Plasma Stability in Tokamaks and Stellarators · • n=0 mode: in tokamaks is solved with wall...

Plasma Stability inTokamaks and Stellarators

Gerald A. Navratil

GCEP Fusion Energy Workshop

Princeton, NJ1-2 May 2006

ACKNOWLEDGEMENTS

Borrowed VGs from many colleagues:J. Bialek, A. Garofalo,R. Goldston, A. Hubbard, R. Lahaye,J. Menard, H. Neilson, M. Okabayashi, E. J. Strait,S. Sabbagh, T. Taylor, M. Zarnstorff,…

MHD EQUILIBRIUM AND STABILITY

• MHD Equilibrium requires: ∇p = J x B

• MHD sets β limit ⇒ Loss of equilibrium

• Sources of free-energy for instability:Magnetic: B2/2µo Pressure: nT

• Three primary limiting β phenomena:Long Wavelength Ideal Modes: n = 0, 1, 2, 3Short Wavelength Ideal Modes: n → ∞

Long Wavelength Resistive Modes: n = 1, 2Magnetic Reconnection: E + v x B = ηJ

β LIMITING MODES: LOW-n• Must deal with long wavelength modes

Shift & Tilt: n = 0 and 1 Kink: n = 1

• n=0 mode: in tokamaks is solved with wallstabilization & active feedback control

• n=1 kink mode: limits determined to 10% to 20%.Tokamaks: solutions in hand using plasma rotation &active control. Stellarators: external magnetic fieldtransform may allow sufficiently high beta below kinklimit.

β LIMITING MODES: HIGH-n• Must deal with short wavelength

ballooning/interchange modes, n → ∞

• Limits well understood and with 10% to 20%accuracy compared with experiments

• Controlled by plasma shapeand magnetic shear profile:

β LIMITING MODES: TEARING MODES• Must deal with long wavelength resistive tearing

modes, n = 1, 2, …

Magnetic field line puncture plot showing island structure• Tearing Modes: In tokamaks: stabilized by current

profile control and active control with local ECH. Instellarators: controlled by tailoring externaltransform.

130–02/TST/wjS A N D I E G O

DIII–DNATIONAL FUSION FACILITY

A COMPACT STEADY STATE TOKAMAKREQUIRES OPERATION AT HIGH βN

High power density⇒ high βT

Large bootstrap fraction⇒ high βp

Steady state ⇒ high βN

1 + κ2

22

Pfus

PCD

cur

nqQ

ss= εeff

1 –

2

( ) B3aκγ

∝βN

ξ √A q βN

βT βp ∝ ( )βN

β

ε 1 + κ2 2

T

βN = 3.5Eq

uilib

rium

Lim

it

q* = 4

βN = 5

Curr

ent L

imit

Pow

er D

ensi

ty

Bootstrap Current

Conventional Tokamak

Advanced Tokamak

Pressure Limit βN ∝ power density ×bootstrap current

βN = βT /(I/aB) εβp

2

PRIMARY LIMITING MODE IN MAGNETICCONFINEMENT SYSTEMS: LOW-n Kink

• Long wavelength global MHD modes driven bypressure & current gradient:

Shift & Tilt: n = 0 and 1 Kink: n = 1

• ‘Classic’ Instability: Ideal conducting wall onplasma boundary stabilizes the kink mode byfreezing magnetic flux value on wall surface.

• Resistive conducting wall stabilization fails onmagnetic field soak-through time scale: τw

δW = 1—23

o2 2 ∫ d x ε c δB + ε c (∇ × B )⋅(ξ × δB)2

o

+ (∇⋅ξ)(ξ⋅∇p )o + γp (∇⋅ξ) 2o

perturbed magnetic energy

plasma compressionpressure driven - destabilizing

current driven - destabilizing

Foundation of Kink Mode Stability Builton Energy Principle δW Stability Analysis

1957 Bernstein, Frieman, Kruskal, Kulsrud

p

δW = – ∫ d x ε c δB 3 2o

212v

vacuum perturbed magnetic energy

If δW + δW < 0 mode is unstablep v

BASIC KINK MODE• Long wavelength mode driven by pressure &

current gradientCylindrical k ~ 2π/L Toroidal: low n = 1

• Unstable when δWp + δW∞v < 0

• Dispersion Relation: γ2K + δWp + δW∞v = 0 ,

where K is kinetic fluid mass• Define Γ∞2 = [δWp + δW∞

v]/K ~ [vAlfvén/L]2

IDEAL WALL STABILIZES THE KINK MODE• Ideal wall traps field in vacuum region and restoring

force stabilizes the kink – EXTERNAL Kink:

• Unstable when δWp + δWdv < 0 Note: δWd

v > δW∞v

• Dispersion Relation: γ2 - Γ∞2 + [δWdv-δW∞

v]/K = 0• Critical Wall Distance, dc, where kink stable for d < dc:

simple [δWdv-δW∞

v]/K parameterization with d:γ2 - Γ∞2[1 – dc/d]/K = 0

KINK MODE IS STABILIZED BY IDEAL WALL

309-04/GAN/rs

γ / Γ∞

Ideal Stability

γ2 – Γ2 (1 – dc)∞0 = d

0.00

0.02

0.04

0.06

0.08

0.0 0.5 1.0 1.5 2.0

IdealInstability

γ γ → → ΓΓ∞∞

Plasma-Wall Separation, d/dc

ideal modeideal mode stable stable

ideal modeideal mode unstable unstable

RESISTIVE WALL ‘LEAKS’ STABILIZING FIELD: τW

• Stabilizing field decays resistively on wall timescale τw ~ L/R: dψw/dt = - ψw/τw

• Quadratic kink: γ2 - Γ∞2[1-dc/d] = 0 coupled to‘slow’ flux diffusion γψw = - ψw/τw : τw >> τAlfvén

• Cubic Dispersion Relation with new ‘slow’root–the RWM: γ2 - Γ∞2[1-(dc/d) γτw/(γτw + 1)] = 0

KINK MODE GROWTH IS SLOWED BY RESISTIVE WALL

309-04/GAN/rs

γ / Γ∞

0.00

0.02

0.04

0.06

0.08

0.0 0.5 1.0 1.5 2.0

ResistiveWall Mode

Plasma-Wall Separation, d/dc

Real ω ≈ 0γ ≈ τw–1

Resistive wall mode (RWM) is unstable

— Mode structure similar to ideal external kink— Mode grows slowly: γ ~ τ–1

w

Resistive Wall

γ τwγ τw + 1⋅

Ideal Stability

γ2 – Γ2 (1 – dc∞0 = d )

ideal modeideal mode unstable unstable

309-04/GAN/rs

RWM STABILIZED IN DIII-D BY ROTATIONFOR MANY WALL-TIMES, τW

Normalized plasma pressure, βN, exceeds no-wall stability limit by up to 40% n = 1 mode grows (γ ~ 1/τW) after toroidal rotation at q = 3 surface has

decreased below ~1 kHz

ROTATION AND DISSIPATION CAN STABILIZE RWM• Rotation Doppler shift: γ ⇒ γ + iΩ where Ω is plasma rotation.• Dissipation represented by friction loss (γ + iΩ)ν, where form

of ν still being actively studied by theory community:

(γ + iΩ)2 - Γ∞2[1-(dc/d) γτw/(γτw + 1)] + (γ + iΩ)ν = 0 (as shown in Chu, et al. Phys. Plasma 1995;

consistent with numerical result of Bondeson & Ward, PRL 1994)• Cubic Dispersion Relation with three roots: in region where

d < dc new ‘slow’ RWM root can be damped with ‘fast’ stable kinkmode roots tied to rotating plasma with usual ordering:

τ w-1 << Ω << vAlfvén/L

• Why is RWM Slow Root Stabilized?kink energy release < dissipation loss of RWM

slowed by wall in flowing plasma

KINK MODE GROWTH IS SLOWED BY RESISTIVE WALLAND STABILIZED BY PLASMA ROTATION

309-04/GAN/rs

ω ≈ τw–1

StableGap

0.00

0.02

0.04

0.06

0.08

γ / Γ∞Resistive Wall

Mode

PlasmaMode

γ → Γγ → Γ∞∞

0.0 0.5 1.0 1.5 2.0Plasma-Wall Separation, d/dc

Plasma DissipationResistive Wall

Γ2 (dc/d)γ τwγ τw + 1

∞ (γ + iΩ) νDIS +

Ideal Stability

(γ + iΩ)2 – Γ2∞0 = +

Resistive wall mode (RWM) is unstable

— Mode structure similar to ideal external kink

— Mode nearly stationary: ω ~ τw–1 << Ωplasma

— Mode grows slowly: γ ~ τ–1w

Dissipation + rotation stabilizes RWM

S A N D I E G O


Feedback control of NBIpower keeps βN belowstability limit (107603)

No other large scaleinstabilities encountered(NTM, n=2 RWM, . . . )

SUSTAINED ROTATION ABOVE CRITICAL VALUE⇒ RELIABLE OPERATION ABOVE THE NO-WALL LIMIT

258–02/EJS/wj

107603

0

1

2

3

4

1000 1500 2000 2500 3000

βN

Time (ms)

0

6

12 Rotation (kHz) at q~2

βNno wall (2.4li)

106535

Ideal n=1 kink observedat the wall-stabilizedβ limit

300

Trot ~1 ms < τwall

200

Toro

idal

Ang

le

100

2111.0 2111.5Time (ms)

2112.0 2112.5

τg ~ 300 µs << τwall

δBp βN ~ 2 βno-wall

N— β = 3.7%

MARS PREDICTIONS OF ΩcritτA IN QUALITATIVE AGREEMENT WITH MEASUREMENTS ON DIII-D AND JET

• Both damping models predict Ωcrit within a factor of 2

• In DIII-D Ωcrit τA ~ 0.02 with weak βdependence

• In JET Ωcrit τA ~ 0.005 with weak βdependence

sound wavesound wave

S A N D I E G O


MODE FREQUENCY AND DAMPING CANNOT BE FIT SIMULTANEOUSLY

• Both damping models • Both damping models predict predict γγ too low too low

• Kinetic damping predicts• Kinetic damping predicts mode frequency mode frequency ωω

• Further work on damping• Further work on damping [e.g. neoclassical viscosity] [e.g. neoclassical viscosity] models being explored models being explored

-4

-2

0

0.0

1.0

2.0

0.0 0.2 0.4 0.6 0.8 1.0Cβ

Mode rotationfrequency ωRWM τW

Growth rate γRWM τW

kinetic

sound wave(κ|| = 0.5)

experiment(Ωτ ~0.02)A

RWMRWM

RWMRWM

NSTX provides crucial data for understanding the dissipation mechanisms that allow rotational stabilization of the RWM

• Insight from drift-kinetic theory:– Trapped-particle effects at finite ε significantly weaken ion Landau damping, but… – Toroidal inertia enhancement modifies eigenfunction when Ωφ / ωA > 1/4q2

• Experimental Ωcrit / ωA suggests scaling ∝ ε / q2 – why?• Is dissipation localized to resonant surfaces, or more global?

– Addressing questions above w/ NSTX / DIII-D similarity experiments, and hi-res CHERS

• ST has uniquely high ωsound / ωA distinguish between ωs and ωA scaling

Needed for predicting control requirements for RWM stabilization in ITER & CTF

(Columbia Univ.)

NSTXDIII-Dshifted &

scaled × 1.1

13

309-04/GAN/rs

FEEDBACK LOGIC FOR RWM FEEDBACK STABILIZATION

Smart Shell Explicit ModeControl

Feedback cancelsthe radial flux from MHD

mode at wall sensor

Feedback cancelsthe flux from MHD mode

at plasma surface

309-04/GAN/rs

DIII–D INTERNAL CONTROL COILS ARE PREDICTED TOPROVIDE STABILITY AT HIGHER BETA

Inside vacuum vessel: Faster time response for feedback controlCloser to plasma: more efficient coupling

Internal Coils(I-coils)

Normalized Growth Rate

1

10

Ideal kink

1.00.80.60.4No-Wall Limit Ideal-Wall Limit

0.20

No Feedback

C

γτw

10

10

10

2

-1

-2

FEEDBACK WITH I-COILS IN DIII-D INCREASES STABLE PLASMA PRESSURE TO NEAR IDEAL-WALL LIMIT

269-04/MO/jyS A N D I E G O


• VALEN code: - DCON MHD stability - 3D geometry of vacuum vessel and coil geometry

• τw is the vacuum vessel flux diffusion time (~ 3.5 ms)Resistive

Wall Mode: Open loop growth rate

β

• VALEN code prediction

• External C-Coil: - Control fields must penetrate wall - Induced eddy currents reduce feedback

• τw is the vacuum vessel flux diffusion time (~3.5 ms)

1

10

Ideal kink


0.20

No FeedbackExternal C-coils

C

10

10

10

2

-1

-2



Accessiblewith

ExternalC-coils


γτw

β

• C-coil stabilizes slowly growing RWMs


• Internal I-Coils: - Improved coil/plasma coupling - Improved spatial match to RWM field structure

• τw is the vacuum vessel flux diffusion time (~3.5 ms)

1

10

Ideal kink


0.20

Internal I-coils

No FeedbackExternal C-coils

C

10

10

10

2

-1

-2

• I-coil stabilizes RWMs with growth rate 10 times faster than C-coils



Accessiblewith

ExternalC-coils

Accessiblewith

InternalI-coils


γτw

β


FEEDBACK EFFICACY DEMONSTRATED BY GATING OFF THE GAINFOR 20 MS AT TIME OF EXPECTED RWM ONSET

NATIONAL FUSION FACILITYS A N D I E G O

DIII–D

104119

Without feedback, slow Ip ramp rate (0.5 MA/s) destabilizes slowly growing RWM

With feedback, beta collapse avoided

n=1 mode starts up during feedback off period, stabilized after feedback is turned back on

n=1 mode detected on poloidalfield probes and SXR arrays,decoupled from driver coils

104119-1

0

1

2

1.0

1.2

1.4

Relative Displacement (SXR)

Feedback Current (C79)

(cm

) (k

A)

0

35Feedback Gain

βN

n=1 δBr 104118 104119

321

20

10

0

(gau

ss)

0

Feedback OFFTime (ms)

15501540153015201510150014901480

Time (ms)

1200 20001400 1600 1800

timeC-COIL

no walllimit

ideal walllimit

NOFEEDBACK

Cβ

1.0

0.00.0 100 15050

Rotation (km/s)

MARS predictionStable (without feedback)

Unstable(without feedback)

FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH CβAT ROTATION BELOW CRITICAL LEVEL PREDICTED BY MARS



• Trajectories of plasma discharge in rotation versus Cβ

• No feedback plasma approaches limit and disrupts

• C-coil feedback plasma crosses limit & reaches higher pressure

timeC-COIL

no walllimit

ideal walllimit

NOFEEDBACK

Cβ

0.00.0 100 15050

Rotation (km/s)

MARS prediction

ZEROROTATION

MARS /VALEN prediction with measured amplifier time responsefor zero rotation

Stable (without feedback)

Unstable


• With near zero Rotation, Cβ is near the maximum set by existing control system characteristics: bandwidth & processing time delay



1.0

I-COIL

• I-coil feedback plasma reaches near zero rotation

timeC-COIL

no walllimit

ideal walllimit

NOFEEDBACK

Cβ

1.0

0.00.0 100 15050

Rotation (km/s)

MARS prediction

ZEROROTATION

MARS /VALEN prediction with measured power supply time responsefor zero rotation

Stable (without feedback)

Unstable


• Combination of low rotation and feedback reaches Cβ is the ideal wall-limits



I-COIL

I-COIL

PlasmaRotation (km/s)

n=1 δBp ModeAmplitude (gauss)

Time (ms)

βn

Estimated no-wall limitFeedback coil current amplitude

F.B on

NoFeedback

With Feedback

119666/119663

Feedbackcurrent (kA)

RWM FEEDBACK ASSISTS IN EXTENDING βn~4 ADVANCEDTOKAMAK DISCHARGE MORE THAN 1 SECOND

1000 2000

4.0

0.0200

0.0

0.0

NoFeedback

With Feedback

1500 2500

30.0



• High performance plasma approaches β ~ 6%

• Without feedback plasma disrupts due to RWM

NATIONAL FUSION FACILITYDIII–D 093-05/EJS/rs

RWM Stabilization Has Opened New High Performance Regimes Above the No-Wall Stability Limit

• Simultaneous feedback control of error fields and RWM

• Additional ECCD power in FY06 will help sustain high qmin

• New divertor will help density control at high triangularity

⟨J||⟩ (A/cm2)

Safety Factor

(105 Pa)Pressure

0.0 0.2 0.4 0.6 0.8 1.0ρ

βN

4li

6li

βT (%)

122004

20001000 3500 40002500Time (ms)

1500 30000

1

2

3

4

5

6

020

40

60

80

0.0

0.5

1.0

1.5122004.2500

0

2

4

6

RWM Coil Concept for ITER

• Baseline RWM coils located outside TF coils

Applying Internal RWM Feedback Coils to the Port Plugs in ITER Increases β−limit for n = 1 from βN = 2.5 to ~ 4

• Integration and Engineering feasibility of internal RWM coils is under study.

VALEN Analysis Columbia University

No-wall limit

7 RWM Coils mounted behind the BSM inevery other port except NBI ports.

(assumes 9 ms time constant for each BSM)

• Internal RWM coils would be located insidethe vacuum vessel behind shield modulebut inside the vacuum vessel on theremovable port plugs.

NCSX

3-Period NCSX Plasmaand Coil Design

Compact Stellarator Low-n Stability

Stellarators provide external magnetic field transform aiming at:

• Steady state without current drive.

• Kink stable at sufficiently high pressure (β > 4%)without feedback control or rotation drive.

Compact Stellarators (CS) improve onprevious designs.

• Magnetic quasi-symmetry:

– good confinement.

– link to tokamak physics.

• Lower aspect ratio.

NCSX Stability Modeling Predicts Kink Stability up to 6%PIES Free-Boundary Equilibrium at β = 4.1%

MCZ 050218 8

0

1

2

3

4

0

5

10

15

20

250

1

2

0

1

2

3

4

0.1 0.2 0.3 0.4Time (s)

<!>

ne

(102

0 m

-3)

PNB

Prad

Mirnov B.

<!>

(%)

Pow

er (

MW

)(A

.U.)

W7AS: 〈β〉 ≈ 3.4 % : Quiescent, Quasi-stationary• B = 0.9 T, iotavac ≈ 0.5• Almost quiescent high- β phase, MHD-activity in early medium-β phase

• In general, β not limited by anydetected MHD-activity.

• IP = 0, but there can be localcurrents

• Similar to High Density H-mode(HDH)

• Similar β>3.4% plasmasachieved with B = 0.9 – 1.1 T witheither NBI-alone, or combinedNBI + OXB ECH heating.

• Much higher than predicted βlimit ~ 2%

54022

Current-carrying Systems are Subjectto Reconnection – “Tearing Modes”

Reversemagneticshear

Normalmagneticshear

• A rational field line can be an O point around which islandsform. (- j for normal shear, + j for reverse shear)

3/2

6

4

2

01.00.80.60.40.20.0

Minor radius (r/a)

MSE data

Sa

fety

Fa

cto

r

µ 0

1.22!nc

dw

dt= " # + a1$

1 2%&Lq

Lp

w

w2 +wc

2

'

( )

*

+ , - a2

.&i2 %& g($ )

w3

Lq

Lp

'

( ) )

*

+ , ,

2

Ohmic current Bootstrap current Finite transport correction

Polarization current

Pressure-driven “Bootstrap Current”is a Boon and a Bane

- In the presence of a pressure gradient,trapped particles entrain a parallel“bootstrap” current. A “neoclassical”effect, i.e. collisional, but including non-local orbit effects.

- May allow steady-state operation ofaxisymmetric toroidal systems.

- Drives “neoclassical tearing modes” innormal shear regions due to currentdepletion in the magnetic islands.

Theory Accurately Predicts Growth ofNeoclassical Tearing Modes (NTM)

Bootstrap current + normal shear drives NTM’s.- Agrees to factor of ~2 with neoclassical resistivity,

over a wide range of plasma parameters.- Important challenge to theory of magnetic reconnection.- Reverse shear stabilizes NTM’s, as predicted.- Strong implications for toroidal system optimization.

3.0 3.5 4.00

1

2

3

4

5

Time (s)

W (

cm

) Theory

Magnetic

24MW NBI

ECE

Shot 18432

Island Width Predicted by Neoclassical Theory

(cm)

time (s)

4

2

0

0.2 0.22 0.24 0.26 0.28 0.30

w (

cm)

0.8

0.4

0.0

!p(meas)

6

8

0.2

0.6!p(meas)

Measured Island Width (cm)

R = 3m, Te ~ 5keV R = 1m, Te ~ 1keV

Replacing Bootstrap Current in IslandsStabilizes Neoclassical Tearing Modes

Steerable ElectronCyclotron CurrentDrive wavelauncher.

ECCD

ITER will have ECCD for NTM control.

NATIONAL FUSION FACILITYDIII–D

DIII–D Demonstrates NTM Active Stabilization with ECCD

• High beta is achieved with pre- emptive stabilization of the 2/1 NTM – Stable operation at the no-wall beta limit for >1 s

• ECCD is applied before the mode appears – Real-time tracking of the q=2 surface maintains current drive alignment

• Tearing mode appears promptly when ECCD is removed

PNB (MW)

⟨PNB⟩ (MW)

Current ×10 (MA)

EC Power (MW)

B (n = 1) (G)~

Div. Dα (au)

βN4 li

0 2000 4000 6000 8000Time (ms)

|B| (T)

0

5

10

15

0

20

40

0

1

2

3

0

2

4

1.4

1.5

1.6

ECCD in ITER Can Reduce the m/n=2/1 NTM Island

• “Cross-machine bench marking...”R.J. La Haye, et.al., submitted to Nuclear Fusion

• Locking condition from 0-D model

wa (1 + 20 )=

3

lock

DIII–D → ITER ...τw = 3 → 188 ms (J. Bialek)

...f0 = 8 → 0.42 kHz (A. Polevoi)

...τE0 = 0.1 → 3.7 s (J. Cordey)

...τA0 = 1.6 → 3.0 μs (Y. Gribov)

... | = 0.081 → 0.025

— w≈ 5 cm in ITER to lock

— w≈10 cm at f0 = 1.4 kHz

wa

253-05/RJL/jy

τwτE0

ω02 τA0

2wa 2* 14–1

-5

0

5

10

15

20

25

0 5 10 15 20 25m/n=2/1 Island full width w (cm)

12 MW ECCD50/50 Modulation(K1 = 0.74, F = 0.5)

NO ECCDSaturated Island

(if Beta Maintained andif Mode Does Not Lock)

NO ECCDUnstableRegion

12 MW ECCDNo Modulation(K1 = 0.38, F=1)

ITER, m/n=2/1, βN = 1.84

Isla

nd g

row

th ra

te (τ

R/r)

dw/d

t

Key Open Issues in Stability

Spherical Torus Compact StellaratorAdvanced TokamakLow-n Kink:• Rotation Stabilization Physics & Scaling• Scale Active Feedback to ITER - n = 1, 2,…• Coil Modularity & Failure of “Mode Rigidity”Low-n Tearing:• NTM Active Control Requirements for ITER• Quantitative Theory: Seeding physics, island rotation, ECCD localization & modulation, small island modeling

Low-n Kink:• Validate no-wall high-beta kink limits• Rotation effects in QS equilibria

Low-n Tearing:• NTM control with external magnetic transform + large bootstrap current• Magnetic Island control as β and Ip vary.

Plasma Stability in Tokamaks and Stellarators · • n=0 mode: in tokamaks is solved with wall...

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Transcript of Plasma Stability in Tokamaks and Stellarators · • n=0 mode: in tokamaks is solved with wall...