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Max-Planck-Institut für Plasmaphysik, EURATOM Association
Computational Plasmaphysics
Ralf Schneider
Max-Planck-Institut für Plasmaphysik, Euratom-IPP Association,
Wendelsteinstrae 1, D-17491 Greifswald, Germany
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Computational physics
Why numerical methods?
Complexity of equations
Example
Simulation of experiments
To test validity of theory
To gain an idea of experimental performance
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Computational physics
Max-Planck-Institut für Plasmaphysik, EURATOM Association
The Computational Stellarator W7-X
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Plasma Wendelstein 7-X
Max-Planck-Institut für Plasmaphysik, EURATOM Association
slow drift of guiding center
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Optimized stellarator
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Plasma in a computational model
• 10 Variables: densities, velocities, temperatures
• 10 billion grid points
• 100 million time steps
• 100 FLoatingPointOPerations/sec / timestep / gridpoint
• or 1 billion teraflop/sec
• Cray T3E with 784 PE (ca. 75 gigaflop) or 500 years computing
• NOT VERY REALISTIC
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Plasma:
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Particle aspect of plasma dominates
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Plasma is treated as one fluid with infinite conductivity
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Max-Planck-Institut für Plasmaphysik, EURATOM Association
MHD is basis for all equilibrium calculations
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Max-Planck-Institut für Plasmaphysik, EURATOM Association
MHD, equilibrium
• Existence in 3D ?
• Theoretical ?
• Experimental ?
• Accessible only by
computational models
• but
not before 1975
• thus Optimization started
with IBM360/91
• W7-AS Design 1978
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Equilibria, VMEC
Stationary states of plasma
energy (fixed boundary)
MHD force balance
p constant on (nested) surfaces labelled by s
Poloidal and toroidal fluxes
are invariant functions, together with m(s)
the mass distribution
• r and z periodic functions (Fourier series)
• Hybrid finite elements in s, (artificial) Time-like iteration
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Vacuum fields - free-boundary - coils
Boundary Value Problems, Greens Function
• Last closed magnetic surface (lcms) defines
completely interior plasma properties
• Search for external current distributions (i.e. coils)
producing a vacuum field B with boundary conditions on
the lcms (n exterior normal)
NESTOR / NESCOIL codes
Iterative combination of VMEC & NESCOIL
allows free-boundary computations
NEMEC
Max-Planck-Institut für Plasmaphysik, EURATOM Association
VMEC
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Plasma configuration given
calculate coils to produce it
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Coils 1-50
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Is a given plasma configuration stable against small pertubations?
Find ways to prevent instabilities
~  g
~
Fg
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Tokamak operation limited by MHD instabilities
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Necessary to design equilibrium with „good“ confinement properties
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Computational Remarks
Speedup of equilibrium codes due to
Peak speed of cpu:
(same parameters)
New Codes:
(same equilibrium)
10 fold
12 fold
16 fold
28 fold
500 fold
IBM 360/91
Cray-1S
Cray-1S
Cray-1S
Cray-1S
24 fold BETA
50 fold MOMCON
30 fold VMEC
Cray-1S
YMP-464(4cpus)
J916 (16)
SX4(2)
T3E-600(784)
1980
1988
1992
1996
1998
MOMCON/FIT
VMEC
VMEC2
1980
1985
1989
Better algorithms gave a speedup of around 30.000 !
New hardware ``only`` 5.000 ...
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Turbulence suppression
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Turbulence suppression
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Gyrokinetic turbulence simulations
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Plasma-edge physics
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Length scales
sputtered and backscattered species and fluxes
Plasma-wall interaction
Molecular
dynamics
Binary collision Kinetic
approximation Monte Carlo
Kinetic
model
impinging particle and energy fluxes
Fluid
model
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Diffusion in graphite
Carbon deposition in divertor regions of JET and ASDEX
UPGRADE
Major topics: tritium codeposition
chemical erosion
JET
Paul Coad
(JET)
ASDEX
UPGRADE
Achim von Keudell (IPP, Garching)
V. Rohde (IPP, Garching)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Diffusion in graphite
Internal Structure of Graphite
Granule sizes ~ microns
Void sizes ~ 0.1 microns
Crystallite sizes ~ 50-100 Ångstroms
Micro-void sizes ~ 5-10 Ångstroms
Multi-scale problem in space (1cm to
Ångstroms) and time (pico-seconds to
seconds)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Multi-scale ansatz
Mikroscales
MC
Mesoscales
KMC
Macroscales
KMC and Monte
Carlo Diffusion
(MCD)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Molecular dynamics – HCParcas code
- Hydrogen in perfect crystal graphite – 960
atoms
- Brenner potential, Nordlund range
interaction
- Berendsen thermostat, 150K to 900K for
100ps
- Periodic boundary conditions
Developed by Kai Nordlund, Accelarator laboratory, University of Helsinki
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Time variation of pressure
• Equilibration of pressure with time
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Molecular dynamics – Simulation at 150K, 900K
150K
900K
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Molecular dynamics results
two diffusion channels
no diffusion across graphene layers (150K – 900K)
Lévy flights?
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Molecular dynamic results
Non-Arrhenius temperature dependence
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Kinetic Monte Carlo – basic idea
0 = jump attempt frequency (s-1)
Em = migration energy (eV)
T = trapped species temperature (K)
Assumptions:
- Poisson process (assigns real time to the jumps)
- jumps are not correlated
Max-Planck-Institut für Plasmaphysik, EURATOM Association
KMC results for transgranular diffusion
D /(cm2 / s)
1000 / T (K 1)
- strong dependence on void sizes and not on void fraction
- saturated H (Tanabe)
0~105s-1 and step sizes ~1Å (QM?)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Multiscale model
Activation energies:
trapping-detrapping 2.7 eV
desorption 1.9 eV
surface diffusion 0.9 eV,
0~1013s-1
jump attempt frequency
,
jump step length ~35Å
for entering the surface for a solute
H atom 2.7 eV
porous graphite structure
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Multiscale model - results
desorption starts
between 900 K and 1200 K
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Multiscale model – 900 K, 0.1 ms
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Multiscale model – 1500 K, 0.001 ms
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Multiscale model – diffusion types
surface
diffusion 0.9 eV
adsorptiondesorption 1.9 eV
Max-Planck-Institut für Plasmaphysik, EURATOM Association
PIC(Particle-in-cell)-method
Principle:
Applications:
Low temperature plasmas (methane, RF discharges)
Complex plasmas (plasma crystals)
Parasitic plasmas in the divertor (radiative ionization)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
PIC simulation: RF capacitive discharge
Model system for chemical sputtering: methane plasma
(2DX3DV PICMCC multispecies)
Collaboration with IEP5, Bochum University (Ivonne Möller)
ne ~ 109-1010 cm-3
nn ~ 1015 -1016 cm-3
fRF = 13.56 MHz
ne = 1010 cm-3, nH2 = 9.2·1014 cm-3,
nCH4 = 7·1014 cm-3, p = 0.085 Torr (11 Pa)
potential
Max-Planck-Institut für Plasmaphysik, EURATOM Association
PIC simulation: RF capacitive discharge
electron and CH4+ ion density
Electrons reach electrode only
during sheaths collapse
CH4+ ion energy distribution
Energetic ions at the wall due
to acceleration in the sheath
Max-Planck-Institut für Plasmaphysik, EURATOM Association
PIC simulation: RF capacitive discharge
electron velocity distribution
electron-impact ionization rate
-3 -1
n'ei , cm c
14
8x10
14
6x10
14
4x10
14
2x10
0
0
32
64
96
128
Y, D
Energetic electrons oscillate between sheaths
Ionization spread over the bulk
Max-Planck-Institut für Plasmaphysik, EURATOM Association
PIC simulation: RF capacitive discharge
electron energy probability function
11
10
10
T1 = 0.39 eV, n1 = 10 cm
9
10
T2 = 3 eV, n2 = 10 cm
9
10
3/2
-3
eepf (eV- cm )
10
8
10
T2
7
10
T1
6
10
0
5
10
15
20
electron energy (eV)
Figure from V.A. Godyak, et al., Phys. Rev. Lett., 65 (1990) 996.
Bi-maxwellian distribution due to stochastic heating
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Dusty (complex) plasmas
Lower electrode
Negative charge due to higher
electron mobility
Levitation in strong sheath
electric field
Max-Planck-Institut für Plasmaphysik, EURATOM Association
PIC simulation: Plasma crystal - full 3D!
Top view
Quasi - ordered 3D structure
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Complex multi-scale physics requires complex computational tools
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Thanks!!
Ralf Kleiber, Ulrich Schwenn, Volodja Kornilov,
Stefan Sorge
Mathias Borchardt, Jörg Riemann,
Alex Runov, Xavier Bonnin
Konstantin Matyash, Neil McTaggart,
Manoj Warrier, Francesco Taccogna
Andrea Pulss, Andreas Mutzke, Henry Leyh
And many contributions from colleagues all over the world