1 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,

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.

7

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.