Island s uppression s imulations for SWIM Physics results and coupling issues

1. Island suppression simulations for SWIM Physics results and coupling issues Thomas G. Jenkins, University of Wisconsin-Madison in collaboration with Dalton Schnack, Carl Sovinec, Chris Hegna, Jim Callen, Fatima Ebrahimi University of Wisconsin-Madison Scott Kruger, Johan Carlsson Eric Held, Jeong-Young Ji Tech-X CorporationUtah State University Bob Harvey CompX SWIM Project Meeting 23 September 2008 Princeton, NJ

2. Goal of Slow MHD campaign – numerically simulate ECCD stabilization of NTM’s Experimental efforts to stabilize neoclassical tearing modes via electron cyclotron current drive have been very successful [La Haye, Phys. Plasmas 13, 055501 (2006)]. • For ECCD, the RF-induced current is relatively small (of the same order as the electric • field) – small expansion parameter • Add an RF term to the kinetic equation: Gyrophase-averaged Fokker-Planck Coulomb collision operator Quasilinear diffusion tensor from RF source

3. RF terms appear in the fluid equations Taking fluid moments in the conventional manner yields Can approximate fα as a local Maxwellian, here – RF perturbations are small Closure calculations for qα, πα are also affected by RF.

4. Much of the basic physics is independent of the problem details Ultimately, the closure scheme and the small parameter expansion yield a self-consistent set of fluid equations for ECCD-influenced MHD. While work proceeds on that front, we can study simpler models (e.g. resistive tearing modes, rather than NTM’s) to gain physical insight. (Neoclassical effects enter the Rutherford equation additively.) Consider the electron momentum equation, which gives rise to the MHD Ohm’s law. RF-induced momentum yields an additional term, A physically reasonable form for the lowest-order effect of the RF is λrf = RF amplitude f(x,t) = space/time dependence

5. Simple RF models can yield sound physics results, even without self-consistency For these simulations, model the RF as a Gaussian function in the poloidal plane – neglect toroidal variation for now. Ramp up on some intermediate timescale (slow compared to Alfvén timescale, fast compared to resistive timescale), neglect closure problem. Parameters: (Rrf,Zrf) = position of RF peak wrf = half-width λrf = amplitude What physics results arise from the inclusion of this term in the MHD model?

6. RF effects modify the saturated island width of resistive tearing modes Marked reduction in the size of saturated islands can be achieved – proof of principle. Island width data obtained from field line traces – expensive.

7. Magnetic energy is a rough measure of island width δEmagnetic = volume integral of δB2/2μ0 Island width ~ √δψ δEmagnetic ~ w4 NIMROD can now make continuous plots of island width (recent diagnostic development by S. Kruger). (Functionality not used here, though.)

8. What is the effect of the RF on the tearing modes? Tearing mode can be influenced by the RF in two major ways: Modification of tearing parameter Δ’ -will affect linear growth rate and saturated island width -easy to diagnose – proportional to linear growth rate Modification of helical current profile μ = μ0 (JB)/(BB) -becomes important as mode saturates nonlinearly -not so easy to diagnose. Conceptual picture:   J0 J0 B0  B0 B J δJ, δB

9. Study Δ’ and current profile modification in DIII-D-like geometry Begin with equilibrium where the dominant instability is the (2,1) resistive tearing mode Pack grid around (2,1) and (3,1) rational surfaces

10. To study Δ’ modification, find growth rate after RF influence on background comes to steady-state Allow only toroidally symmetric (n=0) modes in simulation, ramp RF up to steady state – yields new RF-modified ‘equilibrium’ Then allow n=1 (and higher) modes into simulation, check linear growth For fixed RF current (centered on (2,1) rational surface), the growth rate (and thus Δ’) is reduced as the poloidal cross-section of the RF spot is reduced. In agreement with results of Pletzer/Perkins [PoP 6, 1589 (1999)].

11. Δ’ modification - saturated island width is decreased as RF cross-section is reduced For fixed RF current (centered on (2,1) rational surface), the saturated island width is also reduced for smaller RF poloidal cross-sections. Here, IRF/I0 < 0.04%, so current profile modification is negligible. In agreement with general conclusions of Hegna/Callen [PoP 4, 2940 (1997)]. Pletzer/Perkins find that for IRF/I0 < 4%, offset from rational surface is destabilizing – simulations in progress to verify this effect.

12. Current profile modification – quasilinear flattening occurs at the rational surface even in the absence of RF Full μ profile and closeup. Relaxes to a quasilinearly flattened equilibrium at long times.

13. Current profile modification – same simulation, but with RF added at (2,1) rational surface To study current profile modification, turn on RF after resistive tearing mode has saturated; examine the island width reduction and μ profile (IRF/I0 ~ 1%). Without the RF, quasilinear flattening leads to increased current on outward side of rational surface; decreased current on inward side. RF adds current across the rational surface – destabilizing inside, but stabilizing outside (yielding net stabilization).

14. The additional current is linear in λRF, but island reduction is limited by the RF cross-section The island cannot shrink appreciably below the width of the poloidal cross-section of the RF drive. In agreement with Hegna/Callen When RF is offset from rational surface, investigating Δ’ and current profile modifications relative to non-offset case; compare with Pletzer/Perkins conclusions (in progress). Size of RF cross-section relative to saturated island size also important.

15. Equilibration of RF effects over flux surfaces RF effects entering NIMROD are not assumed to be flux-surface averaged, we rely on the dynamics of the plasma to distribute them over the flux surface. Previous simulations have suggested that equilibration occurs over the flux surface after a few Alfvén times. Here, we have a cylindrical plasma with a z-independent, poloidally localized RF drive. Can we analytically capture any of the physics of equilibration?

16. Heuristically, one can show that the RF launches torsional Alfvén waves Images from C. Sovinec, SWIM Slow MHD campaign meetup, December 2007.

17. Analytically, one can demonstrate this with a textbook MHD model

18. What do sudden, spatially localized RF perturbations do? Most terms die off on resistive timescale, with shorter- wavelength spatial scales damping more quickly

19. z-independent perturbations grow, rather than damp, on resistive timescale (>> Alfvén timescale) In this geometry the unperturbed field lines are straight, so that single field lines in z are the equivalent of a `flux surface’ The z-independent (n=0) magnetic field perturbations grow up from zero on a resistive timescale; γl,m,n = √(n2k2vA2 – Ωl,m,n2) so that

20. The shorter timescale comes about because equilibration over the flux surface is a nonlinear effect In the previous model, boundary conditions in z are periodic – waves launched from the middle of the cylinder propagate in z toward both endcaps, cross the boundary, and begin to interfere with one another Similar effects occur in the periodic θ-coordinate of the previous movie; the RF is uniform in z, so the waves begin nonlinearly interfering after only propagating halfway around poloidally. Bz Bθ

21. Outstanding question – what is the relevant flux-surface- equilibration timescale for nonuniform RF perturbations? In particular, what happens near the (2,1) rational surface (where we will most likely want the RF to be) for a tightly toroidally localized RF perturbation?

22. Good progress achieved on milestones thus far First simulation results with ad hoc JRF term being written up for publication, demonstrating proof-of-principle concepts and agreeing well with existing literature Full coupling of GENRAY and NIMROD is well under way and bugs are rapidly being worked out (beginning phase 3 of our 5-phase plan, as outlined in the proposal) – Scott Kruger’s talk (next).