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
. ss. INST
MgROF MASTR OF
LIBRAREY
SCIg c
LINDGREN
at the
ASSLOCUSETTS INSTITUTE OF
TECHNOLOXI
lOt .oJa
Aut96
1 w en4dAgIphsics, Hy 29 1961
of Gm0
CerLtied by . .
n
.1
Accepted
by.
.
...
*
Thesis Supervisor
-
*
*
*
*
*
*
*
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
are also used to determine temperature and entropy increases in the plasma.
Absi'omt
I*
Intdmitoz
.U,*OsawaI
MI.
Dismusio
2
Ths Infinite On.. Dimmrnanu
IT,* AmIA17tma Intogwation
To
rmwio*
Iwto
sz
can
7
17
20
Thare hasbson 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
Mhis plasmnfield interaetion
has a
B
generated by external sources.
whrth is the subject of this discussion,
ofaspeets 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
properties of the plasse do not change appreciably.
0ao
What is meant by
mations and "strong' fields will be made clearer further on. In
addition to this, ti.
form.
ni c
geometry of the proble
is reduced to its simplest
Doing this naturally limits the applicability of the resultine
data;, st*il
it produces a good approximation of ti-
saoan in a variety of cases.
plasmanef
d inter-
Te formal stateant of tats prebla most amenablo to qualtative
investgati n is in terms of the no*alUd
tion.
seroeeeapt
esentialy these are the equationa that re
equations of
ltfrom averaging
this averaging being done by integration
thie mtions of many particle
y Mseas of this integration
of thn bitUman eqat±*t in velocity spaceo.
as average iSid velocity %#and
pitser(2) defines a ou~r t dnsityn
derives an equation of fluid ation and a generalised tfr
In the case to be
eomposed oa
oansidered bore
the fluid to asamed to be
of sangly chargi d ions of identical
to generalse to ta
ation, but itSt
mber o
asses and of elecbtrone
types of ons does not coplicate the sito
s out that this does not lead
mstin. A mnaoasd" gas as *eeea simply to -*
lalations done later.
eqaol to 5/3 in
of Ohs's law.
say additional Inftr-
the gas oonetant f
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,
the addition of a
Xj
I
This is asi
ly itler's equation with
f raOe.
There are certain assuamptions aready Implicit in this equations
mass of the eleotna
the
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
is never an eleotef
pace charge.
field dua to
of this last assumption, there
Also, viscosity eaffocts
are neglaote
llowig what is actually the stress tensor to be tsan
as a sUcar pressure. Aaoounti
dae to external ma4te"g
ditferent
for direott
al pressure differenes
c fields by wdag a stress tensor with at least
nournese diaoaal terms might seow actualy anessary for this
tip. of problm
differen
howavw, it
rill
S
rn out tat t this
ossawt
e
in the saculations
Also for this case5 the
"J
where 1 IsoSti
=
ensrased Oha's ]A
E
+
reduces to the for
kxB
eectrical reisatvity and e i
twe electric field.
assuaption made here is that the resistivity is a constant scalarl
again, rwiti
it
fields & e not affer
On
once
as a diagonal tensor to acooOunt for external magnetic
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
nall compared to the othersl
terMn is neggibly
here the limitation of
salowaser
is first
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
p
+
there are the continuity equation
(P "
)
=
o
and the equation of .ats
= PRT
here R is the 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
pawebility
ded in the V
ierm in this last equation.
8
The
o is taken equal to that of free space.
?lnUy there is the energy equation
j
+
>t
LN(
f
-
owlm heat mapacity and Y
is the ratio of spoei io
where
, it the oWnsta-to
heats,
This equation implios two vry aportast assumptionst
She or
0
sour
a
t energy is ohbie heating ( viscosity having alread been
n glected )g and the ther,
i*o,
one, that
hat there is no heat transfer in the fluid,
all proeeose are a4abatia.
This lat
asaption is valid when heat
s an
rtci
ditaeio
to be demwibed.
than the preoe
ear
mw tabulating the equatims obtaind
Se(
t)
=
j x7
-v
xg
+
'14
t (p)
9t
+
, =
pRT
=
0
j3
(Vt
V xE
,AAO
Cv
-V
it is seen that tha*,
soy als
ads
seven equations
,
pT
ntaining sevn viaria
e.
It
be ssn that tin #quations re nnlin ar, wvt4ch Iamdlately pro-
anW strn4ahtgbrward analytical integration of them.
with theria
Jbenfw
tio
Thaus proceding
reqUires some additonal sipificateions.
continuing wita thi 9, it is
astractive to rougly sketch seas
pImpertis oof pleas irn etrong magneti* fteids predued by constant
exernal earrents.
ere"strongfmaget i. fields" means
moh larger than sq field met uap 1
order,
gas.
ne mfor
ti
curaents in the plea.
e line tend to be 'fse"
tfieda that are
to tie fLrst
in a highly oudtacting
If a motin is imparted to the gas, the foree lime are wdragged
along* wih it.
resistivit
field lias.
To the next ordsr,
trrent dissipation due to the finite
of she gas allows a relatve aetiet
To see th smea
between the gas and tbe
clearly, one can lmagine a claster of s*
ternally produced parallel magunee tield lines in a plasma with a finite
resittvity.
in it
Beause tbe pleasm
will have to decay if
Iag he fields.
it
is resistiv*e
is not regenerated by externally intaesi4-
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.
ard what the whole proeses 'orks
throug
the crrant around an area
the plans
Therefore, the cluster spreads out
like' is a diffutsion of the field line
WV~*
O2flflI S&"
,p s1 .xn
do.
x
iwyd
tam
WqR 0#UUf*J fruv140z-T
WTfll*~
pu*Vs
s~ou q.'t *%wzr 4n p~jo
0
em
C17
f0U
Wtiwr
01o%
polf tft")
**yp*4 WIWood
CWn'~d P WqSVMWv MonVad np wte sn
am~ 3uoy*^eUT
A0w
n
ppgq~ oqj
**pnc r&"vbu
matvPuwunx. o * i tv 06 Awwwr
cii
% -''
znraw ej;#.r
Mq uy pwpa;4 OffasS wOW'~
'eanclw0*
tzoyvi rawqo"luona AmO saq 0%
PS*ATT IOU wr umoid "q& 'O*pvOg ay~p;.d a"t 2Iwu £^'pp o% .mqd
au
\vli £*TV#wn~
n
MAn
01M
ST'R So fldowo
F,,
pO U*OTfl u
# snpstT
V;
n~
OTuIadS V ERt
VpfoOo #flbmdI
ps
NfliD
^dT*msp :0vv
"
OuMoMua
,x
pufl iri
TMnft
0%tOt.d am n4 %'Ind
fl0
30enw
Sit pwrW U
anThIT
-~,VWM-,
Ot'l"01MTO -
covertfg the x * 0 srfte
of the pl as.
To seek a solation for the behavior of thisyastm for timest
one £irst eonsiders the equation of nation
( )
S+
Jx
0,
-
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 onaslderd in the s-dLrectiou, the not currents in the plasm always
£Low in the
yedirootion. Then the equation of motion becomes
+
r
x
=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
notion of the plasma.
This smea
eslowness" is again applied to the
that the teras on the left are negligibLy
small, or that the ters on the right nearly cancel.
J
ax
Sinee for this configuration the equation
-A o Vx
x
Thbs
becae
s oaly hao a twntional dependoenc
on the xn oPrdnatetela
equatSon
of motion then becomeM
and this may be integrated to give
=
+
Tbe r/2
oy be oonsidered asa ua
pressure at all pott
preseures vary.
in
-
tite pressure0 , and the total
pace remains constant as the manetic and plasm
Sitqly from this equation it
plase leaks alowly into ti
poetAngl
CONSTANT
is
possible to see that the
magnetic field while the magnetic field corre-*
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
to look
l It. ( fIguer 2. ).
Fiare 2,
be expected
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
1
Oha's law then becomes
=
E
+
<B
and differentiating it with respect to x yields
+t
Jle
±
'b
_-
The cont tuilt equation becomes
~e
4
)&±
JN
+
and the other Nswell equation becomes
&Sabstattng
for
from the NaxImal
/
ivrom the continuity equation and frr
equation into the differentAated for
1;
-n
+
-
e
~td
X
of Ohm's las yields
aO
labn
ubeibStUgfor
3 ron the first PanLl equatin yields
xl)
A. AO-
4a
3
u
7
ata
whish v$l be called equenoc 1).
Sw the proe.as
ea
of re*otcen is carried tbroug
starting from the
ecaton
At AxpT
s t jp sd substituting 5/a
Maupliss
4u &tet tutingp
yoidtsaeite
.
W ntsis
the
)
2)
or
Y" txlandj
Y
? fro te
iedn
et
duera
-
a
o
ta
aiig
Ge~,
ex GC)
+
±t
by
-'
asrso
fOr P
in term of b3 deri'ved from the eqUmo
ubettted,
yielding
:(z+
;
t.s$
go G_
±v±>=
P
(273
of
_Y
Y-
whieb will be called equation 2).
Staple subsiaticm produces no further redation after this point,
U4.
tvenaforNUa
*ad tb* pmp.. most be oontiwed b~y thwt1wie
owdmsI
lot
=
thu.
3T3
e
@a
La'
-+
t
)T
DOI
~tu
man of .. ti0~,u
Tm
ad s"bauutdzg
+~
that
+
j
%"tpxaly in .qattons 1) cmd 2) yieldo
" of
13.
- /-I
"-
#AT
using the 0seasionless variables
a aMs tmornntm Uen has served its
variaMel
niv axl
intent by eminatng
but it bae also had other effeots in changing the
is
t
alttf I'tabr
equeeeod*
/is
axis.
The
ompared to the real x- xis when the proporton-
*mal.
Also
X is neaer less than ero Iease the
defining itegral is neaver angatiwr
tive vales on the
as a
What this mnans
rloprrspod to sial
ee
is that large asep*
poetive values on the %Z
is$, wt4le large positie valaes on the saxis arrespond to large po81si
tive va~es on the Xmaos.
tf ions
c
uartherrwe
it
and A are determined as
of 1, tXe invesfm trasortbao
0
des not loate them
4wth
reference to the original saeordate.
reason for this is that the iesttrnnsfmatioaea
The
offetod by nying
14
is also true tha t
tAt it
+
Thus al
_
the inverse tranaformation can do is replot the functions in
At least, however
their proper sca~l.
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
-
.
f
_
~
+
_
Equation 1) then becomes
4
-
-I
t(
ta7
t-tTy
L
r
3
15*
ad eVanding the tmr
_
/2-I
on tho left yields
A
+
I)-
"%C)
-
I
)
4----
~
X4-
A~
4L~+
Tkas equation 1) beuwMes
Orepdingly#r equation 2) beecues
".f-. I
.- %(
Af V4-t )
4)
hfI"
If-
YX
=A
or
+
Thus, writg
primss for
{
Oj
/
- o
+
± Q-~±
~/
#
/
x
it may imedlately be shown that these equations lead to a process
Taking the limiting case
that at least resembles ordinary diffusion.
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
= (LN ')
/
_
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 .ais 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
condActor.
17,
IV.
totQ
fSlb
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
reasonaig.
g
uasyttial
approach values determid from physiqoa
rSmmabriag that tlhe functions are n
in terns of the
transfrrmed ooordinate X, it may be seen tha att large as
one and f approaches
ro.
ono and g appmroahes sa,.
g a proach
Oonweieyp, vWry near the origin tf approaches
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
I
_
)-~1
i-
9. 1
1
18.
Integratng again
+ f
-
Both f' and g' go to sero at large sg
i
+;&
A'
thus it is dffticult to discuss
the behavior of the rightsnd side of the exponent at this limit.
heless,
Never-
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)
SC)
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
Lplian()j howvear, their spli
sed in texts like Ieo(1 ) and
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.
functions, though they
The actual
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 incompressible plasma, f still had the fora of an open integral.
20.
wIaieJ l Interton
m
y
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
and iterating.
i
ty
V
Thus the method is a nrmerical first integrationg and since
appears to the second order In tie equations, it is necessary to intoe
grate t4See to obtain values for it.
S
1(
/ /
The equations are writtn in the fore
/
/
/
-=-
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 aproaoed
e at the origin and both held their asmtotic values at large a.
It
would seem more acurate to start at the origin and look for asyatotic
values of the funotions by iterating away from the origin, but it was
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
xn
= Ax
This was also done numerically.
T
The variables f and g are thus also
plotted as funtions of x and t in ( Figure 4. ).
viously, the origin of the real x-axis cannot
sero-point in ( Fgur 4. ) is arbitrary.
the napeti
As was mentioned pre-
eo redetermined and the
These figwrs thas detAerine
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
T-~--PC
R
22.
Chmaging to dlmsienlAeas variables
(I-
~7 -- o'k
"T=~cop
i
)
5. ) with
wich in plotted in ( Figure
'L
Since tk
prOoess was asummed adiabatic, th
speific entropy was
eJ.ualated from the xmaxis crves of g and f by using the expression
5
-
S o 0.
C-I
e
-*1-
-
Cr
'j '
Changing to daillmelse variables
L-
-
5o
+.
±t
CV
'h
_____
<t, N _
+,+r
r
+
CV
N
(
I-Z
Or, subtratng *,and dividing by c
~
-
~wta)9/
Then the increase in entropy as a
t-
"(I,
(fL
tanction of x is giive
by
vhich was oal"ulatod and plotted in ( Figure 6. ) using
Y
* 51/3.
The
total increase in entropy as a funtion of time was obtained by graphicagly
23.
integrating the Mr"n in ( Pigre 6, ),
nd wa s found to be
All of the caloultdons for these graph
aocuracy.
Because of tie,
are only to s-ldesrule
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.
acuraq would requte the services of a computer.
Gratr
I I
I
-I
m-i
4117±f4I"
_
-l 1 11
-1
++H+ H'
j
- __117
I-~4
_
_
-i"T<-
_ T
Hf_
----
-T
I,4J
T
_
_
_
_
H
4,
L-+
F-TFFITF:T
*11
+
-tV
-
rT.
ITM
i
FF
1
~
fill
-F
_
-h
-
Ii
t
II III p Jill
fill
+
4:
-
LFr
_T
II
TFFr
I
I_
TF
V'
JL T-
E_7
7---I
H4~ H
+4T-rj
-- n
I III
L
U1
U
- iHH
1+ 1
++
-M
:4
ir-::Ft
_H_
4I 4774
FWLH
I AI
1-
r4
-T
+
F
-7-Fl-
r
A
I [ FFI
+
7+
+
I
T
4F T
0
25.
1±]
s
.J
iii-',
KTiRH
K17
1,
HE-
K
-4114 '1K
TEFTi
27.
1.
E. L. Inoe,
Sutiono
OrdiaMy Ifrential
Dvemr Puthications
Ino.
Nw York (1956).
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