The Drift-Kinetic Equation
The long-wavelength limit of gyrokinetics, following guiding centers under slow drifts and collisions.
Following guiding centers
The drift-kinetic equation is the long-perpendicular-wavelength limit of gyrokinetics: it drops finite-gyroradius effects entirely and evolves the guiding-center distribution. It is the natural framework for neoclassical transport, where the relevant scale is the orbit width, not the turbulence wavelength.
df/dt + (v_parallel b + v_d) . grad f + a_parallel df/dv_parallel = C[f]Here v_d collects the grad-B and curvature drifts, and the magnetic moment mu is conserved so velocity space reduces to (v_parallel, mu). Unlike gyrokinetics, the fields are evaluated at the guiding center, not gyro-averaged over a finite ring.
Trapped and passing particles
In a torus the field is stronger on the inboard side, so particles with small parallel velocity mirror and become trapped, executing banana orbits. The drift-kinetic equation naturally separates trapped and passing populations, which is the origin of neoclassical transport and the bootstrap current.
How it is solved numerically
- Neoclassical codes solve the drift-kinetic equation on a flux surface for transport coefficients
- delta-f methods evolve only the perturbation from a Maxwellian to reduce sampling noise
- Continuum solvers use (v_parallel, mu) grids with conservative collision operators
A common technique linearizes about a local Maxwellian and solves for the response that carries the neoclassical fluxes and the bootstrap current fraction.
Where it applies
Neoclassical transport sets the irreducible collisional floor for confinement and predicts the self-generated bootstrap current, which any steady-state scenario relies on. Bootstrap fraction estimates for the Hyperion breeder come from drift-kinetic calculations.