PHYSICS OF PLASMAS 22, 082118 (2015)
Experimental study of a linear/non-linear flux rope
Timothy DeHaas, Walter Gekelman, and Bart Van Compernolle
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
(Received 15 May 2015; accepted 2 August 2015; published online 20 August 2015)
Flux ropes are magnetic structures of helical field lines, accompanied by spiraling currents.
Commonly observed on the solar surface extending into the solar atmosphere, flux ropes are
naturally occurring and have been observed by satellites in the near earth and in laboratory
environments. In this experiment, a single flux rope (r ¼ 2.5 cm, L ¼ 1100 cm) was formed in the
cylindrical, magnetized plasma of the Large Plasma Device (LaPD, L ¼ 2200 cm, rplasma ¼ 30 cm,
no ¼ 1012 cm3, Te ¼ 4 eV, He). The flux rope was generated by a DC discharge between an
electron emitting cathode and anode. This fixes the rope at its source while allowing it to freely
move about the anode. At large currents (I > pr2B0c/2 L), the flux rope becomes helical in structure
and oscillates about a central axis. Under varying Alfven speeds and injection current, the
transition of the flux rope from stable to kink-unstable was examined. As it becomes non-linear,
oscillations in the magnetic signals shift from sinusoidal to Sawtooth-like, associated with elliptical
C 2015
motion of the flux rope; or the signal becomes intermittent as its current density increases. V
AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4928888]
INTRODUCTION
A magnetic flux rope is a tube-like, current carrying
plasma embedded in an external magnetic field. The superposition of a poloidal magnetic field created by the current
and the external magnetic field generates a structure of helical magnetic field lines. This bundle of helical field lines
resembles a rope. Magnetic flux ropes are ubiquitous plasma
structures. For instance, flux ropes are most commonly
observed in solar physics, where X-Ray and UV images from
the Solar Terrestrial Relations Observatory (STEREO),1 the
Transition Region and Coronal Explorer (TRACE),2 and the
Atmospheric Imaging Assembly (AIA)3 show multiple flux
ropes extending up from the solar surface into the solar corona.
The collision and unraveling of multiple flux ropes on the sun
have been theorized to contribute to coronal heating4–6 and the
initiation of Coronal Mass Ejections (CMEs)7 and solar flares.8
Flux ropes have often been generated in a number of experimental devices as well. The impulsive disruption of a stable
arched flux rope by a strong ion flow was studied at UCLA;9
and in the reconnection scaling experiment (RSX) at LANL, a
series of experiments were preformed on the coalescing of
multiple flux rope during collisional reconnection.10,11
Under certain conditions, a flux rope can become unstable. In particular, when its current exceeds the KruskalShafranov limit, the flux rope becomes kink unstable,
causing the tube-like plasma column to tilt and deform.12–14
The growth of this kink instability can then saturate and initiate complex motion. A number of experiments have studied
kink-unstable flux ropes, including experiments in spheromaks,15,16 the rotating wall machine at Madison,17,18 and the
RSX, where experiments similar to the conditions of this
work verified transitions in the kink-mode under varying
boundary conditions.19,20 Previous experiments at UCLA
have been focused on three-dimensional measurements
involving multiple kink-unstable flux ropes.21–23 Typically,
the experimental conditions were held constant while
1070-664X/2015/22(8)/082118/7/$30.00
multiple probes measure the three-dimensional dynamics at
tens of thousands of locations inside the experimental chamber. These investigations focused on the effects of interacting
and reconnecting flux ropes. The dynamics of multiple flux
ropes can become complex depending upon the conditions of
the plasma experiment.24 In such cases, reconstructing threedimensional dynamics becomes difficult when the motion is
chaotic. Similarly, multiple plasma effects, including pressure
effects, J B forces, drive wave instabilities, and current instabilities, can dominate the system, making it difficult to separate
cause-effect behavior without changing the plasma conditions.
In contrast to the previous three-dimensionally motivated experiments, the present experiment involves only a
single kink-unstable flux rope. The experimental conditions
(specifically the input power) were varied to observe transitions in the flux rope motion. In particular, this experiment
was designed to observe transitions from linear to non-linear
behavior, from coherent to chaotic dynamics. The base
behavior of a single flux rope is necessary to interpret the
interaction of multiple flux ropes.
EXPERIMENTAL SETUP
The experiment was performed on the LArge Plasma
Device (LAPD) at UCLA, a 24 m long cylindrical (d ¼ 1 m)
vacuum vessel was designed to generate a highly reproducible, quiescent, magnetized plasma.25 The machine is surrounded by eighty steady state solenoidal magnets, which
are controlled by ten independent power supplies. This
allows for variable control of the axial magnetic field along
the machine. The LArge Plasma Device is fitted with over
four hundred diagnostic ports. These are located approximately every 32 cm axially and every 45 poloidally, which
facilitates a well-diagnosed plasma. The LAPD operates continuously, generating a pulsed plasma at 1 Hz for 15 ms.
The machine runs for approximately four months before the
plasma source has to be cleaned and recoated.
22, 082118-1
C 2015 AIP Publishing LLC
V
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Phys. Plasmas 22, 082118 (2015)
FIG. 1. Representation of the LArge
Plasma Device (LAPD) at UCLA. This
simple schematic demonstrates experimental setup for the single flux rope
experiment. Two pulsed direct current
sources are used. The first source, a
barium oxide coated cathode generates
a magnetized embedding plasma. The
second source, a masked LaB6 cathode
generates a flux rope (r ¼ 2.5 cm,
L ¼ 1100 cm).
In generating plasma in the LAPD, there are typically
three explicit parameters that can be varied in order to implicitly change the statistical and electrical properties of the
plasma. The first parameter is the external, background magnetic field. The second is the gas to be ionized. In this experiment, the gas was He; and the magnetic field was 330 G or
660 G. These two conditions were held constant. The parameter that was varied in this experiment was the input power
associated with the plasma sources: two cathode-anode pairs,
which generated the background plasma and the current
source for the flux rope. These are described below.
There are two plasma sources for the present flux rope
experiment. The first source, or the main cathode, produces
an ambient, uniform plasma in which to embed the flux rope.
The second source, or the LaB6 cathode, generates the flux
rope. An illustration of the LAPD flux rope experiment is
shown in Figure 1.
Considering the main cathode, this oxide-coated cathode
generates a uniform, magnetized, current neutral plasma 18
m in length and 60 cm in diameter (dn/n 3%, 0.05 eV
Te 5 eV, 1010 cm 3 n 2 1012 cm3, Ti < 1 eV).26
The cathode is constructed from a barium oxide coated
nickel plate, which is heated to approximately 900 C. A
transistor switch, in series with a capacitor bank, is used to
pulse a voltage between the negatively biased cathode and a
molybdenum anode located 50 cm away. In this flux rope
experiment, 40 V is applied between the field-aligned cathode and anode. At its sufficient temperature, the main cathode is emissive, resulting in a 4 kA discharge for 12 ms.
The flux rope source is inserted into the LAPD at an
axial position defined to be z ¼ 0 cm. This source is a small,
circular cathode (d ¼ 10 cm) made of lanthanum hexaboride
(LaB6).27 It faces the main cathode, which is 16 m away. A
small portion of the LaB6 material is masked with a carbon
plate such that the cathode only emits from a circular, 5 cm
diameter hole. This is the size of the flux rope. Between the
two plasma sources, a square, mesh, molybdenum anode
(dimensions of thirty-by-thirty centimeters) is placed 11 m
from the LaB6 source (see Figure 1). With an independent
capacitor bank and switch, a voltage is applied between the
LaB6 cathode and its anode. This generates a current channel, or flux rope, down the axis of the LAPD. The current
channel is fixed at z ¼ 0 cm while it is free to move about the
anode at z ¼ 1100 cm, a fixed-free condition that affects the
flux rope stability. The LaB6 cathode emits up to 6 A/cm2
for 5 ms.
Both the discharge current and discharge voltage of the
LaB6 cathode were independently controlled in this experiment. The discharge voltage was controlled explicitly by the
voltage on of a capacitor bank, while the discharge current
was controlled implicitly through the temperature of the
LaB6. To become highly emissive, the LaB6 must be heated
above 1600 C. In this experiment, the LaB6 is indirectly
heated by passing a current through an s-shaped carbon flat,
placed behind (yet electrically isolated from) the cathode.
The carbon heater resistively dissipates between 5 and 7 kW
of power. By varying the dissipated power, the temperature
of the LaB6 changes, hence implicitly controlling the cathode’s emissivity and discharge current. For this experiment,
the discharge current was varied between 20 A and 130 A,
while the discharge voltage was varied between 60 V and
170 V (Table I).
The experimental timing is as follows and is repeated
every second:
•
•
The main cathode-anode pair generates a helium plasma
of 12 ms duration with a background field of 330 G.
Six milliseconds after the initiation of the background
plasma (which will be considered t ¼ 0 s for this paper),
the LaB6 source is activated, creating a flux rope for 5 ms.
The flux rope is allowed to evolve without external interference in space and time for that 5 ms period. During this
period, measurement signals from multiple probes (including
magnetic probes, Langmuir probes, Mach probes, and spectrometers) were digitized with 16-bit analog-to digital converters (dt ¼ 3.2 107 s).
TABLE I. Varying conditions of discharge power that were tested for the
generation of a single magnetic flux rope.
Discharge voltage (V)
100
100
100
100
100
100
100
60
73
120
173
Discharge current (A)
Total power (kW)
20
40
55
75
95
110
130
75
75
75
75
2
4
5.5
7.5
9.5
10
13
4.5
5.5
9
13
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FLUX ROPE DYNAMICS
As long as J B 0 or (J B)/c Dp, the plasma is in
stable equilibrium.28 However, if the current flowing along
the background field is above a certain threshold (given in
Eq. (1)), the flux rope will become kink unstable. This is a
well-studied MHD instability in which a straight plasma column becomes distorted into a helical structure. For the experimental geometry of a single flux rope with a fix-free
boundary condition, an analytic theory has been constructed
and experimentally verified.19,29 In this fixed-free case, the
current threshold above which the flux rope becomes unstable is half the Kruskal-Shafranov limit13,14
IKINK >
pa2 Bz c
:
2L
(1)
With a flux rope length of L ¼ 1100 cm and a radius of
a ¼ 2.5 cm, the current threshold is 30 A for a background
magnetic field of 330 G and 60 A for a background magnetic
field of 660 G.
In this experiment, the flux rope is driven kink unstable,
and the dynamics of this current system are shown to dramatically change after the instability is initiated. For instance,
Figure 2 plots two magnetic signals on either side of the kink
threshold (one at I ¼ 20 Amps, one at I ¼ 75 A) for B0 ¼ 660
G. In the stable case, the perpendicular magnetic oscillations
are small compared to the stable magnetic field. These small
oscillations correspond to randomly excited standing Alfven
waves between boundaries. After the instability threshold,
the magnetic signals observe large-scale oscillations in the
perpendicular direction. These large oscillations correspond
to motion of the flux rope after it has been freed from its
axis. By plotting the real, peaked, frequencies observed in
the system, the transition between a stable flux rope and
unstable flux rope can be observed (Figure 3). It is important
to note that the instability threshold presented in Figure 3
is inconsistent with the 60 A inferred from Eq. (1).
Experimentally, the transition into a kink-unstable regime
can be a gradual transition; and small, coherent oscillations
can be seen below the theoretical threshold. Likewise, reductions in the instability threshold can be obtained by the introduction of strong axial flows. However, this has not been
FIG. 2. A comparison between two magnetic signals between a kinkunstable flux rope and a stable flux rope immersed in a background magnetic
field of 660 G.
FIG. 3. Observed, peaked frequencies in the magnetic signals associated
with a fixed location on the edge of stable flux rope: (x,y,z) ¼ (5, 0, 870) cm.
The plot shows a transition of the flux rope immersed in a background magnetic field of 660 G toward kink-unstable dynamics.
verified directly through axial flow measurements for this
experiment.
After the flux rope has become kink unstable, the dynamics of this current channel are governed by changes in its
current and the voltage difference between its boundaries.
Figure 4 shows several magnetic time traces at the same
location. By increasing the discharge voltage on the LaB6
source, the signals become more coherent. While increasing
the current, the signals become more intermittent.
Using data from a fixed magnetic probe (x, y, z ¼ 5, 0,
870 cm), a Fourier transform was taken and averaged. This
analysis shows a characteristic spectrum that is prevalent in
each kink unstable case studied for this paper. The spectrum
shows coherent modes embedded in an exponential background (Figure 5). The coherent modes are representative of
elliptical motion within a traverse plane. The exponential
spectrum is the manifestation of random intermittent pulses
in the magnetic signals. These two aspects will be discussed
below.
Considering, first, the coherent mode: this mode manifests itself as multiple peaks in the Fourier spectrum. The
lowest peak is the coherent mode while its harmonics are an
effect of the linear Fourier analysis imposed on a global process. By isolating and filtering out these peaks, a mode structure can be identified, corresponding to an m ¼ 1, m ¼ 2, etc.
This gives the anticipation of a flux rope description as the
wave phenomena. However, each mode is in phase with one
another, not being excited randomly as the discharge power
increases. Instead, the collection of these peaks is the manifestation of approximate circular motion of the flux rope in
the transverse plane, whose radius of rotation in a plane is
greater than (or on the order of) the size of the flux rope. The
motion can be seen in Figure 6, where the instantaneous central position of the flux rope is tracked for several periods at
two different axial positions. Translation of the flux rope in
this elliptical motion can increase to four times the radius of
the flux rope.
To explore how the transitory motion manifests itself in
the Fourier analysis, a simple model has been created. A virtual, cylindrical flux rope of 75 A (with a Gaussian current
density profile) was constructed. In the model, this flux rope
is moved in the transverse plane in increasing concentric
circles (from rcircle ¼ 0.5–4 cm). The flux rope is moved
around each circle with constant velocity and a constant period of rotation. A virtual probe is placed at (x,y) ¼ (5, 0)
cm. As the flux rope moves in greater and greater circles, the
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Phys. Plasmas 22, 082118 (2015)
FIG. 4. Magnetic signals for fixed location (x,y,z) ¼ (5, 0, 870) cm. While varying the discharge voltages independently of the current in the flux rope, the
motion of the flux rope becomes more coherent and reproducible. Upon varying current in the flux rope, the motion becomes more intermittent.
virtual probe samples more of a magnetic profile that varies
as 1/r rather than r. This gives an appearance of waveform
steepening on the virtual probe. As seen in Figure 7, the
waveforms from the virtual probe become more ramp-like or
FIG. 5. (a) Frequency spectrum of magnetic signal at a fixed position. I ¼ 75
A, V ¼ 100 V. (b) Frequency Spectrum of magnetic signal at a fixed position. I ¼ 130 A, V ¼ 100 V.
more pulse-like as the circular motion increases. In either
case, the Fourier transform of these time signals will yield
multiple peaks, identical to the ones seen in the experiment.
The properties of coherent motion will depend on the current in the flux rope and the discharge power. With an increase
of discharge voltage at constant current, the motion of the flux
rope becomes more coherent. The increase in power corresponds to an increased displacement of the flux rope in its elliptical motion. The period associated with this mode,
however, scales with current, or current density, in the flux
rope. This can be seen in Figure 8(a), where the frequency
scales with the current. The frequency remains fixed when the
voltage is varied independently of current (Figure 8(b)).
As noted in the previous flux rope experiments, including systems of multiple flux rope, the coherent mode is not
phase locked.21 This requires the data to be conditionally
triggered in order to reconstruct the two-dimensional motion
of the flux rope. For example, Figure 9 plots ten magnetic
signals at the same location. Figure 9(a) contains data that is
triggered at the same time, indicated by the initiation of the
LaB6 source. Figure 9(b) contains the same ten shots but are
conditionally triggered. Because the motion of the flux rope
is non-local, strong correlations will exist between two
probes at different spatial locations. This is true as long as
both probes are located near the path of the flux rope. Spatial
reconstruction is then possible when the phase differences,
determined by the fix probe, are applied to all probes taking
planar measurements.
With increases in current density, the motion of the flux
rope transitions to motion characterized by an exponential
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FIG. 6. Instantaneous central position
of the flux rope plotted over several
rotation periods for 75 A flux rope current and 175 V voltage discharge. This
orbit is associated with the coherent
mode.
background in the frequency spectrum. The exponential nature is easily identified by plotting the Fourier spectrum on a
log-linear scale (Figure 10). This exponential background is
a feature of the Fourier spectrum that consistently presents
itself upon varying conditions in this experiment. It is connected with random intermittent Lorentzian pulses that serve
to modify the orbit of the flux rope, de-correlating the
FIG. 8. (a) Frequency of the coherent mode under varying conditions of discharge current for B ¼ 330 G. (b) Frequency of coherent mode is independent of discharge voltage.
FIG. 7. Virtual magnetic probe measurement for a flux rope traveling in
ever greater and greater circles in the transverse plane. Probe measurement
is at a fixed position in space (x,y) ¼ (5,0) cm. (a) y-component of magnetic
measurements. (b) x-component of magnetic measurements. (c) Fourier
spectrum of By magnetic signals from virtual probe with flux rope orbiting
central axis with a radius of r ¼ 4 cm and constant velocity.
FIG. 9. Ten consecutive “shots” of magnetic signals from the fixed probe.
(a) is the representation of the un-aligned signals with the bold trace being
the averaged signal. (b) is the representation of the aligned signals using
conditional triggering with the bold trace being the averaged signal.
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DeHaas, Gekelman, and Van Compernolle
Phys. Plasmas 22, 082118 (2015)
transverse to the field. Statistical quantities, however, can be
tested. For instance, by fitting an exponential to the spectrum, the width of these pulses may be found. Figure 12 plots
these results for a magnetic field of 330 G. The width of
these impulses is shown to depend upon the discharge current. This is consistent with the flux ropes motion transitioning into a more intermittent regime. As the discharge current
increases the width monotonically decreases till it saturates
at about 10 ls width.
FIG. 10. Frequency spectrum of magnetic signal on a log-linear plot for a
fixed position (x, y, z ¼ 5, 0, 870 cm). I ¼ 130 A, V ¼ 100 V. A line is fit to
the spectrum to show an approximate exponential spectrum.
FIG. 11. Example of Lorentzian pulse recorded by magnetic probe under the
conditions of I ¼ 110 A, V ¼ 100, B ¼ 660 G.
coherent motion in time. This characteristic exponential
background is not uncommon in the LAPD and other experimental devices.30–32 In exploring these Lorentzian pulses,
LAPD experiments have focused on density transport transverse to the magnetic field, an effect which is often associated with non-linearly interacting drift waves through E B
forces.33 In this experiment, by introducing a hot, dense
(nRope/n0 2, TRope/T0 2) filament connected with a current, the exponential spectrum can be observed in both the
magnetic spectrum as well as the density spectrum.
Figure 11 shows an example of a Lorentzian found in
the magnetic signal. Simultaneous measurements along the
axis of the LAPD observe pulses that are well correlated in
space. This suggests that these intermittent pulses are connected to large-scale physics, possibly related to a characteristic orbit of the large-scale flux rope. However, due to
unpredictable intermittency of these pulses, two-dimensional
reconstruction of flux rope motion cannot be observed without simultaneous measurement at multiple positions
FIG. 12. Width of Lorentzian pulses for varying discharge currents
(B0 ¼ 330 G). Width depends on the current density rather than increases in
the discharge current.
SUMMARY
A five centimeter diameter flux rope was generated in
the Large Plasma Device at UCLA. It was created via a DC
discharge between a LaB6 cathode and a mesh molybdenum
anode placed 11 m away. The current in the flux rope was
varied independently from the discharge voltage, while the
background, embedding magnetic field was held constant at
330 G. The flux was driven kink-unstable, and its dynamics
were measured via magnetic probes. The motion of the flux
rope is characterized by two types of motion, most easily
identified by the Fourier spectrum of magnetic time traces.
The first type of motion is associated with elliptical motion
of the flux rope in the transverse plane. This coherent mode
manifests itself as multiple, harmonic, peaks in the spectrum.
The second type of motion is characterized by intermittent,
Lorentzian pulses in the magnetic signals. These pulses modify the coherent motion in time, preventing two-dimensional
spatial reconstruction. As the current in the flux rope
increases, the signals become more intermittent. However, as
the discharge voltage increases with constant current, the signals become more coherent. Further scaling experiments are
necessary to find a suitable cause that sets the characteristics
of the coherent mode and its transition into intermittent
motion.
ACKNOWLEDGMENTS
Funding for this research comes from the Department of
Energy, Office of Fusion Energy Sciences and National
Science Foundation. The work was performed at the Basic
Plasma Science Facility, a user facility funded by DOE
(DOE-DE-FC02-07ER54918:011).
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