Plasma Stability in Tokamaks and Stellarators

Gerald A. Navratil

GCEP Fusion Energy Workshop Princeton, NJ 1-2 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: 2 Magnetic: B /2µo Pressure: nT • Three primary limiting β phenomena: Long Wavelength Ideal Modes: n = 0, 1, 2, 3 Short Wavelength Ideal Modes: n → ∞ Long Wavelength Resistive Modes: n = 1, 2 Magnetic 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 wall stabilization & 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 field transform may allow sufficiently high beta below kink limit. 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 shape and 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. In stellarators: controlled by tailoring external transform. A COMPACT STEADY STATE TOKAMAK REQUIRES OPERATION ATβ HIGH N γ ε β2 Pfus eff N 3 Q = ∝ cur B aκ ss P nq (1 Ðξ √ qβ ) CD A N

High power density ⇒ β high T P Large bootstrap fraction r e Power Density Power s ⇒ highβ A s p d u v r a e ⇒ β Current Limit n L Steady state high β c i N T e m d it β ∝ power density× 2 To N 1 +κ C ka ε o ma bootstrap current n k Equilibrium Limit v e 2 n t β io = 5 n N 2 al 1 +κ 2 To β β ∝ β kam T p N ak ( 2 ) = 4 β N = 3.5 q* β = β 2(I/aB) εβ Bootstrap Current N T / p DIII–D NATIONAL FUSION FACILITY 130Ð02/TST/wj SAN DIEGO PRIMARY LIMITING MODE IN MAGNETIC CONFINEMENT SYSTEMS: LOW-n Kink • Long wavelength global MHD modes driven by pressure & current gradient: Shift & Tilt: n = 0 and 1 Kink: n = 1

• ‘Classic’ Instability: Ideal conducting wall on plasma boundary stabilizes the kink mode by freezing magnetic flux value on wall surface. • Resistive conducting wall stabilization fails on magnetic field soak-through time scale: τw Foundation of Kink Mode Stability Built on Energy Principle δW Stability Analysis 1957 Bernstein, Frieman, Kruskal, Kulsrud perturbed magnetic energy current driven - destabilizing —1 3 2 2 ε 2 ∇ × ⋅(ξ × δ δWp = 2 ∫ d x {εo c δB + o c ( B ) B)

2 + (∇⋅ξ)(ξ⋅∇po ) + γpo (∇⋅ξ) } pressure driven - destabilizing plasma compression = –1 d 3x 2 2 δWv 2 ∫ εo c δB vacuum perturbed magnetic energy

If δWp + δWv < 0 mode is unstable BASIC KINK MODE • Long wavelength mode driven by pressure & current gradient Cylindrical k ~ 2π/L Toroidal: low n = 1

∞ • Unstable when δWp + δW v < 0 2 ∞ • Dispersion Relation: γ K + δWp + δW v = 0 , where K is kinetic fluid mass 2 ∞ 2 • Define Γ∞ = [δWp + δW v]/K ~ [vAlfvén/L] IDEAL WALL STABILIZES THE KINK MODE • Ideal wall traps field in vacuum region and restoring force stabilizes the kink – EXTERNAL Kink:

d d ∞ • Unstable when δWp + δW v < 0 Note: δW v > δW v 2 2 d ∞ • Dispersion Relation: γ - Γ∞ + [δW v-δW v]/K = 0

• Critical Wall Distance, dc, where kink stable for d < dc: d ∞ simple [δW v-δW v]/K parameterization with d: 2 2 γ - Γ∞ [1 – dc/d]/K = 0 KINK MODE IS STABILIZED BY IDEAL WALL

2 2 dc 0 = γ – Γ∞ 1 – } ( d )

Ideal Stability

0.08 Ideal Instability 0.06 γ → Γ∞ γ /Γ ∞ 0.04 ideal mode ideal mode 0.02 stable unstable 0.00 0.0 0.5 1.0 1.5 2.0 Plasma-Wall Separation, d/dc

309-04/GAN/rs RESISTIVE WALL ‘LEAKS’ STABILIZING FIELD: τW • Stabilizing field decays resistively on wall time scale τw ~ L/R: dψw/dt = - ψw/τw

2 2 • Quadratic kink: γ - Γ∞ [1-dc/d] = 0 coupled to ‘slow’ flux diffusion γψw = - ψw/τw : τw >> τAlfvén • Cubic Dispersion Relation with new ‘slow’ 2 2 root–the RWM: γ - Γ∞ [1-(dc/d) γτw/(γτw + 1)] = 0 KINK MODE GROWTH IS SLOWED BY RESISTIVE WALL

2 2 dc γ τw 0 = γ – Γ∞ 1 – ⋅ d γ τw + 1 } ( } ) Ideal Stability Resistive Wall

Resistive wall mode (RWM) is 0.08 Real ω ≈ 0 unstable –1 γ ≈ τw 0.06 — Mode structure similar to γ /Γ ∞ Resistive ideal external kink 0.04 Wall Mode –1 — Mode grows slowly: γ ~ τw 0.02 ideal mode unstable 0.00 0.0 0.5 1.0 1.5 2.0 Plasma-Wall Separation, d/dc

309-04/GAN/rs RWM STABILIZED IN DIII-D BY ROTATION FOR 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

309-04/GAN/rs 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: 2 2 (γ + iΩ) - Γ∞ [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 kink mode roots tied to rotating plasma with usual ordering: -1 τ w << 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 WALL AND STABILIZED BY PLASMA ROTATION 2 (d /d) 0 = (γ + Ω)2 – Γ2 + Γ∞ c γ τw } i ∞ + (γ + Ω) ν } i DIS } γ τw + 1 Ideal Stability Resistive Wall Plasma Dissipation

Stable Resistive wall mode (RWM) is Gap unstable 0.08 — Mode structure similar to –1 ideal external kink 0.06 ω ≈ τw –1 γ /Γ — Mode grows slowly: γ ~ τw ∞ Plasma 0.04 Resistive Wall Mode Dissipation + rotation Mode γ → Γ 0.02 ∞ stabilizes RWM — Mode nearly stationary: –1 0.00 ω ~ τw << Ωplasma 0.0 0.5 1.0 1.5 2.0 Plasma-Wall Separation, d/dc

309-04/GAN/rs SUSTAINED ROTATION ABOVE CRITICAL VALUE ⇒ RELIABLE OPERATION ABOVE THE NO-WALL LIMIT

106535 107603 ● Feedback control of NBI β 4 N power keepsβ below 3 N stability limit (107603) 2 no wall β (2.4l ) 1 N i ● No other large scale 0 instabilities encountered Rotation (kHz) at q~2 (NTM, n=2 RWM, . . . ) 12

6 ● 0 Ideal n=1 kink observed at the wall-stabilized 1000 1500Time (ms) 2000 2500 3000β limit

δB p ● β βno-wall 300 N ~ 2 N — β = 3.7% 200

τ µ τ Toroidal Angle 100 g ~ 300 s << wall τ Trot ~1 ms

sound wave

• In DIII-D Ωcrit τA ~ 0.02 with weak β • In JET Ωcrit τA ~ 0.005 with weak β dependence dependence

• Both damping models predict Ωcrit within a factor of 2 MODE FREQUENCY AND DAMPING CANNOT BE FIT SIMULTANEOUSLY

Growth rate γRWM τW 0

experiment(Ωτ ~0.02) A • Both damping models sound wave -2 predict γ too low (κ|| = 0.5) RWM

• Kinetic damping predicts kinetic mode frequency ω -4 RWM 2.0 Mode rotation • Further work on damping frequency ωRWM τW [e.g. neoclassical viscosity] models being explored 1.0

0.0 DIII–D 0.0 0.2 0.4 0.6 0.8 1.0 NATIONAL FUSION FACILITY S A N D I E G O Cβ 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… 2 – Toroidal inertia enhancement modifies eigenfunction when Ωφ / ωA > 1/4q

(Columbia Univ.)

NSTX 2 DIII-D • Experimental Ωcrit / ωA suggests scaling ∝ε / q –why? shifted & • Is dissipation localized to resonant surfaces, or more global? scaled × 1.1 – Addressing questions above w/ NSTX / DIII-D similarity experiments, and hi-res CHERS

• ST has uniquely high ωsound / ωA Æ distinguish between ωs and ωA scaling Needed for predicting control requirements for RWM stabilization in ITER & CTF 13 FEEDBACK LOGIC FOR RWM FEEDBACK STABILIZATION

Smart Shell Explicit Mode Control

Feedback cancels Feedback cancels the radial flux from MHD the flux from MHD mode mode at wall sensor at plasma surface

309-04/GAN/rs DIII–D INTERNAL CONTROL COILS ARE PREDICTED TO PROVIDE STABILITY AT HIGHER BETA ● Inside vacuum vessel: Faster time response for feedback control Closer to plasma: more efficient coupling Internal Coils (I-coils)

309-04/GAN/rs FEEDBACK WITH I-COILS IN DIII-D INCREASES STABLE PLASMA PRESSURE TO NEAR IDEAL-WALL LIMIT • VALEN code prediction

No Feedback 2 Ideal kink 10 • VALEN code: Normalized - DCON MHD stability Growth Rate 10 - 3D geometry of γτw vacuum vessel and coil geometry 1

• τw is the vacuum vessel -1 10 Resistive flux diffusion time (~ 3.5 ms) Wall Mode: -2 Open loop growth rate 10

0 0.2 0.4 C 0.6 0.8 1.0 No-Wall Limit β Ideal-Wall Limit

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy FEEDBACK WITH I-COILS IN DIII-D INCREASES STABLE PLASMA PRESSURE TO NEAR IDEAL-WALL LIMIT • C-coil stabilizes slowly growing RWMs

No Feedback Ideal kink 2 External C-coils 10 • External C-Coil: Normalized - Control fields must Growth Rate 10 penetrate wall γτw - Induced eddy currents reduce feedback 1

• τw is the vacuum vessel -1 10 Accessible flux diffusion time (~3.5 ms) with -2 External 10 C-coils 0 0.2 0.4 C 0.6 0.8 1.0 No-Wall Limit β Ideal-Wall Limit

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy FEEDBACK WITH I-COILS IN DIII-D INCREASES STABLE PLASMA PRESSURE TO NEAR IDEAL-WALL LIMIT • I-coil stabilizes RWMs with growth rate 10 times faster than C-coils

No Feedback 2 Ideal kink 10 External C-coils Internal I-Coils: Internal I-coils • Normalized - Improved coil/plasma Growth Rate 10 coupling γτw - Improved spatial match to RWM field structure 1

• τw is the vacuum vessel -1 10 Accessible Accessible flux diffusion time (~3.5 ms) with with -2 External Internal 10 C-coils I-coils 0 0.2 0.4 C 0.6 0.8 1.0 No-Wall Limit β Ideal-Wall Limit

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy FEEDBACK EFFICACY DEMONSTRATED BY GATING OFF THE GAIN FOR 20 MS AT TIME OF EXPECTED RWM ONSET

35 Feedback Gain Without feedback, slow Ip ramp rate (0.5 MA/s) destabilizes 0 3 slowly growing RWM 2 βN 1 With feedback, beta collapse 0 20 avoided δ 104118 104119 10 n=1 Br

(gauss) 0 1200 1400 1600 1800 2000 Time (ms)

2 n=1 mode starts up during Relative Displacement (SXR) 104119 1 feedback off period, stabilized after feedback is (cm) 0 turned back on

-1 n=1 mode detected on poloidal 1.4 Feedback Current (C79) 104119 field probes and SXR arrays, 1.2 (kA) decoupled from driver coils

1.0 1480 1490 1500 1510 1520 1530 1540 1550 DIII–D NATIONAL FUSION FACILITY Time (ms) S A N D I E G O Feedback OFF FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH Cβ AT ROTATION BELOW CRITICAL LEVEL PREDICTED BY MARS

• Trajectories of plasma discharge in rotation versus Cβ

ideal wall1.0 limit MARS prediction Stable (without feedback) • No feedback plasma approaches limit Unstable and disrupts (without feedback) Cβ • C-coil feedback plasma C-COIL time crosses limit & reaches higher NO pressure no wall FEEDBACK 0.0 limit 0.0 50 100 150 Rotation (km/s)

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH Cβ AT ROTATION BELOW CRITICAL LEVEL PREDICTED BY MARS • With near zero Rotation, Cβ is near the maximum set by existing control system characteristics: bandwidth & processing time delay

ideal wall 1.0 limit MARS prediction Stable (without feedback) • I-coil feedback plasma Unstable reaches near zero rotation Cβ

MARS /VALEN prediction with measured I-COIL C-COIL time ZERO amplifier time response ROTATION for zero rotation NO FEEDBACK no wall 0.0 limit 0.0 50 100 150 Rotation (km/s)

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH Cβ AT ROTATION BELOW CRITICAL LEVEL PREDICTED BY MARS

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

ideal wall 1.0 limit I-COIL MARS prediction Stable (without feedback) Unstable

Cβ

MARS /VALEN prediction with measured I-COIL C-COIL time power supply time response ZERO ROTATION for zero rotation NO FEEDBACK no wall 0.0 limit 0.0 50 100 150 Rotation (km/s)

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy RWM FEEDBACK ASSISTS IN EXTENDING βn~4 ADVANCED TOKAMAK DISCHARGE MORE THAN 1 SECOND

4.0 119666/119663 • High performance βn With Feedback plasma approaches Feedback Estimated no-wall limit current (kA) F.B on β ~ 6% Feedback coil current amplitude 0.0 200 Plasma With Feedback • Without feedback plasma disrupts Rotation No (km/s) Feedback due to RWM 0.0 30.0 n=1 No δBp Mode Amplitude Feedback (gauss) 0.0 1000 1500 2000 2500 Time (ms)

DIII–D NATIONAL FUSION FACILITY S A N D I E G O 269-04/MO/jy 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

122004.2500 122004 1.5 1.0 6 β (%) T 0.5 Pressure (105 Pa) 5 0.0 β N 4 80 60 6 3 li 40 〈J 〉 (A/cm2) 2 20 || 4li 0 6 1 Safety Factor 0 4 100015002000 2500 3000 3500 4000 2 Time (ms) 0 0.0 0.2 0.4 ρ 0.6 0.8 1.0

DIII–D 093-05/EJS/rs NATIONAL FUSION FACILITY Applying Internal RWM Feedback Coils to the Port Plugs

in ITER Increases β−limit for n = 1 from βN = 2.5 to ~ 4 VALEN Analysis Columbia University

RWM Coil Concept for ITER

• Baseline RWM coils located outside TF coils

• Internal RWM coils would be located inside No-wall limit the vacuum vessel behind shield module 7 RWM Coils mounted behind the BSM in but inside the vacuum vessel on the every other port except NBI ports. removable port plugs. (assumes 9 ms time constant for each BSM) • Integration and Engineering feasibility of internal RWM coils is under study. Compact Stellarator Low-n Stability NCSX 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 on previous designs. • Magnetic quasi-symmetry: – good confinement. – link to tokamak physics. • Lower aspect ratio.

3-Period NCSX Plasma and Coil Design NCSX Stability Modeling Predicts Kink Stability up to 6%

PIES Free-Boundary Equilibrium at β = 4.1% W7AS: 〈β〉 ≈ 3.4 % : Quiescent, Quasi-stationary 4 • B = 0.9 T, iota ≈ 0.5 3 vac

2 • Almost quiescent high- β phase, > (%)

! MHD-activity in early medium-β < 1 phase 54022 0

2 ne • In general, β not limited by any )

-3 detected MHD-activity. m 1 20 • IP = 0, but there can be local (10 0 currents 25 . Mirnov B 20 • Similar to High Density H-mode 15 (HDH) 10 (A.U.) 5 • Similar β>3.4% plasmas 0 achieved with B = 0.9 – 1.1 T with 4 either NBI-alone, or combined 3 PNB NBI + OXB ECH heating. 2 • Much higher than predicted β 1 Prad limit ~ 2% Power (MW) 0 0.1 0.2 0.3 0.4 Time (s) MCZ 050218 8 Current-carrying Systems are Subject to Reconnection – “Tearing Modes”

Normal magnetic 6 shear MSE data

4

Safety Factor 2 3/2

0 Reverse 0.0 0.2 0.4 0.6 0.8 1.0 magnetic Minor radius (r/a) shear

• A rational field line can be an O point around which islands form. (- j for normal shear, + j for reverse shear) 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 of axisymmetric toroidal systems. - Drives “neoclassical tearing modes” in normal shear regions due to current depletion in the magnetic islands.

2 2 ' * µ dw Lq ' w * . % g($) Lq 0 1 2 &i & ) , = # " + a1$ %& ) 2 2 , - a2 3 ) , 1.22!nc dt Lp ( w + wc + w ( Lp +

Ohmic current Bootstrap current Polarization current Finite transport correction Theory Accurately Predicts Growth of Neoclassical Tearing Modes (NTM)

R = 3m, Te ~ 5keV R = 1m, Te ~ 1keV 5 8 0.8 Magnetic Shot 18432 !p(meas) 4 6 0.6 3 Theory 4 0.4 !p(meas)

2 ECE w (cm) Measured Island Width W (cm) (cm) 2 0.2 1 Island Width Predicted by Neoclassical Theory 24MW NBI (cm) 0 0 0.0 3.0 3.5 4.0 0.2 0.22 0.24 0.26 0.28 0.30 time (s) Time (s)

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. Replacing Bootstrap Current in Islands Stabilizes Neoclassical Tearing Modes

Steerable Electron Cyclotron Current ECCD Drive wave launcher.

ITER will have ECCD for NTM control. DIII–D Demonstrates NTM Active Stabilization with ECCD

15 Current ×10 (MA) 10 PNB (MW) 5 〈P 〉 (MW) EC Power (MW) • High beta is achieved with pre- 0 NB emptive stabilization of the 2/1 NTM 40 ~ B (n = 1) (G) – Stable operation at the 20 no-wall beta limit for >1 s 0 3 • ECCD is applied before Div. Dα (au) 2 the mode appears 1 – Real-time tracking of the q=2 surface 0 maintains current drive alignment 4 • Tearing mode appears promptly 4 2 li β when ECCD is removed N 0 1.6 |B| (T) 1.5

1.4 0 2000 4000 6000 8000 Time (ms)

DIII–D NATIONAL FUSION FACILITY ECCD in ITER Can Reduce the m/n=2/1 NTM Island

• “Cross-machine bench marking...” • Locking condition from 0-D model R.J. La Haye, et.al., submitted to 3 2 2 w w ω0 τA0 τw 1 + 20 = Nuclear Fusion a ( a) 2* 14–1 τE0 ITER, m/n=2/1, β = 1.84 25 N NO ECCD ★ DIII–D → ITER Unstable 20 NO ECCD ...τw = 3 → 188 ms (J. Bialek) Region Saturated Island (if Beta Maintained and ...f0 = 8 → 0.42 kHz (A. Polevoi) dw/dt 15

/r) if Mode Does Not Lock)

R ...τ = 0.1 → 3.7 s (J. Cordey) τ 12 MW ECCD E0 10 No Modulation ...τA0 = 1.6 → 3.0 μs (Y. Gribov) (K1 = 0.38, F=1) ... w | = 0.081 → 0.025 5 a lock 0 Island growth rate ( — w≈ 5 cm in ITER to lock

-5 — w≈10 cm at f0 = 1.4 kHz 0 5 10 15 20 25 m/n=2/1 Island full width w (cm) 12 MW ECCD 50/50 Modulation (K1 = 0.74, F = 0.5)

253-05/RJL/jy Key Open Issues in Stability

Advanced Tokamak Spherical Torus Compact Stellarator Low-n Kink: Low-n Kink: • Rotation Stabilization Physics & Scaling • Validate no-wall high-beta kink limits • Scale Active Feedback to ITER - n = 1, 2,… • Rotation effects in QS equilibria • Coil Modularity & Failure of “Mode Rigidity” Low-n Tearing: Low-n Tearing: • NTM Active Control Requirements for ITER • NTM control with external magnetic • Quantitative Theory: Seeding physics, island transform + large bootstrap current rotation, ECCD localization & modulation, • Magnetic Island control small island modeling as β and Ip vary.