Closure relations in various forms are crucial to integrated modeling

1. Neoclassical particle-based closures for NTM islandsD. A. Spong, ORNLSWIM RF/MHD Meeting, University of Wisconsin, Dec. 4-6, 2007 • Acknowledgements to: S. Hirshman, D. Castillo-Negrete, L. Berry, K. Shaing, J. Breslau, H. Sugama, S. Nishimura, E. D’Azevedo, members of SWIM project • Simple 3D effects in NTM islands - DKES code • Particle based closures - DELTA5D Monte Carlo code • Computational characteristics • Physics models • Next steps/Summary

2. Closure relations in various forms are crucial to integrated modeling • Relate microscopic/kinetic phenomena to macroscopic fields - connect disparate scales • Examples • RF tail formation - impact on Ohm’s law • Neoclassical effects in 3D fields - impact on parallel viscosity, bootstrap current • Parallel density/heat transport in magnetic island region • Energetic particles - destabilization by wave/particle resonances • The feasibility and validity of integrated modeling depends strongly on a good balance between comprehensive vs. efficient in the closure relations that are used.

3. A spectrum of closure approches can be considered: Gyro- kinetic ions and electrons MHD + kinetic MHD + kinetic MHD + additional fluid closure relations

4. Characteristic timescales are widely separated: • NTM’s evolve over resisitve timescales • Closure relations equilibrate over pitch angle scattering times • MHD time steps are ~ 10-2 R0/vA • Many collisional equilibration times are completed during growth of NTM, but not during an MHD timestep e/tMHD NTM/e CDX-U DIII-D ITER Possible strategy: run particles continuously on separate processors with frequent updates from MHD and less frequent updates of MHD by particles.

5. NTM islands introduce 3D magnetic field structure into the tokamak: • “stellarator within a tokamak” effect • K. Shaing, Phys. Rev. Lett. 87, 245003 (2001) • K. Shaing, Phys. Plasmas 9, 4633 (2002) • K. Shaing, D. Spong, Phys. Plasmas 13, 22501 (2006) • Direct/indirect modification of |B| • Field line trajectory around island  particles see non-axisymmetric magnetic field even though they are immersed in an axisymmetric field • 3D transport effects: • Neoclassical viscosity enhanced locally by ripple  island rotates relative to surrounding plasma • Plasma confinement can be locally improved by radial E field (“snakes”) • Island induced bootstrap current - modifies NTM evolution

6. Tearing mode islands  broken symmetry in |B| modified bootstrap current and stellarator-like scalings in  transport

7. DELTA5D DELTA5D Monte Carlo code • Full-f and f versions • Alpha particle and NBI slowing down and losses in stellarators • Global energy losses through outer surface • Diffusion coefficient, viscosity, bootstrap current • Collision operator (Langevin model) + Quasilinear RF operator (J. Carlsson)

8. DELTA5D DELTA5D particle equations are based on VMEC flux coordinates, but use cylindrical components of E, B: Hamilitonian orbit eqns. in real space coordinates: • M3D routine (Breslau) ports BR,BBZ from finite elements to uniform s, ,  grid • Filled in data (no edges) - optimal for SVD compression Ref. “Collisional Transport in Magnetized Plasmas,” by P. Helander and D.J. Sigmar

9. Recent tests on the Cray XT4 (Franklin) show close to linear scaling up to 16,384 processors

10. MHD-to-particle coupling issues: • Need for following particles multiple steps between MHD steps • Physics reason: collisional evolution • Computational reason: noise reduction, filtering • Data compression • Minimize memory “footprint” • Improved scatter/gather operations • Assign particles to processors rather than regions • Decouple processors used for particles from those for fluid equations

11. SVD data compression has recently been extended to 3-dimensional data: • GLRA (Generalized Low Rank Approximation) method • J. Ye in 21st International Conference on Machine Learning (2004) • D. del-Castillo Negrete, D. Spong, E. D’Azevedo, S. Hirshman, “Compression of MHD Simulation Data Using SVD,”Journal of Computational Physics 222 (2007) 265-286. • Iterative algorithm for minimizing Frobenius norm between 3D data and GLRA matrix form: • Analogous to 2D SVD (Nk = 1 limit), but iteration required and Dk matrices are not diagonal

12. SVD fits of M3D dataset reproduce more exact fits for rank = 10 - 20 -> compression ratios of 35 - 100: R-Z space

13. SVD fits of M3D dataset reproduce more exact fits for rank = 10 - 20 -> compression ratios of 35 - 100: s-space

14. Significant compressions/timing improvements can be achieved while retaining all significant data features • Better adapted to future multi-core processors - where pins become strangle point • Smaller data size improves effective memory bandwidth

15. SVD techniques can also be used for smoothing particle data prior to MHD coupling Binned particle data SVD smoothed particle data

16. Two components to 3D neoclassical models: Parallel momentum balance ( JBS, parallel heat flux,…) Conversion (stress tensor) of  flows to || flows Parallel friction/viscosity Perpendicular particle/energy fluxes (E, polarization currents,…) Fluxes driven by parallel flows Fluxes driven by radial diffusion

17. Coupling to MHD model Typical fluid MHD model Influenced by neoclassical/RF closures • Closure relations allow inclusion of a variety of kinetic/transport effects in fluid MHD models • Reduced analytic closures often used • Hybrid closure models - fluid + particles • Currently only done for collisionless fast ion tails via momentum equation • Electron contributions to the pressure tensor in Ohm’s law are of crucial importance for tearing instabilities closure relation:  = function of n, T,V, B, E

18. DELTA5D A f partitioning separates out the Maxwellian, u||, q||, and diamagnetic flow parts of f: This approach allows the parallel flow and diamagnetic velocities to fixed consistent with those from the MHD model. Extension of H. Sugama, S. Nishimura, Phys. Plasmas 9, 4637 (2002) to f particle method

19. DELTA5D From these f components, viscosity coefficients can be obtained: • parallel flow and diamagnetic velocities are supplied by the MHD model. • since these velocities are explicit (not embedded in the stress tensors) implicit differencing is possible on these terms

20. DELTA5D For axisymmetric equilibria, viscosity-based closure relations can be benchmarked with the DKES code: || benchmark with DKES || from DELTA5D: running time averages - 1 flux surface 3200 electrons * DKES: Drift Kinetic Equation Solver

21. Distribution function benchmarking with DKES is also underway: v = 1x10-4 Delta5D (not fully equilibrated) DKES   v||/v v||/v

22. DELTA5D Radial scan shows viscosity enhancements related to inner and outer island chains at q = 1 and q = 2, but viscosities outside islands are also modified.

23. Summary, Future Directions • Closure relations are important component of extended MHD models • A different approach is required for kinetic closures of the thermal plasma than for energetic tail populations • Within certain limits a new f particle closure can be developed with the goal of maintaining consistency with MHD variables • Much work remains: • Calculate complete neoclassical stress tensor matrix • Extend to local viscosities and flows • Test with a simple NTM evolution model • Electric field within island - polarization drift currents • Couple to MHD code • Formulation for stochastic regions • Your suggestions and collaboration are welcomed

24. My suggestions: • 0th order • Working model: introduce (ad-hoc) fluctuating JBS, JRF, pressure convection/parallel transport • Ala’ Gianakon, et al., POP’96; Spong, et al., ORNL-TM 10947 • Show computational MHD NTM mode (single helicity, ’ < 0) and RF stabilization • Generate benchmark cases for closures • 1st order • Using 2/1, 3/2, … benchmark cases, improve/test Neo and RF closures offline • Incorporate models of these improved closures into the MHD model • 2nd order • Directly couple closure equations/code with MHD code • Verify NTM stabilization with RF, multiple NTM’s, etc.