Kinetic Approach to microscopic-macroscopic coupling in fusion plasmas

Koichi NoguchiPhysics & Astronomy Dept., Rice Univ.

Giovanni LapentaPlasma Theory Group, Theoretical Division, LANL, USA

Collaborators: J.U Brackbill (Particle Solutions), W. Daughton (U Iowa), S. Markidis (UNM),

P. Ricci (Dartmouth), R. Nebel, E. Evstatiev, J. Park (LANL)

Motivation: simulation of burning plasmas

Lavender Field, Provence, near ITER

Outline

1. Multiscale processes in plasmas, the case of ITER

2. The implicit moment PIC method

3. Benchmarks

4. Applications:– 3D reconnection– Reconnection in low beta plasmas– Fusion applications

1 – Multiple scales

Fusion Devices

Space

Role of micro-macro coupling

Scales involved in plasma physics

10-210-310510-3Pressure tensor

10-310-710-710-7Resistivity

10-810-110410-4Electron inertia

10-610110610-2Hall

Solar interiorSolar coronaEarth magnetotail

ITER (D @ 10keV)

Length scale

Scales where the various terms become important (SI UNITS)

ITER: Multiscale - Multiphysics

Ions: D @ 10keVα: fusion generated

Source: ITER web site

• Eliminates the smaller scales

• Quasineutrality is imposed

• Reduces the velocity space to 2D

• Some high order non-linearity are neglected

Gyrokinetic PIC

ÏpLeq

<<1

Ï „Ï ‰cp

<<1

In ITER D=0.1 cm, He=10 cm and gyroaverage could have problems

Multiscale coupling in space plasmas

• High Collisionless

• Small scales, non gyromotion

• Macro/micro coupling

• Methods developed there could be used for ITER.

Gombosi et al., Univ. Michigan G. Lapenta, AGU Fall meeting, 2004

k Ïp>>1

Example of CELESTE3D

2 – Simulating micro-macro coupling

A possibility: implicit moment PIC

Description of implicit moment PIC

Fundamental Equations 　 (Classical)

• We consider collisionless plasmas

• Vlasov-Poisson model - Vlasov equation

- Maxwell equations

(Newton equations)

Eulerian formulation

Lagrangian formulation

Explicit PIC Computational Cycle

Time step and grid spacing limit:

– Explicit stability constraints

– Implicit accuracy conditions

ωpeÎ ”t 2

Δx λDe

vth ,e

Î ”t

Δxλ

Deω

pe

Î ”t

Δx1

cÎ ”t Δx

Summary of the Stability constraints

• Maxwell equations: implicit second order formulation for the field E

• Newton equations: implicit form

• Solver: Implicit moment method

Implicit formulation (Classical)

Particle mover

Field Solver

Implicit Moment Method

Ïs

p

qpS x x

p

Js

p

qpupS x x

p

Πs

p

qpupupS x x

p

Fluid equations

ITER: Multiscale - Multiphysics

Gyrokinetic

Implicit Moment PIC

GyrokineticD,T

D,T

4 – V&V and applications

1. Reconnection physics in 3D

2. Parallelization & Relativity

3. Inertial Electrostatic Confinement

Explicit:Pritchett, JGR106, 3783 (2001) Implicit:

CELESTE3D

•Explicit [Pritchett, JGR, 106, 3783 (2001)] Grid 512 X 256 grid, 9,000,000 particles, Time step: massively parallel computer

•Celeste3D [Ricci et al., GRL, 29, 2088, (2002)]Grid: 64X64 200,000 particles, Time step: Workstation

TEST: GEM challenge

ωpeÎ ”t 0 .15

ωpeÎ ”t 1 . 5

Electron outflow Ion density

Ion outflow Bz

x x

x xImplicit:

CELESTE3D

x

T=0 T=8

T=16

T=32

T=24

T=48

z

Performance – See Poster

• New PRASEK project:– CELESTE– FLIP (MHD)– DEMOCRITUS (plasma-

material interaction, kinetic)– GLOW (plasma-material

interaction, fluid)– Relativity

• C++ object oriented

• Parallel

0

2

4

6

8

10

12

14

16

1 2 4 8 16

#　processors

PARSEK speed-up

Ideal speed-up

PARSEK efficiency

Ideal efficiency

(logarithmic axis)

• Maxwell equations: New scheme for current prediction

• Newton equations: Implicit form, relativistic

• Solver: Implicit moment method,Newton-Krylov method,Energy conserving method

New Relativistic Formulation

Particle mover

Field Solver

xp

n 1xp

nup

n 1/2

γΠ”t , γ 1

vp

c

2-1

, upvpγ

up

n 1 up

nqsÎ ”t

ms

Ep

n θ xp

n 1/2up

n 1/2

γBp

n xp

n 1/2

Test: relativistic 1D two-stream instability

Growth Rate :

Im(p)

V0=0.9c, 100,000 particles, 128 mesh, Te=0.01eVtp=0.01 (Explicit), 0.2 (Implicit)

0E

2 )/2

t p

2 γ3ω

p

2 1

kv0ω

2

1

kv0ω

2

Can we use CELESTE in low beta, high toroidal field?

Electron acceleration

ExplicitImplicit

BT=0

BT=BP

BT=10BP

Vye

Mi/Me=180

Summary

Question: how can we study burning plasmas kinetically

Possibility: consider implicit moment PIC

Fully kinetic Able to capture micro-macro

modelingExtensive application to space

plasma physics

Conclusions:The method is matureRecent upgrades ParallelizationRelativitySuite of relevant applications

R&D100 prize in 2005

CartaBlanca: A High-Efficiency, Object-Oriented, General-Purpose Computer Simulation Environment

PARSEK

General tool for PIC simulations

Includes:•Implicit kinetic PIC (Celeste)•Implicit fluid PIC (Flip)•Plasma-material interface (Democritus)

Waves

Light

Langmuir

whistlerIon acoustic

ωpeÎ ”t 0 .01 ω

peÎ ”t Ï€

Brackbill, Forslund JCP, 1985

Orbits -Gyromotion

• No averaging, accuracy determined by t, x

• Accurate gyroradius and drift motions at large t

• Valid at all beta

• Valid at all: ρk┴

• Short scales are not eliminated and the energy channel towards them remains open

Drifts and Gyroradius

Implicit corrected for

Method III

Implicit (described above)

Method II

Leap-Frog BorisMethod I

Vu, Brackbill, JCP, 116, 384 (1995)

μ B

3D reconnection: micro-macro coupling

Large scale processes

Small scale processes

Question: Is the small/large scale coupling captured?

Simulation of the small scale processes (LHDI)

• Free energy: diamagnetic drift

• Driving: density gradient

• Stabilization: high beta

• Frequency:

• Wavelength:

• Direction:

• Seen in space and experiments (e.g. MRX)

• Present only on the edges of the sheet

• Requires a kinetic treatment

e>> ω

i

k Ïe

1

k 0

Simulation with L=di

Daughton, Lapenta, Ricci, JCP, 116, 384 (1995)

Effect of microinstabilities captured correctly

.55

1.1

2.2

L/ρi

L

explicit implicit

•Current intensification•Temperature anisotropy

Reconnection isenhanced (Poster:FZ1.00008)

Can we use CELESTE in low beta, high toroidal field?

• We considered reconnection with different:

– B toroidal (guide field)– Using a Harris equilibrium

• We computed:– Reconnection rate– Onset– Ion/electron decoupling

mechanism– Break-up mechanism

• As BT increases we kept the same t, even while the gyrofrequency increased.

Reconnection rateExplicit

Implicit

mi

me

25

mi

me

180

mi

me

1836

BT=0 BT=BP BT=10BP

IEC Simulation – See posters: BP1.137-138 LP1.107