Steinhauer APS poster 2019 Anatomy-revised

Meet “GRUSHENKA,” a wizard that performs
a 2D cat-scan history of an FRC experimental shot
Reconstruct fully-2D equilibria from routine magnetic data
Principles & definitions
• “Tangibles” = inputs to Grushenka: what can be measured Routine magnetic data: wall flux yw(z), wall field Bw(z), excluded
flux radius Rf(z)
• “Intangibles” = outputs from Grushenka: what can’t be directly
measured
Full 2D structure; actual dimensions of FRC, trapped flux, plasma energy, current, ...
• Physics model - moderate sophistication
• Numerics – near instantaneous computation
• Main advantage - immediate post-shot processing
Quick preview
• Equilibria exist only in limited range of Rf-Zf space (“shape domain”)
• Both FRCs and high-b mirror plasmas can be found, depending on {Rf-Zf}
• Realistic plasma rotation has a strong effect
• Outsized role for periphery (scrape-off layer region, “SOL”)
• Conventional formulas: imperfect and sometimes awful
separatrix radius & half-length, trapped flux, plasma energy, ...
• Rotation rate model
L. Steinhauer, T. Roche, and J Steinhauer, and the TAE team
TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
General properties of FRC equilibria
Anatomy of a Field-Reversed Configuration
Rapidly-rotating fast ions + non-rotating bulk ions (bulk ion rotation “stopped” by edge biasing) approximated closely by W » WJ o jq/rqne (bulk “current” drift)
Flexible pressure-like surface function P(y)
Limited shape domain
Plasma energy and current
Energy: Ep = ò(3/2)pdV conventional formula Epf = (3/2)(Be2/2μ0)Vf Current: I = òòjqdrdz
Examples: fixed Zf = 1.1m
•
•
•
Flexibility: two-parameter form P(y ; Ps, ac)
Ps ~ multiplicative factor
ac ~ current structure in core Analytic core and periphery branches connect smoothly at separatrix y = 0
Figure: P¢(y) = jq/r (current density) ac = parameter labeled in figure
ac > 0 current peaked in core ac = 0 flat core current profile ac > 0 current hollow in core
•
•
not just any {Rf, Zf} is “accessible” When fail to compute equilibrium
(a) can’t balance forces
(b) tearing unstable condition
Figure: shape domains, two classes
Traditional FRC: const wall flux; weak end mirror
C-2 family: variable wall flux; strong end mirror
Rf/Rw
•
Rf upper / lower bounds of shape domain Distinguish core and periphery
1/3 or less of the plasma energy is in the core
• Computational domain: see figure ®
R
Solid curve R (z), dashed R (z)
f s mid-
• Modified Grad-Shafranov equation plane
Maschke-Perrin 1980 accounts for rotation
R
Ep,core (kJ)
t (ms)
• Mixed boundary conditions
¶y/¶n = 0 at mid-plane, and end-plane
Radial wall: prescribe wall flux y (z) w
• Flexibility
Two-parameter form of surface function P(y)
• Longer length ® stronger mirror effect ® less trapped flux needed
• Smaller radius ® less trapped flux • Below a certain radius:
Zf = 1.1 1.3
1.5
Adopt reasonable value of rotation rate W
no trapped flux = no FRC
Rf
s
•
.
.
• Conventional formula: f (Wb) » B (T)R 3(m)/R (m). 0efw
• Figure: trapped flux / formula for fixed values of Zf®
f
Computation fast; an entire experimental timeline can quickly be found to track plasma evolution from early to late
After start-up transients plasma energy is sustained as long as neutral beam injection continues
00axisZs ZstrZmz *222
solid curves: with rotation W = W
dashed curve: no rotation
J
f /f T0
Numerical procedure
• Dotted lines ~ boundary of computed shape domain
• Rf vs Zf trajectory evolves (measured)
• C-2U and C-2W shots evolve within shape domain
Rf/Rw
Snapshot
(1) Collect data snapshot { yw & Rf vs z } from data at
a given time
(2) Distill data to extract smooth fit yw,fit(z) and shape
dimensions {Rf,Zf}
(3) Nested algorithm: iterates y(r,z) (relaxation method), and
adjusts parameters {Ps, ac} to target {Rf,Zf} Computation time: a second or so on an ordinary PC
Timeline
• Does the measured diamagnetic object actually contain a
+0.5 0
Z /R f
y = 0 -0.5
-1
y/yR
Core ¬ | ® Periphery (SOL)
w
• Observe
Limited radius band
Lower bound on half-length
Uptilt: mirror effect, strong for C-2 family Rotation effect (blue lines) assumes W = WJ
Field-reversal or not?
R-Z trajectory and shape domain
closed-field region?
• Figure ® C-2 family, same Rf, different Zf
Dashed red lines Rf vs z
Solid red line Rs vs z
• Shorter length: imbedded FRC
• Longer length: mirror plasma; no FRC • Explanation: mirror effect
Trapped flux
Z = 1.1m: FRC f
Zf = 1.7m: mirror
Zf/Rw
1/4 or less of the plasma current is in the core
Bottom line: periphery (SOL) plays an outsized role in equilibrium
Properties of specific FRC experiments
•
-P¢(y) +1
C-2 family
Trad’l FRC
r
Rw
Time evolution of core plasma energy
What’s inside Grushenka?
straight wall
Zf +
end cone
Rend
end- plane
Dy=-μrP¢(y)exp(mWr 2kT) 0 i tot
1.7
mirror effect
mirror