# On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows

## Abstract

This study reports the stability conditions for intense zonal flows (ZFs) and the growth rate γ _{TI} of the corresponding “tertiary” instability (TI) within the generalized Hasegawa–Mima plasma model. The analytical calculation extends and revises Kuo's analysis of the mathematically similar barotropic vorticity equation for incompressible neutral fluids on a rotating sphere [H.-L. Kuo, J. Meteor. 6, 105 (1949)]; then, the results are applied to the plasma case. An error in Kuo's original result is pointed out. An explicit analytical formula for γ _{TI} is derived and compared with numerical calculations. It is shown that, within the generalized Hasegawa–Mima model, a sinusoidal ZF is TI-unstable if and only if it satisfies the Rayleigh–Kuo criterion (known from geophysics) and that the ZF wave number exceeds the inverse ion sound radius. For non-sinusoidal ZFs, the results are qualitatively similar. As a corollary, there is no TI in the geometrical-optics limit, i.e., when the perturbation wavelength is small compared to the ZF scale. Finally, this also means that the traditional wave kinetic equation, which is derived under the geometrical-optics assumption, cannot adequately describe the ZF stability.

- Authors:

- Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences; Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1467162

- Alternate Identifier(s):
- OSTI ID: 1466870

- Grant/Contract Number:
- [AC02-09CH11466]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- [ Journal Volume: 25; Journal Issue: 8]; 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; perturbation theory; geometrical optics; plasma flows; geophysics; hydrodynamics

### Citation Formats

```
Zhu, Hongxuan, Zhou, Yao, and Dodin, I. Y. On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows. United States: N. p., 2018.
Web. doi:10.1063/1.5038859.
```

```
Zhu, Hongxuan, Zhou, Yao, & Dodin, I. Y. On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows. United States. doi:10.1063/1.5038859.
```

```
Zhu, Hongxuan, Zhou, Yao, and Dodin, I. Y. Mon .
"On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows". United States. doi:10.1063/1.5038859. https://www.osti.gov/servlets/purl/1467162.
```

```
@article{osti_1467162,
```

title = {On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows},

author = {Zhu, Hongxuan and Zhou, Yao and Dodin, I. Y.},

abstractNote = {This study reports the stability conditions for intense zonal flows (ZFs) and the growth rate γTI of the corresponding “tertiary” instability (TI) within the generalized Hasegawa–Mima plasma model. The analytical calculation extends and revises Kuo's analysis of the mathematically similar barotropic vorticity equation for incompressible neutral fluids on a rotating sphere [H.-L. Kuo, J. Meteor. 6, 105 (1949)]; then, the results are applied to the plasma case. An error in Kuo's original result is pointed out. An explicit analytical formula for γTI is derived and compared with numerical calculations. It is shown that, within the generalized Hasegawa–Mima model, a sinusoidal ZF is TI-unstable if and only if it satisfies the Rayleigh–Kuo criterion (known from geophysics) and that the ZF wave number exceeds the inverse ion sound radius. For non-sinusoidal ZFs, the results are qualitatively similar. As a corollary, there is no TI in the geometrical-optics limit, i.e., when the perturbation wavelength is small compared to the ZF scale. Finally, this also means that the traditional wave kinetic equation, which is derived under the geometrical-optics assumption, cannot adequately describe the ZF stability.},

doi = {10.1063/1.5038859},

journal = {Physics of Plasmas},

number = [8],

volume = [25],

place = {United States},

year = {2018},

month = {8}

}

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##
On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]

dataset, May 2018

- Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
- Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 files (12.2 kB)

Works referencing / citing this record:

##
On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]

dataset, May 2018

- Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
- Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 Files (12.2 kB)

##
On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]

dataset, May 2018

- Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
- Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 files (12.2 kB)

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