1. YQ Liu, Peking University, Feb 16-20, 2009 Effects of 3D Conductors on RWM Stability and Control Yueqiang Liu UKAEA Culham Science Centre Abingdon, Oxon OX14 3DB, UK

2. YQ Liu, Peking University, Feb 16-20, 2009 Outline 1.Why important? 2.CarMa code 3.CarMa modelling and experiments  RFX  ITER 4.Outlook 5.Overall Summary (selected key notes)

3. YQ Liu, Peking University, Feb 16-20, 2009 Why important ?  RWM is an external mode  External mode produces magnetic field perturbations in vacuum region, leading to interaction with external (3D) conducting structures  n=0 vertical instability is another example  3D structure may couple n>0 RWM with n=0 vertical control system  Geometrical coupling  Realistic prediction for RWM stability and control in ITER

4. YQ Liu, Peking University, Feb 16-20, 2009... in more details  Many present fusion devices have essential 3D feature for the conductors (partial walls, poloidal/toroidal cuts, coils, etc.)  Realistic prediction of RWM stability and control performance in ITER requires 3D modelling of conducting structures  Importance of 3D geometry already demonstrated by comparing passive growth rates of RWM between RFX experiments and CarMa simulations [Villone08]  Accurate simulation of RFA under ac conditions also requires 3D modelling of conducting structures  3D simulations and benchmarks are gaining momentum during recent years [VALEN, STARWALL, CarMa]

5. YQ Liu, Peking University, Feb 16-20, 2009 Outline 1.Why important? 2.CarMa code 3.CarMa modelling and experiments  RFX  ITER 4.Outlook 5.Overall Summary (selected key notes)

6. YQ Liu, Peking University, Feb 16-20, 2009 CarMa code  Couples MARS-F (MHD) and 3D eddy current code CARIDDI (EM)  Formulation  Forward coupling (from MHD to EM)  Backward coupling (from EM to MHD)  Benchmark CarMa  Other RWM codes with 3D conductors  VALEN (Columbia, US)  STARWALL (IPP, Germany)  Typhoon+KINX (Russia)

7. YQ Liu, Peking University, Feb 16-20, 2009 CarMa formulation: overview  MARS-F basically solves single fluid linear MHD  CARIDDI solves 3D eddy current problem (quasi- magnetostatic Maxwell)  Using integral formulation and FEM (curl-conforming edge elements)  State-of-the-art fast computing techniques  Discretized equation eddy current plasma electrode  Other features (rotation, resistivity, feedback, kinetic extentions, etc.) not shown here  Need to couple I to U : U=U(I)

8. YQ Liu, Peking University, Feb 16-20, 2009 S Resistie wall S  The plasma (instantaneous) response to a given magnetic flux density perturbation on S is computed as a plasma response matrix. plasma S Resistie wall  Using such plasma response matrix, the effect of 3D structures on plasma is evaluated by computing the magnetic flux density on S due to 3D currents.  The currents induced in the 3D structures by plasma are computed via an equivalent surface current distribution on S providing the same magnetic field as plasma outside S. Forward coupling procedure S S S Albanese IEEE Trans. Mag. 44 1654(2008) Portone PPCF 50 085004(2008) Liu PoP 15 072516 (2008) Pustovitov PPCF 50 105001(2008)

9. YQ Liu, Peking University, Feb 16-20, 2009 Mutual inductance matrix between 3D structures and equivalent surface currents Induced voltage on 3D structures Equivalent surface currents providing the same magnetic field as plasma Matrix expressing the effect of 3D current density on plasma Modified inductance matrix Dynamical matrix N  h matrix h  N matrix h << N h =DoF of magnetic field on S N =DoF of current in 3D structure Forward coupling procedure

10. YQ Liu, Peking University, Feb 16-20, 2009  Forward coupling has difficulty to include plasma inertia, flow, and kinetic effects  Can be overcomed using backward coupling scheme [Liu PoP 15 072516(2008)]  Start again with replacing plasma current by equivalent surface current Backward coupling procedure eddy current equation: total field at coupling surface S: sensor flux: feedback current:

11. YQ Liu, Peking University, Feb 16-20, 2009  With algebraic combinations of previous equations, it is possible to obtain the following linear relations (w.r.t. the eigenvalue)  Linear boundary condition for MHD code, with computational boundary at coupling surface  Sensor flux for RFA or feedback  Similar BC can be derived even for coupling to a nonlinear MHD code Backward coupling procedure

12. YQ Liu, Peking University, Feb 16-20, 2009 Key features of CarMa  Accurate for RWM calculations. Coupling scheme analytically proven [Liu PoP 15 072516(2008)]. Well benchmarked against MARS-F for 2D walls, and against other similar codes for 3D walls  The coupling matrices assemble responses from all poloidal Fourier Harmonics. Hence the final system contains all unstable/stable RWM (multimode approach)  Capable of treating volumetric conductors (no thin shell approximations)  State-of-the-art fast techniques for solving EM problems allow very detailed modelling of conductor geometry [Rubinacci JCP 228 1562(2009)]  CarMa with backward coupling allows inclusion of inertia, rotation, kinetic effects, and feedback

13. YQ Liu, Peking University, Feb 16-20, 2009  Growth rate calculation  Unstable eigenvalue of the dynamical matrix  Standard routines (e.g. Matlab) or ad hoc computations (e.g. inverse iteration: see the following…)  Beta limit with 3D structures  Controller design  state-space model (although with large dimensions and with many unstable modes)  Time domain simulations  Controller validation  Inclusion of non-ideal power supplies (voltage/current limitations, time delays, etc.) What CarMa can do ?

14. YQ Liu, Peking University, Feb 16-20, 2009 Benchmark coupling scheme and CarMa  Choose a plasma with circular cross section, and aspect ratio =5  Assuming an axi- symmetric complete thin wall (2D wall), run MARS- F to compute growth/ damping rates of unstable/stable RWM  Run CarMa with 3D discretization of 2D wall  Compare results q-profile pressure profile

15. YQ Liu, Peking University, Feb 16-20, 2009 Both growth and damping rates agree MARS-F [s -1 ]Coupling surface 1 [s -1 ]Coupling surface 2 [s -1 ] Unstable eigenvalue292.7 291.2  j 3.6e-4293.3  j 7.5e-4 Stable eigenvalue #1-165.7-159.2-159.3 Stable eigenvalue #2-221.4-216.6 Stable eigenvalue #3-279.4-278.7-278.2 Stable eigenvalue #4-379.0-373.9-374.0 Stable eigenvalue #5-560.5-562.1-561.3 Stable eigenvalue #6-589.1-581.2-581.3  CarMa results independent of choice of coupling surface

16. YQ Liu, Peking University, Feb 16-20, 2009 Benchmark mode structure  Eddy current density distribution along the wall, computed by MARS-F alone (line) and by CarMa (circle)  For the unstable mode

17. YQ Liu, Peking University, Feb 16-20, 2009  Positive comparison with other 3D RWM codes (STARWALL, VALEN) Courtesy of J. Bialek and E. Strumberger Benchmark with other codes

18. YQ Liu, Peking University, Feb 16-20, 2009 Outline 1.Why important? 2.CarMa code 3.CarMa modelling and experiments  RFX  ITER 4.Outlook 5.Overall Summary (selected key notes)

19. YQ Liu, Peking University, Feb 16-20, 2009 RWM study for RFX: equlib. & geometry  RFX upgrade: rw=1.1a, R/a=2m/0.459m  Typical unstable RWM: m=1, |n|=2,...,6 Coils Mechanical structure Vessel Shell

20. YQ Liu, Peking University, Feb 16-20, 2009 RWM stability with MARS-F  RWM growth rates are well measured in RFX experiments  MARS-F with a 2D wall reproduces exp. growth rates for a large range of plasma parameters and various n’s  Including other structures tends to underestimate growth rates  MARS-F computes two unstable RWM for some n’s (=2,3)

21. YQ Liu, Peking University, Feb 16-20, 2009 RWM stability with CarMa (3D structure)  For RFP plasmas, all three codes: ETAW(cylindrical Newcomb solver), MARS-F, CarMa(2D) agree well, as shown below for one equilibrium  Gaps in conducting wall destabilize RWM. However, mechanical structures and outer shells give additional stabilization  3D wall structures (gaps) split two otherwise identical eigenvalues, as well as in tokamak cases γ [s -1 ] Cylinder(ETAW)MARS-FCarMa(2D)CarMa (3D) n=2<0 2.45 0.43 1.81 0.37 1.94 0.45, 0.46 2.40, 2.48 n=31.82 1.90 2.08 2.16 1.91 2.49 2.58, 2.62 2.96, 3.04 n=44.094.044.275.46, 5.53 n=56.816.897.459.62, 9.73 n=611.811.712.917.0, 17.2

22. YQ Liu, Peking University, Feb 16-20, 2009  3D effects are important on growth rate!  Purely axisymmetric estimates of growth rates are largely underestimated on RFX-mod CarMa modelling vs. RFX experiments Villone PRL 100 255005(2008)

23. YQ Liu, Peking University, Feb 16-20, 2009 Eddy current flow modified by wall gaps 2D wall 3D wall with gaps  CarMa computed wall eddy current pattern  For an unstable mode with n=3,m=1

24. YQ Liu, Peking University, Feb 16-20, 2009 ITER modelling: detailed 3D wall geometry  Examples:  mesh A: OTS + bypass  mesh B: OTS+holes+extensions +blanket mesh A outer triangular support (OTS) mesh B

25. YQ Liu, Peking University, Feb 16-20, 2009 Simple hole approximation for ITER walls leads to too pessimistic prediction for RWM stability  Eddy current patterns significantly affected by tubular extensions...  that allow better imagine current flow, hence more stablising effect

26. YQ Liu, Peking University, Feb 16-20, 2009 Major 3D wall effects are holes and tubular extenstions  3D holes roughly double growth rates  Tubular extensions reduce growth rates to a level as with 2D complete walls γ [s -1 ]  N 2.6762.8302.9853.3013.461 MARS-F γ 8.49515.8229.63165.8577.2 Mesh#1:2D γ 8.46215.5628.35127.2334.9 Mesh#2:2D+OTS γ 7.89614.3425.5499.26206.8 Mesh#3:2D+OTS+bypass γ 7.90314.3725.63100.0208.9 Mesh#4:3D+OTS+holes γ 15.8332.6071.94N.A. Mesh#5:mesh#4 refined γ 15.6532.0670.20N.A. Mesh#6:mesh#5 smaller holes γ 13.7127.1355.781230N.A. Mesh#7:3D+OTS+holes+ext. γ 9.36617.6333.22184.4878.7

27. YQ Liu, Peking University, Feb 16-20, 2009 RWM feedback study for new ITER in-vessel coils reveals requirement on the coil current  Consider an ITER plasma with  Use 3x9 ELM control coils for RWM feedback  Multivariable controller based on LQG technique satisfying certain specification requirements  N = 3.17  Actual limiting factor is current  Assuming 20kA current limit (ELM control off), RWM can be stabilised for field perturbation within 300Gauss  Assuming 250A current limit (ELM control on), field perturbation within 5Gauss 20kA current limit 1 sec. settling time

28. YQ Liu, Peking University, Feb 16-20, 2009 Detail: LQG control  Multivariable controller designed by using the LQG technique, based on the following requirements  Obtain a closed loop null controllable region as close as possible to the ideal result (BAP)  Allow to recover from a disturbance (initial condition on the unstable plane) as soon as possible (within current/voltage limits)  Avoid to generate a n  1 magnetic field  Stabilize all the modes with growth rates lower than reference equilibrium (i.e. lower  N )  Use a balanced truncation technique to obtain a sufficiently low order controller (five)

29. YQ Liu, Peking University, Feb 16-20, 2009 Plasma/circuit model V(t) y(t) T IN T OUT y 1 (t) - V 1 (t) K(s) 27 input voltages (3 coils per 9 sectors) 3 voltage Fourier components 144 magnetic outputs (48 measurements per 3 sectors) 48 magnetic Fourier components RWM feedback controller Detail: control diagram

30. YQ Liu, Peking University, Feb 16-20, 2009 The BAP is in the range of perturbations of tens of mT The BAP is in the range of perturbations of fractions of mT ELM control off: current limits 20 kA ELM control on: current limits 250 A B k (t) are N=18 measurements of the vertical magnetic field in the outboard region at equally spaced toroidal angles  k The interior of the polygon corresponds to stabilizable perturbations Another approach of control study: best achievable performasnce (BAP)

31. YQ Liu, Peking University, Feb 16-20, 2009 Control coils voltage-current distribution

32. YQ Liu, Peking University, Feb 16-20, 2009 Outlook  For RWM, geometrical coupling of different n’s (including n=0!), via 3D conductors, is probably more important than physics coupling due to nonlinear MHD  Nonlinear MHD coupling should be more important for other, more localised modes such as TMs and ELMs  Extensive work going on to include 3D geometrical effects in  RFA simulations  RWM feedback stabilisation (in particular for ITER)  Nonlinear (quasi-linear) MHD modelling for RWM (e.g. nonlinear interplay between mode damping and momentum damping)

33. YQ Liu, Peking University, Feb 16-20, 2009 Overall summary (key notes) 1.RWM research important for ITER 2.Sensor optimisation crucial for RWM feedback 3.Understanding RWM physics calls for hybrid MHD-kinetic description 4.RFA tests RWM damping physics 5.State-of-the-art in RWM modelling:  Damping physics + 3D structures + feedback

34. YQ Liu, Peking University, Feb 16-20, 2009 yueqiang.liu@ukaea.org.uk