1. Challenging problems in kinetic simulation of turbulence and transport in tokamaks Yang Chen Center for Integrated Plasma Studies University of Colorado at Boulder

2. Outline Gyrokinetics The δf-method for Particle-in-Cell simulation Problems with kinetic electrons A fluid electron model for energy particles driven modes Problems with long wavelength radial Er Problems with transport time scale simulation

3. The task: understanding and predicting tokamak transport Neo-classical transport: collisional transport enhanced by toroida geometry Anomalous transport: induced by small scale turbulence (drift waves, micro-tearing modes)

4. GEM Is a semi-explicit δf particle-in-cell gyrokinetic turbulence code that includes full electron dynamics and shear Alfvenic magnetic perturbation, general magnetic field geometry and multiple species.

5. Gyrokinetic Equations Gyrokinetic ordering Reduce 6D to 5D Eliminate high frequency waves

6. The δf-method Conventional PIC (full-f): Define weight: δf method reduces the particle number by |δf/f| 2

7. Outline Gyrokinetics The δf-method for Particle-in-Cell simulation Problems with kinetic electrons A fluid electron model for energy particles driven modes Problems with long wavelength radial Er Problems with transport time scale simulation

8. Problems with kinetic electrons Use canonical momentum To avoid time derivative Ampere’s eqn: Quasi-neutrality: Direct implementation is numerically unstable due to Courant condition

9. The split-weight scheme Split-weight scheme to increase Δt: (Manuiskiy and Lee 2000, ε=1). Originally thought to allow large Δt because electrons are almost adiabatic. Wrong!

10. Need an equation for GK Poisson equation Vorticity equation Why is the split-weight scheme mode stable? by solving more equations. These eqns are compatible only for the physical modes. For grid-scale numerical modes the two sets of eqns are not equivalent. Explicit finite difference in t still unstable

15. Outline Gyrokinetics The δf-method for Particle-in-Cell simulation Problems with kinetic electrons A fluid electron model for energy particles driven modes Problems with long wavelength radial Er Problems with transport time scale simulation

16. Mass-less fluid electrons Electron continuity equation: Evolution equation for vector potential: Ohm’s law: Obtain from quasi-neutrality Gyrokinetic thermal ions and alpha particles -> Solve Ampere’s law backwards:

17. N=10 mode poloidal structure in ITER

18. Close the fluid electron model with particles Ion terms The complete Ohm’s equation is obtained by combining Ampere, Faraday’s law and velocity moment of GK equations: Electron pressureElectron inertia --- electron distribution from drift-kinetic equation

19. Closure scheme efficient for shear Alfven Shear Alfven wave ITG For shear Alfven, closure scheme also allows larger Δt However, the split-weight scheme more accurate for drift waves Both solve one additional equation to be stable

20. Outline Gyrokinetics The δf-method for Particle-in-Cell simulation Problems with kinetic electrons A fluid electron model for energy particles driven modes Problems with long wavelength radial Er Problems with transport time scale simulation

21. Problem with long wavelength E r Radial Electric field is important for both neoclassical transport and for regulating turbulence as sheared zonal flows Use MHD to determine the ion polarization density: Quasi-neutrality condition: But the ion gyrokinetic equation is only first order accurate! (Parra and Catto, 2008—2010)

22. To the first order guiding center variables are defined as

23. To the second order: Littlejohn, J. Plasma Phys. 29, 111 (1983), etc.

24. Other problems with ion gyrokinetics GK equation not (completely) derived for -- the transport barriers or at the edge, equilibrium scale length not much larger than ρ i -- cases of strong ExB flow, For ETG,, GK ions are valid but requires many points to do gyro-averaging For small devices (e.g. NSTX), time step of GK simulation It is possible to follow gyro-motion with comparable time steps

25. Solution: Return to Lorentz ions Quasi-neutrality with n p double counts Implicit δf algorithm developed (Chen & Parker POP 2009) Being Used to study tearing mode (J. Cheng, to be submitted) To be implemented in toroidal geometry

26. Outline Gyrokinetics The δf-method for Particle-in-Cell simulation Problems with kinetic electrons A fluid electron model for energy particles driven modes Problems with long wavelength radial Er Problems with transport time scale simulation

27. Challenges in transport time scale simulation Current Transport Scale modeling is based on multiscale expansion --TGYRO (based on GYRO) --TRINITY (besed on GS2) Local fluxtube simulations to get transport coefficients Solve 1D transport equation to evolve profiles M. Barnes, POP 2010

29. But scales are not well separated! For ITER,

30. However, physical particle and energy sources have to be used, and the Coarse Graining Procedure is needed to control noise In principle, profile evolution is modeled if the simulation Is extended to transport time scale.

31. What is the turbulence problem, anyway? The turbulence problem is usually understood as: given density and temperature profiles, determine the anomalous transport coefficients. In practice, the only well-defined problem is the local problem: But if profile effects are important, i.e. no scale separation, how to define the turbulence problem? Or equivalently, what is the right source to use in a global simulation that prevents profile relaxation, so that a steady state is obtained?

32. Summary on code development Current features: unique algorithm for finite- beta/kinetic electrons. General equilibrium profiles and flux surface shapes, a hybrid GK ion/fluid electron model for TAEs, etc. Future work: 2-D domain decomposition, kinetic electron closure for the fluid electron model. Lorentz ion/drift kinetic electron in torus.