Response of a Resistive and Rotating Tokamak to External Magnetic Perturbations Below the Alfven Frequency

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1 Response of a Resistive and Rotating Tokamak to External Magnetic Perturbations Below the Alfven Freuency by M.S. Chu In collaboration with L.L. Lao, M.J. Schaffer, T.E. Evans E.J. Strait (General Atomics) Y.Q. Liu (Culham U.K.) M.J. Lanctot, H. Reimerdes (Columbia U.) Y. Liu (DUT), T.A. Casper, Y. Gribov (ITER) Presented at Twenty-Third IAEA Fusion Energy Conference Daejeon, Republic of Korea October 11-16, 21

2 Introduction and Motivation Experiments in DIII-D [1] and JET showed ELMs stabilized by external fields Vacuum field model [2] predicts outside (> 88%) flux surfaces becomes stochastic Supported by splitting of heat deposition footprint on divertor plate Puzzle: edge electron temperature gradient not reduced indicating that heat transport not much enhanced in direct contradiction with modeling of edge transport[3] Only density gradient reduced. Need improved magnetic field line structure models Examine different plasma models with MARS-F code to show that field lines would not become stochastic This may lead to significantly reduced transport The formulation and approach can be readily applied to ITER and external excitation at higher freuencies [1]Evans et al. Nucl. Fusion, 45, 595 (25) [2]Schaffer et al. Nucl. Fusion, 48, 244 (28) [3]Joseph et al. J.Nucl. Mat., , 591 (27)

3 Outline Plasma responses to perturbation magnetic fields can be conveniently formulated to include ideal, resistive, and rotation effects using an extended energy approach MARS-F provides a comprehensive tool to model plasma response with resistive and rotation effects based on an extended energy approach Non-resonant plasma response varies weakly with resistivity and are similar to vacuum response Resonant plasma response is different significantly from vacuum but similar to ideal response Below Alfven freuency excitation spectrum is independent of coil geometry but excitation strength depends critically on configuration

4 Response of Ideal Plasma is Maximization of Response with Total Energy Conservation Total energy Wg = K +W p +W v +E c = Kinetic energy Plasma potential energy Perturbed vacuum energy Energy supplied by coils For ideal plasma or vacuum response, without the external coils, K +W p +W v = i B + i B i i The external coil excites the external kinks to amplitudes given by B = i a i plasma surface B i + c a i = B i c ds /( 2 ) c is magneto-static potential due to the external coil with c n = at plasma surface Chu et al NF 43, 441,23

5 Response of (Force Free) Resistive Plasma Reveals Specific Plasma Effects Formulation for ideal plasma can be easily extended to resistive plasma W pt ( A +, A)= dv With solution [ A + AA + AsB A] B s = A s = P(B ) 1 A nm (V V nm )(B A) nm exp(in + im ) nm is amount of reconnected flux at the resonant surface and serves as constraint in the maximization. (B A) nm = No resonances,, resistive plasma responds similar to ideal Paramagnetic and higher than vacuum ~ 1//(Aspect Ratio) With resonances, nm limits the response at the rational surfaces Diamagnetic (screening) Chu et al PF B1,62,1989

6 Mars Solves Euler Euations for Free Energy Functional with Inclusion of Resistivity and Rotation E. Of Motion t = ~ = i, ( ~ +in )v 1 = pi = in v 1 pt1+ j1 B + J b1 1 U(v1 ) Ohm s Law Ampere s Law BG Pressure Density E. Anisotropic P. ( ~ +in )b 1 = (v1 B j1 )+(b1 )R 2 j 1 = b1 p 1 = ( 1 ) p p 1 ( ~ +in ) 1 = (v 1 ) v 1 p t1 = p1 I + p b b + p ( I b b ) resistivity Plasma Rotation Perturbed uantities are represented by Fourier components in poloidal direction p and p are non-local anisotropic pressures induced by the MHD motion Liu et al PF 7,3681,2

7 SURFMN Computes Vacuum B Field of Real 3d Coils Showing Edge Stochasticity w mn = 16 ' ~ mn ~ mn = S 2 2 m B mn B mn = 2 B exp(i(m pest n)da da Results from SURFMN have been found to correlate with experiments But imply outer 12% of flux surfaces becomes stochastic Schaffer et al, NF,48,244, 28

8 Analytic Model for A Coil Set Verification of Vacuum Results - Blue discrete coils Z l l - Red analytic model Top current ring Side current panel μ B = 4 dl,ds j dl r ds r 2 Bottom current ring Coordinate system local to the coil Y X l l = 1 r current density j = Kl Current potential K = cos(n)[h (l l 1 ) H (l l 2 )] ( s l l )

9 Analytic Perturbed Magnetic Fields Agree with Both SURFMN and MARS-F n=-3, odd-connection n=-3, even-connection Poloidal harmonic m Pitch-resonant m=n line in white Poloidal harmonic m The vertical axis is suare root of normalized euilibrium flux With vacuum field assumption, the computed perturbed magnetic field has substantial amplitudes at the pitch-resonant surfaces, especially for the even parity connection Computed from analytic model, agrees with SURFMN and MARS-F with vacuum assumption

10 Pitch-Non-Resonant Side Plasma Response Relatively Independent of Resistivity n=3 VACUUM S= B n B n 3 3. m= m= Experimental rotation profile utilized, / A = 2.7% at magnetic axis On the pitch-non-resonant side (positive m s), the distribution of the amplitudes are almost the same for the range of resistivity ( S=1 6 up to ideal)

11 Pitch-Resonant Side Response is Different from Vacuum But Similar to Ideal n=3 IDEAL S= B n B n 3 3. m= m= Experimental rotation profile utilized, / A = 2.7% at magnetic axis On pitch resonant side (-m s), the distribution of the amplitudes is uite similar to the ideal case, showing suppression at the rational surfaces, except at the very edge (< 2-3%) Similar to conclusion by Izzo and Joseph (N.F. 48, 1154, 28)

12 B n Plasma Responses for A Range of Resistivity Remain Close to Ideal Pitch-Resonant-Side =1/3 n=3 Solid m=-1 Dotted m=-12 Blue ideal 6 Magenta S=1 8 6 Yellow S=1 7 Green S=1 6 B n Pitch-Non-Resonant Side Solid m=1 Dotted m=12 S=1 6 to S= Experimental rotation profile utilized, / A = 2.7% at magnetic axis On pitch-resonant side (-m s), amplitudes with different S are similar to that of ideal response. It is suppressed at the resonant surfaces, except at the very edge On pitch-non-resonant side (+m s), amplitudes with different S s all similar, and are just slightly different from vacuum vac vac vac

13 Study and Compare Effect of Euilibrium on Plasma Response Experimental Case Circle Ellipse.6 p.3 / A /

14 Resonant and Non-resonant Circular Plasma Responses Similar to Shaped Geometry Pitch-Resonant Side =8/3 Solid m=-8 Dotted m=-9 Blue ideal 3 Magenta S=1 8 3 Yellow S=1 7 B n Green S=1 6 B n 2 2 S=1 6 vac n=3 Pitch-Non-Resonant-Side Solid m=8 Dotted m=9 vac Experimental rotation profile utilized, / A = 2.7% at magnetic axis On resonant side (-m s), amplitudes with different S are similar to that of ideal response. It is suppressed at the resonant surfaces, except at the very edge. On non-resonant side (+m s), amplitudes with different S s all similar, and are just slightly different from vacuum S=1 6 to

15 Elliptical Plasma Shows Similar Responses Pitch-Resonant-Side =9/3 n=3 Pitch-Non-Resonant Side Dotted m= Solid m=-9 Blue ideal B B n n Magenta S=1 8 4 Yellow S=1 7 vac 4 Solid m=9 Dotted m=13 S=1 6 to 2 2 vac On pitch resonant side (-m s), amplitudes with different S are similar to that of ideal response, suppressed at pitch resonant surfaces. / A = 2.7% at magnetic axis On pitch non-resonant side (+m s), amplitudes with different S s all similar, and are just slightly different from vacuum

16 Application to ITER Perturbations By Error Field Correction Coils The error field correction side, top and bottom coils produce correction fields to counter effect of error fields on the plasma The displacement shows stronger helicity content than perturbed magnetic field.

17 Application to Higher Freuencies Excitation Spectrum Independent of Coil Geometry Villard et al CPR 4,95,1986 Kerner et al JCP 142, 271,1998 K = Kinetic Energy P = Poynting Flux Q = K P Q a =Q for antisymmetric connected Coils Q s =Q for symmetric connected Coils / A The excitation spectrum is independent of configuration of external antenna The strength of excitation depends critically on configuration

18 Conclusion Plasma response to external field can be viewed as a variational problem with constraint of total perturbed energy Without resonances plasma response is paramagnetic and can be approximated by vacuum response At resonances, small resistivity and plasma rotation leads to near ideal plasma conditions and screening and suppression of normal magnetic field Plasma response in ELM suppression experiments is close to being ideal with possible stochasticity at very edge ~2% n Mars-F can be applied to ITER and higher freuencies to study plasma perturbation by external coils

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