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Decay instability of BGK-waves
Volume 44A, number
3
PHYSICS LETTERS
4 June 1973
DECAY INSTABILiTY OF BGK-WAYES V.V. DEMCHENKO*, N.M. EL-SIRAGY and A.M. HUSSEIN Atomic Energy Authority, Cairo, Egypt Received 16 February 1973 Stability of periodical BGK-waves of a small (but finite) amplitude has been investigated. It is shown that BGK-waves are unstable with respect to the “splitting” in a great number of satellite waves. The physical mechanism of the instability and the growth rates of unstable oscillations have been examined.
Exact solutions describing one-dimensional stationary nonlinear electrostatic waves (below to be mentioned as BGK-waves) in a collisionless plasma were obtained in the paper [11. In particular, it was demonstrated that by adding a certain number of particles trapped in the potential through of monochromatic wave of a large amplitude (MWLA), solutions of essentially arbitrary travelling waves can be constructed. However, the authors of paper [1] did not investigate stability of the solutions obtained with respect to various types of perturbations. As far as we know, investigations of the BGK-waves stability are absent at all. At the same time the problem of stability of periodical waves of a large amplitude is of paramount importance. Indeed, as the recent experiments [2—5] showed the “saturation” of various type instabilities, appearing in case of the beam-plasma interaction, takes place due to the trapping of particles in the potential through of monochromatic waves excited in the plasma by instability. In a number of papers [e.g.6, 7] it was demonstrated that the “saturation” mechanism of small-scale high-frequency instabilities whichofappear in a current-driven plasma is the mechanism trapping particles, Basic equations and assumptions. As in ref. [1] we shall consider stationary nonlinear onedimensional electrostatic waves. A nonlinear differential equation describing dependence of the electric field potential on coordinate (x) has the form (see eq.(6) of [1])
*
On leave from the Atomic Energy State Committee,
USSR.
d2~+ ~
(1)
=
d~
~2
where V(~p)= 8ir
f ~f(E~[2(Ee~P)] m~
f
+8ir
dEf(E)[2(E~(P)]
—~
where a sign (±)refers to ions (electrons) respectively; m±is the mass of respective particles, f~are particle distribution functions; E~= m~v2/2±e~pis the particle energy and uis the velocity. From eq. (1) we find in an implicit form dependence of the potential ~oon coordinates X
=
f d~[V(~0)
±
—
V(~)]
1/2
(2)
where V(p0) and x0 are constants (at x = x0, p = Solution of eq. (2) is entirely determined by function V(p). In particular, as it has been noted in [1], one can construct periodic solutions of eq. (2) if the V(~p) 2). Since has a of form of a well (e.g. const pwhen eq. (2) form function V(~p) wasV(p) not=limited was derived one has the right to choose V(p) in the form V(p)
~ ~z ~p2P(p= 1,2,...), p”l
where positive constants j.z ~ 1. Travelling BGK-waves. Solution of eq. (2) for values p> 2 can not be expressed in the form of an analytical function. For this reason, assume p = 2 in an expression for V(~p).In this case solution of eq. (2) has the form 193
Volume 44A, number 3
PHYSICS LETTERS
4 June 1973
(0) =
a÷cn(w;k)
(3)
where a~ \/(u÷--p
exp(-
where q
irK/K’), K’
K(k’). Taking into
1)/2p.2, w ~/~(x x0), ‘~/(o~--ii1)/2u~. cn(w; k) is a Jacobian elliptical
account the presented [9J relation limq(at k -~ 0) = 1/16 0.05 ~ 1 one can conclude that the series (6)
function of variable (w) with period 4K(k) (K(k) is a total elliptic integral of the first kind) solution (3) corresponds to a travelling wave (it will be recalled that in the present paper the consideration is carried
are fast convergent ones. Therefore, in a limiting case
k
out in the system of waves). At first it may seem that solution (3) is a stable
of small k expression (5) isAllowing transformed intoabove 4cos(irw/2K). for the 2~k7kq~ mentioned and a well-known relation 2/ 1 ~ 2(2/) cos[2(/— (2/
[
221 (m0
cos q~---
2P
one and intoappreciably consideration of terms /lP~p (atp >2)taking can not affect the character of the solution obtained. In the nect section of this paper it will be demonstrated that in fact. the situation is different. InstabUity ofBGK-waves. To investigate stability of solution (3) we shall introduce small deviation from the equilibrium state
1 we come to the conclusion that equation “in variations” is transformed into Hill’s generalized equation which at p = 2 is represented as a well-known Mathieu’s equation: + [0~+20~ cos(Kx)1 ~ = 0.
where ~
i~=~
~
~.
Equation for function ~ (equation “in v~iriations”) has the form
~ (2p— l)pp ~2(p-1) ~ =
(4)
Even simple qualitative considerations show the instability of solution (3). In fact, substitution of expression (3) in eq. (4) transforms tl~elatter ii~ Lame’s in elliptical functions [81.with As is known, equation Lame’s equation (as any equatian periodical coefficients) at a certain range of parameter values has unstable solutions. However, to find unstable solutions of eq. (4) one must use rather cumbersome mathematical calculations. The analysis of the problem is considerably simplified if one uses a well-known relation binding Jacobian functions with theta functions: ~/k’t9 2(irw/ 2K) cn(w: k) = ~ (5) 2 Theta functions 19,(irw/2K) are where k’as=~.,/lk defined sums of the following series 02
~K
2
(~)- —
(irw’~
04~K)
=
~
(n— 112)
n=l q
1+2 ~ fl = 1
194
L
cos(2n 2
—
[
1) (imw
(--- l)~q~cos[2n~~)]~ (6)
m)r]+
m
ir~/~i/K(k), Oo
2a~q1/4~/~7~.
(7)
ji
1+3ji2b~,01 3p2b~/2, There are manymethods to
find unstable solutions of Mathieu’s equation. Here we shall make use of Wittaker’s method which was used before for the investigation of the stability of the investigation of the stability of the plasma nonlinear longitudinal oscillations [101. Omitting intermediate calculations we present a final expression for the growth rate of unstable oscillations
(~= exp(~x)sin(l~x—~a).aan unknown parameter) 40o+O1 and at the same time a ~2 (Oo+ l)+ condition necessary for build-up of instability 01 > 0o - I. Consequently, if parameters of the
plasma and of nonlinear periodical wave are in compliance with the written condition, the BGKwave turns out to be unstable. In the next section we shall examine the physical nature of instability and the mechanism of its “saturation”.
Restriction of the amplitude of an unstable Oscillation. Appearing side-bands waves. It not difficult to show that inoffact, the amplitude ofisdisplacement is a limited at any value x. Indeed, writing down the equation “in variations” (7) we confined linear terms over ~. Taking into considerationtononlinear terms over ~ leads to the restriclion of the growth rate of the oscillation amplitude. To the above statement shall write down eq. prove (7) taking into account the we term
~+X2[l÷h
cos(~x)j~+2p
30,
=
2~
(8)
Volume 44A, number 3
where A2
PHYSICS LETTERS
Oo, h
201/60 ~ 1. Solution of eq. (7)
=
‘y”~~(i —kq2Z
2(Z 2/Z1) cos(5Kx+~3)+~kq
in the first approximation, according to the Krilov— Bogaljubov method, has the form (in the case considered ic 2X) =
4 June 1973
X {cos [(~
F~irM/K(k).
(9)
a cos(~Kx+t~),
cos [(~ —f)x+~]}, (15)
Proceeding from relation (15) one can readily understand the physical meaning of instability in the
where functions (a) and (~9)are satisfied by the differential equations 2 ~ Z sin0 (10) dZ = 2A K
paper considered. appearance of satellite waves with a wave Indeed, vectors (~K ±F) besides the basic oscillation with a wave vector ~ follows from relation
,
where C is constant of integration, By excluding variable 6 from relations(10) and (12) one obtains
(15). Therefore, the considered instability has a character of “decay” instability when an original nonlinear wave of a large amplitude “splits” in a great number of satellite waves. Appearance of only one pair of satellite waves follows from relation (15). However, one should keep in mind that when obtaining eq. (8) we confined with 1) value p = 2 in eq. (4) and 2) taking into account only nonlinear term ~ It turns out to be physically evident that in a general case (at a sufficiently large value p) taking into consideration a greater number of nonlinear terms leads to the “splitting” of the original wave in a great
dZ
number of satellite waves.
— =
2~+ 2X2 h
+F)x+~] +
2/.../7~)
— cos6 K
—
6— Z
(11)
,
K
where ~ A— ,‘c/2, Z rua2, 0 2& As one can easily make sure by direct integration the set of eqs.(l0), (11) has the integral 2A2 -~Z co~0= C — 2~.Z+3 -~-~ Z2,
14X4h2Z2 =
L
2
1 /2
(1—cos 0)]
K2
(12)
4
1/2
(Z_Zp)]
= [~
The authors are obliged to Professor Dr. M. ElNadi and Dr. T. El-Khalafawy for encouragement and support. They also wish to thank Mr. V.A. Utenock for his aid in preparation of this manuscript
=
(13)
p=l
for publication.
where ~ are roots of the equation P(Z) = 0. Solution of eq. (13) is presented in [11] and has the form Z = [Z 2(u;k)—QZ 2(u; k)—Q] 1II[sn where 2sn
[1] I.B. Bernstein, J.M. Greene and M.D. Kruskal, Phys. Rev. 108 (1957) 546. [2] J.H.A. Van Wakeren and H.J. Hopman, Phys. Rev. Lett. 18 (1972) 295. [3] E.P. Barbian and B. Jurgens, Proc. IlIrd Int. Conf. on
(14)
Q-plasmas, Elsimore, 1971, p. 317. [4] G.J. Morales and T.M. O’Neil, Phys. Rev. Lett. 28 (1972) 215. [5] R.N. Franklin et a!,, Phys. Rev. Lett. 28 (1972) 1114.
____________________
u
M(x —x 0),
k
~/(Z2—Z3)/Q(z1 —Z3),
M~+[(Z2—Z4XZ3—Z1)]1/2
~
(Z2—Z4)/(Z1_Z4)
sn(u; k) the Jacobian elliptical sine, x0 a constant of integration which can be assumed to be zero. In the peculiar case Z3 Z1 Z2 expression (14) ~“
,
is transformed into
2 Z
—
~1— cos [2(2
K(k))]).
2kZ2q
Taking into account that a
tains expressions for value
=
V’Z, from (9) one ob-
,
[6] D. Biskamp and R. Chodura, IAEI Conf., Madison, 11(1971) 265; Invited paper “Recent Development in Computer Simulations of Plasmas”, Vth Europ. Conf. on Controlled fusion and plasma physics, 11(1972) 93. [7] D. Forslund, R. Morse and C. Nielson, IAEI Conf., Madison, 11(1971) 277. [8] E.T. Whittaker and G.N. Watson, A course of modern analysis (London, 1935). (Dover Pub!., New York, 1945). 9] formulae E. Jahnkeand andcurves, F. Emde, Tables of functions with [10] V.V. Demchenko and L.A. El-Naggar, Phys. Lett. 39A (1972) 205; Cz. J. Phys. 22B (1972) 1194; Proc. Vth Europ. on Controlled fusion and plasma physics,Conf. 1(1972)142. [11] W.F. Ames, Nonlinear ordinary differential equations in transport processes (Academic Press, N.Y., 1968) p. 55. 195
3
PHYSICS LETTERS
4 June 1973
DECAY INSTABILiTY OF BGK-WAYES V.V. DEMCHENKO*, N.M. EL-SIRAGY and A.M. HUSSEIN Atomic Energy Authority, Cairo, Egypt Received 16 February 1973 Stability of periodical BGK-waves of a small (but finite) amplitude has been investigated. It is shown that BGK-waves are unstable with respect to the “splitting” in a great number of satellite waves. The physical mechanism of the instability and the growth rates of unstable oscillations have been examined.
Exact solutions describing one-dimensional stationary nonlinear electrostatic waves (below to be mentioned as BGK-waves) in a collisionless plasma were obtained in the paper [11. In particular, it was demonstrated that by adding a certain number of particles trapped in the potential through of monochromatic wave of a large amplitude (MWLA), solutions of essentially arbitrary travelling waves can be constructed. However, the authors of paper [1] did not investigate stability of the solutions obtained with respect to various types of perturbations. As far as we know, investigations of the BGK-waves stability are absent at all. At the same time the problem of stability of periodical waves of a large amplitude is of paramount importance. Indeed, as the recent experiments [2—5] showed the “saturation” of various type instabilities, appearing in case of the beam-plasma interaction, takes place due to the trapping of particles in the potential through of monochromatic waves excited in the plasma by instability. In a number of papers [e.g.6, 7] it was demonstrated that the “saturation” mechanism of small-scale high-frequency instabilities whichofappear in a current-driven plasma is the mechanism trapping particles, Basic equations and assumptions. As in ref. [1] we shall consider stationary nonlinear onedimensional electrostatic waves. A nonlinear differential equation describing dependence of the electric field potential on coordinate (x) has the form (see eq.(6) of [1])
*
On leave from the Atomic Energy State Committee,
USSR.
d2~+ ~
(1)
=
d~
~2
where V(~p)= 8ir
f ~f(E~[2(Ee~P)] m~
f
+8ir
dEf(E)[2(E~(P)]
—~
where a sign (±)refers to ions (electrons) respectively; m±is the mass of respective particles, f~are particle distribution functions; E~= m~v2/2±e~pis the particle energy and uis the velocity. From eq. (1) we find in an implicit form dependence of the potential ~oon coordinates X
=
f d~[V(~0)
±
—
V(~)]
1/2
(2)
where V(p0) and x0 are constants (at x = x0, p = Solution of eq. (2) is entirely determined by function V(p). In particular, as it has been noted in [1], one can construct periodic solutions of eq. (2) if the V(~p) 2). Since has a of form of a well (e.g. const pwhen eq. (2) form function V(~p) wasV(p) not=limited was derived one has the right to choose V(p) in the form V(p)
~ ~z ~p2P(p= 1,2,...), p”l
where positive constants j.z ~ 1. Travelling BGK-waves. Solution of eq. (2) for values p> 2 can not be expressed in the form of an analytical function. For this reason, assume p = 2 in an expression for V(~p).In this case solution of eq. (2) has the form 193
Volume 44A, number 3
PHYSICS LETTERS
4 June 1973
(0) =
a÷cn(w;k)
(3)
where a~ \/(u÷--p
exp(-
where q
irK/K’), K’
K(k’). Taking into
1)/2p.2, w ~/~(x x0), ‘~/(o~--ii1)/2u~. cn(w; k) is a Jacobian elliptical
account the presented [9J relation limq(at k -~ 0) = 1/16 0.05 ~ 1 one can conclude that the series (6)
function of variable (w) with period 4K(k) (K(k) is a total elliptic integral of the first kind) solution (3) corresponds to a travelling wave (it will be recalled that in the present paper the consideration is carried
are fast convergent ones. Therefore, in a limiting case
k
out in the system of waves). At first it may seem that solution (3) is a stable
of small k expression (5) isAllowing transformed intoabove 4cos(irw/2K). for the 2~k7kq~ mentioned and a well-known relation 2/ 1 ~ 2(2/) cos[2(/— (2/
[
221 (m0
cos q~---
2P
one and intoappreciably consideration of terms /lP~p (atp >2)taking can not affect the character of the solution obtained. In the nect section of this paper it will be demonstrated that in fact. the situation is different. InstabUity ofBGK-waves. To investigate stability of solution (3) we shall introduce small deviation from the equilibrium state
1 we come to the conclusion that equation “in variations” is transformed into Hill’s generalized equation which at p = 2 is represented as a well-known Mathieu’s equation: + [0~+20~ cos(Kx)1 ~ = 0.
where ~
i~=~
~
~.
Equation for function ~ (equation “in v~iriations”) has the form
~ (2p— l)pp ~2(p-1) ~ =
(4)
Even simple qualitative considerations show the instability of solution (3). In fact, substitution of expression (3) in eq. (4) transforms tl~elatter ii~ Lame’s in elliptical functions [81.with As is known, equation Lame’s equation (as any equatian periodical coefficients) at a certain range of parameter values has unstable solutions. However, to find unstable solutions of eq. (4) one must use rather cumbersome mathematical calculations. The analysis of the problem is considerably simplified if one uses a well-known relation binding Jacobian functions with theta functions: ~/k’t9 2(irw/ 2K) cn(w: k) = ~ (5) 2 Theta functions 19,(irw/2K) are where k’as=~.,/lk defined sums of the following series 02
~K
2
(~)- —
(irw’~
04~K)
=
~
(n— 112)
n=l q
1+2 ~ fl = 1
194
L
cos(2n 2
—
[
1) (imw
(--- l)~q~cos[2n~~)]~ (6)
m)r]+
m
ir~/~i/K(k), Oo
2a~q1/4~/~7~.
(7)
ji
1+3ji2b~,01 3p2b~/2, There are manymethods to
find unstable solutions of Mathieu’s equation. Here we shall make use of Wittaker’s method which was used before for the investigation of the stability of the investigation of the stability of the plasma nonlinear longitudinal oscillations [101. Omitting intermediate calculations we present a final expression for the growth rate of unstable oscillations
(~= exp(~x)sin(l~x—~a).aan unknown parameter) 40o+O1 and at the same time a ~2 (Oo+ l)+ condition necessary for build-up of instability 01 > 0o - I. Consequently, if parameters of the
plasma and of nonlinear periodical wave are in compliance with the written condition, the BGKwave turns out to be unstable. In the next section we shall examine the physical nature of instability and the mechanism of its “saturation”.
Restriction of the amplitude of an unstable Oscillation. Appearing side-bands waves. It not difficult to show that inoffact, the amplitude ofisdisplacement is a limited at any value x. Indeed, writing down the equation “in variations” (7) we confined linear terms over ~. Taking into considerationtononlinear terms over ~ leads to the restriclion of the growth rate of the oscillation amplitude. To the above statement shall write down eq. prove (7) taking into account the we term
~+X2[l÷h
cos(~x)j~+2p
30,
=
2~
(8)
Volume 44A, number 3
where A2
PHYSICS LETTERS
Oo, h
201/60 ~ 1. Solution of eq. (7)
=
‘y”~~(i —kq2Z
2(Z 2/Z1) cos(5Kx+~3)+~kq
in the first approximation, according to the Krilov— Bogaljubov method, has the form (in the case considered ic 2X) =
4 June 1973
X {cos [(~
F~irM/K(k).
(9)
a cos(~Kx+t~),
cos [(~ —f)x+~]}, (15)
Proceeding from relation (15) one can readily understand the physical meaning of instability in the
where functions (a) and (~9)are satisfied by the differential equations 2 ~ Z sin0 (10) dZ = 2A K
paper considered. appearance of satellite waves with a wave Indeed, vectors (~K ±F) besides the basic oscillation with a wave vector ~ follows from relation
,
where C is constant of integration, By excluding variable 6 from relations(10) and (12) one obtains
(15). Therefore, the considered instability has a character of “decay” instability when an original nonlinear wave of a large amplitude “splits” in a great number of satellite waves. Appearance of only one pair of satellite waves follows from relation (15). However, one should keep in mind that when obtaining eq. (8) we confined with 1) value p = 2 in eq. (4) and 2) taking into account only nonlinear term ~ It turns out to be physically evident that in a general case (at a sufficiently large value p) taking into consideration a greater number of nonlinear terms leads to the “splitting” of the original wave in a great
dZ
number of satellite waves.
— =
2~+ 2X2 h
+F)x+~] +
2/.../7~)
— cos6 K
—
6— Z
(11)
,
K
where ~ A— ,‘c/2, Z rua2, 0 2& As one can easily make sure by direct integration the set of eqs.(l0), (11) has the integral 2A2 -~Z co~0= C — 2~.Z+3 -~-~ Z2,
14X4h2Z2 =
L
2
1 /2
(1—cos 0)]
K2
(12)
4
1/2
(Z_Zp)]
= [~
The authors are obliged to Professor Dr. M. ElNadi and Dr. T. El-Khalafawy for encouragement and support. They also wish to thank Mr. V.A. Utenock for his aid in preparation of this manuscript
=
(13)
p=l
for publication.
where ~ are roots of the equation P(Z) = 0. Solution of eq. (13) is presented in [11] and has the form Z = [Z 2(u;k)—QZ 2(u; k)—Q] 1II[sn where 2sn
[1] I.B. Bernstein, J.M. Greene and M.D. Kruskal, Phys. Rev. 108 (1957) 546. [2] J.H.A. Van Wakeren and H.J. Hopman, Phys. Rev. Lett. 18 (1972) 295. [3] E.P. Barbian and B. Jurgens, Proc. IlIrd Int. Conf. on
(14)
Q-plasmas, Elsimore, 1971, p. 317. [4] G.J. Morales and T.M. O’Neil, Phys. Rev. Lett. 28 (1972) 215. [5] R.N. Franklin et a!,, Phys. Rev. Lett. 28 (1972) 1114.
____________________
u
M(x —x 0),
k
~/(Z2—Z3)/Q(z1 —Z3),
M~+[(Z2—Z4XZ3—Z1)]1/2
~
(Z2—Z4)/(Z1_Z4)
sn(u; k) the Jacobian elliptical sine, x0 a constant of integration which can be assumed to be zero. In the peculiar case Z3 Z1 Z2 expression (14) ~“
,
is transformed into
2 Z
—
~1— cos [2(2
K(k))]).
2kZ2q
Taking into account that a
tains expressions for value
=
V’Z, from (9) one ob-
,
[6] D. Biskamp and R. Chodura, IAEI Conf., Madison, 11(1971) 265; Invited paper “Recent Development in Computer Simulations of Plasmas”, Vth Europ. Conf. on Controlled fusion and plasma physics, 11(1972) 93. [7] D. Forslund, R. Morse and C. Nielson, IAEI Conf., Madison, 11(1971) 277. [8] E.T. Whittaker and G.N. Watson, A course of modern analysis (London, 1935). (Dover Pub!., New York, 1945). 9] formulae E. Jahnkeand andcurves, F. Emde, Tables of functions with [10] V.V. Demchenko and L.A. El-Naggar, Phys. Lett. 39A (1972) 205; Cz. J. Phys. 22B (1972) 1194; Proc. Vth Europ. on Controlled fusion and plasma physics,Conf. 1(1972)142. [11] W.F. Ames, Nonlinear ordinary differential equations in transport processes (Academic Press, N.Y., 1968) p. 55. 195