1 alan hoffman, h.y. guo, k.e. miller, r.d. milroy redmond plasma physics laboratory university of...
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1
Alan Hoffman, H.Y. Guo, K.E. Miller, R.D. Milroy
Redmond Plasma Physics LaboratoryUniversity of Washington
APS Plasma Physics ConferenceOctober 24-28, 2005
Denver, CO
Principal Attributes of FRCs Sustained by Rotating Magnetic Field Current Drive
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
oetmm BkTnp 2/2
rcrsBo
Be
xs rs/rc
Prolate FRC inside Flux Conserver
ocs
sessp Br
x
xBrx 2
2
32
1FRC 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:
2/1 2sx
External flux e = rc2Bo
Internal flux p 0.31xsrs2Be
Flux conservation: Be = Bo/(1-xs2)
4
Rotating Magnetic Fields (RMF) Applied to Flux Confined FRC
‘Rotating Radial Field ‘Drags’ Electrons– Must have ci < << ce for electrons, but not ions, to follow rotation.
Electrons Magnetized on Rotating Field Lines (ce >> 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.
RMF antennaIz = Iosint
RMF antennaIz = Iocost
Bz field coils
driven electron current rotating field B
5
Flux Build-Up is Key to FRC Formation & Sustainment by RMF
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 Idia < Isync. Higher Tt produce higher diamagnetic currents (for a given ne) and requires
higher RMF frequency.
TTerndt
dRMF
sse
p
22 a
so
sRMF r
rBT
*2 22
ssee rneT 4225.0
/
/*2*
22e
s
so
e
r
re
BnThe balance of TRMF with T determines
the maximum possible electron density:
for */rs < 0.5
{
(*/rs ~ e/)
6
RMF Partial Penetration is Rugged, Natural Phenomenon
25.02
seeo
edia ren
BI
Diamagnetic
line current:
Synchronous line current:
25.0 sesync renI rsrs
*
edgeo
edge
2
*
Near synchronous edge rotation with small = - e allows deep RMF penetration:
As long as ~< 0.5, edge and penetration adjust naturally so that */rs .
2
4
seo
ee
sync
dia
rn
B
I
I
Key current drive parameter:
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.
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TCS Experiment (1/4 view)
Main bias and end coils energized in parallel to serve as flux conserver
0.1
00
0.2
0.3
0.4
0.25 0.50 0.75 1.00 1.25 1.50
Distance to the midplane (m)
Rad
ius
(m)
RMF Antenna
Mirror Coils
Main Bias CoilsEnd Coils
RMF2003.8a - hg2005.alh.f1
rcrs
Be
Bo
csss
oe rrx
x
BB /
1 2
8
Temperature Higher Early Before Impurity Ingestion & Radiation
-20
-10
0
10
20Be
Bint
BRMF
Mag
neti
c F
ield
(m
T)
00.4
0.8
1.2
1.6
2.0
020
40
60
80
100
nd
nd
(10
19 m
-2)
Tem
pera
ture
(eV
)
Tt
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.51.0
1.52.02.5
2.00
1.0
2.0
Pab
s (M
W)
Pra
d (M
W/m
2 )
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) Profiles
0 10 20
Radius (cm)
B (
mT
)n e
(101
8 m
-3)
30hg
2005
.alh
.f4c
40-20
0
1.0
1.5
2.0
0.5
-10
10
20
RR Profile
DRR Profile
Shot 9217f = 152 kHz = 7 kHz
t = 1.0 msTt = 28 eV
t = 0.35 msTt = 41 eV
Bz
BRMF
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.
0 10 20
Radius (cm)
J (
105 A
/m2)
30
hg20
05.a
lh.f
5
400
0.4
0.6
0.8
1.0
0.2
0.5
1.0 DRR Profile
=
e/
DRR Profiler/
2r/
0
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)
0.65 0.70
1.4
37
3811
12
0
0.5
1.0
1.5
13
1.5
1.6
0.850.800.75
Time (msec)0.90
hg20
05.a
lh.f
16
p (
mW
b)
r s (
cm
)B
e (m
T)
T R
MF
(N
t-m
/m)
ffdfr
Very non-uniform resistivity profile is required in calculations to reproduce edm. = i + e/(1+e(rs-r)/) with i = 30 -m, e = 1000 -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.
TCS325 t=454.0TCS325 t=454.0
Calculation Experiment
11
Calculated Profiles During and After edm
0 10 30
hg20
05.a
lh.f
13a
20
Bz
BRMF
Radius (cm)
40
n e (
101
8 m
-3)
B (
mT
)
0
05
1510
-5-10
2
4
6
0 10 30
hg20
05.a
lh.f
14a
20
Bz
BRMF
Radius (m)40
n e (
101
8 m
-3)
B (
mT
)
0
05
1510
-5-10
2
4
6
8
During edm After edm
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
eiso
emeisrm rB
enrenT
29.007.129.0)(29.0 242
Calculated DRR torque can be set equal to ‘measured’ RMF torque, TRMF = 0.8(2B
2rs2/o)(*/rs). For scaling purposes we assume different
resistivities in inner and edge regions, with e = 10i. However, we do not have an independent measure of the fraction of the measured Pabs attributable only to the azimuthal currents, P.
eio
eeisrm
BrenjP 58.0458.029.0
2222
227.0 srmo
e renB
High edge resistivity affects power more than torque since P = eT.
13
Torque Based Resistivity Scaling
Experimental torque based density scaling:
sr
msm r
nrBn
2/1
4/12/1)/*(096.0
DRR calculated density scaling :
2/12/1
2/1)/*(67.0
itsr
sm r
rBn
Inferred resistivity scaling :
m)m10(
503192/1
m
it n
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 ne
3/2Tt1/2 is constant), the experiments display contrary results!
(The 60 eV calculation resistivity profile was chosen to reproduce the 59 eV experimental results.)
0
0.5
1.0
1.5
2.0
2.5
3.0
0 1 2 3 4 5 6 7 8 9
B(*/rs)1/2/(r/0.15)1/2rs
n m (
101
9m
-3)
114 kHz83 kHz
152 kHz
258 kHz
Calc
nm = 0.044{Bw(*/rs)1/2/rsr1/2}4/3
4436
59
25
28
41 24
36
39
3060
Tt
14
Resistivity calculated from total Pabs appears higher due to other contributions to absorbed power besides j2
Experimental power based density scaling:
risr
mabsm
ffr
nPn
/57.01025.0
2
4/12/1
DRR calculated density scaling :
2/12
2/1
/57.0125.0
iprisr
mffr
Pn
Inferred ‘resistivity’ scaling with P assumed equal to Pabs :
m)m10(
100''
3192/1
mip n
0.5
1.0
1.5
2.0
2.5
0
3
0 2 4 6 8 10 120.15Pabs
1/2/rrs2(1+0.57fi/fr)1/2
n m (
101
9m
-3)
114 kHz
83 kHz
152 kHz
258 kHz
Calc
nm = 0.0073{Pabs1/2/rrs
2(1+0.57fi/fr}4/3
44
36
5930
60
25
28
41 24
36
39
Tt
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
P is calculated based on DRR model and resistivities inferred from torque balance
0.5
0.65 0.70 0.850.800.75Time (msec)
0.90hg
2005
.alh
.f12
a
1.0
P (
MW
/m) 1.5
2.0
0
2rwSPoyn
jtot2
j2jz
2
= 30 + 1000/(1+e(a-r)/) -m
Calculated Distributions of Absorbed Power
(Pa
bs-
P)
/P
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(B/Be)2/(*/rs)
1:1
4:1114 kHz83 kHz
152 kHz258 kHz
Calc: 256 kHz
Measurements of Excess Absorbed Power
The ratio of excess absorbed power to that due to j2 appears to scale as (B/Be)2/(*/rs), ranging from 2 (at high densities) to 4 (at low densities) times this value.
Pabs = P + Pz + Pdyn.
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 j2 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’
00
0.1
150
100
50
83 kHz114 kHz152 kHz258 kH
z
0.2
B/Be
0.3
hg20
05.a
lh.f
10
0.4
Pab
s/6.8(
2Be/
o)2 (
-m)
17
Resistivity Scales like Chodura Collision Frequency
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).
scepic fC v/vexp(1Chod Previous -pinch flux decay rates well modeled using Cc = 0.1, fc = 3.
m1)m10(
1050 )eV(/)m()kHz(23.03192/12
ChodChod
2/1
te Trf
ee
e enen
m
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.