1. YQ Liu, Peking University, Feb 16-20, 2009 Resonant Field Amplification Yueqiang Liu UKAEA Culham Science Centre Abingdon, Oxon OX14 3DB, UK

2. YQ Liu, Peking University, Feb 16-20, 2009 Outline 1.Introduction 1)What is resonant field amplification (RFA)? 2)Why interesting and important? 3)How to measure RFA? 2.Basic analytic theory 3.Toroidal modelling vs. Experiments

3. YQ Liu, Peking University, Feb 16-20, 2009 What is RFA ?  RFA: plasma amplifies an external field, which has the same resonant component (same field helicity) as one of the stable eigenmodes present in plasma

4. YQ Liu, Peking University, Feb 16-20, 2009 Why important ?  Error fields strongly affect plasma stability and confinement  Plasma can amplify external (static or LF ac) error fields due to resonance with (meta-) stable MHD modes (RFA).  Known example is RFA due to stable RWM  Causes magnetic braking of plasma rotation, modification of mode stability, etc.  Can also be useful to probe plasma stability boundary (active MHD spectroscopy)  Will be a significant issue for ITER with regard to momentum damping and RWM stability [Hender NF 47 S128(2007)]

5. YQ Liu, Peking University, Feb 16-20, 2009 Apply fields with external saddle coils Measure plasma response Measurement of RFA in JET using saddle loops How to measure RFA in experiment ?

6. YQ Liu, Peking University, Feb 16-20, 2009 Outline 1.Introduction 1)What is resonant field amplification (RFA)? 2)Why interesting and important? 3)How to measure RFA? 2.Basic analytic theory 3.Toroidal modelling vs. Experiments

7. YQ Liu, Peking University, Feb 16-20, 2009 Basic theory  RFA was first proposed by Boozer [Boozer PRL 86 5059(2001)]  As linear response of plasma (stable eigenmode) to external fields  With strongest amplification near stability margin  Theory can be understood from solution of a general ODE, without involving plasma physics  The full solution is

8. YQ Liu, Peking University, Feb 16-20, 2009 Basic theory  Special case A: steady-state linear response to a travelling wave  Special case B: A marginally stable mode does not give an “infinite” RFA response !

9. YQ Liu, Peking University, Feb 16-20, 2009  In our notation Basic theory  RFA amplification factor for RWM normally defined as [Reimerdes NF 2005, PPCF 2007] whereis the vacuum field in the presence of walls but in the absence of plasma  RFA determined by the eigenvalue (damping rate and real frequency) of the stable mode  Maximum amplification if external frequency matches intrinsic frequency of the mode  Experimentally measurable amplification factor helps to deduce the mode eigenvalue (active MHD spectroscopy)

10. YQ Liu, Peking University, Feb 16-20, 2009 Outline 1.Introduction 1)What is resonant field amplification (RFA)? 2)Why interesting and important? 3)How to measure RFA? 2.Basic analytic theory 3.Toroidal modelling vs. Experiments

11. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments  MHD spectroscopy  No-wall ideal kink beta limit  Stable RWM spectrum  Low-n peeling mode induced RFA  RFA measurements test RWM damping physics  RFA inter-plays with NTV-caused momentum damping

12. YQ Liu, Peking University, Feb 16-20, 2009 Resonant field amplification (RFA)  Observed in high-pressure plasmas, where low-frequency error fields are amplified by the plasma response, due to meta-stable low-frequency MHD modes (RWM)

13. YQ Liu, Peking University, Feb 16-20, 2009 RFA as MHD spectroscopy  RFA can be used as a tool to determine  Troyon beta limit  Damping rate and frequency of stable RWM …  … Using an empirical formula [Reimerdes NF 2005, PPCF 2007]

14. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

15. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments JET

16. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

17. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

18. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

19. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

20. YQ Liu, Peking University, Feb 16-20, 2009 RFA mode structure

21. YQ Liu, Peking University, Feb 16-20, 2009 RFA induced by peeling mode  Some of the RFA peaks in experiments correlated with ELM-free period prior to the first ELM, before reaching the RFA threshold [Gryaznevich PPCF 50 124030(2008)]

22. YQ Liu, Peking University, Feb 16-20, 2009  Equilibrium reconstructed from shot 70200  Among other parameters, peeling mode stability sensitive to edge current density, which is somewhat arbitrarily chosen here  Our goal is to reach qualitative understanding. Quantitative prediction requires extremely accurate knowledge of the plasma equilibrium  Choose two rotation profiles, differing slightly at the plasma edge Equilibrium and rotation profile

23. YQ Liu, Peking University, Feb 16-20, 2009  Stability of peeling mode, as an edge current driven mode, is largely controlled by proximity of edge q to an integer number  Unlike the external kink mode, which is mostly driven by beta in advanced tokamaks  For our equilibrium, m=6 is the most unstable peeling mode (in SFL coordinates) Eigenmode structure

24. YQ Liu, Peking University, Feb 16-20, 2009 Effect of rotation on peeling stability

25. YQ Liu, Peking University, Feb 16-20, 2009  With fixed field and total current, scaling plasma pressure leads to change of edge q-value, hence stability of ideal peeling mode  Peeling mode becomes stable for qa just above 6 for these equilibria  Compute RFA response from both stable peeling and RWM  Ratio of contribution from two modes varies with simultaneous increase of betan and qa RFA response from stable modes

26. YQ Liu, Peking University, Feb 16-20, 2009  A better way to track the peeling response is to keep a low beta, and scale qa only.  This requires slight scaling of total plasma current at a fixed magnetic field RFA response from peeling mode

27. YQ Liu, Peking University, Feb 16-20, 2009 RFA tests damping model semi-kinetic damping sound-wave damping

28. YQ Liu, Peking University, Feb 16-20, 2009 Stability also tests damping model

29. YQ Liu, Peking University, Feb 16-20, 2009 RFA & momentum damping

30. YQ Liu, Peking University, Feb 16-20, 2009 RFA & momentum damping

31. YQ Liu, Peking University, Feb 16-20, 2009 RWM couples to momentum confinement  Meta-stable RWM amplifies plasma response via RFA, creating large helical field perturbation inside plasma.  Plasma particles, going through these 3D fields, experience a viscous force (NTV) [Shaing PoP 10 1443(2003)] [Zhu PRL 96 225002(2006)]

32. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

33. YQ Liu, Peking University, Feb 16-20, 2009 Toroidal modelling vs. Experiments

34. YQ Liu, Peking University, Feb 16-20, 2009 Coupling to other MHD modes  DIII-D observes triggering of stable RWM by ELMs or fishbones  Triggering is sporadic, and criteria for ELM not known  Hypothesis: plasma generated n=1 perturbation increases effective rotation threshold, similar to magnetic breaking  Strong evidence in JET and DIII-D suggesting coupling of RWM to tearing modes

35. YQ Liu, Peking University, Feb 16-20, 2009 Summary  Error fields play a key role in stability and performance of fusion plasmas  RFA response by (meta-)stable modes in plasma complicates the matter, and requires re-thinking when designing error field correction coils  Good news is that RFA can be used as an active MHD spectroscopy tool to  detect damping rate and frequency of stable RWM  validate mode damping physics  ( examples of other such tool: using TAE cascad to predict q-profile evolution)  RFA due to low-n, stable MHD modes can be modelled using codes such as MARS-F (MARS-K), IPEC,...