- Email: [email protected]
The interaction of large-amplitude plasma waves with a moving electron plasma
Volume 73A, number 3
PHYSICS LETTERS
17 September 1979
THE INTERACTION OF LARGE-AMPLITUDE PLASMA WAVES WITH A MOVING ELECTRON PLASMA Abraham C.-L CHIAN Instituto de Pesquisas Espaciais, INPE, Conseiho Nacional de Desenvoh’i,nento Cientifico e Tecnolôgico, CNPq, 122OO-Sa~oJosé dos Campos, SP, Brazil Received 25 June 1979
A fluid treatment of the interaction of large-amplitude plasma waves with a moving electron plasma is presented. An exact dispersion relation is obtained. In addition, a condition for the occurrence of wavebreaking for the subluminous waves is determined.
Recently, there has been considerable interest in the interaction of intense electromagnetic waves with plasmas due to applications to laser—plasma interaction and pulsar electrodynamics [1—5]. In this paper we study the interaction of large-amplitude electrostatic waves with a moving electron plasma. An exact (relativistic, nonlinear) dispersion relation is obtained by seeking a travelling wave solution of one-dimensional fluid plasma equations. The case of a stationary plasma has been treated by Akhiezer et al. [4]. In the present paper their work is extended to allow arbitrary plasma stream velocities parallel to the wave propagation. In addition, the problem of wavebreaking for the subluminous waves is analysed. The effect of plasma motion is important in many nonlinear wave problems. Max [3] pointed out that an accurate description of strong waves in a pulsar plasma requires the inclusion of arbitrary plasma stream velocities. Andreev and Silin [6] showed that the dynamic behaviour of large-amplitude waves generated by the resonant absorption process during laser—plasma interaction is very sensitive to plasma motion. The phenomenon of wavebreaking plays an important role in the absorption of nonlinear waves in plasmas. Zakharov et a!. [7] analysed the evolution of wavebreaking of nonlinear plasma waves by computer simulation. They showed that wavebreaking is an effective mechanism by which nonlinear waves dissipate energy to the plasma and plays a crucial role in the 180
kinetics of plasma turbulence. Drake et a!. [8] employed a lagrangian formulation to show that the breaking of a large-amplitude wave due to relativistic electron-mass variation can strongly enhance the anomalous absorption of the wave in the plasma. For a nonlinear travelling wave with phase velocity less than the velocity of light, wavebreaking is expected whenever particles can reach velocities equal to the phase velocity [5]. Although we cannot follow the evolution of the wave after it breaks, we demonstrate in this paper that the travelling wave solution is capable of yielding a condition for the occurrence of wavebreaking. The plasma is taken to be cold, infinite and homogeneous. The average electron charge and current densities are neutralized by the ions, but the ion dynamics is ignored [4]. Since we are considering travelling wave solutions we can write the basic equations in terms of the combined space—time variable 0 = t nz/c [3,4]. For purely longitudinal waves the basic equations are the relativistic equation of motion, the equation of continuity and the equation of Poisson: d( ~ e (1 nv/c) ‘~‘~/ = E, (1) —
—
—
dN n d(Nv) dO c dO n dE —-~- -~-
=
=
4ire (N~~ — N),
(2) (3)
Volume 73A, number 3
PHYSICS LETFERS
where n is the index of refraction (— 00 n ~ 00), ‘y 2 and N = (1 v2/c2)_h/ 0 is the neutralizing back—
ground ion with density. In the laboratory frame the wave propagates a velocity c/n. Note that theSlongitudinal waves can be either superluminous (ml < l) or subluminous (ml > 1). The behaviour of the electron density follows upon integrating eq. (2): N (1_— n Vs/c)N
V~ (NV/(N)
(45)~
~1 —nv/c where —c < V5
‘
dT
2/m. A fIrst where = yv/c, r =can w1,O 4irN0e integralu of eq. (6) be and obtained= by multiplying eq. (6) by d(u n7)/dr and integrating
17 September 1979
eq. (9) recovers the linear dispersion relation [9]
2/w~. (11) P 21r(l nI3~)r’~’ We now turn our attention to the problem of wavebreaking which may occur for subluminous waves. This problem can be greatly clarified by doing the analysis in the stationary wave frame 5~which moves with the wave velocity c/n with respect to the laboratory frame S. In frame S 1 (with the corresponding ties denoted by a subscript 1) there is no time quantidependence, thus the basic equations (l)—(3)become —
v1 (dldz1)(71 v1) = (e/m) E1 N1v1 =constantN0f1(V5—c/n),
(12) (13)
dE1/dz1
(14)
—
=
4ire(N01
,
—
N1),
where r1 = (1 — n _2)_h/2. A first integral corresponding to eq. (7) is obtained by combining eqs. (12)—(14) and ~(dyintegrating 2 =W 1/dr1) 1 —f1 (15) where = ~I (y~ 1)~”~ + (1 ,
—
—
—
—
1 (du)2 2
dr
W =
(1
—
—
Ti =
f
nv/c)2’
f
7
—
~
(7,8)
where W is a constant of integration that characterizes the wave amplitude and 13~= V5/c. Analysis shows that f has a minimum when u = [‘~13S (where r5(l — 13~)1/2), for which the value off is 1/1~.Evidently eq. (7) shows that a periodic wave solution exists for W> I /1’s. The dispersion relation can be obtained from a further u~ 1—nv/c integration du,of eq. (7): (9) —
r~1~w~z
1/c. (16) A typical plot of f1 as a function of y~ is displayed in fig. 1. It can be seen from fig. 1 that the maximum
0.7
_
-
(w- f)h12
where P is the period of oscillation, u1 and u2 are determined by settingf= W:
-
(10) Eq. (9) recovers the result of a stationary plasma [4] if V~is set to zero, and can be obtained from the dispersion relation for a stationary plasma by a Lorentz transformation [21.For W slightly greater than I /I’5
I
I
0.2
I
0 Fig. 1.
1.0
—
-
—
I
1mm
I
I
I
‘~‘r5r1!1-,~35/n ~max
Variation of fi
with ‘yj
~
for V5/c = 0.5 and n
1.1.
181
Volume 73A, number 3
PHYSICS LETTERS
value of W1 for a periodic solution to exist isf1
thus
= 1): W1 max = 1 II~/n. (17) For W 1 exceeding Wimax wavebreaking takes place. The turning points g1 and g2 are given by the equation f1 =W1: —
g1,2 = r’~Ij[(l
—
13
17 September 1979
5/n) W1
2
=
(eEmax) mcw~ 2
2(W— 1/F5).
Substitution of eq. (21) into eq. (8) gives WcF1 (1 —P5/n).
(24)
Combining eqs. (23) and (24) then yields R~=2[I’1(l —13~/n)—i/r’~] .
i~(1/n
—
135)(F~F~W~I)V2/(F5F1)] —
(18)
.
Combining eqs.exists (17) and riodic solution only(18) if 7ithen lies shows in the that rangea pe1
2] (19) 5/n) A condition for the occurrence of wavebreaking can be obtained by transforming eq. (19) back to the laboratory frame 5, which gives —
~3~)2+
(1
13
—
.
(23)
R
(25)
Therefore, oc-the 2 ~for subluminous An analysis waves of eq.wavebreaking (25) shows that curs if R subluminous waves can attain large amplitudes only if the wave phase velocity is close to the velocity of light; as the phase velocity approaches the plasma stream velocity the wave amplitude allowed decreases, and becomes zero if the phase velocity equals the plasma stream velocity.
v~
if V5>c/n,
c/n
[11 A.C.-L. Chian and (1975) 505.
2n135
+
vc/c =
— ______________
(21)
—
2f3~— n (l
+ 132)
‘
max
182
—
=+
w c ~‘
P.C. Clemmow, J. Plasma Phys. 14
12] Chian, (1979), to be published. [3] A.C.-L. C.E. Max, Phys.Plasma FluidsPhys. 16 (1973) 1277. [41 A.I. Akhiezer, l.A. Akhiezer, R.V. Polovin, A.G. Sitenko and K.N. Stepanov, Plasma electrodynamics, Vol. 2 (Per-
Hence for subluminous waves the electron velocity must lie in the range (20) for a solution to exist, otherwise wavebreaking takes place. Using eq. (21) the condition for wavebreaking can be expressed in terms of a dimensionless parameter R that characterizes the wave amplitude. The maximum of E is determined by combining eqs. (1) and (7) and noting that the minimum value off is 1/F~ E
References
2, (W— 1/F5)~
(22)
gamon, 1975) Ch. 1. [5] A. Decoster, Phys. Rep. 47 (1978) 285. [61 N.E. Andreev and V.P. Silin, Soy. J. Plasma Phys. 4(1978)
508. [7] V.E. Zakharov, A.F. Mastrynkov and V.S. Synakh, Soy. J. Plasma Phys. 1(1975) 339.
[81 J.F. Drake, Y.C. Lee, K. Nishikawa and N.L. Tsintsadze, Phys. Rev. Lett. 36 (1976) 196. [91 P.C. Clemmow and J.P. Dougherty, Electrodynamics of particles and plasmas (Addison-Wesley, 1969) Ch. 6.
PHYSICS LETTERS
17 September 1979
THE INTERACTION OF LARGE-AMPLITUDE PLASMA WAVES WITH A MOVING ELECTRON PLASMA Abraham C.-L CHIAN Instituto de Pesquisas Espaciais, INPE, Conseiho Nacional de Desenvoh’i,nento Cientifico e Tecnolôgico, CNPq, 122OO-Sa~oJosé dos Campos, SP, Brazil Received 25 June 1979
A fluid treatment of the interaction of large-amplitude plasma waves with a moving electron plasma is presented. An exact dispersion relation is obtained. In addition, a condition for the occurrence of wavebreaking for the subluminous waves is determined.
Recently, there has been considerable interest in the interaction of intense electromagnetic waves with plasmas due to applications to laser—plasma interaction and pulsar electrodynamics [1—5]. In this paper we study the interaction of large-amplitude electrostatic waves with a moving electron plasma. An exact (relativistic, nonlinear) dispersion relation is obtained by seeking a travelling wave solution of one-dimensional fluid plasma equations. The case of a stationary plasma has been treated by Akhiezer et al. [4]. In the present paper their work is extended to allow arbitrary plasma stream velocities parallel to the wave propagation. In addition, the problem of wavebreaking for the subluminous waves is analysed. The effect of plasma motion is important in many nonlinear wave problems. Max [3] pointed out that an accurate description of strong waves in a pulsar plasma requires the inclusion of arbitrary plasma stream velocities. Andreev and Silin [6] showed that the dynamic behaviour of large-amplitude waves generated by the resonant absorption process during laser—plasma interaction is very sensitive to plasma motion. The phenomenon of wavebreaking plays an important role in the absorption of nonlinear waves in plasmas. Zakharov et a!. [7] analysed the evolution of wavebreaking of nonlinear plasma waves by computer simulation. They showed that wavebreaking is an effective mechanism by which nonlinear waves dissipate energy to the plasma and plays a crucial role in the 180
kinetics of plasma turbulence. Drake et a!. [8] employed a lagrangian formulation to show that the breaking of a large-amplitude wave due to relativistic electron-mass variation can strongly enhance the anomalous absorption of the wave in the plasma. For a nonlinear travelling wave with phase velocity less than the velocity of light, wavebreaking is expected whenever particles can reach velocities equal to the phase velocity [5]. Although we cannot follow the evolution of the wave after it breaks, we demonstrate in this paper that the travelling wave solution is capable of yielding a condition for the occurrence of wavebreaking. The plasma is taken to be cold, infinite and homogeneous. The average electron charge and current densities are neutralized by the ions, but the ion dynamics is ignored [4]. Since we are considering travelling wave solutions we can write the basic equations in terms of the combined space—time variable 0 = t nz/c [3,4]. For purely longitudinal waves the basic equations are the relativistic equation of motion, the equation of continuity and the equation of Poisson: d( ~ e (1 nv/c) ‘~‘~/ = E, (1) —
—
—
dN n d(Nv) dO c dO n dE —-~- -~-
=
=
4ire (N~~ — N),
(2) (3)
Volume 73A, number 3
PHYSICS LETFERS
where n is the index of refraction (— 00 n ~ 00), ‘y 2 and N = (1 v2/c2)_h/ 0 is the neutralizing back—
ground ion with density. In the laboratory frame the wave propagates a velocity c/n. Note that theSlongitudinal waves can be either superluminous (ml < l) or subluminous (ml > 1). The behaviour of the electron density follows upon integrating eq. (2): N (1_— n Vs/c)N
V~ (NV/(N)
(45)~
~1 —nv/c where —c < V5
‘
dT
2/m. A fIrst where = yv/c, r =can w1,O 4irN0e integralu of eq. (6) be and obtained= by multiplying eq. (6) by d(u n7)/dr and integrating
17 September 1979
eq. (9) recovers the linear dispersion relation [9]
2/w~. (11) P 21r(l nI3~)r’~’ We now turn our attention to the problem of wavebreaking which may occur for subluminous waves. This problem can be greatly clarified by doing the analysis in the stationary wave frame 5~which moves with the wave velocity c/n with respect to the laboratory frame S. In frame S 1 (with the corresponding ties denoted by a subscript 1) there is no time quantidependence, thus the basic equations (l)—(3)become —
v1 (dldz1)(71 v1) = (e/m) E1 N1v1 =constantN0f1(V5—c/n),
(12) (13)
dE1/dz1
(14)
—
=
4ire(N01
,
—
N1),
where r1 = (1 — n _2)_h/2. A first integral corresponding to eq. (7) is obtained by combining eqs. (12)—(14) and ~(dyintegrating 2 =W 1/dr1) 1 —f1 (15) where = ~I (y~ 1)~”~ + (1 ,
—
—
—
—
1 (du)2 2
dr
W =
(1
—
—
Ti =
f
nv/c)2’
f
7
—
~
(7,8)
where W is a constant of integration that characterizes the wave amplitude and 13~= V5/c. Analysis shows that f has a minimum when u = [‘~13S (where r5(l — 13~)1/2), for which the value off is 1/1~.Evidently eq. (7) shows that a periodic wave solution exists for W> I /1’s. The dispersion relation can be obtained from a further u~ 1—nv/c integration du,of eq. (7): (9) —
r~1~w~z
1/c. (16) A typical plot of f1 as a function of y~ is displayed in fig. 1. It can be seen from fig. 1 that the maximum
0.7
_
-
(w- f)h12
where P is the period of oscillation, u1 and u2 are determined by settingf= W:
-
(10) Eq. (9) recovers the result of a stationary plasma [4] if V~is set to zero, and can be obtained from the dispersion relation for a stationary plasma by a Lorentz transformation [21.For W slightly greater than I /I’5
I
I
0.2
I
0 Fig. 1.
1.0
—
-
—
I
1mm
I
I
I
‘~‘r5r1!1-,~35/n ~max
Variation of fi
with ‘yj
~
for V5/c = 0.5 and n
1.1.
181
Volume 73A, number 3
PHYSICS LETTERS
value of W1 for a periodic solution to exist isf1
thus
= 1): W1 max = 1 II~/n. (17) For W 1 exceeding Wimax wavebreaking takes place. The turning points g1 and g2 are given by the equation f1 =W1: —
g1,2 = r’~Ij[(l
—
13
17 September 1979
5/n) W1
2
=
(eEmax) mcw~ 2
2(W— 1/F5).
Substitution of eq. (21) into eq. (8) gives WcF1 (1 —P5/n).
(24)
Combining eqs. (23) and (24) then yields R~=2[I’1(l —13~/n)—i/r’~] .
i~(1/n
—
135)(F~F~W~I)V2/(F5F1)] —
(18)
.
Combining eqs.exists (17) and riodic solution only(18) if 7ithen lies shows in the that rangea pe1
2] (19) 5/n) A condition for the occurrence of wavebreaking can be obtained by transforming eq. (19) back to the laboratory frame 5, which gives —
~3~)2+
(1
13
—
.
(23)
R
(25)
Therefore, oc-the 2 ~for subluminous An analysis waves of eq.wavebreaking (25) shows that curs if R subluminous waves can attain large amplitudes only if the wave phase velocity is close to the velocity of light; as the phase velocity approaches the plasma stream velocity the wave amplitude allowed decreases, and becomes zero if the phase velocity equals the plasma stream velocity.
v~
if V5>c/n,
c/n
[11 A.C.-L. Chian and (1975) 505.
2n135
+
vc/c =
— ______________
(21)
—
2f3~— n (l
+ 132)
‘
max
182
—
=+
w c ~‘
P.C. Clemmow, J. Plasma Phys. 14
12] Chian, (1979), to be published. [3] A.C.-L. C.E. Max, Phys.Plasma FluidsPhys. 16 (1973) 1277. [41 A.I. Akhiezer, l.A. Akhiezer, R.V. Polovin, A.G. Sitenko and K.N. Stepanov, Plasma electrodynamics, Vol. 2 (Per-
Hence for subluminous waves the electron velocity must lie in the range (20) for a solution to exist, otherwise wavebreaking takes place. Using eq. (21) the condition for wavebreaking can be expressed in terms of a dimensionless parameter R that characterizes the wave amplitude. The maximum of E is determined by combining eqs. (1) and (7) and noting that the minimum value off is 1/F~ E
References
2, (W— 1/F5)~
(22)
gamon, 1975) Ch. 1. [5] A. Decoster, Phys. Rep. 47 (1978) 285. [61 N.E. Andreev and V.P. Silin, Soy. J. Plasma Phys. 4(1978)
508. [7] V.E. Zakharov, A.F. Mastrynkov and V.S. Synakh, Soy. J. Plasma Phys. 1(1975) 339.
[81 J.F. Drake, Y.C. Lee, K. Nishikawa and N.L. Tsintsadze, Phys. Rev. Lett. 36 (1976) 196. [91 P.C. Clemmow and J.P. Dougherty, Electrodynamics of particles and plasmas (Addison-Wesley, 1969) Ch. 6.