Gyrocenter Gauge Kinetic Theory and Algorithm for Radio-Frequency Waves in Plasmas

Hong QinPrinceton Plasma Physics Laboratory, Princeton University

Numerical Flow Models for Controlled Fusion

16-20 April, 2007Porquerolles, France

http://www.pppl.gov/~hongqin/Gyrokinetics.php

2

Acknowledgement

Thank Prof. Ronald C. Davidson and Dr. Janardhan Manickam for their continuous support.

Thank Drs. John Cary, Peter J. Catto, Bruce I. Cohen, Ronald Cohen, Gregory W. Hammet, Nathaniel J. Fisch, Roman A. Kolesnikov, William M. Nevins, W. Wei-li Lee, Cynthia K. Phillips, David N. Smithe, Edward A. Startsev, William M. Tang, and Ernest J. Valeo for fruitful discussions.

US DOE contract AC02-76CH03073.

3

Classical gyrokinetics: average out gyrophase

Magnetized plasmas → fast gyromotion.

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

Only for low frequency waves.

2

0

1 [ Valsov Maxwell eqs ]2

dπ

θπ

− .∫

, ?f fc

v Bvx p

∂ × ∂⋅ ⋅∂ ∂

Highly oscillatory

dependentθ

Does notwork

3

3

( ) 0

4 1( )

4 ( )

0

π

π

⎡ ⎤⎛ ⎞∂ ∂ × ∂⎟⎜⎢ ⎥+ ⋅ + + ⋅ , , = ,⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠∂ ∂ ∂⎣ ⎦

∂∇× = , , + ,

∂

∇⋅ = , , ,

∇⋅ = .

∫∑

∫∑

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 px p

B v x p E

E x p

B

4

Motivations (presentation outline)

Geometrically generalized Vlasov-Maxwell system

Geometic gyrokinetic theory

Gyrocenter gauge theoryand algorithm

Geometric field theory

Gyro-symmetry

For RF waves

Decouple, not average

Valid for high freq.

5

Modern gyrokinetics = field theory + gyro-symmetry

Gyro-symmetry

Decouple gyromotion

Field theory

Interaction between particle and fields

γ

Poincare-Cartan-

Einstein 1-form

6

1 2 2( ) ( ) xP x p x M p T M g p p m c∗ −= , | ∈ , ∈ , , = −

7D spacetimeInverse of metric

Phase space is a fiber

bundle P Mπ : ⎯→ .

Cotagent bundle

Where does live? -- phase spaceγ

7

0i dτ γ =

Worldline

Inner product

Exterior derivative

Different for different particles:charged particle, neutron,polymer, virus, …

Poincare-Cartan-Einstein 1-form particle dynamicsγ →

Hamilton’s Eq.

8

Geometrically generalized Vlasov-Maxwell system --- A field theory

31

,3

d d dm

γ γ γΩ ≡ − ∧ ∧!

0L f i dfτ τ= = .

( ) ( ) ( ) 0L f L f L fτ τ τΩ = Ω + Ω = .

( ) 0.L i d d iτ τ τΩ = Ω + Ω =

Liouville 6-form

Vlasov Eq.

Liouville Theorem

Conservative form

9

Geometrically generalized Vlasov-Maxwell system --- A field theory

4d dA jπ∗ = ∗ ,

xS L= ,∫

1( )

1( ) 4

2 xL x dA dA f

ππ γ

−= − ∧ ∗ + Ω ∧∫

1( )

( )( ) ( 0 1 2 3)

( )x x

xj x f

A xα

π α

δγα

δ′ − ′

′= Ω ∧ , = , , , .∫ ∫

( ) 0S

E LAδδ

= =

Variationalderivative

Fiber integral

Euler derivative

10

Second order field theory → 4-diamagnetic current

( )

( )

1

1

( )

( )

( )( )

( )

( )( 0 1 2 3)

( )

x x

x x

xj x x x f

A x

xx x f

A xx

απ α

β π α β

γδ

γδ α

′ − ′

′ − ′

′′

′′

,

∂= − Ω ∧

∂⎡ ⎤∂ ∂⎢ ⎥− − Ω ∧ , = , , , .⎢ ⎥∂∂ ⎣ ⎦

∫ ∫

∫ ∫

Axαβ

∂∂

4-diamagnetic current

Valid for any . Exact conservation properties.

Allow physics models, approximations build into .

γ

γ

11

Example: 1-form for charge particles with Lorentz force

2

2

( )2

( ), four vector potential, 1-form

( 2 ), 1-form momentum

vA p d dt

A

p v

γ φ

φ

⎡ ⎤⎢ ⎥= + = + ⋅ − + ,⎢ ⎥⎣ ⎦

≡ − ,

≡ − / ,

A v x

A

p

( )

2

2

12

12

L v

v

φ

φ

= ⋅ + −

⎛ ⎞⎟⎜= + ⋅ − + ⎟⎜ ⎟⎜⎝ ⎠

A v

A v v

dt×

1( )

1( ) 4

2 xL x dA dA f

ππ γ

−= − ∧ ∗ + Ω ∧∫

1( )4

xd dA f

ππ

−∗ = Ω∫

( ) 0 0df i dττ γ= , =

Action

Vlasov

Maxwell

12

What is symmetry?

Coordinate dependent version:

Geometric version:

L dSηγ =

Lie derivative Symmetry vector field

0 0 .

Problem, what is ?

L d Ldtθ θ

θ

∂ ∂⎛ ⎞⎟⎜= , =⎟⎟⎜⎝ ⎠∂ ∂

S. Lie (1890s)

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

Symmetry is group

13

Symmetry is invaraint

( )L d i i d dSη η ηγ γ γ= + =

Noether’s Theorem (1918)

Cartan’s formula

( )

is conversed.

d dS

S

γ η τ τ

γ η

⋅ ⋅ = ⋅

⋅ −

14

What is gyrosymmetry?

1 1x y

y x

v vB x v B y v

η⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎟⎜ ⎟⎜⎟ ⎟⎜= + + −⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜⎟∂ ∂ ∂ ∂⎜ ⎝ ⎠⎝ ⎠

Noether’sTheorem

Gyrophasecoordinate

Coordinate perturbation to find .

Q: How does the dynamics transform?

A: It transforms as an 1-form!

η

2 2

2x yv v

Bµ

+=

ηθ∂

=∂

Amatucci, Pop 11, 2097 (2004).

15

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

γ γ γ γ ε

γ γ γ ε

− ∗ −⎡ ⎤Γ = = = − +⎣ ⎦= − − ⋅ +

0

10

( )Continuous Lie group,

Vector field (Lie algbra), ,

g z Z g z

G dg d

G dg dε

ε

ε

ε

ε=

−=

: = ,

= / |

− = / | .

Zγ in Pullback

Cartan’s formula Insignificant

No need for Poisson bracket

Taylor expansion

16

0th order gyrocenter coordinates

Catto, 1979

[ ]

[ ]

[ ]

[ ]

0

0 0 02

00

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) 1

( ) ( ) ( )

( ) ( ) ( )( )

( )

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( ) ( ) ( )

x x

x x x

x x

x x

xx

x x x

u y y y y y

w y y y y y

y y

y y y

y y yy

B y

y y yy y

B yB y

y y u y y w y

ρ

, ≡ − ⋅ ,

, , ≡ − × × ,

, ⋅ , = ,

, ≡ × , ,

× −, ≡ .

×≡ , ≡ ,

≡ + , + ,

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

E B BD b

v D v b v c( )xy, .v

0 ( ) ( )g z t Z u w tθ: = , , = , , , ,x v X

1

( ) ( ) ( ),

sin ( ) ( ) ( ) ( ) ( )

u u w w

u w

ρ

θ

≡ + , , ≡ , , ≡ ,

≡ − ⋅ , = + + .

x X X v X v X v

c X e X v D X b X c X

17

Lie perturbations

0 1 2

0 1

( )

0, 0,

Noncanonical coordinates:

Z u w θγ γ γ γ

γ γθ θ

= , , ,= + + +...,

∂ ∂= ≠

∂ ∂

X

: ( ) 0 .g Z Z g Z such that γ θ→ = ∂ /∂ =

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 OZ ZZ Z i d Z dS

Z Z L Z

L L Z dS

γ γ γ ε

γ γγ γ γ

γ γ γ

γ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠

= + + ,

= ,

= − −

= −

+ − +

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 1 1

ε ε

ρ ρ ε

ρ ρ

= + , = + ,

× ×, , ,

⎛ ⎞ ⎛ ⎞∇ ∂ ∇ ∂⎟ ⎟⎜ ⎜⎟ ⎟, , ,⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜Ω ∂ Ω ∂⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞∇ ∂ ∇ ∂⎟ ⎟⎜ ⎜⎟ ⎟, , ,⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟Ω ∂ Ω ∂⎝ ⎠ ⎝ ⎠

∼ ∼ ∼

∼ ∼

∼ ∼

c c

E E B BE E t B B t

E E B BE E t B B t

E E E B B Bv B v BE E B B

Cary, Littlejohn 1980s

18

22

0 1( ) ( )2

γ φ εγ γ⎡ ⎤⎢ ⎥= + ⋅ − + = + + ,⎢ ⎥⎣ ⎦

vd dt OA v x

( )22 2 2

0 0002 2

Dw u wA u d d dtB

θ φγ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

+ += + + ⋅ + − + ,b D X

0 110 0

3

0 1 130 0

1 00

1 ( )2 2

1( ) ( )2

1( )2 2

ρ ρ ργ

ρ θ ρ µ

φ ρ ρ ρ ρ

⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥= ∇ ⋅ + + ⋅ × − ∇ ⋅ + + ⋅∇⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+ − ⋅ ⋅ + + ⋅ + + ⋅∇⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤⎛ ⎞∂ ∂⎟⎜⎢ ⎥− + + ⋅ − ⋅∇ ⋅ − + ⋅ .⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠∂ ∂⎣ ⎦

w w wu dB B

ww d dB B w

w wu dtt B t

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 εεε == , , | = , 11 ( ) 0 11( ) ( ) ( ) ( )G ZZ Z i d Z dS Zγ γγ= − + ,

in the 0th order gyrocenter coordinatesγ

19

†01 1 1 1

0

1 11 1 1

0

31

01 1 30

†1 0 1 1

0

1( )2 2

( )

1 ( )2

( )

θ

µ

γ ρ ρ

ρµ

ρθ

ρ θ

⎡ ⎛ ⎞⎟⎜⎢= × − +∇ + ∇ ⋅ + + ⋅ ×∇⎟⎜ ⎟⎜⎢ ⎝ ⎠⎣⎤ ⎡⎡ ⎤∂ ∂⎥ ⎢⎢ ⎥− ∇ ⋅ + + ⋅ + ⋅ + + + +⎥ ⎢⎢ ⎥∂ ∂⎣ ⎦ ⎣⎦⎡⎤ ∂⎢⎥+ + ⋅ + − + − ⋅ ⋅∇⎢⎥ ∂⎦ ⎣

⎤⎥+ + ⋅ + − ⋅ + +⎥⎦

u

u

w wZ G S uB

S Sw d du GB u

S wdw Gw B

w d uG BB

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 10 1 1

00

( )

12 2

µ φ ρ

ρ ρ ρ

⎡ ∂⎢ + − +⎢ ∂⎣

⎤⎛ ⎞∂ ∂⎟⎜ ⎥− ⋅ + ⋅∇ ⋅ + + ⋅ .⎟⎜ ⎟⎜ ⎥⎝ ⎠∂ ∂ ⎦

SGt

w wu dtt B t

X

D c aE b

1 0γ θ∂ /∂ = .

+

1 1, G S

FreedoomGyro-center gauges

under coordinate perturbationγ

20

( )

( )

( ) ( )

( )

0 1

2 2 2 2

0 0 00

2

1 10

2 2

2

Z Z Z

w u w DA u d d dt

B

wZ d H dt

B

γ γ γ

γ θ φ

γ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

= + ,

+ += + + ⋅ + − + ,

=− ⋅ − ,

b D X

R X

( )

( )

† ††

2 2

1 00 200

2 2

0 0 0 1200

0

44

24

w w uH u

BB B

w wR

BB

Rt

ψ

⊥ ⊥= − × ⋅ + ⋅∇×∇

− ∇⋅ − : ∇ − +

∂≡∇ ⋅ , ≡− ⋅ ,

∂

E B b b bB

E bb E

cR c a a

† †

†( ) ( )01 1 1

2

0

12

uu w

B

dπ

ψ φ ρ ρ

α α θ α α απ

⊥⎛ ⎞× ⎟+⎜ ⎟⎜≡ + − + + ⋅ + ,⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

≡ , ≡ − .∫

E b BX b c A X

Gyrocenterdynamics

in the 1st order gyrocenter coordinate, / =0, γ γ θ∂ ∂

0

( )( )

( ) ( )( )

Gyro-gauge invariant

R R XX

R R X tt

δ

θ θ δ

′

′

⎯→ +∇ ,

⎯→ + .= , , = , ,

∇= −∂/∂ ,∇ .

R X

( )†

† .

0

2

0 00 2

u

DB u

t tµ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

≡ ∇× + + ,

∂ ∂≡ −∇ + − −

∂ ∂

B A b D

b DE E

21

( )

( )

[ ]

†

† † †

† †

†

( )( )

( )

( )

21

1 020 0

1 1

0

2

1 1 0200

1 10

21

11 0300

1

2

2

2

u

S w wuu B B B B B

w S

B B B

w wuG

BB

wS

B

S w wG

BB

SG

µ

θ

ρ

ρ

ρθ

⊥

⊥

⎛ ⎞∂ ⎟⎜ ⎟⎜=− + + + ∇ ⋅ ×⋅∇⎟⎜ ⎟∂ ⎜ ⎟⎜⎝ ⎠

∇ + +− ∇ ⋅ × + ×

= × ⋅ ⋅ + ⋅∇ ⋅⋅∇

− ⋅∇ ⋅ − ⋅ ∇ + + ,

∂= − ⋅ + + ,⋅∇ ⋅∂

∂=−

X

X

BG b a b baa B

A XD a b b

B b G c b a ba B

b D a b A X

b Xa B c A

( )11

1w

ρµ

− + .⋅∂Xa A

( ) ( )

† †† ††

† †

† † ( )

01 1 101 00

2

00 3 2 20 0 0

2 2 3

0 02 20 00 0

0

2

2 2

ES u S Su S E B

t uB B

w wu wu B

B B B

w u wu wu wB BB B

wB

θ⊥⊥

⊥ ⊥

⎛ ⎞ ⎛ ⎞⋅∂ × ∂ ∂⎟ ⎟+⎜ ⎜⎟ ⎟⎜ ⎜+ + ⋅∇ + + + =⎟ ⎟⎜ ⎜⎟ ⎟∂ ∂ ∂⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎡ ⎤⎢ ⎥− × ⋅ ⋅∇ + ∇ ⋅ × − ∇ ⋅ ×⎢ ⎥⎣ ⎦

− : − ⋅∇ ⋅ + ⋅∇ ⋅ + ⋅∇ ⋅∇

∂+ ⋅

BE b B b

E B b aa a b b D a b

ca b a b b D a bB a B

a2

0 12002

w uwt B tB

ψ∂

− ∇ : + ⋅ + .∂ ∂D b

E aa a

1 1 and gyrocenter gauge Transformation Sg

22

2nd order

( ) ( ) ( )† †

†

2 2

2 0 1 11 1

11 1 1 1 1 1

00

1 1 1100

11 1

1 12 2

1 22 2

22

dt

u w

G GBB

G GBB

t

µ θ

µ θ

γ ψ

ψ

ψ µµµ

µµ

φ

⊥

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∗

=− ,

⎡ ⎤≡ ⋅ × × − + ⋅ ×⎣ ⎦

⎛ ⎞∂ ⎟⎜+ ⋅ + ⋅ − + ⋅ ,⎟⎜ ⎟⎜ ∂⎝ ⎠

≡ + + ,

∂≡− − .

∂

x

x

E G B b b c G B

aG E E E c

aG G c

AE ∇

23

Perturbation techniques — quest of good coordinates

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

24

Gyrocenter dynamics

0i dτ γ = .

† †

† †

† †

†

( )

0 00 0 2

0

0 0 1 20

2

2

12

0

du

dt B B

dudt B

d d B uB R

dt dt B

B

ddt

µ

θ

ψ ψµ

µ

⎡ ⎤⎢ ⎥⎣ ⎦

×= + ⋅ × −

⋅=

⋅= + ⋅ − + + ⋅ ×

∂− ⋅ − : + +

∂

=

X B b Eb b

B E

X ER b b

E bb E

∇

∇∇

∇ ∇

( ) †† †

†

0

2

0 0 1 2

2

u B

DB u

t tµ ψ ψ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

≡ × + + , = ⋅

∂ ∂≡ − + + + − −

∂ ∂

B A b D B b

b DE E

∇

∇

Banos drift

, driftB× ∇E B

drift by spacetime

inhomogeneties of 0E

Curvature drift

& curvature driftD∇ D

25

Gyrokinetic equations

( )0 0 6

.

∂= , ≤ ≤ .

∂

=

j

j

dZ F jdt Z

F F

0dZdtθ

⎛ ⎞∂ ⎟⎜ = ,⎟⎜ ⎟⎜⎝ ⎠∂

0F d du F

Ft dt dt u

∂ ∂+ ⋅ + = ,

∂ ∂XX

∇

0τ τ= = .L F i dF

26

Gyrokinetic field theory

3

1 ,3

d d dm

γ γ γΩ≡− ∧ ∧!

†

† †

† † †

†

1 2 3

j ii j ij t i j

j ili j i j l t

j ili j

B dx dx dx du d d

A b A u b H dt dx dx du d d

A H A A dx dx dt d d

A bH dx dx dt du dµ

µ θ

µ θ

µ θ

µ

⎡ ⎤⎢ ⎥,, ,⎣ ⎦⎡ ⎤⎢ ⎥,, , ,⎣ ⎦

,,

Ω = ∧ ∧ ∧ ∧ ∧

+ − − ∧ ∧ ∧ ∧ ∧

+ + ∧ ∧ ∧ ∧

− ∧ ∧ ∧ ∧ .

4d dA jπ∗ = ∗ ,

1( )

( )( ) ( 0 1 2 3)

( )x x

xj x F

A xα

π α

δγα

δ′ − ′

′= Ω ∧ , = , , , .∫ ∫

γ

27

Pullback of distribution function

[ ] ( )

pullback from ( ) :

( ) ( ) ( )

F Z

f z g F Z F g z∗= = .

Particle distribution

Don't know ( ).f z

Gyrocenterdistribution

28

( ) ( )

1 01 1

0 0

1F Fu F

t m u

S S S Su t t

t u

ψ

θ φθ

∂ ∂+ ⋅ = ⋅

∂ ∂

∂ ∂ ∂ ∂+ ⋅ + + = + , − ⋅ + ,

∂ ∂ ∂ ∂

b b

X X v A XX

∇ ∇

ρ ρ

Gyrocenter gauge theory for waves in magnetized plasmas

.0 1

0

F F F

const

= +=B

Gyrocenter gauge function

[ ( ) ]( )

( )

( ) ( )( )

( ) ( ) ( ) ( )

01 0 1

1

in

in in

e

e e

zy

z

z

nn

zn z

n nx y

z z

k FF J uA V J A

k u u

HS S

i k u

IS uA

i n k u

nI i I iV A V A

i n k u i n k u

θ

θ θ

φω

ω

λφ

ω

λ λλ ω ω

⊥

∗

=∞∗

=−∞

′

⊥ ⊥

− ∂= − +

− ∂

= −−

= −− +

+ + ,− − + − +

∑

Solution

29

( ) ( ) ( ) [ ] ( )3 6 1 6 10 1 0d v f t d Z F Z t g d Z g F gδ δ− ∗ −, , = , − = −= ∫ ∫ ∫j v r v V X r V X r

†

† †

†

†

[ ( ) ]

[ ( ) ] [ ( ) ]

[ ( ) ]

1 0

00 0

00

GS

g F F L F F t S F FuB B

F S Ft S t

uB

S Ft

θ θ µ

µ µ θ

∗ ∂= + = − × + , + ⋅ − ⋅

∂

∂ ∂ ∂ ∂+ ⋅ + , + + + , ⋅ +

∂ ∂ ∂ ∂

∂ ∂ ∂− + , ⋅ +

∂ ∂ ∂

b BA X

BA X A X

A X

ρ ∇ ∇ ∇

ρρ ∇ ρ

ρρ

( )[ ( )]( ) ( )

[ ( )] [ ( ) ] [ ( ) ]

61 1 0 1 0

1

0 0 01 0 1 1 1 0 0

e u g F F Z d Z

F S Fg F F F S

u

δ

θ θ µ

∗⊥

∗

= ,+ + + −

∂ ∂ ∂ ∂+ = + ⋅ + + + + ⋅ + .

∂ ∂ ∂ ∂

∫j v b X r

b A X A X

ρ

ρρ ∇ ρ

Pullback of current

Lab coord. 0th order gyrocenter

1Pullback of g

30

( ) ( ) ( ) ( )

[ ( ) ]

( ) ( )

[ ( ) ]

061 0 0

0 0 00

03 00

0 0 00

e

4

z

z

z

zX r

x

yp

z

Fr e d Z u A

u

F S Fik S

u

Fe d v u A

u

F S Fik S

u

A

AiA

δ

ρξ θ µ

θ θ µ

ωχ

π

φ

⊥

∗∗

⊥

∗∗

∂= + − + +

∂⎫∂ ∂ ∂ ∂ ⎪⎪+ + + ⋅ + ⎬⎪∂ ∂ ∂ ∂ ⎪⎭

∂− ⋅= + +∂⎫∂ ∂ ∂ ∂ ⎪⎪+ + + ⋅ + ⎬⎪∂ ∂ ∂ ∂ ⎪⎭

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜=− ⋅⎜⎜⎜⎜⎜⎜⎝ ⎠

∫

∫

j X r v b X

A X

v b X

A X

ρ ρ

ρ

ρ ∇ ρ

ρρ

⎟⎟⎟⎟⎟⎟⎟

Pullback of current

31

( )

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

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

( )

20

0

2 2 2 22 2

2

22 2

2

4 12

n

pn z

n z n n z n n z

x

n n z n z n nn n z

n z

e Fv dv du

i m un k u

n J k inJ J k nJ n nJ kv v vc c kc

inJ J k J k J J nv v v iJ J k vc c c

nJ kc

πχ π

ω ω

λ λ λ λ ω λλ λλ

λ λ λ λ λ ω λ λλ

λλ

=∞ ∞ ∞⊥⊥ −∞=−∞

′

⊥⊥ ⊥

′ ′ ′′

⊥ ⊥⊥ ⊥

∂=

Ω ∂− +

− − Ω − Ω

− Ω× −

∑ ∫ ∫

( ) ( ) ( )( ) ( )2

2n n z nV n zV

iJ J k J nv v u J k uc c

λ λ λ ω λ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⊥ ⎟⊥⎜ ⎟⎜⎝ ⎠

,

− − Ω −

( )

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

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

( )

20

0

2 2 2 2 2 2

2

2

2

4 12

n

pn z

n z n n z n nu

n n z n z n nn n

n

e Fv dv du

i m vn k u

n J k u inJ J k u n J n Jv vc cc

inJ J k u J k u inJ Jv v u inJ Jc c c

nJ

πχ π

ω ω

λ ω λ λ ω λ λλ λ λλ

λ λ ω λ ω λ λ λ λλ

λ

=∞ ∞ ∞⊥⊥ ⊥

−∞ ⊥=−∞

′

⊥⊥

′ ′ ′′

⊥⊥

∂=

Ω ∂− +

− − Ω − Ω

− − − − Ω× Ω

∑ ∫ ∫

( )( ) ( ) ( )( ) ( ) 22 2 nz n n z nunJ uk u iJ J k u nJ uu

c vc cv

λω λ λ ω λλ

′

⊥⊥

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟.⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− Ω−⎜ − Ω ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

Gyrocenter gauge susceptibility – same as classical theory

32

Gyrocenter gauge algorithm

0∂ ∂

+ ⋅∇ + = ,∂ ∂

F Fd duFt dt dt uX

X

( ) ( )0 0S S S S

u t tt u

θ φθ

∂ ∂ ∂ ∂+ ⋅ + + = + , − ⋅ + ,

∂ ∂ ∂ ∂X X v A X

Xρ ρ

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

1( )4

xd dA f

ππ

−∗ = Ω∫

Gyrokinetic eq.

Gyrocenter gauge

Pullback

Maxwell’s

33

Gyrocenter gauge data structure – Kruskal ring

Kruskalring

gyrocentergauge

species

[ ]

X

u

s n

µ

⋅

⋅⋅⋅

⋅

gyrocenter

replace gyrophase

replace particle

SciDAC proposal

34

( ), ,0 1 0 =d

z z ut Gdt µµ θ= + + = Ω

Example – stochastic resonant heating

( )B z

ˆ ˆcos sinx yE t E tω ω= +E x y

dSdt

φ= ≈− ⋅ρ Ε

xy

( )Resonant layer

reszω =Ω

Gyrocenter gauge

Gyrocenter

ˆuz

ˆ

1

2

d d d SG

dt dt dtdSdt B

µµ

θµ

θ θ ⊥

∂=− =−

∂∂ ∂

=− = ⋅ = ⋅∂ ∂

E Evρ

35

( )( )

sin ( )cos cos ( )sin

sin ( )cos cos ( )sin

sin

00

0

12

2 2

2 22

2

yx

t yxt

t

y xt res res

Ed Et t t t

dt B B

EEt t t t

B B

E Et

B

µθ ω θ ω

µ

µ µ θ ω θ ω

πµ θ ω

==

= ′

⎡ ⎤⎢ ⎥= − −⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥= | + − −⎢ ⎥⎣ ⎦

−≈ | + −

Ω

∫

( )

( )

sin ( )cos sin cos

cos ( )sin sin cos

0

0

24

24

t

res rest

t

res rest

t t t

t t t

π πθ ω θ ω

π πθ ω θ ω

′=

′=

⎛ ⎞⎟⎜≈− − ⎟⎜ ⎟⎜⎝ ⎠Ω⎛ ⎞⎟⎜≈ + − ⎟⎜ ⎟⎜⎝ ⎠Ω

∫

∫

( )22

2 4y xE Ev π⊥

′

−∆ =

Ω

Asymptotic for large ′Ω average overrandom phase

Stochastic resonant heating rate

36

Conclusions

Geometrically generalized Vlasov-Maxwell system

Geometic gyrokinetic theory

Gyrocenter gauge theoryand algorithm

Geometric field theory

Gyro-symmetry

For RF waves