Dynamics of a fast Maxwellian electron cloud in coronal plasma

Transcription

1 Radio Science, Volume 36, Number 6, Pages , November/December 2001 Dynamics of a fast Maxwellian electron cloud in coronal plasma E. P. Kontar 1 Institute of Theoretical Astrophysics, University of Oslo, Oslo, Norway V. N. Mel'nik Institute of Radio Astronomy, Kharkov, Ukraine V. I. Lapshin National Science Center "Kharkov Institute of Physics and Technology," Kharkov, Ukraine Abstract. Dynamics of a hot electron cloud with an initially Maxwellian electron distribution is considered both numerically and analytically. It is shown that only a small group of the electrons propagates into the plasma but the main group of the electrons is concentrated near the initial location of the cloud. The distribution of the electrons left presents a plateau with the maximum velocity growing linearly with distance. 1. Introduction In the study of active processes occurring in an astrophysical plasma, for example, solar flares, acceleration of particles in shock waves and in strong electrical fields [Benz, 1993; McLean and Labrum, 1985], pulsar magnetospheres [Manchester and Taylor, 1977], planetary foreshocks [Muschietti et al., 1996], etc., there is a problem about behavior of a small group of fast electrons in relatively dense and cold plasma [Sigov and Levchenko, 1996]. The main interaction with the ambient plasma is the electron - Langmuir wave interaction, where electrons and protons of the surrounding plasma play the role of background media for Langmuir waves. In the case of an inequality En << Tn, where n and n are the densities of fast electrons and plasma, respectively, and E and T are the energy of fast electrons and the temperature of plasma, respectively, analysis is usually carried out in the framework of the weak turbulence theory [ Vedenov et al., 1962; Drummond and Pines, Also at National Science Center "Kharkov Institute of Physics and Technology," Kharkov, Ukraine. Copyright 2001 by the American Geophysical Union. Paper number 2000RS / 01 / 2000RS $ ]. If the initial distribution function of electrons fo(v) has positive derivative Ofo(v)/Ov > 0, as is known [Vedenov and Ryutov, 1975], the generation of Langmuir waves takes place as a result of instability. These waves influence electrons in such a manner that the electron distribution function becomes a plateau when the beam is considered to be in every spatial point [Vedenov et al., 1962]. At the same time, the case of a spatially limited beam of particles present special interest for astrophysical applications [Zaitsev et al., 1972]. In a general statement the problem becomestrongly complicated [Melrose, 1990; Muschietti, 1990]. However, there are circumstances allowing us to simplify the problem. First, because of the finite beam size, electrons pass a given point for finite time, and therefore only the first onedimensional stage of electron relaxation can be taken into account [Churaev and Agapov, 1980]. In addition, one dimensionality can be supported by the longitudinal magnetic field, which suppresses waves to spread at an angle to the axis of the beam i Vedenov et al., 1962]. The second simplifying circumstance is connected with the time of quasi-linear relaxation r, which, especially in astrophysical situations, is much less than the time of electron propagation t: r << t. This fact enables us to proceed from a kinetic de- scription to a gas dynamic one [Vedenov and R yutov, 1975; Mel'nik, 1995] and to find the analytical solu- 1757

2 1758 KONTAR ET AL.: DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD tion in the case of an initially monoenergetic beam [Mel'nik, 1995; Kontar et al., 1998; Mel'nik et al., 1999]. According to this solution, electrons propagate in a plasma as a beam-plasma structure, which consists of electrons and plasmons and moves with constant speed [Mel'nik, 1995; Kontar et al., 1998]. If the initial electron distribution function has a negative derivative Ofo(v)/Ov < 0, at first sight the situation is much different. Such an electron distribution is stable, and therefore plasma waves should not be raised, while initially unstable distribution immediately leads to generation of waves. This is valid when this distribution function is presented in all space. However, if the fast electrons occupy a limited spatial region the positive derivative Of/Ov > 0 appears when fast particles overtake slow ones. This results in generation of waves and the establishment of a plateau in the electron distribution function. Thus the fast particle propagation constantly removes the system from an equilibrium state, and quasi-linear relaxation, on the contrary, tends to restore the steady state. In this work on example of initial MaxwellJan electron distribution, quasi-linear electron dynamics in cold plasma is studied. It is shown that inhomogeneity of quasi-linear relaxation causes a small part of the electrons to be separated from the main part. The particles remaining in the injection region are "locked" by the Langmuir waves. 2. Numerical Solution of Kinetic Equations of Quasi-linear Theory In the framework of quasi-linear theory the electron distribution function f(v, x, t) and the spectral energy density of Langmuir waves W(v,x,t) in the one-dimensional case are described by the system of kinetic equations [ Vedenov and Ryutov, 1975] Of Of 4 v 2e 2 0 q W Of O- + v xx = m 2 O v v O v ' (1) ow _ of Ot - 7rø P n v2 W vv' O. = Jp kv. (2) We solve the initial value problem when at the initial time moment a cloud of fast electrons is located in an area with linear size d, and the initial distribution function is Maxwellian f(v,x,t =0) = / vo no exp (-x 2/d 2) exp -, v>0. (3) In (3), no is the density of fast Maxwellian electrons, and v0 is the thermal velocity of these electrons. The spectral energy density of plasma waves at the moment t = 0 is chosen to be thermal and homogeneous in space WT(V, x, t = 0) = 10-5 notevt e. (4) The numerical solution of the system of equations (1) and (2) is presented for the following parameters of plasma and electron cloud: n = 10 s cm -3, VTe = 4 X 10 s cm s -, no = 2 x 103 cm -3, v0 = 4 x 10 ø cm s -i, and d = 3 x 10 ø cm. These values are characteristic for the conditions in the solar corona at the generation of radio bursts by the fast electrons. The basic results of the numerical solution are presented in Figures 1-5. In Figure I the dependence of electron cloud density is shown at various time moments. It is seen that from the moment of time t m 0.8 s the concentration profile has a bend: a group of fast electrons begins to separate from the basic group. Two groups of electrons become more and more obvious with time. The full separation is considered to take place at the distance m 3d at the moment t m 1.5 s. Fastest electrons propagate into the plasma as a separate group. They generate a high level of waves at every spatial point and a plateau is formed at the distribution function with constant maximum velocity Umax. This state is realized at distances x > 3d; thus Umax 2.5v0 (see Figure 2). The other group of more slow electrons is concentrated in the region of initial cloud location. In this area a plateau is also formed at the electron distribution function (Figure 3) because of an quasi-linear relaxation, and the plateau height quickly decreases with x (Figure 4). At the same time, maximum velocity of the plateau grows linearly with coordinate, reaching the maximum value Umax at x 3d (Figure 2). In Figure 5 we see that the maximum of the wave energy density is shifted deeper into the plasma in comparison with that of electron energy density. The numerical solution also shows that starting from t m 1.5 s (and at least up to the end of calculations t 10 s) the decrease of plateau height and maximum velocity of the plateau for x < 3d occurs very slowly. 3. Qualitative Consideration Since the initial distribution is steady and Langmuir waves are at the thermal level, at first, electrons 0 p

3 KONTAR ET AL' DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD 1759 t=0.3 s t:0.75 s 1' t=l.02 s t=l.5 s -3 6 ß I ' 1'2 1' ( 1'2 distance x/d distance x/d Figure 1. Distribution of electron cloud density in space at various moments of time t=1.5 s O.5 / / 0.0, I, I t I I s I distance x/d Figure 2. Maximum velocity of the plateau at various distances x.

4 1760 KONTAR ET AL.- DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD x=1.33d x=1.72d, t=l.o2 s o.oo o.o velocity v/v o Figure 3. Electron distribution function f(v, x, t) in the area x < 3d. move freely, and consequently, from (1) and (3) we have for the electron distribution function f(v,x, t) ' (5) For x > 0 the electron distribution function has positive derivative Of/Or (Figure 6). Fast electrons begin to overtake slow particles. From the condition Of/Or = 0 and (5) one finds the equation of the curve that divides areas with Of/Ov > 0 and Of/Ov < 0 x - t +. (6) In Figure 7 the curve (6) is shown for various velocities vz > v2 > v3. The derivative Of/Ov becomes positive for small velocities and later for faster electrons. At a given point x the maximum velocity when Of/Or > 0 is reached at to = d/vo. The positive t=1.5 s 0.8 x, 0.6,,, Q)..c 0.4. Q) n , I,? ß distance x/d Figure 4. Plateau height for x < 3d as a function of distance. The curve represents the theoretical expression (19).

5 KONTAR ET AL.' DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD ,5 t=1.5 s 0,4 0,3 0,2 0,1 / / \ '"""Electron energy density / Wavenergy density I i// I //! // i//! z// ",/ i:/i! 1 I I / / 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 distance Figure 5. Spatial distribution of energy density: electrons (triangles) and waves (dimonds). The dash curves show the corresponding theoretical results (22) and (23). x/d derivative is the condition for quasi-linear relaxation development. Since only a small part of all particles takes part in the relaxation, the plateau is considered to be established up to the velocity v = u(x, t), which can be found from equality (see Figure 6) f (v - O, x, t) f(u(x, t), x, t) and determined by the expression u(, t) tvø + Vo t. (8) The maximum value of u(x, t) (at the moment of time to - d/vo) grows linearly with distance (compare fig. ) u(x) = vox/d. (9) Taking into account (8), it is possible to estimate the quantity of particles participating in quasi-linear relaxation,(,t) ' f If(v, x, t) - f(0, x, t)] v. 0 (10) The time of quasi-linear relaxation r = (a pn'/n) -1 becomes f(v,x,t) p(x,t) 0 u(x,t) v Figure 6. Unstable electron distribution function during the free electron propagation.

6 1762 KONTAR ET AL.- DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD V 1 V 3 to=d/vo Figure 7. Areas with positive (above the curve (6)) and negative (below curve (6)) derivatives Of/Ov for different velocities Vl > v: > va. far from (x > > d) and near to (x - 0) the place of injection is connected with the fact that in the first case the number of fast (v >> v0) particles is small and in the second case the number of slow (v < v0) electrons is large. As a result in both cases and n' are small values. At large x the curve x(t) has an asymptote where (13) -o- vo /ln (v/ /;o/ro) 3vo. (14) This means that the particles with velocity v > uo (in the tail of the Maxwellian distribution) practically do not take part in quasi-linear relaxation and propagate freely. At the same time, quasi-linear relaxation plays an important role for electrons with v _< vo at distances x > Xmin (Figure 8), where v/d 2 + v t 2 exp (x 2 / d 2 ) 12V1 /;2 -!, (11) X {exp [d2(d2+v /;2)]-1} where r0 = (copn0/n) -1. It is natural to consider that the relaxation influences the dynamics of electron propagation when r is compared to the time of propagation t r(x, t) = t. (12) The curve x(/;) where equality (12) takes place is presented in Figure 8. The long quasi-linear time (11) Xmi n -- X(/;min) d, (15) /;min "(Tod2/v )1/3 For small x we have the border 0.04 s. (16) Xlow --0 dv/tovo/d, (17) behind which the relaxation is also absent. Let us discuss the propagation of electrons in the area (x < 3d) where part of the electron cloud separates (Figure 1). After the initial stage of free spread, a plateau from v 0 up to v u(x, t) is established at the distribution function at /; /;min. Further, 0 tmi n to Figure 8. Curve r(x,t) = t (12) separating the area of free electron propagation (r > t, open area) from the area of quasi-linear relaxation (r < t, hatched area).

7 KONTAR ET AL.: DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD 176:] no coming electrons have a positive derivative Of/Ov for _ --exp (-x2/d2), (19) large velocities, and therefore the plateau becomes v0 wider (equation 8). The largest width is reached at which is found to be in good agreement with the the moment to = d/vo. Thus, as is pointed out in numerical solution (Figure 4). One derives the cor- (9), u(x) = vox/d. The level of Langmuir turbulence responding spectral energy density from (5) and (19) grows and reaches the maximum at to = d/vo. As is seen from Figure 5, the plasma waves are basi- x, to) - cally concentrated in the vicinity x 2d. At this v u(x,to) point, electrons taking part in relaxation have the maximum velocity v u(x - 2d) 2v0. Therefore x/ f f(v'x'tø)dv- u(x, to) f(v,x, to) dv particles with v > 2v0 pass this point not interacting o o with waves. These electrons are involved in the re- laxation when the quasi-linear time r - (a pn,/n) -1, where n, 0 f f(v,x - O,t - O)dv 0.02n0, (18) 2vo becomes less than the time of propagation t. The maximum velocity for these electrons, as was noted above, is equal to u0. The group of electrons separated propagates with constant velocity equal to a half-maximum velocity. These electrons are accompanied by Langmuir waves and form a beam-plasma structure [Mel'nik, 1995; Kontar et al., 1998; Mel'nik et al., 1999]. What occurs with the electrons located in the area x < 3d? These particles appear to be "locked" by Langmuir turbulence. If there were no waves, electrons could propagate further in the plasma. How- ever, there is a significant level of Langmuir turbulence at the point x = 2d and this means that as soon as the fast electrons begin to overtake slow particles, quasi-linear relaxation instantly (since r - W -1 in the presence of turbulence) slows down the fast particles. As a result the energy of the group of fastelectrons is transformed into waves, and this process repeats at the following moment. Therefore the Langmuir turbulence generated by the fast electrons causes the group of electrons to be "locked". It is believed that if other nonlinear processes resulting in conversion of plasma waves into emission (l+i - t+i, l + l - t) or in scattering out of resonance with beam (l + i - l + i) [Muschietti and Dum, 1991] are taken into consideration, the conditions for the formation of a new group of electrons may appear. The basic group of electrons in the area x < 3d is close to a stationary state. The height of the plateau in this area is defined by the expression 1 p(x, to) -- U(X, to) u(x,to) o f (v, X, to)dv " mnøv4 Gp exp (-x2/2d 2) 1 I - - erf 1 ] --, where 0 exp (v/vo) (20) is the probability integral. The energy density distribution of electrons and plasmons follows - f(v, x, to)v2dv o 0'2mnovo exp (-x2/d2), (22) u(x,to) E, - Wp / W(v,x, to)v-2dv o O.lmnov ( )3 x exp (-x /2d ) (23) and is in agreement with the numerical results (Figure 5). 4. Conclusions Thus, despite the fact that two-stream instability does not initially take place, further along, the condition for the instability appears because fast electrons overtake slow ones. As a result, a group of fast electrons separates from the main group of electrons, and a beam-plasma structure [Mel'nik, 1995] propagating into plasma is formed. At the same time there is Langmuir turbulence excited by the electrons, which "locks" the electrons remaining at the site of injection. Beam-plasma structure consists of electrons and plasmons [Mel'nik, 1995; Kontar et al., 1998]. Acknowledgments. Authors are pleased to thank Supercomputing Systems AG, Switzerland and Martin Frey for access to the supercomputer GigaBooster. One of the authors (E.P.K.) thanks Ukrainian Office of Inter-

8 1764 KONTAR ET AL.: DYNAMICS OF A FAST MAXWELLIAN ELECTRON CLOUD national Science Foundation for financial support (grant PSU082125). References Benz, A.O., Plasma Astrophysics: Kinetic Processes in Solar and Stellar Coronae, Kluwer, Dordrecht, Churaev, R.S., and A.V. Agapov, Three-dimensional quasi-linear relaxation of an electron beam in a plasma, Plasma Phys. Rep., 6, , Drummond, W.E., and D. Pines, Non-linear stability of plasma oscillations, Nuclear Fusion, $uppl., part 3, , Kontar, E.P., V.I. Lapshin, and V.N. Mel'nik, Numerical and analytical study of the propagation of a monoenergetic electron beam in a plasma, Plasma Phys. Rep., 2J, , Manchester, R.N., and J.H. Taylor, Pulsars, Freeman, San Francisco, Calif., McLean, D.J, and N.R. Labrum, Solar Radiophysics Cambridge Univ. Press, New York, Mel'nik, V.N., "Gas-dynamic" expansion of a fastelectron flux in a plasma, Plasma Phys. Rep., 21, 89-91, Mel'nik, V.N., V.I. Lapshin, and E.P. Kontar, Propagation of a monoenergetic electron beam in the solar corona, Sol. Phys., l&i, , Melrose, D.B., Particle beams in the solar atmosphere- General overview, Sol. Phys., 130, 3-18, Muschietti, L., Electron beam formation and stability, Sol. Phys. 130, , Muschietti, L., and C.T. Dum, Nonlinear wave scattering and electron beam relaxation, Phys. Fluids B, 3, , Muschietti, L., I. Roth, and R.E. Ergun, On the formation of wave packets in planetary foreshocks, J. Geophys. Res., 101, 15,605-15,614, Sigov, Y.S., and V.D. Levchenko, Coherent phenomena in the relaxation of a diffuse electron beam in open plasma systems, Plasma Phys. Rep., 23, , Vedenov, A.A., and D.D. Ryutov, Quasilinear effects in two-stream instabilities, in Reviews of Plasma Physics, 6, edited by M.A. Leontovich, pp. 1-76, Consultants Bureau, New York, Vedenov A.A., E.P. Velikhov, and R.Z. Sagdeev, Quasilinear theory of plasma oscillations, Nuclear Fusion Suppl., part 2, , Zaitsev, V.V., N.A. Mityakov, and V.O. Papoport, A Dynamic theory of type III solar radio bursts, Sol. Phys., 2J, , E. P. Kontar, Institute of Theoretical Astrophysics, P.O. Box 1029 Blindern, Oslo 0315, Norway. (eduardk@ ast to. uio. no) V. I. Lapshin, National Science Center "Kharkov Institute of Physics and Technology," I Academicheskaya str., Kharkov 61108, Ukraine. (lapshin@kipt.kharkov.ua) V. N. Mel'nik, Institute of Radio Astronomy, 4 Krasnoznamennaya str., Kharkov 61002, Ukraine. (melnik@ira.kharkov.ua) (Received May 14, 2000; revised November 24, 2000; accepted February 8, 2001.)