Now  we  examine  the  parallel  component  of  the  exact  momentum
equation  (44).   Proceding  as  before,  we  would  find  6°E  =  0
to  lowest  order  and  evaluate  B-E   to  first  order  in  terms  of
a  zero  order  left  side  of  the  equation.   But,  if  we  choose  not
to  impose  the  constraint  that  B-u  =  &-u_     to  lowest  order
(temporarily  violating  the  expansion  in   5)  we  find  (approxi-
mating slightly)


(47)      B-  ^  +  ^-  B-div  (P_  =  JT  koB-E


-  35  -


This  should  be  compared  with  (42)  and  also  with

(48)      _Ko^+i-curl  B-  J  •

The  two  equations  (47)  and  (48)  describe  plasma  oscillations
(presumably  very  fast)  about  an  equilibrium  given  by  (42)  for
p"-E  and  curl  B/u.  for  J.   A  large  amplitude  plasma  oscilla-
tion would  interfere  with  our  basic  requirements  for  guiding-
center  motion.   But,  for  slowly  varying  "source"  terms  div  (P
and  curl  if,  the  generated  plasma  oscillations  will  be  small  in
amplitude,  and  the  quasi -equilibrium  for  both  B'E  and  B-J  will
be  set  up  after  a  short  transient  comparable  to  a  plasma
period.
The  situation  is  more  complicated  if  one  includes  dis-
placement current  and  considers  problems  in  which  the  plasma
frequency  is  not  high  compared  to  the  Larmor  frequency.   In
such  a  case  one  must  impose  as  an  additional  requirement  for
the  validity  of  the  guiding-center  theory  that  all  natural
motions  be  slow  compared  to  the  plasma  frequency  as  well  as
the  Larmor  frequency.   Otherwise,  variations  in  the  "source"
terms   div  P   and   curl  B,  will  produce  inappropriately  large

values  of  (3-J  and  pMT,   Thus  plasma  oscillations  must  be

19
expressly  excluded  from  any  guiding-center  theory.
18.  H.  Grad,  "Ohm's  Law",  Institute  of  Mathematical  Sciences
Report  NYO-6486-IV,  August  1956.
