Kinetic Simulation of Sheared
Flows in Tokamaks
J.A. Heikkinen2, S.J. Janhunen1, T.P. Kiviniemi1,
S.Leerink1, M. Nora1, and F. Ogando1,3,
1
Teknillinen Korkeakoulu (TKK), Finland
2 Valtion Teknillinen Tutkimuskeskus (VTT), Finland
3 Universidad Nacional de Educación a Distancia (UNED), Spain
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Outline
I. Background
II. Gyrokinetic full f approach for toroidal fusion plasmas.
III.Examples of Vlasov codes.
IV.Description of a PIC full f code.
V.Testing:
–
Comparison to neoclassical theory.
–
Linear and nonlinear benchmarks of unstable modes.
VI.Influence of noise on results.
VII.Transport simulations in toroidal plasmas.
VIII.Future challenges of full f
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section I
Background
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Turbulence and zonal flows
●
●
●
●
Self-organization and zonal flow generation with
turbulence is important not only in fusion plasmas but also
in other fluids like planetary atmoshpheres and solar
plasma
Transport transients and related transport barriers are
important, e.g., for ITER fusion reactor design
Empirical evidence from a number of tokamak facilities
indicates a complex scaling law for transport transients
The scaling laws for transport barriers in tokamaks are
today to a large extent not understood by physics
principles
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Sheared flows and turbulence
appear in toroidal plasma simulation
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Kinetic simulation required
●
●
All trials up to date around the world to see the
spontaneous confinement transition in an experimental-like
first principles plasma simulation have failed so far.
Reason to this may be that not all relevant physics like
kinetic details of particle motion have been included in the
simulation codes, lack of self-consistency with background
evolution, or model simplifications like using fluid codes
have been too crude.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section II
Gyrokinetic full f approach for
simulation of toroidal plasmas
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Full f simulation of tokamak plasmas
●
●
In full f simulation, one solves for the whole particle distribution function in
phase space and in time either non-perturbatively, with gyrokinetics or
drift-kinetically
Pioneering non-perturbative 5D particle-in-cell (PIC) simulation
of magnetically confined toroidal plasmas (unrealistic me/mi) :
Cheng & Okuda, ’78; Sydora, ’86; LeBrun, ’93, Kishimoto, ’94
●
Eulerian and semi-Lagrangian Vlasov 3D, 4D, and 5D
simulation:
Cheng & Knorr, ’76; Manfredi, ’96; Sonnendruecker, ’99; Xu, ’06; Scott, ’05; Grandgirard, ’06
●
Gyrokinetic 5D particle simulation:
Furnish, ’99; Heikkinen, ’00, 03’; Chang, ’03.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Calculation flux in particle codes
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.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Delta f vs. full f
●
●
Delta f calculates
perturbations from an
assumed background
distribution f0.
Powerful for small f-f0
–
Linear mode analysis
–
“Snapshot” transport analysis
–
Path-breaking global transport
studies for large toroidal
installations
●
●
Full f calculates the whole
particle distribution.
Fitting processes that
perturbate strongly the
particle distribution
–
Strong transient or long time
scale transport in core or edge
plasmas
–
Strong particle/energy sources
–
Full neoclassical equilibrium
–
Edge plasma (sheaths, wall
losses, recycling, separatrix,
flows)
–
Large MHD (sawteeth, ELMs)
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Delta f vs. full f
●
●
Automatic extension of Delta
f to full f calculation may be
possible
–
Non-conservative effects;
collisions, sources
–
Loading optimization
–
Automatic increase of the
number of markers when
required
Few particles (~10-100) per
cell are needed for good
results with small f-f0.
●
●
Full f can be realized both in
PIC and Vlasov (Eulerian,
semi-Lagrangian)
–
Vlasov simulation is free of noise
and has controlled numerical
resolution in phase space.
–
PIC is numerically flexible and
straightforward to implement
Needs many particles per
cell (~1000) or fine grid
discretization of velocity
space for an acceptable
noise level or accuracy.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section III
Examples of full f Vlasov codes
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Full f 5D gyrokinetic code GYSELA Grandgirard,
’06 (CEA, LPMIA-Univ, IRMA-Univ, LSIIT-Illkirch)
●
Semi-Lagrangian GYSELA code:
- fixed grid, follows trajectories
backwards,
t
dtF=0
- global code
- 5D GAM and ITG Cyclone tests
successful
(R-R0)/i
x,v
t- t
Interpolation
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Full f 5D gyrokinetic code TEMPEST Xu ’06
(LLNL, Calif-Univ, GA, LBNL, PPPL)
- Based on modified gyrokinetics valid
for large long-wavelength and small
short-wavelength variations (Qin, 06)
- Fixed grid, equations solved via a
Method-of-Lines approach and an
implicit backward-differencing scheme
using iteration advances the system in
time
- Developed for circular core or divertor
edge geometry
- 4D neoclassical tests successful. 5D
drift wave and ITG benchmarking
ongoing
-
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section IV
The ELMFIRE code; an example of
a full f PIC code
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
The ELMFIRE group
Founded in 2000 at
Euratom-Tekes
Contributors from
Finland
Spain
Holland
Main affiliations
VTT
TKK
... but also ...
CSC
Åbo Akademi
UNED (Spain)
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
ELMFIRE code
●
Full f nonlinear gyrokinetic particle-in-cell approach for global
plasma simulation (present version electrostatic).
●
Magnetic coordinates (ψ,,) Boozer ’81.
●
Guiding-center Hamiltonian White & Chance, ’84.
●
Gyrokinetics is based on Krylov-Boholiubov averaging method
in description of FLR effects (P. Sosenko, ‘01).
●
Adiabatic or kinetic electrons with impurities.
●
Parallelized using MPI with very good scalability.
–
Based on free software: PETSc and GSL for math calc.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
ELMFIRE full f features
●
●
●
●
An initial canonical distribution function avoids the onset of
unphysical large scale ExB flows (Heikkinen, ‘01)
Direct implicit ion polarization (DIP) and electron
acceleration (DEP) sampling of coefficients in the
gyrokinetic equation
Quasi-ballooning coordinates to solve the gyrokinetic
Poisson
Versatileequation
heat (RF, NBI,
Ohmic) sources and particle
sources/ recycling
●
Full binary collision operator
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Poisson equation
●
W.W. Lee proposed “standard” model with polarization drift
included in equation operator.
–
●
Ion density evaluated from ion motion without polarization
drift
P. Sosenko proposes including polarization in the ion
density.
–
Ion density evaluated from ion motion with polarization drift.
Circular gyro-orbits.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Implementation into ELMFIRE
●
●
●
Solve  by isolating ion
polarization drift contribution
to density.
That contribution is
calculated implicitely every
timestep using also previous
values of .
The gyroaveraged electric
field is interpolated from grid
potential values for the ion
polarization drift.
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
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section V
Benchmarking of ELMFIRE
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Testing ELMFIRE
●
●
●
Comparison to neoclassical theory in the presence of
turbulence.
–
Frequency of GAM and Rosenbluth residual.
–
Neoclassical radial electric field.
Comparison to other codes has been done in the Cyclone
Base cases.
–
Linear growth of unstable modes and their phase.
–
Nonlinear saturation of transport in both adiabatic and
kinetic-electron case.
Comparison to experimental results.
–
Collaboration with IOFFE Institute and St. Petersburg
Polytechnic
working
withforthe
FT-2
tokamak.
Numerical
Flow Models
Controlled
Fusion
– Porquerolles,
France, 16-20 April, 2007
Available resources
●
●
●
Gyrokinetic full f computation is very demanding
computationally. Parallel computation is a need.
CSC (The Finnish IT Center for Science) provides shared
use of high-end parallel computers.
–
IBM eServer cluster 1600. 512
processors with 2.2TFlops, 384GB
RAM and High Performance Switch
communication.
–
Cluster of 768 AMD OpteronTM
processors up to 3.2TFlops, 1600GB
RAM, Infiniband network.
Cray XT4 (Hood): 70TFlop, 70TB RAM;
HP ProLiant Supercluster, 10.6 Tflop,
Numerical Flow Models for Controlled Fusion – Porquerolles,
100 TB
France, 16-20 April, 2007
Geodesic Acoustic Modes
●
●
●
Neoclassical theory predicts
GAM frequency and
Rosenbluth residual.
Results show good wide
agreement with theory.
Simulations done on a
plasma annulus.
–
R=0.3-0.9 m, a=0.08 m, B=0.62.45 T, q=1.28-2.91, Ti=90-360
eV, ni=5.1×1019m-3 (r/a=0.75)
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Neoclassical radial electric field
●
Neoclassical radial electric field is
well followed in conventional (Lmode) turbulent simulations both in
radius and in time
–
R=1.1 m, a=0.08 m, B=2.1
T, I=22 kA, parabolic ion
heated n,Ti,e profiles
(r=0.04m).
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Neoclassical benchmarking
Parameters and boundary conditions
●
FT-2 like parameters, R0=0.55 m and a=0.08 m,
●
Itot = 55 kA, T,n,j ~ (1-(r/a)2) ,
●
n0=5*1019 1/m3
●
T0=300 eV in high T case, T0=120 eV in low T
●
no heating, no loop voltage
●
relaxing profiles, cooling by recycling on outer
radius
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Neoclassical benchmarking
Number of particles must be sufficient
as
Er may depend on noise
Er radial dependence fairly well
predicted by standard neoclassical
theory. However, Reynolds stress and
poloidal Mach number can be important.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Linear growth of unstable modes
●
Test based on adiabatic Cyclone Base Case (Dimits PoP
'00)
–
Red points from ELMFIRE, blue line: fit from article.
–
Figures show growth rates and typical time evolution for a
mode with ki=0.3
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Linear growth of unstable modes
●
Test based on adiabatic Cyclone Base Case (Dimits PoP
'00)
–
Red points from ELMFIRE, blue line: fit from article.
–
Figures show growth rates and typicalRegion
time evolution
for a
of linear growth
mode with k  i=0.3
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Linear growth of unstable modes
●
Test based on kinetic Cyclone Base Case (Chen NF '03)
–
Filled circles and squares from ELMFIRE
–
Dashed line: fit for the growth rate  from Chen NF ‘03.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Evolution of thermal conductivity
●
Evolution of i is studied with nonlinear runs of Cyclone Base.
–
●
Measured at r=a/2 (q=1.4). Using kinetic electrons. R/LT=10.
Weak collisionality; T(a/2)=2000 eV, n=5*1017 m-3.
Convergence requires a large number of particles per cell.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section VI
Influence of noise on results
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Influence of noise on results
●
●
●
Particle simulation not only covers physical density
fluctuations, but it produces undesirable noise with fluxes
that perturbate the solution.
Associated diffusivity can be estimated from the radial
particle shift during decorrelation time.
Physical radial ion heat conductivity can be estimated from
mixing-length estimate of the physical level of fluctuations,
being also proportional to T3/2.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Effects on calculated conductivity
●
●
Image shows influence of
strong collisionality and
potential averaging on ion
radial heat conductivity.
–
Collisionless cases show
residual noise
conductivity.
–
Noise is filtered out by
averaging potential over
flux surface.
Scaled Cyclone Base Case with
kinetic electrons; T=100 eV,
n=4.5*1019 m-3
So far noise is reduced by
“brute force” ... higher N!
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Contribution of noise to the heat flux
The convective noise flux prediction gives a fairly good estimate of t
unphysical ion heat conductivity in the simulations
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
From noise to turbulence spectrum
• Wave number spectrum from different time instants demonstrates
how one moves from white noise to physical spectrum in modes.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Problematic levels of noise
●
●
●
●
Figure show exceptionally
bad case regarding noise
effects.
Fluctuations @ r=a/2
It is a kinetic cyclone base
case with scaled parameters
and low T=100eV and
n=4.5*1017 m-3.
Density fluctuations remain
almost constant in time.
Regression shows almost
perfect N-1/2 scaling,
indicating that results are
dominated by noise.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
but not always problematic...
●
In FT-2, fluctuation levels are
much higher (10-40%) than
in scaled Cyclone Base Case
(~1%).
–
●
●
So high perturbation level
warrants the use of a full f
scheme.
Image shows density
fluctuations relative to flux
surface average.
Relative importance of noise
values can be seen in videos
of both cases.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Probability distribution functions
●
●
Fluctuation levels can be
represented by the PDF
graphs.
PDFs show fluctuation
distributions over a middle
flux surface.
–
Density values are
averaged over time.
–
Turbulence in FT-2 takes
fluctuations up to 40% in
start (up) and 15% after
50µs (down).
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section VII
Transport simulations
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Case 1 under study: LH heated FT-2
●
●
Parameters from 100 kW
LH heated 22 kA FT-2
tokamak.
Case shows a rapid growth
of potential and electric
field and reduction of
thermal conductivity
interpreted as ITB
formation.
–
Top figure diagram (r,t) of
flux surface average of
potential.
–
Bottom figure
ion
Numerical Flow Models for Controlled Fusion – Porquerolles,
diffusivity at mid radius.
France, 16-20 April, 2007
Evolution of profiles
●
Strong ExB flow shear is created
at r=0.05 m, where a knee-point
in Ti profile is found
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Calculated spectra
●
Parameters of the FT-2 case, devised to cause the
formation of Internal Transport Barrier.
–
B=2.2T, I=22kA, n=3.5×1019m-3, Ti=250eV, Te=300eV
S(k) at r=7.5cm
S(k) at r=5.1cm
 vs. time
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Case 2 under study: LH heated FT-2
●
Heating phase for 100 kW
LH heated 22 kA FT-2
tokamak (O8+ impurities
included).
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Evolution of large scale fluctuations
●
Density fluctuations plots show the formation and further
destruction of macroscopic structures
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Evolution of diffusivity
●
●
Both particle diffusivity and heat conductivity drop
drastically when poloidal flow shear destroys the turbulent
structures
The figures show values from the middle radius
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Case 3 under study – Ohmic FT-2
Reproduce the FT-2 reflectometer signal I(t) with ELMFIRE
I(t)=∫ w(r,θ) δn(r,θ,t) r dr dθ
●
●
W(r, θ ) = Weighting function calculated by beam tracing code using exp.
data
δn(r,θ,t) = Density fluctuations simulated by ELMFIRE code
Results
●
●
Poloidal velocity calculated by the shift of the power spectrum of I(t) is in
reasonable agreement with the poloidal velocity measured by the Doppler
reflectometer for an Ohmic discharge
Width of the power spectrum is too narrow compared to the experimental
results.
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Frequency broadening is too narrow in
the simulation
v pol
 65000 * 2 * 0,06
m
 f *  
 1361
18
s
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Section VIII
Future challenges
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Unresolved issues in full f
●
●
How to ensure a sufficient number of particles in all grid
cells when density varies strongly in global PIC simulation
Adaptation of good grid resolution for strongly varying f in
Vlasov simulations
●
Full collision operator in Vlasov simulations
●
SOL plasma simulation with sheath boundaries
●
Gyrokinetics with strong fluctuations
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Scalability
IB-new = Sepeli with InfiniBand and optimized parallelization
IB = Sepeli with InfiniBand
LAM = Sepeli with Gigabit Ethernet
IBM-new = IBM with optimized parellelization
Time per time step (s)
10000
1000
IB-new
IB
LAM
IBM-new
IBM
100
10
1
10
100 – Porquerolles,
Numerical Flow Models
for Controlled Fusion
Number
of processors
France,
16-20
April, 2007
1000
Resource scenario
Finnish
Computational
resources for
academic research
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007
Conclusions
●
●
Successful 5D gyrokinetic full f PIC and Vlasov simulations
of tokamak electrostatic turbulence and transport for core
plasma.
Careful benchmarking of the codes is performed in appropriate
limits for the turbulence saturation and neoclassical
characteristics.
–
●
●
Linear and nonlinear benchmarking.
Vlasov approach has less noise and better control of resolution;
PIC code is more flexible to implement
Most urgently neededACKNOWLEDGEMENT
for edge plasma simulations; further
S
development of
nonlinear
terms This
in gyrokinetics
may be needed
CSC: The Finnish IT
project receives
Center for Science
for its computing
facilities
funding from the
European
Commission
Numerical Flow Models for Controlled Fusion – Porquerolles,
France, 16-20 April, 2007