von der Linden - 2016 - CT Workshop

Investigating the Dynamics of Canonical Flux Tubes
US-­‐Japan Compact Toroid Workshop 8/24/2016
Jens von der Linden
University of Washington
Jason Sears, Evan Carroll
Lawrence Livermore National Laboratory
Alexander Card, Eric Lavine, Manuel Azura-­‐Rosales
University of Washington Thomas Intrator
Los Alamos National Laboratory
Setthivoine You
University of Washington
LLNL-­‐PRES-­‐700761
1
Mochi.LabJet -­‐ Study evolution of canonical flux tubes with core-­‐
skin currents and helical flows
1. Study the stability of lengthening magnetic flux tubes with core and skin currents.
𝜓 = ∫ 𝐵 ⋅ 𝑑𝑆⃗
2. Reconstruct canonical flux tubes from magnetic field and ion flow measurements.
Ψ0 = ∫ Ω0 ⋅ 𝑑𝑆⃗ = ∫ 𝑚0 𝜔0 + 𝑞0 𝐵 ⋅ 𝑑𝑆⃗
𝜃&
𝑧̂
𝐽⃗789:
𝐽⃗;<=>
𝑬
𝑬
𝑩
2
Mochi.LabJet -­‐ Study evolution of canonical flux tubes with core-­‐
skin currents and helical flows
1. Study the stability of lengthening magnetic flux tubes with core and skin currents.
𝜓 = ∫ 𝐵 ⋅ 𝑑𝑆⃗
2. Reconstruct canonical flux tubes from magnetic field and ion flow measurements.
Ψ0 = ∫ Ω0 ⋅ 𝑑𝑆⃗ = ∫ 𝑚0 𝜔0 + 𝑞0 𝐵 ⋅ 𝑑𝑆⃗
𝜃&
𝑧̂
𝑬
𝐽? ;<=>
𝐽? 789:
𝑬
3
Mochi.LabJet -­‐ Study evolution of canonical flux tubes with core-­‐
skin currents and helical flows
1. Study the stability of lengthening magnetic flux tubes with core and skin currents.
𝜓 = ∫ 𝐵 ⋅ 𝑑𝑆⃗
2. Reconstruct canonical flux tubes from magnetic field and ion flow measurements.
Ψ0 = ∫ Ω0 ⋅ 𝑑𝑆⃗ = ∫ 𝑚0 𝜔0 + 𝑞0 𝐵 ⋅ 𝑑𝑆⃗
𝜃&
𝑧̂
𝑢B
𝑬
𝑢?
𝑬
𝑢B
4
Mochi.LabJet -­‐ Study evolution of canonical flux tubes with core-­‐
skin currents and helical flows
1. Study the stability of lengthening magnetic flux tubes with core and skin currents.
𝜓 = ∫ 𝐵 ⋅ 𝑑𝑆⃗
2. Reconstruct canonical flux tubes from magnetic field and ion flow measurements.
Ψ0 = ∫ Ω0 ⋅ 𝑑𝑆⃗ = ∫ 𝑚0 𝜔0 + 𝑞0 𝐵 ⋅ 𝑑𝑆⃗
𝜃&
𝑧̂
𝑬
𝐽? ;<=>
𝐽? 789:
𝑬
5
Define a 𝑘L − 𝜆̅ space for flux tubes
WXU
𝑘L =
inverse aspect ratio
Y
Short-­‐Thick
Short-­‐thick flux tube becoming long and thin with increasing current
Long-­‐Thin
Tokamak
Regime
Low Current / Flux High Current / Flux R TU
𝜆̅ = S current to magnetic flux ratio
V
𝜆̅ > crit(kL)
𝜆̅ ~ crit(kL)
You, Yun & Bellan (2003)
𝜆̅ < crit(kL)
6
Analytical 𝑘L − 𝜆̅ space: lengthening current-­‐carrying flux tube crosses the sausage instability boundary
𝜖 = 1 current profile
𝛿 = 0 abruptness
stable
𝛿𝑊
𝜖=0
𝛿=1
𝜖=0
𝛿=0
stable
stable
unstable
unstable
𝑚=1
𝛿U =
cde
cd
𝑎 “abruptness”
unstable
𝑚=1
𝑚=1
𝑚=0
𝑚=0
𝛿U ~1
𝜉 𝜉′
unstable
𝑚=1
𝜆̅ = 2 2
𝑚=1
𝛿U ~0
𝜉
𝜖=
hi
g
hj
g
=
Tklmn
𝜖=0
𝒑 𝒋
𝜉′
𝜖=1
𝒑 𝒋
Tolodp
𝒓
𝒓
7
Analytical 𝑘L − 𝜆̅ space: derived with textbook linear ideal MHD but with both core and skin currents Simplify with Simplify with Bellan (2003)
Newcomb (1960)
analysis of flared flux tubes
analysis of internal stability
𝜹𝑾(𝝃𝒓) = ⇒ 2𝑘L − 𝑚𝜖𝜆̅
𝜹𝑾𝒑𝒍𝒂𝒔
+
𝜹𝑾𝒊𝒏𝒕𝒇
+
Set wall to ∞
Ignore wall effects
>0
𝜹𝑾𝒗𝒂𝒄 Stability Criterion
𝛿 + 1 2𝑘L − 𝛿 − 1 𝑚𝜖𝜆̅
𝑚𝜆̅ − 2𝑘L
W
W
̅
+ 𝜖 − 1 𝜆 − 𝑘L W + 𝑚W
𝑘L
W
𝐾‚ 𝑘L
> 0
ƒ
L
𝐾‚ 𝑘
8
𝛿 can only be determined by integrating Euler-­‐Lagrange equation
Simplify with Bellan (2003)
analysis of flared flux tubes
𝜹𝑾(𝝃𝒓) = 𝜹𝑾𝒑𝒍𝒂𝒔
+
𝜹𝑾𝒊𝒏𝒕𝒇
Set wall to ∞
Ignore wall effects
𝜹𝑾𝒗𝒂𝒄 +
Simplify with Newcomb (1960)
analysis of internal stability
𝛅𝑾𝒑𝒍𝒂𝒔
𝜋𝐿 U
Š 𝑓𝜉 ƒW + 𝑔𝜉 W 𝑑𝑟 + ℎ𝜉 W
=
2𝜇‰ ‰
U
‰
Minimize 𝛅𝑾𝒑𝒍𝒂𝒔 with solutions to Euler − Lagrange: ⇒ 𝛅𝑾𝒑𝒍𝒂𝒔
𝜋𝐿𝜉UW 𝑓U
=
𝛿 + ℎU
2𝜇‰ 𝑎
˜
˜9
𝑓
˜c
˜9
− 𝑔𝜉 = 0 9
Numerical 𝛿U (𝜆̅, 𝑘L) results: current profile dependence and significant sausage unstable region in 𝜆̅ − 𝑘L space
stable
stable
kink
sausage
stable
stable
kink
sausage
𝜖 = 0.3
𝛿𝑊
sausage
kink
𝜖 = 0.5
𝑚=0
stable
stable
𝜖 = 0.7
𝛿𝑊
Internal 𝑚=1
10
Mochi.LabJet -­‐ Study evolution of canonical flux tubes with core-­‐
skin currents and helical flows
1. Study the stability of lengthening magnetic flux tubes with core and skin currents.
2. Reconstruct canonical flux tubes from magnetic field and ion flow measurements.
𝜃&
𝑧̂
𝑢B
𝑬
𝑢?
𝑬
𝑢B
11
Reconstructing the RSX gyrating canonical flux tubes
Plasma Gun
B, u
𝐵̇ , triple probe, M ach measurement p lanes Anode
J
50 cm Kruskal -­‐
Shafranov
crowbar
Gyration frequency is coherent across shots. Conditional sampling aligns traces from 3,000 shots. 300 A
12
Reconstructed 3D Canonical Electron Flux Tubes
∫ Ω: ⋅ 𝑑𝑆⃗ = ∫ 𝑒𝐵 + 𝑚: 𝜔: ⋅ 𝑑𝑆⃗ ~∫ 𝐵 ⋅ 𝑑𝑆⃗
13
Reconstructed 3D Canonical Ion Flux Tubes
∫ Ω= ⋅ 𝑑𝑆⃗ = ∫ 𝑒𝐵 + 𝑚= 𝜔= ⋅ 𝑑𝑆⃗
14
Ongoing Work: Constraining Ion Flow
Mach measurements incomplete, RSX is decommissioned
𝑢=? is measured in 2nd and 4th plane, 𝑢= is measured in 4th plane. Need to constrain 𝑢=¡
and extrapolate 𝑢= in 3D volume.
𝐽⃗ = 𝑛𝑞 𝑢= − 𝑢:
Electrons frozen to magnetic field lines
𝑢= ~
£⃗
>¤n
+𝛼
¦
|¦|
Use the plane measurements of 𝑢= to fit for 𝛼(𝑥, 𝑦) and match the flux rope rotation as extrapolating along z.
Force Balance
Can the ion flows be extrapolated by balancing the centrifugal and Coriolis force terms balance the 𝐽⃗×𝐵 − ∇p ?
15
Summary
Mochi.LabJet is designed to generate canonical flux tubes with skin and core currents, and axial and azimuthal shear flows.
Analytical and numerical studies indicate that a lengthening flux tube may develop a sausage instability on top of a kink.
Reconstructing canonical flux tubes from magnetic field and ion flow measurements.
16
Acknowledgements
This work is supported by DOE Grant DE-­‐SC0010340 and the DOE Office of Science Graduate Student Research Program and prepared in part by LLNL under Contract DE-­‐AC52-­‐07NA27344. 17
18
RSX diagnostic resolution
19
RSX Shot distributions
Normalized Spectral Power
20