1. 1 A Proposal for a SWIM Slow-MHD 3D Coupled Calculation of the Sawtooth Cycle in the Presence of Energetic Particles Josh Breslau Guo-Yong Fu S. C. Jardin Princeton University Plasma Physics Laboratory presented at the SWIM Workshop University of Wisconsin Dec 4th, 2007

2. 2 Overview Sawteeth will occur in ITER with a large energetic particle population present (due to α-particles) –there is presently no comprehensive 3D model of this The M3D (and NIMROD) codes should be able to model this phenomena with minor modifications Recent successes in Validation and Verification of these codes will give credibility to this modeling We propose to perform this as part of SWIM as it involves the coupling of a heating system and MHD –future step could involve RF as well

3. 3 M3D-K Fluid/Particle Hybrid Model MHD Equations CGL Pressure Tensor Gyrokinetic Equations

4. 4 Gyrokinetic equation (for particles) can use either  f or full-f method in  f, distribution function is sum of equilibrium + perturbed “weight function” evolves in time equilibrium distribution is a function of the adiabatic invariants equilibrium evolves (slowly) as the adiabatic invariants for each particle change due to detailed particle trajectory motions

5. 5 M3D Verification and Validation Good agreement between M3D and NIMROD for CDX-U sawteeth simulations; Good agreement between M3D-K, NOVA2, and NIMROD for energetic particle stabilization of internal kink and excitation of fishbone; M3D-K results of beam-driven TAEs are consistent with NSTX observations: mode frequency and its chirping, mode saturation time scale.

6. 6 M3D and NIMROD 3D MHD Codes have successfully completed a major nonlinear benchmark: Sawtooth activity in a small tokamak. Figure shows kinetic energy vs time the first 10 toroidal modes for NIMROD and M3D with same initial conditions, sources, and boundary conditions t1t1 t2t2 t1t1 t2t2 Flux surfaces (Poincaré plots) t=t 1 (NIMROD)t=t 2 (NIMROD) t=t 1 (M3D) t=t 2 (M3D)

7. 7 M3D agrees with NOVA2 for n=1/m=1 mode with energetic particle effects growth rate frequency ideal kink fishbone G. Fu

8. 8 Mode Structure: Ideal Kink v.s. Fishbone G. Fu

9. 9 Non-linear single-mode TAE simulations reproduce many features observed in NSTX data M3D Nonlinear Hybrid simulations: –Mode growth and decay times are approximately 50 - 100  s –Bursting/chirping results from: Non-linear modification of fast-ion distribution Change in mode structure –Chirp  f / f  20% - consistent with experiment Data Simulation n=2 t=0 t=336 G. Fu

10. 10 Recent M3D hybrid simulation results: (1)Alpha particle stabilization of n=1 kink in ITER; (2) Nonlinear frequency chirping of fishbone; (3) Beam-driven TAEs in DIII-D; (4) Beam-driven TAEs in NSTX; (5) Beam-driven GAM in DIII-D G.Y. Fu, 2004 IAEA Fusion Energy Conference G.Y. Fu et al., Phys. Plasmas, 2006 4m 1.2m G. Fu

11. 11 As flattening region of distribution function increases, the mode frequency chirps down. distribution G. Fu

12. 12 Domain decomposition is same for particles as for fluid cells  Toroidal decomposition is ok..scales well Poloidal decomposition is presently only 1D Need to add radial decomposition as well typical run will have 25,000  50 zones: 20 particles/zone = 25M particles should scale well to 1000p

13. 13 Work Plan Goal is to have initial results in the first year using the current capability of M3D-K and to have more complete results by the third year; Preliminary run will just add “test particles” to existing sawtooth simulation as a sanity check. Need to upgrade M3D to have more general particle domain decomposition (to run on > 500 p) and to have sources and sinks of particles. Initial tests will determine if  f (or some improved  f method) is sufficient. Do we need to use full-f method ?

14. 14 Summary Sawteeth will occur in ITER with a large energetic particle population present (due to α-particles) –there is presently no comprehensive 3D model of this The M3D (and NIMROD) codes should be able to model this phenomena with minor modifications Recent successes in Validation and Verification of these codes will give credibility to this modeling We propose to perform this as part of SWIM as it involves the coupling of a heating system and MHD –future step could involve RF as well