On the interplay between neoclassical tearing modes and nonlocal transport in toroidal plasmas

This Letter presents the first observation on the interplay between nonlocal transport and neoclassical tearing modes (NTMs) during transient nonlocal heat transport events in the HL-2A tokamak. The nonlocality is triggered by edge cooling and large-scale, inward propagating avalanches. These lead to a locally enhanced pressure gradient at the q = 3/2 (or 2/1) rational surface and hence the onset of the NTM in relatively low β plasmas (βN < 1). The NTM, in return, regulates the nonlocal transport by truncation of avalanches by local sheared toroidal flows which develop near the magnetic island. These findings have direct implications for understanding the dynamic interaction between turbulence and large-scale mode structures in fusion plasmas.

this study, the interplay between nonlocal transport and the NTM was investigated in about 14 discharges. The maximal ECRH power can reach up to 3 MW. The plasma parameters are the plasma current I p = 150-160 kA, toroidal magnetic field B T = 1.28-1.33 T, line-averaged electron density n e = (0.8-2.0) × 10 19 m −3 and edge safety factor q 95 ≈ 4. The nonlocal thermal transport events were triggered by employing a supersonic molecular beam injection (SMBI) or gas-puffing to generate cold pulses at the plasma boundary 5 . The equilibrium electron temperature (T e ) profile and temperature fluctuations (  T e ) were measured by a 16 channel electron cyclotron emission (ECE) radiometer 22 with 1.25 MHz digitizer. Besides, Mirnov coils, soft-x-ray (SXR) detectors and Doppler reflectometer were used to measure MHD instabilities and turbulence at a sampling rate of 1 MHz. The toroidal plasma rotation was detected by a charge exchange recombination spectroscopy (CXRS) with a spatial resolution ≈ 1.5 cm 23 . Figure 2(a-c) show typical waveforms for a discharge with "nonlocal" transport induced by the SMBI (see vertical yellow bar) in ECRH-heated plasmas. In Fig. 2(d), the time evolution of density fluctuation power is plotted. The ECRH power applied is constant (P ECRH ≈ 1.32 MW) until switch-off at t ≈ 915 ms. After the molecular beam is injected at t ≈ 838 ms, the plasma density slightly increases, due to fueling. From the ECE signals plotted in Fig. 2(b), one can clearly see that after the SMBI, the edge T e (blue curves) drops because of local cooling by the SMBI, but in the core region the electron temperature (red curves) increases unexpectedly. The possible impact by the enhanced ECRH efficiency on the core T e rise has been eliminated as the core plasma density is not dipped after the SMBI. This transient, long-distance and reversed polarity response where the core T e rise on response to edge cold pulses is the so-called "nonlocal" heat transport, a phenomenon widely observed in many fusion devices [1][2][3][4][5][6][7] . However, the physical mechanisms responsible for triggering the nonlocal transport remain unclear. Code simulations on TEXT 24 and TFTR 25 as well as our modelling indicate that the nonlocal effects cannot be reproduced with transport coefficients that are functions only of local thermodynamic variables. At present, several theoretical models have been proposed [8][9][10][11][12] to explain the nonlocal effect. Among them the avalanche paradigm (related to self-organized criticality) attempts to unravel the transport enigma via the interplay between individual turbulent eddies and large-scale transport events 26,27 . Here, several localized eddies cooperate to produce a transient, extended transport events, or avalanche. Avalanches can propagate outward (as local excesses) or inward (as voids). In HL-2A, experimental evidence indicates that this avalanche-like transport may play a crucial role in the distant nonlocal transport events. Typical avalanche features, such as self-similarity and long-range radial correlations in temperature fluctuations (  T e ), are depicted in Figs 3 and 4, for a similar nonlocal shot. In Fig. 3(f), the red curves, calculated by the structure function (SF), and rescaled range (R/S) analyses of the  T e data measured at ρ ≈ 0.45, show a high slope, i. e., the Hurst exponent (H ≈ 0.78) during the first nonlocal phase (before the NTM). Here, the qth-order SF is defined as the qth moment of the increments of X(t) for a time series of X ≡ {X t : t = 1, 2,..., n}, i. e., S x,q (τ) ≡ 〈 |X(t + τ) − X(t)| q 〉 = cτ qH 28 ; and the R/S ratio is defined as the maximal range of the integrated signal normalized by the standard deviation, i. e., 2 and it depends asymptotically on n by

Experimental Results and Discussion
c is approximately constant and H is the Hurst exponent 29 , which equals to 0.5 for a random process and is larger than 0.5 for a sequence with long-term correlations. The radial profile of the Hurst parameter during that period further indicates that the H values are close to 0.8 at all radii (see red points in Fig. 3(g)), suggesting large diffusive propagation. Meanwhile, the cross-correlation function (CCF) of  T e between multi-channel ECE signals exhibits radially long-distance inward propagation in temperature fluctuations (see Fig. 4(a)). All these features are consistent with the avalanche picture. More details on the avalanche characteristics have been described in ref. 30. In Fig. 2, we notice that along with the nonlocal response in T e , an NTM is excited at t ≈ 844 ms. The mode numbers have been identified as m/n = 3/2 with a frequency f ≈ 7 kHz, as seen in the frequency spectrum of the SXR signal in Fig. 2(c). The neoclassical nature of the tearing modes is illustrated in Fig. 2(e), where the variation in the fluctuation amplitude of the radial magnetic field (square root of  B r ∝ island width) around the mode frequency (f = 6-9 kHz) shows approximately a linear relation with β N [ = β(100aB/I p )] in five shots having the 3/2 islands. Another character of the NTM can also be seen in Fig. 2(a,c). After the ECRH is switched off at t ≈ 915 ms, the saturated island width quickly drops on a time scale of ~10 ms, much shorter than the current diffusion time (μ 0 a 2 /η ~ 100 ms) for conventional tearing modes in HL-2A. Similar results have been observed in the TCV tokamak 31 .
For investigating the role of the nonlocality in triggering the NTM, we first inspect the location and onset time of the NTM during the nonlocal transport. In Fig. 2, one can see that between the time of SMBI and the NTM onset (see vertical black dashed line), the T e increases in the core and decreases in the edge, respectively. At the inversion surface of ρ = 0.46, the change in T e is minimal (≈ 0). It is found that the q = 3/2 rational surface, which is determined from the phase jump in T e fluctuations measured by ECE, is located around the inversion surface of the nonlocality. Further investigation on other shots shows that the NTM locations are all located close to the inversion surface. In Fig. 5(a), the radial gradient of the local electron temperature, dT e /dr, between two radial loci around the inversion surface is drawn versus the time delay relative to the SMBI/gas puffing time (Δ t = t − t SMBI/gas−puff ), for five "nonlocal" shots. The open circles mark the onset times of the 3/2 NTMs in these discharges. One can see that the NTMs are always triggered when the local temperature gradient reaches its maximum. These results are consistent with theoretical predictions 13,15 , which state that a large enough temperature (or pressure) gradient, which is linked to a large bootstrap current, is the crucial element that drives the NTM. Concerning the seeding process of the NTM, prevailing theories presume that the NTM is linearly stable 13 . The development of NTMs requires a seed island whose width must exceed a critical island width, w crit . However, in our experiment no visible MHD mode (e. g., sawtooth activities, fishbones, Alfvénic modes and edge localized modes (ELMs) as seed islands) is observed prior to the NTM, as shown in Fig. 2(c). Although a transient increase is observed in the density fluctuation power during the nonlocal phase (see Fig. 2(d)), there is no clear evidence indicating nonlinear energy coupling between electrostatic and magnetic turbulence via the bicoherence analysis 32 . In our case, there are no obvious ambient seeds. However other possible explanations are that the gradual increase of the temperature (or pressure) gradient before the NTM onset may induce small changes in the current profile and hence minor (invisible) tearing modes, to seed the NTM. Similar "seedless" NTMs have also been reported in other fusion devices 31,33 .
In the above experiments, because the NTM might be self-generated with the nonlocality, its onset is mainly attributed to the transient enhancement of the local pressure gradient around the q = 3/2 rational surface. Thus, the critical values of β N to induce the NTMs in nonlocal discharges are substantially lower than those without nonlocal transport, as illustrated in Fig. 5(b). This gives us a serious caution that in fusion devices the control of NTMs could be more challenging than we expect, since strong perturbation of local parameters can trigger NTMs at a relatively low β N . In case of high β N plasmas with internal transport barriers (ITBs), the local perturbation of the ITB (or ELMs and sawteeth) may also induce the NTM. In the present study, a significant influence of the NTM on the truncation of the nonlocal responses has been observed. Figure 3(a-e) plots time evolutions of plasma density, multi-channel ECE signals and the contour-plot of the frequency spectrum of the Mirnov signal in a discharge with two nonlocal transport induced by edge gas-puffing. After the first gas-puffing at t = 570 ms, the nonlocal event appears, and then the 3/2 NTM is excited with f = 6-8 kHz, similar to that depicted in Fig. 2. After the occurrence of the NTM, the second gas-puffing is injected at t = 770 ms, which also triggers a nonlocal transport event. However, for the second nonlocality, the changes of T e in both edge and core regions are much smaller than the first one. These results indicate that the nonlocality is regulated by the self-generated NTM.
To understand the above phenomena, we have first compared the change of avalanche properties between the two nonlocal periods. In Fig. 3(f,g), the Hurst parameter estimated at ρ ≈ 0.45 and the radial dependence of H during the second nonlocal phase (with NTM) are plotted (black points) for a comparison with the first nonlocal phase. With NTM, the Hurst exponents drop abruptly in the inner region, suggestive of a return to diffusive transport, whereas in the outer area they are nearly unchanged. The truncation of avalanche-like transport by the NTM can be seen more clearly in Fig. 4(b), where the contour-plot of CCF between ECE signals in the second nonlocal period is drawn. The figure shows that the radial correlation in  T e remains large in the outer zone (ρ ≈ 0.5-0.8), consistent with high Hurst parameters in that region, as shown in Fig. 3(g). However, in the inner region (ρ < 0.5), the radial correlation of transport events is truncated. Note that the 3/2 NTM island is located close to ρ ≈ 0.45. To study the impact of the magnetic island on the nonlocality in HL-2A, we have compared the radial dependence of the toroidal flow (V φ ) and flow shear (dV φ /dr) measured by the CXRS with (red) and without (black) the conventional magnetic island (m/n = 2/1). The CXRS monitors the carbon line emission (CVI, n = 8 → 7, 529.06 nm). The results are shown in Fig. 6. Without island, the V φ profile is quite smooth with roughly a constant flow shear rate at all radial locations. With the island (~8.6 cm in width), the magnitude of V φ is generally reduced (see Fig. 6(a)). However, in the latter case, the shear rate of V φ is locally enhanced with dual shear layers developed at the edge of the island, due to probably the influence of the magnetic island on local structures of the toroidal flow. Similar results have been observed in LHD, where a poloidally sheared flow occurs in the magnetic island when the island width grows to a certain extent 19 . According to theories and modeling 10,26,27 , such sheared flows   should regulate or truncate the avalanche-like transport events. Hence, the mechanism for the nonlocal transport is weakened. This explains the truncation of the nonlocal transport by NTMs.

Conclusion
In this Letter, we report the first observation on the interplay between nonlocal heat transport and NTMs generated during the transient nonlocal response at the HL-2A tokamak. The nonlocality is induced by edge cold pulses, along with inward propagating avalanche transport events. The enhanced local pressure gradient at the rational surface can trigger the NTM in relatively low β plasmas. With NTMs, locally sheared toroidal flows are developed in the magnetic island. As a result, the nonlocal avalanche transport event is truncated by flow shearing. The results provide significant evidence for understanding dynamical interactions between multi-scale turbulence and large mode structures.