1. Introduction to the Resistive Wall Mode (RWM)
Yueqiang Liu
UKAEA Culham Science Centre
Abingdon, Oxon OX14 3DB, UK
YQ Liu, Peking University, Feb 16-20, 2009

2. Outline What is RWM? Why important? Approaches/tools to study RWM
Analytical
Numerical
Experimental
Status-quo in RWM research
What is known?
Partially understood?
Not understood?
Plans for following lectures
YQ Liu, Peking University, Feb 16-20, 2009

3. What is RWM ?
External ideal kink instability (time scale = microseconds)
Normally pressure-driven (above no-wall beta limit)
Resistive wall slows down kink instability to time scale of wall eddy current decay time  RWM (typically milliseconds)
At high pressure, mode located towards low-field side (kink-ballooning)
Low toroidal mode number n=1,2,3
Similar to vertical instability (RWM with n=0)
Three consequences of slowed down
Still unstable  eventually causes disruption
Time scale feasible for feedback control
Kinetic effects become important
YQ Liu, Peking University, Feb 16-20, 2009 3

4. Why important ?
Important for advanced tokamaks, aiming at steady state, high bootstrap current, high pressure operation
Good microscopic property (internal transport barrier), rather bad macroscopic MHD (low pressure limit due to RWM)
Stabilization of RWM essential for increasing fusion power production of advanced tokamaks
MHD modes in ITER
Causing disruptions
RWM (advanced scenario)
NTM (conventional scenario, mode locking)
Degrading performance
ELM (H-mode)
AE/TAE (alpha-particle destabilized), sawteeth, etc
Possibly stable or not so important
TM, Interchange mode, etc
YQ Liu, Peking University, Feb 16-20, 2009 4

5. In more detail ...
The key for success of AT is to increase normalised plasma pressure by stabilising RWM
fraction of plasma self-generated current
e.g.
Example: for ITER advanced scenario (Scenario-4), successful stabilization of n=1 RWM can increase from 2.5 to 3.5
YQ Liu, Peking University, Feb 16-20, 2009 5

6. The other way to look at it …
YQ Liu, Peking University, Feb 16-20, 2009

7. Outline What is RWM? Why important? Approaches/tools to study RWM
Analytical
Numerical
Experimental
Status-quo in RWM research
What is known?
Partially understood?
Not understood?
Plans for following lectures
YQ Liu, Peking University, Feb 16-20, 2009

8. Analytic approaches
According to ideal MHD description, RWM is ideal kink mode, whose free energy largely dissipated by eddy currents in the wall.
In cylindrical theory, growth rate determined by combining
and
and vacuum solution
Let’s go through a simple analytic example:
cylindrical Shafranov equilibrium
YQ Liu, Peking University, Feb 16-20, 2009

9. Analytic approaches
Consider single fluid, ideal, incompressible plasma, no flow
Perturbed momentum equation
With perturbed quantities
Faraday’s law gives
The z-component of curl of momentum equation (toroidal torque balance) gives …
YQ Liu, Peking University, Feb 16-20, 2009

10. Analytic approaches
Assuming a step density function, we have vacuum-like field everywhere
… and a jump condition across
Vacuum solution + jump condition result in the dispersion relation for ideal (current-driven) external kink
YQ Liu, Peking University, Feb 16-20, 2009

11. Analytic approaches Adding a jump condition across a (thin) wall
… together with the plasma & vacuum solution, we arrive at the dispersion relation for the RWM
Neglecting plasma inertia
YQ Liu, Peking University, Feb 16-20, 2009

12. Analytic approaches
There are enormous literatures covering various analytical aspects of RWM
Probably one of the finest is offered by [Betti PoP (1998)] (as far as analytics can go)
A very useful dispersion relation, valid in toroidal geometry, has been derived by several authors [Haney PF B1 1637(1989), Chu PoP (1995)]
… representing also the energy principle
inertia
plasma
vacuum+wall
kinetic
YQ Liu, Peking University, Feb 16-20, 2009

13. Modelling tools Basic is system of ideal MHD equations
Additional terms/equations for RWM modeling:
Vacuum equations
Equation for resistive wall
Equation for feedback coils
Flow terms
Kinetic terms
Full toroidal codes that are used for RWM study
MARS-F [Liu PoP (2000)], CarMa [Albanese COMPUMAG 2007]
VALEN [Bialek PoP (2003)]
NMA [Chu NF (2003)]
KINX [Medvedev PPR (2004)]
CASTOR_FLOW, STARWALL [Strumberger NF (2005)]
AEGIS [Zheng JCP (2006)]
MARG2D [Tokuda IAEA FEC08]
MARS-F is so far the only code including both feedback and advanced rotational damping physics
YQ Liu, Peking University, Feb 16-20, 2009 13

14. Experimental approaches: identify RWM
Not always easy from experiments. However, several possibilities do exist:
Check beta limit – unstable only if beta exceeds no-wall limit
Use ideal stability code to compute beta limit
Use experimental li-scaling
Resonant field amplification (RFA routinely used on DIII-D and JET)
If possible, measure mode growth rate and frequency
Both proportional to inverse wall time
RWM frequency normally between 0-100Hz, unlocked island a few KHz
RWM growth rate sensitive to plasma-wall separation [JT60-U], unlike internal modes
Mode structure
Global field perturbation and displacement within plasma (ELM, TM)
Ballooning structure at plasma surface
MHD spectroscopy [DIII-D, JET]
Measure resonant field amplification by (marginally) stable RWM
Using either a dc-pulse excited error field
Or traveling/standing waves field perturbation
YQ Liu, Peking University, Feb 16-20, 2009 14

15. Experimental approaches: stabilise RWM
Not easy by local modification of plasma equilibrium profiles, largely determined by transport requirements and properties of AT:
Reversed or flat central q profilebroad current profile  low li
Strong pressure peaking
Stabilization by plasma flow (passive way)
Various damping mechanisms (MHD, kinetic)
Still active research area
Feedback stabilization of RWM (active way)
Using magnetic coils to suppress the magnetic field produced by RWM
Very similar to vertical stability control of elongated plasmas
Difference is helical field perturbation
Also possible to apply feedback + plasma flow
YQ Liu, Peking University, Feb 16-20, 2009 15

16. Active control: one more point …
The fundamental reason that a magnetic feedback system, by suppressing the field perturbation, can stabilise the plasma instability, is that …
for an ideal plasma, the field lines are frozen into the plasma
This is the underlying assumption of many magnetic control of plasmas (vertical instability control, RWM control, etc.)
For this to be successful, plasma
must generate external field perturbations to interact with coil fields
can be treated as ideal (field line frozing)
For the above reasons, tearing mode (TM or NTM) or internal kink (sawteeth) cannot be stabilised by magnetic feedback (fortunately there are other means to stabilise them)
How about ELMs ?
YQ Liu, Peking University, Feb 16-20, 2009

17. Outline What is RWM? Why important? Approaches/tools to study RWM
Analytical
Numerical
Experimental
Status-quo in RWM research
What is known?
Partially understood?
Not understood?
Plans for following lectures
YQ Liu, Peking University, Feb 16-20, 2009

18. Status-quo: critical issues in mode physics
Understanding damping physics of the mode
Requires comparison of experiments with theory and simulations
Alfven continuum damping [Zheng PRL (2005)]
Sound wave continuum damping [Bondeson PRL (1994), Betti PRL (2005)]
Parallel sound wave damping [Chu PoP (1995)]
Damping from plasma inertial and/or dissipation layers [Finn PoP (1995), Gimblett PoP 7 258(2000), Fitzpatrick NF 36 11(1996)]
Particle bouncing resonance damping [Bondeson PoP (1996), Liu NF (2005)]
Particle precession drift resonance damping [Hu PRL (2004)]
Effect of error field – experiments show mode stability very sensitive to error field
Nonlinear coupling of mode stability, error field, and plasma momentum damping
A metastable RWM amplifies external error field, causing toroidal torque which damps plasma flow
Plasma flow below threshold results in unstable RWM
YQ Liu, Peking University, Feb 16-20, 2009 18

19. Status-quo: critical issues in mode control
Two essential components in feedback
Plasma dynamics (P)
Controller (K)
Constructing plasma response models (PRM) describing the mode dynamics [Liu PPCF (2006), Liu CPC (2007)]
Controller design = normally solving nonlinear optimization problem with constraints [Fransson PoP (2000)]
Choice of active coils (u): high priority topic in ongoing ITER design review
Ideally coils should be placed as close as possible to plasma
Physical constraints on space
Choice of sensor signals (pick-up coils) (y) [Liu NF (2007)]
Realistic control design
3D conducting structures (walls, coils)
Noise (v,w,n), ac losses for superconducting coils (ITER)
Power supply constraints (voltages, currents, time delays, etc.)
YQ Liu, Peking University, Feb 16-20, 2009 19

20. Status-quo: mode physics
MHD physics
Ideal kink + resistive wall (well understood)
Fluid continuum resonance damping (understood)
Resistive-viscous damping (understood)
Kinetic physics
Parallel sound wave damping (understood)
Particle bounce resonance (part. understood)
Particle precession drift resonance (part. understood)
Resonant field amplification (RFA) (part. understood)
Coupling to momentum confinement (poor understood)
Coupling to other MHD modes (not understood)
YQ Liu, Peking University, Feb 16-20, 2009

21. Status-quo: mode control
Resembles vertical stability control of elongated plasmas (n=0 RWM)
Magnetic feedback works because of:
External mode magnetic structure
Slow growth rate to allow feedback system to react
Important aspects:
Plasma (RWM) dynamics (part. understood)
Controller design and optimisation (PID, H-infinity, SISO, MIMO, …) (part. understood)
Choice of active coils (understood)
Sensor signal optimisation (well understood)
3D conductors for modelling (part. understood)
Practical issues: noise, power saturation, ac-losses (for SC), … (not well understood)
YQ Liu, Peking University, Feb 16-20, 2009

22. Summary: issues to be resolved
YQ Liu, Peking University, Feb 16-20, 2009

23. Plans for following lectures: topics
Active control of RWM
Damping physics of RWM
Resonant field amplification (RFA)
3D conductor effects on RWM
YQ Liu, Peking University, Feb 16-20, 2009

24. Plans for following lectures: structure
On each topic, try to show three aspects of research:
Analytic theory
Toroidal modelling
experiments
Basic analytic theory (not a comprehensive coverage)
Systematic modelling results
Brief description of some experimental results (to compare with modelling)
YQ Liu, Peking University, Feb 16-20, 2009