Propagation of ion acoustic waves along cylindrical plasma columns - transverse boundary condition

Wong, A. Y.

English

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Propagation of Ion Acoustic Waves 
Along Cylindrical Plasma Columns 
A. Y. WONG 
July 1965 R - 1  
Plasma Physics Group 
Department of Physics 
University of California 
Los Angeles, California 
This research was partially supported by the 
National Aeronautics and Space Administration 
under grant NGR-05-007-066. 
Propagation of Ion Acoustic Waves Along Cyl indr ica l  Plasma Columns 
A. Y .  lrlong 
Department of Physics, University of Ca l i fo rn ia ,  Los Angeles 
I t  has  been found by I’long, Motley and DIAngelo’ t h a t  t h e  propagation 
o f  i on  trcoustic waves along an ex te rna l  axial  magnetic f i e l d  between 
4kg and 14 kg i n  h ighly  ionized plasmas i s  not  a f f ec t ed  by t h e  size and 
shape o f  t he  c y l i n d r i c a l  plasma column, 
agree with a plane wave ana lys i s  even though t h e  plasma rad ius  R was of 
t he  o rde r  of t h e  wavelength A .  Neither changing t h e  plasma diameter  by 
a h a l f  n o r  a l t e r i n g  the  densi ty  p r o f i l e  through varying t h e  a x i a l  magnetic 
f i e l d  (4-141:~) seened t o  have any e f f ec t  on t h e  phase ve loc i ty  and damping. 
On t h e  o t h e r  hand, L i t t l e  has observed t h e  e f f e c t s  of  boundaries on ion 
acous t i c  waves i n  F. xeakly ionized plasma ’ ?  the presence of a weak mag- 
n e t i c  f i e l d  ( l c s s  then 50 gauss) with 1 - R .  
found t h o t  i n  the  propagation o f  e l e c t r o s t a t i c  e l ec t ron  o s c i l l a t i o n s  t h e  
s i z e  of t h e  plasma column nus t  be taken i n t o  account when kR-1. 
e x p e r h e n t a l  r e s u l t s  l ed  us t o  re-examine the  theory o f  i on  acous t i c  waves 
t o  inc ludz  t ransverse  boundary e f f e c t s ,  
The experimental r e s u l t s  indeed 
2 
Malmberg and Wharton’ a l s o  
These 
blaking, t h e  e l e c t r o s t a t i c  approximation f o r  i on  acous t i c  waves - E=-90 
and V*D=O - , we can der ive a wave equation f o r  the  wave p o t e n t i a l  0: 
where €, .and c Z Z  are elements of  the  d i e l e c t r i c  tensor  E: - 
and k = the wave number i n  t h e  d i r ec t ion  of propagation. 
From a theory taking the  ions  as  c o l l i s i o n l e s s  and the  e l ec t rons  as a 
f l u i d  ( a s  appropriate  t o  WD's experiment ) we obtained 1 
>> 1 For w < w << w and - c i  ce 8n KT B2 2 2  ci 2 1 + kD pi  
3 
2 - 4meL where kD - KTi = ion cyclotron rad ius  'i 
and wci w t h e  ion and e l ec t ron  cyclotron frequency respec t ive ly .  
Neglecting the  e l ec t ron  i n e r t i a  i n  t h e  ion wave, it can be shown t h a t  
ce 
where n 
n = t h e  per turbed plasma dens i ty  
= t h e  unperturbed plasma dens i ty  
0 
Equation (1) can thus be w r i t t e n  as 
We s h a l l  make use of the  following experimental observat ion i n  our s e l e c t i o n  
of the  appropriate  so lu t ion  f o r  equation ( 2 ) :  
X l r  
1) no(r )  = Jo (-1, where x = 1st root  of t h e  zeroth order  Bessel func t ion  1 R 
R = plasma radius .  
= constant  along the  r ad ia l  d i r e c t i o n .  2) _1 n 
0 
3) azimuthal symmetry. 
2 €1 X 1 2  with t h e  condi t ion t h a t  k E = - 
R2 z z  
This condi t ion can be r ewr i t t en  as 
1 - -  
k2 
Z '  Ti 
Te 
+ 2 -  - 
k2 
+ 
2 
EL X 1  
k2R2 
= o  
Under the  experimental condi t ions o f  WMD 
k << kD and Ti = Te 
2 
we have 2 (-) = 2 + - 
kvi k t R 2  
1 0  Y 1  
The e f f e c t s  of t ransverse  boundary conditions are important only i f  
2 
2 . 2  x1 
kD2R2 
A similar condi t ion i s  also ar r ived  at  when both ions and e l ec t rons  are 
t r e a t e d  as f l u i d s .  
- 4 -  
In  IVMD’s experiment 
T = Ti = 2500OK 
n = cm/sec 
R = 1.5 cm, 
e 
2 -1 c m  kD : 6 x 10 
2 2  
E = k  p I D i  we have s ince  
2 
’i 2 Condition (3) reduces t o  - = - 
R2 x,? 
>> 1 kDP i 
Thus we come t o  the  conclusion t h a t  t he  t ransverse  boundary condi t ions 
are important i f  t h e  ion cyclotron radius ( r a t h e r  than the  wavelength of 
t h e  ion  wave) is  comparable t o  t h e  plasma rad ius .  
experimental  f ind ing  of I N D :  t he  independence o f  propagation c h a r a c t e r i s t i c s  
This agrees with t h e  
on t h e  s i z e  of the  plasma column i n  t h e  presence of a magnetic f i e l d  such t h a t  
<< A .  Examination o f  condi t ion (3) shows t h a t  t he  t ransverse  boundary e f f e c t s  ’i 
on ion  acous t i c  waves can be v i t i a t e d  by the  app l i ca t ion  of  a s u f f i c i e n t l y  
high magnetic f i e l d  which reduces t h e  boundary cont r ibu t ion  with respec t  t o  
t h e  e l e c t r o n  cont r ibu t ion  i n  ion acoust ic  waves. 
e l e c t r o n  plasma o s c i l l a t i o n s  a r e  sens i t i ve ly  dependent upon the  boundary con- 
In con t r a s t ,  the  longi tudina l  -
d i t i o n s  s i n c e  the  corresponding - ion contr ibut ion t o  t h e  e l ec t ron  - o s c i l l a t i o n s  
is  zero.  
t h e o r e t i ~ a l l y . ~ ’ ’  
The importance of such dependence has been shown experimentally’ and 
In summary, t he  independence of  ion  acous t ic  waves on 
i '  " , '  
- 5  - 
t ransverse  boundary condi t ions i s  a t t a inab le  under t y p i c a l  laboratory condi t ions 
and f u r t h e r  s impl i f i e s  i t s  labora tory  inves t iga t ion  and increases  i t s  r o l e  i n  
plasma diagnosis .  6 
Ackn ow 1 e deemen t s 
It is  a p leasure  t o  acknowledge useful discussions with D r .  D .  Sachs, 
D r .  A. Kaufman and D r .  B. Fr ied.  This work was supported by t h e  National 
Aeronautics and Space Administration under grant  NGR 05-007-066. 
-~ -- ~ ~ 
- 6 -  
References 
1. A. Y.  Wong, R.  IV. Motley and N .  D'Angelo, Phys. Rev. 133, A436 (1964). -
2 .  P. F.  L i t t l e ,  Nature 194, 1137 (1962). -
3. J .  H.  Malmberg and C .  B. Wharton, Phys. Rev. Le t te rs  13, 184. -
4 .  R .  W. Gould, "Propagation o f  E l e c t r o s t a t i c  Naves Along a Plasma Column," 
Culham Laboratory Report (1964) . 
5 .  Knehl, Stewart and Yeh, Phys. o f  Fluids ( t o  be publ ished) .  
6 .  I .  Alexeff and R .  Neidigh, Phys. Rev. 129, 516 (1963). -
A. Y .  IVong, Appl. Phys. Le t t e r s  6,  147, 1965. 


Propagation of ion acoustic waves along cylindrical plasma columns

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