1. 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
Añadir transparencias de separacion entre temas
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

2. Outline
Background
Gyrokinetic full f approach for toroidal fusion plasmas.
Examples of Vlasov codes.
Description of a PIC full f code.
Testing:
Comparison to neoclassical theory.
Linear and nonlinear benchmarks of unstable modes.
Influence of noise on results.
Transport simulations in toroidal plasmas.
Future challenges of full f
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

3. Section I Background
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

4. 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, April, 2007

5. Sheared flows and turbulence appear in toroidal plasma simulation
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

6. 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, April, 2007

7. Section II Gyrokinetic full f approach for simulation of toroidal plasmas
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

8. 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, April, 2007

9. 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
Hablar de quiescent initialization?
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, April, 2007

10. 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, April, 2007

11. 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, April, 2007

12. Section III Examples of full f Vlasov codes
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

13. Full f 5D gyrokinetic code GYSELA Grandgirard, ’06 (CEA, LPMIA-Univ, IRMA-Univ, LSIIT-Illkirch)
x,v t
t- t
dtF=0
Interpolation
Semi-Lagrangian GYSELA code:
- fixed grid, follows trajectories backwards,
- global code
- 5D GAM and ITG Cyclone tests successful
(R-R0)/i
Z/i
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

14. 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, April, 2007

15. Section IV The ELMFIRE code; an example of a full f PIC code
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

16. 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, April, 2007

17. 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, April, 2007

18. 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 equation
Versatile heat (RF, NBI, Ohmic) sources and particle sources/ recycling
Full binary collision operator
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

19. 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.
What are the advantages of Sosenko's model in comparison to Lee's?
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

20. 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 i
py+
px0
px-
xp,yp
px+
py0 ds k
py- x
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, April, 2007

21. Section V Benchmarking of ELMFIRE
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

22. 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 with the FT-2 tokamak.
Que es el Rosenbluth residual? Lo he escrito bien?
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

23. 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 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, 100 TB
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

24. 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= m, a=0.08 m, B= T, q= , Ti= eV, ni=5.1×1019m-3 (r/a=0.75)
Hay menos puntos de Ar porque necesita casos mas largos hasta llegar a equilibrio.
Las oscilaciones GAM se detectan por tener modo m=n=0
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

25. Neoclassical radial electric field
Neoclassical radial electric field is well followed in conventional (L- mode) 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).
A que perfiles se refiere? Que puedo contar de esto?
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

26. 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, April, 2007

27. 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, April, 2007

28. 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, April, 2007

29. 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
Region of linear growth
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

30. 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, April, 2007

31. 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.
Explicacion teorica de resultados
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

32. Section VI Influence of noise on results
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

33. 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, April, 2007

34. 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.
So far noise is reduced by “brute force” ... higher N!
Scaled Cyclone Base Case with kinetic electrons; T=100 eV, n=4.5*1019 m-3
Al principio se ve el crecimiento neoclasico y ese crecimiento se ve tambien en el caso filtrado porque no afecta a las colisiones.
El filtrado se carga la turbulencia. El hecho de que la diferencia filtrado-sin filtro sea constante hace ver que el ruido es mayor que la turbulencia.
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

35. Contribution of noise to the heat flux
The convective noise flux prediction gives a fairly good estimate of the
unphysical ion heat conductivity in the simulations
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

36. 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, April, 2007

37. Problematic levels of noise
r=a/2
Figure show exceptionally bad case regarding noise effects.
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.
Este caso es el mismo que antes
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

38. 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, April, 2007

39. 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, April, 2007

40. Section VII Transport simulations
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

41. 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 diffusivity at mid radius.
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

42. 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, April, 2007

43. 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
Por que la figura de enmedio no tiene un maximo a kr=1
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

44. 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, April, 2007

45. 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, April, 2007

46. 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, April, 2007

47. 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, April, 2007

48. Frequency broadening is too narrow in the simulation
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

49. Section VIII Future challenges
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

50. 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, April, 2007

51. Scalability
IB-new = Sepeli with InfiniBand and optimized parallelization
IB = Sepeli with InfiniBand
LAM = Sepeli with Gigabit Ethernet
IBM-new = IBM with optimized parellelization
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

52. Resource scenario
Finnish Computational resources for academic research
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007

53. 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 needed for edge plasma simulations; further development of nonlinear terms in gyrokinetics may be needed
ACKNOWLEDGEMENTS
CSC: The Finnish IT Center for Science for its computing facilities
This project receives funding from the European Commission
Numerical Flow Models for Controlled Fusion – Porquerolles, France, April, 2007