:ENTERn-1 REPORt' 00 AN EXl\C'l' ANALYSIS OF ALIMITEO PIANE PIASMA m A

MAGNE'l'IC FIEtD

Ifa'C£'VEDLAWRENCE

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Lewi Tonks

on

AN EXACT ANALYSIS OF A LIMITED PLANEPLASMA IN A MAGNETIC FIELD

.. -

STR!

I. INTRODt£TION

Prior analyses of a plasma in a magnetic field have been limited, as far as

the writer is aware, to cases in which the relative change in field over the

orbital distance and the relative change in ion concentration are both small, or

in which the relations have been viewed in a purely hydrodynamical way. The first

approach excludes cases which can be of considerable interest, for it fails for a

plasma edge. The second loses all sight of the structure imposed by the orbital

motions.

II. FORMULATION OF ffiOBLEM

The present analysis attempts to overcome these limitations, although it

may impose others. The problem attacked is that of describing exactly the re la-

tions which exist between a limited plane plasma and a vacuum. It is assumed

that there are no collisions, and, for the present, that positive and negative

,;particles are present in equa1 numbers, have the same mass m and are mono-

energetic with speed Vo in the plane perpendicular to the field.

{ It does not seem possible to start nth an assumed density distribution

because the distribution chosen might well be inconsistent with the existence._~~ .

of orbits, and the problem of distribution in velocity space reDBins open;

certainly at the plasma qoundary there are no particles moving perpendicular to

it. Nor is it possible to begin with an assumed field distribution. Seemingly,

the only approach is to describe the plasma as if it were being built up from

its face inward essentially in the way chosen here •.'

UNCL"~S!FIED !!?

•• I It.

• • •• • t • t• I I . • ' ) J

• • t • I • •...' .'• • • • t J

• I • '" •• . I., 't.• I' I.f::. :':"1,,. . . , '.

•• •• t t)• • •• • •.. t •• ,

••.- 2 -

) , ',~. )

~·)~It.)

• ••'. I' t "1'1'1 .~J ', 1 " ~

•• • ..t'tl'~ ~ "

• · l'I \' IJ'lJ .Il~',.t" '" I "':'rt:.~ .~. tiI!tVL.. ,-.,,\;u l~

Referring to Yig. 1, the vacuum magnetic field Bo is uniform and. in the

z-directioll. The plasma is uniform in the x- and z-directions and extends to the

right from y =O. Consider only positive particles which, in a f~ld 13 have un--an~la.r velocity W = eB/me. We now define a Class J} particle aa; one which, en)

-crosses an element of area at g h, / '

Co'·f..::J , (b) has its velocity lying

v.'ithin an elClncnt of anale d So pointing In the negative x-direction, and (c) crosses

within an elem=nt of tire dt at t = O. The particle distribution in the rlazma

is completely defined when the spat 181 ,listribution of Cla.sl:: 1 particleo is fixed.

Let this be 0 ( J) ) • Then there are

ef(~) de di) d ,~' dt (1)

particles in the angle-space-time element. The angle de embra.ces a bundle of..trajectories whose point::> of' minimum y (y = 0, y >0) which will be called their

apogees, lie very close to d? d S a.nd extend over a range d J of 5 given by

d! = (vol w ('7 » de

These trajectories diverge until at some other locale, x, y, they cover a ranee

of dy given by

dy • h.1 dSas t,'7

where, of course, d yId ! refers to the family of trajectories

x = x( 5, 7' t), y. y( s, 7, t) (1. 5 A,B)

At x, Y the Class I} particles will occupy an x-extent of

•dx = - x dt

By direct substitution 'We now find the number of Class p particles at x, y to be

expressible as

•••• t J •• •• •• ••..) .

• ••

dx dy dz dJ) •·..".. : :. .,• J t • •." .

• • • • J.. ' .., I' •.. " .:1':' I': ..• • •. .

••• •• •• •• •••

I' ) } I" ,

l,)},l .~"J),tllt

-3- ,'1 J ) l'

~::. ':~.\,,' '\ \\j'"~\c\ I\c.'S\f\EO1 , t •• • ,I-t\\), . .

t JI •• ,

This can be simplified by noting that the uniformity with respect to x assures

'that when t is eliminated between Eq (1.5 A) and (1.5 B) Y will be a function

of' x - 5 rather than of either coordinate indiVidually. Accordingly....---x OY/8 5 = x oy/OX = y ( ~, y)

We note further that each particle JTJal".es a double contribution to the density,

once on the outward leg (y > 0) and again on the inward leg (y < 0) of its journey.

It follows that the density of Class ~ particles at x, y is

and the density of all of the positive particles is

n(y) (2)

where, when y is small, ~} (y) =0 because every trajectory encountered between

the plasma edge and y reaches to y; but when y is so great that some trajectories

do not penetrate to it, then >J(Y) is the coordinate of the apogee of the trajec-

tory whose perigee lies at y. A more general point of view is that trajectories

are present throughout the vacuum as well as 'the plasma. but that for IJ < 0,

6" (fJ) =0." With this concept no Juggling of ~ (y) at the lower limit is neceS­

sary. Such juggling can, however, be convenient.

The current density of all positive particles 1s eVidently

A further integration is required to calculate the magnetic field change.

B(y) B8.n-e rY

d,1';(1)W (1) x ('2,Y') d (4)

• 0 -t v J YJ I· ( ·\\ ~coo 1(y') y ,.,!.:.: '"'.. ·:"I.~'('if n.f;~.It:'ED" U'F~vl.. ,'-'--'.iI" '

:':". t't,'• :. • J :. t

, • • J •.. . . .

. ._"4 _

The equations of the particle trajectories are the usual

Y•• - ..' eB(y) i. ; = eB(y) y. ,-- me ' me .-....

(5), (6)

~

and-:-because of symmetry 'We can proceed by letting tf be the sum of ~oth positives

and~negatives •....

This set of equations can be started toward dimensionless form by the sub-

stitution of the dimensionless capitals for the lower-case variables as follows:

w =.n eBo =.n U6me .

(7)

t I: T/Wo; (x, ~, y, y')Vo

= -W o(X,H,I,I') (8),(9)

The new equations are

(10), (11)

dH

= 0 j2

d Y .. .{) (Y) ~dT2 dT

y yt

n (Y) • 1 + 8»e2

Vo f dI' III 6 (vo HI Wo){"l {H) dX/dTmc

2W o

2 j IdI/dT I"""0 H(y t )

~ _{2 (I) dI = O.dT dT

It only remains to put

(12)

to make the field equation dimensionless also:'

11(Y) = 1 +f d;'11

' S(H)J2 en) dX/dT dBdye /dT

. 0 H(Y~

(13)

~:t,now, we apply the same conversion to Eq (2) we find

... -',-;"t"

sen) {l (13) 'Ul. dY/dT (14)

UNG~~SS\FlED.' : : ...

• t' ." • J

't .. t, I • :'• ,.. .' l

't t., .:,:. ::I' • :" '1) ,',,'

• • J J '. ,'I.J : •I' , tj. J • J I I J

.~'.. ',;~.t:.,_._. ~

t' \) "} • t _ ~ J •

1) t '.) ,.l)}l tt 1.. "... 5- - ~ I) • I I ,1' ~ I) ~ "

III t .' J1'I ., t " • J

I • • • , .. I • tt ... • I)' .

' •• ) ,t', ."...~. -~•

,In the absence of collisions there is no natural distribution or particles

to, 80veruthe choice of the build-Up function SOt). '!he Simplest choic:e is

, B(B) n(R) • S,constant,andiu strong plasmas it may be that s(!> '. So is

'1IOreusetul.

III. ,SOLt11'ION FOR WEAK PLASMAS I

The solution or the equations is a machine Job aud is being undertaken on

:' the Univac. Meanwhile, the only order of approximation which it i8 worth while

to 'try to solve analytically is that in ,which the particle density is so low that

the depression of B is so small that t~ deviation of' the trajecto~1es trom

circular 18 negligible. Then the orbit radius i8 unity, and tor the trajectory

and for fixed Y

Y • E + 1 .. cos ~

. dt/dT • sin T

d2y/d'i2 • cos T, • ... OX/dT

au • ... sin T dT

(15)

(16)

(17)

(18)

The t1rst integration in Eq (13) tor S(H} =S, constant, 'is conveniently

carried out in the two regions

(A)O~ yt < 2

yt

I (x'" H .. 1)S dI/dT

o

y' ,

_ '1' H cos T sin T dTdH .. S 8in T :-s s1n [cos-l (l-t t >]

[, 21 1/ 2

,=_-6 1 - (l_yt) -I' ','

•••.. • .. »• • • •. . '.

• .. » t·, '. " ..',) ) -' .

•- .. ."• • •• •. .,

, • J ••.'" ... ' .,~ • J."

, • J J

" ,..1 I') /) I,

dH ·,0

I'I

(B) ~'t: yt'

. . I'si {yt .. B ... 1)'dY/dT

'-2..."

'. ... 6 ~

Using

1,. 008.1 (i-Y); "'-0 '.s, f.5:- 1r

; 'I.. ). ") t»'»J . )} "' 1 .'1 'J ) J 1,1, ) ) .. 1 JI,),1' . )JI)'II af', ')'

.,1 .. It »'.t'. ", ) Jt· , .t • • "

t " • ~l'• • I • I •• • •

i .. ."•(19)

-' ,.

'" 'the" second, 1utegratlO,n,leacis' to

.n (t) ~ l~f'".',(S/!!) (,If-~1n, 2,",~,'/2);. -61f!2J", " "

'. . - .. ", .' " '. :.:

..-..o So t~ 2 .

2~Y(20)

¥hiCh ;oepreser,rts th~ depression 01' the magnet1cfield in dimensionless tOr1ll..

'l'he~uatrtity (2/-rr). ({l-l)!S • J2/1T)AH!(RoS>'l8 plotted against Y as Curve A

~f' Fig. 2.

1'urning to the partlclaconcentration given oy Eq.(14):

(21)

, '/'In Fig. 2, n(Y) 18, ihown &8 Curve B.

Now we can compare the depression in the field to the })Uticle concentration

and it will suttice to do this tor 0 ~ y ~ 2 because forY > 2 relations re-

_in conStant e,

, .

nCT)' eo 1, .B .. Bo ' = 2'ftm2vQ2 (1 _ 8in 2 t)n(y) 'Bo Bo ' 2 f

The best rot.J8h ~Pl."ox1mat:l.on tor the introduction of temperature '1' 1s to put

~aud:Puttlng,

f _. to;. :', ,-, ..,'

, ',".'",. ".

¥;e' bave1', .

,11 .. Bo =~B "

,Bo (B ~ Bo> : ~(B2) /2

, . ,(22)

"

: '

.".. .,'J.. : :. ..., I 1 I •

• J) J. i J J_

••. )' : ~ 1 )

.J l,

.,.r ,I J) 'I1 t } I

} /) /., / .'

I; J

"

; .

~ \. ' ) t l )

)l ~J)).

7 • ) ,- .. " ·Jl't·

_\ :" . • • -, "£ • J .),» I. " ) ~

.:-. '. •1. I, t .' I. t ) 't ~; '.'. It • ). .' t

'. • ......tl

, ,}

'the 'bracketed ejpressiou should 'be rep1fc:.~\'9.Y. ~tlJ" f = 2 value,•

'1IUIely, unity, &nd. 'thus the generally accepted relation is conf'irmed. It is not

'strause tha'tiu the bf>\U1darY.1ayer, Y< 2, 'the accepted relation should be de-

. ,partOdtrom because theret~e plasma is not isotropic. ~is deps.rt~e is shown, ---

'bYC,UrVe C of Fig. 2,·1ibich is a plot of the negati:Ve of the braeke~ in Eq (23) •

•' Using an analysis based on the BQltzmanu Equation Dr.L.· Henrich has recently

shown 'that in general

(24)

were the bar denotes an average'. This is eas11y confirmed for all Y in the

present case of lIeak plasma and constant Sen) •

IV. ':mESSURE Ilfl'EGRAL OF THE EQUAfiONS WITH A SPEED DIS'mIBU'l'ION MID 1Ufi PLASMA STRENC1.I'H

For simplicity we shall make the substitution

and shall use

. WCH) = S(H) .n. (n)

o c>o -?Y: aY/aT, Y =d2Y/a:J.'-

In 'the integrals of Eqs(13) and (14)

l' 0

'Y • Y (H,Y)

.but, alao·• 0

Y • Y (H, Y (H,T»

(25)

(26)

Accordiugly

" .

•• at 0ya: - Yay (27)

, Eq (14) applies to particles of a particUlar speed, s..ay v, When there is a

,distribution of epeeds, and this is indicated by re-writing it:

•• I I".It : : )l,

• . ' j)• •. , • , i I

, »)" . J J

, J ' 1 I I;' ! J

./ I' :) J . I,

. ,

. ..',, lJ~t)"

- 8 - i,,'" tt) 1,.,.1'. ,1tI"' ,t ,"

,1 ) 1 1 •• ,") • t • • , ,(I 1 • )) ) ., .

rY(V) . ;"

j ~ dHCv). Y (B(v), Y(v»)

B(v,y(v»).----

(28)

·tbe dependence of Y and H On v arising through Eq(9) andt~ subscript

v = indicating that the l1umber or quantity ot the variable attributable to

particles 1n the speed range dv is meant. Then, using the lead furnished by

L. R. Henrich, we write:

B 2 vQ

Wo- y- - £y (Y

iCY)

• ;r;.li:~;&..2m""';'vWo~ { Wv (Y) Y(y,Y) - Wv (H) Y(H(Y),Y)

y

+[ Wv (n) (~i/ay) dB 1H(Y)

o

The f1rst and secoad terms within the braces vanish because Y at a trajectory

apogee, B· Y, and at a perigee, H· H(Y), is zero. To the remaining term we

apply Eq (27) 80 that

dn,,¥ I:

dy(29)

S1m1larly it we denote that part ot aD/ay which 18 usociated with v­

speed p8.rticles as .n IV 'We have from Eq (13)

iy (l

£2 ' !..... 3t(F Xv (Y) : w 0

(Y)

.,.t ....

J J ....t,. J J

, t ) ) •t t)>> ....

t 't J t.,

Can be taken outside the integral•. . ."• • •• • •. . . .• • • ) t

I • t •

.. I • t J

) • J .. )

: : ," I t J J,.} J J. t ) I.l, J ) ) 'I t

Using Eq (10) and noting that £2 (Y)

J11 (Y) fl· v (Y) = - !..-

1 ~ .. • •

'liti 1:,t:"".'1 ) '.:::9 '- I J I I I ...:: t: ,. -It: "PI

., 'I,.: .: : 'I,•• I • .1 •• •,I. .. •

•

.-....

(0)

Ttie integrals in ~qB (29) and (':~O) are identical so that it becmies apparent that.-.-

11(Y) .0. tv (y)

.nd m'Jltiplytng by dv, integrating over all speeds,

J ~I f ~ ~fl I V (Y) ,av =..1 L. . (Y), nv y- dv = n y

gives

d-dy

2 :2(B + 8 1T' m n y ) : 0

which is valid for any velocity distribution. For the M!&Xwellian case it becomes,

as expected,

~ (B2 + 8'fr nld') • O.dy

V. ffiOBABIE NATURE OF RESULTS FOR STRONG PLASMAS

As a result of the calculations with finite S(H) we shall establish a

set of curves for magnetic field v.s. distance which I conceive will look like

A and B of Fig. 3 with increasing S(H). For a still larger value, BIBo will

fall asymptotically to zero. For a further increased SeE) we shall find BIBo

cutting zero with a finite slope and a s~trical solution shown by Curve D

beeomes possible in which a plasma of high enough concentration is held between

two regions of oppositely oriented magnetic field of equal magnitude. What my

be the physical reality of such a situation is uncertain, and I am very dubious.

Mathematically, the region shown could be matched by a second 'Plasm layer to the

right which would restore the vacuum. field beyond it to the original Bo value... .

, .But nowhere would the plasma thus "ConSV.,l;'tep" approach uniformity and it will not

. • • ~. '. I. • ,

be possible to establish a self-'~c!ns1sterit 11>188ma with a distribution of velocities.II', 1 '.. • '" t" • I "

','. 11' l • '. 'I I.J 1"j ;

'),.:,} J/.t ••~I, t' .1.

'I - 10':: It: 't. II, '.:

'. • • I •• I'. • • . t' •.,I. • I· I, • .)

. ' "'.'"- .... .. .VI. ELECTRIC Curva:NTS AND MASS MCYl'IONs IN THE BOUNDARt -ragP,.

At a distance Yo from the p~as_ edge where the plasma is assumed

to be essentially Maxwellian and uniform 1JDagine a plane P as shown in Fig. 4.

To the right of P 'trajectories are circular so that there 1s ~o mass drift.-..

, , ~tany point all directiOD$ of motion are equally likely tar ~ velocity classes

40 that there is no net current density.

We shall be interested in the total current per unit s-depth of plas:ma,

and 'We have just seen that all of it lies 1#0 the lef't of P. There we analyze

the electric current not by volume element but by particles in their trajectories.

Some trajectories cross P, many do not. The current, in terms of 6{2, arising

trom the former. viII be denoted by J lI from the latter by J2 , and precise mean­

ing 1s given these quantities by writing

(32)

The cut trajectories are responsible for a current because, as with tra­

jectory t 1 of Fig. 4 the particle makes its contribution to Jl by its trans­

port from A ,via B to C; its transport trom C to A is balanced, as we

have seen, among all the other particles to the right of P. Particles in cir­

cular orbits like t 2 which 11e entirely within the region contribute nothing

to J 2, but for particles in trajectories like C3 their progress from D to E

in each cycle does contribute to J2 •

Now Jl and J2 can be separated out of the double integral in Eq (13).

The area of integration is shown as O~DEO in Fig. 5. Since Y lies in a

~iform region the orbit there is circular, constant in size, and actually ofradiu6

(dimensionless) 1/1l(Y); and H • H(Y) is parallel to H .Y. A little reflec-

tion will show that reversing the order of integration is physically equivalent

(1) to selecting one trajectory (H • constant) and smmnirig its current contri­

butions and then (2), summing over all the trajectories involved. It then be-

comas evident that theI

cut trajectorie~ ~~~those, • t J ) J

., • I J J

". • I, ". • lJ I l) J })

~ It J ') J ') J ) J J S 1

" ," ~) J ) J \ } I.) I •, 1 •

I "

included in the area CDEC. '

For this the double

. · t.: t)' ).~ ,," ,J• • '. t• 11:-: ",' t, t t t.. '.:•• _,I l 'a,

I. t' I ."" • ) I" ) ott

'" J • J )1 • t • t ..• • •

integral for a single 'velocity c'"iAse. 'is

-is necessarily consta~t and the equations of motion are

Y =H + ..11-1 (1 - cos wT) , X. _11 _lily

we must now determine By to be consistent with a Maxwell distribution

Applied to particles of speed v Eq (21) becomes

• 2 nhm e-1Jmv2 V}h. 1/(2kT)

Therefore

00

s· JBy dv • 8 n'e/Bo2

• (Bo2 - B2)/(1rBo

2)

"After Bumming Eq (33) over all v we then have

J •_

1 - 112 (1 -n) (1 +£2 )• - 2 n1 . 2.Cl.- ..l. L

and from Eq (32)

These relations lead to the following comments:

(1) J2 is a pa.re.magnetic current which is more than overbalanced by

the diamagnetic J1 •

(37)

(2) In weak enough plasmas where il 1s only slightly less than unity

(Yery al1ght depression of B) J2 is r:aegligible compared to J l which is a

cOnt1rlatlon of the earlier treatment ot the weak plasma case.

(3) As B is severely depressed,.fl approaching zero, both currents

grow without limit. For a particle of aver~ velocity the orbital radius in.. ' . ... . ,the uniform plasma is au. ~<(b..\ fl.) .~ ~~!,le: .actual current corresponding to Jl

" ,I .1" "',', '.'.· , .' . .., .• J I J t .. , 't', '

, J I) ,) I I J t

1', I I I I'

Al.o

t .. i •

• • •J ': • :••

t ': 1 l t• • ) • I. '. ' .t ' •, '"'- .. -

wUl be of the OII"4er of the product or this Ad -DeVI 'or -au "'". !'be api-

'tude of .12 wU1 be of the order or b Usc Tne where 'b is the thicmea. or

the la,.r throuIh which the paraapetic currot ,utenda, DSC l.f.a .. av.r&ge

~U8ity or 1\1141118 centers, and 'it> 18 the Imtrap paraapet1cZ~lttft1oc1t,..~

,_Iqua'tiq IJ1 I to J2 tar the very .trong plus cue leads to

b~(t)(~) au

JIoy the &'9Wage particle a.tDsit7 18 leas lu tbe boUDC1ar7 la,er aDd beDce the

guiding caner 4eD8ity is less:

-1lo > DSC .- -T 1. an upper 11ll1t which TD e&D 'Deftr Marly reach 80 'that..i ) vD

probab17 'by MWral told. Accordln&l7

(40)

•

and the bouIl4ar7.~ 1s leftral tia. as thick .. the ra41ua or an averap

orbit in the UIl1fona plua. For. weak p1.&lJE. .. haft aJ.read7 .... that 'the

factor II act lea. tban 2. '!'hi. eu-t. couiderab1e 40ubt 011 tM appUcabUity

of eleevt.ca1 lkin effect 1deu to atroq plaIN••

(4) De current J2 18 as.ociated nth a -.• .,-ttoo of 'the charpd

pe.rticlee Which will t:ranealt 41eturbancee in tme regiOD of the 'boaa4ar7 ~r

to adJacent restou, ud suspne that bere _y lie._ cause ot 1D1'tabl11~y•

•.. ' . ..• I •

• • • I, I, ••

t l """ It • "I " • I"

I, I, ", ,'" ,.' It

• I ,t' ,-.',. I,I .f I _f., • I

• 1"', • II ,I I, I.

, • Jl

t J ' •• • •• I •J " • •

I.

..

.-

Tk',AJE

) t,: J

FI (;;':'. ·1 ", ·'. .' ) .:. • • , I ••• • t ..t J I

( T ai<~ 'E.-If: ,I.N 'iP/i... /..tvI) A-I • ,.J. .. .

•

j:, "

" )4·If

:.. /

.....

Ii

".J

--."

; ,

/.

'. ..l....' .•~"•

.~,....oa ..... ,

,<'\1:/.

e.,

; {

I,.... !i'

• I "''N(I;n:'''~1!

(.I i"!

........ ;.._.., ......... "'-

......,".".,

\.

'.\

,-

I"

I •, ', I• 1

1.1

•.. ' . -.~ . ~ ' .. • • , I. •

~". ,.~ II" .•.•J' • ~. • •• , .

" . .. ... .,••• - Jt"",

• J ) J I...I. 1 t, ',' » .• • II " .',

..

.....

.~\ . :

.~ .

..... "

~ .,J 'r.• , .,',

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