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ASCOT: solving the kinetic equation of minority particle species in tokamak plasmas
Eero Hirvijoki, Otto Asunta, Tuomas Koskela, Taina Kurki-Suonio, Juho Miettunen, Seppo Sipilä, Antti Snicker, Simppa Äkäslompolo
TL;DR
The paper addresses the need for a comprehensive and structurally coherent Monte Carlo treatment of fast-ion and impurity kinetics in tokamak plasmas. It develops consistent full-gyro and guiding-center formulations and applies them in a complete ASCOT rewrite, whose results agree well with the previous code. The scope includes three-dimensional geometry, MHD perturbations, impurity physics, and first-principles energetic-particle sources.
Problem
Existing ASCOT development had produced structural weaknesses and inconsistent treatment of guiding-center dynamics and collisions, while broader physics and computing demands required a more complete code.
Method
The paper develops a three-dimensional Monte Carlo framework solving minority-species kinetics with consistent Hamiltonian motion and collisions in full-gyro and guiding-center phase spaces.
Results
The rewritten ASCOT reproduces ASCOT3 reliably in benchmark configurations, with practically identical one-dimensional densities and toroidal current densities and very good agreement in distribution functions.
Takeaways & Limitations
ASCOT provides a comprehensive fusion-oriented test-ion framework covering three-dimensional neoclassical dynamics, fast-ion MHD effects, impurity transport, atomic reactions, and plasma flow.
Takeaways & Limitations
NBI geometry implementation and benchmarking against existing codes remain ongoing, with detailed NBI comparisons deferred to separate work.
Abstract
from arXiv · showhide
A comprehensive description of methods, suitable for solving the kinetic equation for fast ions and impurity species in tokamak plasmas using Monte Carlo approach, is presented. The described methods include Hamiltonian orbit-following in particle and guiding center phase space, test particle or guiding center solution of the kinetic equation applying stochastic differential equations in the presence of Coulomb collisions, neoclassical tearing modes and Alfvén eigenmodes as electromagnetic perturbations relevant to fast ions, together with plasma flow and atomic reactions relevant to impurity studies. Applying the methods, a complete reimplementation of the well-established minority species code ASCOT is carried out as a response both to the increase in computing power during the last twenty years and to the weakly structured growth of the code, which has made implementation of additional models impractical. Also, a benchmark between the previous code and the reimplementation is accomplished, showing good agreement between the codes.
1. Introduction
ASCOT4 is a complete rewrite of ASCOT motivated by demands for more complete physics, modern computing, and structural problems caused by organic code growth. It consistently treats Hamiltonian motion and collisions in the relevant phase space.
- ASCOT’s piecewise growth compromised its original philosophy and structure as physics demands and computer architectures evolved.
- ASCOT4 fully rewrites the code and reviews the model philosophy rather than only modernizing programming conventions.
- The rewrite identifies an inconsistency in applying a particle-phase-space Monte Carlo collision operator alongside guiding-center dynamics.
- ASCOT4 treats collisional and Hamiltonian parts consistently while solving the kinetic equation.
- The paper details full-gyro and guiding-center kinetic-equation solutions, equations of motion, time integrators, and test-particle Coulomb collisions.
2. ASCOT – Monte Carlo code for minority species in tokamak plasmas
ASCOT evolved from an accelerated orbit simulator into a scalable three-dimensional tool for energetic ions and impurity transport. Its applications expanded through realistic magnetic backgrounds, three-dimensional walls, full gyro-orbit following, atomic physics, and plasma flow.
- ASCOT progressed from simple axisymmetric orbit studies to realistic shaped, asymmetric tokamak magnetic backgrounds with X-points and scrape-off layers.
- MPI parallelization enabled simulations using hundreds or thousands of parallel processes as simulation sizes increased.
- Arbitrary three-dimensional magnetic fields and three-dimensional wall surfaces extended ASCOT to non-periodic features and wall-limited simulations.
- Full gyro-orbit following enabled simulations of fusion-born tritium in DIII-D that agreed very well with measurements.
- For impurity transport, ASCOT follows particles from the core to the first wall in an unrestricted three-dimensional domain and includes atomic physics and SOL plasma flow.
- The code’s substantial growth involved many developers, varied programming conventions, and scaling to tens of thousands of parallel processes.
3. Kinetic equation for minority particle species
ASCOT solves the minority-species kinetic equation by combining Hamiltonian orbit following with stochastic collision modeling. It supports both full gyro motion and guiding-center phase spaces, with numerical schemes chosen for their differing dynamics.
- The kinetic equation evolves a test-particle distribution in phase space under deterministic motion and collisional change.
- High dimensionality motivates a Kolmogorov or Fokker–Planck formulation whose solution is obtained by simulating random paths and averaging them statistically.
- ASCOT models either rapid full gyro motion or guiding-center motion when background gradients are large relative to the Larmor radius.
- The code provides separate equations, algorithms, and collision operators for particle and guiding-center phase spaces.
- Full gyro motion: Full gyro-orbit integration uses Runge–Kutta or modified leap-frog methods, with leap-frog explicitly conserving energy when the electric field is zero.
- Guiding-center motion: Guiding-center energy is zero-rate-changing for static backgrounds, and fourth-order Runge–Kutta integration is adequate for its slower orbit dynamics.
- Coulomb collisions: Coulomb collisions are represented through velocity-space friction and diffusion, combined with Hamiltonian motion in stochastic differential equations.
- Guiding-center collisions: The guiding-center transformation must consistently cover the entire kinetic equation, including collisional effects and compatible phase-space coordinates.
4. Fast ion sources
ASCOT models several fast-ion sources, including fusion products, neutral beam injection, and ICRH ions, using probabilistic initialization and reaction-based transport inputs.
- Fusion product source: Fusion-product initialization first computes reaction rates on a cylindrical grid, then generates representative test particles from that distribution.The code supports thermal, beam-target, and beam-beam reaction cases.
- Fusion product source: Weighted and uniform-weight schemes sample particle positions differently while assigning weights to represent the calculated fusion reaction rate.Weighted initialization samples inside the separatrix and uses local reaction rates; uniform initialization accepts positions according to normalized local rates.
- Neutral beam ion source: The NBI model includes beamlet structure and had geometries implemented for eight devices at the time of writing.Adding new device geometries and benchmarking against existing codes remained ongoing work.
- Neutral beam ion source: Neutral beam injection samples a neutral from a random beamlet, assigns a dispersed velocity, and advances it until ionization or wall impact.Ionization is determined by comparing survival probability with a uniformly distributed threshold; wall impact before ionization is classified as shine-through.
- ICRH ion source: ICRH ions are initialized from user-defined radial Gaussian profiles and Maxwellian energies, with optional energy cut-offs and power-scaled weights.The modeled distribution is localized near a resonant major radius rather than spread only in normalized poloidal flux.
5. Magnetohydrodynamic activity for fast ion modelling
ASCOT represents neoclassical tearing modes and Alfvén eigenmodes as electromagnetic perturbations and incorporates them into guiding-center orbit following through modified potentials.
- Perturbation model: The MHD model treats stationary NTMs and rotating AEs as perturbations of the magnetic vector potential, with time dependence also perturbing the electric potential.The perturbations are represented with helical structures and mapped using Boozer coordinates.
- Mode assumptions: NTMs are commonly modeled without an electric perturbation because their rotation frequency is usually low, whereas rapidly rotating AEs impose a relation between their perturbed potentials.AE input can include radial potential functions, mode numbers, and frequencies from MHD codes such as LIGKA.
- Coordinate handling: Unlike previous methods restricted to field-aligned coordinates, ASCOT can follow test-particle guiding centers in any coordinate system.Boozer coordinates remain important for specifying and transforming the perturbative quantities.
- Modified equations: The magnetic perturbation is inserted into the symplectic part of the Lagrangian, producing modified effective potentials used in the equations of motion and energy evolution.Electric perturbations are incorporated through the gradient of the perturbation potential.
- Implementation: The time-dependent method requires α, ∇α, ∇˜Φ, and ∂α/∂t at each location, obtained through transformations between cylindrical or Cartesian and Boozer coordinates.The formulation still yields zero for static backgrounds and time-independent perturbations.
6. Interaction models for impurity particles
ASCOT extends impurity transport modeling by accounting for low-energy impurities, background plasma flow, and charge-state-changing atomic reactions in three-dimensional tokamak domains.
- Impurity transport: Impurity transport can be followed from the core and scrape-off layer through the halo plasma to the wall in a fully three-dimensional environment.This supports applications including trace-element injection, tritium retention, material migration, and plasma-performance studies.
- Background plasma flow: Measured SOL flows with typical velocities of Mach 0.5–1 significantly affect low-energy impurity transport but are absent from purely Maxwellian collision assumptions.The standard Coulomb operators assume a background plasma with no flow.
- Background plasma flow: ASCOT models parallel plasma flow by switching to a locally moving frame before evaluating Coulomb collisions and then transforming velocities back to the laboratory frame.This is equivalent to calculating Coulomb coefficients for drift-Maxwellian backgrounds.
- Atomic reactions: Atomic reactions alter impurity charge states and therefore directly affect transport, especially for heavy impurities such as tungsten with Z = 74.The charge-state dependence enters through the particle equations of motion.
- Atomic reactions: When only charge-state evolution matters, effective ionization and recombination replace explicit modeling of every individual atomic reaction.A particle can undergo effective ionization, recombination, or no net charge-state change during one time step.
- Atomic reactions: Reaction-rate coefficients are taken from ADAS as functions of local electron temperature and density, with data imported for carbon, beryllium, tungsten, and nitrogen.The imported species define the reported scope of the available atomic-reaction data.
7. Wall collisions
ASCOT detects collisions with a three-dimensional first wall represented by triangular elements, while a mixed orbit-following scheme improves collision localization without the cost of full gyro-orbit tracking everywhere.
- Wall representation: The simulation boundary is the vessel first wall, represented by triangular and planar quadrilateral elements; quadrilaterals are split into triangles for collision detection.Accurate wall geometry is required for reliable fast-ion power-load and impurity-deposition simulations.
- Collision detection: Wall-collision candidates are found by intersecting each orbit step with triangle planes and retaining intersections whose parameter satisfies 0 < t_i ≤ 1.The orbit step is parameterized from r0 to r1 = r0 + ∆r.
- Collision detection: A candidate intersection lies inside a triangle when α ≥ 0, β ≥ 0, and α + β ≤ 1; the smallest t_i determines the returned collision point.These conditions provide the parametric test illustrated for the wall triangle.
- Optimization: A distance lower bound and bounding-volume hierarchy reduce the CPU time needed for three-dimensional wall-collision checks.The authors report that this twofold mechanism requires only a small percentage of total simulation time.
- Mixed orbit following: The mixed scheme switches to full gyro-orbit following within one Larmor radius of the wall and returns to guiding-center tracking farther away.It improves wall-collision location estimates while avoiding the computational cost of complete gyro-orbit following.
8. Diagnostic tools
ASCOT records weighted test-particle histories in multidimensional distributions and normalizes them into spatial, velocity-space, and diagnostic profiles. Its diagnostics include species-resolved power and torque deposition, wall-impact data, and post-processed toroidal torque evaluation.
- ASCOT accumulates test-particle weights after each time step into distributions with up to six dimensions, including time and test-particle species.Additional dimensions represent selected phase-space coordinates.
- Weights crossing histogram bins are distributed according to the fraction of time spent in each bin, with linearly interpolated weight changes.
- Implemented radial diagnostics include particle and energy densities, parallel and toroidal currents, power deposition, several torques, and charge-exchange sources and sinks.
- Collisional power and torque are calculated as moments of the kinetic collision term, enabling separate contributions for each background plasma species.This approach makes species partitioning straightforward in ASCOT4.
- The toroidal j × B torque can be calculated after simulation from changes in poloidal flux, using only the toroidal-force component.
- Radial profiles are normalized by bin or flux-surface volumes, while velocity-space distributions use their corresponding differential volume elements.The implementation supports direct and Monte Carlo volume integration depending on whether flux surfaces are closed or open.
- ASCOT records wall-hit location, energy, and direction to evaluate energetic-particle power and particle fluxes on plasma-facing components.
9. Benchmark between the new and old versions of ASCOT
ASCOT4 is benchmarked against ASCOT3 using a JET-like axisymmetric slowing-down simulation of fast neutral-beam ions. The codes show very good agreement across radial profiles and local velocity-space distributions, with ASCOT4 producing smoother collisional torques.
- 200 000 test particles were simulated in a JET-like axisymmetric magnetic background with 3 T toroidal field and 1.8 MA plasma current.Guiding-center motion and Coulomb collisions were followed until particles reached the local background-ion temperature.
- The benchmark compares fast-ion density, toroidal current density, toroidal j × B torque, and collisional power and torque depositions as functions of ρp.
- The 1-D fast-ion densities and toroidal current densities from ASCOT3 and ASCOT4 are practically identical.
- The power depositions match well, with small deviations attributed to ASCOT4 calculating them from the distribution function rather than absolute phase-space-coordinate changes.
- ASCOT4 produces less noisy collisional torques than ASCOT3, while the j × B torques agree apart from a small ASCOT3 core anomaly.
- Local four-dimensional velocity-space distributions from ASCOT4 and ASCOT3 show very good agreement.The comparison provides a more thorough test than agreement between radial moments alone.
10. High performance and high throughput computing
ASCOT combines high-throughput particle independence with HPC and HTC execution mechanisms. Its rewritten implementation scales to thousands of MPI processes and supports distributed workstation computing, HDF5 storage, and analysis libraries.
- ASCOT4 completed a test run using 16 384 parallel processes on the Helios supercomputer.
- Independent particle calculations allow ASCOT inputs to be partitioned across processes, after which distributions and other outputs are merged.
- ASCOT supports HPC through MPI on supercomputers and HTC through Condor execution on idle workstations.
- MPI distributes tasks across up to thousands of processes and uses one process to read files before broadcasting and scattering data.
- HDF5 provides the primary binary input and output format, with MATLAB tools available for analysis and visualization.
11. Summary and future work
ASCOT was comprehensively redesigned as a Monte Carlo test-ion code for consistently solving minority-ion kinetics in full gyro and guiding-center formalisms. The redesign incorporates broad fusion-relevant physics while correcting inconsistencies in conventional collision and orbit integration, although fully consistent numerical integration remains ongoing.
- ASCOT4 provides a consistent formalism for solving minority-ion kinetics with Hamiltonian motion and collisions in both full gyro and guiding-center representations.The code targets energetic ions and impurity species using a redesigned Monte Carlo orbit-following approach.
- Full 3D operation includes neoclassical drifts, a physical first-wall boundary, fast-ion MHD models, plasma flow, and atomic reactions for impurity ions.The code also supplies particle sources, multidimensional distributions, and wall-impact information including energy and direction.
- The redesign corrected mismatches between guiding-center Hamiltonian motion and collision operators, coordinate systems, and numerical integration schemes.Earlier approaches combined guiding-center dynamics with an untransformed full-gyro collision operator and sometimes used Euler integration for deterministic collisions.
- Coulomb collisions are now treated consistently with Hamiltonian motion, while guiding-center modeling adds spatial diffusion alongside velocity-space diffusion.The spatial diffusion term is given in Eq. (42).
- Consistent numerical integration schemes for the collisional contribution remain under development.