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按照章等[Zhang Y Z, Liu Z Y, Mahajan S M, Xie T, Liu J
2017 Phys. Plasmas 24 122304 ]发展的漂移波-带状流理论, 将多重尺度导数展开法应用到电子漂移动理学方程, 零级为描述微观尺度捕获电子模的线性本征模方程, 一级为介观尺度受带状流调制的捕获电子模的包络方程. 其中线性本征模方程已经在谢等[Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y2019 Phys. Plasmas 26 022503 ]的研究中被求解, 利用该文得到的捕获电子模的本征值和二维模式结构计算包络方程中的群速度. 径向群速度由托卡马克磁场的测地曲率贡献, 极向群速度来自逆磁漂移速度和法向曲率, 它们仅是极向角的函数, 后者给出极向角到时间的映射. 径向群速度作为时间的函数, 其周期在毫秒量级, 具有快速过零的特征. 这为研究捕获电子模驱动带状流提供了充实的理论基础.The multiple scale derivative expansion method is used to manipulate the electron drift kinetic equation, following the theoretical framework of drift wave–zonal flow system developed by Zhang et al. [Zhang Y Z, Liu Z Y, Mahajan S M, Xie T, Liu J2017 Phys. Plasmas 24 122304 ]. At the zeroth order it is the linear eigenmode equation describing the trapped electron mode on a mirco-scale. At the first order it is the envelop equation for trapped electron mode modulated by the zonal flow on a meso-scale. The eigenmode equation has been solved by Xie et al. [Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y2019 Phys. Plasmas 26 022503 ] to obtain the eigenvalue and two-dimensional mode structure of trapped electron mode. These are essential components in calculating group velocities contained in the envelop equation. The radial group velocity arises from the geodesic curvature of magnetic field in tokamak. The poloidal group velocity stems from the normal curvature and diamagnetic drift velocity, which yields the mapping between the poloidal angle and time. Since the radial group velocity is also a function of poloidal angle, it is mapped to a periodic function of time with a period of milliseconds. The numerical results indicate the rapid zero-crossing, which is significant in the drift wave – zonal flow system and provides a sound foundation for studying zonal flow driven by trapped electron mode.-
Keywords:
- tokamak /
- trapped electron mode /
- group velocity /
- zonal flow
[1] Diamond P H, Itoh S I, Itoh K, Hahm T S 2005 Plasma Phys. Controlled Fusion 47 R35
Google Scholar
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Google Scholar
[3] Fujisawa A 2009 Nucl. Fusion 49 013001
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[8] 章扬忠, 谢涛 2013 核聚变与等离子体物理 33 193
Google Scholar
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Google Scholar
[9] Xie T, Qin H, Zhang Y Z, Mahajan S M 2016 Phys. Plasmas 23 042514
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[14] Qiu Z Y, Chen L, Zonca F 2015 Phys. Plasmas 22 042512
Google Scholar
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Google Scholar
[16] Catto P J 1978 Plasma Phys. Controlled Fusion 20 719
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Google Scholar
[19] Johnson R S 2005 Singular Perturbation Theory (New York: Springer) Chapter 4
[20] Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y 2019 Phys. Plasmas 26 022503
Google Scholar
[21] Cheng C Z, Chen L 1981 Nucl. Fusion 21 403
Google Scholar
[22] Liu Z Y, Zhang Y Z, Mahajan S M, Liu A D, Xie T, Zhou C, Lan T, Xie J L, Li H, Zhuang G, Liu W D 2021 Plasma Sci. Technol. 23 035101
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图 1 (a) 捕获电子模扰动电势
${\tilde \varphi _n}\left( {r, \vartheta } \right)$ 实部的等高线图; (b) 坏曲率区的放大图Fig. 1. (a) Level contours of the real parts of the perturbed electrostatic potential
${\tilde \varphi _n}\left( {r, \vartheta } \right)$ for trapped electron mode; (b) the close-up view in bad curvature region.
图 2 (a) 捕获电子模扰动密度
$\tilde n\left( {r, \vartheta } \right)$ 实部的等高线图; (b) 坏曲率区的放大图Fig. 2. (a) Level contours of the real parts of the perturbed density
$\tilde n\left( {r, \vartheta } \right)$ for trapped electron mode; (b) the close-up view in bad curvature region.
图 3 (a) 捕获电子模扰动压强
$\tilde \varepsilon \left( {r, \vartheta } \right)$ 实部的等高线图; (b) 坏曲率区的放大图Fig. 3. (a) Level contours of the real parts of the perturbed pressure
$\tilde \varepsilon \left( {r, \vartheta } \right)$ for trapped electron mode; (b) the close-up view in bad curvature region.
图 5 捕获电子模的 (a) 径向群速度
${\upsilon _{\rm{gr}}}\left( \vartheta \right)$ , (b) 极向群速度${\upsilon _{\rm{gy}}}\left( \vartheta \right)$ 和(c) 径向群速度随时间的变化${\upsilon _{\rm{gr}}}\left( t \right)$ Fig. 5. (a) Radial group velocity
${\upsilon _{\rm{gr}}}\left( \vartheta \right)$ , (b) poloidal group velocity${\upsilon _{\rm{gy}}}\left( \vartheta \right)$ , and (c) radial group velocity versus time${\upsilon _{\rm{gr}}}\left( t \right)$ for trapped electron mode.表 1 基本平衡参数
Table 1. Basic equilibrium parameters.
R/m a/m ${r_j}/a$ ${T_{\rm{e}}}$/eV ${\tau _{\rm{e}}}$ q B/T $\hat s$ ${L_n}/R$ ${\eta _{\rm{e}}}$ ${\eta _{\rm{i}}}$ n 1.65 0.4 0.41 250 10 1 1.35 1 0.1 1 0 –44 
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[1] Diamond P H, Itoh S I, Itoh K, Hahm T S 2005 Plasma Phys. Controlled Fusion 47 R35
Google Scholar
[2] Itoh K., Itoh S I, Diamond P H 2006 Phys. Plasmas 13 055502
Google Scholar
[3] Fujisawa A 2009 Nucl. Fusion 49 013001
Google Scholar
[4] Smolyakov A I, Diamond P H, Shevchenko V I 2000 Phys. Plasmas 7 1349
Google Scholar
[5] Chen L, White R B, Zonca F 2004 Phys. Rev. Lett. 92 075004
Google Scholar
[6] Guo Z B, Hahm T S 2016 Nucl. Fusion 56 066014
Google Scholar
[7] Zhang Y Z, Liu Z Y, Mahajan S M, Xie T, Liu J 2017 Phys. Plasmas 24 122304
Google Scholar
[8] 章扬忠, 谢涛 2013 核聚变与等离子体物理 33 193
Google Scholar
Zhang Y Z, Xie T 2013 Nucl. Fusion Plasma Phys. 33 193
Google Scholar
[9] Xie T, Qin H, Zhang Y Z, Mahajan S M 2016 Phys. Plasmas 23 042514
Google Scholar
[10] Tang W M 1978 Nucl. Fusion 18 1089
Google Scholar
[11] Landau L D, Lifshitz E M 1987 Fluid Mechanics (2nd Ed.) (Oxford: Pergamon Press) p263
[12] Miki K, Diamond P H 2010 Phys. Plasmas 17 032309
Google Scholar
[13] Qiu Z Y, Chen L, Zonca F 2014 Phys. Plasmas 21 022304
Google Scholar
[14] Qiu Z Y, Chen L, Zonca F 2015 Phys. Plasmas 22 042512
Google Scholar
[15] Sasaki M, Itoh K, Hallatschek K, Kasuya N, Lesur M, Kosuga Y, Itoh S I 2017 Sci. Rep. 7 16767
Google Scholar
[16] Catto P J 1978 Plasma Phys. Controlled Fusion 20 719
[17] Catto P J, Tang W M, Baldwin D E 1981 Plasma Phys. Controlled Fusion 23 639
[18] Brunner S, Fivaz M, Tran T M, Vaclavik J 1998 Phys. Plasmas 5 3929
Google Scholar
[19] Johnson R S 2005 Singular Perturbation Theory (New York: Springer) Chapter 4
[20] Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y 2019 Phys. Plasmas 26 022503
Google Scholar
[21] Cheng C Z, Chen L 1981 Nucl. Fusion 21 403
Google Scholar
[22] Liu Z Y, Zhang Y Z, Mahajan S M, Liu A D, Xie T, Zhou C, Lan T, Xie J L, Li H, Zhuang G, Liu W D 2021 Plasma Sci. Technol. 23 035101
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