Extended Lumped Parameter Model of Resistive Wall Mode and The Effective Self-Inductance

Extended Lumped Parameter Model of Resistive Wall Mode and The Effective Self-Inductance M.Okabayashi, M. Chance, M. Chu* and R. Hatcher A. Garofalo**, R. La Haye*, H. Remeirdes**, T. Scoville*, and T. Strait* Princeton Plasma Physics Laboratory * General Atomics ** Columbia Univeristy "Active Control of MHD Stability: Extension of Performance" Workshop under the auspices of the US/Japan Collaboration at Columbia University, 18-0 November 00

OUTLINE - Extended Lumped Parameter Model and RWM Effective Self-Inductance MOTIVATION APPROACH AND ASSUMPTIONS OF EXTENDED LUMPED PARAMETER MODEL - Relation to recent other approaches (models) "RWM EFFECTIVE SELF-INDUCTANCE" - Plasma response is conveniently characterized by one parameter APPLICATIONS - Dispersion relation / Initial value time dependent solution - Resonant magnetic braking, - Non-resonant braking, - Resonant Field Amplification

MOTIVATION - Develop simple description of RWM stabilization process, useful for qualitative and semi-quantitative discussion. - M. Okabayashi, N. Pomphrey and R. Hatcher, N. Fusion. 38(1998) 1607 - M. Okabayashi et al, EPS 00 (Plasma Phy. and Controlled Fusion in press) - The model to be consistent with MHD analysis models developed by various groups - A. Bondeson and D. Ward, Phys. Rev. Lett. 7 (1994)709 - J. Bialek, etal., Phys. of Plasmas, 8 (001) 170 - M. Chu et al., (IAEA 00) - R. Fitzpatrick, Phy. Plasmas 9 (00) 3459 - A. Garofalo, Sherwood meeting (00)

RWM AND ITS FEEDBACK CONTROL CAN BE CONSIDERED AS THE n=1 HELICAL INSTABILITY ANALOG TO THE n = 0 VERTICAL INSTABILITY Marginal Stability Parameter Vertical Drift RWM Feedback B r External Equi. Mag. Field Negative Curv. FIXED M t w g t 0 w 1 1 n c Eddy Flux Decay Induced Image Current Displacement Ideal kink Feedback Helical flux Vertical Helical "Mode Rigidity" is the fundamental assumption

RWM AND ITS FEEDBACK CONTROL CAN BE CONSIDERED AS THE n=1 HELICAL INSTABILITY ANALOG TO THE n = 0 VERTICAL INSTABILITY Feedback B r Vertical Drift External Equi. Mag. Field Negative Curv. FIXED M t w g t 0 w Marginal Stability Parameter 1 1 n c Eddy Flux Decay Induced Image Current Displacement RWM Ideal kink Feedback Helical flux no wall Limit 100 g Growth Rate without Feedback t 0 w 10 1.0 0.1 Vertical Position Control g n=0 t w RWM Control = 7-10 RW M t w Ideal wall Limit g = -4 Vertical Helical Mode Strength Higher elongation "Mode Rigidity" is the fundamental assumption Higher b

Approach and Assumptions of Extended Lumped Parameter Model Include essential RWM physics - Pressure balance and B-normal continuity on plasma surface - Rotation dissipation in adhoc manner, however, consistent with existing theories - Assumptions - Rigid displacement: justifiable based on experiments [M. Okabayashi et al., Phys. Plasmas 8,071(001)] - Cylindrical geometry - One mode excitation - No coupling to other [stable] modes

EXTENDED LUMPED PARAMETER RWM MODEL Explicit Usage of Boundary Conditions - on the Plasma Surface, Wall and Coil ( I p, I w, I c ) - Independent parameters: plasma skin current, wall current, feedback coil current PLASMA x r M pw M pc M wc I I I p w c plasma displacement rotational dissipation plasma plasma rotation (1/x r) d(rx r )/dr a (g + i W ) r=a + N f passive shell Pressure Balance and B-normal Continuity on Plasma Surface fixed displacement approximation m - nq Conveniently formulated "RWM effective self-inductance" V =d/dt h 0 L M M eff pw cp M L M wp w wc M M L pc cw c I p I w I c toroidal number active coil external Y = S M Y i,j displacement A + + = (mf / g t +f ))(a y '(a )/ m y(a ) + /f) b = /f -1 n L effi p + M pwi w+ M pci c =0 L eff = (h - b + a(g + inwf)) L 0 n / (h -b - + a(g + inwf)) standard mutual inductance 0 Y n standard mutual inductance (f=1 <--> no wall limit, b n =1) L eff represents all RWM characteristics currently considered j

EXTENDED LUMPED PARAMETER RWM MODEL Explicit Usage of Boundary Conditions - on the Plasma Surface, Wall and Coil ( I p, I w, I c ) - Independent parameters: plasma skin current, wall current, feedback coil current PLASMA x r M pw M pc M wc I I I p w c plasma displacement rotational dissipation plasma plasma rotation (1/x r) d(rx r )/dr a (g + i W ) r=a + N f passive shell Pressure Balance and B-normal Continuity on Plasma Surface fixed displacement approximation m - nq Conveniently formulated "RWM effective self-inductance" V =d/dt h 0 L M M eff pw cp M L M wp w wc M M L pc cw c I p I w I c toroidal number active coil external Y = S M Y i,j displacement A + + = (mf / g t +f ))(a y '(a )/ m y(a ) + /f) b = /f -1 n L effi p + M pwi w+ M pci c =0 L eff = (h - b + a(g + inwf)) L 0 n / (h -b - + a(g + inwf)) standard mutual inductance 0 Y n standard mutual inductance magnetic decay index N''z + Br_ext =0 : vertical instability (f=1 <--> no wall limit, b n =1) L eff represents all RWM characteristics currently considered j

Extended Lumped Parameter Model kinetic dissipation RWM Dispersion Relation Dispersion Relation with Extended Lumped Parameter Model Expresses Explicitly Geomertical Elements plasma beta wall time constant geometrical term [ c ( g + iw) + a(g +iw) +h - b ) ](g + t -1 k n ) - g [c ( g + iw) + a( g + iw) + h - b - ] (M / L L ) =0 w k n pw w p 0 0 M = (r /r ) m ij i j

Extended Lumped Parameter Model kinetic dissipation RWM Dispersion Relation Dispersion Relation with Extended Lumped Parameter Model Expresses Explicitly Geomertical Elements plasma beta wall time constant [ c ( g + iw) + a(g +iw) +h - b ) ](g + t -1 k n ) - g [c ( g + iw) + a( g + iw) + h - b - ] (M / L L ) =0 w k n pw w p 0 0 Fitzpatrick's " Simple Model of RWM... in Phys. Plasmas 9(00)3459 geometrical term M = (r /r ) m ij i j kinetic plasma beta dissipation* wall time constant* geometrical term

ñ Modeling of Feedback and Rotation Stabilization of the Resistive Wall Mode in tokamaks ñ IAEA Meeting, Lyon, France, October 14 to 19, 00 GENERAL FORMULATION INCLUDING PLASMA ROTATION AND DISSIPATION (M. CHU IAEA 00) The ìresistive wall modeî can be described in terms of circuits on the resisitive wall driven by the plasma or the external coils (7) The plasma responds to the magnetic field B lw on the resistive wall through the relationship The response of the RWM to the coil currents is therefore (8) (9)

Schematics of RWM Feedback Analysis with Toroidal Geometry Resistive Wall Thin Shell Approximation Plasma Surface Resistive Shell (eigen modes) Active Coil ï Introducing "skin current stream fluction" I w : j = Z x I w d(z-z w ) ï normal magnetic field continuity dy /dn (-) = dy /dn (+) = Bn Helical Displacement Ip (t) C wp, C pw, y p y- w B p - B w I w + y w + B w C wc I c ï Ampere's Law: Y(+) - Y(-) = I w ï Faraday's Law [h I w ] = (d /dt (B n ), assuming z = 1 s s B n I w z B n I w (q,f) can be expanded in eigenmodes PEST / VACUUM ds Green's theorem - + Green's theorem Resistive Shell

Modeling of Feedback and Rotation Stabilization of the Resistive Wall Mode in tokamaks IAEA Meeting, Lyon, France, October 14 19, 00 THE LUMPED PARAMETER MODEL AND RFA (M.CHU, IAEA00) For the least stable resistive wall mode and assuming the matrix elements of the energy and dissipation factors are constants, we arrive at a lumped parameter equation the total RWM field at wall external helical current (10) where A is the 'complex amplification factor'. Note that A = 1 in the absence of plasma potential energy with ideal wall d W IW (W) + aiw A = (11) d W (W) + aiw nw potential energy without wall In this equation we see that the dissipation on the wall and the dissipation is the plasma are separated 1. 'No External Coil': homogeneous equation, dissipation in plasma and the resistive wal (1). 'Steady State with External Coil': inhomogeneous equation. dissipation in plasma (13)

dw FORMULATION AND LUMPED PARAMETER MODEL M. Chu Formulation (IAEA 00) Extended Lumped Parameter Model growth rate total B-field on wall 1/A = d W (W) + aiw nw d W (W) + aiw iw resonant field amplification

dw FORMULATION AND LUMPED PARAMETER MODEL M. Chu Formulation (IAEA 00) Extended Lumped Parameter Model growth rate total B-field on wall tw I p + [-L / ( L - M pw / L w )] t I p = 0 eff eff L = (h - b + iw) L / (h - b - + iw) eff 0 n p 0 n 1/A = d W (W) + aiw nw d W (W) + aiw iw resonant field amplification

dw FORMULATION AND LUMPED PARAMETER MODEL M. Chu Formulation (IAEA 00) Extended Lumped Parameter Model growth rate total B-field on wall t I p w + [-L / ( L - M pw / L w )] t I p = 0 eff eff gt w L = (h - b + iw) L / (h - b - + iw) eff 0 n p 0 n 1/A = d W (W) + aiw nw d W (W) + aiw iw - marginal stability b ( no wall limit) gt = 0 <----> L =0 : h = b w eff n.w - Ideal wall limit gt = infinity --> ( L ( b ) - M L ) = 0 w eff I.w pw w 0 (b gt = n.w - b n + iaw) (b - b -)/ (-) w (b - b + iaw) n.w I.w I.w n resonant field amplification

dw FORMULATION AND LUMPED PARAMETER MODEL M. Chu Formulation (IAEA 00) Extended Lumped Parameter Model growth rate total B-field on wall t I p w + [-L / ( L - M pw / L w )] t I p = 0 eff eff gt w L = (h - b + iw) L / (h - b - + iw) eff 0 n p 0 n 1/A = d W (W) + aiw nw d W (W) + aiw iw - marginal stability b ( no wall limit) gt = 0 <----> L =0 : h = b w eff n.w - Ideal wall limit gt = infinity --> ( L ( b ) - M L ) = 0 w eff I.w pw w 0 resonant field amplification (b gt = n.w - b n + iaw) (b - b -)/ (-) w (b - b + iaw) n.w I.w I.w n normalization constant

dw FORMULATION AND LUMPED PARAMETER MODEL M. Chu Formulation (IAEA 00) Extended Lumped Parameter Model growth rate total B-field on wall t I p w + [-L / ( L - M pw / L w )] t I p = 0 eff eff gt w L = (h - b + iw) L / (h - b - + iw) eff 0 n p 0 n 1/A = d W (W) + aiw nw d W (W) + aiw iw - marginal stability b ( no wall limit) gt = 0 <----> L =0 : h = b w eff n.w - Ideal wall limit gt = infinity --> ( L ( b ) - M L ) = 0 w eff I.w pw w 0 resonant field amplification (b gt = n.w - b + iaw) (b - b -)/ (-) w (b - b + iaw) n.w I.w I.w L I + M I + M I =0 eff p pw w pc c normalization constant I = p M pc L eff I c

RWM Time Evolution with Extended Lumped Parameter Model Plus Angular Momentum Balance - Example(1): Magnetic Braking (Rotation slowing down with increasing mode amplitude) Steady target RWM amplitude coil current or error field W / t = (W - W) /t - C Im[dI p (t) di (t) )] 0 m vis c. momentum conf. time ~1/ a ( g + iw) -> 1/(iaW) known as rotation dissipation "induction motor model" 5000.0 0.0 15.0 Experiment with Various External Fields 103158 103156 101876 101877 103154 C Coil79 (A) ROT@q= (khz) Reduced static n=1 error field 0.0 30.0 0.0 10.0 0.0 Critical Rotation for onset of RWM n=1b r (gauss) 100 150 1300 1350 1400 1450 1500 1550 1600 1650 1700 Time (ms)

RWM Time Evolution with Extended Lumped Parameter Model Plus Angular Momentum Balance - Example(1): Magnetic Braking (Rotation slowing down with increasing mode amplitude) Steady target RWM amplitude coil current or error field W / t = (W - W) /t - C Im[dI p (t) di (t) )] 0 m vis c. momentum conf. time ~1/ a ( g + iw) -> 1/(iaW) known as rotation dissipation "induction motor model" 5000.0 0.0 15.0 0.0 30.0 0.0 10.0 0.0 103158 103156 101876 101877 103154 C Coil79 (A) Critical Rotation for onset of RWM n=1b r (gauss) Experiment with Various External Fields ROT@q= (khz) Reduced static n=1 error field 100 150 1300 1350 1400 1450 1500 1550 1600 1650 1700 Time (ms) 0. (khz) 0.0 Ip (arb) Extended Lumped Parameter Model Simulation 1.6 Magnitude of External Field: 0.8,1.0,1., 1.4,1.8 (arb.) ROTATION 1.4 1. 0. Time (ms) 500.0 0.8 1.0 AMPLITUDE

RWM Time Evolution with Extended Lumped Parameter Model Plus Angular Momentum Balance - Example() : Active Magnetic Braking (Regulated rotation slowing down using mode amplitude controlled by feedback) Steady target RWM amplitude coil current momentum conf. time W / t = (W - W) /t - C Im[dI(t) p (di(t) + di(t) )] 0 m vis c.fb c.pre.pro di(t) = G / ( 1 - G )di(t) p fb fb c.pre.pro preprogrammed coil current Simulation Actual coil current feedback gain Plasma rotation Mode amplitude time [t w ]

RWM Time Evolution with Extended Lumped Parameter Model Plus Angular Momentum Balance - Example() : Active Magnetic Braking (Regulated rotation slowing down using mode amplitude controlled by feedback) Steady target RWM amplitude coil current momentum conf. time W / t = (W - W) /t - C Im[dI(t) p (di(t) + di(t) )] 0 m vis c.fb c.pre.pro di(t) = G / ( 1 - G )di(t) p fb fb c.pre.pro preprogrammed coil current Simulation Actual coil current n1tc1curr Preprogrammed coil current feedback gain Experiment c79f Actual coil current Plasma rotation Plasma rotation cernrott6 I D Mode amplitude L > amplitude from MPID (Larry s matrix) Mode amplitude time [t w ] time (ms)

STABLE REGIME PREDICTED BY EXTENDED LUMPED PARAMETER MODEL IS CONSISTENT WITH EXPERIMENTALLY ACHIEVED PARAMETERS ideal wall limit bn / (.4 li) no wall limit Unstable Expected with Extended Lumped Parameter Model 106193/106530/10653/106535 High bn shots in FY 00 * Steady State High bn Equilibrium used for RFA study time Toroidal Rotation (km/s) 008/MHDWKSP/00

RFA AMPLITUDE AND PHASE SHIFT INCREASE WITH INCREASE OF b n TO MARGINAL STABILITY b n db r at wall (G) Field component on /1 surface (G) 4 3 1 0 3 1 0 3.0.0 1.0 0.0 n = 1 (plasma response only) 1550 105444 10543 105439 No wall beta limit stable Time (ms) 1750 0.0 gt w Damping rate Unstable Marginal stability.0 1.0 no-wall b n /b n 1.6 0.5 Amplitude 0.0 Toroidal phase shift Mode amplitude (degrees) DIII D NATIONAL FUSION FACILITY SAN DIEGO noise level 0.0 180 0.6 1.0 1.6 0.6 1.0 1.6 no-wall no-wall b n /b n b n /b n 176-0/jy

1.0 s /S Fitting deviation COMPARISON OF MODEL AND OBSERVATIONS SHOWS THAT DAMPING OF RFA ARISES FROM ROTATIONAL DISSIPATION Fit of Dissipation Coeff. Best Fit A h o =1.3 =1.4 =1.5 0.0 0.0 Dissipation coeff a/t 0.0 0.5 Mode amplitude noise level Observed Plasma Response to Amplitude 0.0 0.0-180 (degrees) Toroidal phase shift 0.0 Damping Rate gt w Best fit to the model with the dissipation parameter a /t =5. Mode amplitude (arb.) Amplitude 0.0 A EXP MODEL Toroidal phase shift -.0 1.0 g t W no-wall b n / b n 1.6 0.0 0.6 1.0 no-wall b / n bn 1.6-180 dissipation / damping rate : a W f/ g t ª 0.5 - exp w 0.6 1.0 1.6 no-wall b n / bn DIII D NATIONAL FUSION FACILITY SAN DIEGO

Modeling of Feedback and Rotation Stabilization of the Resistive Wall Mode in tokamaks IAEA Meeting, Lyon, France, October 14 19, 00 COMPARISON OF LUMPED PARAMETER MODEL WITH EXPERIMENT amplitude (model) phase (model) amplitude (model) amplitude (observed) phase (model) phase (observed) N N (arb) amplitude (observed) A = phase (observed) [d W (W=0) + ew ] + aiw IW [d W (W=0)+ ew ] + aiw nw Qualitative agreement is also obtained by using the present lumped parameter model with different forms for A. This indicates more details should be included in future comparisons (14) 70-0/JY

SUMMARY Extended Lumped Parameter Model includes the essence of RWM physics The present approach is consistent with other approaches Comparison with experimental results provides semi-quantitative discussion of the present RWM understandings.