Modeling and Measurement of Toroidal Currents in the HSX Stellarator
J.C. Schmitt, J.N. Talmadge, J. Lore
HSX Plasma Laboratory, Univ. of Wisconsin, Madison, USA
Evolution of Toroidal Current
0.05
0.05
0
0
0.5
1
1800
1600
0
0
0.5
1400
S22
S21
150
0.2
0.15
1-D diffusion equation for rotational transform
1
QHS
QHS Equiv Tok
MirFL14 10%
100
1200
0.1
50
0.05
0
0
0.5

• Pfirsch-Schlüter current rotates with the |B| contours with toroidal angle and is reduced by
a factor of 1/3 compared to a conventional stellarator
300
Boundary conditions
On axis: finite current density
@ LCFS :set by measurement
100
50 kW ECH
0
B
B
-300
0.8

Any non-inductive source
0.83 0.84
sec
0.85
0.86
Pfirsch-Schlüter Current
Rotates Helically
• Evolving current profile is modeled by a diffusion equation with a 3d susceptance matrix
• V3FIT1 calculations of magnetic signals are compared to diagnostic signals
• Special thanks to Steve Knowlton, Jim Hanson, and the V3FIT team
0.4
0.6
0.8
1

-0.6
200
-0.8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1
-100
-0.2
400
0
JJBS
bsi-root
i-root
0
-0.4
-200
Diagnostic System
13 Br
Bθ
9
26
1
Red: JPS IN TO PLANE
18
25
2
-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1
0
2
Br
4
45 ms
15 ms
0
2
-2
8
Br
2
4
6
8
10
12
14
4
6
8
10
12
14
16
Br
18
Bθ
2
1
0.5
0
-0.5
-1
16
Gauss
1
0.5
0
-0.5
-1
≈ 1/2 FP
Gauss
Gauss
Φ=0 (symmetry plane)
Φ=1/2 Field Period
Φ≈ 1/6 Field Period
1
0.5
0
-0.5
-1
≈ 1/6 FP
20
22
24
26
28
30
32
2
4
6
8
10
12
14
0
-4
16
10
8
45 ms
•≈ 1/6 FP
15 ms
Bθ
6
4
2
18
2
• V3FIT for equilibrium reconstruction of HSX plasmas.
• HSX can alter the magnetic spectrum with a set of auxiliary coils
Mirror: The helical symmetry is spoiled by introducing (n,m) = (4,0) component
(and harmonics) into magnetic spectrum, affecting equilibrium, bootstrap currents
Well/Hill: Rotational transform profile may be raised/lowered, adjusting the
location of rational surfaces, vacuum islands
tsim=7 ms
20
22
24
26
28
30
22
24
26
28
30
32
20
22
24
26
28
30
32
Bθ
6
4
2
4
6
8
10
12
14
-2
16
18
Summary + Future Plans
P.S. only
18
20
0
-2
1.3
4/3
1.25
1.2
32
51st Annual Meeting of the Division of Plasma Physics, November 2 - 6, 2009, Atlanta, Georgia
Well 4.4
Well 11
Hill 2.9
1.15
8/7
Hill 4.4
Hill 6.2
Hill 11
1.05
1
4/4
0.95
0.9
0
0.02 0.04 0.06 0.08 0.1 0.12
eff
1
Well 2.9
Well 6.2
r
Bθ
0.06
tsim = 
Br
1.1
Φ≈1/2 Field Period
0.04
-2
-4
10
0.02
sec
4
5
1
0.5
0
-0.5
-1
-420 A
-400
• The direction of the bootstrap current is reversed and reduced by ~1/3 compared to
an equivalent tokamak. This results in a reduction in rotational transform.
• Evolving current profile is modeled by a diffusion equation with a 3d susceptance
matrix.
• The Pfirsch-Schlüter current rotates with |B| contours and is reduced by ~1/3
compared to a tokamak, demonstrating the lack of toroidal curvature.
Blue: JPS OUT OF PLANE
17
16
Gauss
2x Rogowski coils (Itor)
• External -- Low-frequency response ≤ 3kHz
• Internal -- High-frequency ≤ 200kHz
2x Diamagnetic Loops (WKin)
• Internal – High-frequency ≤ 125kHz
2x dB/dt triplet array – 32 external triplets
• MHD and bootstrap currents
• Low-frequency response
High-frequency Mirnov array inside vessel
• 36 coils: 26 poloidal, 10 toroidal
• Frequency response ≤ 200kHz
ChERS Diagnostic
• Ti and Zeff estimate
10-chord Thomson Scattering
• Te(ρ) and ne(ρ) profiles
32
Poloidal Index
-250 A
-300
0
• Pfirsch-Schlüter current exhibits dipole behavior and rotates with |B| contours
• Sign of signals at two toroidal locations demonstrate a phase shift showing the helical rotation.
• Toroidal curvature is effectively eliminated in HSX
50 kW ECH
bs e-root
∆IBS = 170 Amps
0.2
• Te, Ne from Thomson Scattering.
• Ti from ChERS. Zeff ≈ 1 from Bremsstrahlung radiation (ChERS optics).
• PENTA3 calculates the fluxes. Er is determined by ambipolarity.
• The electron-root reverses the direction of the bootstrap current
• Bθ has large unidirectional contribution from <J∙B>
• Br is dominated by the m=2 structure of the vacuum vessel
• Measurement evolves slightly faster than simulated values
• Possible sources of error: Zeff, initial plasma profiles, neoclassical
calculation of bootstrap current
CCW
0.82
0.2
0.4
600
0
CW
0.81
0
1
-100
-200
0.6
Gauss
||
0.5
Multiple
ambipolar
solutions
0.8
800
0
0
200
Amps
• Bootstrap current
• Is in the opposite direction and reduced compared to a tokamak
• Reduces the rotational transform in HSX
• Reverses with B-field direction
• Induces toroidal current with long decay times: τη /μ0 ≥ τEXP
1
0
1000
0
JJBS e-root
Gauss
0.1
7
6
5
4
3
2
1
Gauss
0.1
1
2000
Amps
0.15
x 10
Gauss
0.15
2
0.2
10 18 /m 3
3d susceptance matrix links toroidal and poloidal currents
and magnetic fluxes2
S12 = S21 = 0 for Tokamaks
S11 ≈ S12 ≈ S21 for HSX
5
S12
0.2
J BS A / m
S11
• The Helically Symmetric eXperiment
• Quasi-Helically Symmetric and almost no toroidal curvature
• B0 = 1 Tesla
• 26 kW – 100 kW ECR Heating (1st Harmonic)
• Perpendicular launch
• Little to no current drive seen during 1 Tesla campaign
Bootstrap Current
eV
Overview
1/2
== [/(B )]
0
J.D. Hanson, et al, Nucl. Fusion 49 (2009) 075031.
2 P. I. Strand and W. A. Houlberg, Phys. Plasmas 8, 2782 (2001).
3 D.A. Spong, Phys. Plasmas 12, 056114 (2005).