Bursting core-localized ellipticity-induced Alfvén eigenmodes driven by energetic electrons during EAST ohmic discharges

A series of high-frequency ( 400∼1000 kHz ) bursting core-localized Alfvén instabilities have been observed during ohmic discharges in EAST tokamak. The instability trigger favours the discharge conditions of low toroidal magnetic field and low electron density. The toroidal mode numbers are mainly n=2∼3 and they propagate in the ion diamagnetic drift (co-current) direction. These modes are radially localized in the range of ρtor=0.2∼0.35 based on Doppler BackScatter measurement. They are identified as ellipticity-induced Alfvén eigenmodes (EAEs) occurring at q=1 rational surfaces by magnetohydrodynamics simulations using the realistic geometry and plasma profiles. The EAEs show regular bursts with ∼10 ms duration along with the mode frequency chirping downwards and upwards rapidly. It is also found that sawtooth events can interrupt the growth and evolution of the EAEs, causing the modes to disappear immediately. Passing energetic electrons (EEs) that move much faster than Alfvén velocity are responsible for the destabilization of these EAEs, which attribute to the fact that the large poloidal and toroidal frequencies mostly cancel each other and satisfy the EAE resonance condition with primary energy exchange. These novel experimental results of the wave-particle interaction between EAEs and EEs are helpful for extrapolating alpha particle physics that are characterized by small orbit width with respect to machine size in future fusion reactors.

A series of high-frequency (400 ∼ 1000 kHz) bursting core-localized Alfvén instabilities have been observed during ohmic discharges in EAST tokamak.The instability trigger favours the discharge conditions of low toroidal magnetic field and low electron density.The toroidal mode numbers are mainly n = 2 ∼ 3 and they propagate in the ion diamagnetic drift (co-current) direction.These modes are radially localized in the range of ρ tor = 0.2 ∼ 0.35 based on Doppler BackScatter measurement.They are identified as ellipticity-induced Alfvén eigenmodes (EAEs) occurring at q = 1 rational surfaces by magnetohydrodynamics simulations using the realistic geometry and plasma profiles.The EAEs show regular bursts with ∼10 ms duration along with the mode frequency chirping downwards and upwards rapidly.It is also found that sawtooth events can interrupt the growth and evolution of the EAEs, causing the modes to disappear immediately.Passing energetic electrons (EEs) that move much faster than Alfvén velocity are responsible for the destabilization of these EAEs, which attribute to the fact that the large poloidal and toroidal frequencies mostly cancel each other and satisfy the EAE resonance Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
Alfvén instabilities will affect the confinement of energetic particles, as well as the overall performance of future fusion reactors, so comprehensive understanding of their dynamics is crucial [1].The Alfvén instabilities can be excited by energetic particles (both ions and electrons) via the resonant wave-particle interactions [2,3].Over the past decades, the interaction between Alfvén instabilities and energetic ions has been extensively studied on fusion devices around the world [4][5][6].In addition to energetic ions, energetic electrons (EEs) can also drive Alfvén instabilities, as the trapped particle bounce averaged dynamics depends on energy instead of mass [7].In fact, the research on the interaction between EEs and Alfvén instabilities is also of great significance.It can be extrapolated to the alpha particle interactions with shear Alfvén wave (SAW) instabilities in burning plasmas, since under reactor-relevant conditions alpha particles are characterized by small dimensionless orbit, similarly to electrons [7].Also, the electron fishbones can be specifically extrapolated to low-frequency magnetohydrodynamic (MHD) modes (e.g.fishbones) destabilized by trapped alpha particles through precessional drift resonance on ITER.
For future machines with shaped plasmas including elongation like ITER, the large ellipticity-induced SAW continuum gap width might be a threat to the fast particle confinement since more ellipticity-induced Alfvén eigenmodes (EAEs) exist and are less damped by continuum coupling, especially when the machine is operated with a flat density profile, on-axis safety factor q 0 below one and a low magnetic shear region in the core [27].JT-60U presented the chirping EAEs residing at the q = 1 (where q is the safety factor) surface driven by deeply trapped ions during ion cyclotron range of frequency heating in 2001 [27].In 2020, EAEs (located at the q = 1 surface) with toroidal mode number n being −1 and +1 ∼ +5, driven by MeV-range energetic deuterons and fusion-born alpha particles are reported on JET [28,29].
Recently, a series of high-frequency core-localized (at q = 1 rational surface) EAEs with bursting pattern and frequency chirping nonlinear features were observed in Experimental Advanced Superconducting Tokamak (EAST) ohmic discharges in low thermal plasma density and magnetic field strength configurations.It should be noted that the Alfvén instabilities excited by EEs in reactor relevant plasma may be likely occurring at the plasma periphery with lower density where the Alfvén speed is large, which differs from the core plasma situation on EAST low density discharge here.But the observations on EAST are useful for the operation of present device and for extrapolation to alpha particle physics in future reactor relevant plasma.
The remainder of the paper is organized as follows.Section 2 reports the experimental observations of the EAEs.Section 3 presents the main characteristics of the modes, including the measurement of the radial mode position, calculation of the Alfvén continuum, the nonlinear behaviour of the modes, and the impact of sawtooth events.The destabilization mechanism is discussed in section 4 and summaries are given in section 5.

Experimental setup and observations
EAST is a fully superconducting tokamak with a major radius R 0 ≈ 1.85 m, a minor radius a≈0.45 m and elongation κ ≈ 1.7.A novel electromagnetic probe array (EMPA), with a high sampling rate (2 MHz) and self-resonant frequency, has recently been developed on EAST.The EMPA can provide higher frequency magnetic fluctuation measurements [30].
During ohmic discharges, we observed a series of bursting high frequency (400 ∼ 1000 kHz) magnetic perturbations using the EMPA magnetic probes.These magnetic perturbations are identified to be EAEs later on, so for convenience, they will be directly referred to as EAEs below.Figure 1 illustrates the main plasma parameters in the current flattop stage of shot #113416, which is a typical ohmic discharge with bursting EAEs. Figure 1(a) shows the time evolution of the line-averaged electron density and the voltage of the toroidal electric field.The electron density ramps down, and the loop voltage is relatively high, which is about 0.8 V and remains almost constant.In low-density ohmic discharges, when the toroidal electric field exceeds the critical field, some electrons can be accelerated freely and reach very high energies.In figure 1(b), the ratio of the toroidal electric field to the critical field, E loop /E c ≈ 9 [ U loop (V) /n e ( 10 19 m −3 )] increases with time, and satisfies the condition for abundant EEs generation on EAST [31,32].According to the database of EAST discharge experiments, the threshold value of E loop /E c is in the range of 3 ∼ 5, which is favourable for the generation of abundant EEs.However, this threshold can vary with different experimental conditions such as wall conditioning and impurity state etc.As a result, the proportion of EEs in plasma increases, which is represented by the value of HXR (a.u.) /n e ( 10 19 m −3 ) 2 .The hard x-ray (HXR) intensity is detected by the Bi 4 Ge 3 O 12 detector, representing the thicktarget bremsstrahlung emission when EEs leave the plasma and hit the first wall [33,34].At about 4.8 s, the proportion of EEs starts to increase significantly and exceeds an approximate threshold in this shot represented by the grey dashed line in figure 1(b), beyond which obvious fluctuations appear on the magnetic probe signal as figure 1(c).The frequency spectrum is shown in figure 1(d) and a clearer enlargement of spectrum is given in figure 2. These instabilities show bursting and chirping pattern in the frequency range of f ∼ 500−700 kHz.Since the resonance frequency of the EMPA magnetic probe is about 700 kHz, the toroidal mode number n of EAE was determined with great care.A database containing more than 30 discharges has been established to analyse the toroidal mode number.We focused on modes with frequencies below 600 kHz because modes with relatively low frequencies are less affected by magnetic probe resonance frequency effects and their mode number calculations are more reliable.For modes with higher frequencies, we have developed a reliable and flexible method for obtaining, in the absence of instabilities, the background phase difference caused by the magnetic probe resonance frequency effect, which is then subtracted to obtain a realistic toroidal mode number.This method has been carefully and extensively checked and can reliably draw overall conclusions when determining the mode number of higher frequency (> 600 kHz) modes, although there is still sometimes ±1 uncertainty.The statistical results indicate that the toroidal mode numbers are mainly n = 2 and 3.However, cases with n = 1 or 4 also exist, but the probability is very small, less than 10%, and n > 4 is not observed in experiments.Figure 3 exemplifies instabilities with n = 3 and relatively low frequency.In addition, these instabilities propagate in the ion diamagnetic drift (co-current) direction.In figure 1(d), the mode frequency increases as the electron density ramps down, which matches with 1/ √ n e (grey line).Further, results of toroidal field and electron density scanning indicate that the mode frequency is proportional to B t / √ n e as shown in figure 4, satisfying the Alfvén scaling.The perturbations are observed with a large range of plasma parameters: often at low toroidal field with 1.6 ∼ 1.8 T and occasionally at high toroidal field up to 2.4 T, the plasma current 250 ∼ 550 kA and the electron density 0.6 ∼ 2.2 × 10 19 m −3 , which is relatively low.Furthermore, on the basis of a large number of statistics, it is found that the intensity of the magnetic perturbation is enhanced with lower toroidal field and higher plasma current as shown in figure 5, since the q = 1 surface is more likely to exist under these conditions which is important for EAEs.The higher the plasma current at a constant toroidal magnetic field, the lower the safety factor profile, which may potentially provide more interactions with the q = 1 surface statistically, which is the resonant surface of the observed EAEs (described below).This may explain the significant increase of the magnetic perturbation amplitude with increasing plasma current.

The radial location of modes
Doppler BackScatter (DBS) system has been proved to be a powerful diagnostic for perpendicular propagation of density fluctuation and radial electric field measurements.An eightchannel DBS has been installed on EAST, which can probe the turbulence at eight different radial locations simultaneously by launching eight fixed frequencies probing beam (with frequencies being 55, 57.5, 60, 62.5, 67.5, 70, 72.5, 75 GHz) into plasma [35].However, besides the turbulence, DBS is also sensitive to the oscillation of the local cut-off layer which is caused by the large-scale density fluctuation, named 'phase modulation' in [36].Thus, the spatial position of the EAEs can be inferred from analysing the spectrum of DBS signals.
For shot #112786, the spectrum of EAEs measured by magnetic probe is drawn in figure 6(a).Figures 6(b)-(f ) show the spectrums of 67.5, 62.5, 60.0, 57.5 and 55.0 GHz DBS signals, corresponding to the radial positions of ρ tor ∼ 0.23, 0.44, 0.50, 0.58 and 0.73 on the low field side (LFS), respectively.Here, the radius corresponds to the square root of normalized toroidal magnetic flux as ρ tor = √ ψ t .The spectrum of the 67.5 GHz (ρ tor ∼ 0.23) DBS signal exhibits strong correlation with that of the magnetic probe signal, while such fluctuations have not been observed in the more external DBS signals.The cross-power spectral densities of the DBS signal and the magnetic probe signal were calculated for each of the five radial locations, as shown in figure 6(g), which represent the strength of their correlation in the frequency range of EAEs.The radial distribution indicates that the correlation between the DBS signal at ρ tor ∼ 0.23 and the magnetic signal is significantly stronger than the other channels.Hence, the magnetic surfaces near ρ tor ∼ 0.23 are modulated by EAEs, while those beyond ρ tor ∼ 0.44 are unaffected.
Three time slices are chosen in #112786, which are 2.25 s, 4.5 s and 5.95 s, and the density profiles evolution are shown in figure 7(a).The electron density profile is reconstructed based on the POlarimeter-INTerferometer (POINT) [37].The hollow circle symbols represent positions of DBS channels that detect EAEs, and the cross symbols represent DBS channels without EAEs.The frequency of DBS channels that can detect EAEs (i.e. the hollow circle symbols) are 72.5 GHz, 70.0 GHz and 67.5 GHz for 2.25 s, 4.5 s and 5.95 s respectively.Since the electron density decreases over time as shown in figure 7(b), the position of the reflecting layer for a given fixed frequency moves inward, so the frequency of the DBS with EAEs changes and decreases over time.For all three time slices, the radial position of DBS channels subject to the EAEs modulation are approximately in the range of ρ = 0.2 ∼ 0.35.
The safety factor profiles for these three time slices are shown in figure 7(c), which are obtain by EFIT [38] with constraint by the Faraday effect from the POINT [39].In this discharge, sawtooth events can be observed on the electron cyclotron emission (ECE) signals, which exhibit a positive sawtooth waveform evolution inside the q = 1 surface, and an anti-sawtooth waveform outside the q = 1 surface.Therefore, based on the 32-channel heterodyne radiometer for ECE measurements [40], it is possible to estimate the spatial extent where the q = 1 surface is located, as shown by the horizontal solid lines in figure 7(c).The plasma current at 2.25 s is 400 kA, which is lower than the current in the flat-top stage at 4.5 s and 5.95 s in figure 7(d).As a result, from 2.25 s to 4.5 s, the safety factor decreases and the radius of the q = 1 surface expands.The safety factor profile and the radius of q = 1 surface are almost the same for 4.5 s and 5.95 s.This evolution trend of the safety factor profile is consistent with the evolution of the q = 1 surface spatial extent estimated by ECE signals.Overall, although there was some variation in the safety factor profile along the discharge, the DBS channels detecting EAEs pass through the q = 1 surface for all three time slices.In summary, from the DBS diagnostic data, we can conclude that the high-frequency bursting EAE are core-localized in the region of ρ tor = 0.2 ∼ 0.35.

Alfvén continuum and eigenmodes calculation
To understand these high frequency Alfvén activities observed in EAST experiments, we carry out MHD simulations of #112786@t = 5.95 s by using MAS eigenvalue code, where the radial profiles of plasma density n e and safety factor q are shown in figure 7. MAS code consists of multi-level physics model for plasma stability analysis in general geometry [41], in this work we apply the reduced-MHD model which faithfully captures the SAW physics and removes the redundant acoustic coupling that is ignorable in the Alfvén frequency regime.The physics models used for Alfvén continuum and Alfvén eigenmode (AE) simulations are also consistent with each other, which enable the accurate identification of AEs according to their locations on Alfvén continua.
We focus on studying the unstable modes characterized by n = 1 ∼ 4 according to experimental measurements as introduced in section 2. The Alfvén continua are shown in figures 8(a1)-(d1), and it is seen that EAE gaps appear at the q = 1 rational surfaces in the core region for all n numbers, of which lower and upper continuum accumulation point (CAP) frequencies are f ∼ 600 kHz and f ∼ 800 kHz that cover the range of high frequency perturbations from experimental diagnosis as shown in figure 6.Two discrete EAEs are found inside EAE gap near q = 1 for each n number.The higher frequency one is termed as 'odd EAE' since its mode structure exhibits odd parity as shown in figures 8(a2)-(d2).Its frequency is just below the upper EAE-CAP as indicated by the black solid lines in figures 8(a1)-(d1).The odd EAE has similarities with odd toroidicity-induced Alfvén eigenmode (TAE) that has been predicated theoretically in [42,43] and observed experimentally in [44], such as core-localization, frequency near the upper continuous spectrum, and opposite polarizations of two dominant poloidal harmonics.The lower frequency one is conventional EAE with even parity for m and m + 2 harmonics as shown in figures 8(a3)-(d3), which is terms as 'even EAE' in this work and indicated by the black dotted lines in figures 8(a1)-(d1).
The EAE frequencies with different n numbers and parities calculated by MAS are listed in table 1, the odd and even EAE frequencies are in the ranges of f ∼ 776−790 kHz and f ∼ 682−689 kHz respectively, and all modes are localized around q = 1 surface, which show agreements with experimental observation on both frequency and radial location to a certain degree.According to MHD simulations by MAS code, we identify the observed high frequency bursting modes are EAEs that occur around q = 1 rational surface.

Characteristics of mode frequency
The EAEs have three distinctive features on the spectrum (figure 2(b)), which can be summarized as multi-branch coexisting, bursting and chirping.There are always multiple (1 ∼ 4) modes coexisting at the same time, and the frequency separation between each mode is about 20 ∼ 50 kHz.According to our experiments, the dominant single mode and coexistence of two modes are widely observed on the spectrogram, and the coexistences of three or four modes can also appear less frequently.An interesting phenomenon is that several coexisting modes have the same toroidal mode numbers n.Figures 3(a) and (b) show an example of two-mode coexistence with n = 3 for both modes.Figures 9(c) and (d) display the case of three-mode coexistence, where all three modes are n = 1.We have two conjectures on the multiple-mode coexisting feature.The first conjecture is that multiple EAE eigenstates with the same n number exist at the same radial location of EAE gap, which are discussed in section 3.2.The reason has been analysed theoretically in [45], multiple AE eigenstates with the same m and n numbers but in different radial quantum l numbers (l is an interger number) can exist in a given Alfvén continuum gap, of which mode structure and frequency rely on the specific equilibrium.In diverse experiment scenarios, the safety factor and the electron density profiles have rich characteristics, which give rise to the complexity of continuum structure and diversity of eigenmode state, and make the situations in [45] possible.The other conjecture is the 'pitchfork' splitting of the Alfvén instability due to the nonlinear wave-particle interaction, which has been well-established in theory and confirmed in experiments [46][47][48][49].The specific mechanism on this feature will be explored in the future.
The modes are regular bursts with ∼10 ms duration, rather than steady frequencies.Chen et al firstly applied the preypredator model and successfully explained the repetitive bursting features of fishbone instability in [50].In addition, Berk and Breizman gave various scenarios for the nonlinear evolution of high-energy particles in 1992 [51].They pointed out that depending on the relationships between the source, the background damping, and the classical transport rate, either a steady state or pulsations arise.The process is as follows: the distribution function of high-energy particles in the resonant region was flatten as the wave amplitude grows, and then the resonant particle drive was destroyed.With the background dissipation present, the wave will damp at the rate of dissipation γ d .The classical transport mechanism attempts to reconstitute the unstable distribution function simultaneously at the rate of reconstruction ν eff .If the dissipation rate of the wave is much greater than the reconstruction rate of particles, i.e. γ d ≫ ν eff , then the pulsation scenario arise, and on the contrary, the wave behaves as a steady-state scenario.Relevant calculations and model validation will be carried out in the future.
During a burst, the mode frequency usually evolves, dominated by downward chirping.While some upward chirping, hooked chirping or constant frequency are also observed.In general, the frequency changes of the downward chirping events are much larger than that of the upward ones, as shown in figure 2(b).For the downward chirping modes, the frequency changes 10 ∼ 60 kHz (∆f < 10%) within ∼5 ms, so the chirping rate is about 2 ∼ 12 MHz s −1 .Rapid frequency chirping indicates strong nonlinear interaction between wave and EEs.According to the Berk-Breizman model [51][52][53], frequency chirping is associated with the formation of holes and clumps in the phase space.When collisions are weak (at low electron density), these phase-space structures persist and frequency chirping is possible.The wave frequency locks onto the linear resonance frequency of the resonant electrons, and the dissipation of bulk plasma forces the wave-trapped resonant electrons to move in phase space.

The impact of sawteeth on EAEs
Sawtooth events can happen in many discharges combined with the EAEs, since the safety factor on the magnetic axis is less than one.Figure 9 illustrates the impact of sawtooth events on the EAEs. Figure 9(a) shows the ECE signal, which characterizes the evolution of the electron temperature inside the q = 1 surface [40].Within one sawtooth cycle, there can be 1 ∼ 3-groups of EAEs growing up successively.Figures 9(b)-(d) show the situation where there are twogroups of EAEs in one sawtooth cycle, which are named as group-I and group-II EAEs respectively.The toroidal mode number of group-I EAEs is n = 2, while that of group-II EAEs is n = 1, as illustrated in figure 9(d).As the temperature and safety factor profiles in plasma core recover gradually after the sawtooth collapse, the group-I EAEs grow up firstly and saturate, then decay and disappear spontaneously.The rate of decay is slow as the blue arrows in figure 9(b).On the other hand, the group-II EAEs grow up soon afterwards, but the saturation amplitude of group-II EAEs is much larger than that of the group-I.The group-II EAEs suffered the sawtooth collapse as they grew up, resulting in the sudden disappearance of the modes as indicated by the red arrows in figure 9(b).This could be interpreted as the loss of EEs in the plasma core after the sawtooth crash, or the disappearance of the q = 1 surface.

Destabilization mechanism
Alfvén waves are driven unstable by the free energy in energetic particle distribution function.For finite energy transfer between particles and AEs, the proper wave-particle resonance condition plays a crucial role.Given the fact that the energetic ions are absent in ohmic discharges of current work, these EAEs propagating in the ion diamagnetic drift direction observed in experiments could be excited by EEs accelerated by the strong toroidal electric field.The EAE electromagnetic field perturbation uses the ansats of A ∝ exp (−iωt − imθ + inζ), and the generalized resonance condition for both trapped and passing EEs can be expressed as: where ω ϕ = ∆ζ/τ b and ω θ = 2π /τ b are the toroidal and poloidal frequencies, is the poloidal transit time, and p is an integer.
We apply the test particle module in MAS code [54] to simulate 8000 EE orbits that uniformly distributed on the (P ζ , λ) plane, where P ζ = gρ || − ψ is the canonical angular momentum and λ = µB a /E is the pitch angle.Then the poloidal ω θ and toroidal ω ϕ frequencies can be measured in phase space as shown in figure 10.We use E = 200 keV as typical EE energy for resonance analysis in this section, based on the fact that the populated EE energy range 100 ∼ 300 keV under similar experimental condition on EAST [19].According to the experimental observation that EAE propagates along the ion diamagnetic (co-current) direction (IDD) that corresponds to positive frequency in MAS code convention, we mainly focus on co-passing EEs in our analysis characterized by positive toroidal and poloidal frequencies, which is able to excite IDD-EAE when the resonance condition is satisfied.For convenience, all frequencies are normalized by on-axis Alfvén frequency ω A0 = V A0 /R 0 using deuterium in figures 10-12 in this section.The normalized EAE frequency used in following resonance calculation is ω EAE /ω A0 = 1.22, which is the average of n = 1 to n = 4 even mode in table 1.
From figures 10(a1)-(c1), it is seen that ω ϕ ≫ ω A0 and ω θ ≫ ω A0 for co-passing EEs, and ω ϕ ≪ ω A0 and ω θ ≫ ω A0 for trapped EEs in most area of phase space.Theoretically, both ω ϕ → 0 and ω θ → 0 at the trapped-passing boundary, and the nearby ω ϕ and ω θ are closer to EAE frequency compared to the well-circulating case.However ω ϕ ∼ ω A0 and ω θ ∼ ω A0 do not exist in our calculations since they are in the extremely limited phase space region around the trappedpassing boundary.The region is so limited that it cannot be recognized in figure 10, even though we have used ultra-high phase space resolution, thus it can be ignored in this case.The orbit-averaged safety factor ⟨q⟩ = ¸qdt/ ¸dt is shown in figure 10(d1), which reflects the radial location information of safety factor in phase space.The corresponding high resolution diagrams near the trapped-passing boundary in the core region are shown in figures 10(a2)-(d2).
To illuminate the underlying resonance condition for EE excitation of EAE, we apply equation (1) to calculate the resonance lines indicated by p value (i.e.p = (nω ϕ − ω) /ω θ ) as shown in figure 11, where the even EAE frequency ω and toroidal number n in table 1 are used as examples.It is seen that for EAEs consisting of m and m + 2 harmonics, the primary p = m + 1 resonances of co-passing EEs dominate at the q = 1 surface for each n number, where the core-localized EAEs are located in.The p = m + 1 resonance can occur between EAE and passing EEs with ω θ ≫ ω A0 and ω ϕ ≫ ω A0 in a wide pitch angle domain, of which velocity is much larger than v ∥ = V A /2 for EAE resonance given by early study [1].It should be pointed out that p = 0 resonance line does not exist in the calculations of figure 11 because numerical resolution cannot distinguish the extremely narrow region along the trapped-passing boundary with ω ϕ ∼ ω A0 in figure 10.
The resonance between well-circulating EE and EAE is reminiscent of the runaway electron resonance with MHD modes [55], namely, ω ≪ |nω ϕ | and ω ≪ |pω θ | so that nω ϕ − pω θ ≈ 0. As shown in figure 12 We also notice that the resonance between barelycirculating EE and TAE has been identified as the dominant mechanism of TAE driven by EEs in a recent work [26], for comparison, both barely-and well-circulating EE are resonant with EAE that attribute to m and m + 2 poloidal harmonic coupling at q = (m + 1) /n.In addition, according to figure 11, the resonance lines for the well-circulating EEs can extend in a wide range of λ pitch angle, which indicates that abundant resonant EEs are responsible for EAE excitation, in consistent with early MEGA simulations as shown by figure 12 of [25].Furthermore, the magnetic shear in the core region can affect the EE-EAE resonance strength, since flat q profile is favourable for more EEs to interact with the mode in a wider radial region.
Regarding to the barely-trapped EE resonance with IDD-EAE, the positive precession frequency of E = 200 keV barely-trapped EEs in figure 10(b2) is much smaller than the EAE frequency.Therefore, to meet the resonance condition with the high frequency EAE, barely-trapped EEs require much higher electron energy threshold up to MeV, in order to increase the precession frequency.Although such barelytrapped EEs may exist in experiments, we believe that its contribution on driving the core-localized EAEs is much smaller than that of circulating EEs, since the trapped particle population is small in the core region.
In a short summary, passing EEs with ω ϕ ≫ ω A0 and ω θ ≫ ω A0 can resonate with the core-localized EAE around q = 1 rational surface, which is the dominant destabilizing mechanism according to the phase space resonance analysis using realistic EAST discharge geometry.It is worth mentioning that we also carried out resonance condition analysis using MEGA on a realistic EAST equilibrium of #112786 @t = 5.95 s, by initialising particles on (P ζ , λ) diagram, and following the time evolution of their toroidal angle.And the results are in good agreement with the MAS simulation described above.In the future, the power transfer between EEs and EAEs due to the wave-particle interaction will be computed with MEGA to verify above conclusions.

Summary
In this work, the characteristics of a series of high frequency core-localized EAEs driven by EEs in EAST ohmic discharges are reported.The EAEs usually occur under conditions with low toroidal field and low electron density.They are located around q = 1 rational surface, i.e. ρ tor = 0.2 ∼ 0.35, measured by the DBS diagnostic.It should also be pointed out that the EAST discharges in this work are characterized with plasma density being much lower than ITER and reactor relevant plasmas, and in the latter case the EE-driven EAEs may occur and have an impact on stability at the plasma periphery.The EAEs show obvious bursting and chirping behaviours.From a scientific perspective, the observed nonlinear phenomena of EAEs may contribute to the understanding of the fascinating nonlinear dynamics of Alfvén instabilities, which are still under active investigation [56].In addition, it is believed that chirping modes may affect the fast particle transport in different ways than steady frequency AEs, e.g.convective versus diffusive transport [57].Besides, the reduction in neutron emission correlated with bursts has been reported in several devices [1].Therefore, the anomalous transport caused by the bursting and chirping modes may be important in the future fusion reactors.Preliminary exploration of the destabilization mechanism has been conducted, suggesting these EAEs propagating in the ion diamagnetic drift direction could be driven by circulating EEs accelerated by the strong toroidal electric field in low-density ohmic discharges.

Figure 1 .
Figure 1.The waveforms of main plasma parameters in a typical discharge with bursting EAEs.(a) Electron density (red curve), voltage of the toroidal electric field (blue curve); (b) the proportion of energetic electrons in plasma (HXR (a.u.) /ne ( 10 19 m −3 ) 2 , red curve), the ratio of the toroidal electric field to the critical field (blue curve); (c) the signal of a magnetic probe; (d) the frequency spectrum of the magnetic probe signal.

Figure 2 .
Figure 2. Enlargement of the frequency spectrum of the magnetic fluctuations in #113416.

Figure 3 .
Figure 3. (a) Spectrum of the magnetic probe signal of #112835.(b) The toroidal mode number n calculated from the EMPA.

Figure 4 .
Figure 4.The dependence of the observed mode frequency on the square root of electron density and the toroidal field.

Figure 5 .
Figure 5.The relationship between the magnetic perturbation intensity and the toroidal field and the plasma current.

Figure 6 .
Figure 6.The comparison of (a) the magnetic fluctuation spectrum with (b)-(f ) the spectrums of DBS signals at five different radial positions on the low field side for shot #112786.(g) The radial distribution of the cross-power spectral density of the DBS signals with the magnetic probe signal, representing the strength of the correlation between them.

Figure 7 .
Figure 7. (a) The electron density profile for three time slices of #112786, with the hollow circles represent the radial position where EAEs can be detected by DBS, and the crosses represent the DBS channels without EAEs.(b) The time evolution of electron density.(c) The safety factor profile for three time slices; The spatial extent (the horizontal solid line) where the q = 1 surface is located, estimated based on the 32-channel ECE measurement.(d) The time evolution of the plasma current.

Figure 8 .
Figure 8. MHD simulations of EAST discharge #112786@t = 5.95 s with toroidal mode numbers n = 1, n = 2, n = 3 and n = 4 from the left to the right column.The first row shows the Alfvén continua consisting of different m-harmonics, where the black solid and dotted lines indicate the locations of odd and even EAEs calculated by MAS, respectively.The middle and the bottom rows show the m-harmonic radial profiles of corresponding odd and even EAE mode structures, respectively.The red circles in the middle and bottom rows label the q = (m + 1) /n = 1 surface location where the m and m + 2 modes couple to form EAEs with different n numbers.

Figure 9 .
Figure 9.Time evolutions of (a) the ECE signal which characterizes the sawtooth events.(b) The bursting magnetic perturbation.(c) The frequency spectrum of EAEs.(d) The toroidal mode number of EAEs.

Figure 10 .
Figure 10.The EE characteristic frequency at the E = 200 keV plane of (P ζ , λ) phase space.(a1) The poloidal frequency ω θ for trapped and co-passing EEs.(b1) and (c1) Are the toroidal frequency ω ϕ for trapped EE and co-passing EE, respectively.(d1) The orbit-averaged ⟨q⟩ = ¸qdt/ ¸dt for trapped and co-passing EEs.(a2)-(d2) Are the corresponding enlargements of phase space region indicated by the green box near the trapped-passing boundary in the core.The positive frequency denotes the IDD.

Figure 11 .
Figure 11.The resonance slice at the E = 200 keV plane of (P ζ , λ) phase space.(a1)-(d1) show the resonance lines for n = 1, n = 2, n = 3 and n = 4 even EAEs in table 1.The black dashed line represents the resonance condition being satisffied with integer p value, and the magenta and green dotted lines in the upper and lower rows represents the ⟨q⟩ = 1 location.(a2)-(d2) Are the corresponding enlargements of phase space region indicated by the green box near the trapped-passing boundary in the core.
, the large poloidal and toroidal frequencies of well-circulating EE cancel with each other mostly and the frequency for non-zero energy exchange Ω circ n = nω ϕ − (m + 1) ω θ falls in the EAE frequency range.By carefully choosing P ζ value according to the resonance lines nearest to ⟨q⟩ = 1 in figure 11, the Ω circ n curves can intersect with EAE frequencies for all n = 1 ∼ 4 modes in figures 12(a2)-(d2), indicating that the resonance condition is satisfied.On the other hand, the EAE resonance with wellcirculating EEs can be verified by solving equation (1) analytically, the parameters include p

Table 1 .
Odd and even EAE frequencies with different n numbers calculated by MAS.