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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. Introduction1) 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
)coil to0(
)coil to90(o
r
or
B
BRFA
Measurement of RFA in JET using saddle loops
How to measure RFA in experiment ?
YQ Liu, Peking University, Feb 16-20, 2009
Outline
1. Introduction1) 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 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
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]
where
is 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. Introduction1) 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 50 124030(2008)]
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
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 modePeeling mode becomes stable for qa just above 6 for these equilibriaCompute RFA response from both stable peeling and RWMRatio 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 10 1443(2003)]
[Zhu PRL 96 225002(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, ...