Demonstration of tearing mode braking and lockingdue 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)

Introduction

• Theory introduced for tokamak and RFP in late 1980’s: electromagnetictorque 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 largeamplitude and deceleration

• Has allowed detailed tests of braking theory: theory and experimentagree well [Phys. Plasmas, May ‘04]

Outline

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

Innermost resonant m = 1 mode sometimes becomes large:quasi-single-helicity (QSH) mode spectra

• F = Bφ(a)/<Bφ>

When mode grows large, it also decelerates

• Other m = 1 modesalso decelerate

• Bulk plasmadecelerates as well

• Equilibriumessentially unaffected

The n > nQSH modes also decelerate

• Deceleration of m = 1 modesconsistent with decelerationof bulk plasma

• Plasma and modes decelerateat about same rate

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

Braking due to any pre-established causes?

• <ne> 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 ∝ (berrorbmode), and bmode(QSH) is large• Varied error field to test for error effect...

QSH mode amplitude and velocity vary little comparingsmall and large m = 1 error fields

• Shot-ensembled data fromF = 0 plasmas with similarplasma current and density

Time before locking (ms)-5 -4 -3 -2 -1 0

0500

10001500

b rb θ

(G2

)v φ

(1,5

) (k

m/s

)

0102030

0204060

b r(m

=1)

(G)

∝ braking torquedue to error field

0102030

b θ(1

,5)

(G)

Large errorSmall error

Braking curve not significantly affected by error field variations

So, we tested theory of braking by eddy currents

Some history of braking theory/comparison to experiment

• Theory first proposed to account for locking with single large tearingmode 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 etal., 1993] expt. data reported

• Accounted for "forbidden bands" of rotation in a tokamak [Gates andHender, 1996]

• Theory augmented with inclusion of viscous restoring torque fortokamak [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 brakingdata [Hutchinson, 2001]

Basics of the theory, for initially rotating tearing mode

• Theory differs in detail for tokamak and RFP, but fundamentally generic

• Tearing mode, bmode(m,n) induces eddy currents in conducting shell(s)surrounding the plasma

• Eddy currents cause current sheet, jsheet(m,n) near rs

• Local jsheet x bmode braking torque results

• Local deceleration countered by viscous restoring torque from bulkplasma

• With significant viscosity, j x b torque must brake entire plasma tobrake mode (m,n)

MST provides simple geometry for application of modebraking theory

• Single aluminum shell, 5 cmthick

• Circular poloidal cross section• R/a = 150 cm/52 cm

Theory predicts well the experimental mode deceleration

• Only adjustable parameter intheory is τM ∝ 1/viscosity

• Adjusted such that curvescoincide at locking

• Shape of theoretical curvesdepends on other measured data

Theoretical prediction of τM well constrained

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7

v φ(1

,5)

(km

/s)

bΘ

/B(a) (%)

τm = 0.1 ms

τm = 0.5 ms

τm = 1.0 ms

τm = 2.0 ms

τm = 10.0 ms

Expt. data

Modeled τM’s consistent with experimental data

• Experimentally, τM ~ 1.5 ms in standard H2 MST plasmas (onemeasurement)

• MST standard τΕ ~ 1 - 2 ms over entire range of parameters

• As with many tokamaks, we expect that τΜ ~ 1 - 2 ms as well

• Modeled τΜ(D2) > τΜ(H2) also consistent with experimental expectation:larger central n0 observed with H2, hence larger CX momentum loss

For given τM, (theoretical) braking curve depends on modegrowth rate: the importance of time dependence

• Four different lineargrowth rates (bθ ~ t)

• Slowest rate exhibitsdiscontinuity

• Fastest (expt.) rate has nodiscontinuity

Summary

• Growth to large amplitude of single m = 1 mode in MST leads toglobal braking and locking

• Apparently explained by eddy currents in MST’s shell:• Theory reproduces (dynamical) experimental braking curves• Theoretical and experimental values of τM comparable

• Certainly bolsters confidence for braking theory as applied to RFP, andperhaps the tokamak... as well