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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, Peking University, Beijing, China http://www.pppl.gov/~hongqin/Gyrokinetics.php

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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.

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

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

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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

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

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

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

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

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

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Lie perturbations Gyrocenter gauges

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

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+ Freedoom Gyro-center gauges

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Gyrocenter dynamics

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Gyrocenter dynamics Banos drift Curvature drift Effective potentials

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Gyrokinetic equations

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Pullback of distribution function Particle distribution Gyrocenter distribution

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Physics of pullback transformation — Spitzer paradox

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Finally, gyrokinetic equations Gyrokinetic Maxwell’s equations

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Gyrokinetic Poisson equation Polarization density

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Physics of pullback transformation — polarization density

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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 Decouple. Not average. Pullback

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Gyrokinetic Poisson equation

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

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

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

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

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Curvature drfit

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

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