User:Maxime.lesur

Gyrokinetics is a branch of plasma physics derived from kinetics and electromagnetism. 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,

$$\partial _t f \,-\, [H,f]_{\mathbf{z}} \;=\; 0,$$

where $$H$$ is the Hamiltonian of a single particle, and the brackets are Poisson brackets.

We denote $$\vec{b} \equiv \vec{B}/B$$ the unit vector along the magnetic field.

The first step is to proceed a variable change, from canonical phase-space $$\mathbf{z}\equiv(\vec{q},\vec{p})$$ to guiding center coordinates $$\mathbf{Z}\equiv(\vec{R},p_{\|},\mu,\alpha)$$, 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 momentum, and $$\alpha$$ is the cyclotronic angle.

Classical perturbation theory
A first way to derivate the gyrokinetics equations is to make an average of Vlasov equations over the cyclotronic angle, $$\partial _t F \,-\, \overline{[H,F]}_{\mathbf{Z}} \;=\; 0,$$

where $$F\equiv \overline{f}$$

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 $$\overline{\mathbf{Z}}$$ where the new magnetic momentum is an exact invariant, and the Vlasov equation take a simple form, $$\partial _t \overline{F} \,-\, [\overline{H},\overline{F}]_{\overline{\mathbf{Z}}} \;=\; 0.$$