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# 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 $f\left(vec\left\{q\right\},vec\left\{p\right\},t\right)$ of one particle species in a non collisional plasma,
$partial _t f ,-, \left[H,f\right]_\left\{bold\left\{z\right\}\right\} ;=; 0,$
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 $vec\left\{b\right\} equiv vec\left\{B\right\}/B$.
The first step is to perform a variable change, from canonical phase-space $bold\left\{z\right\}equiv\left(vec\left\{q\right\},vec\left\{p\right\}\right)$ to guiding center coordinates $bold\left\{Z\right\}equiv\left(vec\left\{R\right\},p_\left\{|\right\},mu,alpha\right)$, where $vec\left\{R\right\}$ is the position of the guiding center, $p_\left\{|\right\}equiv vec\left\{p\right\} cdot vec\left\{b\right\}$ 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, $partial _t leftlangle f rightrangle ,-, leftlangle \left[H,f\right]_\left\{bold\left\{z\right\}\right\} rightrangle ;=; 0.$

### 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 $overline\left\{bold\left\{Z\right\}\right\}$ where the new magnetic momentum is an exact invariant, and the Vlasov equation take a simple form, $partial _t overline\left\{F\right\} ,-, \left[overline\left\{H\right\},overline\left\{F\right\}\right]_\left\{overline\left\{bold\left\{Z\right\}\right\}\right\} ;=; 0,$
where $overline\left\{F\right\}\left(overline\left\{bold\left\{Z\right\}\right\},t\right) = f\left(bold\left\{z\right\},t\right)$, and $overline\left\{H\right\}$ 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.
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