托卡马克无碰撞捕获电子模在时空表象中的群速度

刘朝阳; 章扬忠; 谢涛; 刘阿娣; 周楚

doi:10.7498/aps.70.20202003

摘要

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 Y 2019 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.

关键词:

Abstract

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 J 2017 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 Y 2019 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:

作者及机构信息

1.
中国科学技术大学核科学技术学院, 合肥　230026

2.
中国科学院磁聚变理论中心, 合肥　230031

3.
四川轻化工大学, 自贡　643000

通信作者: 刘朝阳, lzy0928@mail.ustc.edu.cn

基金项目: 国家自然科学基金(批准号: U1967206, 11975231, 11805203, 11775222)、国家磁约束聚变能源研发计划(批准号: 2018YFE0311200, 2017YFE0301204)和中国科学院前沿科学重点研究项目(批准号: QYZDB-SSW-SYS004)资助的课题

Authors and contacts

1.
School of Nuclear Science and Technology, University of Science and Technology of China, Hefei 230026, China

2.
Center for Magnetic Fusion Theory, Chinese Academy of Sciences, Hefei 230031, China

3.
Sichuan University of Science and Engineering, Zigong 643000, China

Corresponding author: Liu Zhao-Yang, lzy0928@mail.ustc.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. U1967206, 11975231, 11805203, 11775222), the National MCF Energy R & D Program, China (Grant Nos. 2018YFE0311200, 2017YFE0301204), and the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (Grant No. QYZDB-SSW-SYS004)

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参考文献

[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

施引文献

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.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

图 4 (18)式定义的两个平均量 (a)$\bar E\left( \vartheta \right)$; (b)$\bar \varPhi \left( \vartheta \right)$

Fig. 4. Two average quantities and defined in Equation (18): (a) $\bar E\left( \vartheta \right)$; (b) $\bar \varPhi \left( \vartheta \right)$.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

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

下载: 导出CSV

[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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托卡马克无碰撞捕获电子模在时空表象中的群速度