# Gyrokinetic Statistical Absolute Equilibrium and Turbulence

## Abstract

A paradigm based on the absolute equilibrium of Galerkin-truncated inviscid systems to aid in understanding turbulence [T.-D. Lee, "On some statistical properties of hydrodynamical and magnetohydrodynamical fields," Q. Appl. Math. 10, 69 (1952)] is taken to study gyrokinetic plasma turbulence: A finite set of Fourier modes of the collisionless gyrokinetic equations are kept and the statistical equilibria are calculated; possible implications for plasma turbulence in various situations are discussed. For the case of two spatial and one velocity dimension, in the calculation with discretization also of velocity v with N grid points (where N + 1 quantities are conserved, corresponding to an energy invariant and N entropy-related invariants), the negative temperature states, corresponding to the condensation of the generalized energy into the lowest modes, are found. This indicates a generic feature of inverse energy cascade. Comparisons are made with some classical results, such as those of Charney-Hasegawa-Mima in the cold-ion limit. There is a universal shape for statistical equilibrium of gyrokinetics in three spatial and two velocity dimensions with just one conserved quantity. Possible physical relevance to turbulence, such as ITG zonal flows, and to a critical balance hypothesis are also discussed.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1001683

- Report Number(s):
- PPPL-4592

TRN: US1101048

- DOE Contract Number:
- DE-ACO2-09CH11466

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; DIMENSIONS; HYPOTHESIS; PLASMA; SHAPE; TURBULENCE; VELOCITY; WASTE HEAT UTILIZATION; Equilibrium, turbulence, Statistical

### Citation Formats

```
Jian-Zhou Zhu and Gregory W. Hammett.
```*Gyrokinetic Statistical Absolute Equilibrium and Turbulence*. United States: N. p., 2011.
Web. doi:10.2172/1001683.

```
Jian-Zhou Zhu and Gregory W. Hammett.
```*Gyrokinetic Statistical Absolute Equilibrium and Turbulence*. United States. doi:10.2172/1001683.

```
Jian-Zhou Zhu and Gregory W. Hammett. Mon .
"Gyrokinetic Statistical Absolute Equilibrium and Turbulence". United States. doi:10.2172/1001683. https://www.osti.gov/servlets/purl/1001683.
```

```
@article{osti_1001683,
```

title = {Gyrokinetic Statistical Absolute Equilibrium and Turbulence},

author = {Jian-Zhou Zhu and Gregory W. Hammett},

abstractNote = {A paradigm based on the absolute equilibrium of Galerkin-truncated inviscid systems to aid in understanding turbulence [T.-D. Lee, "On some statistical properties of hydrodynamical and magnetohydrodynamical fields," Q. Appl. Math. 10, 69 (1952)] is taken to study gyrokinetic plasma turbulence: A finite set of Fourier modes of the collisionless gyrokinetic equations are kept and the statistical equilibria are calculated; possible implications for plasma turbulence in various situations are discussed. For the case of two spatial and one velocity dimension, in the calculation with discretization also of velocity v with N grid points (where N + 1 quantities are conserved, corresponding to an energy invariant and N entropy-related invariants), the negative temperature states, corresponding to the condensation of the generalized energy into the lowest modes, are found. This indicates a generic feature of inverse energy cascade. Comparisons are made with some classical results, such as those of Charney-Hasegawa-Mima in the cold-ion limit. There is a universal shape for statistical equilibrium of gyrokinetics in three spatial and two velocity dimensions with just one conserved quantity. Possible physical relevance to turbulence, such as ITG zonal flows, and to a critical balance hypothesis are also discussed.},

doi = {10.2172/1001683},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Mon Jan 10 00:00:00 EST 2011},

month = {Mon Jan 10 00:00:00 EST 2011}

}