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Ponderomotive forces for a Vlasov plasma
Volume 84A, number 2
PHYSICS LETTERS
13 July 1981
PONDEROMOTIVE FORCES FOR A VLASOV PLASMA Hachiro AKAMA and Mitsuhiro NAMBU College of General Education, Kyushu University, Ropponmatsu, Fukuoka 810, Japan Received 17 February 1981 Revised manuscript received 6 May 1981
Forces exerted by high-frequency electrostatic waves on a plasma are derived directly from the Vlasov equation. They consist of forces due to dissipation of waves, forces ascribed to the change in wave momentum, and finally, ponderomotive forces.
During the past years, there has been a profusion of theoretical results on forces exerted by waves on a
ization charge density (2)
plasma medium [1—9],and one of the most important phenomena is now recognized to be ponderomo-
p(k,w)P(k,w)~,
tive forces. However, some works are indifferent on the derivation of the ponderomotive forces [1,2], and
wave, and the expression forP(k, in 2 (~if
others study the detailed mechanism in pinderomotive
P(k, w)
phenomena based on the expressions previously obtamed through static methods [3—6].To the authors’ knowledge, an approach through the Vlasov equation has been made only for one dimension without dissipative forces [7]. The purpose of the present note is to give a general derivation of ponderomotive forces for a Vlasov plasma, valid for three dimensions. The straightforward -
.
method we use on the Vlasov equation gives a ponderomotive force, on the one hand, and forces due to dissipation of waves and to the change in wave momentum on the other. We thus come to the conclusion that these three different kinds of forces have to be considered on an equal footing.
The response of a plasma to the high-frequency
where c~= (k c~)/k is the scalar amplitude of the .
—--4-
w) is given by
0/au) k
f dv
.
. (3) viIn terms of the dielectric response function K(k, w),
=
—
k u
+
we obtain —
P(k, w) — —ie0k [K(k, w) 1] (4) The a-component of the force exerted by a wave —
.
on a unit volume of a plasma is given by *
f
g[p (k w)d
+c.c.]
.
The first term on the right-hand side of this equation becomes, by (2), *
—
*
*
p (k, w)~0 — P (k, w)t~ (6) The function P defined in (2) may be looked upon
as the representation of a polarization operator, P,
electrostatic disturbance
with respect to a plane wave, so that
E
Pei(k~x_wt)=P(k,w)ei(kX_o~~t).
~ exp[i(k -x—wt)]
~f0
f1
(1)
We integrate f1 with respect to v and obtain the polar68
(7)
Similarly, P*(k, w) in (6) can be considered as repre-
is described by the linearized Vlasov equation
q
(5)
senting an operator,P~,adjoint toP, acting on a plane wave.
The discussion above leads us to the conclusion that, for a plasma described by the Vlasov equation
0 031—9163/81/0000—0000/$ 02.50 © North-Holland Publishing Company
Volume 84A,
number 2
PHYSICS LETTERS
(1), the density of the polarization charge is given by the operator P, in other words, the operator P is a solution to the Vlasov equation (1). The question now is whether we may assign the operator P+ to any Vlasov plasma. We shall see below that P~is a solution to the time reversal of the plasma given by eq. (1). If we associate primed variables with a plasma after time reversal, we find the relations = —t , u’ = u, c~’(t’)= ~(_t’)= ~(t) —
f(x,t’,u’)f1(x,—t’,—u’)f1(x,t,v)
,
(8)
where the unprimed variables are relative to the plasma before time reversal. The Vlasov equation is now transformed into q a.r0 + U + E = r~f1 , (9)
at
—
•
ax
m
au
—
~
13 July 1981 =
—a d(t)/at
(13)
-
We now proceed to the case where the amplitude of high-frequency electrostatic disturbances depends weakly on x and t, (14) In calculatingp*(k, w)~ as in (6), we applyP~to wave (14). The amplitude c~(x,r) is developed into a Fourier series, so that ~+ acts directly on plane waves.
E~(x,t)ei(~.~~.)t).
Wethusfind ~ (x, t) ei(k X — ~t) *(k — iV, w —
=p
ia~)~ (x, t) ei(k
X— wt)
,
(15)
where the operators V and a~on the right-hand side act only on the amplitude c~(x, t), and where the sign of the time derivative is determined according to rule
and
(13). If we expandP* into a Taylor series and neglect
E = ~ ei(~’~ ~x+wt)
higher-order terms, then (5) becomes
where primes are omitted since no ambiguities can occur. The expression for the polarization charge is
fo = ~[~ *P*(k w)t~0+ c.c.]
p(k,w)Q(k,w)d,
*(l/i)(ap*/ak~a~/a + c.c.]
+
~
+ ~
[d*(l/i)(aP*/aw)ac~/at + c.c.]
(10)
.
(16)
where Q(k, w) =
f
~fL~ dv (af0lau) k .
(11)
Eq. (10) implies that the function Q(k, ~) can be thought of as representing an operator Q, with respect to the plane wave exp[i(k x + wt)]. If Q and P~are applied to the wave exp[i(k x — wt)], we find Q(k, —w) andP*(k w) for their representations, respectively. Comparison of eq. (l1)with eq. (3) yields P*(k w) = —Q(k, —w), or
Substituting trostatic approximation (4) into (16), is valid, and assuming curl ~ =that 0, we theobtain elec-
fa = k0(2eoK°)kI ~ +
k0e0
2 +
e0(K’
—
1)
~
a~’a ak~aX~4I~I +~-g~,
~I ~ 2 (17)
where K’ and K” denote the real and imaginary parts of K, respectively; g0 is the a-component of the momentum density associated with the wave and is given by 2 (18) g~k0e0(aK’/aw)~IdI The first term on the right.hand side of(l7) stands for a force exerted on resonant particles when they ab•sorb waves. The last term is the rate of change of the momentum of electrostatic waves which are made up of the ordered motions of the nonresonant particles. Increase in the wave momentum implies a force on the plasma medium. In the case of normal modes of oscillation, the term ag 0/at corresponds to the Landau .
(12) It follows from (12) that, except for the sign, the operator P~belongs to the time reversal of the system represented by eq. (1). If an expression for ~5+involves time derivatives, we have to be careful in choosing their sign. From (8), the rate of change of physical quantities per unit time is given by reversing the sign of the time derivatives in the representation for P+, for example,
69
Volume 84A, number 2
PHYSICS LETTERS
damping and exactly cancels the first term, which is the force due to the dissipation of waves. The second term in (17) gives a ponderomotive force, and the third a correction for a hot plasma. These two terms are associated with momentum flow in space. In the case of one dimension and of small k, where
(17) reduces to the result given in eq. (18) of ref. [7]. When the electrostatic approximation is not valid for the amplitude ~ , (17) is supplemented with the a-component of the term —~e 0(K’— 1)[~ X curl d
70
*
+
c.c.]
-
References [ii A. Bers, in: Plasma physics, ads. C. Dewitt and J. Peyraud (Gordon and Breach, New York, 1975) p. 113. [2] JR. Cary and AN. Kaufman, Phys. Rev. A2l (1980) 1660.
[31H.
the effects of resonant particles are negligible, eq.
(19)
13 July 1981
Hora, Phys. Fluids 12 (1969) 182.
[4] Fl. Washimi and V.!. Karpman, Soy. Phys. JETP 44 (1976) 528. [51 J.A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941) p. 137. [6] L.D. Landau and EM. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1960) p. 64. [7( H. Schamel and G. Schmidt, J. Plasma Phys. 24 (1980) 149. [8] J.A. Stamper and D.A. Tidman, Phys. Fluids 16 (1973) 2024. [9] T. Hatori and H. Washimi, private communication.
PHYSICS LETTERS
13 July 1981
PONDEROMOTIVE FORCES FOR A VLASOV PLASMA Hachiro AKAMA and Mitsuhiro NAMBU College of General Education, Kyushu University, Ropponmatsu, Fukuoka 810, Japan Received 17 February 1981 Revised manuscript received 6 May 1981
Forces exerted by high-frequency electrostatic waves on a plasma are derived directly from the Vlasov equation. They consist of forces due to dissipation of waves, forces ascribed to the change in wave momentum, and finally, ponderomotive forces.
During the past years, there has been a profusion of theoretical results on forces exerted by waves on a
ization charge density (2)
plasma medium [1—9],and one of the most important phenomena is now recognized to be ponderomo-
p(k,w)P(k,w)~,
tive forces. However, some works are indifferent on the derivation of the ponderomotive forces [1,2], and
wave, and the expression forP(k, in 2 (~if
others study the detailed mechanism in pinderomotive
P(k, w)
phenomena based on the expressions previously obtamed through static methods [3—6].To the authors’ knowledge, an approach through the Vlasov equation has been made only for one dimension without dissipative forces [7]. The purpose of the present note is to give a general derivation of ponderomotive forces for a Vlasov plasma, valid for three dimensions. The straightforward -
.
method we use on the Vlasov equation gives a ponderomotive force, on the one hand, and forces due to dissipation of waves and to the change in wave momentum on the other. We thus come to the conclusion that these three different kinds of forces have to be considered on an equal footing.
The response of a plasma to the high-frequency
where c~= (k c~)/k is the scalar amplitude of the .
—--4-
w) is given by
0/au) k
f dv
.
. (3) viIn terms of the dielectric response function K(k, w),
=
—
k u
+
we obtain —
P(k, w) — —ie0k [K(k, w) 1] (4) The a-component of the force exerted by a wave —
.
on a unit volume of a plasma is given by *
f
g[p (k w)d
+c.c.]
.
The first term on the right-hand side of this equation becomes, by (2), *
—
*
*
p (k, w)~0 — P (k, w)t~ (6) The function P defined in (2) may be looked upon
as the representation of a polarization operator, P,
electrostatic disturbance
with respect to a plane wave, so that
E
Pei(k~x_wt)=P(k,w)ei(kX_o~~t).
~ exp[i(k -x—wt)]
~f0
f1
(1)
We integrate f1 with respect to v and obtain the polar68
(7)
Similarly, P*(k, w) in (6) can be considered as repre-
is described by the linearized Vlasov equation
q
(5)
senting an operator,P~,adjoint toP, acting on a plane wave.
The discussion above leads us to the conclusion that, for a plasma described by the Vlasov equation
0 031—9163/81/0000—0000/$ 02.50 © North-Holland Publishing Company
Volume 84A,
number 2
PHYSICS LETTERS
(1), the density of the polarization charge is given by the operator P, in other words, the operator P is a solution to the Vlasov equation (1). The question now is whether we may assign the operator P+ to any Vlasov plasma. We shall see below that P~is a solution to the time reversal of the plasma given by eq. (1). If we associate primed variables with a plasma after time reversal, we find the relations = —t , u’ = u, c~’(t’)= ~(_t’)= ~(t) —
f(x,t’,u’)f1(x,—t’,—u’)f1(x,t,v)
,
(8)
where the unprimed variables are relative to the plasma before time reversal. The Vlasov equation is now transformed into q a.r0 + U + E = r~f1 , (9)
at
—
•
ax
m
au
—
~
13 July 1981 =
—a d(t)/at
(13)
-
We now proceed to the case where the amplitude of high-frequency electrostatic disturbances depends weakly on x and t, (14) In calculatingp*(k, w)~ as in (6), we applyP~to wave (14). The amplitude c~(x,r) is developed into a Fourier series, so that ~+ acts directly on plane waves.
E~(x,t)ei(~.~~.)t).
Wethusfind ~ (x, t) ei(k X — ~t) *(k — iV, w —
=p
ia~)~ (x, t) ei(k
X— wt)
,
(15)
where the operators V and a~on the right-hand side act only on the amplitude c~(x, t), and where the sign of the time derivative is determined according to rule
and
(13). If we expandP* into a Taylor series and neglect
E = ~ ei(~’~ ~x+wt)
higher-order terms, then (5) becomes
where primes are omitted since no ambiguities can occur. The expression for the polarization charge is
fo = ~[~ *P*(k w)t~0+ c.c.]
p(k,w)Q(k,w)d,
*(l/i)(ap*/ak~a~/a + c.c.]
+
~
+ ~
[d*(l/i)(aP*/aw)ac~/at + c.c.]
(10)
.
(16)
where Q(k, w) =
f
~fL~ dv (af0lau) k .
(11)
Eq. (10) implies that the function Q(k, ~) can be thought of as representing an operator Q, with respect to the plane wave exp[i(k x + wt)]. If Q and P~are applied to the wave exp[i(k x — wt)], we find Q(k, —w) andP*(k w) for their representations, respectively. Comparison of eq. (l1)with eq. (3) yields P*(k w) = —Q(k, —w), or
Substituting trostatic approximation (4) into (16), is valid, and assuming curl ~ =that 0, we theobtain elec-
fa = k0(2eoK°)kI ~ +
k0e0
2 +
e0(K’
—
1)
~
a~’a ak~aX~4I~I +~-g~,
~I ~ 2 (17)
where K’ and K” denote the real and imaginary parts of K, respectively; g0 is the a-component of the momentum density associated with the wave and is given by 2 (18) g~k0e0(aK’/aw)~IdI The first term on the right.hand side of(l7) stands for a force exerted on resonant particles when they ab•sorb waves. The last term is the rate of change of the momentum of electrostatic waves which are made up of the ordered motions of the nonresonant particles. Increase in the wave momentum implies a force on the plasma medium. In the case of normal modes of oscillation, the term ag 0/at corresponds to the Landau .
(12) It follows from (12) that, except for the sign, the operator P~belongs to the time reversal of the system represented by eq. (1). If an expression for ~5+involves time derivatives, we have to be careful in choosing their sign. From (8), the rate of change of physical quantities per unit time is given by reversing the sign of the time derivatives in the representation for P+, for example,
69
Volume 84A, number 2
PHYSICS LETTERS
damping and exactly cancels the first term, which is the force due to the dissipation of waves. The second term in (17) gives a ponderomotive force, and the third a correction for a hot plasma. These two terms are associated with momentum flow in space. In the case of one dimension and of small k, where
(17) reduces to the result given in eq. (18) of ref. [7]. When the electrostatic approximation is not valid for the amplitude ~ , (17) is supplemented with the a-component of the term —~e 0(K’— 1)[~ X curl d
70
*
+
c.c.]
-
References [ii A. Bers, in: Plasma physics, ads. C. Dewitt and J. Peyraud (Gordon and Breach, New York, 1975) p. 113. [2] JR. Cary and AN. Kaufman, Phys. Rev. A2l (1980) 1660.
[31H.
the effects of resonant particles are negligible, eq.
(19)
13 July 1981
Hora, Phys. Fluids 12 (1969) 182.
[4] Fl. Washimi and V.!. Karpman, Soy. Phys. JETP 44 (1976) 528. [51 J.A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941) p. 137. [6] L.D. Landau and EM. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1960) p. 64. [7( H. Schamel and G. Schmidt, J. Plasma Phys. 24 (1980) 149. [8] J.A. Stamper and D.A. Tidman, Phys. Fluids 16 (1973) 2024. [9] T. Hatori and H. Washimi, private communication.