Download parallel electric fields as acceleration mechanisms

Adv. Space Res. Vol. 13, No.4, ~ (4)249—(4)252, 1993
Printed inGreat Britain.
0273—1177)93 $24.00
1993 COSPAR
PARALLEL ELECTRIC FIELDS AS
ACCELERATION MECHANISMS IN
THREE-DIMENSIONAL MAGNETIC
RECONNECTION
M. Hesse* and J. Birn**
*
~
Code 696, NASA/GoddardSpace Flight Center, Greenbelt, MD,
LosAlamos, NM, U.S.A.
U.S.A.
ABSTRACT
An extension to the theory of General Magnetic Reconnection (GMR) is presented. In particular, a
new analytical relation between the rate of change of newly connected magnetic flux and the maximum
possible value of the parallel component of the electric field integrated along the magnetic field will
be derived. Estimates of the maximum value based on three-dimensional MHD simulations will be
presented also.
INTRODUCTION
Magnetic reconnection is considered to be an important transport process occurring in both space
and laboratory plasmas. Examples include magnetotail reconnection /1,2/, and magnetic reconnection
in the solar corona /3/. Electric fields involved in the reconnection process are possible means of particle
acceleration /4/. Apart from electric fields in regions ofsmall magnetic field, parallel electric fields can
also act as particle acceleration mechanisms /5/ thereby causing the possible occurrence of fields-aligned
accelerated particle beams. A possible link between parallel electric fields and magnetic reconnection
was not established within the frame of magnetic merging /6/, which relied on the existence ofmagnetic
separatrices and separators.
In the recent past the examination of the evolution of a plasmoid flux rope led to a re-investigation of
magnetic reconnection /7,8/. The resulting theory, called “General Magnetic Reconnection” (GMR),
proved to be applicable also in systems without magnetic neutral lines or neutral points /9/. There
magnetic reconnection typically involves the existence of parallel electric fields in a localized diffusion
(sub) region. Furthermore, the integral of E
11 along field lines was shown to yield a nonzero value if
reconnection was to have global consequences, and that gradients in the integrated parallel electric
field play a particularly important role /10/. Here we will extend the theory of GMR to include an
analytic expression for the maximum value of the integrated parallel electric field as a function of the
time derivative of the newly reconnected flux. We will apply the theory to our self-consistent MHD
model of magnetotail dynamics, and extrapolate of solar coronal parameters.
THEORY
The theory utilizes the Euler potential (ct and /3) representation of the magnetic field
B=VcxVI3
(1)
Such functions exist if the magnetic field in the domain under consideration is everywhere different
from zero /8/. The electric field can be written
(2)
where the electrostatic potential ~ is replaced by
(3)
(4)249
M. Hesse and J. Biro
(4)250
A connection between E and B is established by generalized Ohm’s law
E+vxB=R
(4)
with v representing the plasma flow velocity and R a general nonideal term, assumed to be different
from zero only in a localized region DR in space. R can be written as
R
RaVa+RpVI3+R
3Vs
(5)
where the arc length coordinate s is defined by
BVs=B
(6)
Insertion of (1), (2), (5) into (4), and using the linear independence of Va, V/3, and Vs yields /8/
da
8~b
Oa
(7a)
(7!>)
= —R,
=
E11
(7c)
the dynamical evolution of the Euler potentials such as would be measured by plasma elements moving
with the flow velocity v. On each field line traversing Dir, the function ~ in the ideal region outside
and on opposite sides of DR, differs by an integral extended over DR along the field line considered,
i.e.,
I,b2=t,bi_jdsEii
(8)
where the indices refer to the different sides of DR. Thus, by defining
=
(note that
_jdsEii
(9)
can be time dependent) we get
(10)
Clearly, if ~ is different on both sides, the plasma elements on different sides can experience a different
change in a and /3 values even if they had the same values initially, i.e., they were connected by a field
line. This corresponds to ‘global effects’ /7/, i.e., a change in magnetic connections of plasma elements
outside of DR. Naturally, a parallel electric field with this property can lead to particle acceleration
/9/. Using (7), we can now define equations specifying the rate of change of a and /3 by magnetic
reconnection /10/
do
do
~jIree = ~I2
d/3
~iI~
do
—
dfi
= ~iI2
—
-~-Ii=
—~
df3
O~
=
(ha)
(llb)
The Hamiltonian structure of (11) implies that the evolution of the changes of magnetic connections
is divergence-free, i.e., there is exactly the same amount of flux newly connected on both sides of DR.
Furthermore, (11) proves that the rate of new connections is fastest for strongest gradients of in a-/3
space.
Assume now the existence of one isolated localized region in configuration space where E11 is not only
different from zero but also yields a nonvanishing! which exhibits a single isolated maximum. For
clarity, we will concentrate on this simplest case, since more general situations with multiple maxima of
and multiple DR subregions are not physically different but only more tedious to treat. In magnetic
flux coordinate (a, /3) space, a possible shape of the contours of! is sketched in Figure 1.
We can now derive a relation between the maximum value of! and the reconnection rate, defined
by the change of newly connected magnetic flux with time. The procedure is illustrated in Figure 1.
If a flux boundary, given by a flux surface 7 separating two regions of space, traverses the location of
the maximum of!, the rate of change of magnetic flux Q newly connecting region 1 with region 2 and
Parallel Eledric Fields
(4)251
a
Fig. h.
Flux space representation of the contours of! =
f E11ds, and of a magnetic flux
boundary traversing the center of the ! ~ 0 region. The direction of the transport given by
(11) is indicated by the arrows.
vice versa is given by the integral of the normal component of the transport ~ = (da/dilr.c,
along 7
d/3/dtI~~~)
(12)
where 7i is one of the two segments of 7 derived from cutting 7 at the location of the maximum value
of!, and dl is a normalized length differential along 7i~We introduce a parametrization (parameter 1)
=
(a(1),f3(l))
(13)
such that (a(0), /3(0)) is the location of the maximum of!. Then the normal is given by
(14)
By substituting (hh) for ~ we can now evaluate (12) to give
dQ
di
—
=
f (O!d/3
J~,k,~O/3dl
L
OEda
dl
8~dl
~jdl
maxmaxjEiids
(15)
Note that, because of the divergence-free structure of ~, the integration along 71 gives the same result
as an integration along any smooth line connecting the maximum of! to the region = 0.
This result extends in a natural way the well-known result that the integral of the electric field along
an X line equals the temporal change of the reconnected flux Q. We have proven here that in the
general case where the magnetic field does not exhibit neutral lines, there exists at least one field line
which carries all the reconnection electric field as a parallel component to B, which can then act as a
particle acceleration mechanism. Under these conditions, it is also possible to show in a straightforward
fashion that GMR is intimately tied to the formation of flux ropes.
In order to estimate the maximum values of! that can be found in reconnection processes in the
magnetotail or the solar corona we use a previous simulation of magnetotail dynamics /11/ to calculate
the change of reconnected flux with time. The result is shown in Figure 2, together with the associated
value of !max.
We find that typical values of !max range around 40kV, but that higher values up to 100kV are also
possible. A re-scaling to solar corona parameters /10/ leads to significantly higher values, in this case
10-25GV (Note that /10/ contains a typing error, the MeV should be GeV instead).
Whether these energies are actually obtained by one or another particle sub-population is a question
that can only be addressed if other physical processes, such as the adiabaticity/non-adiabaticity of ions
in the magnetotail, and the Dreicer electric field limit /12/, are taken into account also.
~)ASR
13~4-Q
(4)252
M. Hease and 1. Birn
20
120~
..,,,.,,...~...,...,...,...,...
max ir*.~atedparalel etedric field (kVJ
100 120 140 160 180 200 220 240 260
time
Fig. 2 Time evolution of the reconnected flux taken from the simulation of Hesse and Birn
/11/. Also shown is the derivative of the former, yielding the maximum value of the integrated
equivalent field aligned potential drop!.
SUMMARY
We have presented an extension to the theory of General Magnetic Reconnection (GMR), to draw
conclusions relevant to particle acceleration by magnetic field-aligned electric fields. We derived a new
analytical relation between the rate of change of newly connected magnetic flux and the maximum possible value of the parallel component ofthe electric field integrated along the magnetic field, equivalent,
but not identical, to a field-aligned potential drop. The maximum value of this potential drop proved to
be identical to the integrated electric field at a neutral line, in a reconnection process yielding the same
rate of reconnected flux. An estimate of the maximum values of the equivalent potential drop using
three-dimensional MilD simulations resulted in values around 40kV for magnetotail applications, and
10 or more GV for applications to the solar corona. Clearly, more work is necessary to clarify details
of the application of the rigorous theoretical results presented in this paper. Applications will have to
take into account, e.g., other plasma physical properties such as collisional effects related to Dreicer
electric field limits, and particle nonadibaticity.
Acknowledgments. This work was supported by NASA, and DOE/OBES.
REFERENCES
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Monogr. Ser., vol. 30, AGU, Washington, D. C., 1984.
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AGU, Washington, 1984.
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