1
Gyrokinetic Simulation:Introduction and Prospect for
Astrophysics
J.M. Kwon
WCI Center for Fusion Theory, NFRI, Korea
East Asia Numerical Astrophysics MeetingYITP, Kyoto, Japan
Oct. 29 ~ Nov. 2, 2012
2
Micro Turbulence in Fusion Device• aaa
G. McKee et al., Plasma and Fusion Research 2007
3
Micro Turbulence in Fusion Device• Source of trouble and reason… why fusion people want to
build ever larger machine
ITER
4
5
Outline
• Introduction to gyrokinetic theory
• Issues in gyrokinetic simulation
• Numerical methods for gyrokinetic simulation
• Summary
6
Introduction to Gyrokinetic Theory
7
Basic Idea of GK Theory
• GK orderings
– small fluctuation: ~ ~ ~– low frequency: ~– anisotropic fluctuation: ∥ ~, ~1– mild non-uniformity in plasma profiles, background
magnetic field etc.: ~ Free energy to drive turbulence(GK with strong gradient, T.S. Hahm ‘09)
Fast MHD waves and cyclotron waves are ruled out (high freq. GK, H. Qin ’99)
T.S. Hahm, Phys. Fluids 31, 2670 (1988)A. Brizard, T.S. Hahm, Rev. Mod. Phys. 79, 421(2007)
8
Basic Idea of GK Theory (cont’d)
• Guiding center transformationparticle space , ↔ guiding(gyro) center space , ∥, ,
+
: gyrophase-angle → average out = − , = × , Ω = ∥ = ⋅ , = = ∥ +
9
Basic Idea of GK Theory (cont’d)• Schematics of guiding center transformations in GK simulation
ü Solve Vlasov equation in guiding center space and evaluate ( , )ü Transform ( , ) to particle space
ü Solve Maxwell equations to obtain EM fields
ü Transform EM fields to guiding center space
Particle Space Guiding Center Space
Vlasov Equation (6D)
Maxwell Equation (3D)
Vlasov Equation (5D)
Maxwell Equation (3D)
10
Gyrokinetic Vlasov Equation
• Transform original 6D Vlasov equation in particle space into
guiding center space
• Take gyro-angle average to remove → reduction to 5D (, ∥, , ), large time step Δ > 1/Ω + ∥∗ + × + × ⟨⟩ ⋅ + −∗ ⋅ − ∗ ⋅ − 1 ∥ ∥ = 0
= − ∥ ∥∗ = + ∥ × ⋅ ⟨⋅⟩ = gyro-phase averaged fluctuations
11
• Solved in particle space i.e. no guiding center transformation
• Evaluation of , in particle space è pull-back transformation
of sources (i.e. from guiding center space to particle space) is
needed: , ∥, , → (, , )
Gyrokinetic Maxwell Equation
− , = 4(, ) −∥(, ) = 4 (, )
(, ) = ∫ ∗∥ + − + ( − )(, ) = ∫ ∗∥ + − ∥ + ( − )
12
• We need to solve to the following forms:
• If we take long wave length limit, we can simplify these as
Gyrokinetic Maxwell Equation (cont’d)
, = −4 −⋅ 1 − Γ ( − 1 ∥)∥ , = −4 −⋅ 1 − Γ ( − Π∥)
≡ ∫ ∗∥ + − ≡ ∫ ∗∥ ∥ + − (long wave length limit → b = ≪ 1, Γ ≈ 1 − , Γ ≈ 1)−(1 + ) , = 4−∥ = 4
(A. Brizard, T.S. Hahm, Rev. Mod. Phys. 2007)
13
Simple Minded View on GK Theory• Gyrokinetic description of magnetized plasmas
( )BBvmcq
mqv
dtd
vxdtd
rrrr
rr
ddf +´+Ñ-=
=
0
||**
||
*||
1ˆˆ
ˆˆˆ
Atc
bBbvdtd
BbcBb
BbvR
dtd
ddfm
dym
¶¶
-Ñ×-Ñ×-=
Ñ´+Ñ´+=r
Motion of charged particle
Motion of charged ring centered at
14
Elementary Plasma Physics• × drift motion of charged particle
è drift motion of gyration center in × direction
15
( )22
2||
||||
ˆˆˆ
B
Bc
Bq
BbBvm
cB
bAbv
dt
Rd
s
ss
ñÑá´+
Ñ´++
úúû
ù
êêë
é ´ñÑá+@
dfmd
rr
Closer look at GK equations
0)()()( *||
||
** =¶¶
+¶¶
+¶¶ fB
dtdv
vfB
dtRd
RfB
t
r
r
||||1ˆˆ Atc
bBbvdtd ddfm
¶¶
-Ñ×-Ñ×-@
è GK equations of motion are nothing but a combination of familiar drift
motions ensuring phase space volume conservation and making them
Hamiltonian flows
ExB drift
mirror force
Parallel motion along perturbed magnetic field
Grad-B + Curvature drift
Parallel E-field ∗ = + ∥ × ⋅ ≈ + ∥ × ∇/Low beta
16
Simple Minded View on GK Theory• What is gyro-average 〈 〉 and how to compute it?
→ Gyro-averaged field is nothing but field felt by “charged ring”
åò
ò ò
=
+@+=
-+º
N
llii
i
XN
dX
xxXxdd
1
2
0
2
0
))((1))((21
)())((21
qrdfqqrdfp
dfqrdqp
df
p
p
rrrr
rrrrr Xr
)( li qrr
or in Fourier space (as is often done in continuum codes)
( )( )
( ) ( ) òò ò
ò òò
ò ò
×^^×
+×
÷÷ø
öççè
æW
=þýü
îíì=
þýü
îíì
=+=
-+º
^ kdevkJkkdedek
dkdekdX
xxXxdd
Xki
i
Xkiik
Xkii
i
i
i
rrrr
rrrr
rrrrr
rrrr
rrr
03
2
0
cos3
2
0
)(3
2
0
2
0
)(ˆ2
1)(ˆ21
21
)(ˆ2
121))((
21
)())((21
fdp
qfdpp
qfdpp
qqrdfp
dfqrdqp
df
p qr
p qrp
p
Integration can be approximated by a few points sum
17
Simple Minded View on GK Theory
• Charged rings have stronger shielding effect than point particles−(1 + ) , = 4ü Additional shielding by polarization charges carried by charged ringsü Significantly enhanced as compared to Debye shielding
Charge density from charged rings
• Charged rings have no response to parallel direction i.e. no such thing as polarization current in parallel direction−∥ = 4 Parallel current carried by charged rings
18
• Gyrokinetic equation for guiding center distribution
Simple Minded View on GK Theory
0||
|| =¶¶
+¶¶
×+¶¶ f
vdtdv
fRdt
Rdft
r
r
å=Ñ+-s
ssDi
i Nqpdflr 4)1( 2
2
2
Hyperbolic + Elliptic PDEs
- Standard numerical techniques can be employed
- Issue is mainly problem size and computational cost
- Blind choice of scheme can easily end up with practically unsolvable one
• Gyrokinetic Maxwell Equation
å=Ñ- ^s
sJcA ||||
2 4pd
||**
||
*||
1ˆˆ
ˆˆˆ
Atc
bBbvdtd
BbcBb
BbvR
dtd
ddfm
dym
¶¶
-Ñ×-Ñ×-=
Ñ´+Ñ´+=r
19
Where dose it stand?
• Where does it stand?
Boltzmann
Gyrokinetic
Gyrofluid
MHD
- Reduction of GK eqs to fluid moments (3D)- Sophisticated closure to model wave-particle interaction and FLR effects
- ≪ 1, reduction to single fluid model- Simple closures
- Strongly magnetized plasma (e.g. tokamak),low frequency fluctuations ω << Ωi
- Reduction from 6D to 5D kinetic equation
20
Where dose it stand?
• Alfvenic turbulence: ~∥• GS scaling of
anisotropy: ∥ ∝ /è < Ω for → 1è regime of gyrokinetic
G.G. Howes et al., ApJ’06, PRL’08
21 D.S. Ryu, WCI-CNU workshop 2010
22
Merits and Limitations
• Applicable to small-scale (gyro-radius) turbulence in strongly
magnetized plasmas è solar winds, corona, ISM, intracluster etc.
• Describe kinetic cascade in 5D phase space (both space and
velocity) è collisionless Landau damping
• Compressional component can be included
• Recover various MHD results for some appropriate limiting cases
• Limitation: fast MHD waves, cyclotron resonance are ruled out
(extension to high frequency gyrokinetic theory, H. Qin, PoP’99)
23
Issues in Gyrokinetic Simulation
24
Problem Size and Parallelization• Though reduced to 5D, problem size is still challenging!
• Complexity of equations è careful choice of numerical
scheme is required i.e. well balanced between good
numerical property and simplicity for easy implementation
and parallelization
Number of grids: × × × ∥ × ≥ 256 × 256 × 64 × 128 × 32~10Electron-proton mass ratio ~ 1:2000
è time scale disparity ~ 45
25
Phase Space Filamentation
• Collisionless Vlasov equation– Characteristic equations ⇒ Hamiltonian flow
– Phase space volume is conserved along the characteristics
– evolution of distribution function with streaming and interactions
⇒ finer and finer filamentation in phase space down to sub-grid scales
+ + ∥ ∥ = − , = 0
26
Phase Space Filamentation
D.K. Jang and D.K. Lee ’12
27
Phase Space Filamentation• Collision becomes important
This process continues until (neglected) collision can catch up.
(Note that neglected collision term becomes important for smaller velocity space scales) = − , = (, ) , ~ ↑ as Δ → 0
28
Phase Space Filamentation• Issues in numerical simulation
– But sufficient grid resolution all the way down to the collisional scale is
practically impossible!
– Strong gradients in phase space è source of many numerical troubles
– Careful choice of scheme and/or adding artificial dissipation is needed to
minimize non-physical behaviors
Insufficient resolutionè Oscillationè Unphysical instabilities
Adding numerical diffusionè Stabilize simulationè Artificial heating/cooling
29
Numerical Simulation of Gyrokinetic Equations
30
Particle in Cell (PIC) method (Lagrangian)
))(())(())((),,,( |||||| ttvvtRRwtvRf pp
ppp mmdddm ---= årrr
markers"" simulation ofnumber particles real ofnumber
=pw
Phase space is sampled by a fewer number of particles carrying “weight”
0
1ˆˆ
ˆˆˆ
||
||**
||
*||
=
¶¶
-Ñ×-Ñ×=
Ñ´+Ñ´+=
p
p
p
dtd
Atc
bBbvdtd
BbcBb
BbvR
dtd
m
ddfm
dymr
31
Particle in Cell (PIC) method (Lagrangian)
• Almost same with conventional PIC simulations (“particles” è
“charged rings”)
• All previous numerical methods developed for PIC can be
employed
• Issue of small scale noise (~ 1/√)
• ~ 2000 particles per cell as rule of thumb
• Explicit scheme, larger time step etc. possible
• Easier to parallelize
32S. Ku and CPES team, ICNSP’11
33
XGC1 full torus simulation of ITG turbulence (S. Ku et al., EPS’12)
34
Continuum Method (Eulerian)
Setup grid system for entire phase space (5D) and apply standard
method to solve hyperbolic PDE i.e. FDM, FVM, etc
nmlkjif ,,,,
nmlkjif ,,,,1-
nmlkjif ,,,,1+
nmlkjif ,,,1, +
nmlkjif ,,,1, -
nJIJ
nJIJ fNfM =+1
nn fNMf 11 -+ =
Inversion of “huge matrix”
is usually required
35
Continuum Method (Eulerian)
Vlasov simulation of two steam instability (D.K. Jang and D.K. Lee ’12)
36
Continuum Method (Eulerian)• All advanced schemes developed in hydro communities can be employed
• Some operations are very costly e.g. ~∫ • CFL condition often dictates implicit time integration
• Phase space granulation poses high velocity space resolution
è grid scale dissipation is necessary è poorer conservation than PIC
• Parallelization efficiency is often less than PIC
• Low noise high quality simulation is possible if sufficient resolution is
provided
è development of massively parallel supercomputer makes this possible!
37
Semi-Lagrangian Scheme• Invariance of distribution function along characteristics
)0,,()),;(),;((
0
0
0||00||||0
||
||
vRftvtvRtRf
fdtd
fvdt
dvf
RdtRdf
t
rrr
r
r
=DDÞ
=Þ
=¶¶
+¶¶
×+¶¶
• Find at grid point by tracing back characteristic line and interpolating • Relatively free from CFL constraint i.e. larger Δ• Interpolation on Eulerian grid à greatly reduce discrete particle noise
Δ (, ∥)(, ∥)
38
Numerical Simulation of Gyrokinetic Equation
• What is δf scheme?– Write distribution function as a sum of known and – Solve only perturbed part only
0||
|| =¶¶
+Ñ×+¶¶
vf
dtdv
fdtRd
tf
r
||
0||0
||
||
vf
dtdv
fdtRdf
vdtdv
fdtRdf
t ¶¶
-Ñ×-=¶¶
+Ñ×+¶¶
rr
ddd
Gyrokinetic equation in δf form(W.W. Lee CPC’87)
))(())(())((),,,( |||||| ttvvtRRwtvRf pp
ppp mmdddm ---= årrr
))(())(())(()(),,,( |||||| ttvvtRRtwtvRf pp
ppp mmdddmd ---= årrr
úúû
ù
êêë
é
¶¶
+Ñ
×--=||0
0||
0
0)1()(vff
dtdv
ff
dtRdwtw
dtd
pp
r
39
Numerical Simulation of Gyrokinetic Equation
• Why δf scheme?( ) 1/~1 2
01
22 <<º å=
ffwN
wN
pp dFor PIC method
Nw
N11 2 <<µe for
Simulation noise is reduced by δf/f0
But… what if 〈w2〉 increases during simulation?
We lose the merit ⇒ actually, it can become large in
ü Short time by phase space filamentation (granulation)
ü Longer time by equilibrium evolution
)entropy : ( ~ 2222
02 Hdxdvfwwf
xDw
dtd
-µ-÷øö
çè涶
= òdn
• Why not δf scheme?
In collisionless simulation (ν=0), particle weights grow linearly in timeand δf scheme stops to work
40
Summary• Gyrokinetic simulation is a well developed tool to study low
frequency micro-scale turbulence. Serious validation efforts are also ongoing in fusion community.
• With careful examination of parameter regime, it can be applied to astrophysical environments such as ISM, intracluster medium etc.– microscopic turbulence with kinetic processes e.g. collisionless Landau
damping
– sub-grid transport model for global simulation studies
– stimulate the extension of gyrokinetic model beyond present limit
• Careful choice of numerical scheme is absolutely critical– beneficial for both communities to expand present simulation capabilities