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.

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 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 ${\mathbf {z}}\equiv ({\vec {q}},{\vec {p}})$ 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 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, $\partial _{t}\left\langle f\right\rangle \,-\,\left\langle [H,f]_{\mathbf {z}}\right\rangle \;=\;0.$

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

See Also

• GYRO
• Gyrokinetic_ElectroMagnetic