Observation of resonant tearing mode induced by energetic-ion redistribution due to sawtooth collapse in HL-2A NBI plasmas

Recent observations on the HL-2A tokamak give new experimental evidence of resonant tearing modes (r-TMs) caused by the redistribution of energetic ions (EIs) due to sawtooth collapses in neutral beam injection plasmas. The m/n=2/1 (m and n are the poloidal and toroidal mode numbers) r-TMs with frequencies chirping down rapidly are found after strong bursts of fishbone modes and closely followed strong sawtooth collapses. The frequencies of fishbone modes and r-TMs chirp down from 10 to 7 kHz and 5 to 2 kHz rapidly, higher and lower than the 4–6 kHz existing weak resistive TMs. The fishbone modes propagate in ion diamagnetic drift directions in poloidal. In contrast, r-TMs propagate in the electron diamagnetic drift direction, though r-TMs are excited by EIs. It is suggested that counter-passing EIs, which originated from redistribution of EIs due to fishbone modes and combined with sawtooth collapses, excite r-TMs. Simulation results from the M3D-K code reveal that the counter-passing EIs play an important role in the excitation of r-TMs. The results can help broaden understanding of the interaction mechanisms between energetic particles and TMs, and open up a new perspective on the excitation mechanism of r-TMs.


Introduction
Tearing modes (TMs) or neoclassical TMs (NTMs), driven by the free energy of magnetic flux or plasma pressure, are one of the most dangerous low-frequency magnetohydrodynamic (MHD) instabilities in magnetically confined plasmas, which can enhancelocal transport, degrade confinement, and even cause disruption in high-beta plasmas [1,2].Further, experimental and theoretical research shows that energetic particles (EPs), generated by auxiliary heating and fusion reactions, can not only drive EP modes and Alfvén eigenmodes [3][4][5], but also affect the behaviors of TMs/NTMs, and even resonate with them [6].Sawtooth modes can not only be stabilized [7] but also destabilized by EPs [8].Sawtooth modes can enhance the radial transport of EPs [9,10], and may be a significant method to expel the helium ash from the core of plasmas [10,11].The onset threshold of NTMs increases with the power of co-injection neutral beam injection (NBI) on DIII-D [12], ASDEX-U [13] and NSTX [14], while the amplitude of TMs is greatly reduced by co-injection NBI on MST [15].The stabilizing or destabilizing effects of EPs on TMs have been found by theory and simulation, when the EPs are inside or outside of the rational surface [16][17][18].Non-resonant interactions between trapped or passing EPs and TMs/NTMs are studied by theory and simulation [18][19][20].In addition to these non-resonant wave-particle interactions between EPs and TMs/NTMs, resonance between them is found and studied by experiment and theory.The frequency chirping phenomena of TMs/NTMs on TFTR [21], ASDEX-U [22,23], EAST [24], DIII-D [25], KSTAR [26] and HL-2A [27] indicate strong wave-particle resonance between EPs and TMs/NTMs.Direct wave-particle interactions resulting in amplitude-bursting and frequency-chirping fishbone-like m/n = 2/1 (m and n are the poloidal and toroidal mode numbers) resonant TMs (r-TMs) have been found in the minimal safety factor q min ∼ 1.5 NBI plasmas on HL-2A [27].The frequencies of r-TMs decrease from f r-TM ∼ 10 to 2 kHz within ∼1 ms.r-TMs are excited by co-passing energetic ions (EIs) generated by co-injection NBI directly, and propagate in ion diamagnetic drift directions in poloidal.The nonlinear hybrid kinetic-MHD simulation results from the M3D-K code [28] reveal that r-TMs can be excited by the resonance between TMs and co-passing EIs [29].
Recently, a new kind of r-TM closely following sawtooth collapses has been found in high-density NBI plasmas.The r-TMs propagate in the electron diamagnetic drift direction, which is opposite to the direction of EIs generated by coinjection NBI.It is indicated that fishbone modes combined with closely followed sawtooth collapses cause a strong redistribution of EIs.The q min ∼ 1 in this condition.Simulation results from the hybrid M3D-K code prove that counter-passing EIs play an important role in the excitation of r-TMs.

Experimental results
HL-2A is a mid-size circular cross section tokamak with major and minor radii R 0 = 1.65 m and a = 0.4 m [30].The NBI injects into plasma with an angle of 31.9 • with respect to the plasma current on the magnetic axis, and the tangent radius is 1.4 m [31].The energy of the deuterium beam ions (E b ) is ∼38-45 keV, and the power of NBI (P NBI ) can reach up to 1.5 MW [32].Abundant MHD instabilities [33,34], e.g.geodesic acoustic mode induced by EPs [35], TMs/NTMs [36][37][38], fishbone modes [39][40][41], EP modes [42,43] and Alfvén eigenmodes [44][45][46][47], have been observed during NBI heating on HL-2A.The MHD instabilities involved in this paper, e.g.TMs, fishbone modes, sawtooth modes and r-TMs, are mainly detected from the fluctuations in signals and their spectrogram of Mirnov probes and soft x-ray (SXR) arrays.The electron dentistry (n e ) is measured by a HCOOH laser polarimeter interferometer [48].The electron temperature (T e ) is obtained from the Tomson scattering (TS) [49].The ion temperature (T i ) and toroidal rotation frequency (f t ) are based on the charge exchange recombination specgrogram (CXRS) [50].The temporal resolutions of TS and CXRS systems are ∼33.3 and 12.5 ms, respectively.Besides, the poloidal and toloidal mode numbers of the modes are identified by the sets of 18 poloidal and 10 toroidal Mirnov probe arrays.The detailed location and arrangement of Mirnov probe system and SXR arrays are introduced in [36,40,43].There is a typical example of the new r-TMs which are driven by the redistribution of EIs due to the low-frequency MHD instabilities, as shown in figure 1.The discharge parameters are I p = 165 ± 5 kA and B t = 1.17  fishbone modes and sawtooh modes appear after the injection of NBI.The amplitude of the modes (A), which are presented by the envelope of Mirnov signal, become more and more higher during t = 900-1220 ms, and then the amplitude of the modes reach to their tops after t = 1220 ms.The spectrogram of the Mirnov signal is shown in figure 1(c), and the lowfrequency TMs, fishbone modes, sawtooh modes and r-TMs can be found from it.There always exist the weak resistive TMs before and during the NBI heating.Because of effects of the injection of NBI on the toroidal rotation of plasmas, the frequencies of TMs change among the range of f TM ∼ 4-6 kHz.The f TM keeps as ∼5 kHz when the plasma is in equilibrium after t = 1140 ms.The frequencies of fishbone modes and r-TMs chirp down rapidly and are in the range of f FB ∼ 7-10 kHz and f r-TM ∼ 2-5 kHz, which are located on the upper and lower sides of the weak TMs.
The detail and repeating processes among TMs, fishbone modes, sawtooth modes and r-TMs are shown in figure 2 during t = 1235-1300 ms, in which the discharge parameters are in equilibrium.(c).The reversal surface of sawtooth modes can be estimated near d = 14.3-18.2cm, as shown in figure 4(d).The fluctuations caused by the usual resistive TMs and r-TMs can be found from the SX56 and SX57 in figures 4(e) and (f ) with d = 21.8 and 24.9 cm, and the obvious reversed sawteeth can be found before the r-TMs.The existence of fishbone modes and sawtooth collapse prove that there is the safety factor q ∼ 1 rational surface.It is indicated that the positions of fishbone modes or sawtooth modes (q = 1) and r-TMs (q = m/n = 2 rational surface) are near r ∼ 10 and ∼25 cm.The radial transport coefficient (D i ) and radial velocity (V i ) of the EIs can be estimated by the transport distance of EIs (∆r) from the q = 1 or the core (r ∼ 10 or 0 cm) to q = 2 rational surface (r ∼ 25 cm) and transition interval from the sawtooth collapses to appearance of r-TMs (∆T ∼ 0.5 ms), i.e.D i ≈ ∆r 2 /∆T ≈ 80-12 m 2 s −1 and V i ≈ ∆r/∆T ≈ (3-5) × 10 2 m s −1 .The D i and V i in experimental conditions are much higher than those in classical level.
The rough effects of low-frequency modes on the discharge parameters (n e,l and T i ) can be found in figure 5. Figure 5(a) is the poloidal Mirnov signal as mentioned in figures 1(b) and 2(a).Though the strong fluctuations can be caused by the fishbone modes in n e,l , the tendency of n e,l still increases during the strong bursts of fishbone modes.However, the apparent sudden drops of n e,l can be found just after the sawtooth collapses, as shown in figure 5(b).The solid and dashed vertical line pairs represent the time before and after the fishbone combined with sawtooth collapses.Figures 5(c) and (d) present the evolutions of T i and f t in different positions.The horizontal red and blue curves are the T i and f t in the core and near q ∼ 2 rational surface.The T i decrease in the core of plasmas near the outbreaks of sawtooth collapses (combined with the effects of fishbone modes) at t = 1287.5 and 1300.0 ms (labeled by the blue and red dashed vertical lines) compared with that during weak TMs but without the other modes at t = 1249.9and 1287.5 ms (marked by the blue and red solid vertical lines).There are changes of f t caused by the low-frequency MHD instabilities near the core of plasmas.While the T i and f t near the q = 2 rational surface almost do not be influenced by the fishbone modes and sawtooth collapses.Therefore, it can be proved that the change of parameters are mainly caused by sawtooth collapses, and the frequency chirping-down behaviors of r-TMs are not related with f t near q = 2 rational surface.
The profiles of discharge parameters (n e , q, T i and T e ) and their relationship with the sawtooth collapses are presented in figures 6(a) and (b).The n e in the core can reach to ∼(4.8-5.1)× 10 19 m −3 during t = 1256-1266 ms, and the ratio of n e,l /n G ∼ 0.8-0.9.Three pairs of the n e profile before and after the fishbone modes combined with sawtooth collapses are plotted by the solid and dashed curves in figure 6(a).The colors (black, blue and red) of the profiles correspond to the vertical lines in different time in figure 5(b).The n e in the core drops after sawtooth collapse, which is in agreement with the tendency of n e,l in figure 5(b).The deposition position of NBI power will shift to the outer region compared with that in low-density plasmas.Based on the EFIT code and experimental data, R 0 /a ≈ 1.65 m/0.35 m, and the q-profile dose not change obviously.There is the q-profile near t = 1260 ms, and the position of q = 2 rational surface is about r/a = 0.7, which is in agreement with the observed fluctuations caused by TMs and r-TMs from SX56 and SX57 with d = 21.8 and 24.9 cm.There are obvious decrease of T i near the strong bursts of sawtooth modes at t = 1262.5 and 1300.0 ms (blue-circle and red red-square dashed curves) inside q = 2 rational surface, compared with the T i -profiles during the weak TMs at t = 1249.5 and 1287.5 ms (blue-circle and blue-square solid curves), as shown in figure 6(b).It is indicated that the redistribution of EIs can be caused by strong sawtooth collapses.Because of the low temporal resolution of TS system, T e -profile is averaged during t = 1252.3± 16.65 ms, as shown in figure 6(c).

Simulation results from M3D-K code
The wave-particle interactions between EIs and TMs are simulated by M3D-K code.The main parameters in simulations are set as follows, inverse aspect ratio ε = a/R 0 ∼ 0.24, the Alfvén speed v A = B t /(µ 0 ρ 0 ) 1/2 = B t /(µ 0 n e0 m i ) 1/2 , where, µ 0 and n eo are the vacuum permeability and electron density in the core, Alfvén time τ A = ϵR 0 /(v A ) and Alfvén frequency ω A = v A /(ϵR 0 ).The beam ion injection velocity is estimated as v b = 0.7v A , and the corresponding maximum gyro-radius of beam ions is ρ h = 0.07a.The slowing-down beam ion distribution function [51] is adopted, which is formed by collisions between EIs generated by NBI and thermal-particles, with the pitch angle parameter being defined as Λ = µB 0 /E b , where µ is magnetic moment, B 0 is the magnitude of magnetic field at magnetic axis.For the initial distribution of EIs, the central pitch angle is Λ 0 = 0.0, the pitch angle width is ∆Λ = 0.3, the radial width is ∆ψ = 0.3(ψ max − ψ min ), where, ψ is poloidal magnetic flux, and the critical velocity is set as v c = 0.38v A .The normalized resistivity is η N = η( The detailed expression for distribution function can be referenced in the previous works [29,52]. The simulation results show that pure resistive m/n = 2/1 TMs can be destabilized by the pressure gradient but without EIs when the beta of background plasmas β MHD = 0.08%, and the resistivity of plasmas is benefit for the destabilization of TMs.The regular m = 2 mode structures presented by the the perturbed current (δJ) in the poloidal cross section are shown in figure 7(a).The modes propagate in the electron diamagnetic drift directions.The r-TMs will be destabilized when the beta of EIs β h = 0.1% near q = 2 surface.Though the r-TMs are induced by the EIs, the modes propagate in electron diamagnetic drift direction in poloidal.Compared with the mode structures of resistive TMs (without EIs), the mode structures of r-TMs become irregular, broadened, twisty and shifted radially outward due to the contribution of EIs, as shown in figure 7(b).
The frequency chirping-down phenomena of r-TMs can be reproduced by the M3D-K code.The normalized frequencies of r-TMs (ω/ω A ) decrease rapidly form about −8.4 × 10 −3 to −7.8 × 10 −3 during t = (290-570)τ A , as shown in figure 8, where, the negative sign (−) means the modes propagate in electron diamagnetic drift direction.Based on the discharge  parameters, the mode frequencies of r-TMs in simulation (f m ) shift down from −8.5 to −7.9 kHz, and the frequency shift down time δt m ≈ 280τ A = 0.24 ms.The toroidal rotation frequency near q = 2 rational surface is f t ∼ 5 kHz, which is equal to the f TM observed in experiment.The mode frequencies in laboratory (f Lab ), f t and f m should satisfy the relationship f Lab = f m +nf t .The starting and ending frequencies of r-TMs in simulation are f Lab = −8.5 + 1 × 5 = −3.5 and −7.9 + 1 × 5 = −2.9kHz, which are within the experimental range (f r-TM ∼ 5-2 kHz), as shown in the first row in table 1.Because of various sources of uncertainties, e.g. the distribution function of EIs, the real-time parameter profiles and so on, there are some certain deviations between f r-TM and f Lab .Though δf m and δt m in simulation are about 0.6 kHz and 0.27 ms, which are a little smaller than those in experiment, the frequency chirping rate in simulation ˙fm ∼ δf m /δt m = 2.2 kHz ms −1 is close to that in experiment ( ˙fr-TM ∼ 1.0-1.5 kHz ms −1 ).The r-TMs are caused by resonance between TMs and EIs.The rapid frequency chirping of the mode (chirping rate ( ω)) can be caused by the changes of EIs in energy and position [43], i.e. ω = ∂ω ∂E Ė + ∂ω ∂r ṙ.The energy of EIs transfers to r-TMs can be proved by the increasing amplitude of Mirnov and SXR signals.Besides, the changed positions of counter-passing EIs, originating from sawtooth collapses and transporting from the core to the edge (q = 2 rational surface), can also cause the frequency chirping of r-TMs.It is difficult to clarify which one is the dominant reason for the limitation in experiment.
Generally, the detailed wave-particle interactions can be described by the relationship ω = nω ϕ + pω θ , where, ω is the mode frequency, ω ϕ and ω θ are the toroidal and poloidal transit angular frequencies for EIs, and p is an integer.The unperturbed distribution function (F 0 ) and the locations of EIs resonant with TMs in the space of angular momentum and In addition, there is a secondary resonant-line ω − ω ϕ = 0 (the pink curves).The population of the trapped EIs is small.Besides, the q value near q = 2 rational surface raises monotonically without reversal, i.e. magnetic shear s = r q dq dr > 0, so that it is impossible to have resonance between the barely trapped EIs and modes.Therefore, the r-TMs should be excited by the counter-passing EIs, which are generated by the redistribution due to strong fishbone modes and combined with closely followed sawtooth collapses.

Discussions and summary
Two kinds of frequency-chirping r-TMs are found in HL-2A NBI plasmas.The previous m/n = 2/1 r-TMs have been found in the plasma current I p ∼ 140-160 kA, toroidal magnetic field B t ∼ 1.1-1.4T and line-averaged electron density n e,l ∼ 1.0 × 10 19 m −3 (∼0.3nG low density) NBI plasmas [27], where n G is Greenwald density.The frequencies of the r-TMs decrease from f r-TM ∼ 10 to 2 kHz within ∼1 ms.The r-TMs propagate in the ion diamagnetic drift direction in poloidal, and are excited by the co-passing EIs generated by coinjection NBI, directly.The q min ∼ 1.5, which is higher than unity, can also be proved by the experimental observation of no fishbone modes or sawtooth collapses being found.The nonlinear hybrid kinetic-MHD simulation results from the M3D-K code [28] reveal that r-TMs are excited by the resonance between TMs and co-passing EIs generated by NBI, and the wave-particle resonance condition is satisfied by ω − ω ϕ + ω θ = 0 [29].The new r-TMs excited by the counter-passing EIs are found in the n e,l ∼ 0.8-0.9nG high-density NBI plasmas on HL-2A.It is found that r-TMs propagate in the electron diamagnetic drift direction in poloidal, and the mode numbers are confirmed as m/n = 2/1 by Mirnov probe arrays.The q min ∼ 1 in this condition.The fluctuations from Mirnov probe and SXR arrays caused by TMs, fishbone modes combined with sawtooth modes and r-TMs, are clarified by the wave filtering method.Owing to the injection-direction and -angle of NBI, the co-passing EIs are dominant.The origination of counter-passing EIs most likely comes from the redistribution of EIs due to strong fishbone modes combined with sawtooth collapses.Based on the discharge parameters in experiment, the M3D-K simulation results prove that the new r-TMs are mainly excited by the counter-passing EIs, and the main wave-particle resonant condition satisfies ω − ω ϕ − 2ω θ = 0.The E and P ϕ of the counter-passing resonant EIs are in the range of E ∈ [0.7, 1.0]E b , and P ϕ ∈ [−0.3, −0.05].The frequency chirping-down phenomenon is reproduced by simulation, which is in agreement with the experimental phenomenon.The m = 2 mode structures of resistive TM and r-TMs presented by δJ are obtained.The mode structures of r-TMs become irregular, broadened, twisty and shifted radially outward due to the contribution of EIs.
Although the experimental phenomena of the previous and new r-TMs are similar on the surface, e.g.fishbone-like frequency chirping, the same m/n = 2/1 mode numbers and so on, in fact there are lots of differences, e.g.frequency chirpingdown range, frequency chirping rate, mode propagating direction, excitation conditions (q min , n e,l and P NBI ), origination of resonant EIs and resonant condition between EIs and TMs, and so on.Detailed comparisons between the two r-TMs on features and destabilization conditions are shown in table 2. To distinguish the two r-TMs, the previous and new r-TM are written as r-TM 1 and r-TM 2 , respectively.There are obvious features and differences between the two modes.(1) r-TM 1 and r-TM 2 propagate in ion and electron diamagnetic drift directions in poloidal, respectively.(2) r-TM 1 is excited by copassing EIs, while r-TM 2 is excited by counter-passing EIs.
(3) The origination of EIs which induced the two kinds of r-TMs is different.The co-passing EIs which drive r-TM 1 are generated by high-power co-injection NBI directly, while r-TM 2 is excited by counter-passing EIs due to redistribution of EIs caused by fishbone modes combined with sawtooth collapses.(4) The q-profiles for the two r-TMs are different.The values of q min are close to 1.5 and unity for the excitation conditions of r-TM 1 and r-TM 2 .(5) The resonance between co-or counter-passing EIs and TMs is satisfied by ω − ω ϕ + 2ω θ = 0 or ω − ω ϕ − 2ω θ = 0. (6) r-TM 1 and r-TM 2 are found in lowand high-density plasmas, and their corresponding n e /n G ∼ 0.3 and 0.8-0.9,respectively.The P NBI for the two conditions are 1.0 and 0.5 MW.The population for the co-and counterpassing EIs is different.
The new results can help broaden understanding of the interaction mechanism between EPs and TMs.The contribution of fishbone modes combined with sawtooth collapses in the redistribution and transport of EIs is found in experiment, and it may be a significant method to expel the helium ash from the core of plasmas.These findings are of crucial importance

Figure 1 .
Figure 1.Low-frequency MHD instabilities in high density NBI plasmas in shot 38645.(a) Discharge parameters of line-averaged electron density (n e,l ) and P NBI presented by the black and blue curves, respectively.(b) Original signal of poloidal Mirnov probe and (c) its spectrogram.
T. The 0.5 MW NBI with E b ∼ 42 keV switches on at t = 900 ms.The lineaveraged electron density (n e,l ) increases from 2.0 × 10 19 to 3.3 × 10 19 m −3 during t = 900-1220 ms gradually, and then it keeps as n e,l ∼ (3.0-3.3)× 10 19 m −3 after t = 1220 ms, as shown in figure 1(a).Obvious and repetitive bursts of magnetic fluctuations mainly caused by the fishbone modes and sawtooth collapses are found from the poloidal Mirnov signals after the injection of NBI, as shown in figure 1(b).The

Figure 2 .
Figure 2. Detailed evolutions of TMs, fishbone modes, sawtooth modes and r-TMs.(a) Fluctuations from the original poloidal Mirnov signal caused by the low-frequency MHD instabilities and (b) its spectrogram.The filtered waves of poloidal Mirnov signals with the passband of (c) 4-6.5, (e) 6-10 and (g) 2-5 kHz, presented by the pink, red and blue curves, respectively.The corresponding spectrogram of (d) TMs, (f ) fishbone modes combined with closely followed sawtooth collapses and (h) r-TMs.
Figure 2(a) exhibits the fluctuations caused by these low-frequency MHD instabilities in poloidal Mirnov signals.The corresponding evolutions of MHD instabilities can be found from its spectrogram, as shown in figure 2(b).The

Figure 3 .
Figure 3. Mode numbers of TMs and r-TMs identified by the phase shift method.Waveforms of filtered TMs (a1)-(b1) and r-TMs (a2)-(b2) with passbands of 4-6.5 and 1.5-5.5 kHz from poloidal (left) and toroidal (right) Mirnov signals arranged from the bottom to the top in order of plasma current direction.

Figure 4 .
Figure 4. Fluctuations caused by the low-frequency MHD instabilities from SXR arrays.Panels (a)-(h) represent the SXR signals with chord distances d = 5.3, 10.0, 14.3, 18.2, 21.8, 24.9, 27.6 and 30.0 cm.The red, pink and blue curves present the fluctuations caused by fishbone modes combined with sawtooth collapses, TMs and r-TMs, respectively.

Figure 5 .
Figure 5. Effects of sawtooth collapses on discharge parameters.(a) Fluctuations caused by TMs, fishbone modes, sawtooth collapses and r-TMs from poloidal Mirnov signal.Evolutions of discharge parameters of (b) n e,l , (c) T i and (d) ft from the core (top) to the edge (bottom).Red and blue curves in panels (c) and (d) are the T i and ft in the core (q ⩽ 1) and near q ∼ 2 rational surface, respectively.

Figure 6 .
Figure 6.Profiles of discharge parameters and their evolutions related with the modes.(a) ne-profiles before (t = 1256, 1258, 1260 and 1262 ms) and after (t = 1264 and 1266 ms) sawtooth collapse presented by the black, blue, green, pink, red and brown curves, and q-profiles at t = 1260 ms, (b) T i -profiles before (t = 1249.9and 1287.5 ms) and after (t = 1262.5 and 1300 ms) sawtooth collapses presented by blue and red curves, and (c) Te-profiles at t = 1252.3± 16.65 ms.

Figure 9 .
Figure 9.The unperturbed distribution function (F 0 ) and the locations of EIs resonantwith TMs in angular momentum and energy (P ϕ and E) space of the EIs with magnetic moment µ ≈ 0.695 and ∆µ = 0.063 at t = 800τ A .