Download 06_chapter 1

CHAPTER 1
INTRODUCTION
1.1
BRIEF HISTORY
Plasma,
tion
of
atoms.
the fourth state of matter,
positive
A
plasma
ions,
state
negative
can
is a collec­
electrons
exist
not
and
only
at
neutral
higher
temperatures but also at lower temperatures provided there
is a mechanism for ionizing the gas and the density is low
enough
such
essential
that
recombination
is
not
rapid.
The
two
characteristics of a plasma are its tendency to
preserve charge neutrality on a macroscopic scale and the
collective
behaviour
arising
out
of
the
presence
of
2
long-range
Coulomb
force
which
99Z of
particles. More than
operates
between
charged
the entire universe is consti­
tuted by matter in plasma state.
I
When the electrons in a plasma are displaced from
their equilibrium positions, strong el eerie fields are set
up
between
the
negatively
charged
layers
and
background positive layers. These electric fields
restore
the
particles
the
more
initial
neutral
condition
back to their original
mobile
electrons
positions.
oscillate
frequency known as the plasma frequency.
tions
propagate,
known
as the Langmuir
and
is
we
get
the
wave.
by
electron
It
is
tend to
bringing
As
with
the
the
a result,
a
certain
If these oscilla­
plasma
wave,
also
a high-frequency
wave
el ectrostatic in nature ..A1S0 unmagnetized
plasma
can
support the low-frequency ion-acoustic mode and the highfrequency electromagnetic wave. The presence of a magnetic
field,
however,
greatly
increases
the
number
of
modes
<
that the plasma can sustain.
1.2
PLASMA INSTABILITIES AND NONLINEAR EFFECTS
The two important characteristics of a plasma are
nonlinearity
grows
to
and dispersion.
in a plasma,
the
nonlinear
If the
amplitude
immediately it passes from
regime,
nonlinear processes come into
and
as
a
operation,
of
a wave
the linear
result,
various
leading to
the
3
saturation
called
of
of
the
wave
tiie dispersion
propagation
of
a
amplitude.
relation,
wave
and
where
k
relation w=w(k)
The
is
w is
the
wave
wave is called dispersive if it contains
different
wave
numbers
different phase velocities
instability
interrelated
laboratory or found
mag netospheric,
Rather,
under
in
these
nature
or
equilibrium
free
energy
gradients
Generally,
plasmas
not
in
can
state
such
in
in density,
with
two important
either
created
most
The
the
cases
fact
magnetic
equilibrium.
assumed
to
that
plasmas
found
arc
Car
they may ha'ie some
temperature
anisotropy
field
the
be
laboratory
that
in
and
intcrplanatory or
thermodynamical
in
implies
as
The
propagaces
plasmas
space,
equilibrium.
created
number.
(w/^).
in the
are
dynamical
and
and nonlinearity are
terms.
frequency
a number of modes
*
having
the
is
and
from
be
the
sources of
(T^^b:
'1'
temperature
),
(R 7 n ,
V B, V T) or some kind of beam streaming through the plasmas. 'ITus
energy
plasma
may
or
be
into
instability
is
converted
radiation
a
in a collective
small
deviation
from
way.
a
violent
motion
electromagnetic
where
such
This
leads
dynamical
a
waves.
of
Plasma
conversion
to
the
equilibrium
fact
the
takes
that
becomes
a
the
further deviation.
When
grow,
of
process
[dace
cause of
into
the
plasma
instabilities
nonlinear
phenomena
are
that
excited
appear
and
modify
waves
the
plasma
states.
important
Ordinarily,
when
the
wave
the
nonlinear
energy
density
larger than that at the thermal
the
wave
energy
density
and
effects
W
(W/
becomes
equilibrium.
the
thermal
much
The ratio of
energy
1
become
density
1
„,) is used as a measure of turbulence.
n0l
Generally, nonlinear phenomena can be grouped into
two
categories
Coherent
: coherent
and
nonlinear phenomena
the nonlinear development
incoherent
or
turbulent.
refer to circumstances where
of the system
is followed with
all due regard to. phase information carried by the waves.
Nonlinear
examples
periodic
of
waves
coherent
or
soli tons
nonlinear
are
phenomena.
nonlinear waves occupy only a small
But,
obviously
coherent
space in the spectrum
of nonlinear phenomena. Most phenomena in nature are, how­
ever,
found
'turbulent'
to
when
be
incoherent
applied
to
a
or
turbulent.
plasma
usually
The
tern
refers
to
circumstances where a large number of collective oscilla­
tions arc excited by a linear instability. A system can be
weakly
or
strongly
turbulent
depending
on
the
fluctuations of the particular field quantities.
level
of
3
1.3
PLASMA-MASER INTERACTION
As
we
have
already
mentioned,
plasmas
either
created in the laboratory or found in nature are far from
the equilibrium state.
energy
Due
for which
to
the
They may have some sources of free
instabilities
presence
of
an
may occur
instability,
in
the
many
plasma.
nonlinear
processes come into operation and the energy is redistri­
buted
over
flows
from
the
various
degrees
the unstable
of
freedom.
inode to the more
The damping mechanisms will
then extract
The
stable
energy
states.
the energy
from
the stable mode and give it to the plasma particles. This
results
in
plasma
Finally,
a
provided
the
heating,
anomalous
quasi-steady
turbulent
sources
the
and
diffusion
state
sinks
is
of
etc.
attained
energy
are
maintained at constant level.
Consideration of anomalous radiation from a plasma
with enhanced low frequency fluctuations can be treated as
•
2-4
a central problem to interpret numerous astrophysics!
,
viz.,
Pulsar
Auroral
Kilometric
Radiation
etc.
and
Radiation,
laboratory
Jupiter
plasma
Emission,
phenomena,
viz., Nuclear Fusion. Since the Lawson criterion'1 (Lawson,
1957)
for
radiation
a
thermonuclear
loss,
reactor
understanding
the
depends
strongly
on
radiation
mechanism
is
necessary for the success of nuclear fusion.
6
The principal
ionized .plasma
moving
is
charged
collision,
source of
classical
particle
radiation in a hot,
fully
bremsstrahl u n g . Whenever
suffers
acceleration
classical bremsstrahl ung
occurs.
a
due
to
Recently,
it
I
has
been
shown
as
Plasma-Maser^,
turbulent
which
that induced bremsstrahlung^, later termed
plasma.
originates
plays
a
very
important
It is a new kind of
because
of
the
role
in
a
radiation process
electron
acceleration
through the modulation fields. Coupling of electromagnetic
or
electrostatic
turbulence
test
fields
modulation fields
electrons,
wave
produces
fields
the
with
low
modulation
field.
nonlincarly interact with
causing
electrons.
The
radiated,
resulting
a
strong
energy of
in
the
acceleration
the
resonant
the
high
frequency
resonant
of
electrons
frequency
These
these
is
test
then
wave
amplitude to grow. This new type of radiation is known as
induced
bremsstrahl ung.
Obviously,
it
will
vanish
in
absence of the low-frequency turbulence. It is, therefore,
termed
as
anomalous
radiation
associated
with
the
1ow-frequency turbulence.
At
this
stage,
it will
be
relevant
the basic differences between classical
induced
bremsstrahlung. Classical
to point
out
bremsstrahlung and
bromsstrahlung emission
occurs whenever a moving electron suffers acceleration due
to collisions.
originates
On the other hand,
because
of
Che
induced
nonlinear
bremsstrahlung
interaction
between
7
the resonant electrons and the
As
mentioned
above,
these
nodulated electric fields.
modulation fields
appear
as a
result of the coupling between a high-frequency perturba­
tion
field
and
the
low-frequency
fluctuations.
Thus
it
t
shows that the origin of acceleration is quite different
in bo th cases.
The plasma-maser interaction does not require any
frequency
or
satisfied.
wave
That
number
is
mode-conversion
matching
why,
it
processes
is
like
conditions
different
to
from
parametric
be
other
interaction,
nonlinear scattering etc.
According
to
the
(also called quasilinear
recent
weak
theory),
turbulence
the lowest
theory
order
mode-
2
mode coupling processes are composed of three parts . They
are
the
resonant
conditions
are
Landau damping
three-wave
interaction^
[the
matching
+«2 = w^], the nonlinear
+k£ = k ^ ,
[the condition is JL-w = (K-k).V)J and the
A
r
■*-->
*">
,
plasma-maser interaction
[the conditions are w = k.v; fif=
-> •*
->
K.VJ. Here w & k are the frequency and wave vector of the
low-frequency wave and
1L & K those of the high-frequency
wave .
This new maser effect in plasma
the
induced
considerable
basically
brcmsstrahlung
attention
in
turbulence, viz.,
instability,
the
past
few
has
years.
received
This
is
an energy up-conversion process in which there
8
is a flow of energy from a low-frequency mode in a plasma
to a high-frequency
high-frequency
turbulence
mode,
mode.
is
thereby causing
The
necessary
presence
for
of
this
growth of the
a
low-frequency
energy
up-conversion
«
process to take place. This maser effect is effective even
without the electron population inversion.
In
plasma
this
acts
as
amplitude grows
mechanism
the
the
wave.
pump
low frequency wave
The
in the
high-frequency
at the expense of the pump wave
The resonant electrons carry effectively
wave
energy.
the energy from
the pump wave field to the radiation field.
Historically,
the growth of the nonresonant
through the nonlinear interaction with
the
resonant
mode
mode
9
was
first
pointed
out
independently
Tsytovich
et al.^
It has later
2
plasma-maser . The nonlinear effect
nonresonant
electron
modes
by
been
of
Natnbu
termed
the
as
the
resonant
and
simultaneously on the evolution
distribution
function
is
called
and
the
of the
inverse
pii asma-maser U
This
anomalous
high-frequency
radiation
in
the
presence of enhanced low-frequency fluctuations are often
reported
wave
in
laboratory
excitations
from
and
space
ion-sound
plasmas.
waves
were
The
Langmuir
reported
by
Amagishi 12 . High-frequency whistler mode emissions driven
by
Jow-f requency
ion
mode
were observed
by Fujiyama
and
9
Nambu^.
The
solitons,
and the emission of upper hybrid wave solitons
from
Ohya
Langmuir
ion waves
15
were
wave
reported
, respectively.
emission
from
by Boswel 1
14
ion-sound
and
Mori
and
In Z pinch plasma, the simultaneous
observation
of
ion-sound
radiation is
reported"^.
waves
and
electromagnetic
The plasma wave excitation from
MHD modes is observed in Tokamak 17 . In space plasmas,
ULL
modulated
static
bursts
whistler
18
emissions1
, chorus-related
ELF
19
, and
turbulence
Langmuir
wave
in
solar
the
emissions
wind
20
the
electrodriven
have
by
been
observed. More recently, the presence of broadband electro­
static bursts has been reported in the
region of Auroral
kilometric radiation, suggesting possible link between the
electrostatic bursts and the AKR generation^ .
Recently,
it
has
been
shown
22 523
that
the
mechanism of plasma-maser can be best understood in terms
of a high-frequency
nonlinear force.
This high-frequency
nonlinear force arises as a result of the resonant inter­
action
between
the
electrons
and
the
modulation
fields
caused by the coupling between a test high-frequency wave
field with the low-frequency turbulence field. This is in
contrast
to
low-frequency
the
parametric
nonlinear
interaction
force,
viz.,
process
the
where
a
ponderomotive
force is produced to make a low-frequency wave unstable.
On the other hand,
in plasma-maser process,
the nonlinear
force is a high-frequency one and makes the high-frequency
10
wave unstable.
The
Radiation,
related
generation
mechanisms
of
Auroral
ULF modulated electrostatic
electrostatic
I
bursts
in
Kilometric
waves
the
and chorus-
earth's
outer
magnetosphere, whistler mode signals in the solar wind and
type III solar radio bursts are presented on the basis of
t'he
theory 2 A .
plasma-maser
universal
also
theory^.
The
extra-ordinary
extra
25
stabilization
of
. The generation mechanism of X-mode
presented
planets.
r.f.
drift mode based on the plasma-maser theory has
also been proposed
is
The
on
basis
emission
of
mode
a familiar
is
The auroral
ordinary
mode.
radiation
kilometric
radiation
extraordinary mode.
may
27
of
Like
and
also
of
of
in
the
the
radio
radiation is also in the
^
AKR,
the
are
It is probable
one
plasma-maser
wave
feature
the
(bKOM)^
be
the
electromagnetic
kilometric
kilometric
interaction
the
the
the
saturnian
broad-band
found
that
Jovian
to be in the
the
plasma-maser
probable
generation
mechanisms of the X-mode from the radio planets.
In space
the
turbulent
frequency modes
wave
etc.
processes
All
and
astrophysical
fluctuation
energy
such as MHD wave,
of
the
previously
plasmas,
is
the
contained
drift
wave,
studied
most
in
of
low-
ion-sound
mode-coupling
show the energy down-conversion from
the high-
frequency mode to the low-frequency mode. Accordingly, the
11
standard mode-mode coupling processes may not play a signi­
ficant
role
in
astrophysical
plasma.
On
Che other hand,
the most important characteristic of the plasma-maser
is
the
to
energy
up-conversion
from
the
low-frequency
mode
t
the high-frequency mode. Thus, the plasma-maser proces may
play
an
important
role
in
space
and
astrophysical
p,
3a sm as24
in
the
1ow-frequency
either
or
interaction
because
plasma-maser
of
the
the
is
value
function
resonant
hangniui r waves
ion-cyclotron
low
theory,
is assumed
of
interaction
But
the
and such waves is
toe
at the
the
to contain
waves.
between electrons
electron distribution
However,
of
turbulence generally
ion-acoustic
resonant
small
study
slope
resonant
between
of
the
velocity.
electrons
and
Ia rge because of the sufficient! y high
value of the slope of the electron distribution function
at
the
resonant
velocity.
Thus
the
pi asma~maser
inter­
action between Langmuir waves and electrons may give rise
lu s ign if ic an L‘1y enhanced radiation.
This
closely
related
accelerated
physical
nonlinear
by
to
mechanism
the
random
process
viz.,
by
introduced by Tsytovich
30
which
low-frequency
mode for the plasma-maser
.
plasma-maser
particles
oscillation.
interaction
was
is
are
The
first
12
Recently,
effective
in
particle
flow
without
open
and
for
supply,
(plasma-maser)
may
is
is
system
type
originate
of
plasma-maser
where
the
For
supply
There
open
the
31 ’32 .
state
new
that
systems
particle
plateau^.
However,
shown
available
turbulent
quasi-1 inear
particle
is
plasma
is
energy
stationary
loss.
it
energy
closed
from
no
with
enhanced
with
the
the
the
radiation
energy
mode-mode
spontaneously
and
systems
outside,
realized
is
and
couplings
as
anomalous
radiation. Some of the input energy is dissipated through
the
nonlinear
turbulance
forces'^.
comes
from
New
the
dynamic
state
interaction
of
of
plasma
a
given
stationary plasma turbulence with its surroundings.
Quite
energy
recently,
conservation
Nambu has sho w n ^
relation
between
that
the total
particle
kinetic
energy and wave energy is satisfied for the plasma-maser
process. The ManJ ey-Ruwe
relation^
among, plasma waves is
l
violated and as a result an efficient energy up-conversion
from the low-frcquency mode to the high-frequency mode is
possible even for a normal unreversed population in plasma
tu rbulonce.
The plasmn-maser process always co-exists with the
cjuasiJ inear
process
between
electrons
mode. The new mode-mode coupling comes
tivc
nonlinear
force
37 . The
standard
and
the
resonant
from the dissipaquasi!inear
theory
13
neglects
the
validity
phase
of
the
order
assumption
approximation
foundation
of
the
2 which
plasma
to the
1.4
second
in
is
linear
clarified
31.
ref.
field
This
response
(E 2) •
under
the
result
theory
The
random
gives
of
a
the
turbulent
»
neglects
l ow f r e q u e n c y
the second o r d e r e l e c t r i c
field
due
turbulences.
A HR IMF DESCRIPTION OF THE. THESIS
The p r i n c i p a l
gate
some
aspects
theory.
pi a s i n a - ma s e r
is
which
there
purpose o f
of
plasma-maser
in
electric
this
As
this
new
wc
ma s e r
have
an e n e r g y
is
of
energy
to
presence
of
low-frequency
for
energy up-conversion process
It
is
high-frequency
now w i d e l y
effect,
f r om
mode
this
viz.,
the
already,
the
a
field
process
low-frequency
radiation
turbulent
accepted
to i n v e s t i ­
up-eonversion
resonant
a
a
is
me n t i o n e d
basically
n flow
study
field.
is
The
essential
to take p l a c e .
that
p I as 111 a-m as e r c a n be
(
best
understood
in
terms
force^’^ .
This
a result
the re s o n a n t
the
of
modulated
turbulence
nonlinear
and
the
a
high-frequency
fields
field
force
of
with
or e l e c t r o m a g n e t i c
nonlinear
interaction
caused
a
test
accelerates
accelerated
high-frequency
electrons
waves.
or
by
force
arises
between e l e c t r o n s
coupling
high-frequency
decelerates
can
nonlinear
t h e n e mi t
between
field.
the
as
and
the
This
electrons
electrostatic
In
considered
the
the
second
emission
nonlinear
chapter
of
force
of
this
thesis
electromagnetic
discussed
waves
above
in
a
we
have
caused
plasma
by
with
Langmuir turbulence 38 . The Langmuir turbulence is excited
I
by
a weak
electron
beam
drifting
through
plasma of Maxwellian electrons and ions.
a
background
The growth
rate
of the electromagnetic wave through the plasma-maser inter­
action lias been calculated. The expression for the growth
rate
contains
direct
two
coup I ing
parts;
one
contribution
part
corresponds
while
the
other
to
the
part
the
polarization contribution.
These results fully agree with
3 c)
those obtained irom the standard formulation method' . The
essential
feature of the present
formulation
is that
the
nonlinear
force is explicitly calculated and is shown to
drive the high-frequency instability.
In
gene rat ion
the
of
third
electromagnetic
mode in a magnetized
turbulence
chapter
through
plasma
the
we
wave
have
in
the
studied
the
extraordinary
in the presence of 'Langmuir
p 1asma~ma sor
interaction^’. The
nonlinear dispersion relation ofthe X-mode in the presence
of Langmuir turbulence has been calculated and the growth
rate
then
coupling
not
that
obtained.
term
in the
contribute
the growth
to
The
results
show
that
nonlinear dielectric
the growth of
the
is wholly derived
direct
[unction
X-inode.
from
the
This
does
means
the polarization
■ontribution. This is markedly different from the results
15
obtained
in
the
case
of
electromagnetic
unmagnetizecl
plasma 3 9 or in
radiation
a magnetized
in
contributed
both
by
the
case
31
plasma
the direct
emission
of
in
ordinary
where
coupling
the
as
an
mode
growth
is
well
as
the
abundance
of
the
*
polarization
terms.
electromagnetic
planets,
the
generation
view
of
in
the
emission
present
applications.
plasma-maser
In
It
study
is
mechanisms
here
of
X-mocle
seems
probable
considered
the
that
may
in
to
the
emission
radio
have
useful
mechanism
be one of
X-mode
the
of
the probable
from
Lho
radio
planets. One of the primary conditions necessary for this
mechanism to be effective is the presence oL a turbulence
Held
from
which
energy
can
be
transferred
to
the
radiation field via the resonant electrons.
In
studied
Lho
fourth
chapter
the generation
l.angmuir
turbulence
plasma-maser
polarized
of
in
their
this
electron
a
magnetized
40
.
interaction
with
of
The
electric
thesis,
we
have
Bernstain waves
from
plasma
Bernstein
vector
nearly
through
'
modes
field.
They propagate
41
are
parallel
the wave vector and perpendicular to the external
neous magnetic
the
to
homoge­
in frequency
ranges
that lie between harmonics of the cyclotron frequency. The
nonlinear
dispersion
waves
the
in
calculated
and
relation
presence
its
of
growth
of
the
Langmuir
rate
is
electron
turbulence
then
Bernstein
has
obtained,
been
it
is
16
observed that the contribution flrom the direct coupling in
the nonlinear dispersion relation vanishes and the growth
rate is wholly derived from the polarization contribution.
The
emission
of
electromagnetic
^4 -
ordinary and extraordinary inodes^
from
the
nonlinear
respectively,
turbulence
in
nonlinear
between
the
field
Langmuir wave
force
present
in
fie Id driven
arising
electrons
and
and
sixth
the
system
are
the
the
presented
c h ap te rs .
is
by a we ak e 1e c iron
outof
in
in a magnetized plasma
consideration
fifth
forces
2
waves
the
resonant
modulated
'['he
again
the
beam.
The
interaction
fields
caused
by
coupling between the high frequency O-mode and X-mode test
Tit'ids respectively with the Langmuir tu lnilence have been
calculated.
The
nonlinear
forces
thus
obtained
are
then
used in their respective fluid equations to obtain finally
their growths.
Then the results
are compared with those
obtained earlier from the standard formulation.
«
Lastly,
have
in the seventh chapter of the thesis,
summarized
the
results
obtained
in
the
we
previous
cha pte rs.
Before
clear one
the
concluding
thing
interaction
nonresonant
high
more.
this
chapter
we would
We have considered
between
resonant
frequency
wave.
Langmuir
like
in this
wave
Considerable
to
thesis
and
the
attention
17
has
aJready
betwecMi
waves,
been
paid
resonant
the
the
ion-sound
considering
through
to
and
the growth
resonant
plasma-maser
non-resonant
rate of
interaction
interaction
the
between
Langmuir
Langmuir
wave
electrons
and
«
ion-sound
electrons
where
waves.
and
w,k
and
Lon-sound wave
The
ion-sound
v£
are
number
resonant
waves
the
is
interaction
weak
between
because
ion-sound
w/k<<v ,
frequency,
anti the electron thermal
the
velocity,
respectively. Then the growth rate of the Langmuir wave is
sina I1 . On
the
interaction
other
between
hand,
the
we
consider
resonant
Lhe
Langmuir
plasma-maser
wave
and
a
non-resonant high frequency wave. As because the resonant
interaction between the electrons and the Langmuir wave is
strong,
1a rge.
the
growth
rate
of
the
high
frequency
wave
is
IS
RKKKRKNCKS
1. A. Hasegawa, Plasma
Instabilities and Nonlinear
(Springcr-Ver1a g , New York, 1975).
Effects
2. M. Nambu, Laser and Particle Beams 1, 427 (1983).
3. M. Nambu, Phys. Fluids 25, 1196 (1982).
4. D.A. Gurnctt, J. Geophys. Res.
Warwick,
204,
J.B.
995
Pearce
(1979);
Sutherland,
and
M.A.
Astrophys.
J.
79, 4227
A.C.
(1974); J.W.
Riddle,
Ruderman
196,
51
Science
and
P.G.
(1975);
N.C.
Wehrlin, J. Geophys. Res. 86, 1365 (1981).
5.
J.D.
Lawson,
Proc.
Phys.
Soc. London
Ser.
B70,
6
(195 7 ).
6. M. Nambu and P.K. Shukla, Phys. Rev. 20A, 2498 (1979).
7 . V .N . Oraevsky and R.7 . Sagdeev, Sov . Phys-Tech. Phys .
7, 955 (1963).
8 . M.N.
Rosenbluth,
B. Coppi and R.N.
Sudan, Ann.
Phys .
55, 248 (1969).
9. M. Nambu, Phys. Rev. Letters 34, 387 (1975) .
Ni. 1he 1ms son , Phys .
L. Stenflo and 11. 1
10. V .N . Tsytovich,
Scripta 11, 251 (1975).
I1 . V.S.
Krivitsky
and
1
3 .V . V 1ad im i rov , J . PI asm a Phys .
46, 209 (1991).
12 . Y. AmagishL, J. Phys. Soc. J pn . 29, 764 (1970).
13.
H.
Fujiyama
and
M.
Nambu,
Phys.
Letters
105A,
295
( 1.984) .
14. R.W. Boswell, Geophys . Res. Letters
11, 1015 (1984).
15. I. Mori, and K . Ohya , Phys. Rev. Letters 59, 1825 (1987).
16. K.H. Finken and U. Achmann, Physica 113B, 135 (1982).
19
.17. l.H. Hutchinson and S.K. Kissel , Phys.
Fluids 26, 310
(1983 ).
18.
N.
Corni 11eau-Wehr1 in, J.
Geophys.
Res.
86,
1365
(1981 ).
19. L.A.
Reinlditner,
U.A. Gurnett,
and T.E.
Feist man, J.
Geophys. Res. 88, 3079 (1983).
20.
C.F.
Kennel,
H.F.
Petschek,
F.V.
Coronity,
K.W.
Fredricks, D.A. Gurnett and F.J. Smith, Geophys.
Res. Fetters 7, 129 (1980).
21.
R.
Pottelettc,
M.
Malingre,
A.
Bahnsen
and
M.
despersen, Geophys. Res. Fetters 16, 5.15 (1987).
22. M. Nambu and H. Akama, Nuovo Cimento 87B, 176 (1985).
2 3 . S.
Bujarbarua,
29A, 2171
S.N.
Sarma
and M.
Nambu,
Phy s . R e v .
(1984) .
24. M. Nambu, Space Sci. Rev. 44, 357 (1986).
25. M. Nambu, Phys. Rev. 35A, 1953 (1987).
26 . S.N. Sarma , K .K . Sarma and M. Nambu, J. P 1asma , 1
IMiys .
46, 331 (1991).
27. M.F. Kaiser, M.FG Desch, J.W. Warwick and .1.B . Pea rce ,
Science 209, 1238 (1980).
1
28. .J.W . Wa rwick , .1.B. Pearce, D.R. Fvans, T.D . Carr, J .J .
Schauble,
J.K.
Alexander,
M.F.
Kaiser,
M.I).
Desch, M. Pederson, A. Fecacheux, G. Daigne, A.
Boischat
and
C.ll.
Barrow,
Science
212,
239
( 1981 ) .
29. V. Fehlane and G. Daigne, J. Geophys.
Res.
90 , 12073
( I98 5).
30. V.N. Tsytovieh, Sov. Phys. JFTP 6 2 , 483 (1985).
31.
M.
Nambu,
T.
llada , S.N.
Sarma and S. Bujarbarua,
Phys. Soc. Jpn. 6 0 , 3004 (1991).
J.
20
32.
V .N . Tsytovich
and V.N.
Krivitsky,
Comments.
Plasma
Phys. Controlled Fusion 1.1, 119 (19(37).
33. W. F.
Drummond
and
D.
Pines,
Nucl . Fusion Suppl . 3,
1049 (1962); A. A. Vedenov, H.P. Velikhov and
K.Z., Sagdeev,
Nucl . Fusion
Suppl . Pt.
2,
465
(1962 ) .
34. M. Nambu, J. Phys. Soe. Jpn. 53, 1594 (1934).
35. M. Nambu, Phys. Rev. betters (Submitted).
36 . J .M. Manley and II. K. Rowe, Proc. oi 1RF 44, 904 (1956)
37. M. Nambu, Phys. Fluids 31, 1296 (198,3).
33 . K ..K.
Sariiia, S. N.
Rev. 43A,
39. M. Nambu, S.N.
Sarma,
M.
Nambu anti '!'. Ilada, Phy s .
5555 (1991).
Sarma
and S. Bujarbarua,
Phys.
FI uids
B2, 302 (1990).
40.
M. Nambu,
S.N.
Sarma anti K.K. Sarma,
Phys. Rev . 4 5 A ,
7456 (1992).
41 . 1 .B. Bernstein, Phys. Rev. 109, 10 (1958).
42. M. N am bu , S.N. Sarma, K.K. Sarma and U .N . Das,
Rev A (submitted).
Phys .