Fokker-Planck Solvers
Fokker-Planck codes evolve the velocity-space distribution of a particle species under collisions and wave forces, quantifying heating and current drive.
Distributions in velocity space
Heating and current drive are best understood as reshaping the distribution of particle velocities. A Fokker-Planck code evolves this distribution function under the combined action of Coulomb collisions, which relax it toward a Maxwellian, and external forces such as radio-frequency waves or a beam, which push it away.
The collision operator
The Fokker-Planck collision operator represents small-angle Coulomb scattering as diffusion and drag in velocity space. Codes solve the resulting diffusion equation in one, two, or three velocity dimensions, often in a reduced coordinate set (energy and pitch angle) that exploits the symmetry of the magnetized plasma.
Wave-particle interaction
- A quasilinear diffusion term represents the effect of resonant waves
- It flattens the distribution along resonance lines, driving current or heating
- The steady distribution determines the driven current and absorbed power
Bounce averaging and trapping
In toroidal geometry, particles are either passing or trapped in magnetic wells. Bounce-averaged Fokker-Planck codes account for this by averaging over the particle's orbit, which is essential for correctly predicting current drive, since trapped particles carry no net current and reduce drive efficiency.
The workflow
A ray-tracing or full-wave code supplies the wave field; the Fokker-Planck solver turns that into a distortion of the distribution and hence the driven current and heating profile. The pairing is the standard method for predicting current-drive efficiency and the fast-particle populations produced by heating.
These tools underpin estimates of how much non-inductive current a design can sustain and how efficiently, which shapes scenario planning for machines including the Hyperion breeder.