Test particles in a two dimensional, turbulent MHD simulation are found to undergo significant acceleration. The magnetic field configuration is a periodic sheet pinch which undergoes reconnection. The test particles are trapped in the reconnection region for times of order an Alfven transit time in the large electric fields that characterize the turbulent reconnection process at the relatively large magnetic Reynolds number used in the simulation. The maximum speed attained by these particles is consistent with an analytic estimate which depends on the reconnection electric field, the Alfven speed, and the ratio of Larmor period to the Alfven transit time

Ambrosiano, J. J.

Goldstein, M. L.

Matthaeus, W. H.

English

NASA Technical Reports Server

(HASA-TH-86121) PARTICLE ACCELEEATION BY
TOEBOLENT SAGNETOHYDBO-DYNAMIC BECONNECTIpN
(HASA) 16 p HC A02/MF A01 CSCi 201
N84-26409
Dnclas
63/75 19435
NASA
Technical Memorandum 86121
PARTICLE ACCELERATION BY
TURBULENT MAGNETOHYDRO
DYNAMIC RECONNECTION
W. H. Matthaeus
J. J. Ambrosiano
and
M. L. Goldstein
JUNE 1984
National Aeronautics and
Space Administration
Goddard Space Flight Center
Greenbelt, Maryland 20771
https://ntrs.nasa.gov/search.jsp?R=19840018341 2020-03-20T22:03:43+00:00Z
NASA
Technical Memorandum 86121
PARTICLE ACCELERATION BY
TURBULENT MAGNETOHYDRO
DYNAMIC RECONNECTION
W. H. Matthaeus
J. J. Ambrosiano
and
M. L. Goldstein
JUNE 1984
National Aeronautics and
Space Administration
Goddard Space Flight Center
Greenbelt, Maryland 20771
PARTICLE ACCELERATION BY TURBULENT MAGNETOHYDRODYNAMIC RECONNECTION
W. H. -Matthaeus
Bartol "Research Fpundation
University = of Delaware
"'••• Newark, EE 19716
J. J. Ambrosiano
. Berkeley Research Associates
P. 0. Box 852
Springfield, VA 22150
M. L. Goldstein
Code 692, Laboratory for Extraterrestrial Physics
NASA Goddard Space Flight Center
Greenbelt, MD 20771
PACS: 52.65.+Z, 52.35.Ra, 47.65.+a, 52.35.Mw, 95.30.Qd
-2-
-"-.; Abstract
Test particles in a two dimensional, turbulent MHD simulation are found
to undergo significant acceleration. The magnetic field configuration is a
periodic, sheet, pinch which undergoes reconnection. .. The test .particles are
>
trapped in the reconnection region for times of order an Alfven transit time
. in _ the large electric. fields, that .^ characterize the .turbulent: .reconnection
process .at-..the relatively ...large :.-magnetic -Reynolds - number .used in the
simulation. The -maTHim-mi speed attained by these particles is consistent with
. an analytic estimate which depends on the reconnection electric field, the
Alfven speed,, and the ratio of Larmor period to the Alfven transit time.
-3-
Acceleration to .chilli•'senergies^ -bf a small fraction of the particles in a
-plasma is familiar^in space physics, astrophysics and fusion physics. Two
examples are the production of energetic particles in solar flares and in
planetary magnetospheres during magnetic substorms. Giovanelli , Dungey ,
Sweet3, Parker* and others suggested that particles can be accelerated to
high- energies by strong electric fields which might arise near X-type
.magnetic;.neutral points.. .-The question of whether strong electric fields
'.-exist near such neutral points has "led ; to the study of magnetohydrodynamic
(MHD) reconnection. Conclusions about the intensity of the self-consistent
• electric field near the X-point and therefore, the "rate of reconnection"
vary considerably in reconnection modelss. The collisional tearing
instability6,, Parker-Sweet mechanism3'7 and Petschek's model8 predict values
of reconnection rates which decrease at high conductivity.
The^  effects of turbulence in analytical and computational models have
often been excluded by assuming that the background magnetic field is smooth
initially and that perturbations are symmetric and/or infinitesimal. There
is indication9'10'11 that finite-amplitude fluctuations can lead to
turbulence in X-point dynamics,-thereby elevating reconnection rates.
Consequently, turbulent reconnection 'might maintain strong X-point electric
fields (and the possibility::;of significant particle acceleration) at high
conductivity. :*--:"
The question of whether a subpopulation of plasma particles might be
effectively accelerated by the reconnection zone electric field has also been
extensively studied12 17. A typical approach has been to calculate the
trajectories of "test-particles" in model MHD magnetic and electric fields.
The electric field has usually been included parametrically, but there has
-4-
been at least one study of test particle orbits in the dynamic electric
fields of a symmetric, forced non-turbulent MHD simulation. It generally has
been found that test particles do not spend very much time near X-type
neutral points, but can easily be trapped near 0-type neutral points12' l *.
This is a crucial point in the assessment of the importance of reconnection
as a particle accelerator. In nonsteady incompressible simulations, strong
accelerating"• ele*t-r-it:~ fields do not appear near 0-points which are the
centers about which reconnected flux gathers after emerging from the
reconnection zone. A long residence time near this type of ' 0-point is
unlikely to produce significant particle acceleration. The short residence
times of^ . particles.,near the X-point region has been viewed as a limitation on
the effic*encyrxrf---tfteiiX-point acceleration mechanism.
Ih^ tnTs" Tetter, we describe an analysis of particle orbit tracing in MHD
2t - '. • •
fields ~which evolve and reconnect in the presence of finite amplitude
fluctuations. We find that turbulent fluctuations appearing near the
recotfaection zone can trap test particles in the strong electric field region
for long enough times to produce significant acceleration. Consequently the
turbulent X-point mechanism is an efficient accelerator.
The: fi'elds in which the test particles propagate are produced by a
spectral-method, incompressible MHD simulation technique that has been
described in detail elsewhere9'10. The magnetic field B and fluid velocity
-*
field v lie in the x,y plane and vary in that plane, all quantities being
independent of the third coordinate, z. Thus, the electric current density,
magnetic vector potential, fluid vorticity and MHD electric field each have
only single components in the z direction, designated J, a, u and E respec-
tively. The familiar dimensionless units19'9 include an arbitrary length
scale L, a characteristic Alfven speed, V , and transit time of unit
3
-5-
. distance, T =_1/V .
di
A, -12 8-?.: spectral:, truncation .and';am:explicit,: timestep of 1/512 were used.
The _:runs • lasted 10teAlfr7ffl:..±lmes- .,iiJ&exv4lijaens±onJl«ss,..^fliaid and -magnetic
Reynolds numbers (reciprocals of the dimensionless. viscosity, v and
resistivity, W) were each equal to 1000 in these simulations. The values of
•*• •*
u(k,t) and.a(k,t) for all values of 2— d wavevectors retained were stored
-every -AT... 10/512 Alfven times. ~ The reconnection process is modeled as an
unforced, .turbulent .initial value, .problem. , in which, at ..t =0 magnetic.. .energy
vresides, almost entirely .±n. a small, number of Fourier modes that describe two
Oppositely ..directed .neutral sheets. ..Turbulent ...reconnection. .is. triggered by
. adding af t = O.a small but, finite -level v of broad-band random-phased ."noise"
perturbations.
-Test particles were ^ introduced into the .simulation .fields at ~t = 0.
- -»• ••-»•
Their position X and three dimensional velocity U were then advanced in time
-t- -»•
.and space .according to dX/dt « U. and
dU/dt = a(U x B + E) (1)
where a = fit, 8 = ZeB/mcT-Js^the nominal : 'test^parti'de- gyrofrequency, and E =
-v x B + pJ is the MHD electric field.
Eq. (1) , . written with particle velocity in units of Alfven speed,
emphasizes the role of the constant ° in coupling the natural units of
particle motion to those of MHD turbulence. Particle acceleration is
entirely due to the z-component of electric field. The implementation of the
particle-pushing algorithm uses the fact that the z-component of Eq. (1)
implies
-6-
Uz + ota = constant (2)
This is an expression of the constancy of the z-component of canonical
momentum and allows the effect of the electric field to be incorporated
implicitly through the space and time dependence of the vector potential. To
ensure that the particles sample the fields in the detail afforded by the
high resolution MHD simulation, we employed a 2-d cubic spline interpolation
procedure to calculate field values at particle positions. The fields were
also linearly interpolated in .time, hence the electric field is constant
between independent snapshots of the MHD simulation. Time integration of the
x and y components of Eq. (1) was done by a fourth-order Runge-Kutta method.
The timesteps were chosen to be much less than the time between frames of the
MHD fields to give good resolution of the particle gyromotion.
The first run we describe consists of 500 particles randomly distributed
in space, each moving with the unit Alfven speed at 45° pitch angle, but with
randomly chosen gyrophase. The value of ot is 643, so that the characteristic
gyroradius is much less than the unit cell of the MHD simulation. Fig. 1
shows magnetic field line plots in the area around the lower
 ;of the two
current sheets at several times. New magnetic islands are appearing due to
the reconnection process. Furthermore, since the same field lines are
plotted in each frame (same values of a(x,y)), it is clear that magnetic flux
is being removed from the strong field regions above and below the current
sheet.
Figure 1 also shows the position of three of the test particles super-
imposed on the field lines at each time. These particles were selected
because of their significant energy gain, and because their trajectories were
typical of particles gaining energy along either the upper or lower sheets.
-7-
.Throughout .-the run, most of the particles are undergoing relatively
unperturbed gyromotion ..in -the strong magnetic Afield regions away from the
reconnectinn.. zones -•n*»a*-•--•th*v.T7pyi----;-and - ln»»n-.--.» X—rpoi-nj-g^.^However, Jigure 2
shows that the.particle energy distribution changes considerably during the
run in that a small number of particles, including the three in Fig. 1, are
accelerated to thousands of times their Initial energy. It is apparent in
Fig. • 1 that the -particles were trapped -within small "magnetic'bubbles" . in .
: the .reconnection .zone which are turbulent fluctuations .of .the.magnetic field.
.'. Detailed ,-analysis , t>f ~the MHD .simulation'1 * -has suggested --that these
..fluctuations- are -characteristic, of reconnection triggered by finite amplitude
turbulence.. ..The .bubbles produce .multiple X—points-;within the reconnection
zone which .remains .centered. about .a. single . filament of electric current
density. Once .formed,-these bubbles move:outward from .the reconnection.zone
with the exhaust velocity of the fluid jets (-0.85V ).'
di
The .trapped particles move with.the bubbles, cross the strong electric
field region where they are accelerated, and enter the reconnected island
region around an 0-point. Within the large island the particles experience
less acceleration because the reconnection electric field is weak there.
Some particles lose '-energy in the- 0—point region, presumably due to
encounters with field lines;; which-.precede.-:^ as rthe^reconnected island grows.
This behavior,; is illustrated inr/iFig. 3 by the time history of the kinetic
energy of the particles identified in Figure 1.
The physics of the particle trapping and acceleration is easily under-
stood. Each small magnetic.":bubble is a trapping center for test particles.
In this simulation, particles in the reconnection zone experience a fluctuat-
ing electric field -with average magnitude of e = 0.1 (in units of B V ).
O 3
Particles trapped in a., bubble are entrained in the high-E region for about
-8-
one Alfven transit time.
From the conservation law (Eq. 2) it follows that if a particle spends a
unit time in a reconnection electric field, e, its velocity will change by
AV s ect. With o = 643, the upper limit to the velocity is about 64. Note
that in Fig. 3, particle no. 382 reaches a speed of about 60. This is the
highest speed that we have found for particles in this simulation. A
preliminary test of this scaling was performed using a second set of 500
particles with a = 321.5. The highest speed attained was about 30.
Assuming the validity of this scaling and neglecting the initial speed of
the test particles, the maximum kinetic energy is e = m(AV )2/2 = e2azm/2
niflx z
where E is in units of V 2. This is equivalent to
3.
0.5 mVa2 [e(Lo>p/c)(Z/Zo)(mo/m)]2 (3)
where u> , Z and m are the plasma frequency, charge state and ion mass of
the background plasma. Eq. (3) assumes that particle motion in the
z-direction is unbounded. If the acceleration acts only until the particles
have traveled a distance L in the z-direction, then the estimated duration of
the acceleration process becomes T* = l/(2/(EO'))> and the maximum energy is
£max
Note that this is proportional to the charge of the test particles, but
independent of their mass.
These simulations indicate that turbulent fluctuations in a reconnecting
magnetofluid can trap test particles in the strong electric field region and
accelerate them to speeds of order efliv . In a future publication, several
3.
issues will be studied including the range of <* for which trapping occurs,
the dependence of the magnitude" of "the acceleration on Reynolds number and
initial:particle speeds,- and. the. influence of a uniform DC magnetic field in
the z-direction. -* • .
We thank S. L. Lamkin for assisting in the preparation of the MHD field
data. Helpful discussions with F. C. Jones, K. Papadopoulos, and B. U. 6.
' '::
:
.. Somerup are gratefully acknowledged. This work was supported,; in part, by a
..NASA .Solar Terrestrial ...Theory. Program grant ..to ...the Goddard,, Space Flight
v ,r Center,v and; NASA, Grant, NSG 7416 to The .College of,. William ..and. Mary. WHM
., .acknowledges the. Laboratory..for_.Extraterrestrial Physics, .Goddard Space
. Flight, Center,..for support.-during the course, of .this research.
-10-
References
*R. G. GiovanelllV: Mon. Hot. Roy.^ -sAstTon. Soc. / 107, 338 (1947),
2J. W. Dungey, Cosmic Electrodynamics (Cambridge Univ. Press, 1958).
SP. Sweet, in Electromagnetic Phenomena in Cosmical Physics, edited by B.
Lehnert (Cambridge Univ. Press, 1958). P. Sweet, Nuovo Cimento Suppl., VIII,
Ser. X, 188 (1958).
... .*E. ,N. Parker, Astrophys. J.-Suppl. Ser.,,j[,. 177 -(1963).
1
 '••
 SE.-N.:' .Parker, Cosmical Magnetic Fields (Clarendon Press,- Oxford, .1979).
, ,
6H. Furth, J. Killeen, and M.'.Rosenbluth, ;Phys. Fluids, J&, 459.. (1963).
7E. N. Parker, J. Geophys. Res., 62^  509 (1957).
•
8H. Petschek, in AAS-NASA Symposium on the Physics of Solar Flares, NASA
SP-50, 425 (1964).
9W. H. Matthaeus and D. C. Montgomery, J. Plasma Phys., 25, 11 (1981).
.
10W. .H. Matthaeus, Geophys. Res. Lett., j?, 660 (1982).
11W. H. Matthaeus and S. L. Lamkin, Phys. Fluids, submitted (1984).
B. U. 6. Sonnerup, J. Geophys. Res., 76, 8211 (1971).
13T. W. Speiser, J. Geophys. Res., 72_, 3919 (1967).
'"D. P. Stern, J. Geophys.-Res., 84, 63 (1979).
J. S. Wagner, J. RiyiCan", -and S.-I. Akasofu, J. Geophys. Res., 84, 891
(1979). ;'•-. >-
16V. M. Vasyliunas, J. Geophys. Res., 85_, 4616 (1980).
17M. Rusbridge, Plasma Phys., 13_, 977 (1971). M. Rusbridge, Plasma Phys.,
19_, 1087 (1977). ..,; :,
18T. Sato, H. Matsumoto and K. Nagai, J. Geophys. Res., 87, 6089 (1982).
19D. Fyfe and D. C. Montgomery, J. Plasma Phys., Ij6, 181 (1976).
-11-
Figure Captions
Figure 1. Magnetic field line plots in the area of the lower current sheet
at several times.. The positions of three particles are indicated by the
symbols: the triangle is particle number 382, the circle is particle 358,
and the square is particle 281. The contours label the same vector potential
values in each panel.
Figure 2. Particle energy distribution at t = 9, indicating' that most
particles have gained little energy; but a small high energy population has
been formed.
Figure 3. The time history of the kinetic energy of particles 382, 358, 281
(cf. Figure 1).
T=0.02 I
Figure 1
CO
CM
C
C
C
C
C
C
CM
O)
o
C
0)
u
3
60
O)
o
CM
o
co m
KINETIC ENERGY
:3600
2400
1200
0
382
j i
2.5 7.5 10
TIME
Figure 3
BIBLIOGRAPHIC DATA SHEET
1. Report No.
TM 86121
2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
Particle Acceleration
dynamic Reconnect ion"
: Turbulent !Magnei6hydro-
5. Report Date
June 1984
6. Performing Organization Code
7. Author(s) W. H. Matthaeus, J. J. Ambrosiano and
M. L. Goldstein
8. Performing Organization Report No.
9. Performing Organization Name and Address
NASA/GSFC
Laboratory for Extraterrestrial Physics
Interplanetary Physics Branch, Code 692
Greenbelt, MD 20771
10. Work Unit No.
11. Contract or Grant No.
12. Sponsoring Agency Name and Address
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
Test particles in a two dimensional, turbulent MED simulation are
found to.undergo significant acceleration.: The magnetic field configuration
is a periodic sheet pinch which undergoes reconnection. The test particles
are trapped in the reconnection region for times of order an Alfven transit
time in the large electric fields that characterize the turbulent reconnec-
tion process at the relatively large magnetic Reynolds number used in the
simulation. The maximum speed attained by these particles is consistent
with an analytic estimate..which depends, on the reconnection electric field,
the Alfven speed, and the -ratio of Larmor period to the Alfven transit
tine. .-.. .. -.-:
17. Key Words (Selected by Author(s))
Magnetic reconnection, turbulence,
acceleration
18. Distribution Statement
19. Security Classif. (of this report)
U
20. Security Classif. (of this page)
U
21. No. of Pages
15
22. Price*
"For sale by 'he National Technical Information Service, Springfield, Virginia 9?"] GSFC 25-44 (10/7 n


Particle acceleration by turbulent magnetohydro-dynamic reconnection

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