Download Gyrokinetic Plasma Model

Introduction to the Particle In
Cell Scheme for Gyrokinetic
Plasma Simulation in Tokamak
a
Jae-Min Kwon
b,c
b
C.S. Chang , S. Ku , and J.Y. Kim
a
a
Korea National Fusion Research Institute
b
Courant Institute, New York University
c
Korea Advanced Institute of Science and Technology
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
M.R.Wade 2003
S.Either et al, IBM J. RES. & DEV. Vol. 52 2008
Gyrokinetic Plasma Model
Original Vlasov Equation
Gyrokinetic Description
Drift Kinetic Description

 
1
 ( R, v|| ,  , t )   ( R   g , v|| ,  , t )
N g
Gyrokinetic Plasma Model
Gyrokinetic Theory

( X ,V|| ,  ,  )
6-d phase space
 : rapid variation in  i1  10 8 sec

dV|| Fi
Fs dX

  Fs 
CS
t
dt
dt V||

( X ,V|| ,  )
5-d phase space
 : averaged out perturbati vely

Fs  Fs ( X ,V|| ,  )
Basic assumptions
 k||   i
~
~ ~
 O( )
 i k  Te Ln
k i ~ O(1)
Simulation Methods
Eulerian Simulation Method
 
 
F ( x , v , t )   f ijnkl (t ) ijnkl ( x , v )
ijnkl
d
f ijnkl  ...
dt
Lagrangian Simulation Method
 
 
 
F ( x , v , t )   w p (t ) ( x  x p (t )) (v  v p (t ))
p
d 
x p  ... ,
dt
d 
v p  ... ,
dt
d
w p  ...
dt
Gyrokinetic Plasma Model
Equations of motion for gyrocenter
ˆ
d 

b
*
X  V||bˆ  bˆ   B    
dt
B
B

X
d
V||  bˆ *    B  bˆ *   
dt
ˆb *  bˆ  V|| bˆ  bˆ   bˆ
B
Gyro-averaged potential
(potential felt by the charged ring)

1
( X ,  , t ) 
2

2
0
 
 
1 N
( X   , t )d   ( X   g , t )
N g 1
Gyrokinetic Plasma Model
Gyrokinetic Poisson Equation


 
1
 i2 2
b 
~
~
2 2





exp(
i
k

x
)











4

n

n
 1  b k
i  
2 2
i
e
2
1



D
k
i

     4 [ni  ne ] ,
2
i
2

2
D
k   1
2

2
i
Charge density from gyro-ring

 

 6
ni ( x, t )   Fi ( X ,V|| ,  , t ) [(X   i )  x ]d z
Adiabatic Electron Response Model
 e(   s ) 
ne  ni 0 exp 

T


e
Simulation Methods
Delta-f Simulation Scheme

dV|| Fs
Fs dX

 Fs 
CS
t
dt
dt V||



 dV||
dV||  

dX
dX 

( Fs 0  Fs )  

 ( Fs 0  Fs )  

( F  Fs )  C  S

 V|| s 0
 dt

t
dt
dt
dt
0
1

0
1


dV|| 

dX
Fs 0 
 Fs 0 
Fs 0  C  S
Assume Fs 0  FMaxwellian
t
dt 0
dt 0 V||







dV
dV
dV|| 

dX
dX 


dX
||
||



Fs 

 Fs 

F   Fs 0 
 Fs 0 
Fs 0
 dt
 V|| s
 dt

t
dt
dt

t
dt
dt

V
||
0
1
1

0
1
1

We solve this part only !
Noise Reduction by Fs / Fs 0 ( 1 for tokama k core)
Tokamak Geometry



B



Modes tend to be aligned to the magnetic field direction.
Efficient representation in the field aligned coordinate : ( ,   q )
Parallelization
Processor 2
Processor 1
Processor 0
Processor N-1
toroidal direction
• 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
Decomposition by Toroidal Mode Number
+
+
Processor 0
Processor 1
+ …… +
Processor 2
Grid system based on quasi-ballooning coordinate
( , ,  , t )    n ,ij (t )Qi ( )Q j ( ) exp[ in (   i ( ))]
n ,ij
 n,ij (t )   n,ij  N (t ) exp[ in i (2 )]

 i ( )  
0

B  

d 
B     
Quadratic spline representation
of the slowly varying part
i
Spatial Grid Requirements
Radial Direction
: 1/  r ~ kr  1/ i
Poloidal Direction : 1 /  ~ k||  k r
Toroidal Direction : 1 /   ~ k  k / q
Processor N-1
Field Solver
Gyrokinetic Poisson Solver

  
 2


ti2
1
      2      2   4e f i ( R, v|| ,  , t ) ( R  i  x )d 6 z  4eg ( x )
Di
De


 ti2 
 n2

1 
   1  2     n,ijQi ( )Q j ( ) exp[ in(  i ( ))]    2  2 n,ijQi ( )Q j ( ) exp[ in(  i ( ))]  4eg ( x )
De 
n,ij 
n,ij  R
 Di 
Multiply n,ij element and integrate over x



 1
2 
n2 
2   RdRdZ exp[ in  i ( )] Qij  1  2ti    Qij   2  2 QijQij n ,ij exp[ in  i ( )]
ij 
 Di 
 De R 



 4e  Rd dRdZQij exp[ in (   i ( ))] g ( x )  sn ,ij
 ti2 

 1
n2 
LHS  2   RdRdZ 1  2   Qij exp[ in i ( )]   Qij exp[ in i ( )]   2  2 QijQij exp[ in  i ( )   i ( ) ]n ,ij
ij 
 De R 
 Di 

2
  2   Qij   Qij  n QijQij   i     i   1

n2 
ti 





 2   dRdZR 1  2 
  2  2 QijQij n ,ij exp[ in  i ( )   i ( ) ]   M n ,ij,ijij
ij 
i j 
 Di   in  Qij  Qij   i  Qij   i   Qij   De R 

M
ij 

n ,ij ,ij  n ,ij 
 sn ,ij
: solved by sparse matrix solver (multi-grid, umfpack)
fewer grid points, faster computation
Field Solver
Evaluation of Turbulent Electric Field


Qi ( )
Qi ( )

 0,ij
Q j ( )  2 Re   n,ij
Q j ( ) exp[ in (   i ( ))]



ij
ij,n1~ N


Q j ( )
 Q j ( )

 ( )



   0,ij Qi ( )
 2 Re    n ,ij Qi ( ) exp[ in (   i ( ))]
 in i
Q j ( ) 




ij
 

ij,n 1~ N




 2 Re   (in)n,ij Qi ( )Q j ( ) exp[ in(   i ( ))]

ij,n1~ N

Conserved energy

ti2
ti2
1
2
2
2
1
 1  3 
2
3 
Z 2
Z 2
d
z
m
v
f

d
x









d
x










 

   const
2
2
2
  2 i i  8  



Di
De
Di




6

ti2
1
N * n 
2
2
2
*
0
n,l M lln,l 
 d x   2Di   2De    0,l M ll0,l  2 Re 
n 1

3
Simulation Procedure
Ex) 2
nd
order Runge-Kutta)
calculate source : z np
solve field equation :  n
Start
Load initial profiles
Setup Grid System
Load marker particles
push marker particles :
t
z *p  z np   z np ,  n
2


calculate source : z *p
solve field equation : *
Diagnosis
push marker particles :
End

z np1  z np  t z *p , *

ITG Mode Simulation
n  10
n  15
n  20
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
0.6
0.8
1.0
1.2
1.4
0.6
0.8
1.0
1.2
1.4
0.6
0.8
1.0
1.2
1.4
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
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
-0.2
-0.4
-0.4
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
t  10  transit
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
t  20  transit
ITG Mode Simulation
Thermal flux time history
(normalized by local gyroBohm level)
t  21 transit
r / R0  1.7
Electromagnetic Turbulence Simulation
p|| - Formulation, neglecting compressional Alfven modes

f s dR
f

 f s  p || s  0
t
dt
p||
p||  v|| 
q
 A|| 
mc


f s  f s 0 ( R, p|| ,  , t )  f s ( R, p|| ,  , t )

 m 
f s 0 ( R, p|| ,  , t )  n s 0  s 
 2Ts 
3/ 2
 m

 m 
exp  s p||2  v 2   n s 0  s 
 2Ts

 2Ts 




dR 
q
 B* bˆE  c
  p|| 
 A||  

  B E  c  
dt 
mc
 B* B*  q


dp||
B 
q

  *   BE    
dt
B*  m
m

      
p||
c
 A|| 

3/ 2
 1
exp 
 Ts
 m s p||2


 B 
 2



 i2 2   42D [ni  ne ]
  pe 
4
 A|| 
  2 A||  
j||i  j||e
c
c


2


Electromagnetic Turbulence Simulation
Cancellation Problem : Curse of the large terms
  pe
 2 A||  
 c
2

4
 A||  
j||i  j||e
c




analytic skin term
j    d vq p f    d vq p h



3
||e

e

||
e

3
e
||
e 
qe f e 0
Te
p


   || A|| 
c


 qe2

3
2
non ad
  d vqe p||he  
d
vA
p
f
 j||ad
||
||
e
0
e
  j||e

 cTe

3

 2 A||
pe / c  A||
2
~
2s
L2
4
 ad
  A|| 
j||i  j||non
e
c
2



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 !

Radial transports by Coulomb collision and RF heating
RF resonance plane


V V 
Banana width random walk for a
trapped particle by Coulomb collision.
V V
Perpendicular velocity change of a
trapped particle by RF heating at
resonance plane.
Banana tips move to the resonance plane
Velocity space at outer mid-plane
V2m
V||m
Kinetic energy of resonant particle :
E
Increase of kinetic energy by RF heating :
Turning points :
Bt 
E


1
mIV||2  BR
2
E  BR 
E  BR 
 BR
  
C.S. Chang et al, Phys. Fluids B2, 2383(1990)
Critical slowing down speed
RF heating
Slowing down by electron collision
Pitch angle scattering by ion collision
G.D.Kerbel et al, Phys. Fluids B2, 3629(1985)
Coulomb Collision Operator

 

 
 1  2   v v  0
    v  1 0  1  2   v v  1 0 
   v  0
C[f s , f ]  C[ f , f s ]     
f s  
f s     
f s  
f s 
   
   
v  t
2

v

v

t

v

t
2

v

v

t







0
s
0
s
Csls
ps f s0
Csls
MC collision of marker particles
against Maxwellian background
ps f s0
Weight modification (for momentum
and energy conservation)
Monte Carlo implementation of the Coulomb collisions RF scattering

 c
2
1 2
1 2 c
  c
c
c
C  Dc 
 ||1f s 
 1f s 
 ||f s 
 ||2f s 
 2f s
2
2

V



V


2
2



V

||
||
 ||

l
ss












 
c 2 


 ||c c
||

c


V  V  t  2 3 ( R2  0.5)   2  c t  2 3 ( R1  0.5) c  ||2 t

 ||2 
 ||2


2
2
0
c
1
V||  V||0  ||c1t  2 3 ( R1  0.5)  ||c2 t
R1 , R2 : uniform random numbers in [0, 1]
Weight modification ensuring momentum and energy conservation
2


d  V
 Vth    
w  3  ( y)  V  p( x )  3  ( y)   th E
Z.Lin et al, PoP 2, 2975(1995)
2
2
dy  V
V 
2 d
 
2 d 
Average momentum and energy changes of
E  2 V 2  TP
p ( x )  2 V  TP
3Vth dt
Vth dt
marker particles by the test particle collision part
Resonant Ion and RF Interaction Model

  rf V  rf 
qI 
Qrf   V   E   B  f I
mI
c


Quasi linear heating
operator
: RF wave induced velocity space flux
(interaction by the RF field component with right
circularly polarized fundamental harmonics only)
C.F. Kennel and F. Engelmann, Phys. Fluids 9, 2377(1966)
  rf
 rf
2
1  2 rf
1  2 rf 
rf
Qrf ( f )  Drf 
 ||s f 
 s f 
 || f 
 || f 
 f 
2
2

V



V


2
2



V
 ||

||
||

 V|| 
1  
 V 
p 

 rf
v||rfs  2 *
1
Vp
b  k 
Vp 
k||






2
 2

 V||  2

rf
*  J 0
2


v s  2  2  2 1 
J 0  bJ 0 J 0 
J 0  bJ 0 J 0


 V p

 Vp 
Z 2e 2
2
  V2
 *   rf  c  k||V||  Drf  I 2 E
8mI





J 2  V 
v||rf  4 * 0 1  || 
V p  V p 
RF-resonance condition
rfn ( Bn , N h , k|| , En , n )  N h

qBn
2
En  n Bn   rf
 k||
mc
m

J 2
v  2 0 2
Vp
rf
||
*
 V 
v  8 J  1  || 
 V 
p 

rf

*
2
0
2
Neoclassical Radial Electric Field
2.0x10
3
E (V/m)
0.0
-2.0x10
-4.0x10
-6.0x10
-8.0x10
-1.0x10
Simulation Result
Neoclassical Theory
3
3
Vi|| 
cTi
eB
 d ln Ti d ln Pi e 

 Er 
K
dr
dr
Ti 

3
3
4
0.00
0.05
0.10
0.15
r/R0
0.20
0.25
Resonant Ion Distribution Function
HFH
r/R0 = 0.12 (2)
4
HFH
r/R0 = 0.16 (3)
4
t = 34 ms
t = 34 ms
t = 34 ms
3
3
3
2
2
2
1
1
1
0
0
-4
4
-2
0
2
4
HFH
r/R0 = 0.21 (5)
0
-4
4
t = 34 ms
-2
0
2
4
HFH
r/R0 = 0.24 (6)
-4
4
t = 34 ms
3
3
2
2
2
1
1
1
0
-4
-2
0
2
4
-2
0
2
4
HFH
r/R0 = 0.27 (7)
t = 34 ms
3
0
HFH
r/R0 = 0.19 (4)
4
0
-4
-2
0
2
4
-4
-2
0
2
4
Resonant Ion Distribution Function
LFH
r/R0 = 0.12 (2)
4
LFH
r/R0 = 0.16 (3)
4
t = 34 ms
t = 34 ms
t = 34 ms
3
3
3
2
2
2
1
1
1
0
0
-4
4
-2
0
2
4
LFH
r/R0 = 0.21 (5)
0
-4
4
-2
0
2
4
LFH
r/R0 = 0.24 (6)
-4
4
3
3
3
2
2
2
1
1
1
0
0
-2
0
2
4
-2
0
2
4
-2
0
2
4
LFH
r/R0 = 0.27 (7)
t = 34 ms
t = 34 ms
t = 34 ms
-4
LFH
r/R0 = 0.19 (4)
4
0
-4
-2
0
2
4
-4
Future Directions
• 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