[J_ CK 1 0 5 0 5 2 _ MRC/(ABQ-R-1208

--_ Revised October' 1989,. _ Copy /_.

Mission Research Corporation

SIMULATION AND THEORY OF

RADIAL EQUILIBRIUM OF PLASMOID PROPAGATION,FINAL REPORT

Mark M. Campbell

Randy M. Clark1 Michael A. Mostrom

September 1989

41

_

- Prepared for: Lawrence Livermore National LaboratoryJ Post Office Box 808

Livermore, CA 94550-

-=_- 7"'

- Under Contract' B052997 (1_ _- :i',_gg_-

_. Prepared by' MISSION RESEAR,Cft cORPORATION1720 Randolph Road, SE.

_ Albuquerque, NM 87106-4245

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UCRL-CR--105052

DE91 000803

ABSTRACTP

Cases _imulating beam currents to about 29 MA were done. Significant noise reduc-

tion and code speed-up were achieved for these cases. The net current scaling observed at

i lower beam current (I,,t ~ I_/_) continued to hold up well, as did the equilibrium itself.

A_slight deviation in the expected thickness and radialstructure of the current layer was

observed, however. A laminar flow model is developed that appears to give good agreement

with the simulation results. Suggestions for future work are discussed briefly.i

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CONTENTS

L

Section Page

1,0 INTRODUCTION 1

2.0 RESULTS OF 1.8-28.8 MA SIMULATIONS 2

3.0 LAMINAR FLOW EQUILIBRIUM MODEL 26

4.0 CONCLUSIONS 31

5.0 FUTURE WORK 32

q

REFERENCES 33

Appendix

A LOW-MODERATE BEAM-CURRENT SIMULATION RESULTS A-1AT AFWL

iii

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FIGURES

o

Figure Page

1 Periodic simulation results comparing azimuthal magnetic field Be (B3, •0.6 kG) versus radial position X2 = R (X2, cre) at ct = 400 cm for tworuns with h = 1.8 MA, (a) run AA, (b) run AB (hollow beam), NOTECHANGE IN X2 SCALING. 5

2 Periodic simulation results comparing Er (E2, 0.51 ' MeV/cm) versus R •(X2, cre) at ct=400 cm for two runs with h = 1.8 MA. (a) run AA,(b) run AB (hollow beam), NOTE CHANGE IN X2 SCALING. 6

3 Periodic simulation results comparing axially-integrated Jx versus R(X2, cm) at ct 400 cm for two runs with h = 1.8 MA. (a)run AA, @(b) run AB (hollow beam), NOTE CHANGE IN X2 SCALING. 7

4 Time averaged current profile for run AA near end of run withh= 1.8 MA. 8

o5 Time aver_.ged current profile for run AB (hollow beam) near end of

run with h = 1.8 MA. 9

6 Periodic simulation AC, B, (B3, 0.6 kG) versus R (cm) at ct = 500 cm •for h = 7.2 MA. 10

7 Periodic simulation AC, E, (E2, 0.51 MeV/cm) versus R (X2, cm) atct = 500 cm for h = 7.2 MA. 11

®

8 Periodic simulation AC, axially.integrated .T_versus R (X2, cm) atct = 500 cm for h = 7.2 MA. 12

9 Time averaged current profile for run AC near end of run withh = 7.2 MA. 13 •

10 Periodic simulation AD, B0 (B3, 0.6 kG) versus R (X2, cm) atct = 200 crn for h = 28.8 MA. 14

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iv

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11 Periodic simulation AD,/_, (E2, 0..51 MeV/cm) versus R (X2, cm)atct :- 200 cm for I_ = 28.8 MA. 15

r

12 Periodic simulation AD, axially-integrated J, versus R at ct = 200 cmfor/b= 28.8 MA. 16

13 Time averaged current profile for run AD near end of run withOoI_ = .o.8 MA. 17

/

,

14 'rime averaged current profile comparing different I_ for five runs. 18

15 Net currentasitscalestothesquarerootofbeam current. 19

16 Simulationresuitsshowingnetcurrentfordifferentbeam currents. 20

17 Simulation results showing the square root of the beam current fordifferentbeam currents. 21

18 Periodicsimulationresultscomparingtimehistoryand frequencyspectrumofthe netcurrentfortwo runs.Ib= 1.8MA. Currentin

unitsof1.35kA, frequencyinunitsofCOo= 3 x i0l°rad/s.(a)run AA,

(b)run AB (hollow beam). 22

19 PeriodicsimulationAC, timehistoryand frequencyspectrumofthenetcurrent.I_= 7.2MA. Currentinunitsof1.35kA, frequencyinunitsof

tj0= 3 x 10I°rad/s. 23

20 PeriodicsimulationAD, timehistoryand frequencyspectrumofthenetcurrent.I_= 28.8MA. Currentinunitsof1.35kA, frequencyinunits

ofCOo= 3 x i0x°rad/s. 24

21 Particledistributionpha,qe-spaceplotofaxialmomentum (Pl - B,_/)

versusradialposition(X2,cre).Left-handplotissheathelectrons,

right-handplotisbeam electrons.(a)and (b)1.8MA run AB atct=400cm. (c)and(d) 7.2MA run AC atct=500cm. (e)and(f)28.8MA run AD atct= 200 cm. 25

A-I PeriodicsimulationresultscomparingB0 versusr atct= 3000cm for

threerunswithI_= 22.5kA. (a)run AA, (b)run AB, (c)run AL. A-7

TE

@

A-2 PeriodicsimulationresultscomparingE, versusr atct= I000cm for

three, runs with I_ = 22.5 kA. (a) run AA, (b) run AB, (c) run AL. A-8 OI

A-3 Periodic simulation results comparing axially-integrated Jj versus r atct = 3000 cm for three runs with I_ = 22.5 kA. (a) run AA, (b) run AB,(c) run AL. A-9

A-4 Time averaged current profile in steady state for ,runs at I, = 22.5 kA. A-10

A-5 Periodic simulation results comparing B0 versus r at ct = 400 cm for

eight runs with Ib =: 452 kA. (a) run BAl, (b) run BA2, (c) run BC,(d) run BD, (e) run BE, (f) run BF, (g) run BG, (h) run BI. A-11 OI

A-6 Periodic simulation results comparing Er versus r at ct = 400 cm for

eight runs with h = 452 kA. (a) run BAl, (b) run BA2, (c) run BC,(d)run BD, (e)run BE, (f)run BF, (g)run BG, (h)run BI. A-12 OJ

A-7 Periodicsimulationresultscomparingaxially-integratedJ,versusr at

ct= 400 cm foreightrunswithI,= 452 kA. (a)run BA, (b)run BAl,(c) run BA2, (d) runBC, (e) run BD, (e) run BE, (f)run BF, (g) run

BG, (h) run BI. A-13 O_

A-8 Periodic simulation results comparing Be versus r at late times for sixruns with h = 452 kA, (a) run BAl at ct = 2000 cm, (b) run BA2 atct = 2000 cre, (c) run BC at ct = 2000 cm, (d) run BD at ct = 20oo cre,(e) run BE at c,' = 1¢.,00cre, (f) run BF at ct = 1600 cm. A-14 O/

A-9 Periodic simulation results comparing E' versus r at late times for six

runs with h = 452 kA. (a) run BAl at ct = 2000 cre, (b) run BA2atct = 2000 cre, (c) run BC at ct = 2000 cre, (d) run BD at ct = 2000 cm,(e) run BE at ct = 1600 cre, (f) run BF at ct = 1600 cm. A-15 O/

A-10 Periodic simulation results comparing axially-integrated Jj versus r atlate times for six runs with I_ = 452 kA. (a) run BAl at ct = 2000 cre,(b)run BA2 atct= 2000cre,(c)runBC at ct= 2000cm, (d)run BD

at ct = 2000 cm, (e) run BE at ct = 1600 cre, (f) run BF at ct = 1600 cm.A-16 Or

,%-11 Time averaged current profile in steady state for runs at/_ = 452 kA. A-17

O_

vi

Oi

A-12 Periodic simulation BB at late and early times for Ib = 1808 kA,

(a) axlally-integrated Jj versus r at ct = 2000 cm, (b) B0 versus r atc$ = 2000 cm, (c) Er versus ratct = 2000 cre, (d) axially-integrated J,versus r at ct := 400 cre, (e) B0 versus r at ct - 400 cm, (f) E, versus rat ct = 400 cm. A-18

k-13 Time averaged current profile for run BB near end of run withh = 1808 kA. A-19

A-14 Time averaged current profile comparing different I, for three runs. A-20

A'15 Net current as it scales to the square root of beam current. A-21'

A-16 Periodic simulation results comparing time history and frequencyspectrum of the net current for two runs with a different number ofaxial cells, h = 22.5 kA. Current in units of 1.35 kA, frequency in unitsof w0 = 3 × 101° rad/s. (a) run AA with 129 axial cells, (b) run AL withtwo axial cells. A'22

A-17 Periodic simulation BF, time history and frequency spectrum of the netcurrent, h = 452 kA. Current in units of 1.35 kA, frequency in units ofWo = 3 × 101° rad/s. A-23

A-18 Periodic simulation Bn, time history and frequency spectrum of the netcurrent. I_ = 1808 kA. Current in units of 1.35 kA, frequency in units

of w0 = 3 × 10l° rad/s. A-24

vii

b

1.0 INTRODUCTION

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Using our earlier test data (reproduced in Appendix A), _rlorder to reduce noise inthe high-current cases we increased the number of particles per cell to 375 for the sheath,

_ used radial curreltt smoothing Over three adjacent cells and eliminated current inside the

axial cell (R _- 0) to reduce the fictitious spike near the axis. This decreased the fluctuation

amplitude by about a factor of three and made estimation of the average current values

from the plots much easier. f

O

The summary plots shown here are in the same format as those in Appendix A but

continue the beam-current analyzed on to much higher levels. The trends are essentially

as expected.

e

We have completed a laminar-flow equilibrium model that predicts the radial profile

of the axial velocity of electrons and the relationship between azimuthal magnetic field

and radial electric field. Specific predictions are obtained for the net current and ion-beam

Q temperature necessary for equilibrium. In ali cases the a_;:eement between the modelpredictions and the ISiS simulations is excellent. The model also predicts parameter

scallngs which have not yet been tested in the simulations and, therefore, need furtherverification.

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2.0 RESULTS OF 1.8-28.8 MA SIMULATIONSO

Table 1 describes the simulation parameters and selected results for the new runs

carried out in August 1989. We have also included a few older simulations that are used$

in some of the summary plots. In ali these runs, the axial velocity of the ion beam was

_,_ = ,0.62, the annulus between the beam and the conducting pipe wall was filled with

plasma at five times the beam density, and the ion beam temperature was T_ = 43.2 kV

(the same as concluded at the end of the research in Ref. 1).O

Cases were run at 1.8 MA for both a normal configuration and one with the uninter-

esting inner region of the beam, which usually shows (nearly) zero net current, replaced

with a particle reflecting metal boundary at R = 25 cm (see Figs. 1-5). These cases

agreed within ~ 10 percent, so higher currents were run with this feature which decreases •

the simulations time by about 40 percent. Because the CPU run time to steady state for

the net current increases roughly as the square root of the beam current, we were able

to run the 7.2 MA and 28.8 MA cases readily (Figs. 6-9 and 10-13, respectively) but a

115 MA case (which might require 15-20 CPU hours) was not undertaken due to the cost. •

The width of the nonzero net current region, which had previously scaled above 100 kA

as approximately -_ "h,_,_T"l/2,continues its rapid decrease for h,_,_ _> 7 MA, (Figs. 4, 5, 9,

and 13). The radial current profiles of these and some lower current cases run previously

are superimposed in Fig. 14. Ali cases continued to exhibit approximately the same net •

current scalir_g (L,,t ~ 'be_,_rl/2,Figs. 15-17) as observed at lower currents.

The radial structure of the sheath region is more complex than earlier thought, Noise

in earlier run_ masked this structure. A negative current region at the inside edge of O:

the sheath (e.g., Fig. 14, 1.8-18.8 MA) became more obvious at higher currents. This

region is a local over-neutralization of the current at the edge of the beam electrons. The

sheath electron cyclotron orbit about B, imparts a large axial velocity component. At its

innermost radial position there is some overlap with original beam electrons which have O

not yet been lost tothe wall (see Fig. 21). Also, the sheath electrons at their inner radial

position overneutralize the beam charge, form a virtual cathode (reverses the sign of Er),

and loose ali their energy (Figs. 2la, c, and e). In this "mixing" region the sheath electrons

lose energy and axial momentum and the original beam electrons gain it (see Fig. 21d-f). •

2

O

The complicated phase space of the beam electrons (Figs. 21b, d, and f)'as they leaveOradially is due to the field reversal of Er and Be in the sheath region just inside the beam

edge. Er and B0 have very similar radial profiles and go to zero at the same radial position

(Figs. 1 and 2, 6 an,_.7, and 10 and 11).

DNo significant compression or expansion of the ion beam was observed in any of these

runs, so we postulate that the equilibrium ion-beam temperature is relatively independentof current.

OFigures 18-20 show that a steady state has been rea_ed at least in the net current.

The noise has been substantially reduced over the level observed in the earlier simulations

reported on in Appendix A.

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TABLE 1. SIMULATION RESULTS AND PARAMETERS.,,

@_LLNL Plasmotd Propagation Results August 1989

...... i i ,

Cell NumberRun Size Radlal

lD _ Ib (cm) X2MIN X2MAX CellsAA -1.8 MA 0,1 -0,05 53.25 533 0)AB 1.8 MA 0.I 25 51.8 268AC 7,2MA 0.05 25 50.25 505AD 28.8MA 0,025 25 50.25 1010

AL 22.5"'kA Run"doneatAFWL Aug--Oct1988

BE 452 kA Run done at AFWL Aug-Oct 1988 @iBB 1.8 mA Run done at AFWL Aug-Oct 1988

Net CurrentResultsatDifferentRadilrsPositions

Run ID

Radius AL BE BB AA AB AC AD Ot54 i0.96:3:{}9" -35.1452.8 -9.5751 40.1750.3 56.1650.125 -23,38 Oi50 67.84 93.9749 99.77 191.58 372.52

48 10.99 41.05 95.87 86.147.75 67.1547.7 97.81 0147.5 75.34 76.3 36.5344 35,! ii.5140 9.02 12.1 10.82 -20,830 -0.67 -1.78 6,0128 4.06 -I.II -1.73 OI

16 I 0.96 -9.80.6 I -0.02 0,15 .0,03

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o --0 64 1:-2 .5--0. I I 3. 3 26. 6 39. 9 53.2

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e

1 .62

II "" 0 36C::13 •

-0.910

-2. 1825.0 ,51 . 7 58.4 45. 1 51 .8

X2@ (b)

Figure 1 Periodic simulation results comparing azimuthal magnetic field B0 (B3, 0.6 kG)versus radialposition X2 = R (X2, cre) at ct = 400 cm for two runs withA = _.sMA. (4 runAA,(b)r_nAB(hollowb,am), NOTECHANGEINX2

0 SCALING.

5

4.98 --

--2.28 - o,

--..3.62 I I25.0 .31 .7 .38.4 45. 1 51.8

X2 O"(b}

Figure 2. Periodic simulation resultscomparir,g E, (E2,0.51 MeV/cre) versusR (X2, cm)atct=400cm fortworunswithIb= 1.8MA. (a)runAA, (b)runAB (hollow •beam),NOTE CHANGE INX2 SCALING.

0

. 10 -1 d 1-- I NT 1

• 1. 7 o 1_ i t |

1 .04-

• ,._.

, Zo 38I

• -0.29,,_lJ

-0.95 1 I I t _1• O. 0 1._3..3 26. 6 .39 . 9 53. 2X2

(a)

* I(_ -I JI--INTIo 1. I I I.,r

_t0.42 --e

_ '

imI

--0.76 --

o --1 .35 I I I25.0 ,.31 .7 ,38.4 4.5. 1 51 .8

X2(b)

Figure 3. Periodic simulation results comparing axially-integrated J_ versus R (X2, cre)O at ct = 400 cm for two runs with I_ = 1.8 MA. (a) run AA, (b) run AB (hollow

beam}, NOTE CHANGE IN X2 SCALING.

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8

r

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9

0'1

q_, q, , ,

• 1.%2'1 .f5 ......... _I ' i i

-5. 12 125.0 3 1 •3 37.6 4.5.9 50.2 e

X2

Figure 6. Periodic simulation AC, B, (B3, 0.6 kG) versus R (cm) at ct = 500 cm forA = 7;2 MA. •

0_

0

10

AW

• 0 45

i ' 1 '-O 18@

[_O. 81@

--1 .44 --

@

--2 • ,07 ,, , I I 125.0 31.3 37 6 43.9 50.2

x'2@

Figure7. PeriodicsimulationAC, E, (E2,0.51MeV/cm) versusR (X2,cre)atct=500cm forIb= 7.2MA,

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_'1.(_ 7 "1 J l-INTI

I .32 •

_Zo. 43I •

q,-,.,.

-2. 180

-3.9425:0 31 .3 37. 6 43.9 50.2

X2 •

Figure 8. Periodic simulation AC, axially-lntegrated Jx versus R (X2, cm) at ct = 500 cmfor Ib = 7.2 MA.

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_a

0

i

II

0

lS

II

,1_--19. 1

4 90OI

_0 O0

-4 91

9 82 .... I I l25.0 51.5 57.6 45.9 50.2

X2 •

Figure 10. Periodic simulation AD, B0 (B3, 0.6 kG) versus R (X2, cre) at ct - 200 cmfor Ib = 28,8 MA,

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14

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• 1 .02

i

. --0.86

--4.61 --0

1 I I-6.4.8• 25.0 ,31. ,3 ,37.6 4,.3.9 50.2X2

Figure11,PeriodicsimulationAD, Er (E2,0,51MeV/cm) versusR (X2,cm) atct=

@ 200cm forI,= 28,8MA,

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15

jI-INT1 oi1.34 ..... I ..........

--1 .56 I i25.0 51 . 3 ,.57.6 45.9 50. 2 o

X2

Figure 12. Periodic simulation AD, axially-integrated Jj versus r at ct = 200 cm forI_ = 28,8 MA, @i

0_

01

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L_ ,

Z , i0 ................. 8 oi

L£ e

0 __ .

N_ o •

/ 0

_ 0

(va) .LN_-t_n3 ' •

18

0

0

0

19

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(v'4) _LN-a_sno •

20 ¸

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21

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PLASMA SHEATH RUN AAi VB,,m,63, _B_'Oc_2"NP'8,EIO, VTH(2)'=O°30

'_°L_L-' ........_/J,ii___ o,

1oo .o!

lO 9 Sp i ..

1o 8 1°,

"710 .IT' "_

1o ' , ,/_ _^,,, _,_ ,,A _L._,

2 , ' i i100, 0,74O0 L , , 1,47 2,21 2 '-J4

FREO ,Q

(a I TIME- 400.20000

Figure I8. Periodic simulation results comparing time history and frequency spectrumof the net current for two runs. I_ = 1.8 MA. Current in units of 1.35 kA, @frequency in units of _)o = 3 × l01° rad/s. (a) run AA, (b) run AB (hollowbeam).

22

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OPLASMA SHEATH RUN AC, VBI.B3. NB-O,2.NPL32,ElO,VTH(2)"0,30

189, 21480 : ' i

142,

• _ 9s

47, 1,1 , 1Q °'o, _25,' 25o. ' _7_. 5o°,

10 11 1 I I 110 10 i• 1o 9

0 "

oo0_005 _.47 2 95 4 42 590

FREQTIME= 500, 29920

Figure 19. Periodic simulation AC, ttme history and frequency spectrum of the aet; cur-• rent. h = 7.2 MA. Current in units of 1,35 kA, frequency in units of

w0 = 3 × 101° rad/s.

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23

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@PLASMA SHEATH RUN AD, VB =-,63, NB;6Oc_2"NP'I ,3E12,VTH(2)=O. 30

351,

263

@

_175.=

88' f ,q .... t I @I0'0, 50, iO0. 150, 200,

_"/LA="

1o ! @__=

0 5 _,,10 1 ,47 2,94 -- 4,42 5.89F'REQ

TIME,= 200. 04000

OJFigure 20. Periodic simulation AD, time history and frequency spectrum of the net cur-

rent. h = 28.8 MA. Current in units of 1.35 kA, frequency in units of_0 = 3 x 101°_ad/s.

@

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ti

L ..... 1 I .......

X2 ' >'2

{e) (f)

Figure 21. Particle distribution phase-space plot of axial momentum (Pl =_ fl,_) versus

@ radial position (X2, cm). Left-hand plot is sheath electrons, right-hand plot is

beam electrons. (a) and (b) 1.8 MA run An at ct - 4oo cm. (c) and (d) 7.2 MA

run AC at ct = 500 cm. (e) aad (f) 28.8 MA run AD at ct = 200 cm,

25

3.0 LAMINAR FLOW EQUILIBRIUM MODEL @1

r

We propose a laminar flow model (similar to a model that has been used for magnetic

insulation calculations 2) as a first simple description of the electron equilibrium In thee_

plasmold. By "lamlnar" we mean that vr = 0, Nevertheless_ we assume that all electrons

now in the plasmold of radius rb wereborn at zero energy on a zero-potential (¢ = 0)

surface colnctdlng with a conducting pipe at radius R. _ rb. Energy conservation then

gives the usual relativistic factore

, ,y-i= (1)

where e (> 0) and m are the electron charge and mass. The ton-beam density n; and the eaxial velocity v,; are assumed to be free parameters independent of radius for r < rb and

v_ = 0 for r > rb. Conservation of axial canonical momentum for the plasmoid electrons

gives

eLP, - ",/mu, - eA, = 0 (2)

where we have taken the vector potential A_ = 0 at the ¢ = 0 surface.

e_

The radial electric field is defined in terms of the potential by

E, = .- d_¢¢= mc' d'_ (3)dr e dr •

and is given la terms of the plazrn_ sources by Potsson's equation

rd"_ r_r = (0 ' •

Thus, using Eqs. 3 and 4 the electron density is

eornc21d (d'7)n,- e2 rdf r_r +ni , (5) • $

26

e

I

• The azimuthal magnetic field Is defined la terms of the vector' potential from Eq, 2by

d md

B, = -_.rA_, = -----("/u,) . (6)' e dre

Combined with Er from Eq, 3 this gives the radial fore.balance equation E_ - v, Bo = O,

which gives Er _ obe In agreement with the ISIS simulation results, Ampere's law relates

Be to the plasma sources by

e

1_£(_B,)= .o_(_,_.,- .,v.) , (_)r dr

Substituting Be from Eq, 6 Into Eq, 7 givese

ld[d ]/Zoe'-- _ (_..) +_ (_,;..,-.,,..)= o , (s)r dr rr_

• Substituting the electron density n. from Eq, 5 into Eq, 8 gives a second-order radial

nonlinear differential equation for the electron velocity v.:

e

For simplicity we analyze the nonrelativistic limit 3 _ 1, valid for us <":.c, which

llnearlzes the equation, We also ignore cylindrical effects, valid for rwh/c _ 1, Defining

@ y = v. - v._ and z _ (r - rb)Wb/C, Eq, 9 reduces to

d2ydx---_= y (z < O) (10)

I which has the general solution y = c1¢" + c2e-_o To keep y finite for large negative z

(r _. rh), we require c_ = 0, Outside the beam, v.i = 0 and n_ may be different thun inside

the beam, Defining A2 - n_(r > rh)/ni(r < rh), the electron velocity is

@ d_v,dx----T = A_v, (x > 0) (11)

27A

w

0

whichhasthegeneralsolutionv,= 0aeA'+ _4e"_',We assume(r_- rb)Awb/a:_i (many0

plasmasklndepthsbetweenbeam andwall),To keepu,finiteforlargepositivex,we

requireas= O,Thus,thevelocityIsdescribedby

' (z<O)_,= (12) •

_4e"_" (_> o)

Becausethepotentlal¢ and itsderlvatlve,theelectricfield,must becontinuousatJthe

beam edgex = O,thisdefinesthetwoconstantsand Zlves •

A---e' (x < o,r < r_)

VA= i - I + ), (13)v. --I-- -_" (_> o,rl+Ae >rb) •

Thus, at the x = 0 (r = rb) the velocity ts

O_v,(r= rb) i

----- (I,_)v,_ 1 + A

whichIssmallIfA :_i,whilev,---+v,iforA -,O,OJ

Several relationships fbllow from this ann,lysis. From Eq, 6, the magnetic field at the

beam edge Is

mv,_ A O,

Be(r=,'b)= e ,,i+,x_b (I_)

which vanishes for A --+0 and approaches a constant for A _ 1, The net current In thebeam Is •

I A

1'.. (r = rb) - 2___rbBO(r : rb) = _ I._/_.,1 + A (rbCZb/C) (16)

where lA _ 47rmc/el._o= 17 kA, Using the relation O"

28

O

where I_istheIonbeam current,Eq, 16reducesto

" I_ (r+)=I+/_ (_)h ' (ts)

The azlmuthalmagneticfi,_+Idexertsa confiningpressureB_(r+)/2_oon theionbeam

@ which must be counteredInequilibriumby a finiteIon-beampressureP_ = B_T+,Using

Eq. 13 and equatingthesetwo pressures,theionbeam densitycancelsfromtheequation,

leavingonlyan expressionforthe iontemperature

O

= ' (19)

This Impliesthatthe ionbeam tsgenerallyquitehot in equilibriumwith (v_)ih/v._=

ii Al(1 + A),i

These theoretical predictions are specific and In some cases can be compared with

existing ISIS simulations, The square of the net current lint, divided by the beam current

@ I_, gsplotted versus I_ for the simulation results In Fig, 17, and the result ts indeed constant

at approximately 5 kA independent of beam current, For ali of our plasmold simulations

(here and in Ref, 1), we used an Ion drift velocity of J_i = 0,62, For ali but the earliest

' simulations In Ref, 1, we use a plasma density between the wall and beam equal to five

'@ times the beam density, or A = V_, Therefore, Eq. 18 gives a predicted value of 5,03 kA

for the constant of proportionality between I_t(r_) and I,. Moreover, Eq, lg gives a

predicted Ion-beam temperature for equilibrium of T# - 46.9 kV. In the ISIS simulations

In Ref. 1, we varied _ _zld empirically found that 43 kV gave a good equilibrium for beam

@ currents of 22.5 kA, 112,5 kA, and 452 kA. In our present research reported here, this

same value of 43 kV was used for the ion-beam temperature in ali of the simulations and no

noticeable contraction or expansion of the ion beam was observed, These two agreements

are excellent, particularly when the wide range of beam currents I, = 22.,5 kA-28.8 MA Is

@ considered,

29

Aw

The radial profile of the electron axial velocity In Eq, 13 also can be compared with

simulation, In ali of theMRG ISIS simulations, A was large (usually A_ = 5), The $

description of the electron velocity v. in gq, 13 Is consistent with, the sin'ml_tion results in

that v./v.# was small at the beam edge _ = 0 and Increased rapidly over a few beam skirt

depths e/Wb until v. _ v._ tn the beam Interior, The near equality of Er and _B0 In thesimulations also ISconsistent with the laminar flow model used here, •

Severalscalingpredictionsremaintobe verified,One trendapparentfromEq,13,

butnotyetverifiedby computersimulation,isthatvuatthebeam edgeincreasesasA

decreases and vj(r = rb) --* v_ as A _ 0, From Eq, 15, the magnetic field at the beamJ

edgeIs proportionaltoA forA _ I,andsaturatesata constantforA _ I,The constant

ofproportionalityiabetweenl_ei(rh)and Ib,

is linearly proportional to the ion beam axial velocity _r._which was never varied from (},62

in our series of simulations, There is also a strong dependence of I_ on A, For A_ 1, @

Ia Is proportional to A_--that ts, to the ratio of exterior plasma density to beam density,

Combined with Eq, 18, this predicts l..(r_) r #_/_(r > rb), or to the number of e,_terior

plasma skin depths across the beam, For A :_ 1, as in the simulations, I.et(rb) cz r#x/_,

or to the number of beam skin depths across the beam, The Ion-beam temperature for @

equilibrium is extremely sensitive to parameters, It Is proportional to fl_ and, for A ._ 1,

to A2, For A :_ 1, lt saturates at (T_)r._ = 1/2 mv_;, These changes in scaling should be

verified by simulation,@

O

O

3O

O •

4,0 CONCLUSIONSR

Increasingthebeam currentto28,8MA doesnotappeaxtostronglyaffecttheequl-

llbrlumscalingdiscussedinour earlierreports,While thereissome indlcationthatthe

• currentsheaththicknessisnow not decreasingas rapidlyat higherbeam currentas at

lowervalues,thenetcurrentstillscalesasIn.,~ /_/._m,The beam temperaturerequired

forequilibriumappearstoremainconstantat43 kV independentofbeam current,

• To trytoshedllghton theobservedscallngsintheISIScoderesults,we completeda

laminarflowmodel oftheelectrons,Specificpredictionswere obtainedforradialprofiles

aswellasforthenetcurrentand theion-beaxntemperaturenecessaryforequilibrium,In

allcases,thepredictionsagreedverywellwiththeISISsimulations,and new parameter

@ scalingswerebroughtoutwhich shouldbeverified.

m

O

O

O

i

31

A

OI

5,0 FUTURE WORK

There are many beam and background plasma paramvters that affect the plasmotd

equilibrium. In spite of having carried out extensive and complex simulations over the last

two years, some of these parameters have remained fixed. The laminar flow model pre- @l

sented here indicates several interesting parameter scallngs that, if verified, would further

our confidence in this model, We could continue simulations on to higher currsnts, but we

believe that this scaling is adequately verified over the present range of I, - 22,5 kA-

28,8 MA. For instance, by moving the metal Ucore" boundary to larger radius (i.e., OI

R = 32 cre) and limiting the plasma sheath thickness to four Debye lengths we could

probably run a 115 MA case to t = 100 (3,3 ns) In about 15 CPU hours, but we ex-

pect little new information to result. A set of 7 MA cases allcwtng variation in beam Ion

temperature and velocity, sheath electron temperature, and sheath thickness and density @1

would indicate how sensitive the high-current equilibrium is to these parameters and wouldfurther test the laminar flow model in new areas.

If an analytic model can be found that adequately predicts the wavelength of the

fastest growing two-stream-like instability we should be able to resolve one wavelength in

thc _xial direction sufficiently to determine a growth rate at moderately high (_, 2 MA)

current. This would provide a useful data polnt in the regime where the "mixing" region

thickness is much less than the be_n radius and whether or not it stays that way with the •

instability present.

O

O

32

E

O _

O

REFERENCESO

1. M. A. Mostrom and M. M. Campbell, "Equilibrium and Microstability of PlasmoidPropagation," MRC/ABQ-R-1026, Mission Research Corporation, Albuquerque,NM, May 1988.

t2. M. A. Mostrom, M. E. Jones and L. E. Thode, "Magnetically Insulated Plasma

Sheath in Coaxial Transmission lines with an External Magnetic Field," J. Appl.Phys. 52 (3), March 1981.

O

Q

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Q

33

O

APPENDIX A

Q LOW-MODERATE BEAM, CURRENT SIMULATION RESULTS AT AFWI

• FollowingisanexcerptfromthequarterlyprogressreportforSeptember1988-April1989.

A.i SIMULATION RESULTS

O 40) have used "nearly l-D"To reduce computation time (by a factor of _- we a

model to generate equilibria. Cases for plasmoid propagation with ion currents ranging

from 22 kA to 1.8 MA have been run thus far. This corresponds to the number of skin

depths across the beam, wb,r_/c, in the range 2.9 to 26.0, respectively. During the courseQ

of simulating higher currents, we noted that noise levels in the diagnostics increased with

the current to a point that made interpretation very difficult by 1.8 MA. Increasing

the current further to > 7 MA would produce nearly unreadable results. We, therefore,

decided to try reducing the noise to acceptable levels. A series of runs (on the AFWLO

CRAY) weredone changing the number of grid points, the "current smoothing" and/or

the number of particles-per-cell in the axial direction (X1, the dimension that is being

approximately eliminated)and in the radial direction (X2). Most of these tests were done

at 22 kA where the run time was short but the noise level still easily observable. The

• "current smoothing" namelist parameters for X1 and X2 (NSM1, NSM2) and the number

of particles-per-cell were adjusted separately or concurrently. We conclude that having

much more than 10 particles per cell and extensive smoothing with NSM1 = 1, NSM2 = 3

reduced the noise considerably with only two cells (three grid points) in Xl. This set ofO

parameters increased the running time by about a factor of two to four over the original

case, which still extrapolates to a run time for a 7 MA case of about 8 hours, a high but

acceptable cost. We then set up a run at a current of 7.2 MA which had run on the AFWL

CRAY to about one-fourth of the equilibration time when the "free" SD I computer chargeO account was shut down due to lack of funds. Results at that point were not close enough

to equilibrium to provide meaningful values. (Our expectation of further funding of this

account has not materialized after several months.) The following figures describe the

results of our efforts on the other cases to date.Q

A-1

O

A.I.1 LOW CURRENT: h = 22.5 kA (e.g., rb = 49 cm and n# = 109 cm -s) O

We begin our series of runs using the particle-in-cell code, ISIS 5.0, developed at Los

Alamos National Laboratory. Results and parameters used in this series can be found in

Table A-1. Since we were running ISiS at the Weapons Laboratory for the first time, we

felt it was necessary to rerun a previous case to insure the validity of future results. We

chose LANL run AM (h = 22.5 kA) as our comparison case. 1 Changing the run ID to

AA (Table A-l), we reran the simulation and got very good agreement with the previous

LANL case AM (Figs. A-la-A-3a). Since the amount of smoothing used affects the codes'@

run time, run AB was built by turning-off the smoothing in Xl and X2 (NSM1 = 0and

NSM2 = 0). The results from run AB were very noisy and considered unusable (Figs. A-

lb-A-3b). Although the results from run AB were not good, it did show us that smoothing

in the radial direction was crucial and had to be used to reduce noise. Run AL was created$

from run AA by reducing the number of axial cells to NX1= 2 and turning-off the axial

smocthJng (NSM1 = 0), but setting the radial smoothing to NSM2 = 3. While the results

were a bit noisy (Figs. A-lc-A-3c), the run time improved by a factor of 34 as compared

to run AA. Some of the noise in run AL might be attributed to the fact that only oneel

slice was plotted for the field plots, while five slices were plotted for runs AA and AB. In

Fig. A-3, run ALhad only two cells for the axial integration. Runs AA, AB and AL aliI

ran to ct = 3000 cm and appeared to be in a steady state (see Fig. A-16). A time-average

current profile versus radius for the h 22.5 kA cases can be found in Fig. A-4. The

three runs exhibit nearly identical time-averaged behavior. •l

Because of the fast running time of run AL, it was felt a parameter study could be

done quickly and efficiently using this case. Nine runs were created (AC-AK) by varying

different parameters and running each case to ct = 400 cm (Table A-l). The results from •

these runs show little or no change versus run AL at this early time and were not included 2

in this report.

A.1.2 MODERATE-CURRENT: Ib = 452 kA (e.g, rb= 49 cm and n_ = 2 x 101° cm -s) •:

For the h = 452 kA runs in this series, we used run AP (previously ran at LANL) as

our base case. 1 Ali runs in this series (BAl, BC-BG, BI) used two axial cells, except for run

BA2, which used four axial cells. Simulation runs in this series were carried out to different •

O

times.While most runswerecarriedouttoeitherct= 1600cm orct= 2000cre,several

runsran onlyto ct= 400 cre.FiguresA-ha-A-7h show B0 versusr,Er versusr,and

axially-integratedJzversusr forcomparingruns(BAI-BA2, BC-BG, BI)atct= 400cre,

whileFigs.A-8a-A-10fshow thesame typeofplotscomparingruns(BAI-BA2, BC-BF)

atlatertimes(ct= 1600cm and ct= 2000cm). Resultsand parametersfortheserunsO

can be found inTableA-I. For thiscurrent,a steadystateforthenetcurrentisreached

atabout ct_ 800 cm (seeFig.A-17).

Run BAl was the first run in the series and was used to compare with LANL run0

AP. 1 The results of BAl were much noisier than AP (Figs. A-ha-A-10a). Run BA2 was

built from BAl by changing IQ = 2 to IQ = 1 (the IQ parameter is the integer iteration

count for subcyciing the particles; higher numbers resolve cyclotron motion better) and

increasing the number of axial cells to NX1 = 4. Also the radial smoothing was turned-

• off (NSM2 = 0). By comparing BAl (Figs. A-ha-A-10a) to BA2 (Figs. A-hb-A-10b) it

shows the importance of having IQ _>2 and the need for radial smoothing (NSM2 > 0).

Run BC was setup by increasing the number of axial particles per cell. The results from

run BC (Figs. A-hc-A-10c) are quieter than run BAl (Figs. A-ha-A-10a) and shows

@ that increasing the number of axial particles per cell will help reduce noise. Run BD

was created by modifying run BC to become a _hollow-beam" case, that is no particles

were allowed in the first row of cells along the axis. The results of run BD (Figs. A-hd-

A-10d) show little or no improvement using the hollow-beam. Run BE was generated

• by reducing the axial and radial cell size from 0.2 cm to 0.15 cm. The results from run

BE (Figs. A-he-A-10e) are quieter than run BC (Figs. A-hc-A-10c) and indicate the

need for a reduced cell size. Most runs to this point used radial smoothing (NSM2 = 3),

but had the axial smoothing turned-off (NSM1 = 0). Run BF (Figs. A-hf-A-10f) was

Q built from run BE (Figs. A-he-A-10e) by setting NSMI= 1, and the results showed a

slight noise reduction by using axial smoothing. Feeling that the best way to continue to

reduce noise was to increase the number of axial and radial particles per cell, run BG was

created in this manner. By comparing the results of run BG (Figs. A-hg-A-7g) to run BC

• (Figs. A-hc-A-7c), it is apparent that increasing the number of particles per cell does

reduce noise. Run BI (Figs. A-hh-A-7h) was built from run BG (Figs. A-hg-A-7g) by

decreasing the number of axial particles per cell by one half. The results showed an increase

in noise and further proves the importance of having a sufficiently large number of axial

Q particles per cell. Comparing run BI (Figs. A--hh-A-7h) with run BAl (Figs. A-ha through

A-3

O

A-7a) shows the importance of having a sufficiently large number of radial particles perO_

cell. Figure A-11 shows time-average current profiles for runs (BA1-BA2, BC-BF)which

ran to times > 1600. Again, ali the runs exhibit nearly identical time-averaged behavior,

except for run BA2.

A.1.3 HIGH CURRENT: Ib = 1808 kA (e,g., rb =49 cm and nb ----8 × 10_° cm,_. OI

Only one run Was made at h = 1808 kA before the Weapons Laboratory account

expired. Run BB (Figs. A--12a-A-12f) ran out to ct = 2000 cre. The net current appeared

to reach a steady state at about ct _ 400 cm (see Fig. A-18). This run had IO = 1 and

too fewparticles per cells radially, and the results looked similar to runs BAl (Figs. A-5a-

A-10a) and BA2 (Figs. A-5b-A-10b) in quality. Because of the extreme noise, the only

results that were of much use was the time average current profile (Fig. A-13) taken atct = 2000 cm. O/

A.I.4 CONCLUSION

0]!t is apparent that several parameters have a great influence on reducing the amount

of noise in a run. Having the axial and radial smoothing (NSM1 and I_iSM2) turned-on is

crucial to noise reduction, and also IQ (the integer iteration count for subcycling particles)

must be set to a value ,___2. The reduction in cell size was important in improving the

simulation results, Having a sufficiently large number of particles per cell appears to have Oihad the most noticeable effect of all.

The scaling of the current radial-profile and the net current is summarized in two

figures. Figure A-14 shows the time average current profile for different values of the beam •

current, and Fig. A-15 shows the net current as it scales to the square root of the beamcurrent.

Figures A-16a-A-16b show a good comparison of the effect that the number of •

axial cells has on the noise in a simulation. Figure A-16a is a time history and frequency

spectrum of the net current for run AA with 129 axial cells with a beam current of 22.5 kA.

Figure A-16b is a time history and frequency spectrum of the net current for run AL with

two axial cells with a beam current of 22.5 kA. After examination one can see that the •i

A-4

' O

@

number of axial cells has only a slight effect on the noise. Figures A-17 and A-18 show@time history and freqt_ency spectrum of the net current for beam currents of 452 kA (run

BF) and 1808 kA (run BB), These figures show that the noise increases as the beam

current increases unless more cells and particles are used, The net current values used

in Figs, A-14-A-15 were taken from the time history plots shown in Figs. A-16-A-18,@Figures A-16--A-.18 also show that the net current reaches a steady state in a time of order

ct/r_ ~ 200c/wb0r_, Thus, the higher current runs may not have to be run as long. As

a rough estimate, the number of CRAY hours to reach steady state appears to scale as

tru,(hr) --, 2(wb, r,/13c) (NPC/40), where NFC is the number of particles per cell.O

By taking an overall view of the runs done in h = 452 kA series, run BG (Figs. A-

5g-A'7g) was giving the best results followed by run BI (Figs. A-bh-A-7h) at early times

(ct = 400 cm). At later times (ct >_ 1600 cm) run BF (Figs. A..-bf-A-10f) gave verye

good results. Again the major factor in ali cases that returned reasonable results was that:

(a) smoothing was turned-on, (b) IQ was set to at least two, (c) a sufficiently large number

of particles per cell was used (> 40),

In order to achieve results comparable with run BG (Figs, A-bg-A-7g) for 1808 kA,

run BB (Figs. A-12a-A-12f) would have to have IQ = 2, DXl = DX2 = 0_1 cm, six

particles per cell axially aad seven radially (42 total), A simulation with these parameter

settings would take about 4 hours of Cray time to execute out to a time of ct = 400 cm,@

@

O

O

A-5

@

@

@ 2,25c1_1, 10

/

@-0 05 _L.

-1 , ,0 13, 1 27,2 4.1 ,4. 55,5

@ X2(a)

, , , ,,

. i....

. ---1 .0 13, 1 27,2 41 , 4. 55,5

@ x2(b)

_ , ,, . ,, , _., ,,,,,

@ 2.23 - -

_o94 I®

-0, 36

-I 6_ [. 5 5@ X2

(c)

Figure A-I. Periodic simulation results comparing Be versus r at ct --- 3000 cm for three

ru,,,withIb--22.5kA,(a)runAA,(b)runAB,(c)runAL.@ A-7

_-2;._ 9 ..... I ..... Y,,_A' , ....

0,8,3 @I

r"

_0 26 1-0,30 "7 @

/6 I I I..........

--0,8_ "1 , 0 131, 27 , 2 4.1 , 4. 55 ,X2Ca) •

• I 0.,._--"I l ........,

0,57 -- @

-o,41 _ •

, I I I ....-0,9f__O,o 1,3, 1 27.2 4.1 ,4. 55,5

X2Cb} Ii

.,107--21, 6 .... I I I

1A

0,89 - " -- @

u-_O ' 02 i-0,84 @-

--1 ,71 I _--0.5 13,5 27,5 4.1 ,5 55,5

X2(c) •

Figure A-2, Periodic simulation results comparing Er versus r at ct = 1000 cre for three

runs with l b= 22.5 kA. (a) run AA, (b) run AB, (c) run AL,

A-8 @

O j

,_1Q -_ J 1-IIqT 1 .... __._

._,_6[1-_--':' .......--i-_..........'

• 2,23

1 ,20

• o,18,., y-0,8_ I I,g i_,5 2_,5 41,5......._,_X2

• (a) '

•. 1,O9--3 J 1-1NT1

2 2 ......... l ...... I-": ' I .........'i

J

Zl ,02

Z

Ib _0,07 ' I I I-o,8_............ -_--:---,s _3,s 27s ,_1s ss,_X2

• Cb)

_31,Q09'3 JI-INTI-- i I'

I 1 ,99 --

i--

z_o,88 -,I I

® ...% t

-0,23I

_ / . J ,._ ...., ..--_,__,_-__,_ ._,_ _;,._ _,_Q x2

Cc)

Figure A-3, Periodic simulation results comparing axially-Integrated J_ versus r at ct =3000 cm for three runs witt_ Ib = 22,5 kA, (a) run AA, (b) run AB, (c) run

-0 AL,A..9

O_

LLNLPLASMOIDPROPAGATION ._

12

"_ RUNAA 01

10 ................................................................................................................_72_......................i ........... ---t ....RUNAB

O "

0'-- 1 I I [ OII0 20 3O 4O _ 60

RADIUS(cm) °

O

FigureA-4. Time averagedcurrentprofileInsteadystateforrunsat[b = 22,5kA,

O

e

O

O

• ,A-ll -

e

V

O

" LI. L PLASMOIDPROPAGATIOIq

Q , RUN,BAl40 ,_,

RUNBA2

< RUNBCII,5 20 ............................................................................................................+ •F-Z,,, RUNBDtr"rx 10 .............................................................................................................................C3

u RUNBE

• 0 ,. ×,

RUNBF

-10

' II -20 I I I I I0 10 20 39 4O 5O 60

RADtUS(cre)

0

Figure A-1I,. Time averaged current profile in steady state for runs at I_ = 452 kA.

ti

O

II1

• A-17

II

® IdOL PLASMOIDPROPAGATION

100

II

8O

Ib-20

-40 I I I I I

• 0 1,0 20 .30 40 50 60

RADIUS(cre)

QFigure A-13. Time averaged current profilefor run BB near end of"run with A = 1808 kA.

Q

Q

• A-19

[LNL PLASMOIDPROPAGATION •

-40 I i I I i II_,0 10 20 30 40 50 60

RADIUS(cm)

Figure A-14. Time averaged current profile comparing different It, for three runs,

O

O

A-20 Q

O

I

,, LLNLPLASMOIDPROPAGATION

0

2O

• 0 1 I 1

0 500 lE+03 1.5E+0,3 2E+03

Ibecrn(kA)

OFigure A'15. Net current as Lt scales to the square root of beam current.

O

O

O

• A-21

PLASMA SHEATH RUN AA, Vl_,,,,e3, N_,,,O,2,,NP,I1 ,E9, VTH(2)-.C) .30g, _7 9 4_

7,40

* g4

247 -- OI

I 1 I

°'o°0_0 o,7_ _._o _,2_ ,_3oo_ .,olo 8 _ _P , L_M 1

1o 7 ]_ @J

'°:

,o_ _k _ J[,oo_ _.,.,L,_,,L,A._ ,A, _l_,.,. j_, *_• 1 ,g6 .t g3 5,89 7 8.=,F'REQ • 1O - l

(a) T,Mm-299e,ooooo

Figure A-16, Periodic simulation results comparing time history and frequency spec. @.trum of the net current for two runs with a different number of axial

cells, h = 22.5 kA. Current in units of 1..35 kA, frequency in units ofWo = 3 x 101° rad/s. (a) run AA with 12g axial cells, (b) run AL with two_ -._ I 11

- U..,K1_ 1 CellS,-

-;. A-22 @

O

IIPLASMA SHEATN RUN BF', VB-.,,6..3, NB-.O,2,,,NP-,2,E:IO, VTH(2),,.O,30

,_. 32,044., 4. :i ...." '_ ' I...... -_ ""I.....

3,..3,Q

22, JI

O- - ............ .......... , ......0 O0 0 50 1 00 1 50 2 3 0o

,,10

10 10 _ I ............ - - '"' I

® lO g !

10 8 1_m,

e'4 '7

0 6 1

, 0,4.9 O,97 1 ,4.8 1 ,95F'REQ

TIMEu 1999,37000II

FigureA-17, PeriodicsimulationBF, time historyand frequencyspectrumof the netcurrent,Ib= 452 kA. Currentinunitsof 1,35kA, frequencyinunitsof

• CUo= 3 X I0I°rad/s,

0

'@ A.23

OI

PLASMA SHEATH RUN BB VB',e3, XlB"_ 2,NPuS,EtO, VTH(2)'0,30 II° _ 4 P

19g, I j

138,

76,15,

-4(_",00 0 50 1 ,00 1 ,50 2300' ' ;]::,LMI_ ,10........._Pr__1,r_lJM .....

10 11 _'_--:'-.....- .... I ....... -...._......I...............'_

10 10

10

0 7

1o 6 i_

5100,00 0.98 1 ,g6 2,95 3,9.3

FREQTIME- 1999,2OO00

lt

Figure A.-18, Periodic eimuliLtion BB_ time history aild frequency spectrum of the netcurrent, Ib = 1808 kA, Current in units of 1,35 kA, frequency in unita of

wo = 3 × 10I° rad/s,

lJ

IL