1. Association DayIasi, 2 July 20101 Association EURATOM – MEdC MHD MHD STABILITY RESEARCH IN TOKAMAKS C.V. Atanasiu National Institute for Laser, Plasma and Radiation Physics External contributors: S. Günter, K. Lackner (1994-present) : MP-IPP Garching, Tokamakphysics Department (Theory 3) A.H.Boozer (1998), Applied Physics Department of Columbia U, USA L.E. Zakharov(1999-present): PPPL, Theory Department, USA L.E. Zakharov (1983-1991), A.A. Subbotin (1983-2005) V.D. Pustovitov (2007): I.V. Kurchatov, Theory Department, Moscow, Russia S. Gerasimov (2003-2008), M.Gryaznevich (2008): JET, Culham, UK

2. Association DayIasi, 2 July 20102 Association EURATOM – MEdC MHD OUTLINE: PROJECT: Plasma models for feedback control of helical perturbations (since 2000) 1. Tearing modes calculations in tokamaks 1.1. Determination of the influence of the plasma triangularity on the tearing mode stability parameter  ' for ASDEX Upgrade; 1.2. Tearing modes calculations, for specific shots of ASDEX Upgrade 2. Resistive Wall Modes (RWM) calculations in tokamak 2.1. Status of the EKM - RWM problem 2.2. Progress in RWM calculation 2.3 Application of RWMs calculation to ASDEX Upgrade and JET 3. Conclusions and next steps

3. Association DayIasi, 2 July 20103 Association EURATOM – MEdC MHD The present contribution deals with Tearing modes and RWMs investigations for toroidal configurations with direct application to ASDEX Upgrade and JET It represents a continuation of our research activity performed during the last ten years under the frame of a wide range of co-operations with different fusion centers around the world Our research activity was mainly oriented to the “Provision of support to the advancement of the ITER Physics Basis” EFDA coordinated activities, shared under Task Force ITM and Topical Group MHD.

4. Association DayIasi, 2 July 20104 1.Calculation of Tearing modes in a diverted tokamak configuration 1.1. Potential energy calculation and boundary conditions for EKM writing the expression for the potential energy in terms of the perturbation of the flux function, and performing an Euler minimization, we have obtained a system of ordinary coupled differential equations in that perturbation [1]. f, V and G are matrices, and Y is the flux function perturbation vector, with the non-diagonal terms representing both toroidicity and shape coupling effects this system of equations describes a tearing mode or an external kink mode, the latter if the resonance surface is situated at the plasma boundary.

5. Association DayIasi, 2 July 20105 we have considered a "natural" boundary condition just at the plasma boundary [1]. From potential theory we know that a continuous surface distribution of simple sources extending over a not necessarily closed Liapunov surface ∂D and of density σ(q), generates a simple-layer potential at p, in ∂D. for a unit perturbation Y 2/1 (m = 2, n = 1) and Y 3/2 (m = 3, n = 2) the corresponding surface charge distributions are Fig. 1 σ(q) for Y 2/1 and Y 3/2 for shot no. 13476 at 5.2 s ASDEX Upgrade

6. Association DayIasi, 2 July 20106 Δ’ dependences on triangularity δ and ellipticity κ for ASDEX Upgrade (shot no. 13476 at 5.2s)  3 different modes have been considered:m/N=2/1, 3/2 and 4/3;  the stabilising influence of the triangularity is at least in qualitative agreement with measurements on ASDEX Upgrade Association EURATOM – MEdC MHD 1.1 Determination of the influence of the plasma triangularity on the tearing mode stability parameter  ' for ASDEX Upgrade

7. Association DayIasi, 2 July 20107 Association EURATOM – MEdC MHD 1.2 Tearing modes calculations, for specific shots of ASDEX Upgrade: observed islands of type m/n=2/1 for shots ? 1)# 20049 at t=4.79 … 4.82 s 2)# 20823 at t=2.234 … 2.419 s 3)# 20833 at t=3.698 … 4.782 s

8. Association DayIasi, 2 July 20108 Association EURATOM – MEdC MHD

9. Association DayIasi, 2 July 20109 Association EURATOM – MEdC MHD

10. Association DayIasi, 2 July 201010 Association EURATOM – MEdC MHD

11. Association DayIasi, 2 July 201011 Association EURATOM – MEdC MHD

12. Association DayIasi, 2 July 201012 2. Resistive Wall Modes (RWM) calculations in tokamak 2.1 Status of the problem 2.1.1 Stabilization of the external kink instability Stabilizing the external kink instability allows higher β and higher bootstrap current fraction, leading to more economical tokamak power plants; If the instability can not be stabilized higher toroidal fields are necessary to compensate for lower β, however, the lower bootstrap fraction would remain; At present we don’t know if we can stabilize the external kink mode !

13. Association DayIasi, 2 July 201013 2.1.2 Theoretical Predictions It has been shown theoretically that the external kink mode can be stabilized if a conducting shell is present and... – the plasma (or shell) is rotating sufficiently fast (roughly 2- 10% of Alfvén speed) – and there is a dissipation mechanism in the plasma (sound waves, viscosity,…) Or – using feedback coils located behind the shell and sensors in front of the shell (no rotation is required)

14. Association DayIasi, 2 July 201014 2.1.3 Experimental Observations DIII-D seem now to be the best experimental tokamak examining kink stabilization. ASDEX Upgrade with its new wall will be the closest to ITER for RWMs studies; Experiments with plasma rotation (from neutral beams) showed that although the β was increased above the predicted limit with no rotation, the plasma consistently found a way to slow itself down, allowing the kink mode to become unstable; Experiments with feedback coils show that the instability can be affected, holding off the instability temporarily, but the plasma does not appear to be stabilized indefinitely, Until now!

15. Association DayIasi, 2 July 201015 2010/11 201 22222 AUG: 3 x 8 coilsITER: 3 x 9 coilsASDEX Upgrade wall Fig. 3 Internal Control coils of AUG (3 x 8) and ITER (3 x 9) coils: to generate up to n=4 error fields with different poloidal structure. Three poloidal coil sets: flexible m spectrum. Eight toroidal coils: n=4 (small core islands), quasi- continuous phase variation for n =1- 3. 2.2 Calculation of the wall response to the EKM FP7 - ASDEX Upgrade plays the best “step-ladder” role for RWMs control in ITER: [O. Gruber, “Status and Experimental Opportunities of ASDEX Upgrade in 2009”, Ringberg Meeting, 2008.]

16. Association DayIasi, 2 July 201016 poloidal direction toroidal direction Fig.4 A typical toroidal wall structure Association EURATOM – MEdC MHD

17. Association DayIasi, 2 July 201017 Association EURATOM – MEdC MHD 2.1. Fixed plasma: time domain formulation 2.2 Rotating plasma: frequency domain formulation The diffusion eq. in the frequency-domain formulation (all values are complex)

18. Association DayIasi, 2 July 201018 Association EURATOM – MEdC MHD 2. 3. Moving wall and fixed plasma 2.4. Coordinate system for a real toroidal wall - we developed a new curvilinear coordinate system (u,v,w) for wall: - 2 covariant basis vectors (r u, r v ) are tangential to the wall surface and r w is normal to the wall surface - therefore, any wall geometry can be described

19. Association DayIasi, 2 July 201019 2.5. Test problem without holes, there exists an analytical solution: U(x,y)=sin x sin y Fig.5 Wall with 3 holes For the wall case with holes we developed a new very fast solving method to determine the U stream function

20. Association DayIasi, 2 July 201020 Association EURATOM – MEdC MHD 2.6. Comparison between different metods Solving methodNo. of grid pointsRunning time[s] Classical101 x 101 (u,v)103 New meth.101 x 101 (u,v) 3 Classical151 x 151 (u,v)690 New meth.151 x 151 (u,v) 14

21. Association DayIasi, 2 July 201021 METHODNo. of grid points/order of approx. O(h n ) Contour Integral around the wall Contour integral around a hole Scalar potential of the surface current U analytic64.939394022671753.36363861288.57368818753 numeric 51x51 / 264.939394022461753.16399557088.58057153492 51x51 / 364.939394022961753.29706613388.56891335580 101x101 / 264.939386648051753.31270973088.57543866452 101x101 / 364.939394883081753.34666385788.57247069322 151x151 / 364.939120672491753.34045271788.57441239957 Table. 2 Accuracy test. 101 x 101 /3 : 101 grid intervals along u, and v; 3 : O(h 3 ) 2.7. Accuracy check 1) Stokes theorem: by performing some line integrals and verifying the corresponding surface integrals (overlapping up to the 6 th significant digit); 2) simple cases permitting an analytical solution.

22. Association DayIasi, 2 July 201022 Association EURATOM – MEdC MHD 2.3 Application of RWMs calculation to JET a) On the advice of our JET colleagues we considered the shots no. 40523, 40418 and 40183 where the RWMs were present, while at shots performed during the S/T S2-2.3.1 the presence of RWMs was not evident. Considering discharges with low wave numbers m/n at the boundary (46.3676s – 46.993 s for shot # 40523) only, we have found that in the no-wall limit, discharges are not stable (Δ’ ≥ 0) but in the ideal wall limit (superconducting wall) are all stable. b) The influence of the wall on the boundary conditions of the external kink mode equations is now under consideration for a 3D wall with holes [3].

23. Association DayIasi, 2 July 201023 Association EURATOM – MEdC MHD Fig. 1 Time dependencies of different plasma parameters for the considered shot # 40523: a) plasma boundary, b) safety factor q, Other considered parameters: plasma pressure p, plasma current I pl, toroidal magnetic field B t and internal plasma inductivity l i Fig.2 The surface charge distribution along the plasma boundary for a flux perturbation Y3/1. Shot # 40523.

24. Association DayIasi, 2 July 201024 3. Conclusions and next steps: the tearing modes can be stabilized both theoretically and experimentally; the EKMs have not been stabilized experimentally with plasma rotation or feedback control, although theoretical results indicate it is possible ! little can be done more analytically for RWM - main part has to be numerical ! to put some feedback, error field and sensor coils; to run the code on the GATEWAY; to take dissipation into account in our numerical approach. References [1] C.V. Atanasiu, S. Günter, K. Lackner, et al., Phys. Plasmas 11, 5580 (2004). [2] C.V. Atanasiu, A.H. Boozer, L.E. Zakharov, et al, Phys. Plasmas 6, 2781(1999). [3] J. Adamek, C. Angioni, G. Antar, C.V. Atanasiu, M. Balden, W. Becker, Review of Scientific Instruments, 81, 3, 033507 (2010).