摘要
测地声模(geodesic acoustic mode, GAM)是环形等离子体中特有的一支静电振荡,其模结构在环向上对称,并在极向近似对称。测地声模的物理机制是由磁场曲率的测地线分量引起的漂移和极化漂移的平衡。由于测地声模可以通过非线性相互作用调节漂移波湍流,从而可以成为一种潜在的对聚变等离子体的输运进行调制的方法而受到了广泛的关注。在本文中,我们用动理学方法理论的研究了测地声模的各种性质,具体内容包括：
1、我们首先应用流体力学方法推导了GAM的色散关系；并指出,由于径向的等离子体参数的不均匀性,GAM的频谱构成一个连续谱。当考虑动理学效应,如离子的有限拉莫轨道半径效应(FLR)和有限漂移轨道半径效应(FOW)时,GAM的连续谱的奇异性可以被去除。我们用解析和数值的方法研究了由于GAM的连续谱导致的各种物理现象。首先,在不存在外加源场(或汇)时,我们考虑了GAM初始的扰动随时间的演化。在不考虑动理学效应时,GAM的初始扰动会逐渐产生短波结构,并由于相混而与时间成反比衰减。而考虑动理学效应时,初始扰动波包会模转换为短波长的动理学GAM(KGAM)而在径向向外传播。在考虑一个外源(如天线)对GAM的激发时,我们发现：在不考虑动理学效应时,在GAM连续谱频率与外源的频率一致的地方,GAM的模结构具有奇异性,外源可以被等离子体共振吸收,其吸收率与GAM的连续谱的斜率成反比。当考虑动理学效应时,在奇异层附近,外源场被共振模转换为KGAM,并向外传播,其模结构由Airy函数描述。
2、我们使用回旋动理学方法,系统的研究了GAM/KGAM的色散关系,并得到了GAM的包含了FOW/FLR和平行电场修正的模结构和适用于很大波长范围的GAM的色散关系。在大漂移轨道的共振离子的假设下,我们研究了离子的径向磁漂移与GAM的共振导致的朗道阻尼率。我们的公式,与前人在小漂移轨道共振离子假设下的结论结合,可以给出GAM/KGAM在整个参数范围的阻尼率。我们得到的大漂移轨道共振离子极限下的GAM的阻尼率的解析表达式,在适用范围内与数值结果及TEMPEST的模拟结果吻合很好。
3、我们非扰的研究了在GAM连续谱存在的情况下,由高能粒子诱发测地声模(EGAM)的非局域理论;在小漂移轨道近似下,得到了由与高能粒子沿磁力线的通行运动的共振相互作用激发的EGAM的非局域色散关系。在对高能粒子采用一个单一抛射角的慢化分布时,得到了局域的EGAM的色散关系,并解析的给出了EGAM不稳定性的抛射角的阈值。当考虑高能粒子和主成份离子的FLR/FOW时,我们可以得到EGAM的非局域的模方程。当高能粒子的密度在径向高度集中,而远离GAM的连续谱的奇异层时,EGAM与GAM的连续谱的耦合非常弱。在高能粒子集中的区域内部,FLR/FOW由高能粒子主导,我们可以得到EGAM的本征模；在远离高能粒子的区域,主成份离子的FLR/FOW主导,EGAM可以与GAM的连续谱耦合而模转换为KGAM向外传播。通过对内外区进行渐进匹配,我们得到了EGAM的非局域色散关系。我们的解析和数值结果显示,在高能粒子的密度在径向高度集中且远离GAM的奇异层时,EGAM被“自捕获”在高能粒子驱动最强的位置,并通过隧道效应耦合到向外传播的KGAM。这也对EGAM提供了一种移动性的能量损失机制,从而对EGAM的激发产生了一个非局域的阈值。我们的数值计算显示当增强与GAM的连续谱的耦合时,EGAM的激发的阈值变大,并数值的给出了EGAM径向的全区域的本征模结构。
最后,我们总结全文并对未来工作提出展望。
Geodesic Acoustic Modes (GAMs) are toroidally symmetric normal modes unique to toroidal plasmas, and the mode structure is also nearly poloidally symmetric. Their existence is associated with the charge separation effect, due to ion radial geodesic magnetic curvature drift and polarization drift. GAMs have received much attention in magnetic fusion plasma due to their potentially important roles in regulating drift waves, and, hence, transports via nonlinear interactions. In this dissertation, we employ the kinetic-theoretic approaches and study various properties of GAM.
First, we derive the dispersion relation of GAM with fluid equations, and show that GAM constitutes a continuous spectrum due to inherent radial inhomogeneity of fusion plasmas. When kinetic effects due to, e.g., finite ion Larmor radii(FLR) and finite guiding-center drift-orbit-width(FOW) are included, the singularity in GAM con-tinuous spectrum can be removed. We study analytically and numerically various phe-nomenons associated with GAM continuous spectrum. We show that, in the absence of an external source, initial wave packet of GAM will spontaneously generate short-wavelength structures, and decay in time as t-1 due to phase-mixing. When kinetic ef-fects are considered, the initial pulse can be mode-converted to short wavelength kinetic GAM(KGAM), and propagate radially outward. When there is an external source, we show that, at the radial location where GAM continuum frequency matches the driving frequency of the source, the mode structure of GAM becomes singular and the external source can be resonantly absorbed by the plasma. Including kinetic effects, the singu-larity is removed and the external source is then resonantly mode-converted to outgoing KGAM.
Second, we employ gyrokinetic equation and systematically study the dispersion relation of GAM for a wide range of wavelengths; including effects of FLR/FOW and parallel electric field. In the limit when the wavelength of GAM is smaller than the drift orbit width of resonant ions, we study the collisionless damping of GAM due to reso-nant interaction with the ion radial magnetic drift. Our analytical formula, combined with previous analytical works in the small drift orbit limit, thus, provides collision-less damping rate of GAM over a broad range of parameters. Our analytical formula, furthermore, agrees well with TEMPEST simulation in its validity regime.
Third, we investigate nonperturbatively the nonlocal theory of energetic-particle-induced GAM(EGAM) excited via transit resonances with energetic particles, taking into account the coupling to GAM continuous spectrum and the nonlocal dispersion relation of EGAM excited by the transit resonance of energetic particles is derived. We first derive the local dispersion relation of EGAM assuming a single pitch-angle slowing down energetic particle distribution function, and give the critical pitch angle for the local EGAM instability. Including the FLR/FOW terms of both bulk thermal ions and energetic particles, then leads to the eigenmode equation of EGAM. Con-sidering an energetic particle beam radially localized away from the singular layer of GAM continuous spectrum, the FLR/FOW term is then dominated by energetic parti-cles inside the localization domain of energetic particles, and we obtain the bounded EGAM eigenmode. Away from the localized energetic particles, EGAM then couples to the background GAM continuous spectrum and mode-converts to radially outgoing KGAM. Asymptotically matching the solutions in the inner and outer regions, we then drive the corresponding global dispersion relation of EGAM. We show both analyti-cally and numerically that, when the energetic particle beam is localized away from the singular layer of GAM continuous spectrum, the mode is radially self-trapped where the energetic-particle drive is strongest. There exist, moreover, an exponentially small tunneling coupling to outgoing KGAM, which leads to convective damping of EGAM and hence, a finite threshold for EGAM excitation. We show numerically that, as the-oretically predicted, the threshold value increases with enhance coupling to the GAM continuous spectrum.The corresponding global mode structure of EGAM is also given numerically.
Finally, conclusions and suggestions for possible future works on this important topic are also given.
引文
[1]T. J. Dolan, Fusion Reasearch, Pergamon Press (2000).
[2]J. D. Lawson. Some Criteria for a Power Producing Thermonuclear Reactor. Proceedings of the Physical Society. Section B 70 (1957)6.
[3]A. H. Boozer. Physics of magnetically confined plasmas. Rev. Mod. Phys.76 (2004)1071.
[4]J. Wesson, Tokamaks, Clarendon Press, second edition (1997).
[5]F. Wagner, G. Becker, K. Behringer et al. Regime of Improved Confinement and High Beta in Neutral-Beam-Heated Divertor Discharges of the ASDEX Tokamak. Phys. Rev. Lett.49 (1982)1408.
[6]Z. Lin, T. S. Hahm, W. W. Lee et al. Turbulent transport reduction by zonal flows:Massively parallel simulations. Science 281(1998)1835.
[7]P. H. Diamond, S.-I. Itoh, K. Itoh et al. Zonal flows in plasma-a review. Plasma Phys. Control. Fusion 47(2005)R35.
[8]L. Chen, Z. Lin, and R. White. Excitation of zonal flow by drift waves in toroidal plasmas. Physics of Plasmas,7(2000):3129.
[9]L. Chen, R. B. White, and F. Zonca. Zonal-flow dynamics and size scaling of anomalous transport. Phys. Rev. Lett.,92(2004)075004.
[10]L. Chen and F. Zonca. Nonlinear equilibria, stability and generation of zonal structures in toroidal plasmas. Nuclear Fusion,47(2007)886.
[11]F. Zonca, R. B. White, and L. Chen. Nonlinear paradigm for drift wave-zonal flow interplay: Coherence, chaos, and turbulence, Phys. Plasma,11(2004)2488.
[12]M. N. Rosenbluth and F. L. Hinton. Poloidal Flow Driven by Ion-Temperature-Gradient Tur-bulence in Tokamaks. Phys. Rev. Lett.80(1998)724.
[13]F. L. Hinton and M. N. Rosenbluth. Dynamics of axisymmetric ExB and poloidal flows in tokamaks. Plasma Phys. Control. Fusion 41 (1999)A653.
[14]H. Y. W. Tsui, K. Rypdal, Ch. P. Ritz et al, Coherent nonlinear coupling between a long-wavelength mode and small-scale turbulence in the TEXT tokamak, Phys. Rev. Lett. 70((1993))2565.
[15]P. M. Schoch, K. A. Connor, D. R. Demers et al. Zonal flow measurements using a heavy ion beam probe, Rev. Sci. Instrum.74(2003)184.
[16]A. V. Melnikov, L. G. Eliseev, A. V. Gudozhnik et al, Investigation of the plasma potential oscillations in the range of geodesic acoustic mode frequencies by heavy ion beam probing in tokamaks, Czechoslovak Journal of Physics,55(2005)349.
[17]A. Fujisawa, T. Ido, A. Shimizu et al. Experimental progress on zonal flow physics in toroidal plasmas. Nuclear Fusion 47(2007) S718.
[18]G. D. Conway, B. Scott, J. Schirmer et al. Direct measurement of zonal flows and geodesic acoustic mode oscillations in ASDEX Upgrade using Doppler reflectometry. Plasma Phys. Control. Fusion 47(2005)1165.
[19]A. Bencze, M. Berta, S. Zoletnik et al, Observation of Zonal flow-like structures using the autocorrelation-width technique. Plasma Phys. Control. Fusion,48(2006)S137.
[20]A. Fujisawa, K. Itoh, H. Iguchi et al, Identification of zonal flows in a toroidal plasma. Phys. Rev. Lett.,93(2004)165002.
[21]A. Fujisawa, A. Shimizu, H. Nakano et al, Turbulence and transport characteristics of a barrier in a toroidal plasma. Plasma Phys. Control. Fusion,48 (2006)S205.
[22]G. S. Xu, B. N. Wan, M. Song et al. Direct measurement of poloidal longwavelength ExB flows in the HT-7 tokamak. Phys. Rev. Lett.,91(2003),125001.
[23]T. Ido, Y. Miura, K. Hoshino et al. Observation of the interaction between the geodesic acoustic mode and ambient fluctuation in the JFT-2M tokamak. Nucl. Fusion 46(2006)512.
[24]T. Ido, Y. Miura, K. Kamiya et al. Geodesic-acoustic-mode in JFT-2M tokamak plasmas. Plasma Phys. Control. Fusion 48 (2006)S41.
[25]Y. Hamada, A. Nishizawa, T. Ido et al, Zonal flows in the geodesic acoustic mode frequency range in the JIPP T-IIU tokamak plasmas, Nucl. Fusion,45(2005)81.
[26]A. V. Melnikov, V. A. Vershkov, L. G. Eliseev et al. Investigation of geodesic acoustic mode oscillations in the T-10 tokamak. Plasma Phys. Control. Fusion 48 (2006)S87.
[27]K. J. Zhao, T. Lan, J. Q. Dong et al. Toroidal Symmetry of the Geodesic Acoustic Mode Zonal Flow in a Tokamak Plasma. Phys. Rev. Lett.96(2006)255004.
[28]T. Lan, A. D. Liu, C. X. Yu et al. Spectral features of the geodesic acoustic mode and its interaction with turbulence in a tokamak plasma. Phys. Plasmas,15 (2008)056105.
[29]T. Lan, A. D. Liu, C. X. Yu et al. Spectral characteristics of geodesic acoustic mode in the HL-2A tokamak. Plasma Phys. Control. Fusion,50(2008) 045002.
[30]Z. Gao, L. Peng, P. Wang et al,Plasma elongation effects on temperature gradient driven insta-bilities and geodesic acoustic modes, Nucl. Fusion,49(2009)045014.
[31]Z. Gao,P. Wang and H. Sanuki,Plasma shaping effects on the geodesic acoustic mode in toroidally axisymmetric plasmas, Phys. Plasma,15(2008)074502.
[32]Z. Gao, K. Itoh, H. Sanuki and J. Q. Dong, Eigenmode analysis of geodesic acoustic modes, Phys. Plasma,15(2008)072511.
[33]D. Zhou, Electromagnetic geodesic acoustic modes in tokamak plasmas, Phys. Plasma, 14(2007)104502.
[34]Z. Gao, K Itoh, H. Sanuki, and J. Q. Dong, Eigenmode Analysis of Geodesic Acoustic Mode and Zonal Flow, in Proceedings of the 21st IAEA Fusion Energy Conference, Chengdu, China, 16-21 October 2006 (IAEA, Vienna,2007), IAEA-CN-149/TH/P2-5.
[35]H. Zhang et al, submitted to 23rd IAEA FEC.
[36]N. Chakrabarti, R. Singh, P. K. Kaw and P. N. Guzdar, Nonlinear excitation of geodesic acous-tic modes by drift waves, Phys. Plasma,14(2007) 052308.
[37]H. Sugama and T. H. Watanabe, Collisionless damping of geodesic acoustic modes, J. Plasma Physics,72(2006)825.
[38]H. Sugama and T. H. Watanabe, Erratum:"Collisionless damping of geodesic acoustic modes"J. Plasma Physics,74(2006) 139.
[39]X. Xu et al., TEMPEST simulations of collisionless damping of GAM and neoclassical resid-ual in edge plasma pedestal, Phys. Rev. Lett.100(2008) 215001.
[40]Edge gyrokinetic theory and continuum simulations,X. Xu, Z. Xiong,M. Dorr et al, Nucl. Fusion, bf47(2007)809.
[41]X. Xu, Neoclassical simulation of tokamak plasmas using the continuum gyrokinetic code TEMPEST, Phys. Rev. E,78(2008)016406.
[42]G. R. Mckee et al.,Structure and scaling properties of the geodesic acoustic mode,Plasma Phys. Control. Fusion,45(2003)A477.
[43]A. Hasegawa and L. Chen, Kinetic processes in plasma heating by resonant mode conversion of Alfven wave,Phys. Fluids,19(1976) 1924.
[44]L. Chen, Theory of magnetohydrodynamic instabilities excited by energetic particles in toka-maks, Phys. Plasmas,1 (1994) 1519.
[45]H. Grad, Phys. Today, Plasmas,22 issue No.12 (1969) 34.
[46]A. Hasegawa and L. Chen,Plasma Heating by Alfven-Wave Phase Mixing, Phys. Rev. Lett., 32 (1974)454.
[47]L. Chen and A. Hasegawa, Plasma heating by spatial resonance of Alfven wave, Phys. Fluids 17(1974)1399.
[48]F. Zonca and L. Chen, Resonant damping of toroidicity-induced shear-Alfv'en eigenmodes in tokamaks, Physical Review Letters, Phys. Rev. Lett.,68(1992)592.
[49]N. Winsor, J. Johnson, and J. M. Dawson, Geodesic Acoustic Waves in Hydromagnetic Sys-tems, Phys. Fluids,11 (1968) 2448.
[50]X. Garbet et al., in Theory of Fusion Plasmas, edited by Connor J. W., Sauter O. and Sindoni E., Vol. CP871 (AIP, Woodsbury, NY) 2006, p.342.
[51]H. L. Berk, C.J. Boswell, D. Borba et al.,Explanation of the JET n= 0 chirping mode, Nucl. Fusion 46 (2006) S888.
[52]R. Nazikian et al., Intense Geodesic Acousticlike Modes Driven by Suprathermal Ions in a Tokamak Plasma, Phys. Rev. Lett.,101 (2008) 185001.
[53]G. Fu, Energetic-Particle-Induced Geodesic Acoustic Mode, Phys. Rev. Lett.,101 (2008) 185002.
[54]V. B. Lebedev et al., Plateau regime dynamics of the relaxation of poloidal rotation in tokamak plasmas, Phys. Plasmas,3 (1996) 3023.
[55]S. V. Novakovskii et al., The radial electric field dynamics in the neoclassical plasmas, Phys. Plasmas,4(1997)4272.
[56]胡希伟著,等离子体理论基础,北京大学出版社(2006)。
[57]L. Chen, Waves and instabilities in Plasmas, World scientific Press(1987).
[58]H. Sugama and T.-H. Watanabe, Collisionless damping of geodesic acoustic modes in Helical systems, Phys. Plasmas,13 (2006) 012501.
[59]Z. Gao, K. Itoh,H. Sanuki and J. Q. Dong,Multiple eigenmodes of geodesic acoustic mode in collisionless plasmas, Phys. Plasmas,13 (2006) 100702.
[60]L. Chen and F. Zonca,Theory of Alfven waves and energetic particle physics in burning plas-mas, Nucl. Fusion 47(2007) S727.
[61]F. Zonca and L. Chen.Radial structures and nonlinear excitation of Geodesic Acoustic Modes, Europhys. Lett.,83 (2008) 35001.
[62]F. Zonca, L. Chen and Z. Qiu.Kinetic theory of Geodesic Acoustic Modes:radial structures and nonlinear excitations, Proceedings of the 22nd IAEA Fusion Energy Conference, ISBN nn-n-nnnnnn-n/ISSN nnnn-nnnn, CD-ROM file TH/P3-7.
[63]F. Zonca and L. Chen, Destabilization of energetic particle modes by localized particle sources, Phys. Plasmas,7 (2000) 4600.
[64]P. H. Rutherford and E. A. Frieman, Drift instabilities in general magnetic field configurations, Phys. Fluids 11 (1968) 569.
[65]J. B. Taylor and R. J. Hastie, Stability of general plasma equilibrium, Plasma Phys.10 (1968) 479.
[66]Z. Qiu, L. Chen and F. Zonca, Collisionless damping of short wavelength geodesic acoustic modes, Plasma Phys. Control. Fusion,51 (2009) 012001.
[67]F. Zonca, L. Chen, R. A. Santoro,Kinetic theory of low-frequency Alfven modes in tokamaks, Plasma Phys. Control. Fusion,38 (1996) 2011.
[68]F. Zonca, L. Chen, R. A. Santoro and J.Q. Dong, Existence of discrete modes in an unstable shear Alfven continuous spectrum, Plasma Phys. Control. Fusion,40 (1998) 2009.
[69]F. Zonca, L. Chen, J.Q. Dong and R. A. Santoro,Existence of ion temperature gradient driven shear Alfve'n instabilities in tokamaks, Phys. Plasmas,6 (1999) 1917
[70]C. Nguyen, X. Garbet and A. I. Smolyakov, Variational derivation of the dispersion relation of kinetic coherent modes in the acoustic frequency range in tokamaks, Phys. Plasmas 15 (2008) 112502.
[71]R. White, The theory of Toroidally Confined Plasmas, Imperial College Press (2001), P.331.
[72]T. H. Stix, Waves in Plasmas, AIP Press(1992), p.343.
[73]R. Nazikian, private communication,2009.
[74]F. Zonca and L. Chen,Resonant and non-resonant particle dynamics in Alfven mode excita-tions, Plasma Phys. Control. Fusion 48 (2006) 537.
[75]X. Wang, F. Zonca, L. Chen et al, Submitted to 23rd IAEA FEC.
[76]F. Zonca, private communication(2009).
[77]Y. Hamadaa, T.Watari, A. Nishizawa et al, GAM generation by drift waves in JIPP T-11U tokamak core plasmas, Nucl. Fusion 50 (2010) 025001.
[78]T. Zhou et al, presented at APS meeting, division of plamsa physics,2008.
[79]F. F. Chen, Introduction to plamsa physics, NY, Plenum Press,1984.
[80]翟侃等,中国科学(A),27(1997)89.
[81]L. Chen, Waves and instabilities in Plasmas, world scientific publications,1987.
