Introduction to the Particle In Cell Scheme for Gyrokinetic

Plasma Simulation in Tokamak

a Korea National Fusion Research Instituteb Courant Institute, New York University

c Korea Advanced Institute of Science and Technology

Jae-Min Kwona

C.S. Changb,c, S. Kub, and J.Y. KimaC.S. Changb,c, S. Kub, and J.Y. Kima

Feb. 14 ~ 15. 2008

Laboratory, Space/Astrophysical Plasma Workshop, POSTECH

Contents

I. Gyrokinetic plasma model

II. Particle In Cell(PIC) simulation method

III. Delta-F simulation method

IV. Numerical implementations In tokamak geometry

V. Ion Temperature Gradient(ITG) mode simulations

VI. Neoclassical simulation of tokamak plasma

VII.Future directions

Introduction

• Plasma turbulence causes a rapid loss of the plasma energy and particles to the tokamak wall.

• From 1994, numerical simulation of tokamak plasma turbulence has been done with gyrofluid and gyrokinetic approaches.

• The gyrokinetic approach is the fundamental one including all necessary features of the plasma turbulence responsible for the anomalous transports.

Introduction

S.Either et al, IBM J. RES. & DEV. Vol. 52 2008S.Either et al, IBM J. RES. & DEV. Vol. 52 2008

M.R.Wade 2003

Original Vlasov Equation

Drift Kinetic Description

Gyrokinetic Description

g

g tvRN

tvR ),,,(1

),,,( ||||

Gyrokinetic Plasma Model

Gyrokinetic Theory

6-d phase space

),,,( || VX

sec10in variationrapid : 81 i

)1(~ Ok i)(~~~ ||

OLTk

k

n

i

ei

SCV

F

dt

VdF

dt

Xd

t

F is

s

||

||

),,( || VXFF ss

Basic assumptions

),,( || VX

velyperturbatiout averaged : 5-d phase space

Gyrokinetic Plasma Model

Eulerian Simulation Method

Lagrangian Simulation Method

...

),()(),,(

ijnkl

ijnklijnklijnkl

fdt

d

vxtftvxF

... , ... , ...

))(())(()(),,(

ppp

pppp

wdt

dv

dt

dx

dt

d

tvvtxxtwtvxF

Simulation Methods

Equations of motion for gyrocenter

B

bBb

BbVX

dt

d ˆˆˆ*

||

**||

ˆˆ bBbVdt

d

bbbB

Vbb ˆˆˆˆˆ ||*

N

gg tX

NdtXtX

1

2

0),(

1),(

2

1),,(

Gyro-averaged potential

(potential felt by the charged ring)

X

Gyrokinetic Plasma Model

eiD

nn

4~1

2

1 , ][4 22222 ieiDi knn

Gyrokinetic Poisson Equation

e

sie T

enn

)(exp0

zdxXtVXFtxn iii6

|| ])[(),,,(),(

Charge density from gyro-ring

Adiabatic Electron Response Model

Gyrokinetic Plasma Model

2222

22

1)exp(

1~

ii

ik

k

xkib

b

Delta-f Simulation Scheme

SCFFVdt

dV

dt

dVFF

dt

Xd

dt

XdFF

t

SCV

F

dt

dVF

dt

Xd

t

F

ssssss

ss

s

)()()( 0||1

||

0

||0

10

0

||

||

SCFVdt

dVF

dt

XdF

t sss

0||0

||0

0

0

0||1

||0

1

0||1

||

0

||

10

ssssss FVdt

dVF

dt

XdF

tF

Vdt

dV

dt

dVF

dt

Xd

dt

XdF

t

Maxwellians FF 0 Assume

We solve this part only !

core)k for tokama 1( /by Reduction Noise 0 ss FF

Simulation Methods

B

Tokamak Geometry

Modes tend to be aligned to the magnetic field direction.

Efficient representation in the field aligned coordinate : ),( q

• Straightforward domain decomposition beyond the plasma boundary.• Relatively low memory and communication costs.• Hard to apply high order (> 2) time integration scheme

(needed for fast ion species, electrons)

Parallelization

direction toroidal

Processor N-1

Processor 0

Processor 1

Processor 2

Decomposition by Toroidal Mode Number

0)( d

B

B

i

i

Grid system based on quasi-ballooning coordinate

Quadratic spline representation of the slowly varying part

ijn

ijiijn inQQtt,

, ))]((exp[)()()(),,,(

)]2(exp[)()( ,, iNijnijn intt

Spatial Grid Requirements

qkk

kk

k

r

irr

/~/1 : Direction Toroidal

~/1 : Direction Poloidal

/1~/1 : Direction Radial

||

+ + + …… +

Processor 0 Processor 1 Processor 2 Processor N-1

Parallelization

Gyrokinetic Poisson Solver

)(4)(),,,(41 6

||22

22 xegzdxRtvRfe ii

DeDi

ti

)(4))]((exp[)()(1

))]((exp[)()(1,

,22

2

,,2

2

xegniQQR

nniQQ

jinijijin

Dejinijijin

Di

ti

Multiply n,ij element and integrate over x

ijniij

jiijinjiij

Deji

Di

tiiji

sxgindRdZQRde

inQQR

nQQinRdRdZ

,

,2

2

22

2

)())]((exp[4

)](exp[1

1)](exp[2

jijijiijn

jiiijinjiij

Dejiiijijiij

iijiijjiij

Di

ti

jijiniijiij

Deijiiij

Di

ti

MinQQR

n

QQQQin

QQnQQdRdZR

inQQR

ninQinQRdRdZLHS

,,,2

2

2

2

2

2

,2

2

22

2

])()(exp[1

12

])()(exp[1

)](exp[)](exp[12

ijnji

jinjiijn sM ,,,,

: solved by sparse matrix solver (multi-grid, umfpack)

fewer grid points, faster computation

Field Solver

Evaluation of Turbulent Electric Field

Nnijij

iijn

ijj

iij inQ

QQ

Q

~1,,,0 ))]((exp[)(

)(Re2)(

)(

Nnijj

ijiiijn

ij

jiij Qin

QinQ

QQ

~1,,,0 )(

)()())]((exp[)(Re2

)()(

Nnijijiijn inQQin

~1,, ))]((exp[)()()(Re2

constxdxdfvmzd Z

Di

tiZ

DeDi

tiii

2

2

2232

2

2

2

22326 1

8

1

2

1

Conserved energy

N

nln

nlllnllll

DeDi

ti MMxd1

,*,,0

0*,0

2

2

2

2

223 Re2

1

Field Solver

Ex) 2nd order Runge-Kutta) npz : source calculate

n :equation field solve

nnp

npp z

tzz

,

2

: particlesmarker push

*

* : source calculate pz* :equation field solve

**1 ,

: particlesmarker push

p

np

np ztzz

Start

Load initial profiles

Setup Grid System

Load marker particles

End

Diagnosis

Simulation Procedure

0.6 0.8 1.0 1.2 1.4

-0.4

-0.2

0.0

0.2

0.4

0.6 0.8 1.0 1.2 1.4

-0.4

-0.2

0.0

0.2

0.4

0.6 0.8 1.0 1.2 1.4

-0.4

-0.2

0.0

0.2

0.4

10n 15n 20n

ITG Mode Simulation

turbulent potential at t=110 turbulent potential at t=160

ITG Mode Simulation

zonal potential at t=110 zonal potential at t=160

ITG Mode Simulation

Ion Heat Flux (normalized by gyro-Bohm level)

ITG Mode Simulation

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

-0.4

-0.2

0.0

0.2

0.4

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

-0.4

-0.2

0.0

0.2

0.4

transitt 20transitt 10

ITG Mode Simulation

Thermal flux time history

(normalized by local gyroBohm level) 7.1/

21

0

Rr

t transit

ITG Mode Simulation

Electromagnetic Turbulence Simulation

0||

||

p

fpf

dt

Rd

t

f ss

s

Bpm

TT

mnvp

T

m

T

mntpRf

tpRftpRff

s

ss

ss

s

s

s

sss

sss

2

1exp

22exp

2),,,(

),,,(),,,(

2||

2/3

022

||

2/3

0||0

||||0

|||||| Amc

qvp

cB

q

c

B

b

B

BA

mc

qp

dt

RdE

E **

*||||

ˆ

m

qB

mB

B

dt

dpE

*

*||

|||| Ac

p

p|| - Formulation, neglecting compressional Alfven modes

eipe jj

cA

cA ||||||

2

||2 4

][4 222eiDi nn

eipe jj

cA

cA ||||||

2

||2 4

ade

adnonee

e

eee

e

eeeeeee

jjfpvAdcT

qhpvqd

Ac

p

T

fqhpvqdfpvqdj

||||02||||

32

||3

||||0

||3

||3

||

Cancellation Problem : Curse of the large terms

analytic skin term

numerical adiabatic current

The analytic skin term and the numerical adiabatic current should be cancelled very accurately !

The problem gets worse for long wave length modes !

adnonei jj

cA

||||||2 4

2

2

||2

||2

~/ LAc

As

pe

Electromagnetic Turbulence Simulation

Perpendicular velocity change of a

trapped particle by RF heating at

resonance plane.

Radial transports by Coulomb collision and RF heating

Banana width random walk for a

trapped particle by Coulomb collision.

VV

RF resonance plane

VV

Banana tips move to the resonance plane

RI BVmE 2||2

1

RBE

Kinetic energy of resonant particle :

Increase of kinetic energy by RF heating :

Turning points :R

Rt B

BEEB

Velocity space at outer mid-plane

RF heating

Slowing down by electron collision

Pitch angle scattering by ion collision

2mV

mV||

Critical slowing down speed

C.S. Chang et al, Phys. Fluids B2, 2383(1990) G.D.Kerbel et al, Phys. Fluids B2, 3629(1985)

sc

sc

sc

sc

sc

clss ff

Vf

Vff

VDC

22

2

2||2||

2

||||

2

11|||| 2

1

2

1

tRtVV cc 2||11||0|||| )5.0(32

tRtRtVV c

c

c

c

ccc

2||2||

||1

2||

2

||221

20

2 )5.0(32)5.0(32

: , 21 RR uniform random numbers in [0, 1]

Monte Carlo implementation of the Coulomb collisions RF scattering

Weight modification (for momentum and energy conservation)

012

01020

00

2

1

2

1],[],[ ssssssss f

t

vv

vvf

t

v

vf

t

vv

vvf

t

v

vffCffC

0ss fpMC collision of marker particles

against Maxwellian background

lssC

lssC

0ss fp

Coulomb Collision Operator

Weight modification ensuring momentum and energy conservation

EV

V

dy

dyxpV

V

Vyw thth

)(

23)()(

23

2

TPth

Vdt

d

Vxp

2

2)( TP

th

Vdt

d

VE 2

23

2 Average momentum and energy changes of

marker particles by the test particle collision part

Z.Lin et al, PoP 2, 2975(1995)

Irfrf

VI

Irf fB

c

VE

m

qQ

: RF wave induced velocity space flux

Quasi linear heating operator

(interaction by the RF field component with right

circularly polarized fundamental harmonics only)

ffV

fV

ffV

DfQ rfrfrfrfs

rfsrfrf

2

2

||2||

2

||||

2

|||| 2

1

2

1)(

0020

||*|| 1

12 JbJJ

V

V

Vv

pp

rfs

00

20

2

||

2

20* 122 JbJJ

V

V

V

Jv

pp

rfs

pp

rf

V

V

V

Jv ||

20*

|| 14 2

20*

|| 2p

rf

V

Jv

2

||20

* 18

p

rf

V

VJv

kb||k

V rfp

||||

* Vkcrf 2

2

22

8 Em

eZD

I

Irf

2V

C.F. Kennel and F. Engelmann, Phys. Fluids 9, 2377(1966)

Resonant Ion and RF Interaction Model

RF-resonance condition

rfnnnn

hnnhnrfn BE

mk

mc

qBNEkNB

2),,,,( ||||

0.00 0.05 0.10 0.15 0.20 0.25-1.0x104

-8.0x103

-6.0x103

-4.0x103

-2.0x103

0.0

2.0x103 E (V/m)

r/R0

Simulation Result Neoclassical Theory

Neoclassical Radial Electric Field

r

i

iiii E

T

e

dr

Pd

dr

TdK

eB

cTV

lnln||

Resonant Ion Distribution Function Resonant Ion Distribution Function

-4 -2 0 2 40

1

2

3

4

HFHr/R

0 = 0.19 (4)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

HFHr/R

0 = 0.27 (7)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

HFHr/R

0 = 0.24 (6)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

HFHr/R

0 = 0.21 (5)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

HFHr/R

0 = 0.16 (3)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

HFHr/R

0 = 0.12 (2)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

LFHr/R

0 = 0.21 (5)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

LFHr/R

0 = 0.24 (6)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

LFHr/R

0 = 0.27 (7)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

LFHr/R

0 = 0.19 (4)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

LFHr/R

0 = 0.16 (3)

t = 34 ms

-4 -2 0 2 40

1

2

3

4

LFHr/R

0 = 0.12 (2)

t = 34 ms

Resonant Ion Distribution Function Resonant Ion Distribution Function

• Efficient schemes for electromagnetic simulation (including compressional branches)

• Realistic simulation conditions including various sources, correct neoclassical equilibrium

• Full-F simulation for the tokamak edge plasmas

• Transport simulation near the marginality, comprehensive transport model for fusion devices

Future DirectionsFuture Directions