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Propagation of solitary ion acoustic waves in inhomogeneous plasmas
Volume 50A, number 6
PHYSICS LETTERS
13 January 1975
PROPAGATION OF SOLITARY ION ACOUSTIC WAVES IN I N H O M O G E N E O U S P L A S M A S
K. NISHIKAWA* and P.K. KAW** Physics Dept., University of California, Los Angeles, California, USA Received 12 November 1974 A WKB solution for the propagation of a solitary ion acoustic wave in a plasma with a density gradient is obtained.
We first derive the equation describing propagation of nonlinear ion acoustic waves in an inhomogeneous plasma. Let n e = N(z) [1 + ~e (z, t)] and ni = N(z) [1 + h"i (z, t)] be the electron and ion density respectively, where N(z) is the average density and he, i denote the fraction of perturbed density due to the ion wave. Assuming the perturbation to be small, we write n e as an expanded form of the Boltzmann distribution
where we have used eq. (1). The above equations show that the lowest order = O = ni
Re = exp [¢(z, t)] - 1 ~ ¢ + ¢2/2
(1)
where ~b(z,t) is the potential normalized by T/e, T being the electron temperature and - e the electron charge. Ions are assumed to be cold. Writing the ion continuity equation and equation of motion and going over to a frame moving with the ion-acoustic speed = z - t (normalize lengths to Debye length at z = 0, time to ion plasma period at z = 0 and velocities to ion-acoustic speed) one obtains [•--
O+l V2] =0
(2)
(5)
and to the next order av r
viz.
a'-t+
N(z) az 2
.
a
a
N(0)
o +
Iv -
+
o] = 0,
(3)
where g = N -1 (dN/dz) is a measure of the density gradient. The term go denotes the additional ion density fluctuation because of motion of ions along the density gradient. To complete the set of equations, we write Poisson's equation * Permanent address: Faculty of Science, Hiroshima University, Hiroshima, Japan. ** Present address: Physical Research Laboratory, Ahmedabad 380009, India.
a2 v
0.
(6)
Eq. (6) is the modified K-dV equation for an inhomogeneous plasma. Two corrections are included: one is the additional term, gv/2 and the other the modification of dispersion term by a factor N(O)[N(~ + t). This latter modification arises because of density dependence of Debye length. We consider a soliton-like solution of which the spatial width is very small as compared with the density gradient scale length r -1 . We then neglect x~ compared to unity and write
=f(0
N(~ + t) 0-7 +
N(0)
-~+~v*v ~ v + ' ~ N(~+t) aGE 2 =
(7)
where f(t) is a known function of time. Introducing the change of variables
v = be(O]v27,
(8)
and the transformation t r = f [f(t') l adt',
~"= be(t)] t~
(9)
and neglecting terms of order [3r~(a/b~) because of the smallness of K~, one obtains av+,aA+#~ i)~
o
-1
+f
i~3 ~ +3 ½a_ o=o.
(IO) 455
Volume 50A, number 6
PHYSICS LETTERS
This reduces to the usual K-dV form for t~=-/~= 1/4.
(11)
We may thus finally write the WKB solution for the solitary wave as [ 1] u = 12AJ ~A sech 2 { x / ~ - If - ¼ t
-4A
f [f(t')] ~ dt'] ).
(12)
where f ( t ) is defined by eq. (7). As the density increases, f decreases, so do the amplitude, and the propagation velocity of the solitary wave: on the other hand the width normalized to the local Debye length increases. This result can be readily understood physically. As the soliton propagates into higher density region, the dispersive term decreases in magnitude (because of modification of Debye length). Thus a smaller amplitude is required to balance the nonlinear term with the dispersive tenn. Thus as long as a solitary wave solution is maintained, its amplitude will keep decreasing as it goes into higher density regions.
456
13 January 1975
The modifications of the propagation velocity and the width are a direct consequence (in a WKB sense) of the decrease in amplitude. A posteriori we also note that our neglect of Ir~'l with respect to (d~'/dr) is justified if r ,~A 3/2. Finally, it may be mentioned that the WKB method described above may also be used for solving other nonlinear equations for inhomogeneous dispersive media derived by Asano and One [2]. We are grateful to Dr. Y.C. Lee for some valuable comments. This work is partially supported by Atomic Energy Commission contract no. A T ( 0 4 - 3 ) - 3 4 PeA. 157.
References
[ 1] R. Davidson, Methods in nonlinear plasma theory (Academic Press, New York and London, 1972) Chap. 2. [2] N. Asano and H. One, J. Phys. Soc. Japan 31 (1971) 1830.
PHYSICS LETTERS
13 January 1975
PROPAGATION OF SOLITARY ION ACOUSTIC WAVES IN I N H O M O G E N E O U S P L A S M A S
K. NISHIKAWA* and P.K. KAW** Physics Dept., University of California, Los Angeles, California, USA Received 12 November 1974 A WKB solution for the propagation of a solitary ion acoustic wave in a plasma with a density gradient is obtained.
We first derive the equation describing propagation of nonlinear ion acoustic waves in an inhomogeneous plasma. Let n e = N(z) [1 + ~e (z, t)] and ni = N(z) [1 + h"i (z, t)] be the electron and ion density respectively, where N(z) is the average density and he, i denote the fraction of perturbed density due to the ion wave. Assuming the perturbation to be small, we write n e as an expanded form of the Boltzmann distribution
where we have used eq. (1). The above equations show that the lowest order = O = ni
Re = exp [¢(z, t)] - 1 ~ ¢ + ¢2/2
(1)
where ~b(z,t) is the potential normalized by T/e, T being the electron temperature and - e the electron charge. Ions are assumed to be cold. Writing the ion continuity equation and equation of motion and going over to a frame moving with the ion-acoustic speed = z - t (normalize lengths to Debye length at z = 0, time to ion plasma period at z = 0 and velocities to ion-acoustic speed) one obtains [•--
O+l V2] =0
(2)
(5)
and to the next order av r
viz.
a'-t+
N(z) az 2
.
a
a
N(0)
o +
Iv -
+
o] = 0,
(3)
where g = N -1 (dN/dz) is a measure of the density gradient. The term go denotes the additional ion density fluctuation because of motion of ions along the density gradient. To complete the set of equations, we write Poisson's equation * Permanent address: Faculty of Science, Hiroshima University, Hiroshima, Japan. ** Present address: Physical Research Laboratory, Ahmedabad 380009, India.
a2 v
0.
(6)
Eq. (6) is the modified K-dV equation for an inhomogeneous plasma. Two corrections are included: one is the additional term, gv/2 and the other the modification of dispersion term by a factor N(O)[N(~ + t). This latter modification arises because of density dependence of Debye length. We consider a soliton-like solution of which the spatial width is very small as compared with the density gradient scale length r -1 . We then neglect x~ compared to unity and write
=f(0
N(~ + t) 0-7 +
N(0)
-~+~v*v ~ v + ' ~ N(~+t) aGE 2 =
(7)
where f(t) is a known function of time. Introducing the change of variables
v = be(O]v27,
(8)
and the transformation t r = f [f(t') l adt',
~"= be(t)] t~
(9)
and neglecting terms of order [3r~(a/b~) because of the smallness of K~, one obtains av+,aA+#~ i)~
o
-1
+f
i~3 ~ +3 ½a_ o=o.
(IO) 455
Volume 50A, number 6
PHYSICS LETTERS
This reduces to the usual K-dV form for t~=-/~= 1/4.
(11)
We may thus finally write the WKB solution for the solitary wave as [ 1] u = 12AJ ~A sech 2 { x / ~ - If - ¼ t
-4A
f [f(t')] ~ dt'] ).
(12)
where f ( t ) is defined by eq. (7). As the density increases, f decreases, so do the amplitude, and the propagation velocity of the solitary wave: on the other hand the width normalized to the local Debye length increases. This result can be readily understood physically. As the soliton propagates into higher density region, the dispersive term decreases in magnitude (because of modification of Debye length). Thus a smaller amplitude is required to balance the nonlinear term with the dispersive tenn. Thus as long as a solitary wave solution is maintained, its amplitude will keep decreasing as it goes into higher density regions.
456
13 January 1975
The modifications of the propagation velocity and the width are a direct consequence (in a WKB sense) of the decrease in amplitude. A posteriori we also note that our neglect of Ir~'l with respect to (d~'/dr) is justified if r ,~A 3/2. Finally, it may be mentioned that the WKB method described above may also be used for solving other nonlinear equations for inhomogeneous dispersive media derived by Asano and One [2]. We are grateful to Dr. Y.C. Lee for some valuable comments. This work is partially supported by Atomic Energy Commission contract no. A T ( 0 4 - 3 ) - 3 4 PeA. 157.
References
[ 1] R. Davidson, Methods in nonlinear plasma theory (Academic Press, New York and London, 1972) Chap. 2. [2] N. Asano and H. One, J. Phys. Soc. Japan 31 (1971) 1830.