Computational Approaches to Simulating RF Stabilization of Neoclassical Tearing Modes

1. S.E. Kruger, J.D. Callen, J. Carlson, C.C. Hegna, E.D. Held, D.D. Schnack, C.R. Sovinec, D.A. Spong ORNL SWIM Meet October 15, 2007 Computational Approaches to Simulating RF Stabilization of Neoclassical Tearing Modes

2. Where Do We Want To Be? Driving question: How do we optimally user RF sources to mitigate the effect of MHD stabilities? Specific to this project, how much ECCD power does ITER need to stabilize the 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. 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

4. Analytic theorists have shown us the correct way that RF sources enter into the MHD equations (success!) • Modified equations: • Where Frfand Srf are: • Q is the quasilinear operator which is proportional to the square of the RF electric field. • Closure terms are affected as well (discussed later)

5. For Electron Cyclotron Current Drive, Need to Perform Ray Tracing to Get RF Electric Fields Wave field propagates into plasma and is absorbed at resonances: Because of the narrow resonance, ray tracing can be used to calculate the RF electric field With the electric field, we can calculate Qnl With Qnl, we can calculate Frf, Srf

6. Hierarchy of models from simple to complex define our computational approach T. Jenkins discusses progress next • Axisymmetric phenomenological model: • Non-axisymmetric phenomenological model: Real sources are toroidally localized and phased as the tearing mode rotates past. This model will allow studying those effects • Giruzzi model Analytic work: CCH and JDC, PoP ‘97; Zohm PoP ‘97Numeric work: Sovinec thesis; transport modeling - e.g., Jardin (Note: This assumes slowly evolving 3D equilibria Numeric work: Yu et.al., PP 2000, Gianakon, PP 2001

7. Most accurate model of the source requires coupling with the RF • Recall form of sources: • We need from RF codes: • Quasilinear operator • Calculation of quasilinear operator requires: • Calculation of electric fields from ray tracing code • How do we do this coupling? • Save details for discussion session • Note: Accurate coupling means getting the closures correct as well.

8. Another important part of the problem is the effect of the sources on the closures • Issue: Need closure terms to be consistent with sources • Consider the kinetic equation with the quasilinear source used to derive previous fluid equations: • Separate distribution function into Maxwellian and small distortion away from Maxwellian(essentially we are assuming ECCD here): • Equation for distortion: Need consistencyamong these terms

9. How to calculate the effect of RF sources on closures • Generalize present NIMROD capability to compute parallel closuresby integrating quasilinear source term along perturbed field lines: • Implies Eric needs to add sources to his calculation of q, • Solve drift kinetic equation directly (DEKIS) • DEKIS is Monte Carl code • For collisions, Langevin operator used: • Monte Carlo operator for Qnl similarly used Thermodynamic drives from fM thatgives the normal heat flux/stress tensors

10. Integral Closure for Heat Flux Includes More Electron Physics Capture fast parallel dynamics of electrons via a Chapman-Enskog-like method: • Valid for arbitrary collisionality, geometry effects • Use semi-implicit operator fortemporal stability • Inclusion of RF terms in equationbeing studied by Held and Ji • Adding RF terms will involveintegrated Qnl along the magneticfield lines (no bounce averagingof Qnlallowed!) Held, PP 8, 1171 (2001) Held, PP 11, 2419 (2004)

11. Why is the closure consistency important?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 For feedback stabilization of tearing modes, do not need closure consistency. To get all of the physics right in solving the NTM problem, need closure consistency

12. Summary • A hierarchy of models leading from simple phenomenological to rigorously derived is planned: • Axisymmetric Jrf • Non-axisymmetric Jrf • Giruzzi model • RF coupling for sources • RF coupling with closure consistency • Significant amount of physics can be obtained just with the second model (and we are close to having that) • Effects of phasing versus non-phasing, amount of current, degree of localization required, etc. • RF coupling sources will require a fair amount of computational development (TBD) • ITER simulations should be done here • Closure consistency is hard, just like the closure problem is hard • Fortunately for ECCD, seems like it is straightforward extension of current plans