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Computational fusion plasma physics

Computational fusion plasma physics.

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Computational fusion plasma physics

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  1. Computational fusion plasma physics L. Villard1, T.M. Tran1, S.J. Allfrey2, P. Angelino3, A. Bottino4, S. Brunner1, W.A. Cooper1, T. Dannert1, G. Darmet3, X. Garbet3, Ph. Ghendrih3, V. Grandgirard3, J.P. Graves1, R. Hatzky5, Y. Idomura6, F. Jenko4, S. Jolliet1, M. Jucker1, X. Lapillonne1, B.F. McMillan1, N. Mellet1, P. Popovich2, Y. Sarazin3, O. Sauter1, B. Scott4 (1) Centre de Recherches en Physique des Plasmas, Association Euratom - Suisse, EPFL, 1015 Lausanne, Switzerland (2) Center for Multiscale Dynamics, Dept of Physics and Astronomy, UCLA, USA (3) Association Euratom - CEA/DSM/DRFC, Cadarache, France (4) Max-Planck Gesellschaft, Association Euratom - IPP, Garching, Germany (5) Computer Center, IPP, Max-Planck Gesellschaft, Garching, Germany (6) Japan Atomic Energy Agency, Taitou, Tokyo 110-0015, Japan CCP 2007 Brussels, Sept 5-8, 2007

  2. Outline • 1. Introduction • Fusion; ITER; tokamak • 2. Transport • Turbulence [also: talk F. Jenko] • 3. Alternative concepts of magnetic confinement • Equilibrium and stability • 4. Burning plasma physics. RF waves • Heating, current drive, fast particles [also: talk Y. Peysson] • 5. HPC issues • massive parallelism, scalability • 6. Outlook and conclusions CCP 2007 Brussels, Sept 5-8, 2007

  3. Outline • 1. Introduction • Fusion; ITER; tokamak • Magnetised plasmas: time-, length- scales; anisotropy; collective effects; kinetic effects; geometry • 2. Transport • Turbulence • 3. Alternative concepts of magnetic confinement • Equilibrium and stability • 4. Burning plasma physics. RF waves • Heating, current drive, fast particle driven instabilities • 5. HPC issues • massive parallelism, scalability • 6. Outlook and conclusions CCP 2007 Brussels, Sept 5-8, 2007

  4. Fusion • … in space (a nursery of stars: NGC 6357) Credit: NASA, ESA and J. M. Apellániz (IAA, Spain) CCP 2007 Brussels, Sept 5-8, 2007

  5. Controlled fusion on earth: magnetic confinement • Figure of merit: density x Temperature x energy confinement time (nTtE ) • Magnetic confinement : tE~1s, n~1020m-3, T~10keV. ITER: nTtE=3.4 atm.s Source: EFDA-JET JET has produced 16MW of fusion power Source: EFDA and J.B. Lister CRPP CCP 2007 Brussels, Sept 5-8, 2007

  6. The plasma will be there (see next p) You are here ITER: the way to fusion • EU, Japan, USA, China, India, South Korea, Russian Federation • 5.3G€ construction cost (1/3 G € /y = 0.06% of world overall R&D) • Virtually inexhaustible, environmentally benign source of energy: • from Deuterium and Lithium; gives Helium. (Tritium is recycled) Source:ITER CCP 2007 Brussels, Sept 5-8, 2007

  7. Magnetic confinement: tokamak Trapped particle Larmor radiusrL Passing particle particle trajectory CCP 2007 Brussels, Sept 5-8, 2007

  8. transformer induction RF fusion heat current drive heat heat bootstrap operational limits transport equilibrium turbulence macroscopic stability coils microscopic stability Complexity: many nonlinearities • Geometry of magnetic configuration is an essential feature of fusion plasma physics Neutral Beams heat CCP 2007 Brussels, Sept 5-8, 2007

  9. Timescales in the ITER plasma • Physics spans several orders of magnitude • Direct Numerical Simulation (DNS) of “everything” is unthinkable • Need to separate timescales using approximations • How to integrate all phenomena in a consistent manner? machine lifetime energy confinement ion cyclotron electron cyclotron 1 shot turbulence CCP 2007 Brussels, Sept 5-8, 2007

  10. Surfers with velocity too different from the phase velocity of the wave will not ride the wave Surfers with velocity just below the phase velocity of the wave will be accelerated -> momentum and energy transfer Net energy transfer from the wave to the particles if Collisionless Landau damping Kinetic effects: wave-particle interaction • General: distribution function in 6D phase space • To be solved with consistent electromagnetic fields CCP 2007 Brussels, Sept 5-8, 2007

  11. Outline • 1. Introduction • Fusion; ITER; tokamak • 2. Transport • Turbulence [also talk by F. Jenko] • Numerical methods and related issues • 3. Alternative concepts of magnetic confinement • Equilibrium and stability • 4. Burning plasma physics. RF waves • Heating, current drive, fast particles [also talk by Y. Peysson] • 5. HPC issues • massive parallelism, scalability • 6. Outlook and conclusions CCP 2007 Brussels, Sept 5-8, 2007

  12. Transport • Hot plasma core, colder edge: => grad T • If grad T > (grad T)crit => instabilities • small scales, i.e. spatial scales ~rL • Growth, then nonlinear saturation of modes • => turbulent state • => transport (heat, particles, momentum) • “anomalous”, essentially by collisionless processes • Turbulent transport >> collisional transport • In spite of this, typical grad T ~ 106K / cm • Need a theory … and numerical tools! CCP 2007 Brussels, Sept 5-8, 2007

  13. Turbulence – gyrokinetic theory (GK) • Assume • Average out the fast motion of the particle around the guiding center • Fast parallel motion, slow perpendicular motion (drifts) • Strong anisotropy of turbulent perturbations (// vs perp to B) phase space dimension reduction 6D -> 5D Zonal Flows ion-driven electron-driven CCP 2007 Brussels, Sept 5-8, 2007

  14. Collisionless gyrokinetics ? • Core fusion plasmas are very weakly collisional (lmfp,// >> device size) • How can we obtain a turbulence-induced heat flux in a system without explicit dissipative term? • Issue for computations: numerical dissipation must be small enough in order to make correct predictions • in the collisionless limit • and to isolate the physical collisional effects from numerical dissipation effects v Spatial scales bounded by Larmor radius and device size => Direct Numerical Simulations (DNS) are possible, but: Collisionless (Landau) wave-particle interactions lead to filamentation in phase space An infinitesimal dissipation leads to finite power transfer x CCP 2007 Brussels, Sept 5-8, 2007

  15. Solving GK: 3 classes of numerical methods • Lagrangian: Particle In Cell, Monte Carlo • Sample phase space (markers) and follow their orbits • Noise accumulation is difficult to control in long simulations • Semi-Lagrangian • Fixed grid in phase space, trace orbits back in time • Multi-dimensional interpolation is difficult • Eulerian • Fixed grid in phase space, finite differences for operators • CFL condition; overshoot versus dissipation • Common to all three: Poisson (+Ampere) field solver • Various methods: finite differences, finite elements, FFT,… Developing several approaches is not a luxury. The complexity of the system implies that all codes are run near the limits of their capability. Therefore cross-comparisons are a necessity. CCP 2007 Brussels, Sept 5-8, 2007

  16. Lagrange: PIC, df, finite elements: ORB5 Color contours of perturbed electrostatic potential • Ion Temperature Gradient driven turbulence • Resolution: 128x448x320, N = 83 million, WiDt=40 • ~6 h on 1024 processors IBM BG/L@EPFL • df :solve for the perturbed distribution function only (control variates) hot core [S. Jolliet, et al., CPC 2007] large scale Zonal Flows => breakup of turbulent eddies => self-regulation of turbulence cold edge turbulent eddies => radial transport CCP 2007 Brussels, Sept 5-8, 2007

  17. Perturbations on a magnetic surface Small scales perpendicular to B • Gyrokinetic ordering • High k// modes are unphysical => filter them out! Next page Color contours of perturbed electrostatic potential Long scales parallel to B CCP 2007 Brussels, Sept 5-8, 2007

  18. Fighting noise with field -aligned Fourier filter (FAFF) • Turbulent perturbations. Detail on a magnetic surface. [S. Jolliet, et al., CPC 2007] Without FAFF: unphysical small scales in the parallel direction develop With FAFF: unphysical small scales in the parallel direction are suppressed CCP 2007 Brussels, Sept 5-8, 2007

  19. Fighting noise with field -aligned Fourier filter (FAFF) • Heat flux versus time Without FAFF [S. Jolliet, et al., CPC 2007] With FAFF Without FAFF, the noise-generated heat flux ends up dominating the simulation CCP 2007 Brussels, Sept 5-8, 2007

  20. With sources – long simulation Heat transport • Numerical noise is controlled over very long time scales • One should verify that the source term does not affect the physics Without source: turbulence decay, grad T -> close to marginal stability Signal / noise CCP 2007 Brussels, Sept 5-8, 2007

  21. Semi-Lagrange approach • Main motivation: get the better of two worlds • avoid the statistical noise problem inherent of Lagrange-PIC-MC methods • avoid the CFL condition inherent of Euler methods f=const along orbit • GYSELA code: • 5D gyrokinetic • Operator splitting • Leapfrog scheme, with averaging of integer and half-integer timesteps Orbit, backward integrated from grid point Interpolation of f @ foot of orbit CCP 2007 Brussels, Sept 5-8, 2007

  22. Fit from other codes (Dimits) Heat diffusivity Ion temperature gradient Lagrange (ORB5) vs½-Lagrange (GYSELA) Alternative approaches bring useful cross-checks of numerical and physical results Heat diffusivity Ion temperature gradient CCP 2007 Brussels, Sept 5-8, 2007

  23. Eulerian methods [also: GENE code, F. Jenko] • Recent improvement: a skew-symmetric finite difference operator that conserves both L1 and L2 norms [Morinishi, et al., JCP 143, 90 (1998)]. G5D code [Idomura et al., JCP 2007] • Better stability against spurious oscillations • Better robustness for long-time simulations • Exact particle number conservation • Good energy conservation properties: • Lagrange-Euler benchmark: comparable cpu time, memory usage for Euler ~ 5 times larger. Euler L1&L2 Lagrange PIC df CCP 2007 Brussels, Sept 5-8, 2007

  24. Eulerian, semi-Lagrangian and Lagrangian simulations confirm the role of zonal flows Zonal flow velocity profile in a cylindrical plasma Zonal flow velocity profile of Jupiter’s high atmosphere Idomura, et al., JCP 2007 NASA Cassini 2000 CCP 2007 Brussels, Sept 5-8, 2007

  25. The plasma edge • Occurrence of “transport barrier”: “pedestal”-like n and T profiles with steep gradients, leading to overall improved confinement. • An indispensable ingredient for the success of ITER The plasma cross-section is shaped: elongation, triangularity As we approach the edge, the geometry becomes more complex: A. Kendl, B.D. Scott, PoP 13, 012504 (2006) separatrix X-point CCP 2007 Brussels, Sept 5-8, 2007

  26. Edge turbulence • Electron response departs significantly from Boltzmann • Perturbations acquire an electromagnetic character • Collisions play a non negligible role • Gyrofluid ElectroMagnetic model: 6-moment of gyrokinetic equations. GEM code [B. Scott, PoP 2005]. Mechanisms linking geometry and turbulence Plasma cross-section elongation leads to reduced turbulent transport fluxes; small effect of triangularity d. [Kendl Scott PoP 2006] CCP 2007 Brussels, Sept 5-8, 2007

  27. Outline • 1. Introduction • Fusion; ITER; tokamak • 2. Transport • Turbulence [see also talk by F. Jenko] • 3. Alternative concepts of magnetic confinement • Equilibrium and stability • 4. Burning plasma physics. RF waves • Heating, current drive, fast particle driven instabilities • 5. HPC issues • massive parallelism, scalability • 6. Outlook and conclusions CCP 2007 Brussels, Sept 5-8, 2007

  28. Beyond the tokamak, beyond ITER • Unlike the tokamak, the stellarator is an intrisically steady-state magnetic confinement concept (does not need a net plasma current, does not need a transformer), but… Inside the largest stellarator : LHD (NIFS, Japan) CCP 2007 Brussels, Sept 5-8, 2007

  29. Stellarator Provide rotational transform by external coil currents • Axial symmetry breaking implies several challenges: • Does an equilibrium with nested magnetic surfaces exist? • Are charged particles confined? • What are the pressure stability limits? • How to heat? RF? Neutral Beam Injection? • Intensive numerical simulations are required The W7-X stellarator is now under construction in Greifswald (IPP) CCP 2007 Brussels, Sept 5-8, 2007

  30. Neutral Beam Injection heating • Leads to anisotropic fast ion distributions • here an example of computation for LHD Mod-B Hot ion pressure HFS deposition bth=2.6% bhot=1.4% Hot ion pressure LFS deposition CCP 2007 Brussels, Sept 5-8, 2007

  31. Outline • 1. Introduction • Fusion; ITER; tokamak • 2. Transport • Turbulence [see also talk by F. Jenko] • 3. Alternative concepts of magnetic confinement • Equilibrium and stability • 4. Burning plasma physics. RF waves • Heating, current drive, fast particle driven instabilities • 5. HPC issues • massive parallelism, scalability • 6. Outlook and conclusions CCP 2007 Brussels, Sept 5-8, 2007

  32. Alfvén Eigenmode in W7-X • Toroidicity-induced Alfvén Eigenmode • Could be destabilised by fast ions • Global wave code LEMAN: DNS of Maxwell’s equations (finite elements, Fourier) • Solves 4-potential (f,A) in order to avoid spectral pollution f=65.3kHz <Bmag>=4.2T CCP 2007 Brussels, Sept 5-8, 2007

  33. Ion Cyclotron RF heating in LHD f=5.57MHz, <Bmag>=0.8T, Deuterium plasma, n0=4x1019m-3 • # radial finite elements = 200, # Fourier modes = 630, • matrix storage (band) memory size = 490 GB Electric field normal component f=fci Power absorption density CCP 2007 Brussels, Sept 5-8, 2007

  34. Outline • 1. Introduction • Fusion; ITER; tokamak • 2. Transport • Turbulence [see also talk by F. Jenko] • 3. Alternative concepts of magnetic confinement • Equilibrium and stability • 4. RF waves • Heating, current drive, fast particle driven instabilities • 5. HPC issues • massive parallelism, scalability • 6. Outlook and conclusions CCP 2007 Brussels, Sept 5-8, 2007

  35. Parallelization scheme – ORB5 Domain decomposition (toroidal direction) • + Transpose operations (2D Fourier transforms of fields) MPI parmove particles Domain Cloning: field quantities (density, potential) are replicated MPI global sum fields Npes = Nd*Nc CCP 2007 Brussels, Sept 5-8, 2007

  36. 6.4 G particles Massive parallelism • Lagrange ORB5 code. Strong scaling with 8x108 particles (left). Weak scaling with 8x105 particles/proc (right). • IBM BG/L in co-processor mode • 6.4 G particles: enough to simulate the whole core of ITER • Euler GENE code: parallel efficiency 89% in strong scaling, 99% in weak scaling from 2k to 32k procs. CCP 2007 Brussels, Sept 5-8, 2007

  37. Conclusion and outlook • Fusion plasma physics is an extraordinary rich and diverse field • This diversity is reflected in the various theoretical and numerical approaches • Turbulence simulations [talk F. Jenko] and fast particles physics [talk Y. Pesson] remain very challenging topics • Future developments will go in the direction of including more self-consistent nonlinear interactions in multiscale simulations • Numerical codes for the simulation of fusion plasmas are mature for the next generation of HPC platforms CCP 2007 Brussels, Sept 5-8, 2007

  38. Multiscale turbulence with GENE:The role of sub-ion-scales GENE simulation (reduced mass ratio) • motivated by experiments, in • particular, “transport barriers” • extremely resource demanding • (millions of CPU-hours per run • if using realistic mass ratios) • sub-ion-scales (which have • been neglected so far) can • indeed be important CCP 2007 Brussels, Sept 5-8, 2007

  39. Parallelization scheme – ORB5 CCP 2007 Brussels, Sept 5-8, 2007

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