Particle dynamics in reconnection field configurations

P&n&. Space Sci. Vol. 29, No. 4, pp. 391-397, FYinted in Northern Ireland

PARTICLE

0032~633/81/040391~7$02.00/0 0 Pergamon Press Ltd.

1981

DYNAMICS IN RECONNECTION CONFIGURATIONS

I.!3. WAGNER*,

P. C. GRAYZ, J. R RAN’,

(Received

FIELD

T. TAJJMAt and S.-l. ARASOFU’

21 October 1980)

Abstract-Trajectories of test particles are studied numerically in two types of reconnection magnetic field configurations, a single X-line magnetic field configuration and a tearing magnetic field configuration. Both adiabatic and nonadiabatic motions are examined, with special emphasis on net energy gain and time spent in the neutral line regions. They spend typically one characteristic gyroperiod in the X-line region and are ejected predominantly along field lines in the vicinity of the separatrix. Both adiabatic and nonadiabatic test particles in the tearing-type field configuration are channelled into and accelerated along the O-line region. It may be inferred from these test particle results that particle energizations are significant along the O-line region, but not along the X-line region. These results are

in qualitative agreement with those obtained by a self-consistent 1. INTRODUCTION

A numerical study of trajectories of test particles can provide some qualitative information on particle dynamics in complicated magnetic field configurations, such as in the X-line and the tearing geometries. However, since the test particle approach is not self-consistent, the results will be only qualitatively useful in discovering the physics of plasma behavior in self-consistent fields of similar geometry. The purpose of this paper is twofold. The first is to study the behavior of adiabatic and nonadiabatic particles in both a single X-line and tearing magnetic field configurations which are given and static. The second is to combine our results with those obtained by a self-consistent study. We are interested in both the average energy gains and the lifetime of particles in the X-type and O-type neutral line regions. Such information is useful in analyzing plasma energization in the Xline region of the reconnection configuration. In the single X-line model there are only two obvious constants of motion, the total energy and the canonical Z-momentum. It is possible to gain further information about the orbits through the use of adiabatic approximations and simplifying initial conditions as discussed by Sonnerup (1971). Sonnerup (1971) used the test particle approach to show that in the adiabatic limit the displacement along the electric field, parallel to the neutral line,

of Alaska,

Fair-

banks, AK 99701, U.S.A. t Department of Physics, University of California, Angeles, CA 90024 and Institute for Fusion Studies, versity of Texas, Austin, TX78712, U.S.A.

* Geophysical

Institute,

University

~0s Uni-

391

particle simulation.

should be of order e&/w,,,, where wIo is the initial perpendicular energy. Rusbridge (1971, 1977) studied the nonadiabatic motion of particles in the single X-line region. His main emphasis was on particle trapping and the breakdown of the first adiabatic invariant in multipole experiments. In his model, the electric field is absent and the separatrix are at right angles. He found that the magnetic moment of a particle entering the X-line region is changed randomly, by a factor of at most 5.75. Recently Stern (1979) studied adiabatic particle motions in an O-line region, using the analytical method discussed by Sonnerup (1971). Stern showed that in the adiabatic limit particles should convect into the O-line region and then the particles are accelerated along the O-line without limit. 2. TRADXTORIES IN A SINGLE X-LINE CONFIGURATION

2.1 Basic equations Consider the standard “reconnection configuration” in which a single X-line lies along the y-axis: B, = B,, tanh (z/L) B, = Bz,, tanh (x/L) (1) 4

=&cl

E, =E, =B,

=0

where BxO, BzO and I$ are constants which determine the magnitude of the magnetic and electric fields, and L is a constant which determines the thickness of the reconnection region. The field configuration is schematically given in Fig. la. Note that (1) is actually a solution of the MaxwellVlasov equations for the special case Bxo= B,,,, when the separatrix are at right angles.

J. S. WAGNER,P. C. GRAY, J. R. KAN, T. TAJMA and S.-I. AKASOFU

392

x

1

field to the merging rate (E, =Mv,B,) as was done by Amano and Tsuda (1978) so that we can compare our results with other trajectory studies, such as in the tearing model to be discussed later and in the plasma sheet model presented by Wagner et al. (1979). In the Earth’s plasma sheet, we may choose the following values for the parameters with which we are concerned: B,0=5y

T,=lOkeV

E,, = 3 mV m-’

T, = 1 keV.

By using the above values,

we have

vgi = 960 km SC’ poi = 2000 km

for protons

TOi==2.1 s V ,,=1.3x104kms-’ FIG.

1.

THE

TWO

MODEL

MAGNETIC

TRAJECl-ORY

F’IELDS

USED

IN THIS

pee = 15 km

STUDY.

top figure (a) shows the single X-line model. Particles are introduced directly above the X-line, at the arrow. The applied electric field is into the page, along the y-axis. The lower figure (b) shows the tearing configuration comprised of a series of X and 0 type neutral lines. Particles are started at the arrow, above an X-line. The electric field is again into the paper, along the y-axis.

7& = 1.1 x 1o-3 s.

The

Therefore,

Q * and E. * have the following

To simplify the computation, (2) is scaled using the approach of Wagner et al. (1979). All lengths are scaled to a scaling gyroradius p = mu,/qB,, and all times are scaled to a scaling gyroperiod TV= m/qB,,. The dimensionless equations of motion (2) combined with the magnetic field (1) become

(3)

!$=-B*tanh<

G.) vy*

where x*=x/p,, t* = f/ro, B* = Bxo,B,,, and E* = EJvoBro. We prefer not to associate the electric

values:

E~*=E&o~B,o=0.6 E,* = I&/v,,~B~~= 0.5 2.2 Analysis

The equation of motion for a charge particle of mass m and charge q in an electromagnetic field is given by

for electrons

and results

The test particles are started above the X-line at x = 0, y = 0 and z = 3Op, (z* = 30) with various pitch and phase angles. The initial pitch angles are (Y= 90”, 80”, 70” and SO”; the initial phase angles are C$= -120”, 0” and 120”. Two asymptotic magnetic field strengths are used, the first with B* = 2, the other with B* = 5. These two values correspond to angles between the separatrix and the x-axis of 27” and ll”, respectively. Three values of E* are used, E* = 0.1 (relatively weak), 0.25 and 0.5 (relatively strong). Similarly, three values of L* were used, namely L* = 30 (nearly adiabatic), L* = 3 and L* = 0.3 (highly nonadiabatic). We examined approximately 216 trajectories for each set of parameters. Trajectories were computed until they left the region lx*\ or lz*1<75p0. In general, the particles which enter the X-line region leave quickly, after several TV. Particles in the B* = 5 field spent an average of 14.87, with a standard deviation of 12.27, in the accelerating region. Particles in the B * = 2 field spent an average 6.97, with a standard deviation of 5.17,. Not all of the trajectories enter the X-line region. Of the trajectories computed, only 28% of the B,* = 5 and 19% of the B,* = 2

Particle

dynamics

field configurations

in reconnection

40

X-AXIS

X-AXIS

I

40

-4oA4oC -90

-50 X-AXIS

Y-AXIS

FIG. 2. _kN ADIABATIC PARTICLE IN THE SINGLE FIELDMODELwITHL*=3O,E*=0.25ANDB*=5.

X-LINE

The particle is started directly above the X-line with initial pitch angle 80” and phase angle 0”. The final pitch and phase were 81” and 16”. The total elapsed time was 310~~. The final energy was l.SW, (a small gain of energy). The particle leaves the X-line region trapped near the midplane.

trajectories enter the X-line region. The other trajectories left the system by crossing the separatrix under E x B drift at a distance greater than L* from the X-line. Two examples of X-line trajectories are shown. The first, in Fig. 2, is a nearly adiabatic orbit with L* = 30, E” = 0.25 and B* = 5. After leaving the X-line region quickly, the particle drifts along the x-axis with a final energy of 1.8 W,,. The second example, Fig. 3, is a nonadiabatic trajectory with L*=3, E*=OS and B*=5. After leaving the X-line region with a final energy of 8.6W,,, 14% of the B*= 5 trajectories left the X-line region drifting along the x-axis. All others were ejected along the separatrix. Only 9% of the B* = 2 trajectories left the X-line region along the x-axis. The only exception to the cutoff 1x*1 or Iz*l< 75p, occurred for trajectories which ultimately entered into runaway conditions along the y-axis (i.e. those with initial pitch angles of exactly 90”). These runaways were terminated after an energy gain of 500 W, (no relativistic corrections were made). Only those trajectories with an initial pitch angle of 90” gained energy in a runaway mode. The runaways resulting from the category of initialization are always confined to the y-z plane near the X-line. They accelerate along the y-axis with an acceleration directly proportional to E*. Particles

FIG.

393

3. A

-10

NONADIABATIC

I

-1

Y-AXIS

PARTICLE

IN THE X-LINE

MODEL

WITHL*=~,E*=OSANDB*=~. The particle

is started directly above the X-line with initial pitch angle 80” and phase angle 0”. The final pitch and phase were 26” and -35”. The elapsed time of the orbit was 1657,. The final energy was 8.6W,, a substantial gain in energy. The particle leaves the X-line along a separatrix. Wf

.

v=

5

*

B’=

2

.

. a

.

a a

*

I

I

I .25

.l

ELECTRIC FIG.

4.

ENERGY

A

GRAPH

w,,)

OF FINAL

VS ELECTRIC SINGLE

I .5

ENERGY FIELD

X-LINE

+E*

FIELD b%o’ (IN

FOR

UNITS

OF INITIAL

PARTICLES

IN

THE

MODEL.

For small electric fields the final energy may be less than W,. For larger electric fields a considerable range of final energies occurs. The data was phase angle averaged.

J. S.

394

WAGNER,

P. G. GRAY, J. R. m,

with initial pitch angles different from 90”, however, followed much more complicated trajectories and spent little time in the X-line region. Particles starting with identical pitch angles but different phase angles were found to have widely varying orbits, especially for the nonadiabatic cases. Notice in Fig. 4 that as E” increases, the spread of the final energies also increases. The average final energy for the non-runaway trajectories with E* = 0.1 was 1.17 W,,; for E* = 0.25 it was 2.01 W, and for E*=0.5 it was 2.54W,. 3. TRAJECTORIES IN A TEARING MAGNJWIC FIELD CONFIGURATION

3.1 Basic equations The magnetic fieId model used in this case is the two-dimensional analytic solution of the timeindependent Vlasov-Maxwell equations given by Kan (1979). The field configuration is shown in Fig. l(b). An ad hoc uniform electric field in the ydirection is added to represent the induced electric field. The fields are described by

Bx =[(a’+

-&a(cu’1) sin OLX 1) cash (crz)+(ar’- 1) cos (ax)]

BZ =[(a’+

-Boa(a2+ 1) sinh ((~2) 1) cash (az)+(a’1) cos

(ccc)]

(4)

T. TATS

and S.-I. AKnsom

rectly above an X-line at x = 0, y = 0 and z = 60~~. The initial pitch angles used varied from 90” to 0’ in 10” increments. At each pitch angle two phase angles were used, 0” and 180”. Two values of the electric field parameter were used, 0.25 (relatively strong), and 0.1 (relatively weak). Four values of L* were used, ranging from L* =30 (highly adiabatic) to L* = 1 (very nonadiabatic), including L” = 15, 7 and 3. Trajectories were computed until they left the region y >2OOp,. This corresponds to an energy gain of 20 W. when E* = 0.1 and an energy gain of SOW, when E*=0.25, Results from trajectories demonstrate the special importance of the O-line in particle energization. Only those particles with an initial pitch angle of exactly 90° will enter the X-line region (defined as x*< L*, z* < L*) and stay there. The near adiabatic orbits (particularly those with initial pitch angles of 80”, 70” and 60”) enter the X-line region n” < L”, z* < L* but receive no appreciable energy gain. The particles are ejected from an X-line region after 1- ZOT,, similar to those trajectories in Section 2. After leaving the region, the nearadiabatic particles enter the nearest O-line and then accelerate along thC O-line in a runaway fashion. Figures 5 and 6 show two typical examples of such orbits. The nonadiabatic particles with L* = 7, 3 and 1 followed similar orbits, except that the

The parameter a! = 0.5 was chosen in the following trajectory calculation. The same scaling scheme used in Section 2.1 can be applied to (4). Letting Bzc = Boa (a’- l)/(a2 + l), the scaling gyroradius becomes p. = m&qB,,. The equations of motion then become -sin (x*/2L*) (z*/2L*)-0.6~0s (x*/2L*) ’ ‘-“*

dv,* -=cosh dt*

d%”_ dt*

-2.083333 sinh (z”l2L”) 1.25 cash (z*/2L*)-0.75 cos (x*/Z*)

60

6

i

* “*

sin (x*/2L”) *+E* +cos (z*/2L*) - 0.6 cos (x*/2L*) ’ vx (5) dv,* -= dt*

-2.083333 sinh (z*/2L*) 1.25 cash (z*/2L*)-0.75 cash (x*/ZL*) * %*

Equations

(5) were integrated

numerically.

3.2 Analysis and results The initial conditions used for this part of the study are as follows: All test particles started di-

-2

-20

SO

180

-2

~

-210

-110 Y-AXIS

X-AXIS

FIG,5. AN ADIABATIC MODEL WITH

PARTICLE L*

=

IN

m

TE.ARlNG

30 ANDE* = -0.25.

-10

FBLD

The particle is started directly above an X-line with initial pitch angle 80” and phase angle 180”. The final pitch and

phase were 103” and 168”. The total elapsed time was 9007,,, but the time spent accelerating down the O-line was only 517,. The final energy was SOW,.

395

Particle dynamics in reconnection field configurations

I -70

-70 -300

-140 X-AXIS

20

-210

-110

-10

O-line until the cutoff distance was reached. No particle was found to leave or become ejected from an O-line. An example of a nonadiabatic orbit is Fig. 7. Particles trapped around an O-line gain considerable energy both parallel (4,) and perpendicular (W,) to the magnetic field. The radius of the orbit (measured from the O-line) monotonically decreases as the particle becomes energized. As long as the particle motion is adiabatic, the radius of the orbit decreases with ExB drift. As the motion becomes nonadiabatic, the radius decreases at a slower rate. Figure 8 shows the relationship between the initial pitch angle and the ratio WJY,. It shows that the smallest ratio W,/w, occurs for adiabatic particles with smallest initial pitch angles, accelerated along a relatively small electric field.

Y-AXIS

4. SUMMARY OF THE RFSULTS FIG. 6. AN ADIABATICPARTICLEIN THE TEARINGFIE&D MODELwITHL*=3OANDE*=-0.1. 4.1 X-line trajectories The particle is started directly above an X-line with initial (a) Only those particles with an initial pitch pitch angle 10"and phase angle 180”. The final pitch and phase were 75” and -136”. The total elapsed time was angle of exactly 90” directly above the X-line gain 8337,,, but the time spent in the acceleration region was energy in a runaway type fashion. All other partionly 1667,. The final energy was 20 W,. cles follow complicated orbits which may or may time spent in the X-line was less than lag, and the not pass neutral lines, depending on the field particles were not always captured by the nearest parameters and initial conditions. O-line. One trajectory entered the sixth O-line I.0 v from the origin, but the majority entered the first or second. After entering the O-line the .90 nonadiabatic particles were accelerated along the a0

-

.70 60 zz= ‘i 3 50-

\ i i \ i

40 .30 20 .I0 .o ’ -16

-8 X-AXIS

0

-210

-110 Y-AXIS

-10

FIG. 7. A NONADIABATIC PARTICLEIN THE TEARINGFIELD MODELWITHL* = 1 ANDE* = -0.25. The particle is started directly above an X-line with initial pitch angle 80” and phase angle 0”. The final pitch and phase angles were 77” and -175”, respectively. The total elapsed time was 1517,, but the particle spent only 457, accelerating down the O-line.

\ i._

- -L*=7, E*=( -----L*=15,~*=.25 -L*= 30, E*-_25 -L*. 15,E* z., I

SO

I

I

70

60 INITIAL

;

.\

- - L”. 30 E+:.,

I

50

-.

.\

‘\

I

40 PITCH

I

30

I

20

I

IO

L ‘\ ‘\ I

0

ANGLE

FIG. 8. A GRAPHOF THE FINALENERGY RATIO WJW,, vs INITIAL PITCH ANGLE FOR PARTICLESIN THE TEARING MODEL. The data was phase angle averaged. The graph shows that adiabatic particles in a small electric field are most likely to have significant energy parallel to the magnetic field (perpendicular to the O-line).

396

J. S. WAGNER, P. C. GRAY, J. R. KAN, T. TM-

(b) Particles which enter the X-line region x*, y-L*, typically gain around l-2 times their initial energy (see Fig. 3). But some particles lose energy by a factor of at most $. (c) The lifetime of a particle in the X-line region is on the order of rO. Particles may spend more time in the region if the motion is adiabatic. (d) The majority of particles, 91% for B* =2 and 86% for B* = 5, leave the X-line region along field lines near the separatrix with nonzero pitch angle. The other particles leave the region as they drift along the x-axis (see Fig. 2). The implication of these results can be summarized as follows. Due to the small energy gain and short lifetime, current-driven instabilities and plasma heating are unlikely to be intense along the X-line region. The results also indicate the possibility of a field-aligned plasma flow near the separatrix. A field-aligned flow component near the separatrix is present in Sonnerup’s (1970) reconnection model. 4.2 Tearing

trajectories

Conclusions from our study of particle trajectories in the tearing field configuration are: (a) Only those particles with initial pitch angle of exactly 90” directly above an X-line will enter and stay in the X-line region. (b) Particles entering the X-line region with pitch angles other than 90” receive negligible acceleration at the X-line and are quickly channelled into an O-line region. (c) Adiabatic particles are convected into the

and S.-I.

AKASOFU

nearest O-line through E x B drift; nonadiabatic particles may enter into any of the neighbouring O-line regions. (d) Both adiabatic and nonadiabatic particles entering an O-line region are never ejected, and are continuously accelerated. The acceleration along the O-line is proportional to E*. This is in agreement with Stern’s (1979) results. (e) Particles in an O-line may receive significant energy perpendicular to the O-line, but the radius of the particle orbit about the O-line is monotonically decreasing. On the basis of these results, it seems likely that suprathermal particles can be produced along the O-line region. 5.

COMPARISON WlTH Rl?SULTS SIMULATION STUDIES

FROM

As mentioned in the introduction, our test particle approach to the problem is limited by the fact that we ignore effects of particle motion on the electric and magnetic fields. Therefore, it is importent to compare our results with those obtained by a self-consistent study. In this section, we show that our results have qualitative similarity to those obtained by a plasma simulation study. Leboeuf et al. (1980) have recently made a simulation study of the tearing process by using a selfrelativistic consistent electromagnetic particle plasma simulation code. They showed that the momentum distribution function along an O-line (the y-axis) has a long tail, indicating characteristics of runaway particles (see Fig. 9). By choosing some

I--

(a)

I

-1

II I

I

0

1

Py/m,c

1 -1

0 P, /m,c

(b)

I

FIG. 9. DISTRIBUTION FUNCTIONSFROM THE SIMULATION STUDY (Leboeuf et al., 1980)

at t = 3OOq,-‘.

The plasma beta at the external current bar is @ = 0.06 and the X-line geometry is 4: 1 for the x vs z scale lengths. (a) The momentum distribution function in the y-direction. An extremely hot thin ion tail stretching to the right edge of the frame can be seen. The momentum units are in m,c, where m, is the electron mass. (b) The momentum distribution function in the x-direction, in which the island chain and coalescence of islands occur. In this particular case, two islands are coalescing and the two humped distribution is characteristic of the two approaching islands along the x-direction. The tails of the distribution stretch up to the margins of the frame.

Particle X-LINE

0t

0

ELECTRONS

dynamics

X-LINE

I

/

I

20

40

60

I 0

20

in reconnection

397

Their simulation does not allow spatial variations along the y-direction so that the runaways cannot excite plasma waves along that direction. However, * Leboeuf and Tajima (1979) have shown that the runaway electrons and the causal d.c. electric field E, are capable of exciting large amplitude plasma waves which heat the plasma particles in the V, direction; at the same time, these large amplitude waves may clamp the runaway momenta (Leboeuf 1979) and prevent effects of “shortet al., circuiting” of the d.c. field along the O-line.

IONS

40

field configurations

60

X/All

FIG. 10. SELECTED PARTICLE ORBITS FROM THE SIMULATION STUDY (Leboeuf et al., 1980). In this case, fi = 0.2 and the system size Lx :L., = 64A: 32A = 2: 1, where A is the grid length. The cross indicates the cross-section of the original X-line and the circle that of the O-line. The upper left diagram shows the orbit of an electron which started from the X-line region. The upper right shows the orbit of an ion starting from the X-line region. The lower left shows the orbit of an electron from the O-line region, while the lower right shows that of an ion from the O-line region.

of the particles in the plasma and tracing their trajectories, they also showed that particles (both electrons and ions) tend to converge along the O-lines, rather than along the X-lines (see Fig. 10). Thus, the O-line carries much of current and therefore magnetic energy. These findings are consistent with our test particle calculations on this point. In fact, we may be able to infer that the particles with large (v,) shown in the tail of the distribution function, Fig. 9, should consist primarily of those particles which converged to the O-line region and were accelerated along it. It is interesting to note also that Leboeuf et al. (1980) found that no strong heating due to the tearing (which involves X-lines) is observed in the x and z velocity distribution functions for both electrons and ions; in the tearing stage heating is primarily due to adiabatic compression, yielding bulk heating in the v, and v, distributions. On the other hand, when O-lines coalesce, intense heating takes place in the u, and v, distribution function for both ions and electrons (particularly for ions). In this coalescence stage, the 0, and v, heating associated with O-lines much surpasses the o, heating [see Fig. 9(b)].

Acknowledgements-This work was supported in part by the Department of Energy under Contract EY-76-S-062229 Task No. 5 and the National Science Foundation, Atmospheric Sciences Section, Grant ATM77-11371AOl to the University of Alaska. REFERENCES Amano, K. and Tsuda, T. (1978). Particle trajectories at a magnetic neutral point. J. Geomag. GeoeZec. 30, 27. Coroniti, F. V. and Eviatar, A. (1977). A magnetic field reconnection in a collisionless plasma. Astrophys. J., Supll. Ser. 33, 189. Kan, J. R. (1979). Nonlinear tearing structures in equilibrium current sheet. Planet. Space Sci. 27, 351. Kan, J. R., Akasofu, S.-I. and Lee, L. C. (1979). Physical processes for the onset of magnetospheric substorms, in Dynamics of the Magnetosphere, (Ed. S.-I. Akasofu) p. 357. D. Reidel, Dordrecht. Leboeuf, J. N. and Tajima, T. (1979). Enhanced interaction between electrons and large amplitude plasma waves by a d.c. electric field. Phys. Fluids 34, 1485.

Leboeuf, J. N., Tajima, T. and Dawson, J. M. (1979). Excitation of large amplitude plasma waves by runaway electrons. Phys. Rev. L&t. 43, 1321.

Leboeuf, J. N., Tajima, T. and Dawson, J. M. (1980). Tearing, magnetic X-points, and coalescence, to be published in Physics of Aurora Arc Formation Geophysical Monographs Series (Ed. by S.-I. Akasofu). Mitchell, H. G., Jr. and Kan, J. R. (1979). Current interruption in a collisionless plasma by non-linear electrostatic waves. Planet. Space Sci. 27, 933. Rusbridge, M. G. (1971). Non-adiabatic charged particle motion near a magnetic field zero line. Plasma Phys.

13, 977. Rusbridge, M. G. (1977). Non-adiabatic effects in charged

particle motion near a neutral line. Plasma Phys. 19, 1087. Sonnerup, B. U. 6. (1970). Magnetic field reconnection in a highly conducting incompressible fluid. J. Plasma Phys. 4, 161. Sonnerup, B. U. 6. (1971). Adiabatic particle orbits in a magnetic null sheet. J. geophys. Res. 76, 8211. Stern, D. P. (1979). The role of O-type neutral lines in magnetic merging during substorms and solar flares. J. geophys. Res. 84, 63. Wagner, J. S., Kan, J. R. and Akasofu, S.-I. (1979). Particle dynamics in the plasma sheet. J. geophys. Res. 84, 891.