Gyrokinetics

Gyrokinetics is a theoretical framework to study plasma behavior on perpendicular spatial scales comparable to the gyroradius and frequencies much lower than the particle cyclotron frequencies.

These particular scales have been experimentally shown to be appropriate for modeling plasma turbulence. 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 gyromotion. For most plasma behavior, this gyromotion is irrelevant. Averaging over this gyromotion reduces the equations to six dimensions (3 spatial, 2 velocity, and time) rather than the seven (3 spatial, 3 velocity, and time). Because of this simplification, gyrokinetics governs the evolution of charged rings with a guiding center position, instead of gyrating charged particles.

Derivation of the gyrokinetic equation

Fundamentally, the gyrokinetic model assumes the plasma is strongly magnetized (

ρ

i

≪

L

plasma

{\displaystyle \rho _{i}\ll L_{\text{plasma}}}

), the perpendicular spatial scales are comparable to the gyroradius (

k

⊥

ρ

i

∼

1

{\displaystyle k_{\perp }\rho _{i}\sim 1}

), and the behavior of interest has low frequencies (

ω

≪

Ω

i

≪

Ω

e

{\displaystyle \omega \ll \Omega _{i}\ll \Omega _{e}}

). We must also expand the distribution function,

f

s

=

f

s

0

+

f

s

1

+

⋯

{\displaystyle f_{s}=f_{s0}+f_{s1}+\cdots }

, and assume the perturbation is small compared to the background (

f

s

1

≪

f

s

0

{\displaystyle f_{s1}\ll f_{s0}}

). The starting point is the Fokker–Planck equation and Maxwell's equations.