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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,

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Presentation on theme: "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,"— Presentation transcript:

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

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 AC02-76CH  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: Lie derivative Symmetry vector field Poincare-Cartan-Einstein 1-form Symmetry group S. Lie (1890s) Disadvantage: it takes longer to explain. Symmetry is group

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

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

11 Lie perturbations Gyrocenter gauges

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

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14 + Freedoom Gyro-center gauges

15 Gyrocenter dynamics

16 Gyrocenter dynamics Banos drift Curvature 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 Conclusions  Gyrokinetic theory is about gyro-symmetry.  Decouple the gyro-phase, not "averaging out".  Pullback transformation is indispensable.  Decouple. Not average. Pullback

24 Gyrokinetic Poisson equation

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

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

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

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

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31 Curvature drfit

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


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