Diffusion of a plasma across a magnetic field

DIFFUSION OF A PLASIA AC£OS A MAOMNST C FIELD

a~IaD STsPH FIN

B., NOasachusetta Isitute Of Teehnuosq

(1959)

SU~UTTED IN PARTIAL FULFILISW OF THE RSQUIRMWS F THE

MgR OF MASTR OF SCIg c . ss. INST LIBRAREY LINDGREN at the ASSLOCUSETTS INSTITUTE OF TECHNOLOXI

lOt .oJa Aut96

of Gm01w en4dAgIphsics, Hy 29 1961

CerLtied by . .

.1 n

Thesis Supervisor

Accepted

Chairma, Departmental Committee on raduste Students

Abstract

An invetigation of the problem _{of plasma diffusion through a} magnetic field is made through _{the macroscopic fluid equations. The} problem is set up in _{a simple one-dimensional geometry of a semi-infinite}

half-space filled with plasma facing a semi.infinite half-*pace containing a ~unir magnetic field, and the _{equations are used to describe their}

subsequent behavior. _{An attempt is mede to solve the equations} analyti-cally, but their final solution is obtained numerically. The solutions

Absi'omt I* Intdmitoz

.U, *OsawaI Dismusio ₂

MI. Ths Infinite On.. Dimmrnanu can

7

IT,* AmIA17tma Intogwation ₁₇

Thare has bson very reomtly a greaty increased interest in

sientUM _{and engineering applications of tonised gases and plasmas this} has dwsnded ftom known theory a quantitative idea of how pL ase behave under given enditions. An example of these is Vt. problem of plasm

oontainment by a sagnetic fleld; or, stating it more generally, the inter aioGn of a plasma with a magneti k tild generated by external sources.

Mhis plasmnfield interaetion whrth is the subject of this discussion, has a B of aspeets that complicate its analysis; tbe complications resulting in soulinear differential equations. As a mcnsequence, the inmatigation of this interaction is limited to a simple example; this is to make the equations consernd tractable enough to yield quantitative data with a reasonao amount of effort. To illustrate, the processes considered can be crndely deseribed as 'Naaw* _{mations of} plass through 'strong' magnetie fields, in which the electromagnetic and thermo ni c properties of the plasse do not change appreciably. What is meant by 0ao _{mations and "strong' fields will be made clearer further on. In}

addition to this, ti. geometry of the proble is reduced to its simplest form. Doing this naturally limits the applicability of the resultine data;, st*il it produces a good approximation of ti- plasmanef d inter-saoan in a variety of cases.

Te formal stateant of tats prebla most amenablo to qualtative investgati n is in terms of the no*alUd seroeeeapt equations of

tion. esentialy these are the equationa that re lt from averaging thie mtions of many particle this averaging being done by integration of thn bitUman eqat±*t in velocity spaceo. y Mseas of this integration

pitser(2) defines a ou~r t dnsityn as average iSid velocity %# and

derives an equation of fluid ation and a generalised tfr of Ohs's law. In the case to be oansidered bore the fluid to asamed to be

eomposed oa of sangly chargi d ions of identical asses and of elecbtrone

to generalse to t a mber o types of ons does not coplicate the sito ation, but it

St

Inftr-mstin. A mnaoasd" gas as *eeea simply to -* the gas oonetant f

eqaol to 5/3 in lalations done later. 1Fr this caSen then, the equation

of motion take the fram

thare B is the agnetie field strength, p is the seaar pressure, and p is the mass density of the iosn, This is asi ly itler's equation with

the addition of a Xj I f raOe.

There are certain assuamptions aready Implicit in this equations the

mass of the eleotna aW the gravitational fOrce are both negleoted, and the amber densites of the electrs and iLns are taken to be exatly equal at all points in t3e fluid. B ssee of this last assumption, there is never an eleotef field dua to pace charge. Also, viscosity eaffocts

are neglaote _{llowig what is actually the stress tensor to be tsan} as a sUcar pressure. Aaoounti for direott al pressure differenes

dae _{to external ma4te"g} c fields by wdag a stress tensor with at least ditferent nournese diaoaal terms might seow actualy anessary for this tip. of problm howavw, it rill S rn out tat t this os a s wt e

differen in the saculations

Also for _{this case5 the ensrased Oha's ]A reduces to to the for}

"J

=

E

+

kxB

where 1 Iso _{Sti eectrical reisatvity and e i twe electric field. On}

assuaption made here is that the resistivity is a constant scalarl once

again, rwiti it _{as a diagonal tensor to acooOunt for external magnetic}

fields & e not affer t the calculation. It must be admitted that ae. g~eMg its dependence on the current, tSperature, and fluid density considerably limits the -usefless of tU _{solution; nevertheless, this} is the first assrupton that will have to be made strioe1y 9br simV-pi ficaton. Anther eassumptan is that a terMn is neggibly

nall

applied.

The two equations _{above resulted from the "plasma properties of the}

fluid in this proble . The _{reumining equations to be introced are of}

a sore general nature. Flrst i _{there are the continuity equation}

and the equation of .ats

=

PRT

here R is th e gas eonstanat a T ir he Sapertre. In addition, thare are the Mawsell equations

and

J = V-x8

The arrnts due to polariation and the tia rate of change of the electric field have been mngleotd as being smll oapared to the magnetisation

curret watch is incm ded in the

V

?lnUy

+

j

where , it the oWnsta-to a owlm heat mapacity and

Y

heats, This equation implios two vry aportast assumptionst one, that She 0 or sour t energy is ohbie heating ( viscosity having alread been

n glected )g and the ther, hat there is no heat transfer in the fluid,

ditaeio s an rtci ear than the preoe to

be

mw tabulating the equatims obtaind

-'14 9t

Se(

t)

=

+

t (p)

V

-V

,

p

it is seen that tha*, seven equations ntaining sevn viaria e. It soy als be ssn that tin #quations re nnlin ar, wvt4ch Iamdlately pro-ads anW strn4ahtgbrward analytical integration of them. Thaus proceding with ther ia tio reqUires some additonal sipificateions.

Jbenfw continuing wita thi 9, it is astractive to rougly sketch seas

j x7

-v

pImpertis oof pleas irn etrong magneti* fteids predued by constant exernal earrents. ere "strong fmaget i. fields" means tfieda that are moh larger than sq field met uap 1 cur aents in the plea. to tie fLrst order, n mfor e ti e line tend to be 'fse" in a a highly oudtacting gas. If a motin is imparted to the gas, the foree lime are wdragged

along* wih it. To the next ordsr, trrent dissipation due to the finite resistivit of she gas allows a relatve aetiet between the gas and tbe

field lias. To see th smea clearly, one can lmagine a claster of s* ternally produced parallel magunee tield lines in a plasma with a finite resittvity. Beause tbe pleasm is resistiv*e the crrant around an area in it will have to decay if it is not regenerated by externally

intaesi4-Iag he fields. Tias the number of flux limes passing through the area will have to be reduced, and if their total number is oonstant, some of

them mast appear outside of the area. Therefore, the cluster spreads out ard what the whole proeses 'orks like' is a diffutsion of the field line

WV~* O2flflI S&" WTfll*~ tam 0#UUf*J

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fruv

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x

pu*Vs

't

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i

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#

V;

n~

TMnf

iri t

ST'R So

fldowo

fl0

%'Ind

F,,

pwrW

-covertfg the x * 0 srfte of the pl as.

To seek a solation for the behavior of thisyastm for timest 0, one £irst eonsiders the equation of nation

S+

( )

J

x

-Sice the plasm is always uniform in any plane perpendiclar to the x-eais thUe V ater redee to /a _x . _{Also, if the l-field is always} onasld-erd in the s-dLrectiou, the not currents in the plasm always £Low in the yedirootion. _{Then the equation of motion becomes}

+

r

x

New it may be men why it was not neessay to regard the resistivity and the pressure as tensors _{JEven if the tensor form had been retained, only} one term in each aould have been used.

At this point the mriterin of eslowness" is again applied to the notion of the plasma. This smea _{that the teras on the left are negligibLy}

small, or that the ters on the right nearly cancel. Thbs

J _ax

Sinee for this configuration the equation

- Vx x

becae s oaly hao a twntional dependoenc on the xn oPrdnatetela equatSon of motion then becomeM

and this may be integrated to give

+ = CONSTANT

-Tbe r/2 oy be oonsidered asa ua tite pressure0, and the total pressure at all pott in pace remains constant as the manetic and plasm

preseures vary. Sitqly from this equation it is possible to see that the

plase leaks alowly into ti magnetic field while the magnetic field corre-* poetAngl leaks into the plam. Neither the pesma nor the field has a

sharp boundary at t > 0; and the current setting up the magnetic field boomes distributed through the plasma rabter than being eonoertrated in

a shtra . A plot of plame density and field strength might the be expected

to look l It. ( fIguer

2.

10.

The next series of stops in seeking a solution to the equations consiste of redicing than as far as possible. The calculations are first siepli-fled _{b- tusing the facts that the magnetic fields are only in the s-directon} and the curaents and therefore electric fields are only in the y-direction ( there are no external fields )g and that all quantities depend

funotioally on the xdirecbtone Oha's law then becomes

1 = E + <B

and differentiating it with respect to x yields

+t 'b

±

_-The cont tuilt equation becomes

~e 4 ₊

_)&±

JN

and the other Nswell equation becomes

&Sabstattng for / _{ivrom the continuity equation and frr X}

from the NaxImal equation into the differentAated for of Ohm's las yields

e

1; -n +

aO

labn ubeibStUg for 3 ron the first PanLl equatin yields

xl) u 3

A. AO- 4a 7 ata

whish v$l be called equenoc 1).

Sw _{the proe.as of re*otcen is carried tbroug starting from the}

ea ecaton

At AxpT

duera

4-

-

Y

&tet tutingp or ? fro te et o ta aiig by

yoidtsaeite _iedn

)

;

t.s$

±v±>=

_Y

s

t

tx

Maupliss

substituting

Y"

-±t

Ge~,

ex

₊

GC)

-'

(273

. W the _{ntsis ubettted, yielding}asrso fOr P in term of b3 deri'ved from the eqUmo of

2)

:(z+

go G_

P

Y-whieb will be called equation 2).

U4.

*ad tb* pmp.. most be oontiwed b~y thwt1wie tvenaforNUa " of

owdmsI lot =

thu.

e

~ tu man of .. ti0~,u

that

Tm

_+~

13.

#AT

_"-

_-

using the 0seasionless variables

a

a-Ms tmornntm Uen has served its intent by eminatng as a variaMel but it bae also had other _{effeots in changing the axis. The} niv axl is t _{equeeeod* ompared to the real x- xis when the} proporton-alttf I'tabr

/is

tive vales on the ee rloprrspod to sia l poetive values on the %Z

is$, _{wt4le large positie valaes on the saxis arrespond to large po81si}

tive va~es on the Xmaos. uartherrwe it and A are determined as tf ions c of 1, tX e invesf m trasortbao

0

des not _{loate them} 4wth _{reference to the original saeordate. The}

14

tAt it is also true tha t

+ _

Thus al the inverse tranaformation can do is replot the functions in their proper sca~l. At least, however this trasxormation has reduced the problem to two equations in two unloowas, and the remininnt disoussion is involved with trying to solve th".

For further oamvninoe, the two equations will be rartten in tere of one variable

not to be conAzsed with the scasordinate, so thst

and

_ -

A

.

Equation 1) then becomes

t-tTy

4 t( ta7 L r 3

15*

ad eVanding the tmr on tho left yields

_ /2-I

A

4L~+

X4-

4----Tkas

Orepdingly#r equation 2) beecues

".f-. I _{.- %(}

_hfI

If-4)

+

{

Thus, w ritg primss for / - o

~/

± Q-~±

A f

V-4-t )

it may imedlately be shown that these equations lead to a process that at least resembles ordinary diffusion. Taking the limiting case of an incompressible plasma i.e. g -- 1, g' -- 0, and ( -- o0 causes

equation 2) to become indeteraminate and equation 1) to become

Zif/ = -j

Rearrangig this as

I

_ / and integrating it yields

and integrating it again yielde

Thus f in this case is of the form of the error function. What this ma .a is that the plasa sits perfectly still, being incompreaible and

infa-kiE and the magnetic field diffuses into it. This is exactly the process undergone by a magnetic field diffusing into a motionless metal

17,

IV. fSlb totQ Iatgration

Ieomse eqaations 1) and 2) are nonlirar second order differential equations, they cannot be straightforwardly integrated over all values of s. What can be done, however, is to look at the forms of theso equations as f and g uasyttial approach values determid from physiqoa

reasonaig. _{rSmmabriag that tlhe functions are n in terns of the} transfrrmed ooordinate X, it may be seen tha at large t as g a proach one and f approaches ro. _{Oonweieyp, vWry near the origin tf approaches} ono and g appmroahes sa,. Ths sratements may be made in light of

the test that, at large distances from She origin of the real coordinate the magnetic field and the plama shoald each be relatively undisturbed if only a finite tim has el sed.

With this muach knowledge in mind about the behavior of the variables, an attiapt may be made to at least partially understand the equations.

Taking equatton 1)

Integrating

SS/'3

18.

Integratng again

+ f

-

+;&

i

A'

Both f' and g' go to sero at large sg thus it is dffticult to discuss the behavior of the rightsnd side of the exponent at this limit.

Never-heless, it might be expeted to go to sro, since the resmining terms give an expression for f similar to that in the case of the incompressible

fluid. _{This is what might be expected at large s where g is nearly}

un-disturbed.

Ixpressions for the term gf' may be derived from both equations. Equation 1) ay be writt=

whish integrates to

Also from equation 2)

19.

Solving tais quadratic

both of the mqa be integrated to give expressions for f. Unfortunate ly nes of the above yield azy nw informotion about the f netional form of either f or g.

It tarned out that none of these methods ere roally helpfu in un-ooupling the equations to obtain an expression for either f or go They either led to omplicated integral fors or approximations too crude to be tuserl. It is possible that methods for treating nonlinear

differ-ential equatlons that at least lead to further simplifloations in this asoe ight exist. Such methods are dia sed in texts like Ieo(1 ) and Lplian()j howvear, their spli ton is suffciwntl difficult that no attmpt was made to use them on this problem.

Aowther possible method of investigation is to substituto various woll-known fmctions for the variables and see if they balance the

equations at asystotic values. A great deal of time and effort was spent trying this, and it proved unequivoeally fritless. The actual functions, though they ight be wllbehaved, are of aufficient complex* ity that no Olosed funoLon reembles either of thesm This is llus-tnated by aetag Uhaot, in the greatly sres*rotive case of an incompress-ible plasma, f still had the fora of an open integral.

20.

y m wIaieJ l Interton

In order to obtain specific data on the plaia-field interaction, it

was necessary to resort to numerical integration. The prooedre used is

described in KEplan, and simply consists of selecting values of the functions and

their divatives at a points, casulating their values at a point close tyV

and iterating. Thus the method is a nrmerical first integrationg and since

i

S / / / /

1( /

-

=-wher the first equation is equation 1) and the tird is equation 2).

In doing he oaJclation, values for f and g and their derivatives were

eflected at large positive a and the iteration earied toward the origin.

These initial vaLues wre adjusted until f approaohed one and g aproa oed

e at the origin and both held their asmtotic values at large

a.

21.

ezrtrtely dtffioult to oestIate the first derivatives at this point.

The results of the numerical integration are plotted in ( Figure . ). The variables are left in the dimensionless form of f and g and are plotted as functions of the transformed coordinates.

These transformed oordLnates were then reoast in the form of the real coordLnates by means of the inverse transformations

x n = Ax T

This was also done numerically. _{The variables f and g are thus also} plotted as funtions of x and t in ( Figure 4. ). As was mentioned pre-viously, the origin of the real x-axis _{cannot eo redetermined and the}

sero-point in ( _{Fgur 4. ) is arbitrary. These figwrs thas detAerine} the napeti field strength and the plama density in both space and time as the diffusion process takes place. _{y also calculating temperature and} entropy changes in the gas# its total behavior ma be considered ade-quately defined.

The temperature of the gas at each point was calculated from the x-exis _{ourves of f and g as follows# sinco}

them

-r

13

22.

Chmaging to dlmsienlAeas variables

(I-

wich in plotted in ( Figure

5.

Since tk prOoess _{was asummed adiabatic, th speific entropy was} eJ.ualated from the xmaxis crves of g and f by using the expression

-*1- Cr 'j '

5 - S o 0. C-I -e

Changing to daillmelse variables

L-- 5o +. <t, N _

+,+r r

Or, subtratng *,and dividing by c

t- "(I,

(fL

Then the increase in entropy as a tanction of x is giive by vhich was oal"ulatod and plotted in ( Figure 6. ) using Y

total increase in entropy as a funtion of time was obtained

*

51/3.

"T=~cop

23.

integrating the Mr"n in ( Pigre 6, ), nd wa s found to be

All of the caloultdons for these graph are only to s-ldesrule

aocuracy. Because of tie, and because of the errar involved in

s*t-ating the boundary oonditions for the f arn g carves in ( Figu 3. ), the curve shapes and the derivod quantities nay only be oansidered to be carrect to soewhat bettter tan an order of magnitude. Gratr

m-i t Ii - i _ -F 1 FF 4117±f4I" -i"T<-

_

Hf_

++H+

H'

j

r4

T-4F T

+4T-rj

I I I I

U

r

I III p Jill _ITM

L A I [ FFI I I I I fill I fill -- n

1+ 1

U1

_H_

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1,

s

27.

1. E. L. Inoe, OrdiaMy Ifrential Sutiono Dvemr Puthications Ino. Nw York (1956).

2.

gLnom

Publishers, Inc., New York (1956).

3. H. W. Liopmann and A. Roshkos Silents of Ga"nmsa, John Wiley & sons, Inc., Now York (195).

4, Wjilred . lan, rdinary Diffterential quatons dAddison-Wesley Publishing Company, Inc., Reading, Mass. (1958).

5.

s.

Chicago (1960).

6. J, O. Linhart, P Pbi,, NPlano th olland Publishing Company,oor