A delta-f Particle-in-Cell framework for the Vlasov-Poisson bump-on-tail instability. The goal: model resonant transport in stellarator-class 3D magnetic geometries.
Idealized symmetries break down in real fusion devices. We're building a kinetic simulation that handles the full nonlinear physics without resolving every background timescale.
Resonant wave-particle interactions drive transport in two very different settings: magnetic confinement fusion devices like tokamaks and stellarators, and astrophysical plasmas like Earth's radiation belts. The same equations describe both.
The standard simplification in WPI theory is to assume idealized symmetries, but real 3D magnetic configurations break those symmetries in ways that matter. The drift orbits bifurcate. Mode chirping, bursting, and phase-space clumping become important. None of this can be captured in the symmetric approximation.
I want to model the nonlinear physics that actually shapes confinement, without resolving every background timescale. That's what reduced-order kinetic simulation makes possible.
We use the Vlasov-Poisson system as a paradigm for general wave-particle interaction. The bump-on-tail instability gives us the cleanest test bed: well-understood linear theory, rich nonlinear behavior, exact analytic benchmarks for saturation amplitude and frequency.
The electric field amplitude Ê(t) and phase φ(t) are evolved through equations derived from Ampere's law, driven by the current density jb of energetic beam electrons. The distribution function evolves on the kinetic timescale.
Linear theory predicts a growth rate γL. The nonlinear saturation amplitude is set by particle trapping in the wave field, with bounce frequency ωb approximately 3.2γL at saturation. This is our primary code validation target.
Split the distribution function: f = f0 + δf. The equilibrium f0 stays static; only the perturbed part evolves. This dramatically reduces Monte-Carlo sampling noise.
Standard PIC samples the full distribution, so the signal we care about is buried in shot noise. With δf, we directly track only the deviation that contains the physics.
When the mode amplitude û → 0, the polar coordinate φ becomes singular. We integrate in pseudo-Cartesian coordinates instead:
Smooth derivatives all the way to the origin. No special-case handling near small amplitudes.
The Hamiltonian splits naturally into linear (free streaming + field oscillation) and nonlinear (wave-particle coupling) parts. We tested three symplectic integrators of increasing order:
Convergence verification took longer than expected. Yoshida's theoretical 4th-order convergence was masked by particle noise floors at certain parameter regimes. Once we resolved which sub-integrator errors were dominating, the slope measurement matched theory.
Wave Interaction Simulation Platform: Langmuir mode stepping, particle trajectory integration, δf splitting, the three symplectic integrators with proper convergence verification.
Validate linear growth rates and the nonlinear saturation ωb ≈ 3.2γL. Extend to multi-mode interactions where simple analytic predictions break down.
Extend the model to non-symmetric background magnetic fields. Study drift orbit bifurcations and how they reshape phase-space transport in stellarator geometry.