# Analysis of the Hermite spectrum in plasma turbulence

## Abstract

The properties of the Hermite spectrum associated with the linear drift-kinetic equation—as used in studies of gyrokinetic turbulence—are examined. A rigorous uniform asymptotic expression is derived for the steady-state spectrum with a Lenard-Bernstein collision operator. It is found that the spectrum is partitioned into three regions whose boundaries are determined by the ratio of the collision frequency ν to the parallel transit frequency kvth. In the regime of small Hermite index, n, with n ≲ (ν/kv _{th}) ^{2/3}, collisions play no role, and the free energy decays like n ^{–1/2} due to phase mixing. In the previously unexplored large-n regime, n ≥ (ν/kv _{th}) ^{2}, collisions are dominant, and the decay of the free energy spectrum is extremely steep, falling off like (n/e) ^{–n}. Most of the free energy is dissipated in the intermediate regime, (ν/kv _{th}) ^{2/3} ≲ n << (ν/kv _{th}) ^{2}, where the asymptotic spectrum is in close agreement with the exponentially decaying “continuization” estimate. Furthermore, our analysis shows that collisions act as a singular perturbation, giving rise to the intermediate regime, where collisions are significantly altering the spectrum well inside the general large-n asymptotic region.

- Authors:

- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
- Univ. of Texas at Austin, Austin, TX (United States)

- Publication Date:

- Research Org.:
- Univ. of Texas, Austin, TX (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1523377

- Alternate Identifier(s):
- OSTI ID: 1400334

- Grant/Contract Number:
- FG02-04ER54742

- 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

### Citation Formats

```
White, R. L., and Hazeltine, R. D. Analysis of the Hermite spectrum in plasma turbulence. United States: N. p., 2017.
Web. doi:10.1063/1.5000518.
```

```
White, R. L., & Hazeltine, R. D. Analysis of the Hermite spectrum in plasma turbulence. United States. doi:10.1063/1.5000518.
```

```
White, R. L., and Hazeltine, R. D. Thu .
"Analysis of the Hermite spectrum in plasma turbulence". United States. doi:10.1063/1.5000518. https://www.osti.gov/servlets/purl/1523377.
```

```
@article{osti_1523377,
```

title = {Analysis of the Hermite spectrum in plasma turbulence},

author = {White, R. L. and Hazeltine, R. D.},

abstractNote = {The properties of the Hermite spectrum associated with the linear drift-kinetic equation—as used in studies of gyrokinetic turbulence—are examined. A rigorous uniform asymptotic expression is derived for the steady-state spectrum with a Lenard-Bernstein collision operator. It is found that the spectrum is partitioned into three regions whose boundaries are determined by the ratio of the collision frequency ν to the parallel transit frequency kvth. In the regime of small Hermite index, n, with n ≲ (ν/kvth)2/3, collisions play no role, and the free energy decays like n–1/2 due to phase mixing. In the previously unexplored large-n regime, n ≥ (ν/kvth)2, collisions are dominant, and the decay of the free energy spectrum is extremely steep, falling off like (n/e)–n. Most of the free energy is dissipated in the intermediate regime, (ν/kvth)2/3 ≲ n << (ν/kvth)2, where the asymptotic spectrum is in close agreement with the exponentially decaying “continuization” estimate. Furthermore, our analysis shows that collisions act as a singular perturbation, giving rise to the intermediate regime, where collisions are significantly altering the spectrum well inside the general large-n asymptotic region.},

doi = {10.1063/1.5000518},

journal = {Physics of Plasmas},

number = 10,

volume = 24,

place = {United States},

year = {2017},

month = {10}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

_{1}| = 1, and G

_{1000}= 0. The negative sign implies that the numerical solution is larger.

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