俘获离子激发的高频内扭曲模的研究
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摘要
在过去的数十年间,中性束注入作为加热方法在托卡马克中得到广泛使用。无论是切向中性束注入还是垂直中性束注入,伴随着高能离子产生的同时能经常观察到鱼骨不稳定性。鱼骨现象和高能束离子的损失相联系,降低了中性束加热的效率。
随着实验上的发现,人们提出很多理论模型来解释鱼骨振荡的机制。对低频鱼骨不稳定性,目前的观点认为鱼骨振荡的机制是内扭曲模与q=1而内的高能离子发生共振相互作用从而被解稳。其共振频率是在垂直注入的情况下接近俘获高能离子的环向进动频率,在切向注入的情况下接近通行粒子的粒子轨道径向漂移频率。除了占主导的低频模,实验观测中还发现了高频模的存在。对于高频模的解释是内扭曲模和由切向注入产生的通行高能离子共振激发出鱼骨不稳定性。其共振频率接近高能离子的循环频率。另外在垂直注入的情况下也发现有类似的高频鱼骨：,相关的实验背景,理论研究进展以及本课题研究的意义将在第一章介绍。
但是目前对实验中出现双频鱼骨模的现象还没有相应的理论解释,本文的后半部分我们对俘获高能离子的解稳作用进行了研究。首先在第二章介绍了本文所涉及的理论和方法,包括粒子导心轨道理论,稳定性理论,等离子体的描述方法和线性扰动理论等。第三章在土豆轨道模型上计算了类EAST托卡马克中深俘获高能离子扰动分布函数的非绝热项产生的扰动能量,得到相应的色散关系,从而求出高能离子激发内扭曲模的共振频率。计算结果表明高能离子与内扭曲模发生共振作用能同时激发低频鱼骨和高频鱼骨,其中较低的频率接近高能离子的进动频率,而较高的频率接近高能离子的反弹频率。可见高能离子与内扭曲模发生共振作用能同时激发低频鱼骨和高频鱼骨,其中低频模的频率与L.Chen等人的结论一致,主要是由非绝热动能项的零阶分量引起的。而高频模则是我们在引入非绝热动能项的一阶分量的情况下发现的新的模,解释了在垂直注入产生俘获高能离子的情况下发现高频不稳定现象的原因。另外我们还发现,低频模的临界β值(磁压比)比较低,很容易被激发,所以在实验中经常能观察到低频模；而高频模的临界β值较高,实验中β值偏低时,就观察不到高频模。但是当中心高能离子的β值超过高频模的临界β时,实验中能观察到两种频率的鱼骨。最后在第四章对全文进行了总结。
Over the past decades of years, with the widely use of neutral beam injection (NBI) as a heating method in tokamak experiments, fishbone instability has been frequently observed, with the presence of energetic ions produced by both tangential and perpen-dicular NBI. The fishbone events are correlated with particle bursts corresponding to loss of energetic beam particles, reducing the beam heating efficiency.
With the experimental discovery, many theoretical models have been proposed to explain the mechanism of the fishbone oscillation. For low-frequency fishbone insta-bility, the current view is that a fishbone oscillation is an internal kink mode, a mode with dominant poloidal and toroidal wave numbers m=1 and n=1, destabilized by the resonant interaction with the energetic ions trapped inside the q=1 surface at a fre-quency around the toroidal precessional frequency of hot ions in the perpendicular injection case and with the circulating energetic ions arising from the effect of finite radial particle-orbit excursion in the tangential injection case. Besides the dominant modes with a low frequency, there still exist modes with higher frequencies in exper-imental observation. It has been proposed, that the instability observed in tangential neutral-beam-injection discharge is excited by the resonance between the internal kink and circulating hot ions at a frequency around the circulating frequency of hot ions. Such a high-frequency fishbone instability has also been found in the perpendicular injection case, accompanying the low-frequency one. The relative experimental back-ground, theoretical progress and importance of the work are introduced in the first chapter.
However, currently, there is no corresponding theoretical explanation for the double-frequency fishbone instability in the experimental observation. In the rest parts of this paper, we investigate the destabilization effect of the deeply-trapped energetic ions on the internal kink mode in EAST-like tokamak. Firstly, we introduce the relevant theory and methods, including the guiding-center orbit theory, instability, descriptions of plasmas, and the linearized perturbation theory, in the second chapter. Then in the next chapter, we calculate the nonadiabatic part of perturbed energy contributed by the trapped energetic ions in the EAST-like tokamak, using the potato-orbit model. With the dispersion relation, we can get the resonant frequency at which the hot ions interact with the internal kink mode. The result shows that the trapped energetic ions can desta-bilize the internal kink mode through resonance at both a low frequency comparable to the precessional frequency and a high frequency comparable to the bounce frequency of the hot ions. The low frequency mode, which is consistent with the conclusion of L. Chen et al, which is due to the zero-order component of the nonadiabatic distribution. The high frequency mode is a new mode found by keeping the first-order component of the nonadiabatic part, which explains the high-frequency fishbone instability during the perpendicular neutral beam injection observed in experiments. We also get the crit-ical central beta value of the energetic ions for the two frequencies. The critical beta value for the low-frequency mode is quite low, so that the corresponding fishbone can be easily excited and observed in the experiments. However, the high-frequency mode has a higher critical beta value, thus it is hard to observe the high-frequency fishbone if the beta value of hot ions is not high enough. If the beta value exceeds the critical beta value of the high-frequency mode, it is possible to observe double-frequency fishbone. The last chapter makes a conclusion over the whole thesis.
随着实验上的发现,人们提出很多理论模型来解释鱼骨振荡的机制。对低频鱼骨不稳定性,目前的观点认为鱼骨振荡的机制是内扭曲模与q=1而内的高能离子发生共振相互作用从而被解稳。其共振频率是在垂直注入的情况下接近俘获高能离子的环向进动频率,在切向注入的情况下接近通行粒子的粒子轨道径向漂移频率。除了占主导的低频模,实验观测中还发现了高频模的存在。对于高频模的解释是内扭曲模和由切向注入产生的通行高能离子共振激发出鱼骨不稳定性。其共振频率接近高能离子的循环频率。另外在垂直注入的情况下也发现有类似的高频鱼骨：,相关的实验背景,理论研究进展以及本课题研究的意义将在第一章介绍。
但是目前对实验中出现双频鱼骨模的现象还没有相应的理论解释,本文的后半部分我们对俘获高能离子的解稳作用进行了研究。首先在第二章介绍了本文所涉及的理论和方法,包括粒子导心轨道理论,稳定性理论,等离子体的描述方法和线性扰动理论等。第三章在土豆轨道模型上计算了类EAST托卡马克中深俘获高能离子扰动分布函数的非绝热项产生的扰动能量,得到相应的色散关系,从而求出高能离子激发内扭曲模的共振频率。计算结果表明高能离子与内扭曲模发生共振作用能同时激发低频鱼骨和高频鱼骨,其中较低的频率接近高能离子的进动频率,而较高的频率接近高能离子的反弹频率。可见高能离子与内扭曲模发生共振作用能同时激发低频鱼骨和高频鱼骨,其中低频模的频率与L.Chen等人的结论一致,主要是由非绝热动能项的零阶分量引起的。而高频模则是我们在引入非绝热动能项的一阶分量的情况下发现的新的模,解释了在垂直注入产生俘获高能离子的情况下发现高频不稳定现象的原因。另外我们还发现,低频模的临界β值(磁压比)比较低,很容易被激发,所以在实验中经常能观察到低频模；而高频模的临界β值较高,实验中β值偏低时,就观察不到高频模。但是当中心高能离子的β值超过高频模的临界β时,实验中能观察到两种频率的鱼骨。最后在第四章对全文进行了总结。
Over the past decades of years, with the widely use of neutral beam injection (NBI) as a heating method in tokamak experiments, fishbone instability has been frequently observed, with the presence of energetic ions produced by both tangential and perpen-dicular NBI. The fishbone events are correlated with particle bursts corresponding to loss of energetic beam particles, reducing the beam heating efficiency.
With the experimental discovery, many theoretical models have been proposed to explain the mechanism of the fishbone oscillation. For low-frequency fishbone insta-bility, the current view is that a fishbone oscillation is an internal kink mode, a mode with dominant poloidal and toroidal wave numbers m=1 and n=1, destabilized by the resonant interaction with the energetic ions trapped inside the q=1 surface at a fre-quency around the toroidal precessional frequency of hot ions in the perpendicular injection case and with the circulating energetic ions arising from the effect of finite radial particle-orbit excursion in the tangential injection case. Besides the dominant modes with a low frequency, there still exist modes with higher frequencies in exper-imental observation. It has been proposed, that the instability observed in tangential neutral-beam-injection discharge is excited by the resonance between the internal kink and circulating hot ions at a frequency around the circulating frequency of hot ions. Such a high-frequency fishbone instability has also been found in the perpendicular injection case, accompanying the low-frequency one. The relative experimental back-ground, theoretical progress and importance of the work are introduced in the first chapter.
However, currently, there is no corresponding theoretical explanation for the double-frequency fishbone instability in the experimental observation. In the rest parts of this paper, we investigate the destabilization effect of the deeply-trapped energetic ions on the internal kink mode in EAST-like tokamak. Firstly, we introduce the relevant theory and methods, including the guiding-center orbit theory, instability, descriptions of plasmas, and the linearized perturbation theory, in the second chapter. Then in the next chapter, we calculate the nonadiabatic part of perturbed energy contributed by the trapped energetic ions in the EAST-like tokamak, using the potato-orbit model. With the dispersion relation, we can get the resonant frequency at which the hot ions interact with the internal kink mode. The result shows that the trapped energetic ions can desta-bilize the internal kink mode through resonance at both a low frequency comparable to the precessional frequency and a high frequency comparable to the bounce frequency of the hot ions. The low frequency mode, which is consistent with the conclusion of L. Chen et al, which is due to the zero-order component of the nonadiabatic distribution. The high frequency mode is a new mode found by keeping the first-order component of the nonadiabatic part, which explains the high-frequency fishbone instability during the perpendicular neutral beam injection observed in experiments. We also get the crit-ical central beta value of the energetic ions for the two frequencies. The critical beta value for the low-frequency mode is quite low, so that the corresponding fishbone can be easily excited and observed in the experiments. However, the high-frequency mode has a higher critical beta value, thus it is hard to observe the high-frequency fishbone if the beta value of hot ions is not high enough. If the beta value exceeds the critical beta value of the high-frequency mode, it is possible to observe double-frequency fishbone. The last chapter makes a conclusion over the whole thesis.
引文
[1]J. Wesson. Tokamaks[M]. Oxford:Clarendon,1997.
[2]K. McGuire, R. Goldston, M. Bell et al. Study of High-Beta Magnetohydro-dynamic Modes and Fast-Ion Losses in PDX[J]. Phys. Rev. Lett.,1983,50(12): 891-895.
[3]W. W. Heidbrink, K. Bol, D. Buchenauer et al.. Tangential Neutral-Beam-Driven Instabilities in the Princeton Beta Experiment[J]. Phys. Rev. Lett.,1986,57(7): 835-838.
[4]W. W. Heidbrink, R. Kaita, H. Takahashi, G. Gammel, G. W. Hammett, and S. Kaye. Measurements of beam-ion confinement during tangential beam-driven in-stabilities in a bean tokamak experiment[J]. Phys. Fluids,1987,30(6):1839-1852.
[5]J. D. Strachan, B. Grek, W. Heidbrink, D. Johnson, S. Kaye, H. Kugel, B. LeBlanc, and K. McGuire. Studies of energetic ion confinement during fishbone events in PDX[J]. Nucl. Fusion,1985,25(8):863-880.
[6]L. Chen, R. B. White, and M. N. Rosenbluth. Excitation of Internal Kink Modes by Trapped Energetic Beam lons[J]. Phys. Rev. Lett.,1984,52(13):1122-1125.
[7]R. B. White, L. Chen, F. Romanelli, and R. Hay. Trapped particle destabilization of the internal kink mode[J]. Phys. Fluids,1985,28(1):278-286.
[8]B. Coppi and F. Porcelli. Theoretical Model of Fishbone Oscillations in Magneti-cally Confined Plasmas[J]. Phys. Rev. Lett.,1986,57(18):2272-2275.
[9]B. Coppi, S. Migliuolo, and F. Porcelli. Macroscopic plasma oscillation bursts (fishbones) resulting from high-energy populations[J]. Phys. Fluids,1988,31(6): 1630-1648.
[10]Y. Z. Zhang, H. L. Berk, and S. M. Mahajan. Effect of energetic trapped particles on the "ideal" internal kink mode[J]. Nucl. Fusion,1989,29(5):848-853.
[11]B. Coppi, P. Detragiache, S. Migliuolo, F. Pegoraro, and F. Porcelli. Quiescent window for global plasma modes[J]. Phys. Rev. Lett.,1989,63(25):2733-2736.
[12]R. Betti and J. P. Freidberg. Destabilization of the Internal Kink by Energetic Circulating Ions[J]. Phys. Rev. Lett.,1993,70(22):3428-3430.
[13]S. Wang. Destabilization of Internal Kink Modes at High Frequency by Energetic Circulating Ions[J]. Phy. Rev. Lett.,2001,86(23):5286-5288.
[14]S. Wang, T. Ozeki, and K. Tobita. Effects of Circulating Energetic Ions on Saw-tooth Oscillations[J]. Phys. Rev. Lett.,2002,88(10):105004.
[15]T. E. Stringer. Radial profile of a-particle heating in a tokamak[J]. Plasma Phys.,1974,16:651-659.
[16]V. Ya Goloborod'ko, Ya.I. Kolesnichenko, and V. A. Yavorskjj. Neoclassical the-ory of alpha-particle transport in the central region of a tokamak plasma[J]. Nucl. Fusion,1983,23(4):399-406.
[17]V. S. Marchenko. Fishbone mode in "potato" regime[J]. Phys. Plasmas,2002, 9(6):2847-2849.
[18]R. J. Hastie, Y. Chen, F. Ke, S. Cai, L. Chen. Energetic Particle Stabilization of m=1 internal kink mode in tokamks[J]. Chin. Phys. Lett.,1987,4(12):561-564.
[19]D. Zhou, S. Wang, and C. Zhang. Sawtooth stabilization by barely trapped ener-getic electrons[J]. Phys. Plasmas,2005,12:062512.
[20]Y. Sun, B. Wan, S. Wang, D. Zhou, L. Hu, and B. Shen. Excitation of internal kink mode by barely trapped suprathermal electrons[J]. Phys. Plasmas,2005,12: 092507.
[21]D. Zhou. High frequency fishbones excited by near perpendicular neutral beam injection[J]. Phys. Plasmas,2006,13:072511.
[22]胡希伟.等离子体理论基础[M].北京大学出版社,2006.
[23]Francis F. Chen.等离子体物理导论[M].科学出版社,1980.
[24]Robert J. Goldston, and Paul H. Rutherford. Introduction to Plasma Physics[M]. Institute of Physics:Publishing Bristol and Philadelphia,2000.
[25]T. M. Antonsen, Jr., and B. Lane. Kinetic equations for low frequency instabilities in inhomogeneous plasmas[J]. Phys. Fluids,1980,23(6):1205-1214.
[26]P. J. Catto, W. M. Tang, and D. E. Baldwin. Generalized gyrokinetics[J]. Plasma Phys.,1981,23(7):639-650.
[27]F. Porcelli, R. Stankiewicz, H. L. Berk and Y. Z. Zhang. Internal kink stabiliza-tion by high-energy ions with nonstandard orbits[J]. Phys. Fluids B,1992,4(10): 3017-3023.
[28]F. Porcelli, R. Stankiewicz, W. Kerner, and H. L. Berk. Solution of the drift-kinetic equation for global plasma modes and finite particle orbit widths[J]. Phys. Plasmas,1994,1(3):470-480.
[29]R. G. Littlejohn. Variational principles of guiding centre motion[J]. J. Plasma Phys.,1983,29:111-125.
[30]1TER Physics Expert Group on Disruptions, Plasma Control, and MHD. Chap-ter 3:MHD stability, operational limits and disruptions[J]. Nucl. Fusion,1999, 39(12):2251-2389.
[31]Y. Wan. Overview progress and future plan of EAST Project[A].21st IAEA Fu-sion Energy Conference, Chengdu,2006.
[32]金尚年,马永利.理论力学[M].高等教育出版社,2005.
[2]K. McGuire, R. Goldston, M. Bell et al. Study of High-Beta Magnetohydro-dynamic Modes and Fast-Ion Losses in PDX[J]. Phys. Rev. Lett.,1983,50(12): 891-895.
[3]W. W. Heidbrink, K. Bol, D. Buchenauer et al.. Tangential Neutral-Beam-Driven Instabilities in the Princeton Beta Experiment[J]. Phys. Rev. Lett.,1986,57(7): 835-838.
[4]W. W. Heidbrink, R. Kaita, H. Takahashi, G. Gammel, G. W. Hammett, and S. Kaye. Measurements of beam-ion confinement during tangential beam-driven in-stabilities in a bean tokamak experiment[J]. Phys. Fluids,1987,30(6):1839-1852.
[5]J. D. Strachan, B. Grek, W. Heidbrink, D. Johnson, S. Kaye, H. Kugel, B. LeBlanc, and K. McGuire. Studies of energetic ion confinement during fishbone events in PDX[J]. Nucl. Fusion,1985,25(8):863-880.
[6]L. Chen, R. B. White, and M. N. Rosenbluth. Excitation of Internal Kink Modes by Trapped Energetic Beam lons[J]. Phys. Rev. Lett.,1984,52(13):1122-1125.
[7]R. B. White, L. Chen, F. Romanelli, and R. Hay. Trapped particle destabilization of the internal kink mode[J]. Phys. Fluids,1985,28(1):278-286.
[8]B. Coppi and F. Porcelli. Theoretical Model of Fishbone Oscillations in Magneti-cally Confined Plasmas[J]. Phys. Rev. Lett.,1986,57(18):2272-2275.
[9]B. Coppi, S. Migliuolo, and F. Porcelli. Macroscopic plasma oscillation bursts (fishbones) resulting from high-energy populations[J]. Phys. Fluids,1988,31(6): 1630-1648.
[10]Y. Z. Zhang, H. L. Berk, and S. M. Mahajan. Effect of energetic trapped particles on the "ideal" internal kink mode[J]. Nucl. Fusion,1989,29(5):848-853.
[11]B. Coppi, P. Detragiache, S. Migliuolo, F. Pegoraro, and F. Porcelli. Quiescent window for global plasma modes[J]. Phys. Rev. Lett.,1989,63(25):2733-2736.
[12]R. Betti and J. P. Freidberg. Destabilization of the Internal Kink by Energetic Circulating Ions[J]. Phys. Rev. Lett.,1993,70(22):3428-3430.
[13]S. Wang. Destabilization of Internal Kink Modes at High Frequency by Energetic Circulating Ions[J]. Phy. Rev. Lett.,2001,86(23):5286-5288.
[14]S. Wang, T. Ozeki, and K. Tobita. Effects of Circulating Energetic Ions on Saw-tooth Oscillations[J]. Phys. Rev. Lett.,2002,88(10):105004.
[15]T. E. Stringer. Radial profile of a-particle heating in a tokamak[J]. Plasma Phys.,1974,16:651-659.
[16]V. Ya Goloborod'ko, Ya.I. Kolesnichenko, and V. A. Yavorskjj. Neoclassical the-ory of alpha-particle transport in the central region of a tokamak plasma[J]. Nucl. Fusion,1983,23(4):399-406.
[17]V. S. Marchenko. Fishbone mode in "potato" regime[J]. Phys. Plasmas,2002, 9(6):2847-2849.
[18]R. J. Hastie, Y. Chen, F. Ke, S. Cai, L. Chen. Energetic Particle Stabilization of m=1 internal kink mode in tokamks[J]. Chin. Phys. Lett.,1987,4(12):561-564.
[19]D. Zhou, S. Wang, and C. Zhang. Sawtooth stabilization by barely trapped ener-getic electrons[J]. Phys. Plasmas,2005,12:062512.
[20]Y. Sun, B. Wan, S. Wang, D. Zhou, L. Hu, and B. Shen. Excitation of internal kink mode by barely trapped suprathermal electrons[J]. Phys. Plasmas,2005,12: 092507.
[21]D. Zhou. High frequency fishbones excited by near perpendicular neutral beam injection[J]. Phys. Plasmas,2006,13:072511.
[22]胡希伟.等离子体理论基础[M].北京大学出版社,2006.
[23]Francis F. Chen.等离子体物理导论[M].科学出版社,1980.
[24]Robert J. Goldston, and Paul H. Rutherford. Introduction to Plasma Physics[M]. Institute of Physics:Publishing Bristol and Philadelphia,2000.
[25]T. M. Antonsen, Jr., and B. Lane. Kinetic equations for low frequency instabilities in inhomogeneous plasmas[J]. Phys. Fluids,1980,23(6):1205-1214.
[26]P. J. Catto, W. M. Tang, and D. E. Baldwin. Generalized gyrokinetics[J]. Plasma Phys.,1981,23(7):639-650.
[27]F. Porcelli, R. Stankiewicz, H. L. Berk and Y. Z. Zhang. Internal kink stabiliza-tion by high-energy ions with nonstandard orbits[J]. Phys. Fluids B,1992,4(10): 3017-3023.
[28]F. Porcelli, R. Stankiewicz, W. Kerner, and H. L. Berk. Solution of the drift-kinetic equation for global plasma modes and finite particle orbit widths[J]. Phys. Plasmas,1994,1(3):470-480.
[29]R. G. Littlejohn. Variational principles of guiding centre motion[J]. J. Plasma Phys.,1983,29:111-125.
[30]1TER Physics Expert Group on Disruptions, Plasma Control, and MHD. Chap-ter 3:MHD stability, operational limits and disruptions[J]. Nucl. Fusion,1999, 39(12):2251-2389.
[31]Y. Wan. Overview progress and future plan of EAST Project[A].21st IAEA Fu-sion Energy Conference, Chengdu,2006.
[32]金尚年,马永利.理论力学[M].高等教育出版社,2005.