jkwon - EANAM 2012 · –anisotropic fluctuation: ... Vlasov simulation of two steam instability (D.K. Jang and D.K. Lee ’12) 36 Continuum Method (Eulerian) ... and δf scheme stops

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

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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)

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Basic Idea of GK Theory (cont’d)

• Guiding center transformationparticle space , ↔ guiding(gyro) center space , ∥, ,

+

: gyrophase-angle → average out = − , = × , Ω = ∥ = ⋅ , = = ∥ +

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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)

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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

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• 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 (, )

(, ) = ∫ ∗∥ + − + ( − )(, ) = ∫ ∗∥ + − ∥ + ( − )

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• 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)

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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

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Elementary Plasma Physics• × drift motion of charged particle

è drift motion of gyration center in × direction

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( )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

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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

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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

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• 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

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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

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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

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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)

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Issues in Gyrokinetic Simulation

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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

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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

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Phase Space Filamentation

D.K. Jang and D.K. Lee ’12

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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

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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

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Numerical Simulation of Gyrokinetic Equations

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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

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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

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XGC1 full torus simulation of ITG turbulence (S. Ku et al., EPS’12)

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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

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Continuum Method (Eulerian)

Vlasov simulation of two steam instability (D.K. Jang and D.K. Lee ’12)

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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!

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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 (, ∥)(, ∥)

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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

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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

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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