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Based on the critical gradient model , the combination of the TGLFEP code and EPtran code is employed to predict energetic particle (EP) transport induced by Alfvén eigenmodes (AEs). To be consistent with the experimental results, the model was improved recently by taking into consideration the threshold evolution and orbit loss mechanism. The threshold is modified to be the normalized critical gradient ((dn/dr)/(n/a)) instead of the critical gradient (dn/dr), and the new threshold is defined as a function inversely proportional to the EP density as obtained by the TGLFEP code. Additionally, the EP loss cone calculated by ORBIT is added into the EPtran code, which provides an important additional core loss channel for EPs due to finite orbits. With these two improvements, the EP redistribution profiles are found to very well reproduce the experimental profiles of two DIII-D validation cases (#142111 and #153071) with multiple unstable AEs and large-scale EP transport. In addition, a neural network is established to replace TGLFEP for critical gradient calculation, and EPtran code is rewritten with parallel computing. Based on this, a module of EP is established and it is added into the integrated simulation of OMFIT framework. The integrated simulation of HL-3 with AE transported neutral beam EP profile indicates that EP transport reduces the total pressure and current as expected, but under some condition it could also raise the safety factor in the core.
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Keywords:
- energetic particle /
- Alfvén eigenmode /
- critical gradient model /
- transport
[1] Wan Y X, Li J G, Liu Y, Wang X L, Chan V, Chen C G, Duan X R, Fu P, Gao X, Feng K M, Liu S L, Song Y T, Weng P D, Wan B N, Wan F R, Wang H Y, Wu S T, Ye M Y, Yang Q W, Zheng G Y, Zhuang G, Li Q 2017 Nucl. Fusion 57 102009
Google Scholar
[2] Zhuang G, Li G Q, Li J, Wan Y X, Liu Y, Wang X L, Song Y T, Chan V, Yang Q W, Wan B N, Duan X R, Fu P, Xiao B J 2019 Nucl. Fusion 59 112010
Google Scholar
[3] Gaffey J D Jr 1976 J. Plasma Phys. 16 149
Google Scholar
[4] Estrada-Mila C, Candy J, Waltz R E 2006 Phys. Plasmas 13 112303
Google Scholar
[5] Van Zeeland M A, Heidbrink W W, Fisher R K, García Muñoz M, Kramer G J, Pace D C, White R B, Aekaeslompolo S, Austin M E, Boom J E 2011 Phys. Plasmas 18 056114
Google Scholar
[6] Heidbrink W W, Collins C S, Podestà M, Kramer G J, Pace D C, Petty C C, Stagner L, van Zeeland M A, White R B, Zhu Y B 2017 Phys. Plasmas 24 056109
Google Scholar
[7] Garcia-Munoz M, Classen I G J, Geiger B, Heidbrink W W, van Zeeland M A, Äkäslompolo S, Bilato R, Bobkov V, Brambilla M, Conway C D, Schneller M, da Graça S, Igochine V, Lauber Ph, Luhmann N, Maraschek M, Meo F, Park H, Tardini G 2011 Nucl. Fusion 51 103013
Google Scholar
[8] Kiptily V G, von Thun C P, Pinches S D, Sharapov S E, Borba D, Cecil F E, Darrow D, Goloborod'ko V, Craciunescu T, Johnson T, Nabais F, Reich M, Salmi A, Yavorskij V, Cecconello M, Gorini G, Lomas P, Murari A, Parail V, Popovichev S, Saibene G, Sartori R, Syme D B, Tardocchi M, de Vries P, Zoita V L 2009 Nucl. Fusion 49 065030
Google Scholar
[9] Fredrickson E D, Cheng C Z, Darrow D, Fu G, Gorelenkov N N, Kramer G, Medley S S, Menard J, Roquemore L, Stutman D, White R B 2003 Phys. Plasmas 10 2852
Google Scholar
[10] Collins C S, Heidbrink W W, Austin M E, Kramer G J, Pace D C, Petty C C, Stagner L, Van Zeeland M A, White R B, Zhu Y B, DIII-D Team 2016 Phys. Rev. Lett. 116 095001
Google Scholar
[11] White R B, Chance M S 1984 Phys. Fluids 27 2455
Google Scholar
[12] Todo Y, Shinohara K, Takechi M, Ishikawa M, 2005 Phys. Plasmas 12 012503
Google Scholar
[13] Podestà M, Gorelenkova M, Gorelenkov N N, White R B 2017 Plasma Phys. Control. Fusion 59 095008
Google Scholar
[14] Bass E M , Waltz R E 2010 Phys. Plasmas 17 112319
[15] Collins C S, Heidbrink W W, Podestà M, White R B, Kramer G J, Pace D C, Petty C C, Stagner L, van Zeeland M A, Zhu Y B 2017 Nucl. Fusion 57 086005
Google Scholar
[16] van Zeeland M A, Gorelenkov N N, Heidbrink W W, Kramer G J, Spong D A, Austin M E, Fisher R K, García Muñoz M, Gorelenkova M, Luhmann N, Murakami M, Nazikian R, Pace1 D C, Park J M, Tobias B J, White R B 2012 Nucl. Fusion 52 094023
Google Scholar
[17] Heidbrink W W, Ferron J R, Holcomb C T, van Zeeland M A, Chen X, Collins C M, Garofalo A, Gong X, Grierson B A, Podestà M, Stagner L, Zhu Y 2014 Plasma Phys. Control. Fusion 56 095030
Google Scholar
[18] Podestà M, Gorelenkova M, White R B 2014 Plasma Phys. Control. Fusion 56 055003
Google Scholar
[19] He S, Waltz R E, Staebler G M 2017 Phys. Plasmas 24 072305
Google Scholar
[20] Staebler G M, Kinsey J E, Waltz R E 2005 Phys. Plasmas 12 102508
Google Scholar
[21] Chen L, Cheng C Z 1980 Phys. Fluids 23 2242-2249.
Google Scholar
[22] Lang J, Fu G Y, Chen Y 2010 Phys. Plasmas 17 042309
Google Scholar
[23] Waltz R E, Bass E M, Staebler G M 2013 Phys. Plasmas 20 042510
Google Scholar
[24] Waltz R E, Bass E M 2014 Nucl. Fusion 54 104006
Google Scholar
[25] Waltz R E, Bass E M, Heidbrink W W, van Zeeland M A 2015 Nucl. Fusion 55 123012
Google Scholar
[26] He S, Waltz R E 2016 Nucl. Fusion 56 056004
Google Scholar
[27] Betti R, Freidberg J P 1992 Phys. Fluids B Plasma Phys. 4 1465
Google Scholar
[28] Pankin A, McCune D, Andre R, Bateman G, Kritz A 2004 Comput. Phys. Commun. 159 157
Google Scholar
[29] Todo Y, Van Zeeland M A, Bierwage A., Heidbrink W W 2014 Nucl. Fusion 54 104012
Google Scholar
[30] White R B 2014 The Theory of Toroidally Confined Plasmas (3rd Ed.) (Imperial College Press) pp82–90
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图 1 (a)增长率和(b)频率与高斯宽度的关系, 其中橙色和紫色的曲线表示TAE, 蓝色线被认为是一种数值噪声, 作为参考, 黑色实线为MEGA的模拟结果
Figure 1. (a) Growth rate and (b) frequency as a function of Gaussian width. The orange and purple curves represent acceptable TAE calculated using TGLFEP. Because the blue curve has a strong perturbation, the mode is treated as polluted by numerical noise. For comparison, the MEGA results are depicted by the black lines.
图 2 能量相关(橙)/不相关(蓝) Angioni模型对比
Figure 2. Comparison between energy dependent (orange) and energy independent (blue) Angioni diffusion coefficients.
图 3 (a)输运前的AE不稳定区间; (b)在ρ = 0.4, 0.5, 0.6处, a/LnEP与nEP的反比例关系; (c) ρ = 0.4处, 旧模型和改进模型模拟的密度演化轨迹
Figure 3. (a) Unstable AE region of n = 3 without transport. Critical a/LnEP is depicted by dash red curve, and a/LnEP of classical slowing down distribution is depicted by solid black curve; (b) inverse proportional function between critical a/LnEP and nEP at ρ = 0.4, 0.5, 0.6; (c) density evolution trajectories of previous CGM (black) and improved CGM (red) at ρ = 0.4.
图 4 (a)捕获粒子和(b)通行粒子在AE扰动下的运动轨迹示例; (c)图(b)在高场侧的放大图
Figure 4. Representative trajectories of (a) trapped and (b) counter passing particles by including AE perturbation; the high magnetic field side of (b) is enlarged in (c) to reveal trajectory variations.
图 5 (a)无扰动时, 不同能量EP在(ρ, v///v)空间的损失锥; (b)有无AE扰动下, 80 keV的EP损失锥对比图
Figure 5. (a) Loss boundary in (ρ, v///v) space with different energies; the loss region is on the right side of the curve. (b) loss cone without/with AE perturbation. The blue area represents the loss cone without AE, and the additional loss by including AE is highlighted by the red area.
图 6 MEGA模拟结果 (a) n = 1—5的TAEs的能量演化过程; (b) n = 3的TAE的模结构, vr,cos为径向扰动速度的cos分量, 不同的极向谐波用不同的颜色区分
Figure 6. (a) Evolution of energy with n = 1–5 TAEs by MEGA; (b) cosine part of radial velocity for the most unstable n = 3 TAE.
图 7 (a) n = 1—5的TAEs在不同磁面的增长率; (b)单n和多n计算的临界梯度剖面的对比
Figure 7. (a) Growth rate in each flux surface of n = 1–5 TAEs by TGLFEP; (b) comparison between critical a/LnEP profiles with n = 1–4 and n = 3.
图 8 (a) EP密度剖面对比图, 经典慢化分布(黑色)、实验(红色)、旧临界梯度模型(蓝色)、考虑(紫色)/不考虑(绿色)AE扰动对损失锥影响的改进的临界梯度模型、MEGA(黄色); (b) EP在螺旋角空间的分布
Figure 8. (a) Density profile comparison: Black curve represents classical slowing down; the red curve is inferred from experiment data; blue curve represents original CGM without loss cone effect; purple/green curve is improved CGM with loss cone from AE perturbed/unperturbed orbits; yellow curve represents MEGA results. (b) EP redistribution in pitch angle space.
图 9 n = 4的TAE的模结构, 实线和虚线分别表示扰动速度的余弦和正弦分量
Figure 9. Spatial profile of n = 4 TAE, where cosine and sine part of radial velocity are depicted by solid and dash curve, respectively.
图 10 不同能量的EP在(ρ, v///v)空间的损失锥
Figure 10. Loss boundary in (ρ, v///v) space with different energies.
图 11 压强剖面对比图, 经典慢化分布为黑色曲线, 旧临界梯度模型为蓝色曲线, 改进的临界梯度模型为红色曲线, 实验结果为绿色三角并配有误差棒
Figure 11. Pressure profile of classical slowing down (black), previous (blue) and improved (red) CGM. For comparison, experimental data is depicted by green triangles with error bar.
图 12 神经网络流程图, 第1步判断AE是否被激发, 第2步计算系数k1 和 k2
Figure 12. Flow chart of the neutral network (NN). The NN estimates if the AE can be excited at first, and calculates the two coefficients k1 and k2 for AE unstable location.
图 13 (a)训练集和验证集的均方差随学习次数的变化; (b)准确度随学习次数的变化
Figure 13. (a) Loss (mean square error) and (b) accuracy for AE stability estimation versus training epoch.
图 14 预测值与目标值的对比图 (a) k1; (b) k2
Figure 14. Predicted (a) k1 and (b) k2 compared with the targets.
图 15 用来验证神经网络的3个平衡的(a)压强剖面和(b)安全因子剖面
Figure 15. (a) Pressure and (b) safety factor profile of three additional equilibria for NN validation.
图 16 HL-3神经网络预测值(MSE为蓝色, 自定义损失函数为红色)与TGLFEP计算值(黑色)对比图 (a), (b) Case 1; (c), (d) Case 2; (e), (f) Case 3
Figure 16. Coefficients of k1 and k2 predicted by NN with loss function of MSE (blue) and custom loss function (red): (a), (b) Case 1; (c), (d) Case 2; (e), (f) Case 3. For comparison, TGLFEP results are depicted by black curve.
图 17 具有(a)单调递增、(b)芯部弱反磁剪切和(c)强反磁剪切安全因子剖面的平衡位形下, 高能量粒子剖面对比图. 绿色曲线为经典慢化模型计算的初始EP剖面, 黑色、蓝色、红色曲线分别为根据TGLFEP、MSE和自定义损失函数得到的临界梯度计算的EP剖面
Figure 17. EP profile comparison for the equilibrium with (a) monotonic, (b) weak and (c) strong shear q-profile. In each panel, green curve depicts initial EP profile with classical slowing down distribution and black curve depicts EP profile with the critical gradient calculated by TGLFEP. The blue and red curves depict EP profiles by NN with loss function of MSE and custom loss function, respectively.
图 18 OMFIT集成模拟流程图, 蓝色框里是旧OMFIT迭代流程, 红色框中的是高能量粒子模块
Figure 18. OMFIT workflow with MOE. The typical iteration is in the blue border, and MOE is in the red border.
图 19 利用带有MOE的集成模拟计算的(a)磁面、(b)总压强、(c)安全因子剖面, 其中绿线仅考虑了EP输运对总压强的影响, 红线考虑了EP输运对压强和电流的影响, 蓝线则为不包含MOE的集成模拟的结果
Figure 19. (a) Flux surface, (b) total pressure and (c) safety factor profile calculated by the OMFIT integrated simulation with MOE. Green curve only considers pressure modification, red curve considers both pressure and current modification, and blue curve is calculated without MOE.
表 1 神经网络输入变量
Table 1. Variables for NN input
变量 符号 表达式 1 电子密度 ne 2 电子温度 Te 3 电子密度特征长度 rlns_e $ -\displaystyle\frac{a}{{n}_{{\rm{e}}}}\frac{\partial {n}_{{\rm{e}}}}{\partial r} $ 4 电子温度特征长度 rlts_e $ -\displaystyle\frac{a}{{T}_{{\rm{e}}}}\frac{\partial {T}_{{\rm{e}}}}{\partial r} $ 5 磁面对应的小半径 rmin $ {r}/{a} $ 6 安全因子 q 7 磁剪切 q_prime $ \displaystyle\frac{{q}^{2}{a}^{2}}{{r}^{2}}s $ 8 压强梯度 p_prime $ \displaystyle\frac{q{a}^{2}}{r{B}^{2}}\frac{\partial p}{\partial r} $ 9 高能量粒子温度 taus_EP TEP/Te 10 高能量粒子温度特征长度 rlts_EP $ -\displaystyle\frac{a}{{T}_{{\rm{E}}{\rm{P}}}}\frac{\partial {T}_{{\rm{E}}{\rm{P}}}}{\partial r} $ 11 磁场强度 B 12 磁面对应的大半径 rmaj $ {R}/{a} $ 13 拉长比 kappa κ 14 拉长比的剪切 s_kappa $ \displaystyle\frac{r}{\kappa }\frac{\partial \kappa }{\partial r} $ 15 三角形变 delta δ 16 三角形变的剪切 s_delta $ \displaystyle\frac{r}{\delta }\frac{\partial \delta }{\partial r} $ 17 Shafranov位移 drmajdx $ \displaystyle\frac{\partial R}{\partial x} $ 18 小半径 a 
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[1] Wan Y X, Li J G, Liu Y, Wang X L, Chan V, Chen C G, Duan X R, Fu P, Gao X, Feng K M, Liu S L, Song Y T, Weng P D, Wan B N, Wan F R, Wang H Y, Wu S T, Ye M Y, Yang Q W, Zheng G Y, Zhuang G, Li Q 2017 Nucl. Fusion 57 102009
Google Scholar
[2] Zhuang G, Li G Q, Li J, Wan Y X, Liu Y, Wang X L, Song Y T, Chan V, Yang Q W, Wan B N, Duan X R, Fu P, Xiao B J 2019 Nucl. Fusion 59 112010
Google Scholar
[3] Gaffey J D Jr 1976 J. Plasma Phys. 16 149
Google Scholar
[4] Estrada-Mila C, Candy J, Waltz R E 2006 Phys. Plasmas 13 112303
Google Scholar
[5] Van Zeeland M A, Heidbrink W W, Fisher R K, García Muñoz M, Kramer G J, Pace D C, White R B, Aekaeslompolo S, Austin M E, Boom J E 2011 Phys. Plasmas 18 056114
Google Scholar
[6] Heidbrink W W, Collins C S, Podestà M, Kramer G J, Pace D C, Petty C C, Stagner L, van Zeeland M A, White R B, Zhu Y B 2017 Phys. Plasmas 24 056109
Google Scholar
[7] Garcia-Munoz M, Classen I G J, Geiger B, Heidbrink W W, van Zeeland M A, Äkäslompolo S, Bilato R, Bobkov V, Brambilla M, Conway C D, Schneller M, da Graça S, Igochine V, Lauber Ph, Luhmann N, Maraschek M, Meo F, Park H, Tardini G 2011 Nucl. Fusion 51 103013
Google Scholar
[8] Kiptily V G, von Thun C P, Pinches S D, Sharapov S E, Borba D, Cecil F E, Darrow D, Goloborod'ko V, Craciunescu T, Johnson T, Nabais F, Reich M, Salmi A, Yavorskij V, Cecconello M, Gorini G, Lomas P, Murari A, Parail V, Popovichev S, Saibene G, Sartori R, Syme D B, Tardocchi M, de Vries P, Zoita V L 2009 Nucl. Fusion 49 065030
Google Scholar
[9] Fredrickson E D, Cheng C Z, Darrow D, Fu G, Gorelenkov N N, Kramer G, Medley S S, Menard J, Roquemore L, Stutman D, White R B 2003 Phys. Plasmas 10 2852
Google Scholar
[10] Collins C S, Heidbrink W W, Austin M E, Kramer G J, Pace D C, Petty C C, Stagner L, Van Zeeland M A, White R B, Zhu Y B, DIII-D Team 2016 Phys. Rev. Lett. 116 095001
Google Scholar
[11] White R B, Chance M S 1984 Phys. Fluids 27 2455
Google Scholar
[12] Todo Y, Shinohara K, Takechi M, Ishikawa M, 2005 Phys. Plasmas 12 012503
Google Scholar
[13] Podestà M, Gorelenkova M, Gorelenkov N N, White R B 2017 Plasma Phys. Control. Fusion 59 095008
Google Scholar
[14] Bass E M , Waltz R E 2010 Phys. Plasmas 17 112319
[15] Collins C S, Heidbrink W W, Podestà M, White R B, Kramer G J, Pace D C, Petty C C, Stagner L, van Zeeland M A, Zhu Y B 2017 Nucl. Fusion 57 086005
Google Scholar
[16] van Zeeland M A, Gorelenkov N N, Heidbrink W W, Kramer G J, Spong D A, Austin M E, Fisher R K, García Muñoz M, Gorelenkova M, Luhmann N, Murakami M, Nazikian R, Pace1 D C, Park J M, Tobias B J, White R B 2012 Nucl. Fusion 52 094023
Google Scholar
[17] Heidbrink W W, Ferron J R, Holcomb C T, van Zeeland M A, Chen X, Collins C M, Garofalo A, Gong X, Grierson B A, Podestà M, Stagner L, Zhu Y 2014 Plasma Phys. Control. Fusion 56 095030
Google Scholar
[18] Podestà M, Gorelenkova M, White R B 2014 Plasma Phys. Control. Fusion 56 055003
Google Scholar
[19] He S, Waltz R E, Staebler G M 2017 Phys. Plasmas 24 072305
Google Scholar
[20] Staebler G M, Kinsey J E, Waltz R E 2005 Phys. Plasmas 12 102508
Google Scholar
[21] Chen L, Cheng C Z 1980 Phys. Fluids 23 2242-2249.
Google Scholar
[22] Lang J, Fu G Y, Chen Y 2010 Phys. Plasmas 17 042309
Google Scholar
[23] Waltz R E, Bass E M, Staebler G M 2013 Phys. Plasmas 20 042510
Google Scholar
[24] Waltz R E, Bass E M 2014 Nucl. Fusion 54 104006
Google Scholar
[25] Waltz R E, Bass E M, Heidbrink W W, van Zeeland M A 2015 Nucl. Fusion 55 123012
Google Scholar
[26] He S, Waltz R E 2016 Nucl. Fusion 56 056004
Google Scholar
[27] Betti R, Freidberg J P 1992 Phys. Fluids B Plasma Phys. 4 1465
Google Scholar
[28] Pankin A, McCune D, Andre R, Bateman G, Kritz A 2004 Comput. Phys. Commun. 159 157
Google Scholar
[29] Todo Y, Van Zeeland M A, Bierwage A., Heidbrink W W 2014 Nucl. Fusion 54 104012
Google Scholar
[30] White R B 2014 The Theory of Toroidally Confined Plasmas (3rd Ed.) (Imperial College Press) pp82–90
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