1. The influence of non-resonant perturbation fields: Modelling results and Proposals for TEXTOR experiments S. Günter, V. Igochine, K. Lackner, Q. Yu IPP Garching Resistive wall modes and error field amplification Error field amplification and plasma rotation Suppression of neoclassical tearing modes by external helical fields

2. Concept of advanced tokamaks Non-monotonic current profile Turbulence suppression high pressure gradients large bootstrap current f BS =  N  A q  0.8 … 0.9  N  4 … 5 MHD stability ?

3. Ultimate limit to maximum  N is external kink mode External kink mode can be stabilised by ideal walls n·B| wall = 0 For optimised current profiles (avoid double low order rational surfaces of same helicity)

4. Günter et al., NF 2000 External kink mode in AUG advanced scenarios Closeness to rational q a destabilising Good agreement between theory and experiment eigenfunction

5. Stabilising influence of an ideal conducting wall Closed wall in distance r w from plasma can be strongly stabilising, especially for: - broad current and pressure profiles - strong shaping of plasma cross section

6. 3d geometry of ideally conducting walls CAS3D: First code dealing with 3D wall and 3D plasma:

7. Destabilising effect of wall resistivity: RWMs Garofalo et al., PRL 1999

8. Simple model for RWMs and error field amplification Fitzpatrick´s (PoP 9(2002) 3459) analytical (inertial layer) model  : stability parameter  >0: ideal kink mode stabilised by infinitely conducting wall  <0: in absence of rotation plasma is stable Plasma rotation Instability drive of plasma mode increases

9. Effect of rotation for varying wall distance ideal („plasma“) mode unstable detailed shape of marginal curve depends on plasma (dissipation) model torque balance between mirror current forces and viscous drag (or inertia) determines mode rotation frequency  can be modified by: - distance of wall (0 <  <1) at given instability drive d/d c increasing wall distance reduces coupling, perturbation can start slipping with respect to wall  rotation stabilizes mode Re(  ) [wall frame]

10. Effect of rotation for varying instability drive  can be modified by: - variation of the MHD instability drive at given wall distance rotation destabilizes plasma in MHD stable region: electromagnetic coupling to wall opens relative velocity plasma- wall to Kelvin-Helmholtz drive (inertia needed) marginal curve corresponds to error field amplification condition (resistive wall mode can be interpreted to error field amplification of the induced wall-current field) more unstable plasma has larger ratio of field amplitude in plasma to wall => reduced wall coupling allows slip and rotational stabilization

11. Numerical treatment of RWMs anderror field amplification In realistic geometry (coupling to internal resonances): MARS (Bondeson) VALEN (Bialek, Boozer) CASTOR-A (Holties, Kerner) - response to frequency dependent external perturbation field - modified to include differential plasma rotation, viscosity - resistive wall included (so far high resistivity only)

12. Numerical results: Error field amplification Here for comparison with simple analytical theory: frequency dependent external (3,1) perturbation field (q a < 3) no internal resonances, no viscosity Re P   j ant B cos  ~ (torque onto plasma) 1/  towards marginal stability Increasing wall distance 1/   /  A 0 0.01 0.02 0.03 Change in plasma stability by varying distance of ideally conducting wall

13. Numerical results: Error field amplification Good agreement with analytical model for ideal plasma (scan in wall distance) ~ Maximum of absorbed power Here for comparison with simple analytical theory: frequency dependent external (3,1) perturbation field (q a < 3) no internal resonances, no viscosity -  W~  2 pl

14. Numerical results: Error field amplification Good agreement with analytical model for ideal plasma (scan in  N ) Maximum of absorbed power Here for comparison with simple analytical theory: frequency dependent external (3,1) perturbation field (q a < 3) no internal resonances, no viscosity

15. Numerical results: torque on plasma Re P   j ant B cos  ~  tor  Torque on the plasma due to external error fields: 1/  ~ Maximum torque

16. Influence of error fields on plasma rotation reduction in resonant frequency, increasing torque increase in , mode growth  reduction in plasma frequency

17. 59223 Saddle current[A] 3.4li  N (%) b r (0 o ) b r (90 o ) Signal which sees no vacuum (or low  N ) pick-up clearly rises as   approaches ideal limit P NBI [MW] NB due to low field B t =1T and high NBI  alfven ~ 4% Experiments on error field amplification on JET

18. Influence of error fields on plasma rotation

19. Proposals for error field amplification experiments on TEXTOR – comparisons with theory Frequency dependence in error field amplification: discharges with q a 3 for comparison), low l i scan in  N /plasma rotation within one discharge, measure (3,1) amplitude increase compared to vacuum case repeat for different frequency of antenna current comparison with code calculations possible Influence of error fields on plasma rotation: compare torque onto plasma with theory (with and without q=3 surface) for different coil current frequencies and plasma pressures

20. Proposals for resistive wall mode experiments on TEXTOR Develop scenarios with external (3,1) RWM mode vacuum vessel: r w /a = 1.35,  w = 14 ms try to stabilize RWM by rotating external (3,1) perturbation fields (compare required rotation velocity with theory)

21. Physics of neoclassical tearing modes (NTMs) j BS   p Magnetic islands driven by the loss of bootstrap current inside island Helical current parallel to plasma current drives magnetic islands unstable

22. Interaction of NTMs with different helicity No simultaneous large NTMs of different helicities

23. Stabilising effect of additional helical field For finite perpendicular heat conductivity helical field perturbation reduces BS current perturbation caused by single magnetic island Contour plots of BS current perturbation Single magnetic islandwith external perturbation field

24. Stabilization of NTMs by external error fields DIII-D: suppression of (3,2) NTM onset successful, but strong reduction in plasma rotation observed n=3 perturbation field

25. Stabilization of NTMs by external error fields On TEXTOR: rotating perturbation fields possible (3,2) NTM stabilization by external (3,1) fields

26. Stabilization of NTMs by external error fields On TEXTOR: rotating perturbation fields possible NTM stabilization by external (3,1) fields for q a < 3 if perturbation field too small use conditions with error field amplifications Influence plasma rotation by external fields, study effect on NTM stability

27. Conclusions “Rotating” external perturbation fields of a single helicity opens new possibilities for MHD experiments on TEXTOR: error field amplification experiments, comparison with theory - frequency dependence of error field amplification - influence on plasma rotation Resistive wall mode studies Stabilization of NTMs by external perturbation fields

28. Newcomb criterion Cylindrical plasma: pointing vector into vacuum region ~ -  ’| r=a For zero growth rate (ok for RWMs) it describes the energy released from plasma from infinitely slow perturbation (no energy converted to kinetic energy) wall position  r plasma edge 1 0 r(  =0) closer to plasma the larger  ’ |r=a (the more unstable the smaller r(  =0) more unstable

29. Error field amplification influences plasma rotation Error field amplification  reduced plasma rotation  RWM growth Strait et al., IAEA 2002

30. Critical Rotation Scaling Strait et al., IAEA 2002