- Email: [email protected]
Plasma wave collisionless damping and charged particle acceleration induced by deterministic chaos in a strongly curved magnetic field
PhysicsLettersAl59(1991) 170—173 North-Holland
PHYSICS LETTERS A
Plasma wave collisionless damping and charged particle acceleration induced by deterministic chaos in a strongly curved magnetic field M.I. Sitnov Institute of Nuclear Physics, Moscow State University, Moscow 119899, USSR Received 10 July 1991; accepted for publication 26 July 1991 Communicated by V.M. Agranovich
Magnetic field line curvature is known to generate deterministic chaos in particle dynamics, while exactly conserving the particle kinetic energy. Nevertheless, in a collisionless plasmathis chaos is found to result in a sort of linear Landau damping effect due to particle deceleration saturating near the lower energy edge of the chaotic domain.
The deterministic chaos phenomenon (DCH), i.e. separate particle stochastization in a regular external field, is more and more considered as a mechan,ism of charged particle acceleration and of plasma wave damping [1—6].However, an attempt to utilize the DCH in a strongly curved magnetic field [7,81 as a dissipation source providing tearing instability of a quasineutral sheet [6] was found in ref. [9] to face considerable difficulties. The matter is that even an inhomogeneous magnetic field conserves both the kinetic energy of each particle and the particle content in a flux tube, thus keeping the electron cornpressibility sufficient to compensate the electromagnetic free energy excess connected with the tearing mode (the ions are supposed to be nonmagnetized). One of the possible solutions of the problem would be to takedue intotoaccount theof loss-cone, dissipation variation the flux providing tube content, This communication turns attention to another dissipation mechanism, being similar to Landau damping [10] by switching on by the plasma wave itself, while in ref. [6] the dissipation was not connected in any way with plasma perturbation. This mechanism is however based on DCH pitch-angle diffusion features rather than on the inverse Cerenkov effect. Consider the particle dynamics within a current sheet with magnetic field B= (B 0z/L, 0, B~)corresponding to the simplest, i.e. linear approximation 170
of the sign-reversing component of the magnetic field. The electrostatic perturbation potential ~(x, z, t) = —(Eo/k)Ø(IzI(poL)112)sin(kx—wt), with Pa being the Larmor radius in the field B 0, is supposed to be localized near the z= 0 plane with a characteristic scale of the order of the nonmagnetized plasma layer thickness at B~= 0 and to decrease on the outside according to a power law: Ø(—~~, 1 ~y<2. This law is chosen with the sole purpose of eliminating the electric potential induced loss-cone, which, though amplifying the acceleration, reflects no DCH diffusion features. The true eigenrnodes of the quasineutral sheet may differ from the chosen perturbation form both by polarization and by z-profile. Note here that the tearing mode at B~= 0 always has a component of such type, with2the localization scaleasbeing [6,9,11}. Anyway, even at least IzI ~L>> (p0L)~ self-consistent kinetic equilibrium has not been found up to now (because of unknown dynamics), it is difficult to prefer some definite wave type. On the other hand, the estimates made below allow very extensive generalizations of polarization and z-profile. Spatial and temporal variations of the perturbative electric field can be neglected on the scale of partide dynamics. In the opposite limiting case the dominating processes would be linear Landau damping [10,121, nonlinear wave—particle interactions [2— 51, or effective loss-cone formation due to merely Elsevier Science Publishers B.V.
Volume 159, number 3
PHYSICS LETTERS A
perturbation phase variation along the particle orbit [13]. Therefore the Hamiltonian to be investigated may be written in the form 2+~±2+~(~z2—Kx)2—axØ(z)=~ (1) H=3) while the electric field variation (electrostatic oscillations) will be taken into account afterwards by varying the value of a: a=a(E(X,t))=(eE/mv~) x (Lp 2, where E=E 0)” 0cos(k~—at) is the local value of the electric field perturbation and the overbar denotes the averaging along nonperturbed orbits (a=0). off Thethe dimensional x-coordinate in eq. w,~ (1) is counted constant ~+p~,/ mw~,where 0= eB~,0/mc,and p01. is the y-component of the conserved generalized momentum. The dimensionless variables 2,arev—~v/v introduced by the substitution r— r/(p0L)” 0,with ~mv~=H(x=0)=H(x) the second exact integral of motion, and = V0/W0. 2,soPathat K2 is The parameter proportional to K=(B~/B0)(L/p0)” the ratio of the field line curvature radius to the Larmor radius of the particle, The model Hamiltonian (1) at a = 0 has been studied in detail [7,8,14,15]. In particular it has been ,
found that at ic~K1 1.6 the equations ofmotion are completely integrable. The integrals of motion are the magnetic momentu and the second adiabatic invariant, namely the action of bounce-oscillationsbetween mirror points. In a relatively small vicinity K
1).
The effective thickness of the layer ESK may be es-
7 October 1991
timated in the following way. The average kinetic energy variation is determined by the dependence of xØ on the magnetic moment 4u, xØ=(21c)~F(v,K), v=mv2/2~tB~, (2) (v2_ 1)1/2
F=
f
Ø(,cc~)~2(l +~2)
j
[v— (1
(2
( X
d~
!)1/2
f
~
1/2
+~2) 1/2] 1/2
J
~
0 2,
~
—
(1 +~2) (1 + ~2)
1/2 1/2]! /2
d~) (3)
Kp)2’+
and by the law of u-evolution at a << 1 and ~ a -1/2 This evolution can be approximated by Fokker—Planck diffusion with coefficient 3[l—sgn(K—K D,~—~DowBv 1)] (4) which has been inferred from the characteristic interval between DCH “collisions” V2IWB (the detailed bounce-period estimate at v ~ 1 [6,7]) and from the magnetic moment stochastic “collision” jump 4~..~g~l/2 [6,7,14], with D 0 being a constant dimensionless coefficient. The dependence (4) results in the following xØ asymptotics, ~ t—~oo. (5) ,
-~
~
,~
~,
In fact any localized distribution will have such asymptotics after spreading sufficiently wide compared to its original scale. It justifies to a considerable extent the somewhat simplified model of diffusion coefficients [7,141 which is supposed by eq. (4). During a half-period of plasma oscillations corresponding to a definite sign of the x-component of the electric field the particle gains a kinetic energy (ke+,t/2)/w 2
Aff~mv0a(E0)(a/it)
C
—
i
1 (kx—R/2)/w
xØcos(kx—at)dt
2—~E—
mv~a(E0) (~D0w5/w) ~ (~+~)2—~ cos ~‘ d~ =
I
~=
kx
(6)
2
RI
Unless a shift in the K-scale took place during the process of diffusion, this energy gain when averaged over a wavelength 2it/k would be exactly compen171
Volume 159, number 3
PHYSICS LETTERS A
sated during the phase of opposite sign of the electric field. However, the necessity ofunperturbed motion to retain pure magnetic diffusion during the kinetic energy variation (axØ= 0) requires the parameter K to vary in the following way, ~
(7)
As a result the decelerating particles being previously within the K-layer with thickness 2~’ (8) have to 1a(E0)(itD0w~/w) change the type of motion, for they cannot diffuse above the DCH boundary K 1 and are “sticking” in that neighbourhood. These simple considerations about the effect of “deceleration saturation” have been confirmed by direct dynamical simulation and are illustrated in fig. 1. Thus particle acceleration dominates near the DCH boundary notwithstanding the change of sign of the perturbative electric field. The corresponding kinetic energy density variation rate for the plasma with number density N and definite K-distribution f(K), may be estimated as i~K-~ ~tc
dw/dt’-~(K~/27t)Nf(K
0)
(9)
Since the acceleration can be maintained due to only
10
ô—~w(w~/a~)2f(K
2~’,
1
) (1tDowB/w)
(10)
where w~,=(4i~e2N/m)”2. Undoubtedly, the effect considered is nonlinear in essence, since the DCH phenomenon itself is caused by particle dynamics nonlinearity. In the case ofsmall amplitude waves this property may be revealed by the dependence of the result on the DCH boundary structure. In particular, when the boundary curvature takes place K~= K1 Cu), with dK1 /dp <0 and a being sufficiently small, particle drift towards larger K values will be accomplished with so much increase of K1 due to p decrease, that the decelerated particle will not be able to reach the boundary at all. Fortunately according to ref. [7J the Hamiltonian (1) yields dK1 /du> 0. Besides that for typical parameters of geotail electrons, for example, with energy 2—~l0~m (see, e.g., keY and space scale value (p0L)” ref.10[16]), an acceptable of a 0.1 is achieved at electric field amplitude 10 mV/rn which is corn-
~
parable withbeequilibrium It should emphasizedfluctuations however that[17]. the dynamics nonlinearity is connected with an external magnetic field rather than with an electrostatic perturbation. Thus DCH damping is related just to linear Landau damping rather than to nonlinear processes [1—5].That relation is confirmed by the damping decrement in eq. (10) being independent of the perturbation amplitude. The author is greatly indebted to C.F. Kennel, A.P. Kropotkin, H.V. Malova and especially to L.M. Zelenyi for a number of helpful discussions.
/
/
References
I
t
Fig. 1. Temporal evolution ofaveraged and normalized (by a factor a) particle kinetic energy variation xØ near the lower energy edge of DCH (initial ic value K(0) = 1.4) for pure magnetic diffusion, acceleration and deceleration (a=0.0, 0.1 and —0.1) denoted by dotted, dashed and solid curves correspondingly. Averagingtime scale 0.3 x 102; z-profile 0(z) = (1 +z2) 1/2; initial coordinates: x=0.7, z= 10~,v~=0.613,v~=0.366.
172
electrostatic wave energy expenditure with density variation rate (d/dt) (E2/ 8it), the corresponding damping decrement may be obtained directly from eq. (9),
2(E
12~. )a)a x mv~(i~DowB/w)
0
7 October 1991
[1] C.F.F. Karney, Phys. Fluids 22 (1979) 2188. [2] M.A. Mal’kov and G.M. Zaslavsky, Phys. Lett. A 106 (1984) 257. [3] A.A. Berzin, G.M. Zaslavsky, S.S. Moiseev and A.A. Chernikov, Fiz. Plasmy 13 (1987) 592. [4] G.M. Zaslavsky, M.A. Mal’kov, R.Z. Sagdeev and A.A. Chernikov, Fjz. Plasmy 14 (1988) 807.
Volume 159, number 3
PHYSICS LETTERS A
[5] G.M. Zaslavsky and R.Z. Sagdeev, Introduction to nonlinear physics: from pendulum to turbulence and chaos (Nauka, Moscow, 1988). [6] J. Buchner and L.M. Zelenyi, J. Geophys. Res. A 92 (1987) 13456. [7] J. Buchner and L.M. Zelenyi, J. Geophys. Res. A 94 (1989) 11821. [8] H.V. Malovaand M.I. Sitnov, Phys. Lett. A 140 (1989) 136. [9] R. Pellat, F.V. Coroniti and P.L. Pritchett, Prepnnt IPFR (UCLA, 1990, PPG-1332). [10]L.D.Landau,Zh.Eksp.Teor.Fiz. 16(1946) 574. [11] B. Lembege and R. Pellat, Phys. Fluids 25(1982) 1995.
7 October 1991
[12] L.S. Kuz’menkov and M.I. Sitnov, Fjz. Plasmy 12 (1986) 92. [13] M.M. Kuznetsova and L.M. Zelenyi, Magnetic reconnection in collisionless field reversals. The universality ofthe tearing mode, Geophys. Res. Lett. (1991), lobe published. [14] T.J. Birmingham, J. Geophys. Res. A 89 (1984) 2699. [15] B.V. Chirikov, in: Topics in plasma theory, Vol. 13 (1984) p. 3. [16] L.M. Zelenyi, IKP INT VINITI AN SSSR 24 (1988) 58. [l7]L.M. Zelenyi and J. Buchner, IKP INT VINITI AN SSSR 28(1988) 3.
173
PHYSICS LETTERS A
Plasma wave collisionless damping and charged particle acceleration induced by deterministic chaos in a strongly curved magnetic field M.I. Sitnov Institute of Nuclear Physics, Moscow State University, Moscow 119899, USSR Received 10 July 1991; accepted for publication 26 July 1991 Communicated by V.M. Agranovich
Magnetic field line curvature is known to generate deterministic chaos in particle dynamics, while exactly conserving the particle kinetic energy. Nevertheless, in a collisionless plasmathis chaos is found to result in a sort of linear Landau damping effect due to particle deceleration saturating near the lower energy edge of the chaotic domain.
The deterministic chaos phenomenon (DCH), i.e. separate particle stochastization in a regular external field, is more and more considered as a mechan,ism of charged particle acceleration and of plasma wave damping [1—6].However, an attempt to utilize the DCH in a strongly curved magnetic field [7,81 as a dissipation source providing tearing instability of a quasineutral sheet [6] was found in ref. [9] to face considerable difficulties. The matter is that even an inhomogeneous magnetic field conserves both the kinetic energy of each particle and the particle content in a flux tube, thus keeping the electron cornpressibility sufficient to compensate the electromagnetic free energy excess connected with the tearing mode (the ions are supposed to be nonmagnetized). One of the possible solutions of the problem would be to takedue intotoaccount theof loss-cone, dissipation variation the flux providing tube content, This communication turns attention to another dissipation mechanism, being similar to Landau damping [10] by switching on by the plasma wave itself, while in ref. [6] the dissipation was not connected in any way with plasma perturbation. This mechanism is however based on DCH pitch-angle diffusion features rather than on the inverse Cerenkov effect. Consider the particle dynamics within a current sheet with magnetic field B= (B 0z/L, 0, B~)corresponding to the simplest, i.e. linear approximation 170
of the sign-reversing component of the magnetic field. The electrostatic perturbation potential ~(x, z, t) = —(Eo/k)Ø(IzI(poL)112)sin(kx—wt), with Pa being the Larmor radius in the field B 0, is supposed to be localized near the z= 0 plane with a characteristic scale of the order of the nonmagnetized plasma layer thickness at B~= 0 and to decrease on the outside according to a power law: Ø(—~~, 1 ~y<2. This law is chosen with the sole purpose of eliminating the electric potential induced loss-cone, which, though amplifying the acceleration, reflects no DCH diffusion features. The true eigenrnodes of the quasineutral sheet may differ from the chosen perturbation form both by polarization and by z-profile. Note here that the tearing mode at B~= 0 always has a component of such type, with2the localization scaleasbeing [6,9,11}. Anyway, even at least IzI ~L>> (p0L)~ self-consistent kinetic equilibrium has not been found up to now (because of unknown dynamics), it is difficult to prefer some definite wave type. On the other hand, the estimates made below allow very extensive generalizations of polarization and z-profile. Spatial and temporal variations of the perturbative electric field can be neglected on the scale of partide dynamics. In the opposite limiting case the dominating processes would be linear Landau damping [10,121, nonlinear wave—particle interactions [2— 51, or effective loss-cone formation due to merely Elsevier Science Publishers B.V.
Volume 159, number 3
PHYSICS LETTERS A
perturbation phase variation along the particle orbit [13]. Therefore the Hamiltonian to be investigated may be written in the form 2+~±2+~(~z2—Kx)2—axØ(z)=~ (1) H=3) while the electric field variation (electrostatic oscillations) will be taken into account afterwards by varying the value of a: a=a(E(X,t))=(eE/mv~) x (Lp 2, where E=E 0)” 0cos(k~—at) is the local value of the electric field perturbation and the overbar denotes the averaging along nonperturbed orbits (a=0). off Thethe dimensional x-coordinate in eq. w,~ (1) is counted constant ~+p~,/ mw~,where 0= eB~,0/mc,and p01. is the y-component of the conserved generalized momentum. The dimensionless variables 2,arev—~v/v introduced by the substitution r— r/(p0L)” 0,with ~mv~=H(x=0)=H(x) the second exact integral of motion, and = V0/W0. 2,soPathat K2 is The parameter proportional to K=(B~/B0)(L/p0)” the ratio of the field line curvature radius to the Larmor radius of the particle, The model Hamiltonian (1) at a = 0 has been studied in detail [7,8,14,15]. In particular it has been ,
found that at ic~K1 1.6 the equations ofmotion are completely integrable. The integrals of motion are the magnetic momentu and the second adiabatic invariant, namely the action of bounce-oscillationsbetween mirror points. In a relatively small vicinity K
1).
The effective thickness of the layer ESK may be es-
7 October 1991
timated in the following way. The average kinetic energy variation is determined by the dependence of xØ on the magnetic moment 4u, xØ=(21c)~F(v,K), v=mv2/2~tB~, (2) (v2_ 1)1/2
F=
f
Ø(,cc~)~2(l +~2)
j
[v— (1
(2
( X
d~
!)1/2
f
~
1/2
+~2) 1/2] 1/2
J
~
0 2,
~
—
(1 +~2) (1 + ~2)
1/2 1/2]! /2
d~) (3)
Kp)2’+
and by the law of u-evolution at a << 1 and ~ a -1/2 This evolution can be approximated by Fokker—Planck diffusion with coefficient 3[l—sgn(K—K D,~—~DowBv 1)] (4) which has been inferred from the characteristic interval between DCH “collisions” V2IWB (the detailed bounce-period estimate at v ~ 1 [6,7]) and from the magnetic moment stochastic “collision” jump 4~..~g~l/2 [6,7,14], with D 0 being a constant dimensionless coefficient. The dependence (4) results in the following xØ asymptotics, ~ t—~oo. (5) ,
-~
~
,~
~,
In fact any localized distribution will have such asymptotics after spreading sufficiently wide compared to its original scale. It justifies to a considerable extent the somewhat simplified model of diffusion coefficients [7,141 which is supposed by eq. (4). During a half-period of plasma oscillations corresponding to a definite sign of the x-component of the electric field the particle gains a kinetic energy (ke+,t/2)/w 2
Aff~mv0a(E0)(a/it)
C
—
i
1 (kx—R/2)/w
xØcos(kx—at)dt
2—~E—
mv~a(E0) (~D0w5/w) ~ (~+~)2—~ cos ~‘ d~ =
I
~=
kx
(6)
2
RI
Unless a shift in the K-scale took place during the process of diffusion, this energy gain when averaged over a wavelength 2it/k would be exactly compen171
Volume 159, number 3
PHYSICS LETTERS A
sated during the phase of opposite sign of the electric field. However, the necessity ofunperturbed motion to retain pure magnetic diffusion during the kinetic energy variation (axØ= 0) requires the parameter K to vary in the following way, ~
(7)
As a result the decelerating particles being previously within the K-layer with thickness 2~’ (8) have to 1a(E0)(itD0w~/w) change the type of motion, for they cannot diffuse above the DCH boundary K 1 and are “sticking” in that neighbourhood. These simple considerations about the effect of “deceleration saturation” have been confirmed by direct dynamical simulation and are illustrated in fig. 1. Thus particle acceleration dominates near the DCH boundary notwithstanding the change of sign of the perturbative electric field. The corresponding kinetic energy density variation rate for the plasma with number density N and definite K-distribution f(K), may be estimated as i~K-~ ~tc
dw/dt’-~(K~/27t)Nf(K
0)
(9)
Since the acceleration can be maintained due to only
10
ô—~w(w~/a~)2f(K
2~’,
1
) (1tDowB/w)
(10)
where w~,=(4i~e2N/m)”2. Undoubtedly, the effect considered is nonlinear in essence, since the DCH phenomenon itself is caused by particle dynamics nonlinearity. In the case ofsmall amplitude waves this property may be revealed by the dependence of the result on the DCH boundary structure. In particular, when the boundary curvature takes place K~= K1 Cu), with dK1 /dp <0 and a being sufficiently small, particle drift towards larger K values will be accomplished with so much increase of K1 due to p decrease, that the decelerated particle will not be able to reach the boundary at all. Fortunately according to ref. [7J the Hamiltonian (1) yields dK1 /du> 0. Besides that for typical parameters of geotail electrons, for example, with energy 2—~l0~m (see, e.g., keY and space scale value (p0L)” ref.10[16]), an acceptable of a 0.1 is achieved at electric field amplitude 10 mV/rn which is corn-
~
parable withbeequilibrium It should emphasizedfluctuations however that[17]. the dynamics nonlinearity is connected with an external magnetic field rather than with an electrostatic perturbation. Thus DCH damping is related just to linear Landau damping rather than to nonlinear processes [1—5].That relation is confirmed by the damping decrement in eq. (10) being independent of the perturbation amplitude. The author is greatly indebted to C.F. Kennel, A.P. Kropotkin, H.V. Malova and especially to L.M. Zelenyi for a number of helpful discussions.
/
/
References
I
t
Fig. 1. Temporal evolution ofaveraged and normalized (by a factor a) particle kinetic energy variation xØ near the lower energy edge of DCH (initial ic value K(0) = 1.4) for pure magnetic diffusion, acceleration and deceleration (a=0.0, 0.1 and —0.1) denoted by dotted, dashed and solid curves correspondingly. Averagingtime scale 0.3 x 102; z-profile 0(z) = (1 +z2) 1/2; initial coordinates: x=0.7, z= 10~,v~=0.613,v~=0.366.
172
electrostatic wave energy expenditure with density variation rate (d/dt) (E2/ 8it), the corresponding damping decrement may be obtained directly from eq. (9),
2(E
12~. )a)a x mv~(i~DowB/w)
0
7 October 1991
[1] C.F.F. Karney, Phys. Fluids 22 (1979) 2188. [2] M.A. Mal’kov and G.M. Zaslavsky, Phys. Lett. A 106 (1984) 257. [3] A.A. Berzin, G.M. Zaslavsky, S.S. Moiseev and A.A. Chernikov, Fiz. Plasmy 13 (1987) 592. [4] G.M. Zaslavsky, M.A. Mal’kov, R.Z. Sagdeev and A.A. Chernikov, Fjz. Plasmy 14 (1988) 807.
Volume 159, number 3
PHYSICS LETTERS A
[5] G.M. Zaslavsky and R.Z. Sagdeev, Introduction to nonlinear physics: from pendulum to turbulence and chaos (Nauka, Moscow, 1988). [6] J. Buchner and L.M. Zelenyi, J. Geophys. Res. A 92 (1987) 13456. [7] J. Buchner and L.M. Zelenyi, J. Geophys. Res. A 94 (1989) 11821. [8] H.V. Malovaand M.I. Sitnov, Phys. Lett. A 140 (1989) 136. [9] R. Pellat, F.V. Coroniti and P.L. Pritchett, Prepnnt IPFR (UCLA, 1990, PPG-1332). [10]L.D.Landau,Zh.Eksp.Teor.Fiz. 16(1946) 574. [11] B. Lembege and R. Pellat, Phys. Fluids 25(1982) 1995.
7 October 1991
[12] L.S. Kuz’menkov and M.I. Sitnov, Fjz. Plasmy 12 (1986) 92. [13] M.M. Kuznetsova and L.M. Zelenyi, Magnetic reconnection in collisionless field reversals. The universality ofthe tearing mode, Geophys. Res. Lett. (1991), lobe published. [14] T.J. Birmingham, J. Geophys. Res. A 89 (1984) 2699. [15] B.V. Chirikov, in: Topics in plasma theory, Vol. 13 (1984) p. 3. [16] L.M. Zelenyi, IKP INT VINITI AN SSSR 24 (1988) 58. [l7]L.M. Zelenyi and J. Buchner, IKP INT VINITI AN SSSR 28(1988) 3.
173