Princeton Plasma Physics Laboratory Multi-mode analysis of RWM feedback with the NMA Code M. S. Chance, M.Okabayashi, M. S. Chu 12 th Workshop on MHD Stability Control: Improved MHD Control Configurations Columbia University New York, NY Sunday, November 18 Tuesday, November 20, 2007 Work supported by U.S. Department of Contract No. DE-AC02-76-CH03073.
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 1 Introduction The nma code utilizes the full spectrum of mhd modes from dcon. With the resisitve shell included, a complete set of open loop (without feedback) eigenfunctions are calculated from energy integrals that include the dissipation in the shell. These eigenmodes can have varying helicities, both positive and negative. They are included as circuit elements in the circuit equations that describes the feedback process. We have calculated the effects of a variety of feedback coil configurations on iter relevant equilibria, examining the mode structures before, and after the stabilization of the rwm. The efficiency of the feedback depends strongly on the phasing of the coils with respect to each other, and to the rwm. The efficiency is also dependent on the interaction of the modes with each other. The RWM is deformed (non-rigid) during the feedback process. Stable modes can perhaps be driven by imperfect coils. Experimental demonstration of these effects could be interesting.
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 2 Normal Mode Approach to Feedback Stabilization of the RWM is based on the Extended Principle W p + K + W v +D w + E c = 0 Perturbed Plasma Kinetic Perturbed Vacuum Dissipation in Resistive Wall Input from External Coil General static plasma equilibrium Chu, Chance, Glasser, Okabayashi, NF, 43, 441 (2003) General external modes W ithout feedback, the energy principle is reduced to the self-adjoint expression W p + K + W v +D w + E c = 0 Perturbed Plasma DCON Perturbed Vacuum VACUUM Dissipation in Resistive Wall Tank A set of normal modes can satisfy this relationship and they are the the open loop eigenfunctions (R W Ms) 3 chu ITPA Meeting 2007
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 3 A set of Equilibria in ITER Geometry Studied with an up-down Symmeterized Approximate Wall A Set of Equilibira Studied with an up-down Symmetrized Approximate Wall ITLocation E R and Double (L iu) Open Wall Loop Growth Rates Plasma Z A pproximate Symmetrized Wall 4 R The wall is an up-down symmetric approximation to the inner wall of the iter wall structure. The open-loop growth rates computed by the nma code agree with those obtained by mars-f.
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 4 Example of the open Loop Eigenfunction (the Resistive Wall Mode) in ITER T he original proposed (Gribov) coil geometry relative to the plasma is similar to C-Coil (Gribov, private communication) 5 chu
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 5 The Coils External to the Plasma Excite the RWM s Through the Feedback Logic Amplitude of RWM Open-L oop Growth R ate E xcitation Matrix Coil Current T ime Constant of Coils i t I c t + I c c i i = = G c l l Gain Matrix c I c E i c F c l ({ i}, { I c }) Sensor Matrix With feedback, the growth rate is then determined by D(s) = si E GF (s + 1 )I c 6
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 6 Plasma Wall Geometry and Coil Set Models External Port Coils (8.905 1.632) (9.075-0.327) 36 E xternal Coils (not yet studied) Z(m) Internal Port Coils (6.171 4.178)(7.249 3.90) (7.715-3.430)(5.702-4.951) Original Gribov Proposal (11.282 3.913) (11.282-3.013) Internal B lanket Coils (7.736 3.406) (8.633 1.867) (8.797-0.626) (8.293-2.414) R(m) 9 chu ITPA Meeting 2007
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 7 Typical Gain vs. Phase Behavior Modeling RWM Experiments with NMA We have used the NMA code to calculate the marginal gain versus the phasing of the feedback coils that is needed to stabilize the RWM in the ITER device. marginal gain Mid_plane coil only with off_midplane The mid-plane coil is optimized to the mode here off_midplane coil toroidal phase (rad)
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 8 Mode Structures We have also looked at the mode pattern on the resistive shell with and without the feedback, showing the mode deformation due to the coils Bn pattern on the plasma surface Mid_plane coil only at marginal gain (g=6) Mid_plane coil + Off_midplane at marginal gain (g=10) poloidal direction top Eigenmode no-feedback bottom toroidal direction
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 9 Effect of Phase Mismatch to the RWM - RWM Feedkack with with Mid-Plane Coil-only (ITER port-plug)- Interaction between Normal RWM and Shallow Stable Branches Destabilzes Overall Feedback when the Feedback Phase Slipps Feedback phase at optimized location Feedback phase -62 deg shift Feedback phase shift -72 deg shift Normal RWM 0.0 Negative Helicity branch Other stable branches Stable 10 40 10 40 10 40 Negative Helicity branch remains stable Negative Helicity branch acts as the positive feedback in phase relation and becomes unstable /u3/okabay/mm_code/iter.30b_all/iter.30b_c_w/090
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 10 Modes Coupling with the C-coil # 199: RWM Mode non-rigidity in low/high rotation plasmas (M. Chance, M. Chu,A. Garofalo..) growth rate 1.5 1.0 0.5 0.0-36 - ITER Mid-PLANE COIL RWM FEEDBACK - The mode-coil phase mismatch requires higher gain. The negative helicity mode contribution is inhibitive 0 φ=-72 +48-60 -48-0.5 0 10 20 30 40 50 gain Mode pattern on plasma surface tor. angle normal RWM branch coupled with Negative helicity branch φ (degrees) : Coil pattern toroidal shift relative to the optimized toroidal coil geometry pol. angle
Multimode analysis of RWM feedback with the NMA code Columbia University, New York, NY, November 18 20, 2007 11 Conclusions The efficiency of the feedback depend strongly on the phasing of the coils with respect to each other and to the rwm. The efficiency is also dependent on the interaction of the modes with each other. The RWM is deformed (non-rigid) during the feedback process. This can be minimized by optimizing the coupling to the rwm Stable modes can perhaps be excited by imperfect coils and overdriving the feedback system. Experimental demonstration of these effects could be interesting.