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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)
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Motivation: simulation of burning plasmas
Lavender Field, Provence, near ITER
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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
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1 – Multiple scales
Fusion Devices
Space
Role of micro-macro coupling
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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)
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ITER: Multiscale - Multiphysics
Ions: D @ 10keVα: fusion generated
Source: ITER web site
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• 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
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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
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2 – Simulating micro-macro coupling
A possibility: implicit moment PIC
Description of implicit moment PIC
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Fundamental Equations (Classical)
• We consider collisionless plasmas
• Vlasov-Poisson model - Vlasov equation
- Maxwell equations
(Newton equations)
Eulerian formulation
Lagrangian formulation
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Explicit PIC Computational Cycle
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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
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• 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
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Implicit Moment Method
Ïs
p
qpS x x
p
Js
p
qpupS x x
p
Î s
p
qpupupS x x
p
Fluid equations
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ITER: Multiscale - Multiphysics
Gyrokinetic
Implicit Moment PIC
GyrokineticD,T
D,T
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4 – V&V and applications
1. Reconnection physics in 3D
2. Parallelization & Relativity
3. Inertial Electrostatic Confinement
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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
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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)
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• 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
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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
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Can we use CELESTE in low beta, high toroidal field?
Electron acceleration
ExplicitImplicit
BT=0
BT=BP
BT=10BP
Vye
Mi/Me=180
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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
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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)
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Waves
Light
Langmuir
whistlerIon acoustic
ωpeÎ ”t 0 .01 ω
peÎ ”t Ï€
Brackbill, Forslund JCP, 1985
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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
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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
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3D reconnection: micro-macro coupling
Large scale processes
Small scale processes
Question: Is the small/large scale coupling captured?
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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)
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Effect of microinstabilities captured correctly
.55
1.1
2.2
L/ρi
L
explicit implicit
•Current intensification•Temperature anisotropy
Reconnection isenhanced (Poster:FZ1.00008)
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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
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IEC Simulation – See posters: BP1.137-138 LP1.107