Introduction to the Particle In Cell Scheme for Gyrokinetic Plasma Simulation in Tokamak a Korea National Fusion Research Institute b Courant Institute, New York University c Korea Advanced Institute of Science and Technology Jae-Min Kwon a C.S. Chang b,c, S. Ku b, and J.Y. Kim a Feb. 14 ~ Laboratory, Space/Astrophysical Plasma Workshop, POSTECH

Contents I.Gyrokinetic plasma model II.Particle In Cell(PIC) simulation method III.Delta-F simulation method IV.Numerical implementations In tokamak geometry V.Ion Temperature Gradient(ITG) mode simulations VI.Neoclassical simulation of tokamak plasma VII.Future directions

Introduction Plasma turbulence causes a rapid loss of the plasma energy and particles to the tokamak wall. From 1994, numerical simulation of tokamak plasma turbulence has been done with gyrofluid and gyrokinetic approaches. The gyrokinetic approach is the fundamental one including all necessary features of the plasma turbulence responsible for the anomalous transports.

Introduction S.Either et al, IBM J. RES. & DEV. Vol M.R.Wade 2003

Original Vlasov Equation Drift Kinetic Description Gyrokinetic Description Gyrokinetic Plasma Model

Gyrokinetic Theory 6-d phase space Basic assumptions 5-d phase space Gyrokinetic Plasma Model

Eulerian Simulation Method Lagrangian Simulation Method Simulation Methods

Equations of motion for gyrocenter Gyro-averaged potential (potential felt by the charged ring) Gyrokinetic Plasma Model

Gyrokinetic Poisson Equation Charge density from gyro-ring Adiabatic Electron Response Model Gyrokinetic Plasma Model

Delta-f Simulation Scheme We solve this part only ! Simulation Methods

Tokamak Geometry Modes tend to be aligned to the magnetic field direction. Efficient representation in the field aligned coordinate :

Straightforward domain decomposition beyond the plasma boundary. Relatively low memory and communication costs. Hard to apply high order (> 2) time integration scheme (needed for fast ion species, electrons) Parallelization Processor N-1 Processor 0 Processor 1 Processor 2

Decomposition by Toroidal Mode Number Grid system based on quasi-ballooning coordinate Quadratic spline representation of the slowly varying part Spatial Grid Requirements …… + Processor 0Processor 1Processor 2 Processor N-1 Parallelization

Gyrokinetic Poisson Solver Multiply n,ij element and integrate over x : solved by sparse matrix solver (multi-grid, umfpack) fewer grid points, faster computation Field Solver

Evaluation of Turbulent Electric Field Conserved energy Field Solver

Ex) 2 nd order Runge-Kutta) Start Load initial profiles Setup Grid System Load marker particles End Diagnosis Simulation Procedure

ITG Mode Simulation

turbulent potential at t=110turbulent potential at t=160 ITG Mode Simulation

zonal potential at t=110zonal potential at t=160 ITG Mode Simulation

Ion Heat Flux (normalized by gyro-Bohm level) ITG Mode Simulation

Thermal flux time history (normalized by local gyroBohm level) ITG Mode Simulation

Electromagnetic Turbulence Simulation p || - Formulation, neglecting compressional Alfven modes

Cancellation Problem : Curse of the large terms analytic skin term numerical adiabatic current The analytic skin term and the numerical adiabatic current should be cancelled very accurately ! The problem gets worse for long wave length modes ! Electromagnetic Turbulence Simulation

Perpendicular velocity change of a trapped particle by RF heating at resonance plane. Radial transports by Coulomb collision and RF heating Banana width random walk for a trapped particle by Coulomb collision. RF resonance plane

Banana tips move to the resonance plane Kinetic energy of resonant particle : Increase of kinetic energy by RF heating : Turning points : Velocity space at outer mid-plane RF heating Slowing down by electron collision Pitch angle scattering by ion collision Critical slowing down speed C.S. Chang et al, Phys. Fluids B2, 2383(1990)G.D.Kerbel et al, Phys. Fluids B2, 3629(1985)

uniform random numbers in [0, 1] Monte Carlo implementation of the Coulomb collisions RF scattering Weight modification (for momentum and energy conservation) MC collision of marker particles against Maxwellian background Coulomb Collision Operator Weight modification ensuring momentum and energy conservation Average momentum and energy changes of marker particles by the test particle collision part Z.Lin et al, PoP 2, 2975(1995)

: RF wave induced velocity space flux Quasi linear heating operator (interaction by the RF field component with right circularly polarized fundamental harmonics only) C.F. Kennel and F. Engelmann, Phys. Fluids 9, 2377(1966) Resonant Ion and RF Interaction Model RF-resonance condition

Neoclassical Radial Electric Field

Resonant Ion Distribution Function

Efficient schemes for electromagnetic simulation (including compressional branches) Realistic simulation conditions including various sources, correct neoclassical equilibrium Full-F simulation for the tokamak edge plasmas Transport simulation near the marginality, comprehensive transport model for fusion devices Future Directions