Plasma Stability in Tokamaks and Stellarators Gerald A. Navratil GCEP Fusion Energy Workshop Princeton, NJ 1- May 006
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: B /µ o Pressure: nt Three primary limiting β phenomena: Long Wavelength Ideal Modes: n = 0, 1,, 3 Short Wavelength Ideal Modes: n Long Wavelength Resistive Modes: n = 1, 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 0%. 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 0% 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,, 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 AT HIGH β N P fus γ ε cur Q ss = eff β N B 3 aκ P CD nq ( 1 ξ A q β ) N β Power Density T ε 1 + κ DIII D NATIONAL FUSION FACILITY SAN DIEGO Current Limit q* = 4 Advanced Pressure Conventional Tokamak Limit Tokamak β N = 5 β N = 3.5 Equilibrium Limit εβ p Bootstrap Current High power density high β T Large bootstrap fraction high β p Steady state high β N β N power density bootstrap current ( 1 + κ β T β p )β N β N = β T /(I/aB) 130 0/TST/wj
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: τ w
perturbed magnetic energy δw = 1 3 d x {ε c δb + ε c ( B ) (ξ δb) p Foundation of Kink Mode Stability Built on Energy Principle δw Stability Analysis 1957 Bernstein, Frieman, Kruskal, Kulsrud o } + ( ξ)(ξ p o ) + γp ( ξ) o pressure driven - destabilizing 1 δw = d x ε c δb current driven - destabilizing o plasma compression 3 o v vacuum perturbed magnetic energy If δw + δw < 0 mode is unstable p v
BASIC KINK MODE Long wavelength mode driven by pressure & current gradient Cylindrical k ~ π/l Toroidal: low n = 1 Unstable when δw p + δw v < 0 Dispersion Relation: γ K + δw p + δw v = 0, where K is kinetic fluid mass Define Γ = [δw p + δw v]/k ~ [v Alfvén /L]
IDEAL WALL STABILIZES THE KINK MODE Ideal wall traps field in vacuum region and restoring force stabilizes the kink EXTERNAL Kink: Unstable when δw p + δw d v < 0 Note: δw d v > δw v Dispersion Relation: γ - Γ + [δw d v-δw v]/k = 0 Critical Wall Distance, d c, where kink stable for d < d c : simple [δw d v-δw v]/k parameterization with d: γ - Γ [1 d c /d]/k = 0
KINK MODE IS STABILIZED BY IDEAL WALL 0 = γ Γ (1 d c) } Ideal Stability d γ / Γ 0.08 0.06 0.04 0.0 ideal mode stable Ideal Instability γ Γ ideal mode unstable 0.00 0.0 0.5 1.0 1.5.0 Plasma-Wall Separation, d/d c 309-04/GAN/rs
RESISTIVE WALL LEAKS STABILIZING FIELD: τ W Stabilizing field decays resistively on wall time scale τ w ~ L/R: dψ w /dt = - ψ w /τ w Quadratic kink: γ - Γ [1-d c /d] = 0 coupled to slow flux diffusion γψ w = - ψ w /τ w : τ w >> τ Alfvén Cubic Dispersion Relation with new slow root the RWM: γ - Γ [1-(d c /d) γτ w /(γτ w + 1)] = 0
KINK MODE GROWTH IS SLOWED BY RESISTIVE WALL 0 = ( γ τ γ Γ 1 d w ) c d γ τ w + 1 } Ideal Stability } Resistive Wall γ / Γ 0.08 0.06 0.04 0.0 Real ω 0 γ τ w 1 Resistive Wall Mode ideal mode unstable 0.00 0.0 0.5 1.0 1.5.0 Plasma-Wall Separation, d/d c Resistive wall mode (RWM) is unstable Mode structure similar to ideal external kink Mode grows slowly: γ ~ τ 1 w 309-04/GAN/rs
RWM STABILIZED IN DIII-D BY ROTATION FOR MANY WALL-TIMES, τ W Normalized plasma pressure, β N, exceeds no-wall stability limit by up to 40% n = 1 mode grows (γ ~ 1/τ W ) 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: (γ + iω) - Γ [1-(d c /d) γτ w /(γτ w + 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 < d c new slow RWM root can be damped with fast stable kink mode roots tied to rotating plasma with usual ordering: τ w -1 << Ω << v Alfvén /L Why is RWM Slow Root Stabilized? kink energy release < dissipation loss of RWM slowed by wall in flowing plasma
γ / Γ 0.08 0.06 0.04 0.0 KINK MODE GROWTH IS SLOWED BY RESISTIVE WALL AND STABILIZED BY PLASMA ROTATION (γ + iω) Γ 0 = + } Ideal Stability Stable Gap ω τ w 1 Resistive Wall Mode Plasma Mode γ Γ 0.00 0.0 0.5 1.0 1.5.0 Plasma-Wall Separation, d/d c Γ (d c /d)γ τ w γ τ w + 1 } Resistive Wall + (γ + iω) ν DIS } Plasma Dissipation Resistive wall mode (RWM) is unstable Mode structure similar to ideal external kink Mode grows slowly: γ ~ τ 1 w Dissipation + rotation stabilizes RWM Mode nearly stationary: ω ~ τw 1 << Ω plasma 309-04/GAN/rs
SUSTAINED ROTATION ABOVE CRITICAL VALUE RELIABLE OPERATION ABOVE THE NO-WALL LIMIT β N 4 3 106535 107603 Feedback control of NBI power keeps β N below stability limit (107603) 1 no wall β N (.4l i) 0 1 Rotation (khz) at q~ 6 0 1000 1500 000 500 3000 Time (ms) No other large scale instabilities encountered (NTM, n= RWM,... ) Ideal n=1 kink observed at the wall-stabilized β limit Toroidal Angle 300 00 100 δb p β N ~ β no-wall N β = 3.7% τ g ~ 300 µs << τ wall DIII D NATIONAL FUSION FACILITY SAN DIEGO 111.0 111.5 11.0 11.5 Time (ms) T rot ~1 ms < τ wall 58 0/EJS/wj
MARS PREDICTIONS OF Ω crit τ A IN QUALITATIVE AGREEMENT WITH MEASUREMENTS ON DIII-D AND JET sound wave In DIII-D Ω crit τ A ~ 0.0 with weak β dependence In JET Ω crit τ A ~ 0.005 with weak β dependence Both damping models predict Ω crit within a factor of
MODE FREQUENCY AND DAMPING CANNOT BE FIT SIMULTANEOUSLY 0 - -4.0 1.0 Growth rate γ RWM τ W experiment(ωτ ~0.0) A kinetic sound wave (κ = 0.5) Mode rotation frequency ω RWM τ W Both damping models predict γ too low RWM Kinetic damping predicts mode frequency ω RWM Further work on damping [e.g. neoclassical viscosity] models being explored DIII D NATIONAL FUSION FACILITY S A N D I E G O 0.0 0.0 0. 0.4 0.6 0.8 1.0 C β
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 Ω φ / ω A > 1/4q (Columbia Univ.) Experimental Ω crit / ω A suggests scaling ε / q 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 ω sound / ω A distinguish between ω s and ω A scaling NSTX DIII-D shifted & scaled 1.1 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 the radial flux from MHD mode at wall sensor Feedback cancels the flux from MHD mode 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 Normalized Growth Rate γτ w 10 10 1 No Feedback Ideal kink VALEN code: - DCON MHD stability - 3D geometry of vacuum vessel and coil geometry -1 10-10 Resistive Wall Mode: Open loop growth rate τ w is the vacuum vessel flux diffusion time (~ 3.5 ms) 0 0. No-Wall Limit 0.4 C β 0.6 0.8 1.0 Ideal-Wall Limit DIII D NATIONAL FUSION FACILITY S A N D I E G O 69-04/MO/jy
FEEDBACK WITH I-COILS IN DIII-D INCREASES STABLE PLASMA PRESSURE TO NEAR IDEAL-WALL LIMIT C-coil stabilizes slowly growing RWMs Normalized Growth Rate γτ w 10 10 1-1 10-10 0 0. No-Wall Limit No Feedback External C-coils Accessible with External C-coils 0.4 C β 0.6 Ideal kink 0.8 1.0 Ideal-Wall Limit External C-Coil: - Control fields must penetrate wall - Induced eddy currents reduce feedback τ w is the vacuum vessel flux diffusion time (~3.5 ms) DIII D NATIONAL FUSION FACILITY S A N D I E G O 69-04/MO/jy
I-coil stabilizes RWMs with growth rate 10 times faster than C-coils Normalized Growth Rate γτ w FEEDBACK WITH I-COILS IN DIII-D INCREASES STABLE PLASMA PRESSURE TO NEAR IDEAL-WALL LIMIT 10 10 1-1 10-10 0 0. No-Wall Limit No Feedback External C-coils Internal I-coils Accessible with External C-coils 0.4 C β 0.6 Accessible with Internal I-coils Ideal kink 0.8 1.0 Ideal-Wall Limit Internal I-Coils: - Improved coil/plasma coupling - Improved spatial match to RWM field structure τ w is the vacuum vessel flux diffusion time (~3.5 ms) DIII D NATIONAL FUSION FACILITY S A N D I E G O 69-04/MO/jy
FEEDBACK EFFICACY DEMONSTRATED BY GATING OFF THE GAIN FOR 0 MS AT TIME OF EXPECTED RWM ONSET (gauss) 35 0 3 1 0 0 10 Feedback Gain β N n=1 δb r 104118 104119 0 100 1400 1600 1800 000 Time (ms) Without feedback, slow Ip ramp rate (0.5 MA/s) destabilizes slowly growing RWM With feedback, beta collapse avoided (cm) 1 0 Relative Displacement (SXR) 104119 n=1 mode starts up during feedback off period, stabilized after feedback is turned back on (ka) -1 1.4 Feedback Current (C79) 1. 1.0 1480 1490 1500 1510 150 Feedback OFF 1530 Time (ms) 1540 104119 1550 n=1 mode detected on poloidal field probes and SXR arrays, decoupled from driver coils DIII D NATIONAL FUSION FACILITY S A N D I E G O
FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH C β AT ROTATION BELOW CRITICAL LEVEL PREDICTED BY MARS Trajectories of plasma discharge in rotation versus C β No feedback plasma approaches limit and disrupts ideal wall limit C β C-coil feedback plasma crosses limit & reaches higher pressure no wall limit 1.0 Unstable (without feedback) C-COIL MARS prediction Stable (without feedback) NO FEEDBACK time 0.0 0.0 50 100 150 Rotation (km/s) DIII D NATIONAL FUSION FACILITY S A N D I E G O 69-04/MO/jy
FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH C β 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 I-coil feedback plasma reaches near zero rotation C β Unstable MARS prediction Stable (without feedback) MARS /VALEN prediction with measured amplifier time response for zero rotation no wall limit I-COIL ZERO ROTATION C-COIL NO FEEDBACK time 0.0 0.0 50 100 150 Rotation (km/s) DIII D NATIONAL FUSION FACILITY S A N D I E G O 69-04/MO/jy
FEEDBACK WITH INTERNAL CONTROL COILS HAS ACHIEVED HIGH C β AT ROTATION BELOW CRITICAL LEVEL PREDICTED BY MARS Combination of low rotation and feedback reaches C β is the ideal wall-limits ideal wall limit 1.0 Unstable I-COIL MARS prediction Stable (without feedback) C β MARS /VALEN prediction with measured power supply time response for zero rotation no wall limit I-COIL ZERO ROTATION C-COIL NO FEEDBACK time 0.0 0.0 50 100 150 Rotation (km/s) DIII D NATIONAL FUSION FACILITY S A N D I E G O 69-04/MO/jy
RWM FEEDBACK ASSISTS IN EXTENDING β n ~4 ADVANCED TOKAMAK DISCHARGE MORE THAN 1 SECOND 4.0 βn Feedback current (ka) 0.0 00 Plasma Rotation (km/s) 0.0 30.0 n=1 δb p Mode Amplitude (gauss) 0.0 F.B on No Feedback No Feedback With Feedback Estimated no-wall limit 119666/119663 Feedback coil current amplitude With Feedback 1000 1500 000 500 Time (ms) DIII D NATIONAL FUSION FACILITY S A N D I E G O High performance plasma approaches β ~ 6% Without feedback plasma disrupts due to RWM 69-04/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 q min New divertor will help density control at high triangularity 1004 1.5 1004.500 6 5 4 3 β T (%) β N 6l i 4l i 1 0 1000 1500 000 500 Time (ms) 3000 3500 4000 1.0 0.5 0.0 80 60 40 0 0 6 4 Pressure (10 5 Pa) J (A/cm ) Safety Factor 0 0.0 0. 0.4 0.6 0.8 1.0 ρ DIII D NATIONAL FUSION FACILITY 093-05/EJS/rs
Applying Internal RWM Feedback Coils to the Port Plugs in ITER Increases β limit for n = 1 from β N =.5 to ~ 4 RWM Coil Concept for ITER VALEN Analysis Columbia University 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.
NCSX 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 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 (100 m-3) <!> (%) Power (MW) (A.U.) 4 3 1 0 1 5 0 0 15 10 <!> 540 ne Mirnov Ḃ 5 0 4 3 P NB 1 P rad 0 0.1 0. 0.3 0.4 Time (s) MCZ 05018 8 B = 0.9 T, iota vac 0.5 Almost quiescent high- β phase, MHD-activity in early medium-β phase In general, β not limited by any detected MHD-activity. I P = 0, but there can be local currents Similar to High Density H-mode (HDH) Similar β>3.4% plasmas achieved with B = 0.9 1.1 T with either NBI-alone, or combined NBI + OXB ECH heating. Much higher than predicted β limit ~ %
Current-carrying Systems are Subject to Reconnection Tearing Modes Normal magnetic shear Safety Factor 6 4 MSE data 3/ Reverse magnetic shear 0 0.0 0. 0.4 0.6 0.8 Minor radius (r/a) 1.0 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 nonlocal 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. µ 0 dw 1.! nc dt = #" + a L 1$ 1 q % & L p ' ) ( w w + w c *., - a &i % & g($) ' + w 3 ) ( L q L p *, + Ohmic current Bootstrap current Polarization current Finite transport correction
Theory Accurately Predicts Growth of Neoclassical Tearing Modes (NTM) W (cm) 5 R = 3m, T e ~ 5keV Magnetic 4 3 Theory ECE 1 4MW NBI 0 3.0 3.5 4.0 Time (s) w (cm) 8 6 4 R = 1m, T e ~ 1keV time (s)! p(meas) Shot 1843 0.8 0.6 Measured Island Width (cm) 0. 0 Island Width Predicted by Neoclassical Theory (cm) 0.0 0. 0. 0.4 0.6 0.8 0.30 0.4! p(meas) Bootstrap current + normal shear drives NTM s. - Agrees to factor of ~ 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 ECCD Steerable Electron Cyclotron Current Drive wave launcher. ITER will have ECCD for NTM control.
DIII D Demonstrates NTM Active Stabilization with ECCD 15 10 P NB (MW) Current 10 (MA) 5 0 40 0 P NB (MW) ~ B (n = 1) (G) EC Power (MW) High beta is achieved with preemptive stabilization of the /1 NTM Stable operation at the no-wall beta limit for >1 s 0 3 1 0 4 Div. D α (au) 4 l i β N ECCD is applied before the mode appears Real-time tracking of the q= surface maintains current drive alignment Tearing mode appears promptly when ECCD is removed 0 1.6 1.5 B (T) 1.4 0 000 4000 6000 8000 Time (ms) DIII D NATIONAL FUSION FACILITY
ECCD in ITER Can Reduce the m/n=/1 NTM Island Cross-machine bench marking... R.J. La Haye, et.al., submitted to Nuclear Fusion Locking condition from 0-D model w 3 w ω 0 τ A0 a (1 + 0 a)= * 14 1 τ w τ E0 Island growth rate (τ R /r) dw/dt 5 0 15 10 5 0 ITER, m/n=/1, β N = 1.84 NO ECCD Unstable Region 1 MW ECCD No Modulation (K 1 = 0.38, F=1) NO ECCD Saturated Island (if Beta Maintained and if Mode Does Not Lock) -5 0 5 10 15 0 5 m/n=/1 Island full width w (cm) 1 MW ECCD 50/50 Modulation (K 1 = 0.74, F = 0.5) DIII D ITER...τ w = 3 188 ms (J. Bialek)...f 0 = 8 0.4 khz (A. Polevoi)...τ E0 = 0.1 3.7 s (J. Cordey)...τ A0 = 1.6 3.0 μs (Y. Gribov)... w = 0.081 0.05 a lock w 5 cm in ITER to lock w 10 cm at f 0 = 1.4 khz 53-05/RJL/jy
Key Open Issues in Stability Advanced Tokamak Spherical Torus Low-n Kink: Rotation Stabilization Physics & Scaling Scale Active Feedback to ITER - n = 1,, 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 Compact Stellarator 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.