1. Demonstration of tearing mode braking and locking due to eddy currents in a toroidal magnetic fusion device B.E. Chapman (University of Wisconsin, USA) R. Fitzpatrick (University of Texas, USA) D. Craig (University of Wisconsin, USA) P. Martin and G. Spizzo (Consorzio RFX, Italy)

2. Introduction Theory introduced for tokamak and RFP in late 1980’s: electromagnetic torque from eddy currents brakes mode rotation Theory later expanded: viscous restoring torque resists braking torque Possibly important in present & future devices But have been few tests of the theory Some plasmas in the MST RFP exhibit m = 1 tearing mode with large amplitude and deceleration Has allowed detailed tests of braking theory: theory and experiment agree well [Phys. Plasmas, May ‘04]

3. Outline Mode braking data Examination of previously established causes of locking in MST Application of eddy-current braking theory

4. Innermost resonant m = 1 mode sometimes becomes large: quasi-single-helicity (QSH) mode spectra F = B  (a)/

5. When mode grows large, it also decelerates Other m = 1 modes also decelerate Bulk plasma decelerates as well Equilibrium essentially unaffected

6. QSH mode velocity a relatively simple function of QSH mode amplitude when mode amplitude becomes large

7. Braking due to any pre-established causes? well below the usual slow-rotation/locking threshold Nonlinear mode coupling plays no role [Phys. Plasmas, May ‘04] Error field? Vertical cut in MST’s shell can be significant source of error Error torque  (b error b mode ), and b mode (QSH) is large Varied error field to test for error effect...

8. QSH mode amplitude and velocity vary little comparing small and large m = 1 error fields Shot-ensembled data from F = 0 plasmas with similar plasma current and density

9. Braking curve not significantly affected by error field variations

10. So, we tested theory of braking by eddy currents

11. Some history of braking theory/comparison to experiment Theory first proposed to account for locking with single large tearing mode in tokamak and RFP [Nave and Wesson, EPS 1987; Hender, Gimblett, and Robinson, EPS 1998] Consistency with tokamak [Snipes et al., 1988] and RFP [Brunsell et al., 1993] expt. data reported Accounted for "forbidden bands" of rotation in a tokamak [Gates and Hender, 1996] Theory augmented with inclusion of viscous restoring torque for tokamak [Fitzpatrick, 1993] and RFP [Fitzpatrick et al., 1999] Mode amplitude locking threshold in RFP consistent with theory [Fitzpatrick et al., 1999; Yagi et al., 1999 & 2001; Malmberg et al., 2000] Theory without viscosity did not account for recent tokamak braking data [Hutchinson, 2001]

12. Basics of the theory, for initially rotating tearing mode Theory differs in detail for tokamak and RFP, but fundamentally generic Tearing mode, b mode (m,n) induces eddy currents in conducting shell(s) surrounding the plasma Eddy currents cause current sheet, j sheet (m,n) near r s Local j sheet x b mode braking torque results Local deceleration countered by viscous restoring torque from bulk plasma With significant viscosity, j x b torque must brake entire plasma to brake mode (m,n)

13. MST provides simple geometry for application of mode braking theory Single aluminum shell, 5 cm thick Circular poloidal cross section R/a = 150 cm/52 cm

14. Theory predicts well the experimental mode deceleration Only adjustable parameter in theory is  M  1/viscosity Adjusted such that curves coincide at locking Shape of theoretical curves depends on other measured data

15. Theoretical prediction of  M well constrained

16. Modeled  M ’s consistent with experimental data Experimentally,  M ~ 1.5 ms in standard H 2 MST plasmas (one measurement) MST standard   ~  1 - 2 ms over entire range of parameters As with many tokamaks, we expect that   ~  1 - 2 ms as well Modeled   (D 2 ) >   (H 2 ) also consistent with experimental expectation: larger central n 0 observed with H 2, hence larger CX momentum loss

17. Summary Growth to large amplitude of single m = 1 mode in MST leads to global braking and locking Apparently explained by eddy currents in MST’s shell: Theory reproduces (dynamical) experimental braking curves Theoretical and experimental values of  M comparable Certainly bolsters confidence for braking theory as applied to RFP, and perhaps the tokamak... as well