| Citation: |
Zhihui ZOU, Ping ZHU, Charlson C KIM, Wei DENG, Xianqu WANG, Yawei HOU. Frequency multiplication with toroidal mode number of kink/fishbone modes on a static HL-2A-like tokamak[J]. Plasma Science and Technology, 2022, 24(12): 124005. DOI: 10.1088/2058-6272/aca00a
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In the presence of energetic particles (EPs), the long-lived mode (LLM) frequency multiplication with n = 1, 2, 3, or higher is often observed on HL-2A, where n is the toroidal mode number. Hybrid kinetic-MHD model simulations of the energetic particle (EP) driven kink/fishbone modes on a static HL-2A-like tokamak using NIMROD code find that when the background plasma pressure is relatively high, and the EP pressure and the beam energy are relatively low, the mode frequency increases almost linearly with EP pressure, and the frequency is proportional to n ('frequency multiplication'), even in the absence of any equilibrium plasma rotation. In addition, the frequency multiplication persists as the safety factor at the magnetic axis q0 varies. In the absence of EPs, the growth rate of the 1/1 mode is the largest; however, as the EP pressure increases, the growth rate of 2/2 modes or 3/3 modes becomes dominant, suggesting that higher-n modes are more vulnerable to EPs. These results may shed light on the understanding of the toroidal mode number dependence of kink/fishbone modes in the advanced scenarios of tokamaks with weak or reversed central magnetic shear.
Confinement improvement and higher beta limit have been achieved in advanced tokamak (AT) operation scenarios, in which the q-profile is reversely or weakly sheared in the core region [1–3], and the AT operation scenarios have been proposed for the steady-state operation of ITER [4, 5]. With weak shear and plasma pressure gradient exceeding a critical value in the core region, besides 1/1 internal kink mode [6–8], higher-harmonics modes, which are usually called "infernal modes" [9, 10], also become unstable. Fishbone modes, which are believed to be driven by the resonant interaction between the internal kink mode and the energetic particles (EPs) [11–14], have been observed on many tokamaks with auxiliary heating since the 1980s [15–19]. During the fishbone burst events, which usually last about 1 ms, a significant loss of EPs has been observed, and they degrade the heating efficiency and limit the beta achieved in the experiments. Besides, the lost EPs can cause damage to the first wall. The time interval between two adjacent fishbone bursts is usually several ms, and the q-profile is monotonic with finite shear in the core region. Similar to the energetic particle (EP) driven fishbone modes, other kinds of EP-driven modes are observed in auxiliary heating experiments with flat q in the core plasma. The modes last 100 ms or longer after the saturation, which is called long-lived modes (LLMs) for the long-lasting feature. The LLMs often cause loss of EPs and breaking of the plasma toroidal rotation. Unlike the bursty fishbone modes, the LLMs may serve as an option for the AT operation scenarios because of the gentle and continuous energy loss in the core plasma of tokamaks.
Because of the potential for AT operation scenarios, many experiments have been performed to study LLMs. During the NBI heating of plasmas with a flat central safety factor profile, LLMs are observed on HL-2A that can last for several hundred ms, and the mode frequency is approximately proportional to the toroidal mode number n, a phenomenon we refer to as frequency multiplication (FM), as shown in figure 1 of [20], where all 1/1, 2/2, 3/3 and 4/4 modes are displayed in the frequency range of 10–60 kHz. The LLMs can be suppressed by ECRH or SMBI (supersonic molecular beam injection) which is related to the changing of the q profile and pressure gradient [20]. On MAST, the FM for LLMs is also observed during NBI heating, not only for weakly sheared, but also for slightly and reversely sheared q-profiles, where the critical plasma pressure, above which the mode becomes unstable, increases with n [21]. For KSTAR plasma with ECRH and NBI, the pressure-driven LLMs last up to 40 sec along with the FM, when the q-profile is above 2 with broad weak shear in the core region [22]. As a characteristic feature of LLMs, the FM has not been well understood.
Several analytical or numerical studies have investigated the properties of LLMs. Solving an analytical dispersion relation developed for LLMs based on the HL-2A equilibrium, Zhang et al found that the growth rate of the 2/2 mode is greater than that of the 1/1 mode, and the frequency is proportional to n (n = 1, 2, 3) [23]. However, they have not considered the FM in detail, and the 3/3 mode is stable in their calculations, different from the experimental observations. Including the equilibrium rotation in the dispersion relation, Xie reported that fishbone mode can turn into LLM with low magnetic shear, when the EP pressure and the plasma rotation frequency both exceed certain critical values [24], however, neither the n > 1 mode or the FM is discussed. Using the kinetic-MHD hybrid code M3D-K for simulations of a circular cross-section tokamak with central weakly sheared q-profile, Ren et al obtained the FM of LLMs in the presence of EPs, where the growth rates of higher harmonics (n > 1) can be greater than that of 1/1 mode [25]. Their study is based on the analytical equilibrium profile for a model tokamak, and the conditions in which the FM is broken remain unclear.
In this work, we study the FM of LLMs, with a focus on their onset conditions. Based on the equilibrium generated from the LLM experiments on HL-2A, using the hybrid kinetic-MHD model implemented in the NIMROD code [26–28], our simulations show that the EP-driven fishbone mode frequency is almost proportional to n in the absence of any equilibrium toroidal flow, and the FM appears when the background plasma pressure gradient is above a certain threshold, but neither the beam energy nor the EP β fraction (the ratio between EP pressure and total plasma pressure) is too high. The 1/1 mode is the most unstable when the EP pressure is not too high [29]. However, higher-n kink/fishbone modes are found more dominant when the EP pressure becomes relatively high.
The rest of the paper is organized as follows. The simulation model is reviewed briefly in section 2. The simulation setup is described in section 3, which is followed by the report on the main results in section 4. First, the q0 effect on kink/fishbone mode is studied in both the absence and presence of EPs, and then the effects of EP β fraction, beam energy, and EP pressure gradient are also investigated. The summary and discussion are provided in section 5.
The simulation model is based on the following hybrid kinetic-MHD equations implemented in the NIMROD code [26, 27].
| ∂ρ∂t+∇⋅(ρV)=0 | (1) |
| \rho\left(\frac{\partial \mathbf{V}}{\partial t}+\mathbf{V} \cdot \nabla \mathbf{V}\right)=\mathbf{J} \times \mathbf{B}-\nabla p_{\mathrm{b}}-\nabla \cdot \boldsymbol{P}_{\mathrm{h}} | (2) |
| \frac{n}{\Gamma-1}\left(\frac{\partial T}{\partial t}+\mathbf{V} \cdot \nabla T\right)=0 | (3) |
| \frac{\partial \mathbf{B}}{\partial t}=-\nabla \times \mathbf{E} | (4) |
| \nabla \times \mathbf{B}=\mu_0 \mathbf{J} | (5) |
| \mathbf{E}=-\mathbf{V} \times \mathbf{B} | (6) |
where ρ, V, J, pb, n and T are the mass density, center of mass velocity, current density, plasma pressure, number density, and temperature of the main species plasma, E and B are the electric and magnetic fields, Γ and μ0 are the specific heat ratio and the vacuum permeability, respectively. Static equilibrium is considered in order to exclude effects from the equilibrium rotations. For nh ≪ nb and βh ~ βb, where nb (nh) is the main species plasma (energetic particle) number density, and βb (βh) is the ratio of main species background plasma (energetic particle) pressure to magnetic pressure, the pressure tensor Ph in the momentum equation can couple the kinetic effects from EPs, which are governed by the drift-kinetic equation [27]. Here Ph = Ph0 + δPh, where Ph0 is assumed isotropic, and δPh is defined as
| \delta \boldsymbol{P}_{\mathrm{h}}=\left(\begin{array}{ccc} \delta p_{\perp} & 0 & 0 \\ 0 & \delta p_{\perp} & 0 \\ 0 & 0 & \delta p_{\|} \end{array}\right), | (7) |
δp⊥ is the pressure due to hot particle motions perpendicular to the magnetic field, and δp║ is the pressure due to the hot particle motions parallel to the magnetic field.
An EFIT equilibrium reconstruction based on HL-2A discharge #16074, is used in our simulation [20]. The equilibrium flux surfaces and the mesh in magnetic flux coordinates within the last closed flux surface (LCFS) are shown in figure 1. Although the LCFS is up–down asymmetric, the flux surfaces are close to circles in the core region. The internal kink/fishbone mode locates in the region of 0 <\sqrt{\psi / \psi_0}<0.2, where the safety factor q-profile is close to unity along with weak magnetic shear (figure 2(a)). Here ψ is the poloidal magnetic flux and ψ0 is the total poloidal magnetic flux within the LCFS.
The EPs from NBI are initialized with the slowing-down distribution function [27],
| f_0=\frac{P_0 \exp \left(\frac{P_\zeta}{\psi_{\mathrm{n}}}\right)}{\varepsilon^{3 / 2}+\varepsilon_{\mathrm{c}}^{3 / 2}} | (8) |
where P0 is the normalization constant, Pζ = gρ║ - ψp is the canonical toroidal momentum, g = RBϕ, ρ║ = mv║/qB, ψp is the poloidal flux, ψn = hψ0, ψ0 is the total flux and the parameter h is used to match the spatial profile of the equilibrium, ε is the particle energy, and εc is the critical slowing down energy [30]
| \varepsilon_{\mathrm{c}}=\left(\frac{3}{4}\right)^{2 / 3}\left(\frac{{\rm{ \mathsf{ π} }} m_{\mathrm{i}}}{m_{\mathrm{e}}}\right)^{1 / 3} T_{\mathrm{e}} | (9) |
with mi being the ion mass, me the electron mass, and Te the electron temperature. Beam ions collide dominantly with the background electrons (ions) for ε > εc (ε < εc). The anisotropic angle distribution may lead to the off-diagonal terms in the equilibrium pressure tensor that would be inconsistent with the reconstructed equilibrium from the experiments. For this reason, only an isotropic distribution in pitch angle is used in loading the particles into the velocity space initially. This also allows a pressure moment to be calculated and the overall normalization for the particles to be set without recomputing the equilibrium for each point in the parameter scan. This is a choice of convenience, as previously adopted and explained in the appendix of [27].
The solid line in figure 2(b) is the total equilibrium plasma pressure profile reconstructed from the experiments, which includes the contribution from beam ions. The dashed line in figure 2(b) is the fitted curve for the EP plasma pressure profile with p / p_0=\exp \left(-\psi / h \psi_0\right), where p0 is the EP pressure at the magnetic axis. The total equilibrium pressure is kept fixed when the EP pressure fraction βf is varied in the parameter scan study to ensure that the MHD equilibrium remains the same. For linear simulations, we set resistivity η = 0, and a total of 106 simulation particles are prescribed in the poloidal plane with 64 × 64 finite elements. Other main parameters are input from equilibrium [23], with the major radius R = 1.65 m, the minor radius a = 0.40 m, the toroidal magnetic field B0 = 1.37 T, and the number density is set to be constant in the radial direction with n = 2.44 × 1019 m-3. The case with q0 = 0.9, βf = βh/β0 = 0.1, h = 0.25 and beam energy εb = 10 keV is specified as the standard reference case for comparisons. From the typical parameters of the HL-2A experiments [31]: n = ni = ne = 2.44 × 1019 m-3, T = Ti =Te = 2 keV and B0 = 1.37 T, we get background plasma beta \beta_{\mathrm{b}}=2 \mu_0\left(n_{\mathrm{i}}+n_{\mathrm{e}}\right) T / B_0^2=2.1 \%, and the typical EP beta βh = 0.35% [23], then βf = βh/(βh + βb) = 0.14, which is closed to standard reference βf.
LLMs are observed in the experiments with NBI heating, and they can be controlled by ECRH and SMBI, subject to the influences from magnetic shear, pressure gradient, and NBI beam properties [20]. In order to investigate these effects in our simulations, we scan q0 to study the effects of magnetic shear on FM for LLMs, and the EP beta fraction βf, beam energy εb and EP pressure gradient to study the effects of EPs on FM. We choose two representative time moments of discharge #016074, one is at 420 ms (M420), and another is at 452 ms (M452). The M420 case is at an early stage of the mode growth, and the M452 case is near the saturation of the mode, as shown in figure 1 of [20]. The pressure gradient profiles of the M420 and M452 cases are shown in figure 3. The pressure gradient of the M420 case is almost two times that of the M452 case. We use the two pressure profiles to study the effects of background plasma pressure gradient on FM.
First, we scan q0 to study its effects on the modes without EP. As shown in figure 4, varying q0 along shifts the q-profile up or down entirely with minimal change in its shape.
For the two equilibriums from the M420 and M452 cases, linear NIMROD calculations show that the m/n = 1/1, 2/2 and 3/3 kink modes are unstable when q0 < 1 in the absence of energetic particles, where m is the poloidal mode number. The linear growth rate of kink modes decreases with n, namely, γ1/1 > γ2/2 > γ3/3. As q0 increases, the growth rate increases first and then decreases to zero as q0 approaches unity. The q0max, the value of q0 at which the growth rate reaches the maximum, increases with n (figure 5). The contour plots of the plasma pressure perturbation for the M452 case (figure 6) show that the mode structure shrinks in size as q0 approaches unity. The contour plots for the M420 case are similar, and thus are not repeated here. For the 1/1 kink mode, the mode structure is global within the q < 1 region. For 2/2 and 3/3 kink modes, the mode structures become more localized around q = 1 surface, which is consistent with the theory prediction [32].
In the absence of EPs, these MHD modes are purely growing instabilities without real frequency or FM phenomenon. This is in agreement with the observation that LLMs only occur in presence of NBI heating or other auxiliary heating.
Now we study the q0 effects on the modes in presence of EPs, and set βf = 0.1. For the M420 case in presence of EPs, the overall growth rates become higher (figure 7(a)). The mode frequency increases slightly as q0 increases, and more importantly, the FM exists for n = 1, n = 2 and n = 3 (figure 7(b)), which is consistent with the experimental observation in HL-2A [23]. For the M452 case, as shown in figure 7(d), the 1/1 and 2/2 kink modes are more stable in the presence of EPs, whereas only the 3/3 mode is driven more unstable by EPs. The FM remains with n = 1 and n = 2. However, for the 3/3 mode, the mode frequency difference from the 2/2 mode is significantly larger than the frequency interval between the 2/2 and 1/1 modes (figure 7(e)). Comparing the results of M420 and M452, we find that a stronger background plasma pressure gradient maintains the FM for n = 1, 2, 3 (as in the M420 case), which becomes weakened with a weaker background plasma pressure gradient (as in the M452 case). This is consistent with experimental findings that LLMs are characteristic of pressure-driven modes [20]. Comparing the results from section 4.1, we have found FM in the presence of EPs, and the FM is weakened or broken with a reduced background plasma pressure gradient.
Comparing figure 7 in this paper and figure 1 in [20] carefully, we find that the frequency range does not match, although FM exists in both figures. The main reason is that we have not considered plasma rotation. According to f_{\exp }=f_{\mathrm{EP}}+n f_{\mathrm{rot}}, where fexp is the frequency measured in the experiments, fEP is the frequency caused by EPs, and frot is the frequency of plasma rotation. Adding the frequencies from the simulations (f_{1 / 1}^{\mathrm{sim}} \simeq 2.5 \mathrm{kHz}, f_{2 / 2}^{\operatorname{sim}} \simeq 6 \mathrm{kHz} and f_{3 / 3}^{\mathrm{sim}} \simeq 9 \mathrm{kHz}) and frequency of plasma rotation (frot ≃ 7 kHz as measured in the experiments [20]), we get f1/1 ≃ 10 kHz, f2/2 ≃ 20 kHz and f3/3 ≃ 30 kHz, which are close to the frequencies measured in the experiments.
For the M420 case, the mode structures shown in figure 8 are similar to the cases in the absence of EPs as shown in figure 6, except that they are now twisted by EPs, which become more significant as n increases and q0 approaches unity. For the M452 case, the poloidal mode structures twisting by EPs are more apparent in comparison to the M420 case. For the n = 3 mode, the structure of mode coupling appears (figures 9(a3), (b3) and (c3)), and the mode frequency is mostly located on the Alfvénic continuum, as shown in figure 7(f), which is indicative of an energetic particle mode (EPM).
We set q0 = 0.9, and scan βf to study EP pressure effects on the modes. For the M420 case, as the EP βf increases, the growth rate of the higher-n mode increases more rapidly. As the EP pressure increases, the growth rates of the higher-n mode approach that of 1/1 mode, and for βf > 0.25, the growth rate of 3/3 mode becomes the largest (figure 10(a)). This indicates that higher-n modes are more vulnerable to EP β effects. When the EP pressure is relatively low (βf < 0.2), the mode frequency increases almost linearly with βf, and is roughly proportional to n. As βf increases to 0.25, the frequency of the 3/3 mode jumps from ~10 kHz to ~70 kHz and then decreases as βf increases further (figure 10(b)).
For the M452 case, the 1/1 modes and 2/2 modes are suppressed slightly by EPs when the EP pressure is relatively low (βf < 0.1); however, there is no suppressing effect on the 3/3 mode. As the EP pressure increases further (βf > 0.2), the growth rate of the 3/3 mode becomes the largest (figure 10(c)), which is similar to the M420 case. For the 1/1 and 2/2 modes, the mode frequency increases almost linearly with βf, and is roughly proportional to n, which is also similar to the M420 case. The frequency of the 3/3 mode jumps to a higher branch (~60 kHz) when βf increases from 0 to 0.1, and the frequency jumping occurs with βf smaller than that of the M420 case (figure 10(d)). For the M420 case, the mode structure is twisted by EPs when βf < 0.2, and there exists poloidal mode coupling for the n = 3 blue mode when βf = 0.3 (figures 11(a1), (b1) and (c1)). For the M452 case, the mode structure is twisted by EPs, and there exists poloidal mode coupling for the n = 3 mode when βf ≥ 0.1 (figures 11(a2), (b2) and (c2)).
Comparing the results of the M420 and M452 cases, we can see that, with a stronger background plasma pressure gradient (M420), the suppressing effects of EPs on kink modes disappear, and the driving effects of EPs on higher frequency modes become weaker. Furthermore, the FM persists for n = 1, 2, 3 up to βf < 0.2 for the stronger background plasma pressure gradient (M420 case), while it is broken for n = 3 with βf > 0.1 for the weaker background plasma pressure gradient (M452 case). This is consistent with the results in section 4.2 that the FM of LLMs becomes more apparent with a stronger pressure gradient.
Now we scan beam energy εb to study its effects on FM, and we set q0 = 0.9 and βf = 0.1. For the M420 case, the growth rate decreases with the beam energy εb when the mode frequency is low (~10 kHz) (figure 12(a)). When beam energy εb is less than 15 keV, the mode frequency is proportional to n approximately. However, for the 3/3 mode, as the beam energy εb increases from 15 to 20 keV, the frequency jumps from 10 kHz to 86 kHz, and the growth rate increases as well suggesting the onset of another branch of mode. For the M452 case, the dependence of growth rate decreases on the beam energy is similar (figure 12(c)). The FM is broken with 5 keV < εb < 10 keV for n = 3, and 15 keV < εb < 20 keV for n = 2. Both 2/2 and 3/3 mode frequencies jump to a higher mode branch above certain but different beam energy levels.
Comparing the results of the M420 and M452 cases, we can see that, FM can be more easily lost with a weaker background plasma pressure gradient (figures 12(b) and 12(d)), which is consistent with the results of sections 4.2 and 4.3.
In the end, we study the effects of EP pressure gradient on FM. For the M420 case, as the EP pressure gradient coefficient h increases, the radial profile of EPs becomes more flat with a smaller EP pressure gradient, and the growth rate decreases (figure 13(a)). The mode frequency decreases slightly with h, and remains to be proportional to n clearly (figure 13(b)). For the M452 case, the effects of h on the growth rate are similar, but are much weaker on the mode frequency. The mode frequency keeps almost constant as h increases. For n = 1, 2, the mode frequencies are low (~10 kHz), and proportional to n approximately. For n = 3, the mode frequency (~60 kHz) can stay on the higher branch for a wide range of EP pressure gradients (figure 13(d)). Based on the above results, the pressure gradient-driven nature of FM for LLMs has been further confirmed and the FM is not much influenced by the EP pressure gradient, which is probably due to the fact that EP pressure is relatively small compared to the background plasma pressure.
The (n = 1, 2, 3) internal kink/fishbone mode driven by EPs on HL-2A tokamak is investigated through kinetic-MHD simulations. In the absence of plasma rotation, the internal kink mode would acquire a finite mode frequency by its resonant drive from the EPs at that finite frequency [11–14, 33, 34]. The mode frequency is found to be proportional to toroidal mode number n (frequency multiplication, FM) even in the absence of equilibrium flow, when the background plasma pressure gradient is strong, and neither the beam energy nor the EP β fraction is too high. Above a certain threshold for the EP beam energy or the EP β fraction, the FM becomes broken, and the higher-n modes can transform to the AE/EPM branch, and such a transition tends to be facilitated by weaker background plasma pressure. The q profile with varied q0 (0.85 < q0 < 0.95) has a weak influence on the FM. Although in the absence of EPs, the growth rate of the 1/1 mode is greater than that of the higher-n mode, the growth rate of the higher-n mode increases more rapidly with βf than that of the 1/1 mode, suggesting the dominance of higher-n modes in experiments with higher EP fraction.
The frequency range we calculated does not match that from experiments with a lack of plasma rotation in our simulation model. After adding the frequency of plasma rotation, we get frequencies close to experimental measurements. Although we perform the simulations in the circular cross-section tokamak, our results may also apply to the non-circular cross-section tokamaks, such as DIIID and EAST, because the LLMs are localized in the core region, where the magnetic flux surface is near circular for non-circular cross-section tokamaks. In addition, the simulation study is limited to the linear regime, whereas the LLMs observed in the HL-2A experiments are clearly in the nonlinear regime. Nonlinear simulation should be performed in the future to find out if the findings in this work remain viable in the nonlinear stage. In the future, we plan on performing nonlinear simulations in order to reveal more physical details of LLMs (such as the long-lasting feature) in the advanced tokamaks with weak or reversed central magnetic shear.
We thank Prof Lu Wang and Dr Da Li for their helpful discussions and suggestions. We appreciate the assistance from Dr Haolong Li. We are grateful for the support from the NIMROD team. This work was supported by the National Magnetic Confinement Fusion Program of China (No. 2019YFE03050004), National Natural Science Foundation of China (Nos. 11875253, 11775221, 51821005 and 11875018), the Fundamental Research Funds for the Central Universities (Nos. WK3420000004 and 2019kfyXJJS193), the Collaborative Innovation Program of Hefei Science Center, CAS (No. 2019HSC-CIP015), the U S Department of Energy (Nos. DE-FG02-86ER53218 and DE-SC0018001). This research used computing resources from the Supercomputing Center of University of Science and Technology of China.
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| 1. | Liu, S., Ren, Z., Wang, W. et al. High-precision prediction of non-resonant high-order harmonics energetic particle modes via stacking ensemble strategies. Machine Learning: Science and Technology, 2025, 6(1): 015069. DOI:10.1088/2632-2153/adbfdb |
| 2. | Liu, S., Ren, Z., Wang, W. et al. Hybrid simulation of q = 1 high-order harmonics driven by passing energetic particles in tokamak plasmas. Plasma Science and Technology, 2023, 25(12): 125101. DOI:10.1088/2058-6272/ace7ae |
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