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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
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