Transcript Advanced methods in global gyrokinetic full f particle simulation of

Plasma Simulations using
Particle Codes
J.A. Heikkinen
Valtion Teknillinen Tutkimuskeskus (VTT), Finland
Contributors: C.-S. Chang1, S. Janhunen2, T.P. Kiviniemi2, S. Leerink2, T.
Makkonen2, M. Nora2, F. Ogando9, S. Sipilä2
1Courant
Institute, New York University, USA, 2Teknillinen korkeakoulu, Espoo, Finland, 3UNED,
Madrid, Spain
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Outline

Introduction to Particle-in-Cell (PIC) Codes
- first principles fluid dynamics, plasma and beam physics, and
astrophysics and cosmology
- solid state device design, accelerator physics, galaxy formation,
fusion reactors, fluid flows, and magnetohydrodynamics (MHD)
- PIC applications are now at a threshold where they can be used for
precision prediction rather than just as quantitative indicators of system
behaviour.

Gyrokinetic Particle Simulation of Magnetized Plasmas
- required for microturbulence simulation
- neoclassical effects, macroscopic plasma evolution?
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
The challenge for modelling: span the
large separation in length and time
scales I
LASER FUSION PLASMA
IMPLOSION
LASER
COLLISIONS
ION MOTION
PLASMONS,
Particle simulations
Hybrid
methods
Hydrodynamic
νDT
<<
ναe
ωpi
<<
DT fusion
101
<<
Alpha slowing-down
102
103
104
105
ωpe
Electron oscillations
Ion oscillations
106
107
108
Rate of variation (nanoseconds-1)Separation of time scales
requires
long, large-scale PIC
simulations
→ Cells, EF-scale machines
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Fast ignition integrated simulation
code
from K. Mima, Osaka University
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
The challenge for modelling: span the
large separation in length and time
scales II
TOKAMAK PLASMA
MICROTURBULENCE
MHD EVOLUTION
TRANSPORT
Fluid transport
τA
Gyrokinetic particle
simulations
<<
τs
Alfvén transit time
10-6
<<
τevol
<<
MHD evolution time
Sound transit time
10-5
Extended
MHD
10-4
10-3
10-2
Problem time (sec.)
τtrans
τres
Transport time Resistive time
10-1
100
101
Separation of time scales
requires
long, large-scale PIC
simulations
→ Cells, EF-scale machines
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Multiscale code Integration in CPES
in Kepler framework
C.S. Chang, New York Univ.
XGC1
Short time
gyrokinetic edge
turbulence/neocl
Core Gyrokinetic
code
ELITE, etc
Linear ELM
Stability check
XGC0
Experimental
time evolution of
kinetic edge
equilibrium
M3D & NIMROD
Nonlinear ELM
crash
Collab.
GA
Joint,
CEMM
Collab.
GPS +
DEGAS2
Neutrals
Atomic physics
Plasma-Surface
Interaction
code
RF-edge interaction,
energetic particles, impurity,
radiation, 3D magnetic
perturbation physics, and
core MHD effects are to be
built into XGC0
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Macroscopic flows and turbulence
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
XGC Gyrokinetic
particle motions
in the plasma edge
(poloidal plane)
ions
electrons
V|| >0
V|| <0
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Basis of PIC plasma simulation
Acceleration and
increment of
velocity
Calculation of
forces from fields
and velocity
Computation of
electric field.
Magnetic is given.
Initial step
with =0
Displacements and
new positions.
Boundary conditions.
Calculation of
density. Current
profile fixed.
Resolution of Poisson
equation for the
electrostatic potential.
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
vj
n ½
 vj
n ½

q n
E j t
m
1D Electrostatic case with Poisson
Eq
GRID
Leap-frog algorithm for particle coordinates
vj
n ½
xj
n 1
 vj
n ½
q n
 E j t
m
 xj  vj
n
n ½
t
i-2
j’th particle (q,m)
cell
d
i-1
i
 (i )  q(1  d )
CIC
 (i  1)  qd
Ej  q(1  d ) E(i)  dE(i  1)
Poisson solver
Ei ½  Ei ½   i x /  0
i+1
NGP
Ei  ½( Ei ½  Ei ½ )
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
 (i )  q
Ej  E (i )
Superparticles
●
●
It is not practical to simulate all particles in laboratory
(1020-1033 m-3) or in space (1018 km-3) plasmas.
Therefore, each simulation particle (a superparticle)
represents many particles of a real plasma (grid spacing ~
λ D)
Collisional effects due to Coulomb field are not modelled
correctly with superparticles
Remedy:
Use finite size particles to
smear out collisions: With the
size ~ several λD, this method
retains collective interaction
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Finite number of simulation particles
creates noise
●
●
●
Physical solution in any PIC
simulation is always
accompanied by a discrete
particle noise
3D solution, potential and density fluctuation
amplitudes at a given position
In the picture, regression
shows almost perfect N-1/2
scaling, indicating that
results in this case are
dominated by noise.
Discreteness may, however,
be beneficial in finding
proper asymptotics, e.g., in
nonlinear Landau damping
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
From noise to turbulence spectrum
• Wave number spectrum from different time instants demonstrates
how one moves from white noise to physical spectrum in modes.
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Finite size filtering
●
CIC provides filtering which creates a transfer function to noise, too.
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Stability and accuracy
●
●
Differencing gives time constraint ωpeΔt<2, while retaining
accuracy for the plasma waves requires ωpeΔt<0.2.
Numerical instabilities may arise due to (aliasing)
- discrete spatial grid
- finite-sized time steps
●
●
●
High frequency modes appear not correctly on the mesh
and modes k and k+2πn/Δx cannot be distinguished.
If k+2πn/Δx resonates with any of the natural modes of the
plasma, system can become numerically unstable.
Remedy: Make either aliased modes Landau damped
(λD>Δx/π) or use finite size particles (size ≥ 0.7Δx).
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Landau damping
●
●
●
Truely dissipationless Vlasov-Poisson simulation cannot be
realized due to the effects of numerical gridsize
With Δx<λD, numerical collision time however longer than ωpe1nλ
D
When the size of phase-space filaments reaches the
interparticle separation, envelope oscillations disappair, and
wave goes on propagating at constant amplitude as in O’Neil’s
analysis (1965)
Discretness in PIC simulations helps
to describe the true asymptotics of
nonlinear Landau damping.
The continuum Eulerian simulations
in principle fail in this task
Carbone, Valentini,
& Veltri, 2007
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
PIC-MCC: Binary collisions I (Takizuka,
1977)
●
●
Scattering between two
test particles randomly
selected from a grid (left).

vr '

vr
After that, the relative
velocity vector (top-right) is
rotated
   t ln(1  R )
1
  2 R2
e4 (Z A Z B )2
NB 
4  0

m m
3
( A B ) 2 vr
mA  mB
●
The new velocity vectors
for both particles are
recalculated so that
momentum is conserved
vr '
A
vr
B
vA '
mB
(vr ' vr )
mA  mB
mA
vB '  vB 
(vr ' vr )
mA  mB
vA '  vA 
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
vB '
Binary collisions II
●
●
●
Pairing of particles for binary collisions is done within a grid
cell
For A-B collision, loop all the particles of species A, and for
every particle, find a random partner from species B
This works if the particles have a same weight, i.e., each
simulation particle corresponds to the same amount of true
particles.
A-B collisions vs. B-A collisions
Species A
Species A
nA
nA
nB
nB
Species B
Species B
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Binary collisions III
●
The problem of different weight factors can be solved by
applying the binary collision operator between groups of
test particles that represent an equal amount of actual
particles
KANA= KBNB
All particles from the first
group collide with an
imaginary particle
representing the state of the
second group, and all
particles from the second
group collide with and
imaginary particle
representing the state of the
first group
NA, NB are the weights
A
1
.
1
.
2
.
A
B
2
.
3
.
1
.
B
1
.
Binary Collisions
2
.
A
B
2
.
3
.
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Collision effects easy to test
Maxwellian energy distribution; all the test
particles started with an energy of 10 keV
Both particles started with a very low kinetic
energy in an environment of average energy
of 10 keV (electron thermalizes faster)
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Laboratory plasmas are collisional
●
ES, PD (Plasma Device), and OOPIC (relativistic,
electromagnetic, 2d3v, collisions) code family (C.K. Birdsall, PTSG,
US Berkeley)
●
Plasma-material boundaries and SOL an important
BIT1 code result (Tskhakaya, 2005) for the Mach number and
application in magnetized
charge fusion
density along the SOL magnetic field up to the plasma-wall
• MP and DS regions cannot be modelled in
interface
fluid or in gyrokinetic approximation.
•
Binary collisions for plasma particles.
Null-collision model for charged-neutral
particle collisions. Self-consistent neutrals
(Matyash, 2003)
•
•
Either gyro-averaged or Boltzmann electrons
•
Either 1d3v or 2d3v
•
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
Sometimes PSI included
for controlled Fusion – Nice, France, 8-12 September, 2008
Simplifications for enhancing code
performance
●
Take electromagnetic field as given (standard particle
code)
●
Solve only for perturbations δf
●
Advance only orbit invariants, not track particle orbits
●
Either freeze ions or use adiabatic approximation for
electrons or ions
●
Solve for quasineutrality
●
Advance particle co-ordinates and fields implicitly
●
Take drift-kinetic limit (guiding-center code)
●
Apply gyrokinetic approximation
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
GYROKINETIC PARTICLE
SIMULATION
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Gyrokinetic (GK) method
Standard nonlinear GK ordering:
Distribution function F can evolve
arbitrary far from its initial value.
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Gyrokinetic model
●
●
●
With the proper averaging process, FLR effects are still
kept.
Coordinates change to gyrocenter. Angle of gyration goes
by averaging.
This introduces changes in equations that use real space
coordinates (i.e. Poisson)
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Larmor ring discretization
●
●
Four-point averaging procedure (W.W. Lee, 1987) often
used for estimates of gyroaverages <..>
Sampling of density ñi on the grid from each particle uses
the same four-point averaging
∂f/∂μ is difficult to evaluate with particle methods; usually f
is taken as a Maxwellian
∫<Φ> ∂f/∂μ dv can then be evaluated with either one, two, or
three Larmor rings with the corresponding weights from the
Maxwellian
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Direct implicit solver for ion polarization
●
●
Ion polarization density can
also be calculated directly from
simulation particles (requires
different guiding-center
transformation, JCP 2008)
Isolate implicitly ion polarization
drift contribution to density.
Larmor circle
i’th subparticle cloud
y
k
py+
i
px
-
px0
xp,yp px+
py0 k ds
pyx
p’th point on the Larmor
orbit of the k’th ion
gyro-center of the k’th ion
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Implicit solver for electron parallel
nonlinearity
A.B Langdon et al
(1983)
●
E field in ∆xa is calculated at advanced time but at
the position after free streaming
–
We demand that |∆xfs|>>|∆xa|. It is a constraint in
Δt
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Gyrocentre motion
• Gyrokinetic model deals
with particle gyrocenters,
which present smoother
transversal trajectories.
0.8030
0.8025
4th order Runge-Kutta,
no error monitoring
t = b / 100
t = b / 25
ITER case #585
0.8020
0.8015

0.8010
0.8005
0.8000
0.7995
0.7990
0.000
0.001
0.002
0.003
time (s)
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
0.004
0.005
0.006
Magnetic background
Simple magnetic configuration for rapid calculations
BT=BT0R0/R
Bp=(0Ip/r) R0/R
Ip
B
r

BT
Bp
q(r) = (r/R0)BT/Bp
J=(R/R0)(r/ Bp)

dlrd R/R0
p| = BpR
• p,  are so called White-Chance coordinates
in which the GC-equations are solved.
• Usually adopted “quasitoroidal” coordinate
system does not allow a consistent coordinate
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Initialization
●
●
●
●
●
Loading particles for local Maxwellian velocity distribution
and for wished density and temperature profiles usually
satisfactory
However, in tokamaks neoclassical equilibrium conditions
difficult to initialize
Finite orbit effects can be accounted for by initialising
particles according to orbit invariants (Hively, 1981)
Electrons to be loaded on ion gyro-orbits
In full f calculation, use equal weights for particles among
a given species
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Simplified boundary conditions
Boundary conditions on  and particles in a torus

1) i0  ij=0
2) i 0  ij=1j+’(i-1)r
Ghost particle
r=rl
Simulation particle
• ij is calculated only for i1 (green grid points).
• Both the ghost and simulation particles are
sampled to find particle density and
coefficient matrix for ij.
• Gc-points are reflected at the boundaries
reflection
rl
rr
ij
i=0
i=1
r
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Plasma-wall boundary problems
1) Initialization on OL orbits
2) Reinitialization by pairing the trapped ones and recycling
them back to the plasma
3) Bohm sheath with a discrete grid
4) Radiation losses/neutral collisions to seek for proper T(r)
5) Zero current on plates?

B
ni
ne
neutral
ions
electrons
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Nonlinear GK (kinetic or gyro-fluid)
studies
●
●
●
●
●
●
●
Zonal flows, geodesic acoustic modes, neoclassical
equilibrium
Most drift-wave-type fluctuations like ITG, TEM, TIM,
ETG, universal and dissipative drift instabilities.
Interchange turbulence
Tearing and internal kink instabilities
Microtearing and drift-tearing mode
Drift-Alfvén and kinetic shear-Alfvén waves
Compressional Alfvén waves
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Gyrokinetic particle codes for
simulation of magnetized
toroidal plasmas
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
∂f and full f particle codes for
magnetized fusion
●
●
●
A number of ∂f PIC codes (GEM,GT3D, GYGLES, GTC,
EUTERPE, ORB5, UCAN, and many others) have paved
the way over almost 20 years for precision prediction for
turbulent transport in tokamaks and stellarators
Full f PIC codes (ELMFIRE, TPC, XGC1, and some
others) and full f semi-Lagrangian code (GYSELA)
approach precision prediction for turbulent transport and
macroscopic plasma evolution in tokamaks
A number of full f particle codes (ASCOT, SELFO, XGC0,
and many others) capable for consistent calculations of
fields and particles in tokamaks and stellarators
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
3d2v particle-in-cell full f code ELMFIRE
J.A. Heikkinen, S. Janhunen, T. Kiviniemi, S. Leerink, M. Nora, and F. Ogando
(TKK,VTT,UNED)
●
●
●
●
Global full-F GK code
Drift-kinetic electrons and gyrokinetic ions
and impurities.
Heat (RF, NBI, Ohmic) and particle sources +
recycling
Full binary collision operator
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Neoclassical radial electric field in
turbulent FT-2 tokamak
Simulation can agree with the neoclassical radial electric field in Lmode turbulent simulations (Vφ from simulation)
R=1.1 m, a=0.08
m B=2.1 T, I=22
kA, n(0)=41019
m-3
Ti,e(0)=180 eV
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
ELMFIRE benchmarking
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Influence of noise on results
●
●
●
Particle simulation noise not only masks physical density
fluctuations, but it produces undesirable fluxes that
perturbate the neoclassical equilibrium.
Associated diffusivity can be estimated from the radial
particle shift during decorrelation time.
Radial ion conductivity can be calculated from mixinglenght estimate of the physical level of fluctuations, being
also proportional to T3/2.
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Effects on calculated conductivity
●
Image shows influence of
collisions and potential
averaging on ion radial
conductivity.
–
–
●
Scaled Cyclone Base Case with
kinetic electrons
Collisionless cases show
residual noise
conductivity.
Noise is filtered out by
averaging potential over
flux surface.
So far noise is reduced by
“brute force” ... higher N!
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
χiNC
Saturation with adiabatic electrons I
Evolution of i is studied with nonlinear runs of
Cyclone Base (Dimits et al., PoP ’00).
Measured at r=a/2 (q=1.4). R/LT=8.28. No
collisions; T(a/2)=2000 eV, n=5.5*1019 m-3.
Adiabatic electrons: ne=n0(r)*(1+e(Φ<Φ>)/kT(r))
χi (m2/s)
Why turbulent transport
saturates at such a low
value?
<χ>i << χgB
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Saturation with adiabatic electrons II
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
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Saturation with adiabatic electrons
III
Due to turbulence growth, ion temperature
profile
gets relaxed.
In regions where Ti decreases, ion orbits
narrow, with an opposite behaviour where Ti
increases.
As a consequence, ion polarization has to
compensate the resulting perturbation in ion
density → Electrostatic potential gets a
bipolar perturbed structure
This leads to suppressed turbulence by
sheared
zonal flows.
Cooled
Heated
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
FT-2 simulations in L-mode
Turbulent damping of zonal flows
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
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Steep pressure profile causes spinup
Strong heating (100 kW)
causes radial electric field to
deviate from neoclassical
Sheared poloidal flow suppresses
transport
R=1.1 m, a=0.08 m
B=2.1 T, I=22 kA,
n(0)=41019 m-3
Ti,e(0)=250 eV
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
3d2v Particle-in-cell full f code XGC1
C.S. Chang (New York University, Center for Plasma Edge Simulation (CPES))
Kinetic XGC1
- Global full-f GK code
- short time simulation of turbulence transport
- consistently with neoclassical physics
Kinetic XGC0
- long time evolution of neoclassical ion-electronneutral equilibrium (full-f, PIC)
•
•
Includes 3D magnetic perturbation effects
Turbulence transport to be imported from XGC1 coupling
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
First 3D electrostatic solution across entire edge
Early time 3D 
(Neoclassical +
Fluctuations)
XGC1
Equilibrium 
Negative potential
well is forming in
H-layer (dark blue)
Positive potential
hill in scrape-off
(yellow/brown)
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Strongly sheared ExB flow profiles in the entire Hmode edge from XGC-1
(eV)
Wall
N
GAM
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
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Bootstrap current calculation across separatrix with XG
Good agreement with analytical models
n = 1.5 cm
Agreement not so good with steeper ped
n = 0.75 cm
Simulation with 150000 ions and electrons in XGC0
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Turbulence capability in XGC1 is under careful verification against
core cyclone results.
Afterwards, XGC1 will return to edge turbulence.
Linear ITG growth rates
have been verified.
Nonlinear ITG is under rigorous
verification
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
CONCLUSIONS
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
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Present status in GK PIC
●
Global GK PIC appears to predict neoclassical effects and
large variations in F together with turbulence. But is there
enough ordering in the theory? A.N. Simakov & P. Catto, Phys.
Plasmas 12 (2005) 012105
●
Global GK is starting to simulate edge plasma flows and
transport therein
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Near future
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GK simulation prepared for transport transient simulations
GK, extended MHD tests for kink, ballooning, drift turb. +
tearing, and resistive-g + tearing possible.
Hybrid GK+extended MHD most attractive, but what are
the criteria for low/high k separation in including
microturbulence –in density, momentum (Reynolds stress),
energy fluid moment equations?
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008
Long term
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Fast and slow MHD benchmarks between extended MHD
and global GK
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ELM crash simulation with the global GK
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True transport time scale (0.1 s) with global GK
CEA-EDF-INRIA school in Numerical Analysis; Numerical Models
for controlled Fusion – Nice, France, 8-12 September, 2008