IOP PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. T142 (2010) 014032 (12pp)
doi:10.1088/0031-8949/2010/T142/014032
Magnetic field line reconnection in
the current systems of flux ropes and
Alfvén waves
W Gekelman1 , E Lawrence1 , A Collette1 , S Vincena1 ,
B Van Compernolle1 , P Pribyl1 , M Berger2 and J Campbell2
1
2
UCLA Department of Physics and Astronomy, Los Angeles, CA 90095, USA
University of Exeter, Exeter, Devon, UK
E-mail: gekelman@physics.ucla.edu
Received 31 May 2010
Accepted for publication 5 June 2010
Published 31 December 2010
Online at stacks.iop.org/PhysScr/T142/014032
Abstract
Magnetic field line reconnection is still considered, by some, to be one of the most important
topics in plasma physics. It has been in this category for close to 30 years and the ‘problem of
reconnection’ has still not been solved. Magnetic field topologies are part and parcel of the
current systems within a plasma whatever their source. Plasma currents may initially be
induced or injected but they soon become entangled or part of the currents of plasma waves,
flows and structures. We first present experimental results of undriven reconnection, which
occurs when two magnetic flux ropes are generated from initially adjacent pulsed current
channels in a background magnetoplasma (length 18 m, diameter 60 cm). The second example
presented is the three-dimensional (3D) magnetic fields and currents associated with colliding
laser-produced plasmas. The currents in this situation are those of shear Alfvén waves. The
wave magnetic field is a small fraction of the background field; nevertheless, reconnection
regions, multiple magnetic ‘X’ points (which are 3D) and induced electric fields are observed.
The first involves the interaction of magnetic flux ropes and the second localized reconnection
sites in the current system of Alfvén waves.
PACS numbers: 96.60.lv, 47.32.cf, 52.35.Vd
(Some figures in this article are in colour only in the electronic version.)
to occur locally when an instability raised the local current
density, became large and triggered reconnection which
then propagated around the torus [6]. Three-dimensional
reconnection has been observed in an experiment in which
two magnetic flux ropes, produced by plasma guns, without
the presence of a background plasma, merge in three
dimensions [7] and recently in the magnetic fields of Alfvén
waves [8]. A glance at the structure of flux ropes as mirrored
in satellite x-ray photographs emerging from the solar
surface is enough to convince one that the reconnection
there must be 3D. It is thought that reconnection leads to
the dissipation of magnetic energy in coronal mass ejections
(CMEs) [9], flare loops [10], compact flares [11], and x-ray
bright points [12]. Systems undergoing reconnection such as
CMEs are conjectured to have a fully 3D nature [13] as well
as the magnetic fields near binary stars [14] and accretion
1. Introduction
Magnetic field line reconnection has been an open topic
for many years and considered by some to be one of the
unsolved mysteries in plasma physics. It was first recognized
as an important candidate of the rapid heating of the solar
corona [1] and has been seen and observed directly in
dedicated laboratory experiments [2], and indirectly in
fusion experiments [3]. There have been many review
papers on reconnection, the most recent by Zwibel and
Yamada [4]. To date, most reconnection experiments have
been inherently two-dimensional (2D) because of boundary
conditions or in the manner that the reconnection was forced
to occur [5]. Some recent experiments were designed to
allow for spontaneous, localized reconnection to occur. For
example, in a toroidal geometry, reconnection was observed
0031-8949/10/014032+12$30.00
1
© 2010 The Royal Swedish Academy of Sciences
Printed in the UK
Phys. Scr. T142 (2010) 014032
W Gekelman et al
Figure 1. The LAPD device. The plasma column is produced with
a dc discharge. The plasma parameters are column length 18 m,
n 6 2.5 × 1012 cm−3 , 200 G 6 B0z 6 2.5 kG, 0.25 eV 6 Te 6 7 eV,
Ti = 1 eV, Ar, He, Ne or controlled mixtures. The device has 450
access ports, many with pump down ports allowing the removal or
introduction of probes and beam while the device is running.
Figure 2. Schematic diagram of the LAPD source and device. The
yellow and purple rings are solenoidal electromagnets, which make
the axial magnetic field Bz . The cathode is a Ni sheet indirectly
heated to emission temperature: 900 ◦ C. A Mo anode is located
30 cm from the cathode. The discharge current 1 kA 6 ID 6 10 kA
between the cathode and anode is initiated with a 4 farad capacitor
bank and transistor switch. The bulk of the plasma carries no net
current and is quiescent.
discs [15]. These and many other cases are discussed by
Priest and Forbes [14] in their comprehensive book ‘Magnetic
Reconnection’.
In this paper, we report on two very different experiments
that are fully 3D and in which reconnection occurs. The first
experiment involves the interaction of magnetic flux ropes in
a background, magnetized plasma. In the second experiment,
localized reconnection events are embedded within the current
system of Alfvén waves. Both experiments are well diagnosed
with plasma parameters and magnetic field measured as a
function of time at thousands of spatial locations.
Figure 3. Cartoon of the magnetic field lines at four radial positions
in a flux rope. The pitch of the magnetic field varies as a function of
the radial position.
2. Experimental setup
field lines within them are just like the twisted fibers that
comprise an ordinary rope. A cartoon of a flux rope is
shown in figure 3. Flux ropes are produced whenever current
flows along a background magnetic field. Flux ropes need
not be straight. For example, the curved solar arcades and
prominences are thought to be flux ropes [22]. Flux ropes
observed by the satellites close to the earth are thought to have
their origin in coronal mass ejections that reach all the way to
the earth [23].
In the first experiment [24] discussed here, two magnetic
flux ropes [25] are created by drawing current from two
cathode emitters (2.6 × 2.6 cm). Each cathode emits 30 A
of current, and both are biased with respect to a mesh
anode (Vbias = 100 V, dia = 16.5 cm) which is 9 m away. The
cathodes are pulsed on when steady state background plasma
had been established. The experiment is highly reproducible
E
and magnetic field data, d B/dt,
is acquired every 4 µs for
2 ms at 20 000 spatial locations throughout the volume. The
magnetic probes are calibrated using a known source and
a vector network analyzer for their frequency response.
E
The signals are numerically integrated to yield B(x,
y, z, t).
Figure 4 shows some of the field lines derived from the
experimental measurements as well as some features of the
flux rope generation.
The flux rope currents heat the background plasma as
shown in figure 5. The electron temperature was measured
The experiments were performed in the upgraded Large
Plasma Device (LAPD) at UCLA. (The original LAPD device
was half the length [17].) A photograph of the device is shown
in figure 1. The plasma is produced with a dc discharge using
an oxide-coated cathode [18] as shown in figure 2.
The plasma is switched on for 15 ms and operated at a
repetition rate of 1 Hz for up to 4 months. The plasma is
stable and experiments, can be repeated millions of times if
necessary. In these experiments a single probe, for example,
E
three-axis differentially wound magnetic pickup (d B/dt)
probe (typically several mm in size) is moved over 1000–2000
locations in a plane transverse to B0z (the background constant
magnetic field) removed from the vacuum system and then
placed in a port at a different z location. This is repeated until
the data sets sampled at up to 30 000 locations are acquired.
We describe two very different experiments in which
magnetic field line reconnection occurs. In both cases, the
current systems and the reconnection sites are fully 3D.
2.1. Experiment 1: interaction of magnetic flux ropes
Magnetic flux ropes are aptly named, the first observation of
flux ropes in a planetary ionosphere was made at Venus [19].
Flux ropes were since seen in the Martian ionosphere [20] as
well as in the lower ionosphere of Titan [21]. The magnetic
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Phys. Scr. T142 (2010) 014032
W Gekelman et al
Figure 4. The two LaB6 electron emitters are located at z = 0. Magnetic field lines, which originate at the guns, are shown in yellow (light
gray) and blue (dark gray) to differentiate them. The magnetic field in a plane (z = 64 cm) is shown on the right. A magnetic ‘X’ point is
clearly visible. The snapshots shown are at t = 1.699 ms (the currents are pulsed on at t = 0).
Figure 5. Temperature profile associated with a single current channel/flux rope. The electron current more than doubles the electron
temperature of the background plasma. The figure insert on the lower right is the electron temperature profile along the white line (y = 5)
through the data plane.
with a swept Langmuir probe at 2601 spatial locations on the
data plane. Analysis of the swept probe also determined the
plasma density, which is also observed to increase by 40% in
the center of the current channel (the background plasma is
approximately 50% ionized).
The self-magnetic fields are of the order of 2% of
the background field, which is enough for them to interact
over their length. The currents of the ropes exert mutual
JE × BE forces causing them to twist about each other and
merge. The resulting magnetic field has both twist (180◦ )
and writhe (180◦ –270◦ ) components. The currents are kink
unstable [26] with q = 2πa B z /L B θ ' 0.7, where a is the
diameter of the current channel and L its length. The
presence of flow can further destabilize the kink mode [27];
however, in this experiment the Mach number of the measured
parallel flow, M = vflow /cs ' 0.5, makes the correction small.
The predicted frequency of the
√ mode with rotation
√ kink
at criticality is f ' (vflow /2L 2) 1 − M 2 = 300 Hz. The
observed frequency is 5 kHz and increases as the discharge
current goes down. This disagrees with the model in [28]. The
E 0 ) is pinned to
current density (calculated from Ej = ∇ × B/µ
the sources at z = 0 but further away the centers of the current
channels rotate in the X–Y plane. The motion of the two flux
ropes is presented in figure 6.
When they move towards each other, magnetic field line
reconnection is triggered. At a distance of z > 600 cm, the
currents have filamented after merging, and reverse currents
are observed. This is a sign of reconnection. In addition,
the currents are observed to filament after merging. The
magnitude of the current density on the same three planes is
shown in figure 7(a). The integral of the current density over
the plane at z = 63 is within a few percent of that measured in
the wires leading to the cathode, but the discrepancy increases
to 20% for the furthest planes. This indicates that some current
leaves the measurement planes.
When the background field is added to the field lines
shown in figure 4, the reconnection regions become elongated;
there is no separatrix. The transverse magnetic field in a
single plane 6.6 m from the current sources is shown in
figure 8. A magnetic sheet is clearly seen in between the
current channels. A times series, or movie, of BE ⊥ (E
r , t) clearly
shows merging as field lines join, reconnect and move away
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Phys. Scr. T142 (2010) 014032
W Gekelman et al
3.5
Δ s (cm)
3.0
2.5
z = 507 cm
collisions
2.0
z = 830 cm
1.5
1.65
1.70
t (ms)
1.75
Figure 6. Separation of the flux ropes as a function of time. This is obtained from the determination of the center of each current channel
from data on 11 planes. Close to the current sources the flux ropes do not move. Some motion is observed 6 m from the sources and
significant motion at δz = 8.3 m.
Figure 7. (a) Current density at t = 1.615 ms. Far from the source (z = 830 cm) reverse currents are observed. (b) Field lines puncturing
planes at three axial locations. The puncture points are colored in order to identify them. The blue (dark gray) lines in the center are the
intersection of the QSL (discussed later) with the data plane.
at nearly right angles. Inspection of the transverse field also
reveals multiple reconnection sites throughout the volume
with time-dependent locations.
The next step is to add the measured axial component, Bz ,
of the flux ropes. This is displayed in figure 9. The field lines
start at 6.6 m from the flux ropes as in figure 8 and rapidly
wander out of the plane. The fully 3D nature of a reconnection
site without the background field, in the region where flux
ropes are reconnecting, is shown in figure 10. When the 270 G
background field is added, the field lines are as depicted in
figure 4.
Does the fact that reconnection apparently shows up in
the transverse field and can be recognized when the axial field
of the currents is present have any significance at all? When
the guide field is added, the field lines and presumably the
reconnection region get stretched out. For strong guide fields,
does the reconnection region disappear entirely? Theoretical
studies of reconnecting plasmas with oppositely directed,
merging magnetic field lines and with arbitrary guide are
prone to a tearing-type instability [28] however, once the
plasma beta exceeds (m e /m i )1/4 the guide field ceases to have
any effect on the growth rate. Computer simulations with
simple X point or neutral sheet geometries have been done
with guide fields of varying strengths. A 2.5D Hall code found
the reconnection rate to drop by a factor of 2.5 when the
guide field was increased to 4 times the transverse field [29].
A 3D particle in cell computer simulation of a Harris sheet
with a guide field [30] shows that the reconnection time scale
increases with increasing guide field and the reconnection rate
is a bit smaller (∼ 20%) for simulation parameters in which
the guide field is of order of the transverse field and is a
factor of 2 smaller for simulations in which Bguide /Bt = 5.
There was a long-standing belief that the presence of a
strong guide field would slow down the growth rate of
instabilities that affect the reconnection rate [31], but a recent
simulation [32] with large system size has shown that this is
4
Phys. Scr. T142 (2010) 014032
W Gekelman et al
4.1
log10(Q) (z = 64 cm)
9
3.4
8
y (cm)
2.7
7
2
6
1.4
5
0.68
-2
-1
0
x (cm)
1
0
Figure 10. The QSL in the flux rope experiment 64 cm from the
current sources that make the flux rope. Contours of the current
density inside the ropes are shown as white lines. The S-shaped
QSL contains field lines, which are close to one another on this
plane; however, their separation diverges hyperbolically as they
approach the anode nearly 8 m away.
Figure 8. B⊥ δt = 1.7 ms, δz = 6.6 m. The background field of
600 G and Bz , the axial component of field of the flux tubes, are not
included in the field line calculation. The transverse extent of the
plane is 12 cm in both the x- and y-direction. The z-direction (which
is the direction of the background magnetic field) points out of the
page.
A method used by the solar physics community to
identify the volume in which reconnection is operative is
the identification of a quasi-separatrix layer [33] (QSL). Two
closely spaced field lines, which enter the QSL, wind up
at very different spatial separations at finite distances along
the current channel. Outside the QSL, neighboring field lines
remain close by. The QSL can be found by moving the
footprint of a field line a small distance from its original
position (x, y) on a plane (z = z 0 ) and measuring where it
winds up (X, Y) on a plane far away (z = z 1 ). Consider the
quantity
v"
#
u u ∂ X 2 ∂ X 2 ∂Y 2 ∂Y 2
t
+
+
+
.
(1)
N=
∂x
∂y
∂x
∂y
Figure 9. Field line approximately 6.6 m from the start of the flux
ropes and (δt = 1.7 ms) after their onset. The ‘X point’ is fully 3D.
The background field is not shown in the diagram and stretches out
the field lines as depicted in figure 4.
If N 1, then the field lines used to generate it are said to
be on a QSL. The concept of the QSL has been extended to
encompass situations in which there is a guide field [34]. The
slip squash factor Q is defined by
not the case and that the relevant waves were kinetic Alfvén
waves.
Evidence for the influence of reconnection in the flux
rope experiment is in the observed return currents, which
evolve after τ >1.0 ms. The currents close to their source are
tethered to the emitting cathodes and always appear as in
figure 7(a). The currents are collected on a remote anode
(δz = 9 m, dia = 16 cm). Return currents become prevalent
at δz > 6 m from the cathodes. When the current density is
integrated
over each of the data planes, the total current,
R
I = JE · n̂ dA, is conserved over the 8 m between the sources
and anode. The return currents are sheet like (figure 7(a)) and
form loops. The return currents are thought to be the result of
the inductive electric field produced during reconnection.
N2
.
Q=
Bz (z0 ) Bz (z1 ) (2)
The squashing degree may be used as a measure of 3D
reconnection [35]. In the case of this experiment, the guide
field is much stronger (99.95%) than the field due to the
currents and Q ' N 2 .
In order to compute Q, many field lines must be computed
very accurately. In the magnetic field data set, there are
regions where the divergence of B is small but does not
vanish. This can come from small errors in the probe
head angle and shot-to-shot variation, but even interpolation
between grid points can introduce a divergence. These errors
5
Phys. Scr. T142 (2010) 014032
W Gekelman et al
begins when two dense, laser-produced plasmas collide in
a background helium magnetoplasma [40]. Electrons from
the dense (n lpp > 103 n 0 ) plasmas stream away from the
carbon targets and a return current in the background plasma
neutralizes the space charge. The details of the experiment are
given in [8]; what is germane is that the interaction produces
current channels that merge and subsequently filament before
merging again. The background magnetic field, Bz0 , is 600 G
and data were acquired at 30 000 spatial locations and at 10 ns
intervals. The current systems have multiple reconnection
sites, one of which is shown in figure 13.
The three-axis magnetic probe used to collect the data
E
is sensitive to d B/dt
and the constant background field does
not show up when field lines are calculated after integration
of the raw data. The same probes were used in the flux
rope experiment. Each component of the field is integrated
E r , z, t) and the current derived JE = (1/µ0 )∇ × B.
E
to get B(E
E r , t) is calculated from a volumetric
The vector potential A(E
integral over the current
Z
µ0
JE(E
r 0)
(3)
d3r 0 .
AE =
4π
|(E
r − rE0 )|
Figure 11. The QSL for Q = 200. The field lines start at the rear
plane (z = 64 cm) within the two current channels and the endpoints
are at z = 830 cm and t = 1.688 ms. The red (x) arrow points in the
x-direction and the blue (z) along the background magnetic field.
can accumulate in the field line integration. Therefore, we
employ a technique typically used in simulations and space
observational data known as ‘divergence cleaning,’ [36].
Briefly, the magnetic vector potential A is calculated from
the magnetic field using Fourier transforms and then fitted to
splines, and finally taking the curl of A analytically gives us
the divergence-free magnetic field.
The distribution of Q at z = 64 cm during the first
collision at t = 1.688 ms is shown in figure 10. The most
salient feature is the S-shaped layer between the two current
channels. If we extend field lines at a particular Q contour,
we can visualize the 3D nature of the QSL. In figure 11,
we show the surface comprising field lines seeded along the
Q = 200 contour. The QSL is bordered by the flux tubes
initially associated with the two current channels. The area
bounded by the QSL is the same in every plane; however, its
character is different in the center plane. The surface is, in fact,
a hyperbolic flux tube [37]. This is obvious from the cut planes
showing the QSL in figure 7(b). The Hyperbolic flux tubes
have been postulated to exist in the solar corona when merging
occurs between upwelling the magnetic field associated with
two or more bundles of field lines, which emerge from the
solar surface. The prominences are nothing but curved flux
ropes that are anchored (or line tied) to the solar surface. The
QSL observed does not lie either in the center of flux rope or
within the main current channels associated with them. The
maximum value of Q peaks when the flux ropes become close
to one another in the course of their writing about, and when
reverse current sheets develop. Figure 12(a) shows the current
system as well as the Q = 1000 when the return currents have
developed. Figure 12(b) is a close-up of the return currents
in the axial region past the region where the QSL has the
smallest radial extent (figure 7(a)). Note that the return current
is close to the QSL and at or nearby the HFT as predicted [38].
Computer simulations [39] of two merging flux tubes have
shown that the reconnection region between them is a region
where current is induced and is the same spatial region as the
QSL just as in this experiment.
The background magnetic field does not contribute to the
E
induced field ( EE I = −∂ A/∂t).
In figure 13, the induced field
is shown as sparkles, which are larger in regions of large
| EE ind | > 1 V cm−1 . The field is largest at this instant of time
in the reconnection region; however, induced fields also show
up in the center of the current channels associated with the
process. It has been established that this entire system is that
of shear Alfvén waves. In a magnetoplasma that supports
Alfvén waves, all low-frequency current systems f < f ci are
those of Alfvén waves. The currents must have associated
return currents, which are diffuse and outside the main current
channels. The cross-field current is carried by ions [41].
As mentioned, the guide field of 600 G is not included in
figure 13. To show the effect of a guide field, it is instructive
to superimpose the guide field in steps. In figure 14, a small
guide field of 0.5 G has been added. The field lines are
stretched along the axis of the device, and the separatrix
geometry in figures 1 and 4 no longer exists. When the full
600 G background field is added, the field lines look nearly
straight. The induced electric field does not change.
The experiments raise several questions. In both cases,
reconnection occurs in the presence of a guide field; there
is no separatrix. Induced electric fields occur in both cases,
sometimes easily associated with reconnection regions but
also occurring in current channels. The reconnection rate is
nothing but the electric field generated by flux annihilation. It
has been argued that this is due only to the induced magnetic
E
field. The total field is EE = −∇8 − ∂ A/∂t.
In an early
reconnection experiment [42] and a recent one also involving
flux ropes [7], it was discovered that both of these terms exist
and have opposite signs thereby greatly reducing the actual
electric field in the plasma. The electric field and resistivity
are part of the generalized Ohm’s law and those working in
the field tend to studiously ignore most of the terms. In some
cases [43], the resistivity is assumed to be Spitzer, which
is derived from a measurement of the electron temperature.
There is no a priori reason that this is the case in a plasma
undergoing reconnection in which localized currents exist.
2.2. Experiment 2: interaction of Alfvén wave currents
The flux rope experiment is contrasted with 3D reconnection
arising in a very different set of circumstances. The event
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Phys. Scr. T142 (2010) 014032
W Gekelman et al
Figure 12. (a) Volumetric data of magnetic field, current and the QSL at t = 2.5 ms (The currents that make the flux ropes are switched on
at t = 0). The QSL is shown as a magenta (gray) surface. Magnetic field lines are rendered in red (dark gray) and yellow (light gray) (the
background field is included). These field lines are seeded from the end of the volume nearest to the LaB6 cathodes. The arrows are the
local current density. Magnetic field lines in a plane transverse to the background field are drawn for δz = 1.8 m. The marker plane (x–z) on
the bottom is located at y = −6 cm. The grid lines occur at δx = 2 cm and δz = 30 cm. (b) Close-up of the return current region at the same
time in figure 12(a). Here the magnetic field lines are drawn in a plane at δz = 7.6 m. An ‘X’ point is visible in the center (see figure 8). The
QSL is drawn in blue (light gray surface). The arrows of current density are reversed close to the transverse plane. In figures 12(a) and (b)
the 3D field lines start close to the two emitters. They go through the ‘O’ points in the closer plane (figure 12(a)) but not the further one.
This reflects the complex current structure seen in planes in figure 7(a).
Other assumptions are used to evaluate pressure gradients and
additional terms in the Ohm’s law. Measurement of all terms
on a spatial grid commensurate with a reconnection event is
difficult and generally not done.
In this case, we do not claim to have measured the plasma
resistivity of the total electric field either in 3D or even in a
plane. However, ∇φ was not measured in this experiment.
It is instructive, however, to look at the inductive part
E
as
of the electric field. Figure 12 displays EE ind = −∂ A/∂t
‘sparkles’ with size and brightness being maximum close to
the reconnection region. There are also induced currents in
the main channels, ‘O points’, of the wave. These are shown
E
in figure 14 in which isosurfaces of |∂ A/∂t|
= 3.5 V m−1 are
shown where the induced field points towards the targets (gold
isosurfaces) and a return EMF (purple) points in the other
direction. To show the 3D character, a background magnetic
field of 0.5 G is added to the 3D time-varying field measured
by the probe.
The QSL may be evaluated for the Alfvén waves using
the same methodology used for the flux ropes. The volume
over which the data were acquired was on one side of the
targets and extended for 6 m (of the order of two Alfvén
wavelengths) along the background magnetic field and most
of the plasma in the transverse direction. The QSL is shown
for the case without the 600 G guide field in figure 15. The
Q = 100 contour is concentrated in the region between the
two current channels of the Alfvén waves and in the region
where the induced electric field peaks (see figure 13).
When a background magnetic field of 100 G is used, the
QSL becomes smaller, of order 3. When the full background
field is used in the calculation the value of Q drops below
2; it all but disappears. Does it make sense, therefore, to
discuss a QSL in this context? The QSL will vanish when
the field lines become completely straight, what matters
is how much they twist over the length of the interaction
and if magnetic reconnection occurs. In this experiment, the
interaction region was too short. If the LAPD were 100 m
long (and collisionless so the waves are not damped), then
there would be sufficient field line twisting to have a robust
QSL. In space, for example, where distances are thousands
7
Phys. Scr. T142 (2010) 014032
W Gekelman et al
Figure 13. Magnetic field lines from data acquired 5.12 µs after
two targets (drawn to scale in the rear of the figure) are struck by
lasers. The field lines in the data plane shown start at 85 cm from the
targets. The sparkles are drawn to mark regions of large
E
EE ind = −∂ A/∂t
' 1 V cm−1 , the largest fields observed. The
contribution from the background magnetic field is not shown.
Figure 15. Alfvén wave QSL. The background magnetic field is
left out of the calculation. t = 5.2 µs. The plane shown is 16 cm on
a side. The plasma column is 60 cm in diameter. The plane shown is
85 cm from the targets as in figure 13.
Figure 14. Magnetic field lines at t = 5.12 µs shown with a
superimposed guide field of 0.5 G. Two representative lines are
drawn in red (light gray) and blue (gray) so that they may be
distinguished. The gold isosurface within the field lines is the
induced electric field, and the purple outer isosurfaces show an
induced field of the same magnitude but in the other direction. One
of the two carbon targets (to scale) that are the sources of the dense
plasmas appears on the right. The transverse extent of the data is a
square of 40 cm on a side. The axial extent is 6 m.
Figure 16. Alfvén wave QSL. t = 5.2 µs. The plane shown is
16 cm on a side. The plasma column is 60 cm in diameter. The plane
shown is 54.5 cm from the targets. The uniform background
magnetic field used in this calculation is 100 G.
field. In this case, a gauge-invariant and physically meaningful
helicity is defined by setting the helicity of the vacuum
field (i.e. the constant background field) to zero [46]. This
procedure is analogous to setting a ground-state voltage
to zero. Letting P be the constant background field, BE 0z ,
BE the total field ( BE ropes + BE 0z ), AE p the vector potential of
the background field and AE the total vector potential, the
generalized helicity can be written [47] as
Z
E dV.
H = ( AE + AE p ) · ( BE − P)
(5)
of wavelengths, it would make sense to consider the QSL
associated with Alfvénic current systems. In essence, it does
not matter if the currents in a system undergoing reconnection
are driven (flux rope experiments) or if they are the currents
of waves (laser plasma collision experiment) as long as the
flux is destroyed and the magnetic topology is sufficiently
changed.
2.3. Helicity and resistivity
The magnetic helicity is a measure of the linkage, twist
and kinking of magnetic field lines [44]. In a closed simply
connected volume V (bounded by a flux surface), the magnetic
helicity is defined [45] by
Z
E · BE dV .
H= A
(4)
This is always gauge invariant, except for fields inside
periodic volumes (as found in some numerical simulations)
with net flux [48]. In the flux rope experiment, the relevant
fields are all located in the volume between the cathode
emitters and the anode 9 m away. The fields in this volume
have been carefully probed in that volume so that a good
estimate of the generalized helicity is possible. The helicity
was evaluated in the flux rope experiment using the divergence
cleaned data and equation (5). Figure 17 shows an isosurface
V
In our case, the field lines cross the boundary of the probed
volume at the ends of the cylinder due to the background
8
Phys. Scr. T142 (2010) 014032
W Gekelman et al
Figure 17. Several magnetic field lines of the full magnetic field of
the flux ropes and a surface of constant helicity density. The field
lines are started at δz = 1 m from the sources. The constant
background magnetic field is 270 G and points from left to right.
The diagram is generated at τ = 1.68 ms.
Figure 18. The winding number between two field lines measures
how much they rotate about each other between the end plates. For
the field lines pictured, labeled x and y, the winding number
w = 1/(2π)(θb − θa ) + 1.
of constant helicity density as well as magnetic field lines,
which are started near the two emitters.
We can give a geometric interpretation of the helicity
in terms of winding numbers. A pair of field lines will
wind about each other through some angle (θ) as they
travel between the cathode and anode (see figure 18). If we
perform a double sum (or integral) over all pairs of field
lines, we obtain the generalized helicity of equation (5). Let
w = θ/(2π ) be the winding number between two lines. We
cannot simply compare the orientation of the endpoints of the
lines, in case the change in angle is more than 2π. Instead, the
winding number between two curves labeled a and b may be
written as
Z
1
1
dE
r ab
,
wab =
ẑ · rEab ×
(6)
2
2π
dz
rab
edge (figure 5) also cause flow. Reconnection can often be
accompanied by resistive heating due to a change in the
plasma resistivity. It is very difficult to measure all the
terms in Ohm’s law; this has been done in a reconnection
experiment years ago [49]. Often when resistivity is measured,
assumptions are often made such as ignoring the scalar part
of the potential or assuming the resistivity to be classical.
The same hold true for estimations of the helicity that rely
on unproven symmetry assumptions. In this experiment, the
volumetric data set allows for evaluation of the time-varying
magnetic energy as well as the helicity. There is an inequality
that ties the rate of change of magnetic field in the plasma
volume, the change of the helicity and the resistivity [50]
dH 2
Z
1
dt
dM ; M =
B 2 dV,
(8)
η>
2µ0
2M dt where rEab is the relative position vector pointing from one
curve to another.
We split the field into n flux elements following n
field lines starting from constant regularly spaced points on
the cathode, each representing 1/n times the total flux 8.
Summing over every pair of flux elements gives
H=
n
n
82 X X
wab .
n 2 a=1 b=1
where M is the volume magnetic energy and H the helicity.
The lower limit on the plasma resistivity determined from
equation (8) is shown in figure 20. As this contribution is due
to the time-varying helicity the background plasma Spitzer
resistivity is added.
The helicity in the laser plasma experiment was evaluated
using the same method. Figure 21 rendered from data acquired
at time τ = 0.6 µs after the targets are struck. Field lines with
the background field excluded are shown and two partially
transparent isosurfaces of helicity density (calculated using
the full magnetic field, equation (5)) are shown as well. The
helicity density is greatest in the current channels and the area
between them where reconnection is observed to occur. As the
current system from the reconnecting Alfvén wave dissipates
the magnetic field and the helicity dies away or leaves the
volume measured. The time behavior of the helicity density
integrated over the volume that the data were acquired was
observed to oscillate in time [8]. The magnetic field data were
acquired on only one side of the targets and the Alfvén waves
are radiated both upstream and downstream with respect to
the targets. It is possible that the total volume helicity may not
oscillate if the helicity on the side of the targets that was not
probed oscillates one-half period out of phase.
(7)
The self-helicities (e.g. w11 ) due to twist within each
flux element are ignored; their contribution to the total is
negligible.
Calculating helicity by summing winding numbers
allows us to concentrate on any subset of field lines in a
gauge-invariant and geometrically meaningful way. For the
entire field, it gives an answer equivalent to the generalized
helicity of equation (5), but will still be well defined even if
some field lines have not been probed. For this reason, we
have used this method to calculate the helicity as a function
of time, although equation (5) gives almost identical results.
Figure 19 shows the time history of helicity, exhibiting regular
oscillations.
When reconnection occurs, there is a transfer of energy
from magnetic fields to plasma flow and heating. It is difficult
to isolate heating and flows as the current within the ropes
heat the electrons and the pressure gradients on the current
9
Phys. Scr. T142 (2010) 014032
W Gekelman et al
Figure 19. The magnetic helicity (a) during one oscillation between t = 1600 µs and t = 1825 µs and (b) over an entire run. (c) The time
derivative of the helicity.
Figure 20. Resistivity estimated from equation (5) normalized to
the classical Spitzer resistivity due to electron ion collisions
ηSpitzer = A(kTe /m e )− 3/2, where A is a constant. The resistivity is
averaged over the volume employed for the helicity calculation. The
resistivity peaks at the times that the two current channels are
moving towards each other as shown in figure 6.
Figure 21. Field lines and surfaces of helicity density (red
Hd = −1000, blue Hd = −500 and pink Hd = 500 G2 cm) at
τ = 0.6 µs after the targets are struck. The field lines are started 1 m
from the targets, which are shown as gray cylinders in the
background.
3. Summary and conclusions
in the flux rope experiment. In both cases, a QSL has been
identified; however, in the colliding plasma experiment the
lower R makes it difficult to compute the pertinent QSL
because of the ‘shorter’ axial extent. In both cases, an induced
electric field and helicity generation are observed. The time
The two experiments discussed involve 3D reconnection in
the presence of a guide field. If one considers the ratio
R = B⊥ /B0 , where B⊥ is the transverse field due to currents
be they due to waves or drifting electrons, R is much larger
10
Phys. Scr. T142 (2010) 014032
W Gekelman et al
rate of change of magnetic helicity was used to put a lower
bound on the plasma resistivity in the flux rope experiment.
The resistivity exceeds the classical Spitzer resistivity by a
factor of 5 when the current channels move towards each
other and reconnection occurs. This was done without trying
to measure all the terms in Ohm’s law. In a very different
reconnection experiment [51], the transverse resistivity was
obtained (by measuring the time-varying magnetic field and
electron temperature in a plane and using assumptions about
symmetry) and coincidentally is of the same order as η/ηSpitzer
in the flux rope case. It is suspected that anomalous resistivity
is the result of wave particle scattering and a variety of
waves have been considered to be candidates [52]. In the
colliding plasma experiment, one surmises that reconnection
is occurring as it appears in the transverse fields and induced
electric fields that are observed. However, this is not a
certainty as the QSL is very weak.
A well-described phenomenon that has a similarity in
this respect is tearing modes in tokomak devices [53]. Here
a helical instability causes local reconnection to occur which,
in turn, affects internal transport and the plasma stability.
The change in magnetic field is far smaller (less than 1%)
than the strong toroidal field that confines the plasma, and
if the electrons carrying the current went around the device
only once their orbits would be insignificantly perturbed. As
fusion plasmas are nearly collisionless, the current-carrying
electrons make thousands of transits before a collision and the
reconnection is a cumulative effect. This in essence makes the
plasma much longer and if a QSL could be calculated for a
tokomak it also would be weak in the single-orbit case.
Every circumstance in which reconnection occurs in
nature has different plasma parameters and boundary
conditions, and there is no single-wave candidate that can be
invoked to explain anomalous resistivity for all of them. It is
also possible that nonlinear structures such as electron phase
space holes could contribute to the resistivity. These have been
observed in a variety of different regions in space including
those in which reconnection occurs [54]. They have also been
seen in a reconnection experiment [55] and a reconnection
computer simulation [56].
One constantly hears that reconnection is still one of the
great unsolved problems in plasma physics. The experimental
measurements presented here lead one to suspect that
reconnection is not an independent topic which can be studied
in isolation. It is not a problem at all. Magnetic field line
reconnection is part of a variety of phenomena associated with
the much broader subject of 3D current systems in plasmas.
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Acknowledgments
The UCLA authors acknowledge support by the Office of
Fusion Energy Science at the Department of Energy and the
ATM and PHY divisions of the National Science Foundation.
The work was carried out at the Basic Plasma Science Facility
at UCLA. The BaPSF is funded by DOE and NSF. We also
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