# Foundations of nonlinear gyrokinetic theory

## Abstract

Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic theory are based on three pillars: (i) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotivelike terms, (ii) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms, and (iii) an exact energy conservation law for the gyrokinetic Vlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on rigorous application of Lagrangian and Hamiltonian Lie-transform perturbation methods in the variational derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry are discussed.

- Authors:

- Department of Chemistry and Physics, Saint Michael's College, Colchester, Vermont 05439 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 21013706

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Reviews of Modern Physics; Journal Volume: 79; Journal Issue: 2; Other Information: DOI: 10.1103/RevModPhys.79.421; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOLTZMANN-VLASOV EQUATION; ENERGY CONSERVATION; EVOLUTION; HAMILTONIANS; LAGRANGIAN FUNCTION; LIE GROUPS; MAGNETIZATION; MAGNETOHYDRODYNAMICS; NONLINEAR PROBLEMS; PERTURBATION THEORY; PLASMA; POLARIZATION; TURBULENCE; VARIATIONAL METHODS

### Citation Formats

```
Brizard, A. J., Hahm, T. S., and Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543.
```*Foundations of nonlinear gyrokinetic theory*. United States: N. p., 2007.
Web. doi:10.1103/REVMODPHYS.79.421.

```
Brizard, A. J., Hahm, T. S., & Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543.
```*Foundations of nonlinear gyrokinetic theory*. United States. doi:10.1103/REVMODPHYS.79.421.

```
Brizard, A. J., Hahm, T. S., and Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543. Sun .
"Foundations of nonlinear gyrokinetic theory". United States.
doi:10.1103/REVMODPHYS.79.421.
```

```
@article{osti_21013706,
```

title = {Foundations of nonlinear gyrokinetic theory},

author = {Brizard, A. J. and Hahm, T. S. and Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543},

abstractNote = {Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic theory are based on three pillars: (i) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotivelike terms, (ii) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms, and (iii) an exact energy conservation law for the gyrokinetic Vlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on rigorous application of Lagrangian and Hamiltonian Lie-transform perturbation methods in the variational derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry are discussed.},

doi = {10.1103/REVMODPHYS.79.421},

journal = {Reviews of Modern Physics},

number = 2,

volume = 79,

place = {United States},

year = {Sun Apr 15 00:00:00 EDT 2007},

month = {Sun Apr 15 00:00:00 EDT 2007}

}