Gyrokinetics

Gyrokinetics is a branch of plasma physics derived from kinetics and electromagnetism used to describe the low-frequency phenomena in a plasma. The trajectory of charged particles in a magnetic field is a helix that winds around the field line. This trajectory can be decomposed into a relatively slow motion of the guiding center along the field line and a fast circular motion called cyclotronic motion. For most of the plasma physics problems, this later motion is irrelevant. Gyrokinetics yields a way of describing the evolution of the particles without taking into account the circular motion, thus discarding the useless information of the cyclotronic angle.

1 Derivation of the gyrokinetics equations
- 1.1 Classical perturbation theory
- 1.2 Modern gyrokinetics
2 References
3 External links
- 3.1 See also

Derivation of the gyrokinetics equations

The starting point is the Vlasov equation that yields the evolution of the distribution function $f(\vec{q},\vec{p},t)$ of one particle species in a non collisional plasma,
$\partial _t f \,-\, [H,f]_{\bold{z}} \;=\; 0,$
where $H$ is the Hamiltonian of a single particle, and the brackets are Poisson brackets.

We denote the unit vector along the magnetic field as $\vec{b} \equiv \vec{B}/B$ .
The first step is to perform a variable change, from canonical phase-space $\bold{z}\equiv(\vec{q},\vec{p})$ to guiding center coordinates $\bold{Z}\equiv(\vec{R},p_{\|},\mu,\alpha)$ , where $\vec{R}$ is the position of the guiding center, $p_{\|}\equiv \vec{p} \cdot \vec{b}$ is the parallel velocity, $\mu$ is the magnetic moment, and $\alpha$ is the cyclotronic angle.

Classical perturbation theory

A first way to derive the gyrokinetics equations is to take the average of the Vlasov equation over the cyclotronic angle, $\partial _t \left\langle f \right\rangle \,-\, \left\langle [H,f]_{\bold{z}} \right\rangle \;=\; 0.$

Modern gyrokinetics

A more modern way to derive the gyrokinetics equations is to use the Lie transformation theory to change the coordinates to a system $\overline{\bold{Z}}$ where the new magnetic moment is an exact invariant, and the Vlasov equation take a simple form, $\partial _t \overline{F} \,-\, [\overline{H},\overline{F}]_{\overline{\bold{Z}}} \;=\; 0,$
where $\overline{F}(\overline{\bold{Z}},t) = f(\bold{z},t)$ , and $\overline{H}$ is the gyrokinetic Hamiltonian.

References

A.J. Brizard and T.S. Hahm, Foundations of Nonlinear Gyrokinetic Theory, Rev. Modern Physics 79, PPPL-4153, 2006.
T.S.Hahm, Physics of Fluids Vol 31 pp. 2670, 1988.
R.G.LittleJohn, Journal of Plasma Physics Vol 29 pp. 111, 1983.
J.R.Cary and R.G.Littlejohn, Annals of Physics Vol 151, 1983.

External links

GS2: A numerical continuum code for the study of turbulence in fusion plasmas.
AstroGK: A code based on GS2 (above) for studying turbulence in astrophysical plasmas.
GENE: A semi-global continuum turbulence simulation code, for fusion plasmas.
GEM: A particle in cell turbulence code, for fusion plasmas.
GKW: Local continuum gyrokinetic code, for turbulence in fusion plasmas.
GYRO: A semi-global continuum turbulence code, for fusion plasmas.
GYSELA: A semi-lagrangian code, for turbulence in fusion plasmas.
ELMFIRE Particle in cell monte-carlo code, for fusion plasmas.
GT5D: Global particle in cell code, for turbulence in fusion plasmas.
ORB5 (Or NEMORB): Global particle in cell code, for turbulence in fusion plasmas.
(d)FEFI: Homepage for the author of continuum gyrokinetic codes, for turbulence in fusion plasmas.
GKV: A local continuum gyrokinetic code, for turbulence in fusion plasmas.
GTC: A global gyrokinetic particle in cell simulation for fusion plasmas in toroidal and cylindrical geometries.