Transverse electromagnetic wave propagation in cold plasma by external electric field oscillatio

Montgomery, D.

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ff 653 July 65 
Department of Physics and Astronomy 
THE UNIVERSITY OF IOWA 
Iowa City, Iowa 
https://ntrs.nasa.gov/search.jsp?R=19660010250 2020-03-16T22:44:59+00:00Z
PARAMETRIC EXCITATION 
OB TRANSVERSE 
WAVES I N  A PLASMA* 
David Montgomery 
Department of Physics and Astronomy 
Universi ty  of Iowa 
Iowa City,  Iowa 
February 1966 
* 
Research supported i n  p a r t  by t h e  Nat ipna l  Aeronautics 
and Space Administration unde r ,  g ran t  NsG-233-62. 
ABSTRACT / yS3 7 
It i s  shown t h a t  an externally-imposed, o s c i l l a t i n g  e l e c t r i c  
f i e l d  e x c i t e s  t r ansve r se  electromagnet ic  waves propagat ing  pe r -  
pend icu la r ly  t o  it, i n  a cold plasma. The mechanism i s  c l o s e l y  
r e l a t e d  t o  t h e  parametr ic  e x c i t a t i o n  of l o n g i t u d i n a l  plasma o s c i l l a -  
t i o n s  r e c e n t l y  p red ic t ed  by Al iev  and S i l i n .  The problem provides  
an a p p l i c a t i o n  of non-secular p e r t u r b a t i o n  methods, when t h e  equat ions  
of motion a r e  expanded i n  powers of t h e  e x t e r n a l  e l e c t r i c  f i e l d .  
A r b i t r a r i l y  small pe r tu rba t ions  which a r i s e  spontaneous1.y i n  t h e  
plasma a r e  amplif ied b y  t h e  a c t i o n  of t he  e l e c t r i c  f i e l d ,  f o r  a 
c e r t a i n  range of t h e  d r iv ing  frequency.  The growth r a t e  of  t he  
o s c i l l a t i o n s  i s  cal-culated.  
, 
It' sonic parameter of a system capable  of' o sc i l l a t in{ ;  at, 
frequency R i s  forced  t o  vary wi th  frequency 2R, it i s  known t h a t  
t h e  sys1,em w i  11 o f t e n  spontaneously break  i n t o  o s c i l l a t i o n  a t  t h e  
frc,quen?y R .  'Thp namr usua l ly  a t t ached  t o  t h i s  ef ' fect  i s  "parametr ic  
1 cxci t,at,ioii" or "par-miet,ric resonance'' ( s e e ,   CY.^;., Minorsky o r  Rogolyubov 
ant1 M i  t , t -~ l~ )o l sk i  i').
Ii l ? ~ ~ r e n t , l y  A I  it3v and S i l i n 3  and S i l i n  have dcmonstrated the  
possibi  I i Ly of p:irarn~-i,r.ic e x c i t , n t i c > n  of l c i t i ~ i  tuclinal e l r c t r o s t a t i c  
w a v c s  i n  :i co I I is  i on I ( ' s s  ~1 asma. The ~ x c  i t ing mprhanism was R s p a t i a l l y -  
urii t'orni, nxtc'r.tiaI ly-i lr iven, e l e c t r i c  f i e l d .  The wave v w t o r s  of t he  
4 
J 
I n  any a p p l i c a t i o n  of t he  paramet r ic  e x c i t a t i o n  technique 
to ,  say,  t h e  t u r b u l e n t  hea t ing  of a plasma, t h e  e x c i t a t i o n  of t rar is-  
verse  waves must be  considered as p o t e n t i a l l y  a source of energy 
loss. The t r ansve r se  o s c i l l a t i o n s  w i l l  be exc i t ed  a t  a frequency 
g e n e r a l l y  above w t h e  plasma frequency, and w i l l  be free t o  l eave  
t h e  plasma. 
p' 
Equations governing the  time development of  t h e  f i e l d  v a r i a b l e s  
a r e  der ived  i n  Sec.  11, b u t  without  t h e  e l e c t r o s t a t i c  assumption. 
They a r e  then  s p e c i a l i z e d  t o  the  case  i n  which t h e  d i s tu rbances  
propagate  pe rpend icu la r ly  t o  t h e  app l i ed  e l e c t r i c  f i e l d .  I n  See.  111, 
a p e r t u r b a t i o n  t rea tment ,  i n  which t h e  expansion parameter measures 
t h e  s t r e n g t h  o f  t h e  ex te rna l ly -app l i ed  e l e c t r i c  f i e l d ,  i s  given f o r  
these  equa t ions .  
Krylov-Bogoliubov-Mitropolskii-Frieman methods2' which have been 
The pe r tu rba t ion  theo ry  i s  an a p p l i c a t i o n  o_ft +%;e 
appl ied  elsewhereb t o  cold-plasma problems. 
which c h a r a c t e r i z e  t h e  p re sen t  problem are: (1) t h e  "fast" t i m e  
s c a l e ,  measured by t h e  frequencies  o f  t h e  o s c i l l a t i o n s  e x c i t e d ;  and 
(21 
fed  i n t o  t h e  o s c i l l a t i o n s ,  and which depends upon t h e  s t r e n g t h  of 
the e x t e r n a l  e l e c t r i c  f i e l d .  The condi t ions  under which the  two 
t i m t  s c a l e s  w i  I I. be q u i t e  d i f f e r e n t  can be e x p l i c i t l y  g iven .  
The two " t i m e  sca les ' '  
t h e  "slow" s c a l e  which measures t h e  r a t e  a t  which energy i s  
11. F;QURJ!IONS FOX TlE PEl ITUKl~EI l  
FIELD Q U A N T I T I E S  
We have i n  mind an axially-symmetric plasma extending t o  
i n f i n i t y  i n  t h e  - + z d i rec t ions .  
vary  i n  space,  and which i s  understood t o  be ex te rna l ly -d r iven ,  
i s  given by 
An e l e c t r i c  f i e l d  which does not  
4 
s i n  w t .  z Eex t .  0 s i n  w t = e 
4 0 )  - -+ 
Eex t .  0 E 
The p1.asma i s  understood t o  be co ld ,  spa t ia l ly-uni form,  and 
t<) con ta in  no d . c .  magnetic f i e l d .  The p a r t i c l e s  of spec ie s  i 
(charge e 
v e l o c i t i e s  
mass m.) o s c i l l a t e  about  t h e i r  equi l ibr ium p o s i t i o n s  wi th  
i' 1 
4 
4 
cos w t = -e, V cos w o t  . i Eex t .  -+ (0) = - -
i 
e 
v 
i m 0 o j. 
wO 
F i n a l l y ,  w e  complete the  d e s c r i p t i o n  of t h e  "equi l ibr ium" s t a t e  by 
spec: f y i n g  
n .  ('I = number dens i ty  of t h e  i t h  spec ie s  of p a r t i c l e  = cons tan t ,  
1 
%(") = magnetic f i e l d  = 0, 
. 
Such a s t a t e  i s  riot an exac t  s o l u t i o n  of the  f u l l  s e t  of' 
co ld  plasma equat ions and Maxwell's equa t ions .  It is ,  however, 
a qood approximation f o r  an  axially-symmetric plasma out  t o  r ad ius  
r from t h e  axi.s o f  symmetry, where 
0 
with  
region clcfinecl by (j). 
of Al.ic.v and Silin. 
- hrr t i  ( " )P ' : / I~ I~ .  Hereaf te r ,  our remarks apply  only t o  t h e  
i 
'The same l i m i t a t i o n  a p p l i e s  a l s o  t o  t h e  work 
Wc now l i n e a r i z e  t h e  cold plasma equat ions  about t h i s  ( t ime-  
The p e r t u r b a t i o n s  on t h e  f i e l d  q u a n t i t i e s  dependent) s t eady  s t a t e .  
a r e  w r j  t,t,en wi thout  s u p e r s c r i p t s  o r  s u b s c r i p t s  : 
7 
(’Id) 
I- 1 
-.I 
Spt>cial iz ini :  E q s .  ( 4 )  to the  case 4 X E 0 and l e t t j n g  
d x  
c 4 Lm Icatis 11) tlic equat ions of Aliev and Si  l i n .  
En:;. (11) d i f f e r  from the  usual l ineayized  cold-plasma equat ions  
on ly  by t,he presence of t h e  time-dependent terms ;!‘I. We now 
1 
s 
-+ (4 ( u l t i m a t e l y ,  e + 1) i n  f r o n t  o f  each v. , eettiq:: 
1. 
i'T; * - 4 r r ~ . e . n ( i )  = o 
1 1  
+ - i  
i k ' B = O  
From t h i s  p o i n t  on, t he  problem i s  t h e  t e c h n i c a l  one of doing 
a p e r t u r b a t i o n  expansion i n  E on Eqs. ( 5 ) .  
9 
111. PERTURBATION EXPANSION OF EQS . ( 7 )  
By s e t t i n g  E = 0 i n  Eqs. ( 5 ) ,  we recover  the usual s e t  of 
I f  we were t o  make a naive f i e l d - f r e e  cold-plasma normal modes. 
p e r t u r b a t i o n  expansion i n  E, 
( t h e  s u b s c r i p t  now ind ica t e s  t h e  o rde r  i n  E ) ,  s u b s t i t u t i n g  (6) i n t o  
( 5 )  and equat ing  c o e f f i c i e n t s  of equal  powers of  c ,  the  p e r t u r b a t i o n  
series would be poorly-behaved. 
t o  the  normal mode so lu t ions  ( n l ( i ) ,  v l ( i ) ,  El, B1, e t c . )  would soon 
By t h i s ,  we mean t h a t  t h e  "co r rec t ions"  
- D +  4 
become l a r g e r  t han  n o ( i ) ,  + v o ( i ) ,  xo,  fro, f o r  c e r t a i n  c r i t i c a l  values  
o f  t h e  frequency wo. 
This  i s  a breakdown o f  t h e  s tandard  p e r t u r b a t i o n  techniques which 
has become famil iar ,  and there  i s  a by-now s tandard  procedure f o r  
avoidink: it. 
concerned wi th ,  the  most usefu l  formula t ion  i s  that of' Frieman.? 
seeks an expansion o f  t h e  form ( 6 ) ,  b u t  w i th  t h e  assumption t h a t  the 
For t h e  pure i n i t i a l - v a l u e  problem, which w e  s h a l l  be 
One 
10 
4 + - b  
v a r i a b l e s  n o ( i ) ,  v o ( i ) ,  Eo, Bo depend on time through e x p l i c i t  
f u n c t i o n a l  dependences on the arguments t, E t ,  E t ,  .... As long as 2 
are not invol-ved, t o  lowes t  order  i n  E .  It t u r n s  o u t  t h a t  t h i s  a r b i t -  
r a r i n e s s  can be e l imina ted ,  and t h e  dependences determined, by t h e  
requirement t h a t  t he  higher-order  co r rec t ions  i n  Eq.  (6) r e a l l y  s h a l l  
remain small compared t o  the terms which a r e  fo rma l ly  of lower order  
i n  E .  
We need, here ,  r e s u l t s  on ly  through O ( E ) .  
This procedure has been d iscussed  i n  d e t a i l  elsewhere.2'5'6 
Modes i n  which "k % = 0 and "k x % = 0 t u r n  out  t o  be coupled 
by Eqs .  (5 ) ,  s o  t h a t  t h e  waves which a r e  dr iven  a r e  of a mixed type,  
n e i t h e r  pu re ly  l o n g i t u d i n a l  nor p u r e l y  t r a n s v e r s e .  It i s  convenient 
t o  p i ck  a d i r e c t i o n  €or 2 and c a l l  it the  x -d i r ec t ion .  
be r e a d i l y  shown from E q s .  (5)  t h a t  waves which have E perpendicular  
t o  the  XZ p lane  a r e  unaffected by t h e  ;(') d r i v i n g  terms, and so  we 
may l i m i t  cons ide ra t ion  t o  waves which have on ly  x and z compurieriis 
of 2.  B then  has only  a y-component, and we may reduce E q s .  (5)  t o  
Then it may 
+ 
i 
-+ 
a s e t  of s c a l a r  equat ions ,  by de f in ing  
. 
11 
4 4  4 
B = B  = B  e , 
T T Y  
+ -. A 
v ( i )  = v + “ v  = v e + v 2  
L T L x  T z ’  
L ’  n ( i )  = n 
(wi th  t h e  i t h  species understood i n  (7c) and ( 7 d ) ) .  
t r i v i a l  members of  Eqs. (5 )  become: 
Now, t he  non- 
e V  B 
E i o i T  
cos w o t  , JVL - (ei/m,) = m. c 
1 
% - - (ei/mi) ET = 0 , J t  
ikEL - 4nC.e % = 0 , 
‘ 0 ,  - i U T  + l/c at 
i i  
BT 
cos w o t  , i i t  c 
‘0 i 
= -E brrC e - 
12 
We now seek a solution of  (8) of t he  form (6):  
where t h e  zero th-order  values  are 
+ i.R t -iRkt ETn - 5 ,  e k + 5 -  T e , 
... , 
P 
I .  
w i t h  
The only d i f f e r e n c e  ( so  fa r )  from a c l a s s i c a  pe r tu rba t ion  
+ + 
expansion i s  t h a t  we consider 5- and 5- t o  be a t  t h i s  s t a g e  a r b i t r a r y  L T 
func t ions  of E t ,  . . ., known only a t  E t  = 0. Thei r  dependence upon E t  
w i l l  emerge, i n  t h e  usua l  way, as a consequence of t h e  requirement 
t h a t  t h e  O ( E )  p a r t s  of Eq. (9) s h a l l  remain small compared t o  t h e  
O(1) p a r t s .  Now, no t ing  t h a t  i n  Eqs. ( 7 ) ,  must be i n t e r p r e t e d  as A t  
\ w e  may pass on t o  t he  ~ ( t )  p a r t  of tile eqa i i s io i i .  
. 
14 
A small amcult u f  a lgebra ic  jugglink: with E q s .  ( t3)  shows 
t h a t  
We s u b s t i t u t e  (9) and (10) i n t o  (Z), and make use of (111, g e t t i n g  
2 
= i k  li "pi w: 7 ) ~  'oi J ('i eiwPt + 2 e - i W  P t ) 
T 
k 2 + iRkt -iR t -i.w t - 5; e k ) (eiwot + e o ) , (E .  w VOi)  (ST e .- - - -  I. p i  
"k 
( 1 4 )  
upon equat ing c o e f f i c i e n t s  of  E .  
I ' .  
Most of t h e  s o l u t i o n s  t o  E q s .  (13) and ( 1 4 )  have t h e  prciperty 
t h a t  E and E grow q u i t e  l a rge ,  even i f  we s t a r t  them from zero,  
f o r  c e r t a i n  c r i t i c a l  va lues  of W . They soon grow t o  dominate E 
and E 
exp [ +  i (Wo - w ) t ]  d r i v e  ET1 t o  l a r g e  va lues  when wo - wp ," ilk, and 
those  on t h e  r i g h t  of (14.) conta in ing  exp [I? i (w0 - Rk) t] d r ive  
T 1  L 1  
To 0 
I n  d e t a i l ,  t h e  terms on t h e  r i g h t  of (13) which con ta in  L 0. 
P - 
EL1 t o  l a r g e  va lues  when the  same cond i t ion  is met. 
The only  way w e  can avoid t h i s  ca t a s t rophe  i s  t o  use t h e  (as 
+ + 
y e t  a r b i t r a r y )  terms J ~ X  / J ( E t )  and Jc, / d ( c t )  on t h e  l e f t  o f  Eqs. (13) 
and (14.) t o  cance l  t he  t rouble-causing terms on t h e  r i g h t .  One can 
then r e a d i l y  so lve  f o r  a well-behaved E and EL1, once t h i s  has  T1 
been done. 
Suppose we de f ine  
where Ak (we assume it i s  
measures t h e  depar ture  from p e r f e c t  resonance.  
and so  write an E i n  f r o n t  of i t) 
Note t h a t  f o r  (15) 
t o  be s a t i s f i e d ,  we must have W0 2 2 W . The cance la t ion  of t h e  
P 
aforementioned resonant  terms g ives  us: 
16 
and 
Eqs. (16) and (17) are  an  autonomous p a i r  of  d i f f e r e n t i a l  
+ +  
equat ions  (wi th  p e r i o d i c  c o e f f i c i e n t s )  f o r  t h e  amplitudes S?, 5; , as 
func t ions  of t h e  slow" time v a r i a b l e  E t .  On t h e  ques t ion  o f  whether 
or  not  t h e y  have growing so lu t ions  hinges t h e  ques t ion  of whether 
o r  n o t  t h e  waves can be pa rame t r i ca l ly  e x c i t e d  by  t h i s  mechanism. 
' 11  
That growing so lu t ions  do ex i s t  can be r e a d i l y  demonstrated; 
+ 
e l i m i n a t i n g  5 -  between Eqs. (16) and (17) l eads  t o  L 
17 
h i s  r e a l  and p o s i t i v e ,  and i n  any plasma, t h e  C i ' s  w i l l  be 
dominated by the  e l e c t r o n  con t r ibu t ion .  Therefore  
or, p ick ing  h > 0 f o r  de f in i t eness ,  
A l l  t h e  s o l u t i o n s  of (18) a r e  of t h e  form 
where 
E q s .  (20)  imply t h a t  t h e r e  always e x i s t  exponen t i a l ly  
2 
growing s o l u t i o n s  which occur whenever A* k 
o f  p e r f e c t  matching, we have Ak = 0, and a maximum growth ra te  
which i s  j u s t  h . 
< 4 h . For t h e  case 
18 
IV. DISCUSSION 
For any w > 2 w there  w i l l  always e x i s t  a range of wave 
0 - p,  
numbers and f requencies  f o r  which t h e  parametr ic  resonance condi t ion  
(15) i s  met, and f o r  which Eqs. (20) p r e d i c t  growing o s c i l l a t i o n s .  
The growth r a t e  f o r  the most r a p i d l y  growing waves, A, must 
be << t he  o ther  f requencies  of t h e  problem, f o r  t h e  mul t ip l e  time 
s c a l e  approach t o  be app l i cab le .  For t h i s ,  i t  s u f f i c e s  t h a t  
The waves w i l l  be o s c i l l a t i n g  a t  f requencies  R 
w 
( t r a n s v e r s e  p a r t )  and k 
( l o n g i t u d i n a l  p a r t ) ,  both of which a r e  much l a r g e r  t han  the .  
P 
maximum growth r a t e  X .  
The foregoing theory  does not  provide an express ion  f o r  t h e  
l i m i t i n g  amplitude of t h e  o s c i l l a t i o n s ,  s ince  no such express ion  can 
come from a l i n e a r  theory,  such as t h a t  of Eqs. (41, o r  of Aliev and 
S i l i n .  7 
I n  a l a b o r a t o r y  plasma, e x c i t a t i o n  i n  t h i s  manner can be 
expected t o  l e a d  t o  t ransverse  electromagnet ic  waves which leave  t h e  
plasma, perpendicular ly  t o  the  appl ied  e l e c t r i c  f i e l d .  They should be 
and occupy a bandwidth which i s  %’ peaked i n  frequency about wo - 
roughly p ropor t iona l  t o  t h e  s t r e n g t h  of t h e  appl ied  e l e c t r i c  f i e l d .  
Thei r  appearance should become less  r a p i d  a s  wo i nc reases .  
some f a i r l y  simple and s t ra ight forward  experimental  p r e d i c t i o n s  can 
be drawn from t he  foregoing theory .  
Thus, 
20 
V . ACKNOWLEDGEMENTS 
Thanks are expressed to Dr. I. Alexeff, for a conversation 
which led to the above calculation. This work was supported in 
part by the National Aeronautics and Space Administration grant 
NsG-233-62. 
21 
'N. Minorsky, Nonlinear O s c i l l a t i o n s  (Princeton,  N .  J., D .  van Nostrand 
2 
CO., I n c . ,  1962). 
N .  N .  Bogolyubov and Y. A. Mit ropo l sk i i ,  As.ymptotic Methods i n  t h e  
Theory of  Nonlinear O s c i l l a t i o n s  (New York, Gordon and Breach Science 
Pub1.i shers ,  I n c  . , 1(961) . [ t r a n s l a t e d  from t h e  Russian.  3 
c f .  pp. 267-298. 
'Yu. M. Al iev and V. P. S i l i n ,  Sovie t  Physics JETP 5, 601 (1965). 
'V. P. S i l i n ,  Sovie t  Physics JETP - 21, 1127 (1$5). 
'E. A. E'rieman, J. Math. Phys. - 4 ,  410 (1963). 
bD. Montgomery and D.  A. Tidman, Phys. F lu ids  I ,  242 (1964). 
D. A. Tidman and H. Sta ine r ,  Phys. F lu ids  8, 345 (1965). 
F. S l u i j t e r  and D. Montgomery, Phys. F l u i d s  8, 551 (1965). 
D.  Montgomery, J. Math. Phys . 5, 1788 (1964); 
D. Montgomery, Physica - 31, 693-(1965). 1 
[See a l s o :  
70ne o the r  s i g n i f i c a n t  d i f f e r e n c e  between t h i s  case  and t h a t  of  
Aliev and S i l i n  i s  worth not ing a t  t h i s  p o i n t .  
as f ixed ,  and thus  f ind  only a narrow band of f requencies  wg f o r  
which t r ansve r se  waves of wavelength 2n/k a r e  pa rame t r i ca l ly  exc i t ed .  
However, for  wo > 2 wp, t he re  w i l l  always e x i s t  some k ' s  f o r  which 7, 
mul t ip l e  r e l a t i o n s h i p  t h a t  must o b t a i n  between wo and w 
Longitudinal  c&se of Befs: 3 and h . 
t h e  f a c t  t h a t  f o r  l o n g i t u d i n a l  waves, all k ' s  o s c i l l a t e  a t  w 
i n  t h e  cold-plasma approximation. 
We a r e  regard ing  k 
have p o s i t i v e  r e a l  p a r t s .  This i s  i n  c o n t r a s t  t o  the i n t e g e r -  - 
i n  t he  
It. i s  only  a r e f l e g t i o n  of  
P' 


Parametric excitation of transverse waves in a plasma

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