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Slightly anisotropic energetic electron fluxes in a turbulent space plasma
Planet. Space Sci., Vol. 41, No. 10, pp. 785-790, 1993
~
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/93 $7.00+0.00
Pergamon
Slightly anisotropic energetic electron fluxes in a turbulent space plasma P. A. Bespalov ~ and V. G. Efremova 2 qnstitute of Applied Physics, Russian Academy of Science, 46 Uljanova st., 603600 Nizhny Novgorod, Russia 2Nizhny Novgorod State University, 23 Gargarin st., 603600 Nizhny Novgorod, Russia Received 14 May 1993 ; revised 27 September 1993 ; accepted 30 September 1993
Abstract. The problem of dispersion of an energetic electron stream in a turbulent space plasma without magnetic field is considered. The basis of calculation is a self-consistent system of quasi-linear equations, which take into account non-one-dimensional diffusion in velocity space. It is shown that for a rather dense electron cloud, the particle distribution function is nearly isotropic. Solutions for the electron distribution function and for the spectral density of plasma wave turbulence are found. These results were compared with the solution for the case of a constant level of plasma turbulence.
1. Introduction The problem of the dynamics of the expansion of fast charged particles in a turbulent plasma is rather important both for the physics of laboratory flux experiments and as a space phenomenon in star coronas, star wind and interplanetary magnetospheres. Expansion of energetic particles from local sources is typical of all hot space objects such as the solar corona, for example, in a rarefied space plasma, collisions are often insignificant and the dynamics of the particle flux is determined by the collective effects of interaction with plasma turbulence leading to anomalous frequency of the collisions. Under known conditions, turbulent dispersion of a fast particle cloud in a dense plasma has a diffusive character in velocity space and can be described by a set of quasilinear equations. The diffusion efficiency is controlled by the noise-wave intensity in a background plasma. For rather weak fluxes, the level of plasma turbulence is practically independent of its dynamics. For more dense fluxes, the self-consistent dependence of the waves on the particles become more significant. The particle distribution function evolution both in a
plasma with a constant level of turbulence independent of the particles and in the self-consistent case have been considered many times in the literature. For example, a comparatively simple system of quasi-gasdynamic equations for self-consistent turbulent dispersion of fast particles in a strong magnetic field (o92 << o92) was found and analysed in the paper by Rjutov et al. (1970). Here, both the plasma wave spectrum and the particle velocity-space diffusion are assumed to be one-dimensional. But in many cases, for example in a weak magnetic field, the fast particle diffusion becomes significantly nonone-dimensional. Such a problem was studied in papers by Bespalov and Trakhtengerts (1974, 1979) and Bahareva et al. (1983). A simplifying factor in this problem was the fact that for sufficiently dense fluxes, energetic particles excite plasma oscillations with phase velocity much lower than their own average velocity. As a result, the particles experience elastic collisions with plasma oscillations, and their velocity distribution function remains almost isotropic. The difficulties in such an analysis were connected with the necessity of a more correct definition of lowenergy particle diffusion. In this work we find a self-consistent set of equations determined by the distribution function or its moments (flux density, average values of velocity and square of velocity) for electron flux turbulent dispersion in a plasma. The determined solution has a rather wide practical interest for the study of matter, pulse and energy transfer in space plasma, for example for diffusive fluxes in solar wind (Lin et al., 1973; Kurt et al., 1975) after explosive particle injection, or for photoelectrons in the Earth's magnetosphere (Galperin et al., 1973). For comparison, a solution is also proposed for the electron flux in isotropic plasma with a level of turbulence independent of the particles. This is useful for a clearer determination of the effects owing to self-consistent events.
2. Basic equations Correspondence to : P. A. Bespalov
We assume that at the initial moment of time, the energetic electron cloud was formed on the plane z = 0 in a uniform
786
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space
background plasma without magnetic field. Langmuir plasma oscillations with a sufficient level of turbulence effectively scatter charged particles. The level of turbulence can be maintained both by an external source in background plasma and by a flux instability. Their own flux evolution is determined by the collective processes of interaction with Langmuir turbulence. One-dimensional and symmetric (in coordinate space) dispersion of the electron cloud is described by the following set of quasi-linear equations (Vedenov et al., 1972 ; Galeev et al., 1973 ; Bespalov and Trakhtengerts, 1979) :
e and m are the charge and the mass of an electron, vg= = Oto/Okz, Vph = to~k, Vk is the linear decrement includ-
ing both the Landau damping in the background plasma and the collisional damping. Note that the initial system (1) can be used for describing the particle dispersion in a plasma with weak magnetic field when 7k > to2 (Mikhailovskii, 1975), where tOBis the electron cyclotron frequency. We first use equations (1)-(4) for analysis of the fast electron cloud dispersion by plasma oscillations which are independent of the electron cloud spectrum.
Of + v x ~f 1 0 2( OJ" 1 c~/~) ~t Oz - v 2 Ov t' D''~vv +D~'xv
+v~-x
3. Dispersion of particle fluxes by plasma turbulence with known spectrum and constant level
D ' " ~ v + D . . . v. . . ~k
~k
& +v.:~z = 2(~k--V0~.~,
(1)
with diffusion coefficients : D~x
=
--
Such a situation is realized, for example, in weak electron fluxes in disturbed solar wind, or in turbulent transition regions of planetary magnetospheres. For simplicity, we assume that the wave energy spectrum density ek is constant and isotropic in the wavevector space function with small dispersion of k, and hence :
Re
gk = g , 3 ( k , -
D ,: ,:
ekk dk dy v{1 - x 2 _ y2 _ (to/kv)[(to/kv) - 2xy]} 1/2
1),
(5)
where : (2)p Uph* ~ t . << U0'
x
kv
Y-
,
(2)
( y_ tox 2
(6)
Here, v0 is the average velocity of energetic electrons. For the spectrum (5) the diffusion coefficients (2) given by: D,.,, ~D 2 U3 /'5p h*
Di,x = 0, and growth rate : =nD }'k
=
np
rr 2
fo f , IO
X2
I
/-)ph, 2
(7)
dv dx where D = D , E [ ( v / V p h , ) - 1 ] , D , = (4ne/m)2k2,eo, and:
1/
kv "X a-]
× R e {1 - x 2 - y 2 - ( t o / k v ) [ ( t o / k v )
-2xy]}
t, ~xx x=+, = 0 ,
{
1,x~> 0,
1/2"
E(x) = O,x < O.
(3)
The distribution functionf(t, z, x, v) satisfies the boundary conditions : D"xbx+
1
(4)
which provide the absence of particle flux through the boundary of the distribution function domain. In formulas (1)-(4), v = ]vl is the velocity of the electrons, x = Vz/V is the cosine of the angle between the velocity and the direction of the dispersion, et(t, z) is the spectral energy density of the plasma oscillations, k is the wavevector, y = k : / k is the cosine of the angle between the wavevector and the direction of dispersion, tOp is the plasma frequency, np is the background plasma density,
For a sufficiently high level of plasma turbulence that satisfies expression (6), the distribution function is maintained as quasi-isotropic. This permits us to write the distribution function in the form of a sum average over the angle main part F and small anisotropic • : f = F + ~,
(8)
where : F = (1/2)
£
fdx,
IOI << F.
From the initial set (1) we can get an equation for the isotropic part F, taking into consideration equations (7) :
787
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space ~zD 0F
O2F
v 2 c32F
~v ~t + t3t2
3 ~z 2
=
ltD ',, ~ / I Lat +
2 ev Y &, t,2jj~ -
t,' L \ 7 , / 0 v
~vv 0v
-~t
"
(9)
/
According to condition (6) for the main stock of particles, the ratio Vph,/Vis rather small. Thus, in the zeroth order of the perturbation theory, for that parameter we have instead a simpler equation, disregarding energy diffusion" t~2 - 0,
(10)
where T = (2nD,/v3)t, ~ = (2x/3rtD,/v4)z. The solution of equation (10) for the initial problem has the form"
1 r +' f i s h ( 2 x/l-2q~)+ch( 2 x//ll_~)l~(q, v)
r sh(2 +2
5
}
lx/1-q ) ~ ~k(q,v)
e x p ( - 2 ) c o s ( q ~ ) d q.
(11)
x/l~q 2
Here, 4~(q, v) and q~(q, v) is a Furje transform of the functions ~b(~, v) and qJ(~, v), which are determined by the initial conditions" ~b(¢,
v)= F
qJ(¢, v) =
6 z/z0
632F
t3F ~2F c3z + ~z 2
t = 0, 2,v/~tD * 4, v ,
I,=0,~P/2,/s~t~,)~,,, .
(12)
For example, we represent the solution of this problem by the following initial distribution :
Fig. 1. The space dependence of distribution function (14) for two moments of time
where again z = (2rtD,/v3)t, ~ = (2x/3~zD,/v4)z. The space dependence of the solution (14) for fixed velocity and several successive moments of time is presented in Fig. 1.
4. Self-consistent turbulent dispersion of a dense fast particle cloud
We now consider the problem of a one-dimensional (in coordinate space) self-consistent expansion of an energetic electron cloud for the case of turbulence excited by the cloud itself. As was mentioned in the Introduction, one important example of such a process is relaxation of postburst electron fluxes in solar wind. We use the set of equations (1) as initial equations describing such a dispersion. After averaging of the first equation over x and v, taking into account the boundary conditions, we find the following formulas for gasdynamic values" ~n
~(~,v) -- 0.
c~
~ + & (nu) = 0,
qS(~,v) = 6(~)Fo(v), (13)
In this case, the evolution of the distribution function is determined by the expression :
e
fft(nu) + ~zz(nu2) =
/eVnpf
--321t~m ) ~303
3t~t (nu2)+ ~z 4rt
dx 1
F(t,z,v) -
F0(v)
4
/'1 ~ ' ~
I°~2 ~ / ~ - ~ )
=-64n
1~_~
! exp(--2)'
~/~-~
J
(14)
m
'f[
1 dy
dkgk])kyk 3,
f0 dvxvSf ) fo dkekYkk2.
-~60p
1
dy
(15)
j ' f d y is the concentration of energetic elecHere, n trons, u = (1/n) Jvzfdv is the average velocity of cloud particles in the direction of dispersion and u 2 = (l/n) Sv2f dv is the average square of velocity in the direction of dispersion. To acquire a closed set of equations for the isotropic
788
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space
part of the distribution function, we use the following simplifying conditions. First, the velocity distribution of a rather dense electron cloud, produced by effective angular scattering, remains almost isotropic and can be represented by equation (8) (Bespalov and Trakhtengerts, 1979). Second, according to the expression for the growth rate (3), slightly anisotropic fluxes of energetic particles generate waves with relatively low phase velocities :
O(t, z, x, 0 is related to the function F by : 1
O(t, z, x, () - 1 - x 2 ~g(t, z, (), *~)(t,z, () = ~' [5Vph
where v0 is a characteristic particle velocity. Under these conditions, the character of the dependence Ofgk from value k does not produce a significant influence on the diffusion coefficients in the set of equations (1), and we can use for the calculation a wave spectrum in a form like equation (5) with variable e, and k,. Third, for a sufficiently large value of growth rate (for anisotropies close to unity the growth rate 7k is much higher than Vk), the intensity of plasma wave can be determined at a quasi-stationary level (Bespalov and Trakhtengerts, 1979) :
5Vph
~zz
]
(20)
For solving equation (19) it is necessary to use the following from the equation (17) conditions: 27
(21)
F(Uph) ~ -•. Uph
Expression (21) has the sense of the boundary conditions for equation (19). We shall search for a solution of equation (19) in the form of a series : F=
-c 1 ~=on!a.(t,z)(~yo)"exp(--(7o),
( = v 5.
(22)
~k -- Yk ~--"0,
(17a)
- - (7k--Vk) = 0, ~Uph ~2 aUph (Tk--Yk) < 0.
(lYb)
Using expression (22) we can write equation (19) as a system for coefficients a.(t, z) :
(17c)
,0tan
Equation (17a) with the expression (3) for the growth rate represents an Abelian integral equation in terms of y, and enables one to find the angular dependence of the anisotropic part of the distribution function • (Bespalov and Trakhtengerts, 1979):
a¢ 0x
o; -
(18)
1 - x 2'
4 c3z L 4
ao, phsign(Z \,,= 0 0 z / J
~/azl~-zSlgn/2_, ~-//Vph 2.,~zz- / \,=o eZ / L ,,=0 d i 1 5 271 , = o n! a,(Vph)'O)" exp ( -- VpSh)-- Vp3h.
a,(t = 0,z) =
(23)
~ bm(z) ( - 1)" ~=.~ (m-n)!n! '
(24)
where bm are coefficients of the Lagerr polynomial series represented by the function F at the moment t = 0 :
¢ ; = axx x=o
Taking into consideration the simplification mentioned above, we can reduce formulas (1), (15), (17) and (18) to the equation for the isotropic part of the distribution function F: c~F 3F//2
)=0,
Initial conditions for the functions a, are determined by the following expressions :
where
5
d~
(16)
Uph << U0,
c~F
sign
I,~'C~Fd#'] ' ]
bm(2 ) =
f;
F(t = O, z, ~)Lm(#7o) d(.
(25)
For example, we consider the problem of electron cloud dispersion with the following initial distribution :
/
1
F(t = O,z, () = n!a,(z, t = 0)((y0)"exp (-(70). (26) 4
+
For this case we have a simpler system for a,(t,z) and vph(t, z) :
~zz+2~zz ~
aa. c~t
h
+ ::z(VphffFd#)][ff OFd#lJ -'}oz= 0.
(19)
3 0( ~a.~ 4 Oz a"Vohsign 3z ]
3+5n 20 r/ x ~z vph2sign
= 0,
Here, Yon is the phase velocity of plasma waves, q = Vnp/Z2~p,
( = v ~.
The anisotropic part of the distribution function
l 5 2r/ n! a,(Vph?0)"exp (-- VpShT0)= _.~. Uph
(27)
789
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space U fSn+3)l/s, l' Vphmax: ( 570 J
\ I "O O
i
0 Space coordinate
Fig. 2. The qualitative space dependence of energetic electron density for self-consistent expansion for three successive moments of time
At points with nonzero values of equations (27) is defined by :
(Oa,/Oz), the solution of
a, = 2tln!y~OVphCSn+3) exp (VpShY0), Uph = I'I'/[Z"t- V(Vph)t].
/' 5e5 ~
a,min =
2k~)n!tlYo3/5.
If a, < anmin, then the particle dispersion becomes close to that for the free case, increasing the loss Q-factor in expression (30). In the case of an arbitrary initial function F the interaction between energetic modes is significant. For this case we limit ourself to a qualitative consideration. It is _apparent from expressions (23) that the integrals j a, dv and I v2a, dv, corresponding to the whole number of particles and the kinetic energy of a definite energetic mode are conserved. Thus, only a space redistribution of energetic modes takes place. The characteristic velocity of certain energetic mode particles' dispersion rises with an increase of their number. Turbulent "walls" therefore are established in the main by highly energetic modes and expanding plateaus in the centre of the cloud-by a zero mode. The equation (19) describing the evolution of the distribution function can be used for the investigation of matter, pulse and energy transfer. The wave spectrum generated by the flux of particles is also of significant interest. The question about the plasma wave spectrum was considered in the paper by Bespalov and Trakhtengerts (1979). Taking into consideration only angle diffusion, which is correct for rather dense cloud, the authors find an expression for energy density ek :
(28)
ek- 3n3e2k.Y2D(t,z)6(k-k.),
Here
V(vph) = a415n+ 2-- 5,oV~h
~:k=0,
s , exp (-- VpSuT0)1 + ~3 + 5 n (VphT0)
)<
VphSign \ OZ J"
The function U?(z) is an initial distribution of phase velocity Vphand is related to a,(t = 0, z) by :
a,(t = O, z) = 2rln!7"o[Ud(z)]-~5,+ 3)exp [70~5(z)].
(29)
The qualitative pattern of cloud expansion is shown in Fig. 2. An expanding plateau is formed at the centre of the cloud. Its boundaries z = z* are defined from the condition of the preservation of the whole number of particles : Q+2
F(t,z)dz+F(t,z*)2z* =
F(t =
(31)
0, z) dz,
for
for
y
yffv4~zdV>O.
v4
dv < 0,
(32)
Here function D(t,z), determining the intensity of plasma waves according to equation (17), is proportional to the gradient of integral I Fv4 dr. In conclusion we mention that equation (19) is correct only for a rather high density of particle concentration. The description considered above stops being correct when the density of particles in the central part of the cloud approaches its density in the turbulent "walls". The last remark determines a limit for the longitudinal size of the cloud L* :
f ~ no(z) dz L* ~< COp
,
(33)
np
where nc is the energetic electron density in the cloud centre. For size L > L*, dispersion of particles has the character of free expansion.
(30) where Q takes into account the possible losses of particles. The typical feature of self-consistent dispersion of an energetic charged particles cloud is the formation of turbulent "walls" on its front. It is explained by the fact that there is a minimum value of particle concentration limiting an interaction by plasma oscillation. The minimum value can be found from the condition :
5. Discussion
Let us compare results of calculations for two cases : particle cloud dispersion in the plasma with a constant level of turbulence and particle cloud self-consistent expansion. The typical feature of a turbulent dispersion in both cases
790
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space
is the at first comparatively low velocity of cloud expansion, u0. Second, the velocity of interacting by waves particles is limited by the value :
of electromagnetic turbulence, leading to effective angle scattering of particles in the velocity space by whistler (Trakhtengerts, 1983 ; Bespalov and Trakhtengerts, 1986) and Alfv6n (Bespalov et al., 199l) plasma turbulence.
1 - x 2/> (Vph/V)2.
Acknowledgement. This work was supported in part by the Russian Academy of Science under research grant No. 93-022850.
(34)
In the self-consistent case for a rather high cloud density, the first condition is executed automatically : References b/0 "~ F'ph <~<~U0,
(35)
where v0 is the characteristic velocity of the particles. For electron dispersion on waves with a constant level of turbulence, the average particle velocity is determined by the energy of plasma turbulence (proportional to D) and is approximately equal to : Uo ~- D L '
(36)
where L is the characteristic longitudinal size of the cloud. For rather high values of D, the velocity of cloud expansion is much lower than the velocity of the energetic particles. A small part of the energetic electrons not satisfying the second condition (34) practically does not interact with the waves. As a result, the energetic particles flying almost along the dispersion axis can "evaporate" from the relatively slowly expanding cloud, taking a role of its harbinger. This effect increases the loss term Q in equation (30). Probably, this harbinger is responsible for a solar type III radio burst (Lin et al., 1981 ; Potter et al., 1980). The velocity of the cloud expansion depends on the regime of wave-particle interaction: in plasma with a constant level of turbulence, the velocity of cloud expansion decreases in time, while this velocity increases in time for cloud self-consistent expansion. The space dependence of the energetic electron cloud is also very different for both regimes of dispersion. In the self-consistent case there are turbulent boundaries and plateaus in the cloud centre. There are no such peculiarities for electron dispersion on waves with a constant level of turbulence. Moreover, we have in this case more significant space dispersion of the electron cloud as a result of a strong dependence of distribution functionfevolution on particle velocity. In conclusion, it is necessary to note that we find a solution for the self-consistent turbulent dispersion with a slightly anisotropic initial distribution function. But even strong initial anisotropy reduces rapidly as a result of scattering of particles by wave turbulence, and the distribution function evolution can also be described by equations (19). The main results of this paper are useful for investigation of turbulent dispersion in plasma with other types
Bahareva, N. M. and Trakhtengerts, V. Yu., Automodel solution in a turbulent model of the plasma cord, Fizika Plazmy 9, 1027-1033, 1983. Bespalov, P. A., Quasilinear relaxation of the system "beamisotropic plasma". Fizika Plazmy 3, 1118-1127, 1977. Bespalov, P. A. and Trakhtengerts, V. Yu., Turbulent dispersion of fast particle cloud in plasma. Zh. Eksp. Teor. Fiz. 67, 969978, 1974. Bespalov, P. A. and Trakhtengerts, V. Yu., Turbulent expansion of fast particles in the magnetosphere and solar wind, in Researches in Geomagnetism, Aeronomy and Solar Physics, Vol. 48, pp. 158-169. Nauka, Moscow, 1979. Bespalov, P. A. and Trakhtengerts, V. Yu., The cyclotron instability in the Earth radiation belts, in Reviews of Plasma Physics (edited by M. A. Leontovich), Vol. 10, pp. 155-292. Plenum, New York, 1986. Bespalov, P. A., Zaitsev, V. V. and Stepanov, A. V., Consequences of strong pitch-angle diffusion of particles in solar flares. Astrophys. J. 374, 369 373, 1991. Galeev, A. A. and Sagdeev, R. Z., Nonlinear plasma theory, in Voprosy Teorii Plazmy, Vol. 7, pp. 3-145. Atomizdat, Moscow, 1973. Galperin, Ju. I., Vernik, A., Dymek, M., Kutiev, I., Mnlyarchik, T. M., Serafimov, K. B., Shuiskaya, F. K. and Shylenina, R. V., Investigation of geoactive corpuscles and photoelectrons by "'Kosmos-261" spacecraft. Kosmicheskie Issledovaniya 11, 101-112, 1973. Kurt, V. G., Logachev, Ju. I. and Pisarenko, N. F., Propagation of solar electrons with energy >30 keV in interplanetary space. Kosmicheskie Issledovaniya 13, 222-235, 1975. Lin, R. P., Evens, L. G. and Fainberg, J., Simultaneous observation fast solar electrons and type III radio burst emissions near 1 a.u. Astrophys. Lett. 14, 191-198, 1973. Lin, R. P. and Potter, D. W., Energetic electrons and plasma waves associated with a solar type III radio burst. Astrophys. J. 251,364 373, 1981. Mikhailovskii, A. B., Theory ~f Plasma Instabilities, Vol. 1. Atomizdat, Moscow, 1975. Potter, D. W., Lin, R. P. and Anderson, K. A., Impulsive 2-10 keV solar electron events not associated with flares. Astrophys. J. 236, L97-L100, 1980. Rjutov, D. D. and Sagdeev, R. Z., Quasi-gasdynamical description of cloud of hot electrons in cold plasma. Zh. Eksp. Teor. Fiz. 58, 739 746, 1970. Trakhtengerts, V. Yu., Relaxation of a plasma with anisotropic velocity distribution. Basic Plasma Physics, Vol. 2, pp. 519552. North-Holland, Amsterdam, 1983. Vedenov, A. A. and Rjutov, D. D., Quasilinear effects for flux instabilities, in Voprosy Teorii Plazmy, Vol. 6, pp. 3-69. Atomizdat, Moscow, 1972.
~
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/93 $7.00+0.00
Pergamon
Slightly anisotropic energetic electron fluxes in a turbulent space plasma P. A. Bespalov ~ and V. G. Efremova 2 qnstitute of Applied Physics, Russian Academy of Science, 46 Uljanova st., 603600 Nizhny Novgorod, Russia 2Nizhny Novgorod State University, 23 Gargarin st., 603600 Nizhny Novgorod, Russia Received 14 May 1993 ; revised 27 September 1993 ; accepted 30 September 1993
Abstract. The problem of dispersion of an energetic electron stream in a turbulent space plasma without magnetic field is considered. The basis of calculation is a self-consistent system of quasi-linear equations, which take into account non-one-dimensional diffusion in velocity space. It is shown that for a rather dense electron cloud, the particle distribution function is nearly isotropic. Solutions for the electron distribution function and for the spectral density of plasma wave turbulence are found. These results were compared with the solution for the case of a constant level of plasma turbulence.
1. Introduction The problem of the dynamics of the expansion of fast charged particles in a turbulent plasma is rather important both for the physics of laboratory flux experiments and as a space phenomenon in star coronas, star wind and interplanetary magnetospheres. Expansion of energetic particles from local sources is typical of all hot space objects such as the solar corona, for example, in a rarefied space plasma, collisions are often insignificant and the dynamics of the particle flux is determined by the collective effects of interaction with plasma turbulence leading to anomalous frequency of the collisions. Under known conditions, turbulent dispersion of a fast particle cloud in a dense plasma has a diffusive character in velocity space and can be described by a set of quasilinear equations. The diffusion efficiency is controlled by the noise-wave intensity in a background plasma. For rather weak fluxes, the level of plasma turbulence is practically independent of its dynamics. For more dense fluxes, the self-consistent dependence of the waves on the particles become more significant. The particle distribution function evolution both in a
plasma with a constant level of turbulence independent of the particles and in the self-consistent case have been considered many times in the literature. For example, a comparatively simple system of quasi-gasdynamic equations for self-consistent turbulent dispersion of fast particles in a strong magnetic field (o92 << o92) was found and analysed in the paper by Rjutov et al. (1970). Here, both the plasma wave spectrum and the particle velocity-space diffusion are assumed to be one-dimensional. But in many cases, for example in a weak magnetic field, the fast particle diffusion becomes significantly nonone-dimensional. Such a problem was studied in papers by Bespalov and Trakhtengerts (1974, 1979) and Bahareva et al. (1983). A simplifying factor in this problem was the fact that for sufficiently dense fluxes, energetic particles excite plasma oscillations with phase velocity much lower than their own average velocity. As a result, the particles experience elastic collisions with plasma oscillations, and their velocity distribution function remains almost isotropic. The difficulties in such an analysis were connected with the necessity of a more correct definition of lowenergy particle diffusion. In this work we find a self-consistent set of equations determined by the distribution function or its moments (flux density, average values of velocity and square of velocity) for electron flux turbulent dispersion in a plasma. The determined solution has a rather wide practical interest for the study of matter, pulse and energy transfer in space plasma, for example for diffusive fluxes in solar wind (Lin et al., 1973; Kurt et al., 1975) after explosive particle injection, or for photoelectrons in the Earth's magnetosphere (Galperin et al., 1973). For comparison, a solution is also proposed for the electron flux in isotropic plasma with a level of turbulence independent of the particles. This is useful for a clearer determination of the effects owing to self-consistent events.
2. Basic equations Correspondence to : P. A. Bespalov
We assume that at the initial moment of time, the energetic electron cloud was formed on the plane z = 0 in a uniform
786
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space
background plasma without magnetic field. Langmuir plasma oscillations with a sufficient level of turbulence effectively scatter charged particles. The level of turbulence can be maintained both by an external source in background plasma and by a flux instability. Their own flux evolution is determined by the collective processes of interaction with Langmuir turbulence. One-dimensional and symmetric (in coordinate space) dispersion of the electron cloud is described by the following set of quasi-linear equations (Vedenov et al., 1972 ; Galeev et al., 1973 ; Bespalov and Trakhtengerts, 1979) :
e and m are the charge and the mass of an electron, vg= = Oto/Okz, Vph = to~k, Vk is the linear decrement includ-
ing both the Landau damping in the background plasma and the collisional damping. Note that the initial system (1) can be used for describing the particle dispersion in a plasma with weak magnetic field when 7k > to2 (Mikhailovskii, 1975), where tOBis the electron cyclotron frequency. We first use equations (1)-(4) for analysis of the fast electron cloud dispersion by plasma oscillations which are independent of the electron cloud spectrum.
Of + v x ~f 1 0 2( OJ" 1 c~/~) ~t Oz - v 2 Ov t' D''~vv +D~'xv
+v~-x
3. Dispersion of particle fluxes by plasma turbulence with known spectrum and constant level
D ' " ~ v + D . . . v. . . ~k
~k
& +v.:~z = 2(~k--V0~.~,
(1)
with diffusion coefficients : D~x
=
--
Such a situation is realized, for example, in weak electron fluxes in disturbed solar wind, or in turbulent transition regions of planetary magnetospheres. For simplicity, we assume that the wave energy spectrum density ek is constant and isotropic in the wavevector space function with small dispersion of k, and hence :
Re
gk = g , 3 ( k , -
D ,: ,:
ekk dk dy v{1 - x 2 _ y2 _ (to/kv)[(to/kv) - 2xy]} 1/2
1),
(5)
where : (2)p Uph* ~ t . << U0'
x
kv
Y-
,
(2)
( y_ tox 2
(6)
Here, v0 is the average velocity of energetic electrons. For the spectrum (5) the diffusion coefficients (2) given by: D,.,, ~D 2 U3 /'5p h*
Di,x = 0, and growth rate : =nD }'k
=
np
rr 2
fo f , IO
X2
I
/-)ph, 2
(7)
dv dx where D = D , E [ ( v / V p h , ) - 1 ] , D , = (4ne/m)2k2,eo, and:
1/
kv "X a-]
× R e {1 - x 2 - y 2 - ( t o / k v ) [ ( t o / k v )
-2xy]}
t, ~xx x=+, = 0 ,
{
1,x~> 0,
1/2"
E(x) = O,x < O.
(3)
The distribution functionf(t, z, x, v) satisfies the boundary conditions : D"xbx+
1
(4)
which provide the absence of particle flux through the boundary of the distribution function domain. In formulas (1)-(4), v = ]vl is the velocity of the electrons, x = Vz/V is the cosine of the angle between the velocity and the direction of the dispersion, et(t, z) is the spectral energy density of the plasma oscillations, k is the wavevector, y = k : / k is the cosine of the angle between the wavevector and the direction of dispersion, tOp is the plasma frequency, np is the background plasma density,
For a sufficiently high level of plasma turbulence that satisfies expression (6), the distribution function is maintained as quasi-isotropic. This permits us to write the distribution function in the form of a sum average over the angle main part F and small anisotropic • : f = F + ~,
(8)
where : F = (1/2)
£
fdx,
IOI << F.
From the initial set (1) we can get an equation for the isotropic part F, taking into consideration equations (7) :
787
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space ~zD 0F
O2F
v 2 c32F
~v ~t + t3t2
3 ~z 2
=
ltD ',, ~ / I Lat +
2 ev Y &, t,2jj~ -
t,' L \ 7 , / 0 v
~vv 0v
-~t
"
(9)
/
According to condition (6) for the main stock of particles, the ratio Vph,/Vis rather small. Thus, in the zeroth order of the perturbation theory, for that parameter we have instead a simpler equation, disregarding energy diffusion" t~2 - 0,
(10)
where T = (2nD,/v3)t, ~ = (2x/3rtD,/v4)z. The solution of equation (10) for the initial problem has the form"
1 r +' f i s h ( 2 x/l-2q~)+ch( 2 x//ll_~)l~(q, v)
r sh(2 +2
5
}
lx/1-q ) ~ ~k(q,v)
e x p ( - 2 ) c o s ( q ~ ) d q.
(11)
x/l~q 2
Here, 4~(q, v) and q~(q, v) is a Furje transform of the functions ~b(~, v) and qJ(~, v), which are determined by the initial conditions" ~b(¢,
v)= F
qJ(¢, v) =
6 z/z0
632F
t3F ~2F c3z + ~z 2
t = 0, 2,v/~tD * 4, v ,
I,=0,~P/2,/s~t~,)~,,, .
(12)
For example, we represent the solution of this problem by the following initial distribution :
Fig. 1. The space dependence of distribution function (14) for two moments of time
where again z = (2rtD,/v3)t, ~ = (2x/3~zD,/v4)z. The space dependence of the solution (14) for fixed velocity and several successive moments of time is presented in Fig. 1.
4. Self-consistent turbulent dispersion of a dense fast particle cloud
We now consider the problem of a one-dimensional (in coordinate space) self-consistent expansion of an energetic electron cloud for the case of turbulence excited by the cloud itself. As was mentioned in the Introduction, one important example of such a process is relaxation of postburst electron fluxes in solar wind. We use the set of equations (1) as initial equations describing such a dispersion. After averaging of the first equation over x and v, taking into account the boundary conditions, we find the following formulas for gasdynamic values" ~n
~(~,v) -- 0.
c~
~ + & (nu) = 0,
qS(~,v) = 6(~)Fo(v), (13)
In this case, the evolution of the distribution function is determined by the expression :
e
fft(nu) + ~zz(nu2) =
/eVnpf
--321t~m ) ~303
3t~t (nu2)+ ~z 4rt
dx 1
F(t,z,v) -
F0(v)
4
/'1 ~ ' ~
I°~2 ~ / ~ - ~ )
=-64n
1~_~
! exp(--2)'
~/~-~
J
(14)
m
'f[
1 dy
dkgk])kyk 3,
f0 dvxvSf ) fo dkekYkk2.
-~60p
1
dy
(15)
j ' f d y is the concentration of energetic elecHere, n trons, u = (1/n) Jvzfdv is the average velocity of cloud particles in the direction of dispersion and u 2 = (l/n) Sv2f dv is the average square of velocity in the direction of dispersion. To acquire a closed set of equations for the isotropic
788
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space
part of the distribution function, we use the following simplifying conditions. First, the velocity distribution of a rather dense electron cloud, produced by effective angular scattering, remains almost isotropic and can be represented by equation (8) (Bespalov and Trakhtengerts, 1979). Second, according to the expression for the growth rate (3), slightly anisotropic fluxes of energetic particles generate waves with relatively low phase velocities :
O(t, z, x, 0 is related to the function F by : 1
O(t, z, x, () - 1 - x 2 ~g(t, z, (), *~)(t,z, () = ~' [5Vph
where v0 is a characteristic particle velocity. Under these conditions, the character of the dependence Ofgk from value k does not produce a significant influence on the diffusion coefficients in the set of equations (1), and we can use for the calculation a wave spectrum in a form like equation (5) with variable e, and k,. Third, for a sufficiently large value of growth rate (for anisotropies close to unity the growth rate 7k is much higher than Vk), the intensity of plasma wave can be determined at a quasi-stationary level (Bespalov and Trakhtengerts, 1979) :
5Vph
~zz
]
(20)
For solving equation (19) it is necessary to use the following from the equation (17) conditions: 27
(21)
F(Uph) ~ -•. Uph
Expression (21) has the sense of the boundary conditions for equation (19). We shall search for a solution of equation (19) in the form of a series : F=
-c 1 ~=on!a.(t,z)(~yo)"exp(--(7o),
( = v 5.
(22)
~k -- Yk ~--"0,
(17a)
- - (7k--Vk) = 0, ~Uph ~2 aUph (Tk--Yk) < 0.
(lYb)
Using expression (22) we can write equation (19) as a system for coefficients a.(t, z) :
(17c)
,0tan
Equation (17a) with the expression (3) for the growth rate represents an Abelian integral equation in terms of y, and enables one to find the angular dependence of the anisotropic part of the distribution function • (Bespalov and Trakhtengerts, 1979):
a¢ 0x
o; -
(18)
1 - x 2'
4 c3z L 4
ao, phsign(Z \,,= 0 0 z / J
~/azl~-zSlgn/2_, ~-//Vph 2.,~zz- / \,=o eZ / L ,,=0 d i 1 5 271 , = o n! a,(Vph)'O)" exp ( -- VpSh)-- Vp3h.
a,(t = 0,z) =
(23)
~ bm(z) ( - 1)" ~=.~ (m-n)!n! '
(24)
where bm are coefficients of the Lagerr polynomial series represented by the function F at the moment t = 0 :
¢ ; = axx x=o
Taking into consideration the simplification mentioned above, we can reduce formulas (1), (15), (17) and (18) to the equation for the isotropic part of the distribution function F: c~F 3F//2
)=0,
Initial conditions for the functions a, are determined by the following expressions :
where
5
d~
(16)
Uph << U0,
c~F
sign
I,~'C~Fd#'] ' ]
bm(2 ) =
f;
F(t = O, z, ~)Lm(#7o) d(.
(25)
For example, we consider the problem of electron cloud dispersion with the following initial distribution :
/
1
F(t = O,z, () = n!a,(z, t = 0)((y0)"exp (-(70). (26) 4
+
For this case we have a simpler system for a,(t,z) and vph(t, z) :
~zz+2~zz ~
aa. c~t
h
+ ::z(VphffFd#)][ff OFd#lJ -'}oz= 0.
(19)
3 0( ~a.~ 4 Oz a"Vohsign 3z ]
3+5n 20 r/ x ~z vph2sign
= 0,
Here, Yon is the phase velocity of plasma waves, q = Vnp/Z2~p,
( = v ~.
The anisotropic part of the distribution function
l 5 2r/ n! a,(Vph?0)"exp (-- VpShT0)= _.~. Uph
(27)
789
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space U fSn+3)l/s, l' Vphmax: ( 570 J
\ I "O O
i
0 Space coordinate
Fig. 2. The qualitative space dependence of energetic electron density for self-consistent expansion for three successive moments of time
At points with nonzero values of equations (27) is defined by :
(Oa,/Oz), the solution of
a, = 2tln!y~OVphCSn+3) exp (VpShY0), Uph = I'I'/[Z"t- V(Vph)t].
/' 5e5 ~
a,min =
2k~)n!tlYo3/5.
If a, < anmin, then the particle dispersion becomes close to that for the free case, increasing the loss Q-factor in expression (30). In the case of an arbitrary initial function F the interaction between energetic modes is significant. For this case we limit ourself to a qualitative consideration. It is _apparent from expressions (23) that the integrals j a, dv and I v2a, dv, corresponding to the whole number of particles and the kinetic energy of a definite energetic mode are conserved. Thus, only a space redistribution of energetic modes takes place. The characteristic velocity of certain energetic mode particles' dispersion rises with an increase of their number. Turbulent "walls" therefore are established in the main by highly energetic modes and expanding plateaus in the centre of the cloud-by a zero mode. The equation (19) describing the evolution of the distribution function can be used for the investigation of matter, pulse and energy transfer. The wave spectrum generated by the flux of particles is also of significant interest. The question about the plasma wave spectrum was considered in the paper by Bespalov and Trakhtengerts (1979). Taking into consideration only angle diffusion, which is correct for rather dense cloud, the authors find an expression for energy density ek :
(28)
ek- 3n3e2k.Y2D(t,z)6(k-k.),
Here
V(vph) = a415n+ 2-- 5,oV~h
~:k=0,
s , exp (-- VpSuT0)1 + ~3 + 5 n (VphT0)
)<
VphSign \ OZ J"
The function U?(z) is an initial distribution of phase velocity Vphand is related to a,(t = 0, z) by :
a,(t = O, z) = 2rln!7"o[Ud(z)]-~5,+ 3)exp [70~5(z)].
(29)
The qualitative pattern of cloud expansion is shown in Fig. 2. An expanding plateau is formed at the centre of the cloud. Its boundaries z = z* are defined from the condition of the preservation of the whole number of particles : Q+2
F(t,z)dz+F(t,z*)2z* =
F(t =
(31)
0, z) dz,
for
for
y
yffv4~zdV>O.
v4
dv < 0,
(32)
Here function D(t,z), determining the intensity of plasma waves according to equation (17), is proportional to the gradient of integral I Fv4 dr. In conclusion we mention that equation (19) is correct only for a rather high density of particle concentration. The description considered above stops being correct when the density of particles in the central part of the cloud approaches its density in the turbulent "walls". The last remark determines a limit for the longitudinal size of the cloud L* :
f ~ no(z) dz L* ~< COp
,
(33)
np
where nc is the energetic electron density in the cloud centre. For size L > L*, dispersion of particles has the character of free expansion.
(30) where Q takes into account the possible losses of particles. The typical feature of self-consistent dispersion of an energetic charged particles cloud is the formation of turbulent "walls" on its front. It is explained by the fact that there is a minimum value of particle concentration limiting an interaction by plasma oscillation. The minimum value can be found from the condition :
5. Discussion
Let us compare results of calculations for two cases : particle cloud dispersion in the plasma with a constant level of turbulence and particle cloud self-consistent expansion. The typical feature of a turbulent dispersion in both cases
790
P. A. Bespalov and V. G. Efremova : Slightly anisotropic electron fluxes in space
is the at first comparatively low velocity of cloud expansion, u0. Second, the velocity of interacting by waves particles is limited by the value :
of electromagnetic turbulence, leading to effective angle scattering of particles in the velocity space by whistler (Trakhtengerts, 1983 ; Bespalov and Trakhtengerts, 1986) and Alfv6n (Bespalov et al., 199l) plasma turbulence.
1 - x 2/> (Vph/V)2.
Acknowledgement. This work was supported in part by the Russian Academy of Science under research grant No. 93-022850.
(34)
In the self-consistent case for a rather high cloud density, the first condition is executed automatically : References b/0 "~ F'ph <~<~U0,
(35)
where v0 is the characteristic velocity of the particles. For electron dispersion on waves with a constant level of turbulence, the average particle velocity is determined by the energy of plasma turbulence (proportional to D) and is approximately equal to : Uo ~- D L '
(36)
where L is the characteristic longitudinal size of the cloud. For rather high values of D, the velocity of cloud expansion is much lower than the velocity of the energetic particles. A small part of the energetic electrons not satisfying the second condition (34) practically does not interact with the waves. As a result, the energetic particles flying almost along the dispersion axis can "evaporate" from the relatively slowly expanding cloud, taking a role of its harbinger. This effect increases the loss term Q in equation (30). Probably, this harbinger is responsible for a solar type III radio burst (Lin et al., 1981 ; Potter et al., 1980). The velocity of the cloud expansion depends on the regime of wave-particle interaction: in plasma with a constant level of turbulence, the velocity of cloud expansion decreases in time, while this velocity increases in time for cloud self-consistent expansion. The space dependence of the energetic electron cloud is also very different for both regimes of dispersion. In the self-consistent case there are turbulent boundaries and plateaus in the cloud centre. There are no such peculiarities for electron dispersion on waves with a constant level of turbulence. Moreover, we have in this case more significant space dispersion of the electron cloud as a result of a strong dependence of distribution functionfevolution on particle velocity. In conclusion, it is necessary to note that we find a solution for the self-consistent turbulent dispersion with a slightly anisotropic initial distribution function. But even strong initial anisotropy reduces rapidly as a result of scattering of particles by wave turbulence, and the distribution function evolution can also be described by equations (19). The main results of this paper are useful for investigation of turbulent dispersion in plasma with other types
Bahareva, N. M. and Trakhtengerts, V. Yu., Automodel solution in a turbulent model of the plasma cord, Fizika Plazmy 9, 1027-1033, 1983. Bespalov, P. A., Quasilinear relaxation of the system "beamisotropic plasma". Fizika Plazmy 3, 1118-1127, 1977. Bespalov, P. A. and Trakhtengerts, V. Yu., Turbulent dispersion of fast particle cloud in plasma. Zh. Eksp. Teor. Fiz. 67, 969978, 1974. Bespalov, P. A. and Trakhtengerts, V. Yu., Turbulent expansion of fast particles in the magnetosphere and solar wind, in Researches in Geomagnetism, Aeronomy and Solar Physics, Vol. 48, pp. 158-169. Nauka, Moscow, 1979. Bespalov, P. A. and Trakhtengerts, V. Yu., The cyclotron instability in the Earth radiation belts, in Reviews of Plasma Physics (edited by M. A. Leontovich), Vol. 10, pp. 155-292. Plenum, New York, 1986. Bespalov, P. A., Zaitsev, V. V. and Stepanov, A. V., Consequences of strong pitch-angle diffusion of particles in solar flares. Astrophys. J. 374, 369 373, 1991. Galeev, A. A. and Sagdeev, R. Z., Nonlinear plasma theory, in Voprosy Teorii Plazmy, Vol. 7, pp. 3-145. Atomizdat, Moscow, 1973. Galperin, Ju. I., Vernik, A., Dymek, M., Kutiev, I., Mnlyarchik, T. M., Serafimov, K. B., Shuiskaya, F. K. and Shylenina, R. V., Investigation of geoactive corpuscles and photoelectrons by "'Kosmos-261" spacecraft. Kosmicheskie Issledovaniya 11, 101-112, 1973. Kurt, V. G., Logachev, Ju. I. and Pisarenko, N. F., Propagation of solar electrons with energy >30 keV in interplanetary space. Kosmicheskie Issledovaniya 13, 222-235, 1975. Lin, R. P., Evens, L. G. and Fainberg, J., Simultaneous observation fast solar electrons and type III radio burst emissions near 1 a.u. Astrophys. Lett. 14, 191-198, 1973. Lin, R. P. and Potter, D. W., Energetic electrons and plasma waves associated with a solar type III radio burst. Astrophys. J. 251,364 373, 1981. Mikhailovskii, A. B., Theory ~f Plasma Instabilities, Vol. 1. Atomizdat, Moscow, 1975. Potter, D. W., Lin, R. P. and Anderson, K. A., Impulsive 2-10 keV solar electron events not associated with flares. Astrophys. J. 236, L97-L100, 1980. Rjutov, D. D. and Sagdeev, R. Z., Quasi-gasdynamical description of cloud of hot electrons in cold plasma. Zh. Eksp. Teor. Fiz. 58, 739 746, 1970. Trakhtengerts, V. Yu., Relaxation of a plasma with anisotropic velocity distribution. Basic Plasma Physics, Vol. 2, pp. 519552. North-Holland, Amsterdam, 1983. Vedenov, A. A. and Rjutov, D. D., Quasilinear effects for flux instabilities, in Voprosy Teorii Plazmy, Vol. 6, pp. 3-69. Atomizdat, Moscow, 1972.