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

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

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

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)]

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 ?

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

YQ Liu, Peking University, Feb 16-20, 2009 Basic theory  RFA was first proposed by Boozer [Boozer PRL (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

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 !

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)

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

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

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)

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]

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

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

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

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

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

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

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

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 (2008)]

YQ Liu, Peking University, Feb 16-20, 2009  Equilibrium reconstructed from shot  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

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

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

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

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

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

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

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

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

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 (2003)] [Zhu PRL (2006)]

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

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

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

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,...