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

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.

Classical gyrokinetics: average out gyrophase

Magnetized plasmas fast gyromotion.

“Average-out" the fast gyromotion– Theoretically appealing– Algorithmically efficient

2

0

1[ Valsov Maxwell eqs ]

2d

pq

p- .ò

3

3

( ) 0

4 1( )

4 ( )

0

p

p

é ùæ ö¶ ¶ ´ ¶÷çê ú+ × + + × , , = ,÷ç ÷çê úè ø¶ ¶ ¶ë û

¶Ñ´ = , , + ,

¶

Ñ × = , , ,

Ñ × = .

òå

òå

Vlasov-Maxwell Equations

j j

j jj

j jj

e f tt c

e d p f tc c t

e d pf t

v Bv E x p

x p

B v x p E

E x p

B

, ?f f

c

v Bv

x p

¶ ´ ¶× ׶ ¶

Highly oscillatory characteristics

dependentq

Does not work

Modern gyrokinetics: all about gyro-symmetry

Gyro-symmetry

Decouple gyromotion

What is symmetry?

Coordinate dependent version:

Geometric version:

L dShg=

Lie derivative

Symmetry vector field

Poincare-Cartan-Einstein 1-form

210 0

2

Problem, what is ?

L d LL v

dtf

q q

q

æ ö¶ ¶ ÷ç= , = , = × + -÷ç ÷çè ø¶ ¶A v&

Symmetry groupS. Lie (1890s)

.Advantage: general, stronger, enables techniques to find q

Disadvantage: it takes longer to explain.

Symmetry is group

What is Poincare-Cartan-Einstein 1-form?

2

2

( )2

( )

( 2 ),

, four vector potential, 1-form

1-form momentum

vA p d dt

A

p v

g f

f

é ùê ú= + = + × - + ,ê úë û

º - ,

º - / ,

A v x

A

p

( )

2

2

1

21

2

L v

v

f

f

= × + -

æ ö÷ç= + × - + ÷ç ÷çè ø

A v

A v v

dt´

On the 7D phase space

What is the phase space?

1 2 2{( ) ( ) }xP x p x M p T M g p p mc* -= , | Î , Î , , = -

7D

spacetime inverse of metric

Phase space is a fiber

bundle P Mp : ¾® .

Hamilton's Eq. 0 .i dt g =

World-line

Symmetry is invaraint

( )L d i i d dSh h hg g g= + =

Noether’s Theorem (1918)

Cartan’s formula

( )

is conversed.

d dS

S

g h t t

g h

× × = ×

× -

What is gyrosymmetry?

1 1x y

y x

v vB x v B y v

hæ ö æ ö¶ ¶ ¶ ¶÷ç ÷ç÷ ÷ç= + + -ç÷ ÷ç ç÷ ÷ç÷¶ ¶ ¶ ¶ç è øè ø

Noether’s Theorem

Gyrophasecoordinate

Coordinate perturbation to find .

Q: How does the dynamics transform?

A: It transforms as an 1-form!

h

2 2

2x yv v

Bm

+=

hq

¶=

¶

Amatucci, Pop 11, 2097 (2004).

Dynamics under Lie group of coordinate transformation

[ ]

1 1 2( )

2( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) .

G Z

G Z

Z g z g Z Z L Z O

Z i d Z d G Z O

g g g g e

g g g e

- * -é ùG = = = - +ë û= - - × +

0

10

( )

Continuous Lie group,

Vector field (Lie algbra),

,

g z Z g z

G dg d

G dg d

e

e

e

e

e=

-=

: = ,

= / |

- = / | .

a

Zg in Pullback

Cartan’s formula Insignificant

No need for Poisson bracket

Taylor expansion

Lie perturbations

0 1 2

0 1

( )

0, 0,

Noncanonical coordinates:

Z u w q

g g g g

g gq q

= , , ,

= + + +...,

¶ ¶= ¹

¶ ¶

X

: ( ) 0 .g Z Z g Z such that g q® = ¶ / ¶ =

1

1

1 2

20 1

0 0

1 1 ( ) 0 1

2 2 ( ) 1

2( ) ( ) 0 2

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ,

( ) ( ) ( )

1( ) .

2

G Z

G Z

G Z G Z

Z Z Z O

Z Z

Z Z i d Z dS

Z Z L Z

L L Z dS

g g g e

g g

g g g

g g g

gæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

= + + ,

= ,

= - -

= -

+ - +

1 2 1 2, , , G G S S

Gyrocenter gauges

0 1 0 1

0 00 1 1 1 1 0

0 0 0 00

0 0 0 0

1 1 1 1

1 1 1 1

1 1

1 11

e e

r r e

r r

= + , = + ,

´ ´, , ,

æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷ç çW ¶ W ¶è ø è ø

æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷W ¶ W ¶è ø è ø

: : :

: :

: :

c c

E E B B

E E t B B t

E E B B

E E t B B t

E E E B B B

v B v BE E B B

Perturbation techniques — quest of good coordinates

Peanuts by Charles Schulz. Reprint permitted by UFS, Inc.

20 1 ( )Og eg g= + + ,

( )22 2 2

0 0002 2

Dw u wA u d d dtB

q fgæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø

+ += + + × + - + ,b D X

0 110 0

3

0 1 130 0 0

1 0

0

1( )

2 2

1( ) ( )

2

1( )

2 2

w w wu d

B B

ww d dwB B B

w wu dt

t B t

r r rg

r q r

f r r r r

é ùæ ö÷çê ú= Ñ × + + × ´ - Ñ × + + ×Ñ÷ç ÷çê úè øë ûé ù é ùê ú ê ú+ - × × + + × + + ×Ñê ú ê úë û ë ûé ùæ ö¶ ¶÷çê ú- + + × - ×Ñ × - + × .÷ç ÷çê úè ø¶ ¶ë û

ca b D a A X XB

a b A X c A X aB

D c aX E b

11 0 1( ) ( )

dgZ g Z G Z

d eee == , , | = , 11 ( ) 0 11( ) ( ) ( ) ( )G ZZ Z i d Z dS Zg gg= - + ,

under coordinate perturbationg

01 1 0 1 1

0

1 11 1 1

0 0

31

01 1 30 0 0

1 0 1

0

1( )

2 2

( )

1( )

2

( )

u

w

w wZ G S u

B

S Sw wd du G

B u B w

Sw wdw G

B B B

wd

B

q

g r r

r

rq

r q

é æ ö÷çê= ´ - +Ñ + Ñ × + + × ´Ñ÷ç ÷çê è øëù éé ù¶ ¶ú êê ú- Ñ × + + × + × + + + +ú êê ú¶ ¶ë ûû ë

ù é ¶ú ê+ + × + - + - × ×Ñú ê ¶û ëùú+ + × + - × +úû

X

X

X

cG B b a b B

D a A X X G b

A X a a bB

A X c E G 11 1 1

0

0

( )

1

2 2

u w

SuG wG

t

w wu dt

t B t

f r

r r r

é ¶ê + + - +ê ¶ë

ùæ ö¶ ¶÷ç ú- × + ×Ñ × + + × .÷ç ÷ç úè ø¶ ¶ û

X

D c aE b

1 0g q¶ / ¶ = .

+

1 1, G S

FreedoomGyro-center gauges

under coordinate perturbationg

( )

( )

0 1

2 2 2 2

0 0 0

0

2

1 10

( ) ( )

2 2

( )2

Z Z Z

w u w DA u d d dt

B

wZ d H dt

B

g g g

g q f

g

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø

= + ,

+ += + + × + - + ,

=- × - ,

b D X

R X

( )

2 2

01 0 130 0

2 2

0 0 020 0

0

2 4

4 2

w w uH

B B

w wR

B B

Rt

y= × + ×Ñ´ +Ñ

- Ñ × - :Ñ - ,

¶º Ñ × , º - × ,

¶

E b bB

E bb E

cR c a a

°

011 1 1

0

2

0

( ) ( ) ( )

1

2

wB

d

E bX A X Xc A

p

y f r r r

a a q a a ap

^´º + - × + - + ,×

º , º - .ò

Gyrocenterdynamics

in gyrocenter coordinate, / =0g g q¶ ¶

0

( )

( )

( ) ( )

( )

Gyro-gauge invariant

R R X

X

R R X t

t

d

q q d

¢

¢

¾® +Ñ ,

¾® + .

= , , = , ,

Ñ = - ¶/ ¶ ,Ñ .

R X

Gyrocenter dynamics

0i dt g= .

† †

† †

† †

†

0 00 0 2

0

1 2 0 00

2

0

( )2

2

1

2

02

du

dt

du

dtBd d u

B Rdt dt B

B

d w

dt B

m

q

y ym

mm

é ùê úë û

´= + ×Ñ´ - ,

× ××

= ,×

×Ñ= + × - + + ×Ñ´

¶+ + - Ñ × - :Ñ

¶

= , º ,

X B b Eb b

b B b B

B E

B bEX

R b b

E bb E

( )†0

2†

0 0 1 2

2†

0 0 1 2

†0

†† † † †

2

2

u

DB u

t t

DB

u

t

m y y

ff m y y

f

é ùê úê úê úê úê úë û

º Ñ´ + + ,

¶ ¶º - Ñ + + + - - .

¶ ¶

º + + + + ,

º + + ,

¶=Ñ´ , =- Ñ - .

¶

B A b D

b DE E

A A b D

AB A E

Banos drift

, driftB´ ÑE B

drift by spacetime

inhomogeneties of 0E

Curvature drift

& curvature driftDÑ D

Effective potentials

Gyrokinetic equations

( )0 0 6

.

¶= , £ £ .

¶

=

j

j

dZ Fj

dt Z

F F

0dZ

dtq

æ ö¶ ÷ç = ,÷ç ÷çè ø¶

0¶ ¶

+ ×Ñ + = ,¶ ¶

F Fd duF

t dt dt uX

X

Pullback of distribution function

[ ] ( )

pullback from ( ) :

( ) ( ) ( )

F Z

f z g F Z F g z*= = .

Particle distribution

Don't know ( ).f z

Gyrocenterdistribution

Physics of pullback transformation — Spitzer paradox

3 30

2 3

2 3

( ) ( )

( ) ( ) ( )

1

y y y

y

yy

j v g F Z dv v F x v dv

v F x F x O dv

F pv dv b

x B B

r

r e

* é ù= = + ,ë û

é ù= + ×Ñ +ë ûæ ö¶ Ñ ÷ç= = ´ .÷ç ÷çè ø¶

ò ò

ò

ò

Finally, gyrokinetic equations

0¶ ¶

+ ×Ñ + = ,¶ ¶

F Fd duF

t dt dt uX

X

( )

( )

0

0

22 31 1 2

( )

22 31 1 2

( )

14 ( ) ( ) ( ) ( )

2

14 ( ) ( ) ( ) ( )

2

ss Z g z

ss Z g z

q F Z G F Z G F Z G F Z d

q F Z G F Z G F Z G F Z d

f p

p

®

®

é ùê úÑ =- + ×Ñ + ×Ñ + ×Ñ .ê úë û

é ùê úÑ = + ×Ñ + ×Ñ + ×Ñ ,ê úë û

òå

òå

v

A v v

Gyrokinetic Maxwell’sequations

Gyrokinetic Poisson equation

( ) ( )

( ) ( )

0 1

0

2

0

1 1 2 220

( ) 4

( ) 2

1( ) ( ) ( )

ff

f

f p

p r

é ùê úê úê úê úë û

^

Ñ =- + + ,

º Ñ , , ,

é ùº + Ñ + ×Ñ ,ê úë û

å

ò

P

ss

q N N N

N wdwdu I F w u

N n VB

x

x x

x e e e e x D x b D

( )

( )( ) ( )

1

2

1 2 221 0

2 2

12 220 0 0

0

2( ) ( ) ( )

2

2( ) ( )

2 2

i

ii

i j

i ji ji j

iN M

i

i jM

i j

f f

f

¥^

-=

¥^ ^

+ -, =+ ¹

æ öÑ ÷ç ÷çº - ÷ç ÷ç ÷W! è ø

é ùæ ö æ ö+ Ñ Ñê ú÷ ÷ç ç÷ ÷ç ç+ ,ê ú÷ ÷ç ç÷ ÷ç ç÷ ÷ê úW W!! è ø è øê úë û

å

å

x x x

x x

( ) 4 412

0

1n O

Bf ræ ö÷ç ÷ç ÷÷ç^ ^ ^è øÑ × Ñ + Ñ

Polarization density2 2 1r ^Ñ =

Physics of pullback transformation — polarization density

1Xx X Gr= + + ,

01 ( ) 22 X Z g z

e ep G dq f

p ® ^

-= = Ñ .

Wò

1 1

1 XG S

B= ´ Ñ .

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.

PullbackPullbackPullback

Gyrokinetic Poisson equation

( ) ( ) ( )0 1N N N= + ,x x x

20 0( ) 4 s

s

q Nf pÑ =- ,åx

0 1

21 1( ) 4 s

s

q N N Nfff pé ùê úê úê úê úë û

Ñ =- + + .åx

1-form, 2-form, 3-form and all that …

Vector ii

v vx

¶=

¶

1-form iip p dx= 2-form j ii

j

pdp dx dx

x

¶=

¶

d

Exterior

derivative

X

iip v Î ¡

Vector ii

v vx

¶=

¶

X

*j iijj i

ppv dx T M

x x

æ ö¶¶ ÷ç ÷- Îç ÷ç ÷ç¶ ¶è ø

Inner product

Pullback

Push forward

vL g

Vector field

Any tensor

Contra-variant base

Vector field and flow

®: ( )g z Z t

zZ

( )G z TgVector field Î

Lie symmetry group

Phase Space

Lie algebra is the Infinitesimal generator of Lie group

Lie perturbations

0 1 2

0 1

( )

0, 0,

Noncanonical coordinates:

Z u w q

g g g g

g gq q

= , , ,

= + + +...,

¶ ¶= ¹

¶ ¶

X

: ( ) 0 .g Z Z g Z such that g q® = ¶ / ¶ =

1

1

1 2

20 1

0 0

1 1 ( ) 0 1

2 2 ( ) 1

2( ) ( ) 0 2

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ,

( ) ( ) ( )

1( ) .

2

G Z

G Z

G Z G Z

Z Z Z O

Z Z

Z Z i d Z dS

Z Z L Z

L L Z dS

g g g e

g g

g g g

g g g

gæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

= + + ,

= ,

= - -

= -

+ - +

1 2 1 2, , , G G S S

Gyrocenter gauges

0 1 0 1

0 00 1 1 1 1 0

0 0 0 00

0 0 0 0

1 1 1 1

1 1 1 1

1 1

1 11

c c

E E B B

E E t B B t

E E B B

E E t B B t

e e

r r e

r r

= + , = + ,

´ ´, , ,

æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷ç çW ¶ W ¶è ø è ø

æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷ç çW ¶ W ¶è ø è ø

E E E B B B

v B v BE E B B: : :

: :

: :

Time dependentPedestal dynamics

Large Er shearingOrbit squeezing

Micro turbulence ELM

Leading order gyrocenter coordinates

[ ][ ]

[ ]0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) 1

( ) ( ) ( )

( ) ( ) ( )( )

( )

x x

x x x

x x

x x

xx

u y y y y y

w y y y y y

y y

y y y

y y yy

B yr

, º - × ,

, , º - ´ ´ ,

, × , = ,

, º ´ , ,

´ -, º .

v b v D b b

v c v v D b b

c v c v

a v b c v

b v Dv

[ ]

0

1

0 0 02

00

( ) ( )

( ) ( ) ( ) sin ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )( )

x x x x

g z t Z u w t

u u w w t t

y y u y y w y y

y y yy y

B yB y

q

r q

: = , , = , , , ,

º + , , º , , º , , º - × , º ,

º + , + , , .

´º , º ,

x v X

x X X v X v X v c X e X

v D v b v c v

E B BD b

a

( ) ( ) ( )u w= + + .v D X b X c X

Lab phase space coords

Time-dependentBackground EXB drift

( )

[ ]

21

01 3 2 20 0 0

1 1

0

2

01 1 120 0 0

20 1

101 20

0 111

( )2

( )

( )2

( )2

1( )

u

w

S w wu w

u B B B

S

B

w wu wG S

B B B

B S wG

w B

B SG

w w wq

r

r

rq

r

¶=- + + Ñ × ´ - Ñ × ´×Ñ

¶

Ñ + ++ ´

= × + ×Ñ × - ×Ñ × - ×Ñ + + ,×Ñ

¶= - × + + ,×Ñ ×

¶

¶=- - + .×

¶

XG b a b b D a baa B

A Xb

c b a b b D a b A Xa B

b Xa B c A

Xa A

±

( ) °

² ±

201 1 1

1 0 0 0 030 0

2 2

02 2 20 0 0 0

3 2

0 1 02 20 0 0 0 0

2

( )2

2 2

S S S wu S E B B

t B u B

wu w w u wu

B B B B

wu w w w uw

B B B t B B t

q

y

éêêê^ êêë

æ ö´¶ ¶ ¶÷ç ÷+ + ×Ñ + + = × ×Ñç ÷ç ÷ç¶ ¶ ¶è ø

ùú+ Ñ × ´ - Ñ × ´ - : - ×Ñ ×Ñúû

¶ ¶+ ×Ñ × + × + × + - Ñ : + × .×Ñ

¶ ¶

E bb E aa

a b b D a b ca b a bB

D bb D a b a E aa aa B

P

1 1 and G S

( )

2 2

† † † †2 0 1 1 1 1 1 1

†1 1 1 1

0 0

† 11 1 1

1 1

2 2

w

dt

u w

wG G

B B

t

q

g y

y

f y

æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç ç^ è ø è ø

= ,

é ùº × ´ ´ - + × ´ + ×ê úë û

º + + ,

¶º - Ñ - - Ñ .

¶

x

E G B b b c G B E G

a cG G

AE

2 1 2 1: ( ) ( ) g g Z Z g g Z® = o

†22 2 1 1

0 0

†2 2 1 1

†0 22 1 1

†0 22 1 1

1 1

2

1

21

21

2

u

w

SS

u B B

G S

B SG

wB S

Gw wq

q

æ ö÷ç ÷ç ÷÷çè ø

æ ö÷ç ÷ç ÷÷çè ø

æ ö÷ç ÷ç ÷÷çè ø

æ ö÷ç ÷ç ÷÷çè ø

¶=- + Ñ ´ + ´ ´ ,

¶

= ×Ñ + × ´ ,

¶= + × ´ ,

¶¶

=- + × ´ ,¶

XG b b G B b

b b G B

c G B

a G B

²02 2 21 0 0 2

0

S S Su S E B

t B uy

q

æ ö´¶ ¶ ¶÷ç ÷+ + ×Ñ + + = .ç ÷ç ÷ç¶ ¶ ¶è ø

E bb P

2 2 2, ,Second order, and g G S

1

1 2

2 2 ( ) 1

2( ) ( ) 0

( ) ( ) ( )

1( ) .

2

G Z

G Z G Z

Z Z L Z

L L Z

g g g

gæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

= -

+ -

2 0gq

¶=

¶

Curvature drfit

( )

† 22 2

†

2 2 2

2 2

1 ( )

( )

( )

B Bu u b b b bu ub u u O

b B B ub b B B

b b bub u b u O

B Bb

ub u OB

e

e

e

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

^

æ ö+ Ñ´ Ñ´ ×Ñ´ ÷ç= = + - +÷ç ÷çè ø× + ×Ñ´

Ñ´ ×Ñ´= + - +

Ñ´= + + .

0D case=

Pullback of distribution function

3( ) ( ) ( ) [ ( ) ( )] [ 1 ]

( ). ( ).

,

Problem: don't know Solution: pullback from

j x f z v d j x n x x v

f z F Z

= , = - , º - ,ò v j v

[ ] ( )( ) ( ) ( )f z g F Z F g z*= = .

( )

( )0

0 1 2 0 2 1

2

1 10

32

13

2

1 2( )

( ) ( ) ( ) ( ( ))

1( ) ( ) ( )

2

( ) ( )

( ) ( )( )1

( ) ( )2 Z g z

f z g F Z g g g F Z g F g g Z

F Z G F Z G F Zg

G F Z O

F Z G F ZO

G F Z G F Z

e

e

* * * * *

*

®

é ù= = = ë û

é ùê ú+ ×Ñ + ×Ñê ú=ê úê ú+ ×Ñ +ë û

é ù+ ×Ñê úê ú= + .ê ú+ ×Ñ + ×Ñê úë û

o o

Needed for Maxwell’q eqs.

Definition of pullback2 1 0

0

2 1

g g g g

g z Z

g g Z Z

= ,

: ,

: ,

o o

a

o a

0Pullback of

nonperturbative

g