# Energetic particle modes of q = 1 high-order harmonics in tokamak plasmas with monotonic weak magnetic shear

## Abstract

Linear and nonlinear simulations of high-order harmonics q=1 energetic particle modes excited by trapped energetic particles in tokamaks are carried out using kinetic/magnetohydrodynamic hybrid code M3D-K. It is found that with a flat safety factor profile in the core region, the linear growth rate of high-order harmonics (m=n>1) driven by energetic trapped particles can be higher than the m/n=1/1 component. The high m=n>1 modes become more unstable when the pressure of energetic particles becomes higher. Moreover, it is shown that there exist multiple resonant locations satisfying different resonant conditions in the phase space of energetic particles for the high-order harmonics modes, whereas there is only one precessional resonance for the m/n=1/1 harmonics. The fluid nonlinearity reduces the saturation level of the n=1 component, while it hardly affects those of the high n components, especially the modes with m=n=3,4. The frequency of these modes does not chirp significantly, which is different with the typical fishbone driven by trapped particles. Lastly, in addition, the flattening region of energetic particle distribution due to high-order harmonics excitation is wider than that due to m/n=1/1 component, although the m/n=1/1 component has a higher saturation amplitude.

- Authors:

- Dalian Univ. of Technology, Dalian (China). Key Lab. of Materials Modification by Laser, Ion and Electron Beams, School of Physics
- Zhejiang Univ., Hangzhou (China). Inst. for Fusion Theory and Simulation and Dept. of Physics; Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
- Chinese Academy of Sciences (CAS), Hefei (China). Inst. of Plasma Physics

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE; National Magnetic Confinement Fusion Science Program of China; Fundamental Research Funds for the Central Universities

- OSTI Identifier:
- 1378441

- Grant/Contract Number:
- 2013GB111000; 2014GB124000; 11322549; 11605245; 11675038; DUT15YQ103

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 5; 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; Particle distribution functions; Magnetohydrodynamics; Tokamaks; Chirping; Hybrid methods

### Citation Formats

```
Ren, Zhen-Zhen, Wang, Feng, Fu, G. Y., Shen, Wei, and Wang, Zheng-Xiong.
```*Energetic particle modes of q = 1 high-order harmonics in tokamak plasmas with monotonic weak magnetic shear*. United States: N. p., 2017.
Web. doi:10.1063/1.4981935.

```
Ren, Zhen-Zhen, Wang, Feng, Fu, G. Y., Shen, Wei, & Wang, Zheng-Xiong.
```*Energetic particle modes of q = 1 high-order harmonics in tokamak plasmas with monotonic weak magnetic shear*. United States. doi:10.1063/1.4981935.

```
Ren, Zhen-Zhen, Wang, Feng, Fu, G. Y., Shen, Wei, and Wang, Zheng-Xiong. Mon .
"Energetic particle modes of q = 1 high-order harmonics in tokamak plasmas with monotonic weak magnetic shear". United States. doi:10.1063/1.4981935. https://www.osti.gov/servlets/purl/1378441.
```

```
@article{osti_1378441,
```

title = {Energetic particle modes of q = 1 high-order harmonics in tokamak plasmas with monotonic weak magnetic shear},

author = {Ren, Zhen-Zhen and Wang, Feng and Fu, G. Y. and Shen, Wei and Wang, Zheng-Xiong},

abstractNote = {Linear and nonlinear simulations of high-order harmonics q=1 energetic particle modes excited by trapped energetic particles in tokamaks are carried out using kinetic/magnetohydrodynamic hybrid code M3D-K. It is found that with a flat safety factor profile in the core region, the linear growth rate of high-order harmonics (m=n>1) driven by energetic trapped particles can be higher than the m/n=1/1 component. The high m=n>1 modes become more unstable when the pressure of energetic particles becomes higher. Moreover, it is shown that there exist multiple resonant locations satisfying different resonant conditions in the phase space of energetic particles for the high-order harmonics modes, whereas there is only one precessional resonance for the m/n=1/1 harmonics. The fluid nonlinearity reduces the saturation level of the n=1 component, while it hardly affects those of the high n components, especially the modes with m=n=3,4. The frequency of these modes does not chirp significantly, which is different with the typical fishbone driven by trapped particles. Lastly, in addition, the flattening region of energetic particle distribution due to high-order harmonics excitation is wider than that due to m/n=1/1 component, although the m/n=1/1 component has a higher saturation amplitude.},

doi = {10.1063/1.4981935},

journal = {Physics of Plasmas},

issn = {1070-664X},

number = 5,

volume = 24,

place = {United States},

year = {2017},

month = {4}

}

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