PHYSICS OF PLASMAS 19, 102102 (2012)
Morphology and dynamics of three interacting kink-unstable flux ropes
in a laboratory magnetoplasma
B. Van Compernollea) and W. Gekelman
Department of Physics, University of California, Los Angeles, 1000 Veteran Avenue, Suite 15-70,
California 90095, USA
(Received 24 August 2012; accepted 12 September 2012; published online 3 October 2012)
Flux ropes are ubiquitous in space and solar plasmas. Multiple adjacent flux ropes are commonly
observed both in the solar corona and in the earth’s magnetotail. The interaction of adjacent flux ropes
is often dynamic and can lead to magnetic reconnection. In this paper, the interaction of three flux
ropes is studied in a low b background laboratory magnetoplasma. The magnetic structure of the flux
rope is produced by the poloidal field of a field-aligned finite sized current which adds to the guide
magnetic field and creates the typical helical field line structure. Each rope produces magnetic fields
on the order of a few percent of the guide field. Volumetric magnetic field data were acquired and the
magnetic field structure and dynamics of the flux ropes can thus be reconstructed. The flux ropes are
found to propagate at the Alfven speed. Merging and bouncing of the flux ropes have been observed.
The ropes twist and writhe as they propagate through the plasma. They are line tied and clearly
separate at the cathode end but further away they merge into one extended rope. The steady state of
the flux ropes is characterized by a rotation of the three flux ropes as a whole. At the same time, the
flux ropes are twisting around each other. Time resolved density and temperature measurements from
C 2012 American
Langmuir probe data exhibit the same dynamics as the magnetic field data. V
Institute of Physics. [http://dx.doi.org/10.1063/1.4755949]
I. INTRODUCTION
Flux ropes are magnetic structures consisting of helical
field lines. The field line pitch increases with distance from
the center, resembling a rope. Flux ropes are ubiquitous in
solar, space, and laboratory plasmas. X-ray images from
STEREO1 or UV images from TRACE2 show the 3D structure of multiple adjacent flux ropes in the solar corona. Models of coronal mass ejections invoke reconnection of flux
ropes for the acceleration of the solar plasma into space.3–5
In the earth’s magnetotail flux ropes and magnetic reconnection are a hot topic, where in situ measurements of flux ropes
are available, e.g., by the Cluster mission.6–8 They enjoy
great interest because of their role as a building block of 3D
magnetic reconnection.9 In 3D reconnection, a current sheet
can break up into small islands called plasmoids which are
none other than magnetic flux ropes.
Flux ropes can be created in the laboratory by a finite
size field-aligned current in a guide magnetic field. The
poloidal field of the current channel adds to the guide field
and results in helical field lines. The poloidal field increases
linearly from the center to the edge of the current channel
resulting in field lines with increasing pitch. Both curved
geometries as straight geometries have been studied in the
past. Laboratory experiments allow for detailed in situ measurements of the properties of flux ropes, a great advantage
over space plasma observations. Furthermore, laboratory
experiments are controlled and configurable. Single flux
ropes or multiple flux ropes can be studied in straight or
curved field geometries in a broad range of parameter space.
a)
Electronic mail: bvcomper@gmail.com.
1070-664X/2012/19(10)/102102/11/$30.00
In the past, several laboratory experiments on flux ropes
have been done. Highly kink-unstable disrupting flux ropes
have been studied in spheromaks.10,11 The impulsive disruption of a stable arched flux rope by a strong ion flow was
shown in a recent experiment at UCLA.12 At the reconnection
scaling experiment (RSX),13 a series of experiments involving
reconnecting flux ropes have received wide attention.14 The
dynamics of the ropes was extensively studied15–17 and compared to theoretical predictions.18 More recently, equilibria of
kink unstable flux ropes were studied at the rotating wall
machine at Madison for both low and high flux rope currents,
through which the transition from diamagnetic to paramagnetic
flux ropes was observed.19 Two long-lifetime, moderately
kink-unstable flux ropes were observed in experiments at the
Large Plasma Device (LAPD)20 and their reconnection was
linked to the concept of quasi-separatrix layers.21–23
In this paper, the interaction of three flux ropes is studied
in a background magnetoplasma. The flux ropes are created
using a hot cathode and remote anode. The magnetic field
structure is measured in a volume covering the full extent of
the flux ropes. The dynamics of the startup phase and the
steady state of the magnetic field evolution and the derived
current density are presented. The magnetic field measurements are complemented by Langmuir probe data of the density and temperature evolution in the flux ropes.
The paper is outlined as follows. Section II describes
the experimental setup and plasma parameters. Section III
examines the magnetic field structure, the current density
profiles and the density, temperature profiles at a given
instant of time. Section IV moves on to describe the dynamics of the system, both during the startup phase, and during
the steady state.
19, 102102-1
C 2012 American Institute of Physics
V
102102-2
B. Van Compernolle and W. Gekelman
Phys. Plasmas 19, 102102 (2012)
II. EXPERIMENTAL SETUP
20,24
The experiment is performed on the upgraded LAPD
at the Basic Plasma Science Facility (BaPSF) at UCLA. The
LAPD is a 21 m long cylindrical device. The plasma is created
through collisional ionization of the neutral fill gas by an energetic electron beam. The electron beam is produced through a
pulsed discharge between a Barium oxide coated cathode,
heated to 850 C, and a 50% transparent wire mesh anode.
Electromagnets provide a uniform axial magnetic field. The
cylindrical plasma column is 18 m long and 60 cm diameter.
In this experiment, the fill gas was Helium at a pressure
of 5:4 105 Torr as measured by an ion gauge. The LAPD
cathode operated at a discharge current of 4 kA and discharge
voltage of 70 V. The machine discharges 24 h a day at 1 Hz
and the discharge length was 15 ms. The density in the
machine was 2:5 1012 cm3 , electron temperature was 4 eV.
The LAPD is readily diagnosable with probe access
every 32 cm along the machine. Probes are inserted in the
machine through ball valves which allow for 3D movement.25
Probes are mounted on an external probe drive system and
can be moved to any (x,y) position with sub millimeter accuracy. The data acquisition system is fully automated and programmable and controls the digitizers and the probe drive
system. Typically, the probe moves through a series of user
defined (x,y) positions at fixed z. At each position, data from
several plasma shots are acquired and stored, before moving
to the next position on the grid. Since the LAPD plasma is
highly reproducible, an ensemble measurement of the plasma
parameters can thus be obtained in a 2D plane at fixed z location. In this experiment, magnetic field measurements were
obtained in 16 2D planes, each 64 cm apart in z, resulting in a
volumetric dataset containing 32 400 positions.
In the experiment, a second cathode and anode are
introduced at opposite ends of the machine, as depicted in
Figure 1. The cathode26 is a 8.25 cm diameter disk of lanthanum hexaboride (LaB6 ), heated to temperatures above
1700 C. At high temperatures, LaB6 is capable of providing
FIG. 1. Cartoon of the experimental setup. The main LAPD cathode-anode
operates at 1 Hz, with 15 ms pulses. After 8 ms, when the LAPD plasma
reaches steady state, the LaB6 cathode-anode is pulsed on for 5.5 ms. The
LaB6 cathode is masked off, essentially creating three equidistant 2.54 cm
diameter electron emitters. Probe access through ball valves at multiple z
locations allows for three dimensional probing of the plasma.
large emission current densities. The 8.25 cm diameter disk is
partially blocked by a carbon mask which has three holes cut
out. The holes are 2.54 cm diameter and are spaced equidistant with center-to-center spacing of 3.81 cm. The carbon
mask effectively reduces the LaB6 disk to three 2.54 cm diameter LaB6 cathodes. Cathode emission will therefore result
in three closely spaced current channels. The LaB6 cathode is
inserted in the machine at the end opposite to and facing the
LAPD source. An anode is inserted 10.86 m away. The anode
is a molybdenum wire mesh with 70% transparency, 25 cm
high by 28 cm wide. Both secondary cathode and anode are
electrically floating with respect to the plasma. A 125 V discharge voltage is applied between cathode and anode, starting
at 8 ms into the LAPD discharge, a time at which the LAPD
plasma has reached steady state. The LaB6 cathode is pulsed
on for 5.5 ms. Throughout the paper, the start of the LaB6
pulse will taken to be t ¼ 0 and the location of the LaB6 cathode as z ¼ 0. The discharge current from the LaB6 cathode is
90 A or about 30 A per current channel. The discharge current
density is 6 A=cm2 assuming uniform emission. Fig. 2 shows
a cutaway of the LAPD with the LaB6 cathode in place and
the orientation of the three holes in the carbon mask. Both
anodes and the LAPD cathode are to the left and out of view.
A low axial magnetic field was chosen in order to ensure
a strong interaction between the flux ropes by maximizing
Bropes =B0 , the ratio of the flux rope poloidal magnetic fields
to the background magnetic field. A field of 330 G was determined to be the minimum field necessary to obtain a large
stable discharge current from the LaB6 cathode. The LaB6
cathode was able to operate at 6 A=cm2 emission current
density at 330 G, which exceeds the threshold for kink instability. Lowering the field further resulted in a substantial
decrease in discharge current from the LaB6 . Fig. 3 shows
the magnetic field profile in the machine. The position of the
LAPD cathode and anode, as well as the LaB6 cathode and
anode is indicated. The LAPD cathode-anode did require a
FIG. 2. Cutaway view, drawn to scale, of a section of the LAPD with the
LaB6 cathode inserted in the machine, and the orientation of the three emitters.
The LaB6 cathode is inserted through a rectangular valve and its axial location
is defined as z ¼ 0. Access to the machine is available every 32 cm axially and
every 45 poloidally, through circular or rectangular ports. Electromagnets
around the machine, drawn in yellow, provide the axial magnetic field.
102102-3
B. Van Compernolle and W. Gekelman
FIG. 3. Magnetic field profile with location of respective cathodes and anodes. The field is 330 G uniform in the probed region between LaB6 cathode
and anode. The LaB6 cathode and anode are positioned at opposite ends of
the machine, 10.86 m apart.
field of 450 G for stable operation, but the field in the region
of interest between LaB6 cathode and anode is flat at 330 G.
III. MORPHOLOGY
Magnetic fields were probed in a 3D volume measuring
13.2 cm by 13.2 cm by 958.5 cm. This consists of 16 (x,y)
planes, each separated 63.9 cm in z, with the first plane at a
distance of 63.9 cm from the LaB6 cathode and the last plane
63.9 cm away from the anode. The radial extent of the planes
is contained well inside the main plasma column so that
edge effects should not play a role. The probe is moved in
each plane on a Cartesian grid consisting of 45 points by 45
points, with Dx ¼ Dy ¼ 0:3 cm. The grid resolution is
smaller than the ion larmor radius (qi ¼ 0:6 cm) and comparable to the skin depth (c=xpe ’ 0:4 cm). The same Bdot
probe was used for all 16 planes. It consists of three orthogonal coils wound on a cube measuring 3 mm on the side. The
Phys. Plasmas 19, 102102 (2012)
coils are differentially wound to reduce electrostatic pickup.
Signals on the probe are proportional to dB/dt, are numerically integrated, and are corrected for the probe angle with
respect to the horizontal, resulting in a simultaneous measurement of Bx ; By , and Bz .
The profile of the measured magnetic field close to the
cathode, at z ¼ 0.64 m, is displayed in Figure 4. In the figure,
the LaB6 cathode positioned at z ¼ 0 is in back of the paper
and the background field points out of the paper. Panel (a)
shows a vector plot of the perpendicular components Bx and
By . The colors correspond to the magnitude of B? with a
maximum value of 7.1 G. Overall, the vectors circle around
clockwise, indicating the presence of a current into the plane.
Two distinct current channels can be clearly made out at
(x,y) ¼ (2.5, 0.5) and (4., 0.5). The Bz component depicted
in panel (b) shows a decrease in several G in approximately
the same locations. The change in Bz is small compared
to the uniform background field of 330 G, but it does indicate
the presence of perpendicular currents associated with each
current channel. The perpendicular currents are diamagnetic
and act to decrease the background magnetic field. The paramagnetic contribution to the perturbed Bz field is negligible
and can be estimated10 by the solenoid formula, Bz ¼ l0 NI=L
< 0:1 G. Panel (c) shows line cuts of the Bx ; By , and Bz on a
horizontal line at y ¼ 1.3 cm, similar to data which would be
observed in space if a satellite were to fly through the fields of
Figure 4. Moving from the left to the right one sees By flip sign,
while Bx is nearly constant, and Bz shows a decrease in the
region where By is changing sign. The 330 G background field
is subtracted out in panel (c).
The volumetric set of magnetic field data enables us to
follow field lines throughout the probed plasma. The magnetic field was measured on a set of discrete grid points. In
general, the probed field will not be divergence free, i.e., it
will not satisfy r B ¼ 0. Small measurement errors or
probe misalignment errors can cause a non-zero divergence,
FIG. 4. Magnetic field profiles at fixed time
at z ¼ 0.64 m. (a) Vector plot of perpendicular
fields, (b) contour plot of Bz , (c) line cuts of
Bx ; By , and Bz with background field subtracted at y ¼ 1.3 cm, (d) colorbar for panels
(a) and (b).
102102-4
B. Van Compernolle and W. Gekelman
FIG. 5. Field lines of the total magnetic field at one instant in time. The
z-axis is compressed by a factor of 10. The coordinate axes are each displayed 2 m apart. Field lines are started at Dz ¼ 0:64 m on circles of radius
1 cm around the center of each current channel. The twist of the field lines
and the twist of the current channels are left handed.
which in turn can lead to non-physical results in the calculation of the field lines, such as non-conservation of the total
flux enclosed by a flux tube. A mathematical technique called
“divergence cleaning”27 was used to ensure r B ¼ 0. In
short, instead of working with the magnetic field vectors B,
one works with the magnetic vector potential A. A is obtained
~ ¼ ik2 B~
from the measured magnetic field in Fourier space, A
k
(using Coulomb gauge, i.e., r A ¼ 0). The inverse Fourier
transform gives us the magnetic vector potential A(r, t), which
is then interpolated using tri-cubic splines. The tri-cubic
splines ensure the C2 continuity of A necessary for r B 0.
In the last step, the magnetic field vector B is recovered from
the splined vector potential through B ¼ r A.
Field lines of the divergence cleaned magnetic field are
displayed in Figure 5 at a time when the discharge current
Phys. Plasmas 19, 102102 (2012)
has reached steady state. In the figure, the ^z axis is compressed by a factor of 10. Coordinate axes are placed every
2 m to aid with perspective. Field lines were started on a
circle of 1 cm radius around the center of each current channel and are followed throughout the measurement volume.
Field lines with different colors emanate from different holes
in the carbon mask. The field lines of a single current channel both twist and writhe. The twist measures the torsion of a
field line about the central axis of the current channel and the
writhe measures the twisting of the central axis of the current
channel.28 In this experiment, the twist is about 270 and the
writhe is about 180 . The twist and writhe of the field lines
are left handed consistent with Jz Bz < 0.10,17 Indeed, electrons flowing from cathode to anode along the positive
z-direction results in a negative Jz , which produces a poloi^ direction. Adding the 330 G background
dal field in the /
field directed along þ^z results in a left handed helix.
The magnetic field data were used to compute the current density from c=4p r B in the 16 planes. As a consistency check,
the parallel current was integrated over each
Ð
plane ( dxdyJz ) and was found to be constant to within a
few percent and equal to the discharge current as measured
by a Rogowski coil at the LaB6 discharge pulser. The profile
of the current density at z ¼ 0.64 m, close to the cathode, is
displayed in Figure 6. Panels (a) and (b) show the perpendicular and parallel current density, respectively. The three flux
ropes are easily identified in both panels. A substantial perpendicular current (J? =Jz ’ 10%) surrounds each current
channel. The diamagnetic nature of this current will be
discussed in the next paragraph. The profile of the current density shows strong similarities with the profile of the perturbed
magnetic field displayed in Figure 4. In both figures, the flux
ropes can be located by examining the parallel component and
the perpendicular component circles around each flux rope.
The main difference between the magnetic field profile and the
current profile lies in the strength and direction of the parallel
FIG. 6. Current density profiles at fixed
time at Dz ¼ 0:64 m. (a) Vector plot of perpendicular current density, (b) contour plot
of Jz , (c) line cuts of Jx ; Jy , and Jz with
background field subtracted at y ¼ 1.3 cm,
(d) colorbar for panels (a) and (b).
102102-5
B. Van Compernolle and W. Gekelman
FIG. 7. Field lines of the both the magnetic field and the current density at
one instant in time. Magnetic field lines are colored blue, red, and green, and
current field lines are colored yellow. The z-axis is compressed by a factor
of 10. The current field lines have strong poloidal twist due to the diamagnetic current. The twist of the current field lines is right handed, opposite to
the twist of the magnetic field lines.
component. The magnetic field has a strong positive parallel
component (Bz =B? ’ 50), while the current has a weaker negative parallel component (Jz =J? ’ 7). Figure 7 shows this
clearly. Since the current density is a vector field, one can calculate and plot field lines for the current density, as well as for
the magnetic field. Magnetic field lines are displayed in blue,
red, and green as in Figure 5, while the current field lines are
displayed in yellow. The reader is positioned behind the LaB6
cathode and looks towards the anode. The magnetic and current field lines have opposite handedness, lefthanded for the
magnetic field, right handed for the current, and the twist of
the current field lines is much stronger due to the relatively
strong J? components. However, the writhe of the current field
lines is lefthanded, just as the writhe for the magnetic field,
since the parallel electron current follows the perturbed magnetic field.
Density and temperature measurements were obtained
with a planar Langmuir probe in a transverse plane at Dz
¼ 319:5 cm from the LaB6 cathode. The plane measured
20 cm by 20 cm with Dx ¼ Dy ¼ 4 mm. 10 shots were
recorded at each location. The probe was swept multiple times
during the last 2 ms of the LaB6 discharge. Due to the jitter in
the phase of the dynamics of the three flux ropes, the voltage
sweeps were not phase locked with the rotation of the flux
Phys. Plasmas 19, 102102 (2012)
ropes. A stationary Langmuir probe measuring ion saturation
current was used as a reference probe to provide a phase reference for the swept Langmuir probe. Figure 8 shows a snapshot of the temperature and density profile of the flux ropes.
The individual ropes cannot be made out in the picture. Both
density and electron temperature are elevated due to the presence of the ropes. Density increases from 2.0 1012 cm3 in
the background plasma to 5 1012 cm3 where the flux ropes
reside. Electron temperature increases from 3-4 eV to close to
10 eV. This is not unexpected and can be understood from a
simple discharge power density comparison between the main
LAPD discharge and the LaB6 discharge. The main LAPD
cathode-anode was operating at 4 kA-70 V for an approximately 60 cm diameter plasma, while the LaB6 cathode operated at 90 A-125 V for its three 2.54 cm diameter current
channels. The corresponding discharge power densities are
100 W=cm2 for the main LAPD discharge and 760 W=cm2
for the LaB6 discharge. It is therefore no surprise that the three
flux ropes have substantial temperature and density increases
associated with them. The local increase in plasma pressure
will generate cross field diamagnetic current J? ¼ rpB
B2 ,
which in return induces a parallel magnetic field opposite to
the guide field. The magnitude of the diamagnetic current
J? ne kTe =Lp B0 is estimated to be 1:2 A=cm2 , with
ne ¼ 5 1018 m3 , Te ¼ 10 eV; Lp ’ 2 cm, and B0 ¼ 0:033 T,
in good agreement with Figure 6. Similarly, the induced diamagnetic field can be estimated from Ampere’s law. For a axisymmetric pressure profile, one finds for the induced field at
the center of the flux ropes, DBz ¼ Bl00 ne kTe ’ 3:0 G, again
in agreement with the measured axial magnetic field reduction.
IV. DYNAMICS
The three flux ropes constitute a very dynamical system.
The evolution of the three flux ropes at the start of the LaB6
pulse will be discussed separately from the steady state.
Fig. 9 shows three components of the magnetic field at
x ¼ 3.0 cm, y ¼ 0.2 cm, z ¼ 3.20 m. t ¼ 0 corresponds to the
turn-on time of the LaB6 cathode. Also shown in Fig. 9 is the
discharge current flowing to the LaB6 cathode, measured
with a Rogowski coil. The magnetic field traces show the development of strong periodic spikes. The amplitude of the
spikes in the magnetic field can be in excess of 100% of its
DC value. The amplitude of the discharge current oscillations compared to the DC discharge current is about 20%.
The dynamics of the first couple hundred microseconds will
FIG. 8. (a) Electron temperature Te , (b) electron density ne from swept Langmuir probe
measurements. Both Te and ne show elevated
levels due to the presence of the flux ropes.
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B. Van Compernolle and W. Gekelman
Phys. Plasmas 19, 102102 (2012)
FIG. 11. Discharge current of the LaB6 source during the first ms shows
overshoots at first and then settles down at a constant level. The flux rope
dynamics during the overshoot is presented in Figure 10 and the steady state
of the flux ropes is shown in Figure 15.
FIG. 9. Top four panels: sample time traces at x ¼ 3.0 cm, y ¼ 0.2 cm,
z ¼ 3.20 m, respectively, discharge current, Bx ; By , and Bz . Lower panel:
Power spectrum of the Bx component, averaged over 10 plasma shots.
be discussed in Sec. IV A. The nature of the oscillations will
be discussed in Sec. IV B.
A. Current ramp up
The dynamics of the three flux ropes during the ramp up
phase is displayed in Figure 10 as a series of snapshots of the
axial current density Jz at different axial locations. Jz is
obtained from c=4pr B, where B is the measured magnetic field averaged over the number of data acquisitions at
each location (typically 10 plasma shots). Time progresses
from left to right, axial distance from the LaB6 cathode from
top to bottom. The frames are each 12:8 ls apart, except for
the last frame which is 150 ls later. For reference, the
discharge current during these early stages is displayed in
Figure 11.
The flux ropes emanate from the LaB6 cathode at z ¼ 0
and propagate at speeds near the Alfven speed. They are
seen after 4:5 ls at z ¼ 0.6 m and after 42:9 ls at z ¼ 8.3 m.
They therefore move at a speed of 2:0 107 cm=s, which is
slightly below the Alfven speed for the background plasma,
vA ¼ 2:5 107 cm=s ðn ¼ 2 1012 cm3 ). For comparison,
the electron thermal velocity is 8:4 107 cm=s and the ion
sound speed is 106 cm=s, both nearly an order of magnitude
different than the observed propagation speed. It is likely
that the propagation speed is almost exactly the Alfven speed
since the flux ropes have been observed to increase the
plasma density, which will lower the Alfven velocity. No
density measurements at these early times are available
though. The propagation speed of the flux ropes being equal
FIG. 10. Axial current density Jz , obtained from c=4p r B at different axial locations, as a series of snapshots of the first 100 ls after the source is turned
on. The flux ropes propagate at the Alfven speed. Close to the cathode, the flux ropes are nearly stationary due to line-tying at the cathode. At z ¼ 2.6 m, as the
discharge current ramps up, the ropes can be seen to kink as the ropes shift in a clock wise direction, corresponding to a left handed kink. Further downstream,
the ropes are observed to merge and bounce. Some data point to the existence of oppositely directed currents but the spatial resolution of the data may not have
resolved these.
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B. Van Compernolle and W. Gekelman
Phys. Plasmas 19, 102102 (2012)
FIG. 12. Isosurfaces of axial current Jz
¼ 2:0 A/cm2 at different times during the
start of the experiment. The flux ropes
propagate down the machine at the Alfven
speed. They kink as the current increases
and are observed to merge and bounce.
Coordinate axes are plotted every 2 m to
aid the eye with the perspective.
to the Alfven speed could be understood if one views a flux
rope as an Alfven wave in the limit of x ! 0 and kk ! 0.
Both have similar current structures, i.e., a central fieldaligned current channel surrounded by return currents carried
by the background plasma electrons.
As time progresses, the flux ropes close to the LaB6
cathode, at z ¼ 0.6 m, remain clearly separated because they
are line tied to their source. At z ¼ 2.6 m, the ropes remain
separated during this early time evolution, but a shift in their
x–y location is observed in the clockwise direction or lefthanded with respect to the magnetic field. After t ¼ 68 ls,
there is no further shift. This shift is believed to be the development of a left handed kink for each of the ropes, as in
Figure 5. Once the discharge current gets close to its maximum value, around t ¼ 68 ls, the kinking does not get stronger anymore and the flux ropes are more or less stationary.
At larger z locations, the flux ropes merge and bounce. This
is clearest at z ¼ 4.5 m where 3 distinct ropes are seen to
merge from t ¼ 55 ls to t ¼ 68 ls and then become separated again at t ¼ 100 ls. The bouncing and merging should
produce reverse currents due to reconnecting magnetic
fields14 but this was not observed. Possibly, the return currents have spatial scales too small to resolve with the probe
and the spatial resolution of the measurement grid. However,
quasi-separatrix layers which indicate the presence of reconnection were observed.23
During the initial phase of the LaB6 discharge,
t < 200 ls, the center of mass of the three flux ropes is field
aligned. Later on, however, the center of mass shifts
markedly, especially at large z. This is due to the onset of a
FIG. 13. Dominant frequency in the magnetic field signal spectra is constant
within error bars along the length of the flux ropes.
rigid body rotation of the kinked flux ropes, as will be discussed in Sec. IV B.
Figure 12 shows the isosurfaces of Jz ¼ 2:0 A/cm2 at
different times. The figure shows the 3D picture and aids in
the understanding of the 2D snapshots of Figure 10. In the
figure, the cathode is in front and the anode is in back. In
the first two frames, the flux ropes are propagating down the
machine at the Alfven speed, and the onset of a left handed
kink can be seen. Next, the kinked flux ropes merge and
bounce. The left handed kink is clearly visible.
B. Steady state
After the first 200 ls of the LaB6 discharge, the discharge
current reaches steady state. The probe signals have periodic
behavior as shown in Figure 9. The averaged frequency spectrum of the Bx component at z ¼ 383 cm, shown in the lower
panel of Figure 9 shows a strong dominant frequency. The amplitude of the oscillations in the probe signals can be larger
than 100% of the mean value. The ratio of oscillation amplitude to mean value increases with distance from the cathode.
The dominant frequency in the probe signal spectrum however
is constant along z, as shown in Figure 13.
The analysis of the steady state requires the use of conditional averaging techniques. The oscillations during the
steady state were not phase locked. They were observed to
have random phases. The top panel of Fig. 14 shows magnetic field data from 12 plasma shots at the same location.
Also plotted are the time traces from a light diode, located
FIG. 14. Spontaneous oscillations are not phase locked. Signals from a light
diode at fixed location are used as a phase reference to phase align the time
traces for each plasma shot. Top panel: time traces of light intensity from
diode and Bx from Bdot probe (12 plasma shots at fixed (x,y)) before phase
aligning. Lower panel: phase aligned time traces.
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B. Van Compernolle and W. Gekelman
Phys. Plasmas 19, 102102 (2012)
FIG. 15. Axial current density Jz calculated from c=4p r B, at 10 times during one oscillation period, and at 3 different z-locations, i.e., from top to bottom
z ¼ 0.64 m, z ¼ 3.20 m, and z ¼ 7.03 m.
outside the machine and recording plasma light through a
vacuum window. The signals from the light diode show the
same oscillatory behavior as the magnetic field data and can
be used as a phase reference for the magnetic field data. For
each plasma shot, a local maximum was searched in a time
interval between 4 ms and 4.3 ms after turn-on of the LaB6
source. Let t0 be the time of the local maximum, then the
magnetic field data of 640 ls before t0 , up to 640 ls after t0
were retained and saved. The retained data trace is 1280 ls
long and the middle of the time series is the time of a local
maximum in the light diode. This effectively phase aligns
the magnetic field data. The lower panel of Fig. 14 shows the
result of the phase alignment. It shows that the signals
remain phase locked for several periods. It also shows that
the experiment is very reproducible from shot to shot. All 16
planes of magnetic field data were thus phase aligned to the
light diode signals. The light diode remained fixed at the
same location during the acquisition of the 16 planes. Therefore, the conditionally averaged data from those planes are
mutually phase aligned as well. The phase aligned data were
then averaged over all acquired plasma shots at each probed
location.
Fig. 15 shows a few snapshots of the parallel current
density Jz during one oscillation period at 3 different z locations, from top to bottom z ¼ 0.64 m, z ¼ 3.20 m, and
z ¼ 7.03 m. Close to the LaB6 cathode, the three current
channels can clearly be identified. Peak current densities are
on the order of 10 A=cm2 . Further away, the three current
channels merge into one extended blob of current. The currents are not stationary in time. They are observed to rotate
counter clock wise in time in a right handed fashion, i.e., in
the electron gyration direction (B0 comes out of the plane).
The rotation is best visible in the bottom row of snapshots, at
locations far away from the LaB6 cathode. At the same time,
the current channels are twisting about themselves. The radius of rotation increases with increasing distance from the
LaB6 cathode. Close to the cathode, at z ¼ 64 cm the current
channels are almost line tied to the cathode. At the anode
however, they are not line tied and they can end at any point
on the 25 cm by 28 cm anode. The breaking of the line tying
condition at the anode can be understood in terms of the finite
sheath resistivity at the anode sheath.18 The parameter18
j ¼ cvzs ðaxc pi Þ2 be1=2 , a dimensionless parameter characterizing
the sheath resistance, is estimated for our parameters to be on
the order of 100, with cs =vz ¼ 2, a ¼ 1 cm, fpi ¼ 230 MHz,
be ¼ 0:02. Our parameters are therefore well described by
the perfect non-line tying boundary condition j 1,
described in the paper by Ryutov et al. vz was not measured
in this experiment but it has been measured in a similar
experiment involving two flux ropes,21,22 and it was found
that vz =cs ¼ 0:5 and directed towards the cathode.
The strong oscillatory behavior of Figure 9 is understood as an identifying feature of the rotation of the flux
ropes. The snapshots of Fig. 15 show that the time to complete one full poloidal rotation is the same as the period of
the oscillations in Fig. 9. Close to the cathode, the amplitude
of the oscillations seen on the magnetic field probe is small
since the radius of rotation is small. Far away from the cathode, the amplitude becomes larger than the DC value. The
radius of rotation of the flux ropes is so large that the probe
only measures strong fields when the flux ropes passes its
position, resulting in spiky signals. The radius of rotation is a
nearly linear function of the distance from the cathode, as
shown in Figure 16.
The rotation of the flux ropes is visualized in 3D in
Figure 17. Field lines were started at the center of each flux
ropes at z ¼ 64 cm, the closest measurement plane to the
FIG. 16. Radius of rotation for each of the flux ropes. The flux ropes rotate
with increasing radius as they move away from the cathode. Close to the
cathode, 64 cm away, the radius is on the order of 1 mm, whereas close to
the anode, the radius increases to more than 5 cm.
102102-9
B. Van Compernolle and W. Gekelman
Phys. Plasmas 19, 102102 (2012)
FIG. 17. Hodograms of the 3 flux ropes.
Each colored surface represents the surface
that the center of each current channel
traces out as it rotates. A grid was added to
each surface to aid the eye. Each line of
the grid is 50 cm apart. The radius of rotation is seen to increase from a few mm
close to the cathode to several cm close to
the anode.
cathode. These field lines were then followed throughout the
measured volume. Over one oscillation period, these field
lines will map out a surface. That surface basically shows
where the center of each flux rope was during one oscillation
period. In the figure, the anode is in the front of the picture
and the cathode in the back. Two inserts show the hodograms
of the three flux ropes. The hodograms are color coded so the
time evolution can be followed. At z ¼ 3.20 m, there is already
a substantial radius of rotation, but the hodograms do not
overlap. At z ¼ 8.31 m, where the hodograms of the three flux
ropes overlap due to the large rotation radius. Typically, at
these large distances, the flux ropes themselves are not individually distinguishable anymore.
Another interesting observation is that the current density in each of the flux ropes oscillates at the same frequency
as the rotation frequency. Figure 18 shows the total current
in each of the flux ropes versus time. This was calculated
from the current density at z ¼ 0.64 m, where the three current channels are clearly separated. The current density in
each flux rope at each moment of time was integrated over
2D space to obtain Figure 18. The current density in a given
flux rope can change in time by as much as a factor of two.
The conjecture is that this is caused by the flux ropes experiencing different plasma conductivities as they rotate. The
parallel plasma resistivity rk Te3=2 changes by a factor of 4
across the plasma column, due to temperature variations, as
seen in Figure 8. When a flux rope gets swept through a
region of colder plasma, the plasma resistivity would be
higher for that flux rope and the resulting current in the rope
would be lower.
The dynamics of the flux ropes was also captured with a
planar Langmuir probe at z ¼ 3.20 m. Both ion saturation
measurements and swept Langmuir measurements were
obtained. Fig. 19 shows
pffiffiffiffiffi representative traces of the ion saturation current (ne Te ) at three locations in the plane. The
top panel shows a trace near the center of the LaB6 cathode.
The middle and lower panel show traces in the edge and far
edge of the plasma from the LaB6 cathode. The ion saturation current in the edge exhibits a series of sharp positive
pulses. These sharp positive pulses can be understood from
the rotation of the flux ropes. Every time the three flux ropes
pass the probe a large positive spike would be picked up in
the ion saturation current. The general shape of the spikes in
FIG. 18. Axial current Iz in each of the current channels, measured at
z ¼ 0.64 m. The axial current in each channel oscillates in time at a frequency of 4.5 kHz. The phase between the oscillations of the current is p=3.
The total axial current piercing the plane is approximately constant in time.
FIG. 19. Ion saturation current traces from a planar Langmuir probe biased
to 75 V at Dz ¼ 3:20 m. The probe tip is facing away from the LaB6 cathode. Top panel: center, x ¼ 2.8 cm, y ¼ 2.2 cm. Middle panel: edge,
x ¼ 7.2 cm, y ¼ 2.2 cm. Lower panel: far edge, x ¼ 11.6 cm, y ¼ 2.2 cm.
102102-10
B. Van Compernolle and W. Gekelman
Phys. Plasmas 19, 102102 (2012)
FIG. 20. (a) Instantaneous electron temperature, (b) instantaneous electron density, (c) instantaneous plasma potential,
(d) time evolution of line integrated density. Panels (a,b,c) are local measurements, from a swept Langmuir probe.
Panel (d) shows data from a microwave
interferometer. The individual flux ropes
are not resolved.
the ion saturation current is asymmetric, in that they have a
sharp rise and a gentler decay. This is consistent with data
from the swept Langmuir probe presented in Figure 8, and
repeated in Figure 20. The swept data show density and temperature increase which are moving counter-clock wise in
the picture and have a sharp gradient at the leading edge
and a slow decay of density in the trailing edge. Ion saturation signals would therefore have an asymmetric shape. Note
that the swept Langmuir measurements had to be conditionally averaged just as described for the magnetic field data
in Figure 14.
Figure 20(c) shows the plasma potential at one instant in
time, as obtained from the swept data. The flux ropes are
associated with an increase in the plasma potential. This
increase extends out to where the ropes were just before.
Note that the rotation of the ropes cannot be explained by an
E B rotation. The electric field calculated from Figure
20(c) would be large at the leading edge of the rotating flux
ropes, and in the picture points down and to the right. The
magnetic field points out of the plane. The E B drift would
therefore point down and to the left, which would make the
ropes spin clockwise. A counter clock wise rotation is
observed, however. Figure 20 shows that the density fluctuations associated with the rotating flux ropes are also picked
up on the line-averaged density as measured on with an
interferometer.
In addition to the Langmuir probe at z ¼ 3.20 m, ion saturation current data were taken with a similar Langmuir
probe at z ¼ 3.52 m. This second probe was kept at a fixed
location in the edge of the LaB6 plasma and was used as a
reference probe for cross correlation analysis of the moving
probe. The real part of the cross spectral density is shown in
Figure 21 and shows the mode structure of the ion saturation
current fluctuations. The different panels display the mode
structure at the 4 dominant peaks in the spectrum. The peaks
are harmonics and correspond to m ¼ 1, m ¼ 2, m ¼ 3, and
m ¼ 4 modes. The fundamental mode is the m ¼ 1 mode at
4.2 kHz, which is just a signature of the rotating flux ropes.
The higher m-number modes at the harmonics are due to the
non-sinusoidal character of the pulses.
The rotation of the flux ropes was compared with theoretical predictions from the paper by Ryutov et al.18 on kink
unstable slender plasma columns. The theory is applicable to
ropes with a L, with a the rope radius and L the rope
length. The theory is further limited to cases where the displacement n of the flux rope due to kinking is gentle, i.e.,
j @n
@z j 1. Both of these restrictions are fulfilled in the
experiment. The limit of “perfect sliding” (also called
“perfect non-line tying”) at the anode is applicable to the
experiment, since the parameter j ¼ cvzs ðaxc pi Þ2 be1=2 1 as
mentioned above. Key features of the theoretical predictions
FIG. 21. Mode structure of ion saturation current fluctuations from cross
spectral analysis of a moving probe and a stationary reference probe. (a)
m ¼ 1 at 4.2 kHz, (b) m ¼ 2 at 8.4 kHz, (c) m ¼ 3 at 12.6 kHz, (d) m ¼ 4 at
16.8 kHz.
102102-11
B. Van Compernolle and W. Gekelman
are as follows: (a) constant frequency of rotation along the
plasma column, (b) increasing displacement of the column
with distance from the cathode, (c) the kink screws into (out
of) the anode if plasma flow is directed towards the anode
(cathode). Both (a) and (b) are observed in the experiment in
Figures 13 and 16. As to (c), the rotation of the left handed
kink is clockwise when looking towards the anode, meaning
that the left handed kink screws out of the anode. This is
consistent with the theory as the plasma flow in a similar previous experiment22 was measured to be in the direction of
the cathode. The predicted frequency of rotation can be estimated from the theory. The formula for the real part of the
frequency was derived for the perfect line-tying case in the
paper by Ryutov and can easily be found for the perfect nonline tying case assuming Bh =aBz x=vA and assuming
vz vA which are both satisfied in the experiment. One can
derive the rotation frequency in a similar manner as given
for the line-tied case and finds f ¼ vz Bh =4paBz , which is the
same as for the line-tied case for arbitrary vz . For our
numbers, vz 0:5 cs 1:0 cs , cs ¼ 1:6 106 cm=s; a ’ 1 cm,
Bh ¼ 10 G, Bz ¼ 330 G, one finds f ’ 1:9 3:9 kHz in
fairly good agreement with the measured frequency of 4 kHz.
The theory predicts the correct order of magnitude for the
rotation frequency. To rule out other instabilities as proposed
in the paper by Paz-Soldan et al.,19 a detailed parameter scan
is needed and is planned in the near future.
ACKNOWLEDGMENTS
The authors wish to thank T. A. Carter for helpful discussions, and Zoltan Lucky and Marvin Drandell for their
technical help in the construction of the LaB6 source. The
research was funded by the Department of Energy and the
National Science Foundation and was performed at the Basic
Plasma Science Facility at UCLA.
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