Introduction to the Resistive Wall Mode (RWM) Yueqiang Liu UKAEA Culham Science Centre

1. Introduction to the Resistive Wall Mode (RWM) Yueqiang Liu UKAEA Culham Science Centre Abingdon, Oxon OX14 3DB, UK

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

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

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

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

6. The other way to look at it …

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

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

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 …

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

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

12. Analytic approaches • There are enormous literatures covering various analytical aspects of RWM • Probably one of the finest is offered by [Betti PoP 5 3615(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 2 2236 (1995)] • … representing also the energy principle plasma vacuum+wall kinetic inertia

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 7 3681(2000)], CarMa [Albanese COMPUMAG 2007] • VALEN [Bialek PoP 8 2170(2003)] • NMA [Chu NF 43 441(2003)] • KINX [Medvedev PPR 30 895(2004)] • CASTOR_FLOW, STARWALL [Strumberger NF 45 1156(2005)] • AEGIS [Zheng JCP 211 748(2006)] • MARG2D [Tokuda IAEA FEC08] • MARS-F is so far the only code including both feedback and advanced rotational damping physics

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

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

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 ?

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

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 95 255003(2005)] • Sound wave continuum damping [Bondeson PRL 72 2709(1994), Betti PRL 74 2949(2005)] • Parallel sound wave damping [Chu PoP 2 2236(1995)] • Damping from plasma inertial and/or dissipation layers [Finn PoP 2 3782(1995), Gimblett PoP 7 258(2000), Fitzpatrick NF 36 11(1996)] • Particle bouncing resonance damping[Bondeson PoP 3 3013(1996), Liu NF 45 1131(2005)] • Particle precession drift resonance damping [Hu PRL 93 105002(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

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 48 969(2006), Liu CPC 176 161(2007)] • Controller design = normally solving nonlinear optimization problem with constraints [Fransson PoP 7 4143(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 47 648 (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.)

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)

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)

22. Summary: issues to be resolved

23. Plans for following lectures: topics • Active control of RWM • Damping physics of RWM • Resonant field amplification (RFA) • 3D conductor effects on RWM

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)