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 a charged particles in a magnetic field is an 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.

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,

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 to guiding center coordinates , where $vec\{R\}$ is the position of the guiding center, $p\_\{\backslash equiv\; vec\{p\}\; cdot\; vec\{b\}$ is the parallel velocity, $mu$ is the magnetic momentum, 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,

Modern gyrokinetics

A more modern way to derivate the gyrokinetics equations is to use the Lie transformation theory to change the coordinates to a system where the new magnetic momentum is an exact invariant, and the Vlasov equation take a simple form,
where , 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.