Rui-Jie Zhou(周瑞杰)

Institute of Plasma Physics,Chinese Academy of Sciences,Hefei 230031,China

Keywords: runaway electrons,bremsstrahlung emission,tokamak

1.Introduction

Major plasma disruptions are a well-known threat to the safety of magnetic confined nuclear fusion reactors of the tokamak type, and continue to be an important and unsolved scientific problem until now.[1,2]In particular, the generated high-energy runaway electrons(REs)is one of the serious issues during this phase.[3,4]It is important to reduce the energy and population of REs and mitigate their effects on the first-wall materials in tokamaks,especially for larger tokamak devices.[5,6]

REs are accelerated mainly by the toroidal electric field in plasma.Because of the weak Coulomb collision of REs with other plasma particles, the collisional radial transport of REs is much less than the thermal transport.It is widely recognized that the anomalous transport of REs in tokamak plasma is important.The transport induced by electrostatic fluctuations has an inverse dependence on the electron velocity.So, for REs,this transport is very small.Conversely, the transport of REs is very sensitive to the magnetic fluctuations.[7,8]

The tearing modes (TMs) instability is driven by the radial gradient of the toroidal current density, accompanied by the formation of magnetic islands,which can induce magnetic fluctuations with large amplitude.So, the study of the effect of TMs on the confinement of REs is of considerable interest and important.We report in this paper the results from EAST tokamak.It reveals that the magnetic fluctuations can enhance the radial diffusion of REs.In the same time,magnetic islands with large size can possibly confine the high energy RE beams inside the islands,preventing the REs from loss.

2.Experimental results

A typical discharge is shown in Fig.1.The TMs with frequencyf ≈4 kHz are excited at 2 s.Its frequency decreases rapidly tof ≈2 kHz following the growth of its amplitude.Its frequency remains roughly stable from approximately 2.08 s,although its amplitude is still evolving.The mode number of the TMs ism/n= 2/1, withmandnare the poloidal and toroidal mode numbers, respectively.It locates at the radial position ofρ2/1(r/a)≈0.38.The second harmonic exists in the frequency spectrum of the TMs, which indicates possibly deformation of the magnetic island structure.The period of interest is divided into three phases.Phase I is the amplitude increasing phase after its frequency is stable.Phase II is the amplitude decreasing phase.Phase III is the phase when its amplitude has decreased to very low level.

In the same time, loss of REs is observed through the measurement of the high energy hard x-ray(HXR)signal.The HXR signal is from the thick-target bremsstrahlung emission when the REs are lost and then hit the limiter or first wall of the EAST device,with energy range from approximately 0.5 MeV to 10 MeV.As can be seen in the frequency spectrum of HXR signal, loss of REs can hardly be observed from 2 s to 2.06 s when the TMs amplitude is small.Then, loss of REs can be observed clearly in HXR signal with the same frequency response, except more harmonics exist probably owning to the response of its electric system.

It can be seen clearly in Fig.2 the relation between the TMs and REs loss.The time evolving relation between the TMs and REs in a very narrow time window is shown in Fig.2(a).Peaks in theand HXR signal are extracted, and the time evolving of those peaks is shown in Fig.2(c).The general tendency is clear that the REs are lost with the same frequency comparing to the rotate frequency of the TMs,and the loss intensity of REs is approximately proportional to the amplitude of the TMs.

Fig.1.(a)Time evolution of the normalized magnetic fluctuations =/B0 deduced from Mirnov signal.(b)Time evolution of the frequency spectrum of ?b.Panels(c)and(d)are for the HXR signal.

3.Confinement characters

The lose of REs is used widely in tokamaks to derive its radial diffusion coefficients.[9–12]For REs,it has

where ?tis the time delay between the peaks in ?bsignal and the corresponding peaks in HXR signal.?ris the inversion radial position of the TMs.Dris the radial diffusion coefficient of REs.So,for the case shown in Fig.1,it can be modified to

The value of ?tcan be deduced easily during sawteethlike situations when the oscillation period is large.The time interval is about several ms in that case.However, for TMs case, the time interval is about 0.5 ms, and the particle confinement time of REs is about the order of several ms in this case.This means there will have several peak delays between the magnetic peaks and the corresponding HXR peaks.So,it is hard to identify which ?bpeak should responsible for one specific HXR peak in TMs case.In this case,dedicated procedure is performed.Firstly,the correspondence between the peaks ofand HXR is aligned when their amplitudes are reached their maximum around 2.12 s,as shown in Fig.2(c).Also,it can be confirmed by the cross-correlation relation between the peaks ofand HXR signal,as shown in Fig.2(b).It indicates there is an 8 peaks delay.As shown in Fig.2(a), the 8 peaks delay corresponding to a time delay of approximately ?t ～4 ms,and a diffusion coefficient of approximatelyDr～3.1 m2·s?1.Then, matrix of theand HXR peaks is build in which they have the corresponding relations.On the other hand, the frequency response of the two signals should correspond with each other.The inverse of the time interval between two adjacent peaks will represent the main frequency character of the signal.As shown in Fig.2(d), the main frequency characters ofand HXR can well represent the frequency spectrum in Figs.1(b)and 1(d),and the two signals are aligned with each other.Moreover,the time delay ?tdeduced in this way is reasonable in physics.However, considering the phase shift of the magnetic fluctuations of TMs from plasma center to edge region, correction of the ?tis still needed.It will be quite complicated to perform this correction properly.So, the diffusion coefficient derived from Eq.(2)in this paper will have large errors, and it is used only to study the tendency of the response of the REs diffusion to the TMs amplitude.

In theory, the diffusion coefficient of REsDris proportional to.The well-known Rechester–Rosenbluth coefficientDRR=πqv‖Rholds when the magnetic field is assumed to be fully stochastic.[13]A shielding factor?should be included considering the correction ofDrdue to the displacement of the runaway electron orbits from the magnetic surfaces (orbit-averaging) and their large gyroradii (gyroaveraging)DRR=?πqv‖R.[14,15]So, it is important to analyze the relation between the diffusion coefficient of REsDrand the magnetic fluctuation amplitude of TMs.This is also why we choose only the time window when the frequency of TMs is stable.It is suitable to analyze only the effect ofonDrwithout affected by other parameters.The results are shown in Fig.3.

It can be seen in Fig.3(a),Dris decreasing following the increasing ofduring phase I,Drincreasing following the decreasing ofduring phase II,andDris decreasing following the decreasing ofduring phase III.A clear separation exists around≈0.8×10?9, which is corresponding to a magnetic island width of approximatelyωcrit～1 cm based on the semi-empirical formula(Gauss).[16]It has to point out the analyze from ECE signals indicates this critical width is larger than 1 cm.The exact value of this island width is hard to be derived accurately from the experimental data.Nevertheless,it indicates that,the magnetic fluctuations of TMs will enhance the radial diffusion of REs when the magnetic island is small.However, when the magnetic island is larger,it can possibly confine the RE beams inside the islands,preventing the REs from loss.A critical magnetic island width exists which can separate this two situations.

Fig.2.(a)The time evolving relation between the TMs and REs,in a very narrow time window.(b)The cross-correlation relation between the peaks of normalized amplitudes of magnetic fluctuations ?b and the HXR signal.(c)The time evolving relation between the ?b and HXR signal.(d)The inverse of the time interval between two adjacent peaks for the ?b and HXR signal.

Fig.3.(a)The relation between the diffusion coefficient of REs Dr and magnetic amplitude of TMs ?b2 during phases I,II,and III.The arrows indicate the direction of the time evolution.(b)The relation between the diffusion coefficient Dr and the HXR intensity that originated from the REs loss.(c)The relation between the HXR intensity and magnetic amplitude ?b2.

Figure 3(b) is the relation between the diffusion coefficientDrand the HXR intensity that originated from the REs loss.The REs loss is governed by dN/dt=S?N/τp,in whichNis the number of the REs,Sis the source term related with the plasma region where REs are lost, andτpis the particle confinement time of REs related with the diffusion coefficientDr.So, HXR intensity is related with both the source termSand the diffusion coefficientDr.Considering phase III,whenDris decreasing,HXR is almost constant.It indicates that the plasma region where REs are lost is decreasing, related with the decreasing of theduring this phase.The HXR intensity can be very sensitive to the plasma region affected by TMs.

Figure 3(c)is the relation between the HXR intensity and the magnetic fluctuation amplitude of TMsIt indicates that the REs loss is enhanced firstly during phase I following the increase of.Then,the REs loss is stable and decreases whenis large.At last, REs loss is very weak during phase III.With the same amplitude of,REs loss is weaker in phases II and III,comparing to the phase I.

4.Confinement of RE beams

The energy spectrum of the thick-target br emsstrahlung emission produced by REs when they are lost from plasma is continuous.The spectrum has a maximum energyE≤E, in whichEis the maximum energy of the generated HXR,andEis the maximum energy of REs.Besides this bremsstrahlung emission, high energy REs that confined inside plasma can also produce synchrotron radiation in the infrared or visible light range.[17,18]When REs with energy high enough to generate synchrotron radiation lost from plasma and then hit the limiter or first wall,they will also generate thicktarget bremsstrahlung emission in the low energy hard x-ray range which can be measured by the HXR system.

Figure 4(a) is the synchrotron radiation image from the RE beams around 2.1 s.The intensity of the synchrotron radiation is higher in the high field side,which is related with the large orbit shift of RE beams.[19–21]The pixel marked by circle A is located around the location of the TMsρ2/1(r/a)≈0.38.Clearly, it indicates the RE beams are trapped by the magnetic islands, and then confined inside the islands.There is also another location nearρ3/2(r/a)≈0.19 marked by circle B where RE beams exist, which is still being analyzed.The pixel marked by circle C is located near the plasma center.

Time evolution of the pixel intensity for those three pixels is shown in Fig.4(b).The intensity of pixel A is at its maximum around 2.1 s when the magnetic amplitude ?bis also at its maximum.It is coincident with the results that RE beams are confined by the magnetic island when the island grows to large size.The intensity of pixel B and C grows to stable after 2.1 s.RE beams are confined stably inside the locationρ2/1(r/a)≈0.38.This is coincident with the results that REs loss is weaker in phases II and III comparing to phase I.

Fig.4.(a) The synchrotron radiation image from the RE beams around 2.1 s, measured by a visible light camera.(b) The time evolution of the pixel intensity for the three pixels that marked in panel(a).

5.Summary

The effect of TMs on the confinement of REs is studied in EAST tokamak.The loss of REs is observed with the same frequency response comparing to the rotate frequency of the TMs when the magnetic fluctuation amplitude grows to a certain level.The HXR intensity can be very sensitive to the plasma region that affected by the TMs and the diffusion coefficientDr.Based on the time response relation between TMs and REs, the general tendency of the radial diffusion coefficient of REsDrcan be derived.The results indicate that, the magnetic fluctuations of TMs will enhance the radial diffusion of REs when the magnetic island is small.Following the increasing of the magnetic fluctuations of TMs,the formed large magnetic island may weaken the radial diffusion of REs.Combining with the synchrotron radiation originated from the high energy REs that confined inside plasma,it indicates that large magnetic island may confine the RE beams inside the islands or inside the plasma center,and weaken the radial diffusion of REs.

At last, it is possible that the critical magnetic island width may related with the large displacement size of REs orbits from the magnetic surfaces.Also, with energy substantially higher than the thermal electrons,REs can carry current effectively in plasma.So, the discrete spatial location of the confined RE beams inside the plasma is possible to change the monotone current distribution of the plasma,which in turn can potentially affect the evolution of the TMs.This two topics are worthy to be studied further,and they will be studied in our future works.

Acknowledgments

Project partly supported by the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No.2021445), the Science Foundation of Institute of Plasma Physics of Chinese Academy of Sciences (Grant No.DSJJ-2022-05), and partly supported by the Comprehensive Research Facility for Fusion Technology Program of China(Grant No.2018-000052-73-01-001228).