I ntrnat. J. Math. Math. Si.
Vol. 2 No. 2 (1979) 325-336
325
NONLINEAR INTERACTION OF WAVES IN A HOT
INHOMO.GENEOUS MAGNETIZED PLASMA
TARA PRASAD KHAN
Physics Department
Dinabandhu Andrews College
Dist- 24 Parganas, West Bengal, India
MAHADEB DAS
Physics Department
Serampore College, Serampore
Dist
Hooghly, West Bengal, India
LOKENATH DEBNATH
Departments of Mathematics and Physics
East Carolina University
Greenville, N.C. 27834, U.S.A.
(Received August 7, 1978 and
ABSTRACT.
in revised form May i, 1979)
This study is concerned with the theory of parametric coupling of
waves in a hot inhomogeneous magnetized plasma in which the temperature gradient
has been taken into account.
The general dispersion relation and the polarization
of the ordinary and the extra-ordinary wave modes are discussed.
The eigen-mode
solutions of the coupled differential equations for the wave amplitudes are obtained
in the terms of the so called three wave interaction matrix elements.
The theory
of nonlinear wave-wave interactions, which has been extended to the case of an
inhomogeneous magnetized plasma, is used to determine the threshold value of the
electric field and the frequency shift.
The results of this paper are also compared
NONLINEAR INTERACTION OF WAVES
326
with the other known results.
It is shown that the findings of this study are in
excellent agreement with the results of earlier investigators.
KEY WORDS AND PHRASES. Parametric instaby, Waves in inhomogeneo plasma,
wave-wave interactions, Threshold electric field and frequency shift.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
76X05, 76E30.
INTRODUCTION.
In recent years there has been considerable interest in the theory of
parametric instability in an inhomogeneous plasma [1-8] because of its fundamental
role in the study of weak plasma turbulence, and of many important physical
applications.
henomena
In their recent research-expository and survey article on parametric
in a plasma, Galeev and Sagdeev [9] reviewed the theory of parametric
instabilities in an inhomogeneous plasma using the generalized Mathleu equation as
a model equation.
They have also presented the latest advances of the nonlinear
theories of parametric instabilities based on the ideas of weakly turbulent plasma.
Perkins and Flick [I0] have made an interesting study of parametric instabilities
in an inhomogeneous plasmas, and then calculated the threshold electric field.
It
is shown that the value of the threshold electric field increases in an inhomogeneous
plasma because energy propagates away from the unstable region by electron plasma
waves.
Thus an additional energy loss occurs and is solely responsible for the
increase of the threshold electric field not observed in a homogeneous plasma.
Eubank [ii] experimentally confirmed the theoretical prediction of Perkins and
Flick.
In a recent paper, Kroll, Ron and Rostoker [12] have suggested that the
nonlinear resonance of two transverse lectromagnetic waves whose frequencies differ
slightly by the electron plasma frequency can be applied to excite longitudinal
electron plasma oscillations. Montgomery [13] observed certain mathematical
inaccuracy and physical limitations of the work of Kroll, Ron and Rostoker, and then
analyzed the problem of nonlinear wave interactions in plasma with laser beams.
327
T.P. KHAN, M. DAS AND L. DEBNATH
Using the perturbation method of Krylov-Bogliubov-Mitropolskii, Montgomery obtained
the amplitude-dependent frequency shift and wave number shift with physical
significance.
It was shown that the resonant excitation of longitudinal plasma
oscillations is possible by the transverse electromagnetic waves.
Etievant, Ossakow,
Ozizmir and Su [14] have investigated the nonlinear wave-wave interactions of
electromagnetic waves in an infinite homogeneous plasma.
It is interesting and
important to take into account the effects of density and temperature gradients on
the above problem.
The present study deals with the theory of parametric coupling of waves in a
hot inhomogeneous magnetized plasma in which the temperature gradient has been
The general dispersion relation and the polarization of
taken into consideration.
The eigen-mode solutions of the coupled
two different wave modes are investigated.
equations for the wave amplitudes are obtained in terms of the three wave interaction
matrix elements.
The theory of nonlinear wave-wave interactions is used
to determine the threshold electric field and the frequency shift.
The results
of this analysis are found to be in excellent agreement with those of earlier
workers.
2.
BASIC EQUATIONS FOR TWO PLASMA MODEL.
In two plasma model, the equations of motion and the continuity equation for
each kind of component are in the usual notations given by
Dv_n
Ze
Dt
m
V
E
(-Tap )
m
O
DO
Dt + 0
V
Z
--v-
-v
v
-n
(2.1)
(2.2)
0
eB
where
cm
is the cyclotron frequency, v
a
is the collision frequency of each
328
NONLINEAR INTERACTION OF WAVES
component and B
B
O
z is the external magnetic field
The subscript u stands for
e or i corresponding to electrons and ions respectively, Z e
electron or ion and
is the charge of an
is the Boltzmann Constant.
The Maxwell equations are
curl
_E
E
c t
1
curl B
(2.3ab)
4e(n-no)
dlv_E
c t
4r nev
dlv B
c
(2.4ab)
0
We consider a plasma model whose density dlstrlbution varies as 0
a
i
with the density gradient
(d__)
and
neglect the time dependence of
UO
due to any external electric field.
<
Vo x
(t)e
Voy
t)e
T
The
e
e
KT
fl e Ze
m
e2_e 2
(2.6)
(2.7)
0
3.
(2 5)
=x=
with
0
THE FIRST ORDER EQUATIONS.
_
In the first order approximation, equations (2.1)
dropping the subscript
-
where
+
v
--o
.VTg) i
(’----
Vv +
I
--Vl
Vv
o
Z__e E
m--o
Pao
zeroeth order solution of (2.1) gives
me e2-)e 2)
___e
pao(1-x)
Zfi x
-o
v--i
represents the first order terms in
(2.2) can be written as,
V’(Tp)
m
.VTg)
(--
_Zl
x v
--o
V_l
(3.1)
329
T.P. KHAN, M. DAS AND L. DEBNATH
Assuming the density gradient is small, we take the space and time dependent
_
exp[i(k
of the first order quantities as
r-
tot).
The equation (3.1) can
then readily be solved to obtain
where
vii =_ _E1
(3.2)
Is the mobility tensor given by
Ze
[(
-Pc
_k
when I is the unit dyadic and
(
c
V_o) I +--
1
_c
_k
-oV 1(+
G
KT
m(’- k
A
(to- k
(3.3/
to
is
yy)
Gyy
.+(_
v’+ iZao
k
o
with
-(k
v
v
--K
)2
#,
+ 2i(l-()k
(to- k
2
v
(3.6)
0
--0
is the ratio of the two specific heats.
and
Using equations (3.2) and taking the Maxwell equations (2.3ab)
(2.4ab),
it turns out that
with
D
--
=(k to2)
2
i
I
c
c
where
to
2
o
(3,7)
toct2
v
(+
k
0-
to
c
4epa
(3.9)
m
In the case of wave propagation along with x-direction (k
k),
we obtain
330
NONLINEAR INTERACTION OF WAVES
the dispersion relation
D(k, m)
Det
(3.10)
o,
There are two independent solutions of equation (3.10) in the form
D
(3.11)
0
ZZ
which corresponds to the ordinary wave, and
D
xx
D
yy
-D
xy
D
yx
(3.12)
=0
which corresponds to the extra-ordinary wave, where
Dxx, Dyy,
D.
are the elements of the matrix
The first order fields for the different modes (ordinary and extra-ordinary)
are obtained as
g
--O
E
with
a
a
a
A
0
0
r-
t)]
(3.13)
exp[i(k
r- m t)]
(3.14)
0
Ae
e
e.xp[i(k
--e
0
e
I
a
o
e
(i b x +
E
__x__
and
b=-ig
y
i D
)/(1 +
b2) 2
yy_
(3.15ab)
(3.16)
D
yx
where
2
YY
Dyx
4.
c
7
iz kv
o oy
(3.17)
c
o
z
(3.8)
THE SECOND ORDER EQUATIONS.
In the second order approximation, we obtain from equations (2.1)
(2.2)
T.P. KHAN, M. DAS AND L. DEBNATH
8v
2
---+--oV
Vv
2
+ vI
Vv
I + v2
Ze
VVo
--E--2m
-v
Z
v
v2-
2
-o
Z
--o
x
ZR
v2
x
1
Vl
(4.1)
+ vp)
m
0/ 2
with
(4.2)
2
o
V_l + (k-
(k
PO
)
i
_v2
(to- k
/
(4.3)
v
--0
Using the space and time dependence of all second order quantities in the
form
exp[i(_k
Z_e
v2
with
Q
mt)]
r
c
m
H =-v
I
KTm
I
pl(-k
we find
(i
k
Vv
I
I
v
-o
o
V_ l)((m-k’--i
y_kpo (Y_oV)-
) I+-k
v
-o
E_2
m
+
c
H
(4.4)
i Q
Vl
(4.5)
2
2)i
+
{yk- i(y- 2) i}
Po
+iyp
I
2
]
(4.6)
Using the Maxwell equations and eliminating the second order magnetic field
follows that
v
x
(v
xE)
whence
J2
=-
@
Po
+
I
c
--v2
2E2
4
3t
c
2
P2
B2,
it
(4.)
@t
o + Pl --Vl
(4.8)
Substituting the values of v in equation (4.4), we obtain
2
V x (V x
4im
2
c
E2)
+
1
c
o
32E2 +
@t
I +
2
"’----.L
-/
c
m
p
Po +
c
H +
-
v
k
so--ok
e
o J
1 Pal
o
3E
2
(4.9)
332
NONLINEAR INTERACTION OF WAVES
EQUATION FOR THE INTERACTING MODES.
5.
We assume a plane wave solution with an amplitude varying in the direction
of propagation and write
_E2
where A
A (x,t)
a
exp[i(k
(5.1)
mt)]
r
is a slowly varying function of x and t because the nonlinearity is
assumed to be weak.
k2A and m2A,
We next drop the second derivatives of A in comparison to
+
(Ul x
(u2
(u
3
-
)AI A3 A2 V123 (mllm21m3)
+
)A 3
where Ak
A
I
A
V321 (-m31m21-ml)
2
(k_2 + kl)
k
3
We write the matrix elements for a
and a
VI2 3
e
2Ti
t A
exp{i(-x A k
+
(5.2)
m)}
m)},
t A
(5.3)
exp{i(-x A k + t A m)}
(5.4)
m3- (ml + m2)
Am
and
0 to obtain the final result
exp{i(x A k
AI A3 V213 (-m21-mllm3)
x +)A2
E
2
D
then use the linear dispersion relation
I
a
and a
a
3
(5.5ab)
ae
2
where
are defined in (3.15ab).
(" PO {-i(al *
3
c
2
a3)(3
(
Ze
a2)
m(
0
3
2
* 3
(a I
12 I
and
Ze m
ms(m
2
3a
(k 2
a3)(k3
c
i6)
Vox)(m 2
k3
+
Z
o
+
k2. Vox)A2
(m 2
2
2) .+
(al "(’a3) (k--2-i) "c
(2 k2 o
11I k2Vx)m (2
-k2 V_ox)
m2
a
+
k2Vox)
2
k
2
V
or)
1
a2
b2
(5.6)
T.P. KHAN, M. DAS AND L. DEBNATH
V321
is obtained by interchanging m
V231
2i
[
PZe (a2 , + a 2
m
2
=__.i
We consider
,
V
-o
m
and
I
k
2
A2 mZe i+b22
,
k, Ak real and A
3
V123
and
c 2{k3(
33 e
2
(--2_k2V_ox) b2+Z flo)
-(
in
m3
333
k3
3(i --klVox)
"al)
+
+
kl
kl
(5.4).
(5.7)
)l(3-k3Vox
is the pump of fixed amplitude.
amplitudes are described by equations (5.2)
3
l(al’e "a3)}]
We take
A’s
The small
to be space
independent so that
t
A
A dt}]
[ep{i
1
(5.8)
o
and then we find
[22 +
t
where
-
9
2
I[AI
I
exp{i
(V123
e
Neglecting i A m
t
assumes the form
t + (A4m)2]
i A m
V
2 13
A
e
2@
2
(5.10)
3
it follows that the solution of
g(t)dt} + B exp{-i
2
o
(5.9)
(59) by W.K B
method
g(t)dt}]
(5.11)
o
with
i
g
IV123
V213
A32
(A
24i
(5.12)
When the wave packet drifts along x the time-increment dt can be written
as dt
(dx)
)
to obtain
AI
xt
g(x) dx]
[exp
)
(5.13)
o
where the integration is limited to the instability zone.
It is interesting to
note that the threshold value of the wave can be calculated when g < 0
334
NONLINEAR INTERACTION OF WAVES
Am
A3=
(5.14)
2/V123V213
Thus the value of the frequency shift can be obtained from (5.14) as
i
Am
2A
3
IV123V213
-{(m 2-
mA2
k2Vox)b2-
+
m2 (m2-k2Vox) b 2-o
2
k3
(m3-k3Vox) (m2-k2Vox)
m
2
where the assumption
o}
3
(ml-klVox)
>> m
m /
e
kl
+
2
m
I (m3-k3Vox)
I]
(5.15)
is invoked.
j
/
/m
With the following numerical values
m
k
w
2
k
3
of Montgomery
Am
1015
sec
=5 6x
i011
sec
k
2
a3-
al
it turns out that
e
2.7 x
108
-i
-i
-i
19 cm
I
106
3
sec
e.s.
-i
U/cm
which is in good agreement with that
13]
Similarly, considering the space dependence only, one can calculate the
wave number shift from equations (5.2)
6
(5.4).
DISCUSSION.
The general features of density gradient and magnetic field in the nonlinear
interactions of plasma oscillations have been investigated.
This is important
in connection with its use as an optical density probe suggested by Kroll, Ron and
335
T.P. KHAN, M. DAS AND L. DEBNATH
Rostoker [13] or as a controlled source of plasma oscillations conceived by
Montgomery [13] the frequency shift will deviate with the increased value of
the density gradient.
However, strong magnetic field will have only influence
on the frequency shift.
ACKNOWLEDGMENT.
This work was partially supported by a summer grant from the
East Carolina University Research Committee.
Authors express their sincere
thanks to Mrs. Lela Skinner for typing the final manuscript.
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Liu, C. S. and Rosenbluth, M. N. Parametric decay of electromagnetic
Waves into two plasmons and its consequences, Phys. Fluids. 19
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2.
Liu, C. S., Rosenbluth, M. N., and White, R. B. Raman and Brillouin
scattering of electromagnetic waves in inhomogeneous plasmas,
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3.
Nicholson, D. R. Parametric instabilities in plasma with sinusoidal density
modulation, Phys. Fluid___s 1__9 (1976) 889-895.
4.
Thomson, J.J.
Finite-Band width Effects on the Parametric Instability on
the Parametric Instability in an inhomogeneous Plasma, Nuclear Fusion
15 (1975) 237-247.
General Formulation,
5.
Nishikawa, K. Parametric Excitation of Coupled Waves, I.
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6.
Gorbunov, L. M., Domrin, V. I., and Ramazashvili.
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Ikemura, T. and Nishikawa, K. Parametric Coupling of Langmuir Waves in an
inhomogeneous plasma, J. Phys. Soc. Japan_, 3__2 (1972)1368-1376.
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created by an intense pump field, J. Phys. Soc. Japan 41 (1976) 281-291.
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Galeev, A. A. and Sagdeev, R. Z.
13 (1973) 603-621.
Stimulated Raman scattering
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plasma, Sov. Phys. JETP 43 (1976) 1128-1138.
Parametric Phenomena in a plasma, Nucl. Fusion
NONLINEAR INTERACTION OF WAVES
336
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Perkins, F. W. and Flick, J. Parametric instabilities in inhomogeneous plasmas,
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Eubank, H. P.
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Kroll, N. M. Ron, A. and Rostoker, N. Optical Mixing as a plasma density
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Montgomery, D. On the resonant excitation of plasma oscillations with
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Threshold electric field for excitation of parametric
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