kA}$ for shot 172 326 (figure 10b), a stochastic layer forms in the region ${\varPsi _N} = (0.8\unicode{x2013} 1)$. These regions are not fully stochastic as remnants of island core structures are still visible in both cases. However, the stochastic exterior of these former island chains now surrounds all of them in the edge region. These results are further confirmed from SURFMN simulations, which show that an overlap of the edge island chains occurs at ${\varPsi _N} = 0.84$ for ${I_{\textrm{RMP}}} =- 6\ \textrm{kA}$ in shot 172 330 (figure 10c) and at ${\varPsi _N} = 0.80$ for ${I_{\textrm{RMP}}} = 5.7\ \textrm{kA}$ in shot 172 326 (figure 10d). Comparison of TRIP3D histograms and SURFMN plots for the rest of the examined experiments reveals that a stochastic edge region is also observed to appear in shots 172 324–172 328 at ${I_{\textrm{RMP}}} = 5.7\ \textrm{kA}$ and in shot 172 329 at ${I_{\textrm{RMP}}} =- 6\ \textrm{kA}$, suggesting that the phenomenon is repeatable and observable with both positive and negative currents.

Figure 10.

TRIP3D histograms of B-line spread at $3100\ \textrm{ms}$ for (a) shot 172 330 and (b) shot 172 326. SURFMN plots of island width at $3100\ \textrm{ms}$ for (c) shot 172 330 and (d) shot 172 326 Shaded regions in (c) and (d) indicate the width of stochastic region, where island surfaces overlap.

For the $n = 3$ perturbations used here, the six I-coils are energized in $+- +- +-$ or $-{+} -{+} -{+}$ patterns, and what is shown here for ${I_{\textrm{RMP}}}$ is the value in the first of these coils (the one located at 30 degrees toroidally in DIII-D). Thus, the difference between positive and negative ${I_{\textrm{RMP}}}$ is shifting the perturbation by 60 degrees toroidally, which results in shifted toroidal locations of the island O-points and X-points, but no difference in the number, width or radial location of the island chains. The phase of the ${I_{\textrm{RMP}}}$ (0 or 60 degrees) changes how that perturbation couples to the intrinsic and EFC fields (which are fixed toroidally).

Examination of similar histograms calculated for the rest of the experiments suggests that the RMP current threshold needed to transition from individual islands with small stochastic exteriors (figure 9d) to island remnants surrounded by common stochastic region (figure 10a,b) varies with the q profile for each shot. The present experiments were not performed with small enough ${I_{\textrm{RMP}}}$ steps to pinpoint the exact threshold value for each q profile. However, for a fixed type of perturbation (here $n = 3$ perturbation from RMP coil), shots with smaller ${q_{95}}$ values are observed to form wider stochastic regions in the edge plasma. This makes sense as the smaller ${q_{95}}$ means that fewer but larger islands are present close to the edge and small island growth can lead to overlap. However, cases with higher ${q_{95}}$ value can support more perturbation modes in the edge, i.e. more island chains. In these cases, a transition to stochasticity can be achieved by a combination of perturbations from different coils, which can be desirable as it reduces the threshold current requirement for each coil. Finally, we note that once the stochastic regions form (figure 10a,b), the locations of the $q = 1$ and $q = 2$ islands shift towards smaller ${\varPsi _N}$. This suggests that if the observed electron transport is dependent on the location of $q = 1$ island chains with respect to the ECH/ECCD pulses (as discussed in § 3.1), the controlled formation of a stochastic region in the edge plasma can be used as a method to manipulate electron transport.