S.E. Kruger, D.D. Schnack, C.C. Hegna, E.D. Held

1. S.E. Kruger, D.D. Schnack, C.C. Hegna, E.D. Held A Collection of Slides To Stimulate Discussion on the Simulation of Feedback Stabilization of Neoclassical Tearing Modes

2. Where Do We Want To Be? Driving question: How much RF power does ITER need to stabilize the expected performance-limiting NTMs? What do we need to answer that question? • Codes that can simulate these nonlinear instabilities in realistic geometry • Fluid models to handle the disparate time scales of NTMs • Accurate and reliable closures for tokamak problems • Codes to accurate model RF driving fields and how the energy and momentum is deposited into the plasma • Integration of RF sources into fluid model

3. What is Needed To Model NTM’s?Current Understanding from Analytic Theory • Experimental/Theory Comparisons Framed In Terms of Modified Rutherford Equations: Hegna PoP 6 (1999) 3980 Lutjens et.al. PoP 8 (2001) 4267 Waelbroeck MHD eOnly valid with Aniso Heat Conduction Two-fluid I (both neo.,gyro. Anisotropic Heat Conduction Hard part (theoretically): e Minimal requirement: Get same physics as analytic theories Desired goals: Include kinetic effects in fundamental way at realistic parameters To date: Simulations have only included e and anisotropic heat conduction

4. Features of NTMs Make It Challenging Computationally • Mode is inherently nonlinear • Need to simulate as initial-value problem • ITER-relevant experiments are in “long-pulse” regime: • Plasma is near marginality at all times • Modes are slow growing: ~100 msec to saturation • “Near marginality” is difficult for initial-value codes • Sensitive to equilibrium • Computationally expensive • Simple cases have been done since 1996. Most challenging case attempted to date: • Try and obtain seeded island from sawtooth with approximate closure scheme • (don’t need to go all the way to saturation).

5. Simulations of DIII-D #86144 Show Resistive MHD Computations Cannot Explain Observations 2/1 3/2 1/1 • Secondary islands are much smaller than experiment • Wexp ~ 6-10 cm • 3/2 island width decreases with increasing S • Sexp ~ 108

6. Plasma Parameters Severely Constrained By Ordering of Length Scales • Need Wd >> dv (lin. layer) High S • High S means smaller secondary island Need smaller threshold • To get smaller threshold, need higher anisotropy • Anisotropy also slows 1/1 growth rate • Quickly leads into realistic plasma parameters Nonlinear NTM calculations are extremely challenging!

7. Lessons Learned From All Previous Simulations • Heuristic closure works • Parallel heat conduction works • Simulations of DIII-D Experiments • Parameters needed are aggressive • Cannot easily decouple 1/1 physics from seed island physics • Classical tearing mode studies • Experience using experimental equilibria is robust, and can be used to help guide how to choose model equilibria • Many experiments displaying NTM’s are near ideal limit • ’~0 is important

8. Near-Term Development for NTM Physics • Improved Fluid Models • Two-fluid to get the effect of Dpol term • Focus of current CEMM SciDAC • Almost there! • Improved closures • Local closures • Landau fluid heat flux • Heuristic closure • Non-local closure • Improve heat flux calculation • Develop non-local stress tensor that contains trapped particle effects • DEKIS Briefly discussed next Not discussed

9. “Heuristic Closure” Meets Minimal Requirements • Simplified model captures most neo-classical effects • (T. A. Gianakon, S. E. Kruger, C. C. Hegna, Phys. Plasmas 9 (2002) 536) • Has “neo-classical” effects relevant for long mean-free-path regime: • Poloidal flow damping • Enhancement of polarization current • Bootstrap current • Neoclassical resistivity • Has other nice features: • Dissipative and energy conserving • No toroidal damping • To Do List: • Repeat previous MHD/NTM calculations with updated algorithm • Investigate “2x2” moment version of heuristic closure Benchmarked against modified Rutherford Equation:

10. Non-local Closure for Heat Flux More Rigorously Includes Electron Physics • Parallel dynamics of electrons is fast - on MHD time scale they have farther than an MHD length scale • Capturing this physics via a Chapman-Enskog-like method: If K(L,L’)=1, • Gives arbitrary collisionality, geometry effects To Do: Improve numerical implementation Implement Landau-like closure Held, PP 8, 1171 (2001) Held, PP 11, 2419 (2004) Landaufluid closures

11. Non-local Closure Can Be Extended To Calculate Parallel Stress Tensor Held, PP 10, 4708 (2003) • In a manner similar to parallel heat flux: To Do: Complete inclusion of trapped particle effects Implement numerically

12. RF Feedback Stabilization of NTM’s a Critical Part of ITER’s Planned Operation • Localized electron momentum deposition differential on the particles • Currents can be induced • Local: produces helical current that affect island region physics • Global: counteract island drive (D′, q) • Not much current required (IRF/Iplasma ~ 3%) Center island current out of plane Rutherford equation shows that effects can be treated independently: Can study RF stabilization of islands without NTM physics Probably easiest MHD mode to study effects of RF sources See: Rutherford, Varenna 86; Kurita, NF 94; Hegna, PP 97; Lazzaro, PP 96; Perkins, EPS 97; Giruzzi NF, 99

13. Work Already Exists on Computational Modeling of RF Stabilization of Tearing Modes • Simplest model: • Slightly more complicated (developed by Giruzzi): • Models fit well into fluid approximation • Numerical analysis poorly understood, but in general time scale of QL diffusion is slow (explicit OK?) • Where do we go from here? • How do we integrate with CQL3D or other RF code? • Do the hot particles modify the neoclassical closures Yu et.al., PP 2000 Gianakon, PP 2001

14. Where Do We Go From Here?Short Term (Post-doc) • Produce a more rigorous Giruzzi-like current source equation • Implement simplified analytic model into NIMROD • Source from ECCD modeling code? (TORAY, …) • Analytic source term? Needs to be function of time and 3D space • Perform simulations of feedback stabilization of classical tearing mode (“JCP case”)

15. Where Do We Go From Here?Longer term • Understand relationship of RF hot particle distribution and NTM closures in more detail. • What is best way of adding RF sources to fluid equations? SRF(x,t), SRF(x), JRF(x,t) • Integrate RF/Classical TM studies with CEMM NTM studies • RF and ions? (Much harder. Not necessary for NTM, but …) • Computer science aspects • What is the best way of interfacing RF codes to MHD codes? • Applied Math • What is numerical stability properties of coupled system? • Fluid codes could always use better solvers Dalton is leading effort to write white paper on an NTM roadmap. If you are interested in participating, please email.

16. Backup Slides

17. Computational Studies Aim to Answer Experimental Questions Why is the 3/2 seen instead of the 2/1 mode? Why does the 3/2 mode appear on the nth sawtooth? Why does the 3/2 mode appear without an obvious “seed”?

18. Modeling of Instabilities Use Extended MHD Models • Momentum Equation • Generalized Ohm’s law: • Temperature Equations:

19. Experimentally RF Successfully Stabilized NTM’s and Other MHD Instabilities ICRF modifies ELM behavior in JETG.P. Maddison et al PPCF 2002 ICRF modifies sawteeth in C-ModS.J. Wukitch et al., PP, 2005 ECCD suppresses NTM’s in DIII-DR. Prater et al., NF, 2003

20. Anisotropic Heat Conduction Needed For Experiments Possible With High-Order Elements • Perpendicular thermal conduction can compete with parallel thermal conduction at rational surfaces where parallel gradient = 0. • Analytic predictions from balancing two terms give scale length that scales as (k||/kperp))-1/4[Fitzpatrick, PoP 2, 825 (1995)] • Fit of computational results is k||/kperp=3.0x103 (wd /a)-4.2. • Result is for toroidal geometry. • Run with biquartic elements.