YQ Liu, Peking University, Feb 16-20, 2009 Introduction to the Resistive Wall Mode (RWM) Yueqiang...

YQ Liu, Peking University, Feb 16-20, 2009

Introduction to the Resistive Wall Mode (RWM)

Yueqiang Liu

UKAEA Culham Science Centre

Abingdon, Oxon OX14 3DB, UK


Outline

1. What is RWM?2. Why important?3. Approaches/tools to study RWM

Analytical Numerical Experimental

4. Status-quo in RWM research What is known? Partially understood? Not understood?

5. Plans for following lectures


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=1,2,3Similar to vertical instability (RWM with n=0)

Three consequences of slowed down Still unstable eventually causes disruptionTime scale feasible for feedback controlKinetic effects become important

What is RWM ?


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 ITERCausing disruptions

RWM (advanced scenario)NTM (conventional scenario, mode locking)

Degrading performanceELM (H-mode)AE/TAE (alpha-particle destabilized), sawteeth, etc

Possibly stable or not so importantTM, Interchange mode, etc

Why important ?


The key for success of AT is to increase normalised plasma pressure by stabilising RWM

Example: for ITER advanced scenario (Scenario-4), successful stabilization of n=1 RWM can increase from 2.5 to 3.5

2.43/,7.1%,80%,5

11.0

bs

22/1

2bs

N

N

aRf

R

af

e.g.

N

fraction of plasma self-generated current

In more detail ...


The other way to look at it …


Outline






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

)2('

mr

kink

rw

w

rw

r 2'

and and vacuum solution

Let’s go through a simple analytic example: cylindrical Shafranov equilibrium


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 …


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


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


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

01 *

*2

kw

bvwv

p WWW

WKin

inertia plasma vacuum+wall kinetic


Basic is system of ideal MHD equationsAdditional 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 studyMARS-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

jb

vv

Bvb

bJBjv

0

PPt

pt

pt

Modelling tools


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

Experimental approaches: identify RWM


Not easy by local modification of plasma equilibrium profiles, largely determined by transport requirements and properties of AT:

Reversed or flat central q profilebroad current profile low liStrong 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 plasmasDifference is helical field perturbation

Also possible to apply feedback + plasma flow

Experimental approaches: stabilise RWM


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 ?


Outline






Understanding damping physics of the modeRequires comparison of experiments with theory and simulationsAlfven 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 dampingA metastable RWM amplifies external error field, causing toroidal torque which damps plasma flowPlasma flow below threshold results in unstable RWM

Status-quo: critical issues in mode physics


Two essential components in feedback

1)Plasma dynamics (P)

2)Controller (K)

Constructing plasma response models (PRM) describing the mode dynamics [Liu PPCF 48 969(2006), Liu CPC 176 161(2007)]

Realistic control design3D conducting structures (walls, coils)Noise (v,w,n), ac losses for superconducting coils (ITER)Power supply constraints (voltages, currents, time delays, etc.)

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 reviewIdeally coils should be placed as close as possible to plasmaPhysical constraints on space

Choice of sensor signals (pick-up coils) (y) [Liu NF 47 648 (2007)]

Status-quo: critical issues in mode control


Status-quo: mode physicsMHD 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)


Status-quo: mode controlResembles 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)


Summary: issues to be resolved


Plans for following lectures: topics

1. Active control of RWM

2. Damping physics of RWM

3. Resonant field amplification (RFA)

4. 3D conductor effects on RWM


Plans for following lectures: structureOn 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 Introduction to the Resistive Wall Mode (RWM) Yueqiang...

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