Observation and Simulation of $n=1$ Reversed Shear Alfv\'{e}n Eigenmode on the HL-2A Tokamak

P. W. Shi$^{1}$, Y. R. Yang$^{2}$, W. Chen$^{1}$, Z. B. Shi$^{1\hyperlink{s}{*}}$, Z. C. Yang$^{1}$, L. M. Yu$^{1}$, T. B. Wang$^{1}$,\\ X. X. He$^{1}$, X. Q. Ji$^{1}$, W. L. Zhong$^{1}$, M. Xu$^{1}$, and X. R. Duan$^{1}$

doi:10.1088/0256-307X/39/10/105201

⇦Chinese Physics Letters, 2022, Vol. 39, No. 10, Article code 105201⇨ Observation and Simulation of $n=1$ Reversed Shear Alfvén Eigenmode on the HL-2A Tokamak P. W. Shi1, Y. R. Yang2, W. Chen1, Z. B. Shi1*, Z. C. Yang1, L. M. Yu1, T. B. Wang1, X. X. He1, X. Q. Ji1, W. L. Zhong1, M. Xu1, and X. R. Duan1 Affiliations 1Southwestern Institute of Physics, Chengdu 610041, China 2State Key Laboratory of Intense Pulsed Radiation Simulation and Effect, Northwest Institute of Nuclear Technology, Xi'an 710024, China Received 6 July 2022; accepted manuscript online 1 September 2022; published online 20 September 2022 ^*Corresponding author. Email: shizb@swip.ac.cn Citation Text: Shi P W, Yang Y R, Chen W et al. 2022 Chin. Phys. Lett. 39 105201 Abstract A branch of high-frequency Alfvénic modes is observed on the HL-2A tokamak. The electromagnetic mode can be driven unstably in the plasma with an off-axis neutral beam heating. Its mode frequency keeps almost unchanged or presents a slow-sweeping behavior, depending on the detail current evolution. The poloidal and toroidal mode numbers are $m/n=1/1$. The mode has a quite short duration ($\leq$20 ms) and usually appears 5–10 ms after the neutral beam being injected into the plasma. Hybrid simulations based on M3D-K have also been carried out. The result suggests that co-passing energetic particles are responsible for the mode excitation. The simulated mode structures are localized nearby location of minimum safety factor ($q_{\rm min}$) and agree with the structures obtained through tomography of soft x-ray arrays. Further, the modes are localized in the continuum gap and their frequencies increase with variation of $q_{\rm min}$ in a wide range. Last but not least, the characteristic of unchanged frequency on experiment is also reproduced by the nonlinear simulation with a fixed safety factor. All those evidences indicate that the $n=1$ high-frequency mode may belong to a reversed shear Alfvén eigenmode.

DOI:10.1088/0256-307X/39/10/105201 © 2022 Chinese Physics Society Article Text Interactions between shear Alfvén wave (SAW) and energetic particles (EPs) have become a major concern in plasma physics.[1,2] Because the wave-particle interaction usually gives rise to energetic particle driven instability, such as toroidal Alfvén eigenmode (TAE),[3-6] reversed shear Alfvén eigenmode (RSAE),[7-11] beta induce Alfvén eigenmode (BAE),[12-14] and energetic particle mode (EPM).[15,16] Among those Alfvénic modes, TAEs are most famous due to their spacial mode structure, which may enhance plasma transport. RSAE is one of typical core localized electromagnetic instabilities in reversed magnetic shear plasma, where improved confinement is expected.[17] As such, the RSAEs characterized by frequency sweeping slowly due to slight change of minimum safety factors have also captured a great deal of attention. The RSAEs play a positive or negative role on the transport and confinement of thermal or energetic particles. It was firstly reported on ASDEX Upgrade that the TAE causes fast ion losses with a convective mechanism while overlap of TAE and RSAE results in a diffusive transport.[18] Thereafter, more and more evidences reveal similar effects of EPs driven instability on energetic particle transports.[19-21] Actually, the SAW fluctuation can also be regarded as energy channels to transfer the $\alpha$ particles energy to the thermonuclear plasma,[22] as kinetic spectroscopy to determine the minimum safety factor or as a monitor of ratio of deuterium and hydrogen.[23] Meanwhile, RSAEs are weakly damped and can easily be excited by EPs. It is ugly to take effective measurements to mitigate and suppress the macro instability. However, before the technical engineering has been performed, systematic and scientific acknowledges for those active modes are needed via actual experiments, analytic theories as well as numerical simulations. With development of external auxiliary heating technology and multiple advanced diagnostics, series of SAW fluctuations have been observed and confirmed on the HL-2A tokamak.[24] Recently, high-frequency Alfvénic modes with poloidal/toroidal mode numbers of $m/n=1/1$ are observed on the HL-2A tokamak. The electromagnetic modes are driven unstably at the earlier stage of neutral beam injection (NBI), and appear alone without other concomitant sidebands in a same frequency region. It is proved to be an RSAE, which shows a different feature to that of traditional RSAEs. Though no undesirable effects of those modes have been found now, much attention should also be paid to the special Alfvénic modes. Since many plasma events may be caused by such an $n=1$ mode. It was claimed that larger-amplitude $n=1$ modes result in stronger toroidal localization of the ballooning modes and lead to high $\beta$ disruption on TFTR.[25] It was found that the $n=1$ EPMs contribute to the abrupt large-amplitude event (ALE), which causes severe losses of both neutron and energetic ion on JT-60U.[26] To better understand the $n=1$ modes, scientific researches have been carried out through combination of experiment and simulation. In this Letter, we first describe the basic experimental characteristics of $n=1$ Alfvénic modes. Then, hybrid simulations by the M3D-K code are provided. A summary of relevant issues in this work is presented finally. Experimental Observations. The experiments were performed in divertor configuration on the HL-2A tokamak with a predominantly circular cross-section and major/minor radius of $R/a = 1.65$ m/0.4 m. Figure 1 shows the basic parameters in the duration of 500–600 ms at two typical discharges. The plasma current is about $I_{\rm p}\simeq160$ kA, toroidal magnetic field is $B_{\rm t}=1.37$ T, and the line-averaged electron density ranges $n_{\rm e}$ = (1.65–2.0) $\times 10^{19}$ m$^{-3}$. A 0.8–1.0 MW neutral beam is tangentially injected into plasma with a co-current going direction and a fixed angle of $58^{\circ}$ at $t=500$ ms. The electron temperature measured by electron cyclotron emission (ECE) radiometer keeps lower than 0.8 keV during the 29743 discharge and it is slightly higher in the 27032 discharge. The so-called critical energies are determined by electron temperatures, i.e., $E_{\rm crit}=(0.75\sqrt{\pi})^{2/3}(m_{\rm i}/m_{\rm e})^{1/3}\frac{m_{\rm e}}{m_{\rm i}}T_{\rm e}$, and evaluated as 18 keV and 24 keV during the early phase of neutral beam injection, which are lower than the accelerated particle energy of 40 keV. In those cases, the energy loss of beam ions is mainly due to collisions with plasma electrons. Here, $m_{\rm i}$ and $m_{\rm e}$ are the ion and electron masses, both thermal ions and beam ions are deuterium particles. Though electrons heating is dominant, beam injection can also give birth to a number of fast ions. The slowing-down time of beam ions[27] on thermal ions can be described by $\tau_{\rm s}=\frac{m_{\rm i}}{m_{\rm i}+m_{\rm b}}\frac{3\sqrt{2\pi}T_{\rm e}^{3/2}}{\sqrt{m_{\rm e}}m_{\rm b}}\frac{2\pi\varepsilon_0^2m_{\rm b}^2}{n_{\rm e}e^4 \ln \varLambda}$, where $m_{\rm b}$ is beam ion mass, $\ln\varLambda=20$, and $\varepsilon_0$ is the permittivity of vacuum. The slowing-down time is about 10.5–12.8 ms and 20–25 ms for the 29743 and 27032 discharges, respectively. The ion temperature obtained from charge exchange recombination spectroscopy (CXRS) increases gradually during the temporal fragment of 505–525 ms, where there are Alfvénic modes with frequency range 70–90 kHz, as shown in Fig. 2. The modes appear within 5–10 ms after the neutral beams are injected into plasma, and can be measured by multiple diagnostics, such as Mirnov coil probe at edge and microwave interferometer at core. Note that only one primary mode ($n=1$) has been measured by magnetic probe during the 29743 discharge, its second and third harmonics are also detected by microwave diagnostic. However, there are no other primary Alfvénic modes with adjacent mode numbers ($n=2, 3,\ldots $) can be found during the experiments. The mode frequency (86–89 kHz) is far from the TAE gap center, which usually has a frequency as $f_{\rm TAEgap}=0.13\times10^{13}B_{t}/(qR\sqrt{\langle n_{\rm e}\rangle})=180$–200 kHz,[28] and it is higher than the BAE continuum accumulation point (CAP, which is an extreme point in Alfvénic continuum gap) with frequency $f_{\rm BAE\text{-}CAP}=\frac{1}{2\pi R}\sqrt{7/4+T_{\rm e}/T_{\rm i}}\sqrt{2T_{\rm i}/m_{\rm i}}$, indicating that the electromagnetic mode is neither a TAE nor a BAE mode. Further, the mode frequency is higher than the ion diamagnetic frequency defined by $f_{\rm *ip}=(T_{\rm i}/eB)k_{\theta}(\nabla \ln n_{\rm i})(1+\eta_{\rm i})/2\pi=40$–60 kHz, where $\eta_{\rm i}=\nabla \ln T_{\rm i}/\nabla \ln n_{\rm i}$ (the ion density $n_{\rm i}\simeq n_{\rm e}$). Thus, kinetic ballooning mode[29,30] with typical frequency range $f_{\rm *ip}/2 < f < f_{\rm *ip}$ is also excluded from the candidate of mode identify. The electromagnetic modes can also be driven unstably in the plasma with ramp-up currents, as shown in Fig. 2(d) for a typical example. With the plasma current increasing from 140 kA to 145 kA, the mode frequency changes from 75 kHz to 90 kHz. This phenomenon displays a similarity to the conventional RSAEs with mode numbers of $n=2, 3, 4 $ and frequencies changing in $\Delta f=35$–50 kHz at time scale of 20–30 ms on the HL-2A tokamak.[31,32] Thus, a close relationship between the mode frequency and safety factor is expected.

Fig. 1. Temporal evolutions of (a) plasma current and toroidal magnetic field, (b) line-averaged electron density detected using a laser interferometer, (c) power of tangentially injected neutral beam, electron (black) and ion (red) temperatures. Subscripts 1 and 2 denote different plasma discharges, i.e., 29743 and 27032 on the HL-2A tokamak.

Fig. 2. (a) Magnetic fluctuation induced by Alfvénic mode (AM, 86–89 kHz) and (b) spectrogram of magnetic signal during the 29743 discharge. The black curve shows the total of BAE continuum accumulation point frequency ($f_{\rm BAE\text{-}CAP}=\frac{1}{2\pi R}\sqrt{7/4+T_{\rm e}/T_{\rm i}}\sqrt{2T_{\rm i}/m_{\rm i}}$) and plasma ration frequency ($f_{\rm rot}$). (c) Magnetic fluctuation induced by tearing mode (TM, 2–4 kHz) and (d) spectrogram of magnetic signal during the 27032 discharge. To make Alfvénic mode more clearer, the $y$ axis of (b) and (d) is limited in the range of 20–140 kHz.

Before the 2–4 kHz tearing mode is completely stable due to the possible effects of energetic particles or current drive induced by neutral beam injection, several Alfvénic sidebands can also be observed. The phase dynamic on magnetic signal induced by Alfvénic modes can be obtained from a numerical filter. It is found that the phase change at rate of $4\pi$, $2\pi$, and $0$ when numerical filters with filtering frequencies of 77–79 kHz, 74–76 kHz, and 71–73 kHz are used to deal with magnetic signal, i.e., the corresponding toroidal mode numbers of AMs are $n=2$, $n=1$, and $n=0$. Interestingly, the frequency and mode number differences of two adjacent AMs are comparable to frequency and toroidal mode number of tearing mode. Further squared bicoherence analysis suggests that there is a nonlinear mode coupling process between AMs and tearing mode, as shown in Figs. 3(a)–3(b). It means that those Alfvénic sidebands are driven by nonlinear wave-wave interaction but not by wave-particle resonance. Usually, nonlinear mode coupling takes place among multiple modes with spacial overlapped structures. But a magnetic islands in the toroidal plasma will provide a much more complex channel, which enables nonlinear interaction between the local modes locating at $q=1$ and $q=2$ surfaces. Actually, such a process is quite similar to nonlinear mode coupling between BAEs/TAEs and tearing modes,[33,34] and we do not discuss in detail here.

Fig. 3. (a) The squared bicoherence and (b) summed squared bicoherence of Mirnov coil signal during 500–520 ms in the 27032 discharge. The phase evolutions of toroidal Mirnov coil arrays obtained from a numerical filter with a filtering frequency region of (c) 77–79 kHz, (d) 74–76 kHz, and (e) 71–73 kHz.

Fig. 4. Mode structures of Alfvénic mode obtained from tomography of soft x-ray arrays.

Soft x-ray arrays with sampling rate of 1 MHz are available during the experimental campaign and the two-dimensional system is utilized to measure poloidal mode number via tomography. There are a cold spot and a hot spot in the core region of 1.55 m$\,\leq R \leq1.80$ m, i.e., a typical $m=1$ mode structure, which can be seen in Fig. 4. The $m/n=1/1$ mode propagates in ion diamagnetic drift direction. Figure 5 presents a statical result about the mode frequencies of TAE, $n=1$ Alfvénic mode and BAE, and line-averaged electron densities where the three energetic particle driven instabilities are excited. Usually, the BAE frequency increases with increasing ion temperature[35] while the TAE frequency shows a reversed relation with the electron density.[36] Great changes have been found in both BAE and TAE frequencies, whereas the mode frequencies of $n=1$ Alfvénic modes vary slightly during multiple discharge environment. Here, the great error bars in TAE frequencies mainly come from the change of electron density or frequency chirping behavior related to the phase-space dynamics of energetic particles. Moreover, the $n=1$ Alfvénic mode can only be observed in the discharges with a moderate line-averaged electron density of $n_{\rm e}$ = (1.2–2.0) $\times 10^{19}$ m$^{-3}$, unlike the BAE or TAE appearing in low ($n_{\rm e}\leq1.2\times10^{19}$ m$^{-3}$) or high ($n_{\rm e}\geq2.0\times10^{19}$ m$^{-3}$) density plasma. Figure 6 shows the basic profiles during (520 ms) and after (528 ms) excitation of $n=1$ mode in the 29743 discharge. Both electron temperatures and densities remain almost unchanged at the two given moments while the ion temperatures and plasma rotation increase significantly. Since the low-$n$ modes always have a radial profile extending over a large fraction of the plasma cross-section. The great changes in plasma rotation and its shear are expected to have an impact over the mode damping.[37] It may be one of factors that contributes to short duration of $n=1$ mode. However, more attention should be paid to the radiation profiles. It is found that the peaks of radiation profiles move from $\rho=0$ to $\rho=0.3$ during the time slice of 504–528 ms. The change indicates that there is an outward transport in carbon impurity.[38] The underlying mechanism of impurity transport is beyond the scope here. It is well known that core localization of impurity usually contributes to the formation of reversed shear plasma.[39] In turn, the outward transport may be unfavorable for maintain of reversed shear, thus lead to the stabilization of Alfvénic modes driven only in reversed shear plasma. The energy depositions of NBI on electrons and ions can be given by the combined calculation between ONETWO and NUBEAM codes based on parameters given in Figs. 6(a)–6(e). The calculated results are arranged at the last subgraph in Fig. 6 and suggest that electron heating becomes dominant, which agrees with the previous evaluation based on critical energy. Further, peaks of deposition profiles are not at the magnetic axis and reveal an off-axis heating process during neutral beam injection. It may cause a non-monotonic safety factor and provide an appropriate environment for excitation of RASEs.

Fig. 5. Statistical results about the mode frequencies of TAEs, $n=1$ Alfvénic modes and BAEs, and electron densities, where the three energetic particle driven instabilities are excited. Note that the data is obtained from multiple discharges.

Fig. 6. Profiles of (a) electron temperatures from ECE, (b) electron densities from laser interferometer, (c) ion temperatures and (d) toroidal rotation frequencies from CXRS, (e) plasma radiations from bolometer, (f) energy depositions of NBI on electrons (qbeame) and ions (qbeami) from ONETWO and NUBEAM calculation. Note that the blue and black curves are corresponding to basic parameters at the moments of 520 ms and 528 ms. There are two more profiles given at 504 ms (red) and 512 ms (green) plotted in (b) and (e).

Hybrid Simulations. To explain the $n=1$ Alfvénic modes, hybrid simulations based on the experimental data have been carried out via the M3D-K code. Figure 7 presents the input equilibrium parameters in the 29743 discharge. The pressure profile ($\times\mu_0/\epsilon B_0^2$) is given by EFIT code and safety factor ($q$) profile with a weak reversed shear is obtained from the current profile fitting code.[40] The dotted curve comes from the calculated $q$-profile with a total up-shift of $\Delta q=0.24$. It is worth noting that the temporal $q_{\rm min}$ evolution is unavailable due to the great uncertainty of reconstruction and absence of motional Stark effect (MSE) spectroscopic measurements. Since the energetic particle diagnostics are absent on the HL-2A tokamak, the fast ion profiles can not be measured from experiment. Thus, an anisotropic slowing-down distribution is used for fast ions and given as \begin{align} f=\,&cH(v_0-v)\frac{{\exp}({-\langle\varPsi\rangle}/{\Delta\varPsi})}{v^3+v_{\rm c}^3}\notag\\ &\cdot\frac{{\exp}(-(\varLambda-\varLambda_0)^2/{\Delta\varLambda}^2)}{\Delta\varLambda\Big({\rm erf}\Big(\frac{1-\varLambda_0}{\Delta\varLambda}\Big) +{\rm erf}\Big(\frac{\varLambda_0}{\Delta\varLambda}\Big)\Big)},\tag {1} \end{align} where $c$ is a normalization factor, $H$ is the step function, $v_0$ is the fast ions cutoff velocity, and $v_{\rm c} = 0.5v_0$ is the critical velocity. $\varLambda \equiv \mu B_0/E$ is the pitch angle, $\varLambda_0 = 0.6$, $({\Delta\varLambda})^2 = ({\Delta\varLambda_0})^2+0.33(1-\varLambda_0){\ln}[(v^3+v_{\rm c}^3)/(v^3+v^3v_{\rm c}^3/v_0^3)]$, $\Delta\varLambda_0 = 0.3$, ${\Delta\varPsi} = 0.3$, $\langle\varPsi\rangle$ is $\varPsi$ average over the particle orbit.

Fig. 7. The input equilibrium parameters for M3D-K hybrid simulations: pressure profile ($\mu_0/\epsilon B_0^2$) given by EFIT code and safety factor profile obtained from the current profile fitting code. The dotted curve comes from the calculated $q$-profile with a total up-shift of $\Delta q=0.24$.

Fig. 8. Real frequency and growth rate of $n=1$ Alfvénic mode as functions of $q_{\rm min}$. Here, the red and black curves are calculated with different fast ion injection velocities of $v_0/v_{\scriptscriptstyle{\rm A}}=0.6$ (experimental value) and $v_0/v_{\scriptscriptstyle{\rm A}}=1.2$, respectively. All the figures given in the following are calculated with injection velocities of $v_0/v_{\scriptscriptstyle{\rm A}}=0.6$.

Referring to the experimental data, we set the ratio of fast ions beta to total plasma beta ($\beta_{\rm frac}$) and normalized gyroradius ($\rho_{\rm h}/a$) as 0.5 and 0.05, respectively. Figure 8 shows the growth rate and mode frequency of $n=1$ Alfvénic mode as a function of $q_{\rm min}$. Here, the red and black curves are calculated with different fast ion injection velocities of $v_0/v_{\scriptscriptstyle{\rm A}}=0.6$ (experimental value) and $v_0/v_{\scriptscriptstyle{\rm A}}=1.2$, respectively. Different tendencies have been found in growth rates and mode frequencies when the minimum safety factor changes with a small scale. The normalized frequency ranges in 20–120 kHz with $v_0/v_{\scriptscriptstyle{\rm A}}=0.6$ while it is 20–82 kHz with $v_0/v_{\scriptscriptstyle{\rm A}}=1.2$. The mode frequencies in the two cases of different injection velocities are comparable, with $q_{\rm min}=1.0$–1.28, though the growth rates are higher and decline much more quickly in the lower injection velocity case. It should be pointed out that the modes are stable when $q_{\rm min}$ is below the unity. This may be explained by the fact that damping rate of $n=1$ Alfvén eigenmode for $q < 1$ reaches a value as four times as that for $q>1.1$.[41] Further, the frequency evolution is different from that of typical up-sweeping RSAE,[7] whose frequency chirps up until its value is approaching the TAE frequency when $q_{\rm min}$ decreases from the fraction of $m/n$ to $(m-1/2)/n$. It looks more like the so-call down-sweeping RSAE with $m < nq_{\rm min}$.[42] The mode usually sweeps down from the TAE frequency when $q_{\rm min}$ decreases from the fraction of $(m+1/2)/n$ to $m/n$.[31,32] In the case of $q_{\rm min}$ being fully out of conventional up-sweeping RSAE region, the RSAE can also exhibit an up-sweeping feature if $q_{\rm min}$ increases from $m/n$ to $(m+1/2)/n$. Note that $q_{\rm min}$ will decrease gradually in time during a current ramp-up phase. However, the $n=1$ high-frequency mode appears at the early phase of tangentially injected neutral beam, which usually provides a current drive effect and improves the safety factor. Thus, the $q_{\rm min}$ dynamics in our experiment is mainly determined by both current with a ramp-up phase and current drive effect. It is possible that the $q_{\rm min}$ increases from $m/n$ to $(m+1/2)/n$ if current drive effect becomes dominant in the HL-2A plasma.

Fig. 9. Two-dimensional structures of $n = 1$ mode with different minimum safety factors: (a) $q_{\rm min} = 1.0$, (b) $q_{\rm min} = 1.08$, (c) $q_{\rm min} = 1.16$, (d) $q_{\rm min} = 1.24$. The white curves indicate the locations of circles with radius of $\rho=0.35$.

The mode structures in the cases of $q_{\rm min} = 1.0, 1.08, 1.16$, and 1.24 are shown in Fig. 9. Note that the white circles indicate the locations of circles with radius of $\rho=0.35$. The $m=1$ poloidal harmonics are found to be dominant in all the cases and the mode structures are localized inside the area $Z/a < \rho$, which agrees well with the core localized mode structure measured by soft x-ray arrays. Figure 10 presents the $n=1$ Alfvén continuum and radial mode structure calculated by Alfvén mode code[43] with the four different minimum safety factors. Here, a finite poloidal beta of $\beta_{\rm i}=0.75\%$ is taken into account, the red and black horizontal lines indicate the mode frequencies given by M3D-K and AMC, respectively. Eigen-solutions can be found at the continuum accumulation point with minimum frequency. Note that CAPs with maximum frequency are unavailable here because $q_{\rm min}$ is larger than one. The radial mode structures are made up of a single $m=1$ poloidal harmonic and highly localized in the core region. Moreover, the corresponding real frequencies increase with upwards shift of CAP due to the growth of $q_{\rm min}$, and matches well with M3D-K simulation. Since the kinetic finite Larmor radii effect, which is important for BAE excitation, has not been added into the AMC. Therefore, the core localized high frequency $n=1$ Alfvénic mode may belong to an RSAE rather than an EPM or a BAE.

Fig. 10. The corresponding Alfvén continuum and radial mode structure calculated by Alfvén mode code with different minimum safety factors: (a) $q_{\rm min} = 1.0$, (b) $q_{\rm min} = 1.08$, (c) $q_{\rm min} = 1.16$, (d) $q_{\rm min} = 1.24$. Subscripts 1 and 2 denote the Alfvén continuum and radial mode structure. The red and black horizontal lines indicate the mode frequencies given by M3D-K and AMC, respectively.

Fig. 11. Histograms of the $p$-values and $\omega_\phi$ for different minimum safety factors: (a) $q_{\rm min} = 1.0$, (b) $q_{\rm min} = 1.08$, (c) $q_{\rm min} = 1.16$, (d) $q_{\rm min} = 1.24$. The black, blue and red curves are the contribution from co-passing, counter-passing and trapped particles, respectively. When $q_{\rm min}>1.16$, there is no contributions from counter-passing and trapped particles. Here $N$ is the corresponding number of fast ion and $N_{\rm total}$ is the total number.

Fig. 12. Nonlinear evolution of mode amplitude $U_{\cos}$ and the corresponding spectrogram for the $n=1$ Alfvénic mode with $q_{\rm min}=1.2$.

General, wave growth is usually driven by free energy in the spatial gradient via wave-particle interaction. The linear resonance condition for shear Alfvén wave and energetic particles can be given as[1] \begin{eqnarray} \omega=n\omega_{\phi}-p\omega_{\theta},\tag {2} \end{eqnarray} where $\omega$ is the mode frequency, $n$ is the toroidal mode number, and $p$ is an arbitrary integer. For passing particles, $\omega_{\phi}$ includes the toroidal procession drift frequency and the toroidal transit frequency, and $\omega_{\theta}$ is the poloidal transit frequency. For trapped particles, $\omega_{\phi}$ and $\omega_{\theta}$ are the toroidal precession drift frequency and the bounce frequency. In the M3D-K code, we have $\omega_{\theta} = 2\pi/{\Delta t}$, $\omega_{\phi} = {\Delta \phi}/{\Delta t}$. Here, $\Delta{t}$ and ${\Delta \phi}$ are the time difference and toroidal angle difference between particle passing through the middle plane twice in a row. Resonance condition can be easily found out by the following steps. We firstly map out the energy perturbation of fast ions in phase space, and then pick out the particles with relatively large energy perturbation to calculate the condition that their characteristic frequencies satisfy. Here, we only show the histograms of the $p$-values and $\omega_\phi$ for the four cases in Fig. 11. The black, blue and red curves are the contributions from co-passing, counter-passing and trapped particles, respectively. In addition, $N$ is the corresponding number of fast ion and $N_{\rm total}$ is the total number. For the case with $q_{\rm min}=1.0$, contributions from co-passing and trapped particles are comparable. The resonance condition for co-passing particles is $\omega=\omega_\phi$ and it is $\omega=\omega_\phi+\omega_\theta$ for trapped particles. Contributions from counter-passing and trapped particles are quite weak and those from co-passing particles play a dominant role during the mode excitation in the case of $q_{\rm min}=1.08$. In the cases with higher $q_{\rm min}$, there are no contributions from counter-passing and trapped particles. The $p$-values are also evaluated as 0. In other words, the resonance conditions remain $\omega=\omega_\phi$ for the higher safety factors though $\omega_\phi$ increases gradually. Simulation results indicate that the co-passing particles are responsible for excitation of the $n=1$ electromagnetic modes. The M3D-K code can also reproduce nonlinear characteristic of the high-frequency modes. Nonlinear evolution of mode amplitude $U_{\cos}$ and the corresponding spectrogram for the $n=1$ Alfvénic mode with $q_{\rm min}=1.2$ are plotted in Fig. 12. Here, $U_{\cos}$ represents ${\cos}(\phi)$ component of the stream function $U$. The mode grows at $800 \tau_{\scriptscriptstyle{\rm A}}$, becomes saturated at $1100 \tau_{\scriptscriptstyle{\rm A}}$, and then its amplitude decreases gradually thereafter. The mode frequency is about 70 kHz and it is comparable to the experimental frequency when the toroidal rotation frequency of 10 kHz is added. More interesting, the mode frequency keeps almost unchanged during the total duration, revealing a similar characteristic to experimental observation in Fig. 2(b). Discussion and Conclusion. The $n=1$ Alfvénic modes with frequencies ranging in 75–90 kHz are observed by multiple diagnostics on the HL-2A tokamak. The modes are electromagnetic and can be driven in the plasma with both fixed and ramp-up currents, where the mode frequencies keep almost unchanged and present a slow-sweeping behavior. Soft x-ray arrays suggest that the fast ions driven modes are localized in the core region and have an $m=1$ mode structure. The modes have a quite short duration ($\leq20$ ms), and there may be a direct relationship between the mode stabilization and outward impurity transport or plasma rotation and its shear. However, due to the absence of MSE and poor temporal resolution of CXRS, a quantitative analysis for the mode damping mechanism is difficult. To explain damping mechanism of the $n=1$ mode, plasma diagnostics, such as CXRS, MSE and collective Thomson scattering (CTS) with high resolution are needed. Both linear and nonlinear hybrid simulations for the $n=1$ Alfvénic modes have also been carried out on M3D-K. It is found that the modes are mainly excited by co-passing particles. The growth rate and mode frequency are closely related to the $q_{\rm min}$. Both the simulated mode structures and nonlinear dynamic agree well with experimental observations. Last but not least, the modes are localized in the continuum gap. Thus, the high-frequency Alfvénic modes may belong to reversed shear Alfvén eigenmodes. However, there are some questions remained unresolvable. Firstly, why do not other primary Alfvénic modes with higher $n$ exist when the $n=1$ modes are unstable. In the experiments without nonlinear wave-wave interaction, only the $n=1$ modes can be observed, but other modes with $n=2,3, \ldots $ seem to be stable. Further detailed comparison of the $n=1$ modes for other up-sweeping or down-sweeping RSAEs with different mode numbers should be needed through either theoretical analysis or numerical simulation. Then, simulation indicates the $n=1$ modes can also be driven by super-Alfvénic particles and range in a wide frequency region, i.e., from the fishbone mode frequency to TAE frequency, as shown in Fig. 8. Such a wide band indicates a cascade and may provide an energy channel in the future burning plasma. Thus, much more attentions should be paid to those modes, especially for their effects on plasma confinements. Acknowledgments. The authors would like to thank the HL-2A team for technical assistances and tokamak operations, and thank professor Z. X. Wang and professor Z. Y. Qiu for valuable suggestions. This work was supported by the National Key R&D Program of China (Grant No. 2019YFE03020000), the National Natural Science Foundation of China (Grant No. 12125502), the Sichuan Science and Technology Program (Grants No. 2020JDJQ0070), and the Natural Science Foundation of Sichuan (Grant No. 2022NSFSC1823). 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ProfilesAlfvén wave cascades in a tokamakDynamics of reversed shear Alfvén eigenmode and energetic particles during current ramp-upInternal Alfvén eigenmode observations on DIII-DElectron cyclotron heating can drastically alter reversed shear Alfvén eigenmode activity in DIII-D through finite pressure effectsObservation of beta-induced Alfvén eigenmodes in the DIII-D tokamak

β

-Induced Alfvén Eigenmodes Destabilized by Energetic Electrons in a Tokamak PlasmaExcitation of the beta-induced Alfvén eigenmodes by a magnetic islandBeam-driven energetic particle modes in advanced tokamak plasmasExperimental evidence of nonlinear avalanche dynamics of energetic particle modesImproved Confinement with Reversed Magnetic Shear in TFTRConvective and Diffusive Energetic Particle Losses Induced by Shear Alfvén Waves in the ASDEX Upgrade TokamakFast ion loss in a ‘sea-of-TAE’Fast-ion energy loss during TAE avalanches in the National Spherical Torus ExperimentAvalanche transport of energetic-ions in magnetic confinement plasmas: nonlinear multiple wave-number simulationUtility of extracting alpha particle energy by wavesMHD spectroscopyEnergetic Particle Physics on the HL-2A Tokamak: A ReviewBallooning instability precursors to high β disruptions on the Tokamak Fusion Test ReactorRole of convective amplification of n = 1 energetic particle modes for N-NB ion dynamics in JT-60UChapter 5: Physics of energetic ionsPerturbative and non-perturbative modes in START and MASTObservation of ballooning modes in high-temperature tokamak plasmasCore-localized Alfvénic modes driven by energetic ions in HL-2A NBI plasmas with weak magnetic shearsDestabilization of reversed shear Alfvén eigenmodes driven by energetic ions during NBI in HL-2A plasmas with q_min ∼ 1Hybrid simulations of reversed shear Alfvén eigenmodes and related nonlinear resonance with fast ions in a tokamak plasmaObservation and theory of nonlinear mode couplings between shear Alfvén wave and magnetic island in tokamak plasmasObservation of off-axis sawtooth oscillations during the presence of nonlinear mode couplings in HL-2A NBI heated plasmasBeta induced Alfvén eigenmode driven by energetic ions on the HL-2A tokamakDestabilization of toroidal Alfvén eigenmode during neutral beam injection heating on HL-2AEffects of toroidal rotation shear on toroidicity-induced Alfvén eigenmodes in the National Spherical Torus ExperimentAccumulation of impurities in advanced scenariosStudy of plasma equilibrium reconstruction on HL-2AExperimental study of the dependence of the damping rate of n = 1 TAEs on the on-axis safety factor and toroidal rotation shearPlasma pressure effect on Alfvén cascade eigenmodesParallel equilibrium current effect on existence of reversed shear Alfvén eigenmodes

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