1. Principal Attributes of FRCs Sustained by
Rotating Magnetic Field Current Drive
Alan Hoffman, H.Y. Guo, K.E. Miller, R.D. Milroy
Redmond Plasma Physics Laboratory
University of Washington
APS Plasma Physics Conference
October 24-28, 2005
Denver, CO

2. Abstract
Field Reversed Configurations (FRC) sustained by Rotating Magnetic Fields (RMF) are distinctly different from the decaying FRCs formed in theta-pinches. The RMF drive reverses particle diffusion, producing very long particle lifetimes, low separatrix densities, and complete reversal of the external confinement field. The density is set by torque balance between the RMF drive and resistive drag on the electrons. An FRC will increase in poloidal flux and expand radially inside a flux conserver until the compressed external field pressure balances the product of density times temperature. Higher temperatures, which are determined by a balance between RMF produced heating and various loss mechanisms, will automatically result in higher diamagnetic currents and poloidal magnetic fields, without requiring any increase in RMF parameters, and with very little increase in absorbed RMF power. Current drive performance thus increases dramatically with increasing plasma temperature. Temperatures in present TCS experiments are limited primarily by radiation and conduction/convection. Recent experiments show that conduction/convection losses can be greatly reduced using anti-symmetric RMF drive, and extensive modifications are being made to TCS to reduce impurities and radiation losses, so large increases in overall performance can be expected.

3. Flux is Major Determinate of Compact Toroid (CT) Lifetime
Prolate FRC inside Flux Conserver rc rs Bo Be
External flux e = rc2Bo
Internal flux p  0.31xsrs2Be
xs  rs/rc
Flux conservation: Be = Bo/(1-xs2)
FRC radius xs = xs(p/e) set by ratio of internal and external fluxes:
Peak plasma pressure set by compressed bias field:
Average beta governed by axial equilibrium:

4. Rotating Magnetic Fields (RMF) Applied to Flux Confined FRC
RMF antenna
Iz = Iosinwt
Iz = Iocoswt
Bz field coils
driven electron current
rotating field Bw
‘Rotating Radial Field ‘Drags’ Electrons
Must have wci < w << wce for electrons, but not ions, to follow rotation.
Electrons Magnetized on Rotating Field Lines (wcet >> 1)
Necessary for efficient current drive.
Absolutely necessary for rotating field penetration.
Resultant RMF Torque Increases FRC Flux and Pressure
Process continues until RMF torque is balanced by resistive electron-ion frictional torque.

5. Flux Build-Up is Key to FRC Formation & Sustainment by RMF{
for */rs < 0.5
The balance of TRMF with T determines the maximum possible electron density:
(*/rs ~ e/)
Flux build-up will continue and Be will increase until ne equals ne*.
Be  (neTt)1/2, so higher temperatures will result in higher magnetic fields, currents, and FRC fluxes as long as Idia < Isync.
Higher Tt produce higher diamagnetic currents (for a given ne) and requires higher RMF frequency.

6. RMF Partial Penetration is Rugged, Natural Phenomenon
Diamagnetic line current: *
Synchronous line current:
Key current drive parameter:
Near synchronous edge rotation with small  =  - e allows deep RMF penetration:
Partial penetration is desirable to maximize torque and minimize Br.
It is ci = eBr/mi which must be kept small to avoid ion drive.
Br component of RMF tends to open up field lines.
Larger B aids radial confinement and stabilizes interchange modes.
As long as  ~< 0.5, edge and penetration adjust naturally so that */rs  .

7. TCS Experiment (1/4 view)
RMF Antenna
Main Bias Coils
End Coils
0.4 Be
Mirror Coils
0.3
Radius (m) rs rc
0.2
0.1 Bo
RMF2003.8a - hg2005.alh.f1
0.25
0.50
0.75
1.00
1.25
1.50
Distance to the midplane (m)
Main bias and end coils energized in parallel to serve as flux conserver

8. Temperature Higher Early Before Impurity Ingestion & Radiation
-20
-10 10 20 Be
Bint
BRMF
Magnetic Field (mT)
0.4
0.8
1.2
1.6
2.0 40 60 80
100
nd
nd (1019 m-2)
Temperature (eV) Tt
-0.2
0.2
0.6 1
1.4
1.8
0.5
1.0
1.5
2.5
Pabs (MW)
Prad (MW/m2)
Pabs
Prad
TIME (msec)
early time
late time
Higher Tt at early time results in higher Be, p, and I with same BRMF and only marginally higher absorbed RMF power.
Thus, the average plasma resistivity is lower at the higher temperature.

9. Typical Double Rigid Rotor (DRR) Profiles10 20
Radius (cm)
B (mT)
ne(1018 m-3) 30
hg2005.alh.f4c 40
-20
1.0
1.5
2.0
0.5
-10
RR Profile
DRR Profile
Shot 9217
f = 152 kHz
 = 7 kHz
t = 1.0 ms
Tt = 28 eV
t = 0.35 ms
Tt = 41 eV Bz
BRMF 10 20
Radius (cm)
J (105 A/m2) 30
hg2005.alh.f5 40
0.4
0.6
0.8
1.0
0.2
0.5
DRR Profile
 = e/
r/
2r/
The profiles are approximately rigid rotor, but with lower central electron rotation speed leading to reduced j, shallower dBz/dr, and slightly broader ne(r) near the field null.
Higher temperature leads to higher j and e.

10. Partial penetrations with lower central electron rotation can lead to some trapped RMF rotating at lower speed, which we call ‘edge driven mode’ (edm)
Very non-uniform resistivity profile is required in calculations to reproduce edm.
 = i + e/(1+e(rs-r)/) with i = 30 -m, e = -m, and  = 1 cm gives best fit to experiments. Numerically, inner structure decays away at rate determined by i.
Inner structure rotates at r and tearing and oscillating torque occurs at d =  - r. Many experimental measurements showing oscillation at d indicate the presence of an oscillating torque.
Calculation
Experiment
0.65
0.70
1.4 37 38 11 12
0.5
1.0
1.5 13
1.6
0.85
0.80
0.75
Time (msec)
0.90
hg2005.alh.f16
p (mWb)
rs (cm)
Be (mT)
(Nt-m/m)
TRMF f fd fr

11. Calculated Profiles During and After edm
During edm
After edm 10 30
hg2005.alh.f13a 20 Bz
BRMF
Radius (cm) 40
ne (1018 m-3)
B (mT) 5 15 -5
-10 2 4 6 10 30
hg2005.alh.f14a 20 Bz
BRMF
Radius (m) 40
ne (1018 m-3)
B (mT) 5 15 -5
-10 2 4 6 8
These profiles are more characteristic of experimental profiles. The low central resistivity allows high azimuthal current flow near the field null with only weak edm current drive.
A tendency toward these profiles is seen on only a few experiments. The central current must be very low, despite low i, since there is little if any RMF drive there.

12. DRR Model can be used to Calculate Effective Resistivities from Torque or Power Balance
Calculated DRR torque can be set equal to ‘measured’ RMF torque, TRMF = 0.8(2B2rs2/o)(*/rs). For scaling purposes we assume different resistivities in inner and edge regions, with e = 10i. However, we do not have an independent measure of the fraction of the measured Pabs attributable only to the azimuthal currents, P.
High edge resistivity affects power more than torque since P = eT.

13. Torque Based Resistivity Scaling
0.5
1.0
1.5
2.0
2.5
3.0 1 2 3 4 5 6 7 8 9
B(*/rs)1/2/(r/0.15)1/2rs
nm (1019m-3)
114 kHz
83 kHz
152 kHz
258 kHz
Calc
nm = {Bw(*/rs)1/2/rsr1/2}4/3 44 36 59 25 28 41 24 39 30 60 Tt
Experimental torque based density scaling:
DRR calculated density scaling :
Inferred resistivity scaling :
The overall resistivity scales approximately as ne-1/2. All experiments also show a resistivity decreasing with temperature, although there is not enough of a temperature spread to determine an accurate temperature scaling. Whereas calculations at constant resistivity show lower peak density at higher temperature (since TRMF = T  ne3/2Tt1/2 is constant), the experiments display contrary results! (The 60 eV calculation resistivity profile was chosen to reproduce the 59 eV experimental results.)

14. nm = 0.0073{Pabs1/2/rrs2(1+0.57fi/fr}4/3
Resistivity calculated from total Pabs appears higher due to other contributions to absorbed power besides j2
Experimental power based density scaling:
0.5
1.0
1.5
2.0
2.5 3 2 4 6 8 10 12
0.15Pabs1/2/rrs2(1+0.57fi/fr)1/2
nm (1019m-3)
114 kHz
83 kHz
152 kHz
258 kHz
Calc
nm = {Pabs1/2/rrs2(1+0.57fi/fr}4/3 39
DRR calculated density scaling : 36 41 24 28 Tt 25
Inferred ‘resistivity’ scaling with P assumed equal to Pabs : 30 60 59 36 44
Again, the overall resistivity scales approximately as ne-1/2. ‘ip’ is higher than it by about a factor of 2 since Pabs is about double P. At higher temperatures the density falls above the above scaling line since the ratio of Pabs/ P is lower.

15. Contributions to Total Absorbed Power
Calculated Distributions of Absorbed Power
Measurements of Excess Absorbed Power
0.5
0.65
0.70
0.85
0.80
0.75
Time (msec)
0.90
hg2005.alh.f12a
1.0
P (MW/m)
1.5
2.0
2rwSPoyn
jtot2
j2
jz2
 = /(1+e(a-r)/) -m
(Pabs-Pq)/P
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9 1
(Bw/Be)2/(d*/rs)
1:1
4:1
114 kHz
83 kHz
152 kHz
258 kHz
Calc: 256 kHz
P is calculated based on DRR model and resistivities inferred from torque balance
Pabs = P + Pz + Pdyn.
The ratio of excess absorbed power to that due to j2 appears to scale as (B/Be)2/(*/rs), ranging from 2 (at high densities) to 4 (at low densities) times this value.

16. Pabs Scaling with B/Be Ratio
The ratio of Pabs to Be2, representing the effective resistivity for power absorption, scales as B/Be.
At a fixed value of B, Pabs only increases linearly with Be, or driven current, due to both the ratio of excess absorbed power to j2 decreasing, and the actual resistivity  decreasing with increasing density or temperature.
Higher temperature operation, at a fixed B, should result in significantly higher magnetic fields and FRC currents without requiring large power increases.
The decrease in the Pabs/Be2 ratio with lower RMF frequency is due to the increasing plasma density and decreasing actual resistivity as  decreases.
Effective ‘ip’
0.1
150
100 50
83 kHz
114 kHz
152 kHz
258 kHz
0.2
B/Be
0.3
hg2005.alh.f10
0.4
Pabs/6.8(2Be/o)2 (-m)

17. Resistivity Scales like Chodura Collision Frequency
Previous -pinch flux decay rates well modeled using Cc = 0.1, fc = 3.
Near separatrix fe ~ f ~ 150 kHz and for Tt = 50 eV, ve/3vs ~ 2 so that the Chodura resistivity is very large and of the same order as the edge resistivity used in the numerical calculations.
The resistivity drops rapidly toward the FRC interior as er decreases sharply, also in agreement with our inferred resistivity profiles.
Chodura resistivity will decrease with temperature (seen experimentally in TCS) and also with increased size since ve will decrease for a given B (also seen in comparisons with the smaller 20-cm radius STX experiments).

18. Summary
RMF current drive of FRCs, with partial penetration, is natural and optimal for many reasons.
RMF parameters determine the FRC density, but the temperature responds to overall power balance.
Current drive performance is seen to improve rapidly with increasing temperature, leading to higher FRC magnetic fields, currents, and fluxes.
Detailed behavior with edms is best modeled using a highly non-uniform resistivity profile characteristic of the Chodura formula.
A new facility, TCS/upgrade is being built with asymmetric RMF drive and control of recycling impurities to greatly increase plasma temperatures and take advantage of the above results.