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The axisymmetric toroidal electrostatic mode discussed in this paper refers collectively to the nearly ideal electrostatic fluid mode with zero toroidal mode number in magnetically confined toroidal plasmas like tokamak, including geodesic acoustic mode, sound waves and the so-called nearly zero-frequency zonal flow. Use is made of cold ion fluid model in the toroidal coordinate system with a circular cross section to develop the theory of parametric excitation for the three above mentioned modes systematically to the first order of inverse large aspect ratio, which ends up with the four following observations: (1) The density zonal flow is only associated with the excitation of the first harmonic cosine sound wave and is independent of the potential zonal flow. (2) The geodesic acoustic mode is the high frequency branch of the dispersion in the form of coupling between the first harmonic sine sound wave and the nearly zero-frequency zonal flow due to geodesic curvature, while the low frequency branch of the same dispersion is identified to be the ‘toroidally modified nearly zero-frequency zonal flow’. (3) Only a weak coupling exists between the second harmonic sine sound wave and the nearly zero-frequency zonal flow. (4) All cosine sound waves and sine sound waves beyond the second harmonic are decoupled to the nearly zero-frequency zonal flow. A Gaussian type of drift wave energy spectrum with only a few parameters is introduced for calculation. Emphasis is laid on the effects resulting from the finite radial spectrum width such as double Landau-singularity, which reveal a significant modification to the δ -spectrum, thus resulting in serious restriction to the parametric excitation of geodesic acoustic mode and nearly zero-frequency zonal flow. Also discussed is the possibility of excitation of density zonal flow in the high q region. Numerical results are presented graphically and discussed in the reasonable physical regime. It is indicated that the geodesic acoustic mode and the nearly zero-frequency zonal flow cannot be parametrically excited at the same radii, and that if the geodesic acoustic mode is parametrically excited, the density zonal flow is expectedly to be observed.
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Keywords:
- geodesic acoustic mode /
- zero-frequency zonal flow /
- sound wave /
- tokamak
[1] Diamond P H, Itoh S I, Itoh K, Hahm T S 2005 Plasma Phys. Control. Fusion 47 R35
[2] Fujisawa A 2009 Nucl. Fusion 49 013001
[3] Itoh K, Itoh S I, Diamond P H 2006 Phys. Plasmas 13 055502
[4] Smolyakov A I, Diamond P H, Shevchenko V I 2000 Phys. Plasmas 7 1349
[5] Chakrabarti N, Singh R, Kaw P K, Guzdar P N 2007 Phys. Plasmas 14 052308
[6] Hillesheim J C, Peebles W A, Carter T A, Schmitz L, Rhodes T L 2012 Phys. Plasmas 19 022301
[7] Conway G D, Angioni C, Ryter F, Sauter P, Vicente J, the ASDEX Up-grade Team 2011 Phys. Rev. Lett. 106 065001
[8] McKee G R, Gohil P, Schlossberg D J, Boedo J A, Burrell K H, deGrassie J S, Groebner R J, Moyer R A, Petty C C, Rhodes T L, Schmitz L, Shafer M W, Solomon W M, Umansky M, Wang G, White A E, Xu X 2009 Nucl. Fusion 49 115016
[9] Zhang Y Z, Xie T, Mahajan S M 2012 Phys. Plasmas 19 020701
[10] Gao Z 2013 Phys. Plasmas 20 032501
[11] Guo W, Wang S, Li J G 2010 Phys. Plasmas 17 112510
[12] Qiu Z Y 2010 Ph. D. Dissertation (Hefei: University of Science and Technology of China) (in Chinese) [仇志勇 2010 博士学位论文(合肥: 中国科学技术大学)]
[13] Hong W Y, Yan L W, Zhao K J, Dong J Q, Cheng J, Qian J, Luo C W, Xu Z Y, Huang Y, Yang Q W, Lan T, Yu C X, Liu A D 2008 Acta Phys. Sin. 57 962 (in Chinese) [洪文玉, 严龙文, 赵开君, 董家齐, 程均, 钱俊, 罗萃文, 徐征宇, 黄渊, 杨青巍, 兰涛, 俞昌旋, 刘阿娣 2008 57 962]
[14] Peng X D, Qiu X M, Lu H L, Wang S J 2009 Acta Phys. Sin. 58 6387 (in Chinese) [彭晓东, 邱孝明, 陆赫林, 王顺金 2009 58 6387]
[15] Lan T, Liu A D, Yu C X, Yan L W, Hong W Y, Zhao K J, Dong J Q, Qian J, Cheng J, Yu D L, Yang Q W 2008 Plasma Phys. Control. Fusion 50 045002
[16] Zhao H L, Lan T, Liu A D, Kong D F, Xie J L, Liu W D, Yu C X, Zhang W, Chang J F, Wan B N, Li J G 2010 Plasma Sci. Technol. 12 262
[17] Kong D F, Liu A D, Lan T, Zhao H L, Sheng H G, Xu G S, Zhang W, Wan B N, Li J G, Chen R, Xie J L, Li H, Liu W D, Yu C X 2013 Nucl. Fusion 53 113008
[18] Kong D F, Liu A D, Lan T, Cui Z Y, Yu D L, Yan L W, Zhao H L, Sheng H G, Chen R, Xie J L, Li H, Liu W D, Yu C X, Hong W Y, Cheng J, Zhao K J, Dong J Q, Duan X R 2013 Plasma Phys. 53 123006
[19] Hasegawa A, Mima K 1978 Phys. Fluids 12 87
[20] Zhang Y Z, Xie T 2013 Nucl. Fusion & Plasma Phys. 33 1 (in Chinese) [章扬忠, 谢涛 2013 核聚变与等离子体物理 33 1]
[21] Braginskii S I (edited by Leontovich M A) 1965 Reviews of Plasma Physics 1 (New York: Consultants Bureau) pp205–311
[22] Abramowitz M, Stegun I 1965 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Publications) 20.2.27
[23] Winsor N, Johnson J, Dawson J 1968 Phys. Fluids 11 2448
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[1] Diamond P H, Itoh S I, Itoh K, Hahm T S 2005 Plasma Phys. Control. Fusion 47 R35
[2] Fujisawa A 2009 Nucl. Fusion 49 013001
[3] Itoh K, Itoh S I, Diamond P H 2006 Phys. Plasmas 13 055502
[4] Smolyakov A I, Diamond P H, Shevchenko V I 2000 Phys. Plasmas 7 1349
[5] Chakrabarti N, Singh R, Kaw P K, Guzdar P N 2007 Phys. Plasmas 14 052308
[6] Hillesheim J C, Peebles W A, Carter T A, Schmitz L, Rhodes T L 2012 Phys. Plasmas 19 022301
[7] Conway G D, Angioni C, Ryter F, Sauter P, Vicente J, the ASDEX Up-grade Team 2011 Phys. Rev. Lett. 106 065001
[8] McKee G R, Gohil P, Schlossberg D J, Boedo J A, Burrell K H, deGrassie J S, Groebner R J, Moyer R A, Petty C C, Rhodes T L, Schmitz L, Shafer M W, Solomon W M, Umansky M, Wang G, White A E, Xu X 2009 Nucl. Fusion 49 115016
[9] Zhang Y Z, Xie T, Mahajan S M 2012 Phys. Plasmas 19 020701
[10] Gao Z 2013 Phys. Plasmas 20 032501
[11] Guo W, Wang S, Li J G 2010 Phys. Plasmas 17 112510
[12] Qiu Z Y 2010 Ph. D. Dissertation (Hefei: University of Science and Technology of China) (in Chinese) [仇志勇 2010 博士学位论文(合肥: 中国科学技术大学)]
[13] Hong W Y, Yan L W, Zhao K J, Dong J Q, Cheng J, Qian J, Luo C W, Xu Z Y, Huang Y, Yang Q W, Lan T, Yu C X, Liu A D 2008 Acta Phys. Sin. 57 962 (in Chinese) [洪文玉, 严龙文, 赵开君, 董家齐, 程均, 钱俊, 罗萃文, 徐征宇, 黄渊, 杨青巍, 兰涛, 俞昌旋, 刘阿娣 2008 57 962]
[14] Peng X D, Qiu X M, Lu H L, Wang S J 2009 Acta Phys. Sin. 58 6387 (in Chinese) [彭晓东, 邱孝明, 陆赫林, 王顺金 2009 58 6387]
[15] Lan T, Liu A D, Yu C X, Yan L W, Hong W Y, Zhao K J, Dong J Q, Qian J, Cheng J, Yu D L, Yang Q W 2008 Plasma Phys. Control. Fusion 50 045002
[16] Zhao H L, Lan T, Liu A D, Kong D F, Xie J L, Liu W D, Yu C X, Zhang W, Chang J F, Wan B N, Li J G 2010 Plasma Sci. Technol. 12 262
[17] Kong D F, Liu A D, Lan T, Zhao H L, Sheng H G, Xu G S, Zhang W, Wan B N, Li J G, Chen R, Xie J L, Li H, Liu W D, Yu C X 2013 Nucl. Fusion 53 113008
[18] Kong D F, Liu A D, Lan T, Cui Z Y, Yu D L, Yan L W, Zhao H L, Sheng H G, Chen R, Xie J L, Li H, Liu W D, Yu C X, Hong W Y, Cheng J, Zhao K J, Dong J Q, Duan X R 2013 Plasma Phys. 53 123006
[19] Hasegawa A, Mima K 1978 Phys. Fluids 12 87
[20] Zhang Y Z, Xie T 2013 Nucl. Fusion & Plasma Phys. 33 1 (in Chinese) [章扬忠, 谢涛 2013 核聚变与等离子体物理 33 1]
[21] Braginskii S I (edited by Leontovich M A) 1965 Reviews of Plasma Physics 1 (New York: Consultants Bureau) pp205–311
[22] Abramowitz M, Stegun I 1965 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Publications) 20.2.27
[23] Winsor N, Johnson J, Dawson J 1968 Phys. Fluids 11 2448
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