# The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak

## Abstract

The two-dimensional (2D) kinetic ballooning theory is developed for the ion temperature gradient mode in an up-down symmetric equilibrium (illustrated via concentric circular magnetic surfaces). The ballooning transform converts the basic 2D linear gyro-kinetic equation into two equations: (1) the lowest order equation (ballooning equation) is an integral equation essentially the same as that reported by Dong et al., [Phys. Fluids B 4, 1867 (1992)] but has an undetermined Floquet phase variable, (2) the higher order equation for the rapid phase envelope is an ordinary differential equation in the same form as the 2D ballooning theory in a fluid model [Xie et al., Phys. Plasmas 23, 042514 (2016)]. The system is numerically solved by an iterative approach to obtain the (phase independent) eigen-value. The new results are compared here to the two earlier theories. We find a strongly modified up-down asymmetric mode structure, and non-trivial modifications to the eigen-value.

- Authors:

- Sichuan Univ. of Science and Engineering, Zigong (China). Dept. of Physics
- Chinese Academy of Sciences (CAS), Hefei (China). Center for Magnetic Fusion Theory
- Univ. of Texas, Austin, TX (United States). Inst. for Fusion Studies
- Southwestern Inst. of Physics, Chengdu (China)
- Univ. of Science and Technology of China, Hefei (China). Dept. of Modern Physics

- Publication Date:

- Research Org.:
- Univ. of Texas, Austin, TX (United States); Sichuan Univ. of Science and Engineering, Zigong (China); Chinese Academy of Sciences (CAS), Hefei (China); Univ. of Science and Technology of China, Hefei (China); Southwestern Inst. of Physics, Chengdu (China)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24); Key Research Program of Frontier Sciences CAS (China); National Magnetic Confinement Fusion Energy Research Project (China); National Natural Science Foundation of China (NNSFC); Foundation of Sichuan Univ. of Science and Engineering (China); Scientific Research Fund of the Sichuan Provincial Education Dept. (China)

- OSTI Identifier:
- 1523476

- Alternate Identifier(s):
- OSTI ID: 1394010

- Grant/Contract Number:
- FG02-04ER54742; QYZDB-SSWSYS004; 2015GB104004; 2015GB111003; 2013GB112009; 11575185; 11575186; 2016RCL21; 17ZA0281

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 10; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; integral transforms; plasma confinement; integral equations; spectral methods; tokamaks; functional equations; Newtonian mechanics; weight function; order theory; turbulence simulations

### Citation Formats

```
Xie, T., Zhang, Y. Z., Mahajan, S. M., Hu, S. L., He, Hongda, and Liu, Z. Y. The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak. United States: N. p., 2017.
Web. doi:10.1063/1.5003652.
```

```
Xie, T., Zhang, Y. Z., Mahajan, S. M., Hu, S. L., He, Hongda, & Liu, Z. Y. The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak. United States. doi:10.1063/1.5003652.
```

```
Xie, T., Zhang, Y. Z., Mahajan, S. M., Hu, S. L., He, Hongda, and Liu, Z. Y. Fri .
"The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak". United States. doi:10.1063/1.5003652. https://www.osti.gov/servlets/purl/1523476.
```

```
@article{osti_1523476,
```

title = {The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak},

author = {Xie, T. and Zhang, Y. Z. and Mahajan, S. M. and Hu, S. L. and He, Hongda and Liu, Z. Y.},

abstractNote = {The two-dimensional (2D) kinetic ballooning theory is developed for the ion temperature gradient mode in an up-down symmetric equilibrium (illustrated via concentric circular magnetic surfaces). The ballooning transform converts the basic 2D linear gyro-kinetic equation into two equations: (1) the lowest order equation (ballooning equation) is an integral equation essentially the same as that reported by Dong et al., [Phys. Fluids B 4, 1867 (1992)] but has an undetermined Floquet phase variable, (2) the higher order equation for the rapid phase envelope is an ordinary differential equation in the same form as the 2D ballooning theory in a fluid model [Xie et al., Phys. Plasmas 23, 042514 (2016)]. The system is numerically solved by an iterative approach to obtain the (phase independent) eigen-value. The new results are compared here to the two earlier theories. We find a strongly modified up-down asymmetric mode structure, and non-trivial modifications to the eigen-value.},

doi = {10.1063/1.5003652},

journal = {Physics of Plasmas},

number = 10,

volume = 24,

place = {United States},

year = {2017},

month = {9}

}

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Web of Science

#### Figures / Tables:

_{θ}ρ

_{s}

_{0}= -0.53, ŝ = 1, q

_{0}= 2, η

_{i0}=more »

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*