Geometric Gyrokinetic Theory 几何回旋动力论

1. Geometric Gyrokinetic Theory几何回旋动力论 Hong Qin秦宏 Princeton Plasma Physics Laboratory, Princeton University Workshop on ITER Simulation May 15-19, 2006, Peking University, Beijing, China http://www.pppl.gov/~hongqin/Gyrokinetics.php

2. Acknowledgement • Thank Prof. Ronald C. Davidson and Dr. Janardhan Manickam for their continuous support. • Thank Drs. Peter J. Catto, Bruce I. Cohen, Andris Dimits, Alex Friedman, Gregory W. Hammet, W. Wei-li Lee, Lynda L. Lodestro, Thomas D. Rognlien, Philip B. Snyderfor, and William M. Tang for fruitful discussions. • US DOE contract AC02-76CH03073. • LLNL’s LDRD Project 04-SI-03, Kinetic Simulation of Boundary Plasma Turbulent Transport.

3. Classical gyrokinetics: average out gyrophase • Magnetized plasmas  fast gyromotion. • “Average-out" the fast gyromotion • Theoretically appealing • Algorithmically efficient Highly oscillatory characteristics Does not work

4. Modern gyrokinetics: all about gyro-symmetry Gyro-symmetry Decouple gyromotion

5. What is symmetry? • Coordinate dependent version: • Geometric version: Poincare-Cartan-Einstein 1-form Lie derivative Symmetry group S. Lie (1890s) Symmetry is group Symmetry vector field Disadvantage: it takes longer to explain.

6. What is Poincare-Cartan-Einstein 1-form? On the 7D phase space

7. What is the phase space? 7D spacetime inverse of metric World-line

8. Symmetry is invaraint Noether’s Theorem (1918) Cartan’s formula

9. Amatucci, Pop 11, 2097 (2004). What is gyrosymmetry? Noether’s Theorem Gyrophase coordinate

10. Dynamics under Lie group of coordinate transformation Taylor expansion Pullback No need for Poisson bracket Cartan’s formula Insignificant

11. Lie perturbations Gyrocenter gauges

12. Perturbation techniques — quest of good coordinates Peanuts by Charles Schulz. Reprint permitted by UFS, Inc.

14. + Freedoom Gyro-center gauges

15. Gyrocenter dynamics

16. Gyrocenter dynamics Curvature drift Banos drift Effective potentials

17. Gyrokinetic equations

18. Pullback of distribution function Particle distribution Gyrocenter distribution

19. Physics of pullback transformation — Spitzer paradox

20. Finally, gyrokinetic equations Gyrokinetic Maxwell’s equations

21. Gyrokinetic Poisson equation Polarization density

22. Physics of pullback transformation — polarization density

23. Decouple. Not average. Pullback Pullback Pullback Conclusions • Gyrokinetic theory is about gyro-symmetry. • Decouple the gyro-phase, not "averaging out". • Pullback transformation is indispensable. • http://www.pppl.gov/~hongqin/Gyrokinetics.php

24. Gyrokinetic Poisson equation

25. 1-form, 2-form, 3-form and all that … Pullback X X Inner product Push forward Contra-variant base Any tensor Vector field

26. Vector field and flow Lie symmetry group Lie algebra is the Infinitesimal generator of Lie group Phase Space

27. Lie perturbations Time dependent Pedestal dynamics Large Er shearing Orbit squeezing ELM Micro turbulence Gyrocenter gauges

28. Leading order gyrocenter coordinates Lab phase space coords Time-dependent Background EXB drift

31. Curvature drfit

32. Pullback of distribution function Needed for Maxwell’q eqs. Definition of pullback