Overview of MHD and extended MHD simulations of fusion plasmas

1. Overview of MHD and extended MHD simulations of fusion plasmas Guo-Yong Fu Princeton Plasma Physics Laboratory Princeton, New Jersey, USA Workshop on ITER Simulation, Beijing, May 15-19, 2006

2. Outline • Introduction • Extended MHD Model • 3D Nonlinear Simulations: Recent results from M3D and NIMROD • Future Direction

3. Introduction/Summary • Motivation: MHD modes are important for fusion plasmas such as ITER. • MHD and extended MHD equations are very difficult to solve. (multiple time and spatial scales, extremely anisotropic heat transport). • Need advanced numerical methods: implicit, high-order finite elements, field aligned coordinates, adaptive mesh refinement, good pre-conditioners, efficient parallel schemes. • Significant progress in nonlinear 3D simulations of MHD modes in fusion plasmas (i.e., tokamaks). • Future direction: important physics problems, fluid and kinetic closures, efficient numerical methods and integrated simulations

4. MHD modes are important for fusion plasmas • Center: sawtooth and fishbone (central plasma profiles and induce seed island for NTM); • Core: ballooning modes ( beta limit), NTM (soft beta limit), TAE/EPM ( alpha particle transport). • Edge: external kink modes (beta limit), resistive wall mode, edge localized modes (H-mode pedestal width and height >> boundary condition for core confinement !)

5. Extended MHD Equations

6. C.R. Sovinec

7. Implicit method enables long time simulations of tearing modes (NIMROD) C.R. Sovinec et al., Phys. Plasmas 10, 1727 (2003)

8. 3D domain decomposition and MPI enable massively parallel computation (M3D) 3D domain decomposition

9. M3D: a 3D nonlinear extended MHD code • Multi-level of physics: ideal and resistive MHD, two fluids (drift ordering), MHD/particle hybrid model for energetic particles, electron fluid/kinetic ion hybrid model. • 2D finite elements (linear, 2nd and 3rd order) on unstructured mesh and 1D finite difference in toroidal direction; • Uses Petsc libraries for parallel data and solver. • M3D team: J. Breslau, J. Chen, G.Y. Fu, S. Jardin, S. Klasky, H.R. Strauss, L.E. Sugiyama, W. Park

10. Recent Results from M3D and NIMROD • Sawtooth oscillation in CDX-U (M3D, J. Breslau ); • Fast ion-driven fishbone in a tokamak (M3D, G.Y. Fu ); • Major disruption in DIII-D tokamak (NIMROD, S. Kruger); • NTM simulations (NIMROD, Giannakon); • ELM in DIII-D tokamak (NIMROD, D.P. Brennan ); • ELM in ITER (M3D, H.R. Strauss )

11. Characteristics of the Current Drive Experiment Upgrade (CDX-U) • Low aspect ratio tokamak (R0/a = 1.4 – 1.5) • Small (R0 = 33.5 cm) • Elongation ~ 1.6 • BT ~ 2300 gauss • Ip~ 70 kA • ne ~ 41013 cm-3 • Te~ 100 eV  S  104 • Discharge time ~ 12 ms • Soft X-ray signals from typical discharges indicate two predominant types of low-n MHD activity: • sawteeth • “snakes”

12. Equilibrium: q0 < 1 • Equilibrium taken from a TSC sequence (Jsolver file). • qmin 0.922 • q(a) ~ 9 toroidal current density • Questions to investigate: • Linear growth rate and eigenfunctions • Nonlinear evolution • disruption? • stagnation? • repeated reconnections? 10 q 1 0 1.0

13. Poloidal Mesh for CDX • 89 radial zones, up to 267 in  in unstructured mesh • Linear basis functions on triangular elements • Conducting wall; current drive applied by adding a source term in Ohm’s law. • Finite differences toroidally; 24 planes

14. n=1 Eigenmode Incompressible velocity stream function U Toroidal current density J A = 8.61 10-3 growth time = 116 A

15. Higher n Eigenmodes Incompressible velocity stream function U n = 2 n = 3 ... m  5 A = 1.28  10-2 m  7 A = 1.71  10-2

16. Nonlinear Sawtooth History10 Modes Retained 1st sawtooth crash 2nd sawtooth crash 3rd sawtooth crash Sawtooth period 1  395 A  100 s; Sawtooth period 2  374 A Reference CDX sawtooth period  125 s

19. Recent Results from M3D and NIMROD • Sawtooth oscillation in CDX-U (M3D, J. Breslau ); • Fast ion-driven fishbone in a tokamak (M3D, G.Y. Fu ); • Major disruption in DIII-D tokamak (NIMROD, S. Kruger); • NTM simulations (NIMROD, Giannakon); • ELM in DIII-D tokamak (NIMROD, D.P. Brennan ); • ELM in ITER (M3D, H.R. Strauss )

20. Fishbone in PDX (McGuire et al, 1983)

21. Excitation of Fishbone at high bh C B A

22. Nonlinear evolutionof fishbone instability

23. Distribution evolution

24. MHD nonlinearity changes mode structure significantly Linear MHD Nonlinear MHD

25. MHD nonlinearity reduces mode saturation level(case C)

26. Recent Results from M3D and NIMROD • Sawtooth oscillation in CDX-U (M3D, J. Breslau ); • Fast ion-driven fishbone in a tokamak (M3D, G.Y. Fu ); • Major disruption in DIII-D tokamak (NIMROD, S. Kruger); • NTM simulations (NIMROD, Giannakon); • ELM in DIII-D tokamak (NIMROD, D.P. Brennan ); • ELM in ITER (M3D, H.R. Strauss )

27. Nimrod Disruption Simulations S.E. Kruger et al., Phys. Plasmas 12, 056113 (2005)

28. Nimrod Disruption simulations

29. Recent Results from M3D and NIMROD • Sawtooth oscillation in CDX-U (M3D, J. Breslau ); • Fast ion-driven fishbone in a tokamak (M3D, G.Y. Fu ); • Major disruption in DIII-D tokamak (NIMROD, S. Kruger); • NTM simulations (NIMROD, Giannakon); • ELM in DIII-D tokamak (NIMROD, D.P. Brennan ); • ELM in ITER (M3D, H.R. Strauss )

30. NTM simulations (NIMROD)

31. Recent Results from M3D and NIMROD • Sawtooth oscillation in CDX-U (M3D, J. Breslau ); • Fast ion-driven fishbone in a tokamak (M3D, G.Y. Fu ); • Major disruption in DIII-D tokamak (NIMROD, S. Kruger); • NTM simulations (NIMROD, Giannakon); • ELM in DIII-D tokamak (NIMROD, D.P. Brennan ); • ELM in ITER (M3D, H.R. Strauss )

32. NIMROD simulations of ELM D.P. Brennan et al, 2005 APS invited talk

33. Recent Results from M3D and NIMROD • Sawtooth oscillation in CDX-U (M3D, J. Breslau ); • Fast ion-driven fishbone in a tokamak (M3D, G.Y. Fu ); • Major disruption in DIII-D tokamak (NIMROD, S. Kruger); • NTM simulations (NIMROD, Giannakon); • ELM in DIII-D tokamak (NIMROD, D.P. Brennan ); • ELM in ITER (M3D, H.R. Strauss )

34. M3D simulations of ELM in ITER H.R. Strauss et al., 2006 Sherwood Fusion theory meeting.

35. Future Direction • Important MHD problems; • Numerical discretization method; • Mesh configuration; • Fluid and kinetic closures; • Integrated simulations

36. Important MHD Problems • Sawtooth simulation with 2 fluid model and energetic particles; • NTM with kinetic closure; • Alpha particle transport with multiple TAEs; • ELM dynamics; • Resistive wall modes with kinetic effects.

37. Important questions for future: discretization • Lagrangian finite elements v.s. spectrum elements • C0 v.s. C1

38. Important questions for future:mesh configuration • Can AMR be effective for global toroidal problems ? • Can field aligned coordinates be used for global modes ? • Can field aligned coordinates evolve nonlinearly for implicit method to work ?

39. Important questions for future:closure problem • Can we find good closures for MHD modes in high temperature fusion plasmas ? • Is it appropriate and feasible to do kinetic closure for main species ? • Is it appropriate and feasible to do pure kinetic simulations for global MHD modes ?

40. Important questions for future:integrated simulations • How to couple MHD with plasma micro-turbulence, RF/NBI heating, energetic particles • Need proper fluid and kinetic closures !!!