Solar Phys
DOI 10.1007/s11207-013-0257-0
Dynamics of an Erupting Arched Magnetic Flux Rope
in a Laboratory Plasma Experiment
S.K.P. Tripathi · W. Gekelman
Received: 1 June 2012 / Accepted: 12 February 2013
© Springer Science+Business Media Dordrecht 2013
Abstract A laboratory plasma experiment has been built to study the eruption of arched
magnetic flux ropes (AMFRs) in the presence of a large magnetized plasma. This experiment simulates the eruption of solar AMFRs in two essential steps: i) it produces an AMFR
(n = 6.0 × 1012 cm−3 , Te = 14 eV, B ≈ 1 kilo-gauss, L = 0.51 m) with a persistent appearance that lasts several Alfvén transit times using a lanthanum hexaboride (LaB6 ) plasma
source, and ii) it generates controlled plasma flows from the footpoints of the AMFR using laser beams. An additional LaB6 plasma source generates a large magnetized plasma in
the background. The laser-generated flows trigger the eruption by injecting dense plasma
and magnetic flux into the AMFR. The experiment is highly reproducible and runs continuously with a 0.5 Hz repetition rate; hence, several thousand identical loop eruptions are
routinely generated and their spatio-temporal evolution is recorded in three-dimensions using computer-controlled movable probes. Measurements demonstrate striking similarities
between the erupting laboratory and solar arched magnetic flux ropes.
Keywords Coronal mass ejections, initiation and propagation · Flares, dynamics · Waves,
plasma
1. Introduction
The dynamics of the magnetic structures that confine plasma plays an important role in
constituting the solar atmosphere and affecting its energetics (Lang, 2001). Solar magnetic
structures can be broadly divided into two categories – i) open magnetic structures that
extend beyond the solar corona from darker extreme ultra-violet (EUV) regions that are
called coronal holes and continuously transport material into interplanetary space (Krieger,
Timothy, and Roelof, 1973; Timothy, Krieger, and Vaiana, 1975), and ii) closed magnetic
S.K.P. Tripathi () · W. Gekelman
Physics & Astronomy, University of California at Los Angeles, Los Angeles, CA 90095, USA
e-mail: tripathi@physics.ucla.edu
W. Gekelman
e-mail: gekelman@physics.ucla.edu
S.K.P. Tripathi, W. Gekelman
structures that more efficiently confine plasma and magnetic energy from days to weeks
and then suddenly erupt (Abbot, 1911; Rosner, Tucker, and Vaiana, 1978). The pre-eruption
phase of the closed magnetic structures is relatively stable and lasts for several Alfvén transit
times (time taken by the shear Alfvén wave to travel from one footpoint to the other). In this
paper, we use the term arched magnetic flux rope (AMFR) for closed magnetic structures
that are essentially current-carrying plasmas confined by an arched magnetic field. The self
magnetic field generated by the current gives the characteristic rope-like twisted magnetic
structure to the AMFR. The presence of an electrical current in solar magnetic flux ropes
has been implied in vector magnetograph measurements, which show significant twist and
associated helicity in the magnetic structures emanating from the photosphere (Alfvén and
Carlqvist, 1967; Pevtsov, Canfield, and Metcalf, 1995; Leka et al., 1996; Wheatland, 2000;
Burnette, Canfield, and Pevtsov, 2004). We also use the term stable AMFR to signify the
persistent appearance of the AMFR structure without any significant changes in its morphology. This term does not indicate the absence of waves or microscopic plasma instabilities in the AMFR. Erupting solar AMFRs lead to an impulsive release of the stored
energy and trigger solar energetic events such as coronal mass ejections (CMEs) and flares
(Dennis and Schwartz, 1989; Kosovichev and Zharkova, 1999; Lang, 2001; Low, 2001;
Cremades and Bothmer, 2004; Chen and Kunkel, 2010; Hudson, 2011).
The exact nature of the instability that drives the CME flux rope eruption is a matter
of intense debate. Contemporary models of the CME flux rope eruptions rely on either
the storage-and-release or on the magnetic-flux-injection paradigm. The storage-and-release
models of CME flux ropes have a two-phase process in which magnetic energy slowly builds
up in the corona during the pre-eruption phase. Once the stored magnetic energy reaches a
certain threshold, the configuration becomes unstable and erupts impulsively. Three examples of the candidate mechanisms related to the storage-and-release paradigm are the loss of
the flux-rope equilibrium (Forbes and Isenberg, 1991), the onset of breakout reconnection
(Antiochos, DeVore, and Klimchuk, 1999), and the ideal kink-instability of AMFRs (Török,
Kliem, and Titov, 2004). The magnetic-flux-injection model of the CME eruption (Chen,
1996; Chen and Kunkel, 2010) relies on the dynamical injection of magnetic flux associated
with the AMFR current to drive the CME eruptions. Other examples of solar AMFRs are
coronal loops, filaments, and small-scale structures in the lower solar atmosphere. The twist
in the magnetic field lines in coronal and photospheric loops may not be always significant,
which is why the term magnetic-flux-tube is associated with these structures. However, the
term magnetic-flux-rope can still be used in a general sense for such structures because magnetic flux ropes become magnetic flux tubes in the limit of low magnetic-field-line twist. It is
unlikely that there is a universal mechanism of solar AMFR eruptions because solar observations reveal a great variety of AMFR structures that cover wide spatial and energy scales.
Although our laboratory experiment can simulate both drive mechanisms of solar AMFR
eruptions, this work focusses on the dynamical injection of magnetic flux and plasma into
an AMFR to trigger its eruption.
In the recent past, improvements in the remote sensing and imaging capabilities of spacecraft and ground-based observations have advanced our knowledge of the evolution of solar
plasma structures and energetic particle release. However, it is still not possible to capture
the complex events in the solar atmosphere with a good spatio-temporal resolution or to
directly measure solar plasma parameters (e.g., the magnetic field in the upper solar atmosphere) to develop a clear understanding of solar AMFR eruptions. A laboratory study of
AMFR eruptions has unique advantages since it enables us to systematically vary plasma
parameters, control the boundary conditions, and perform a detailed study of the AMFR
Magnetic Flux Rope in a Laboratory Experiment
dynamics. When used in association with solar observations and numerical codes, laboratory experiments can test theoretical models and identify processes relevant to solar AMFR
eruptions.
To the best of our knowledge, the first laboratory experiment that simulated solar AMFRs
was reported by Bostick (1956). Bostick’s experiment was primitive, yet it demonstrated that
by driving an electrical current along an arched magnetic field one can create a laboratory
AMFR. The next generation of laboratory plasma experiments on AMFRs was developed
several decades later by the Caltech group (Hansen, 2001; Hansen, Tripathi, and Bellan,
2004; Tripathi, Bellan, and Yun, 2007), followed by AMFR experiments conducted at the
FlareLab (Arnold et al., 2008). The kink instability of an AMFR that does not erupt due to
dominant and axially-uniform arched magnetic field has been studied by the MRX group
(Oz et al., 2011). Interaction of multiple straight flux ropes has been studied in a great detail
in laboratory experiments at the UCLA and LANL (Gekelman, Maggs, and Pfister, 1992;
Intrator et al., 2009). The UCLA group has identified a quasi-separatrix layer in the reconnection region (Lawrence and Gekelman, 2009). Traditionally, a magnetized plasma arc is
produced in laboratory experiments to mimic the behavior of solar AMFRs. The electrical current in such plasma arcs rises to several tens of kilo-amperes within a few Alfvén
transit times (tA ). Hence, a relatively stable pre-eruption phase of AMFRs (characteristic
feature of solar AMFRs) is not observed in the magnetized plasma arc experiments that
simulate AMFR eruption. Moreover, solar AMFRs evolve in the presence of a background
magnetized-plasma that plays an important role in wave excitation and energy transport. To
capture the essential features of solar AMFR eruptions, a laboratory plasma experiment has
been constructed at UCLA (Tripathi and Gekelman, 2010, 2011), which introduces a new
approach that uses two independent plasma sources to produce the AMFR and background
plasma, and a laser source to generate controlled plasma flows from the footpoints of the
AMFR to trigger the eruption. The experiment has the following unique features:
i) The laboratory AMFR erupts in the presence of a background magnetized-plasma. The
relative magnitude of parameters in the AMFR and background plasma can be varied
and the magnetic field direction can be reversed.
ii) Plasma flow from the footpoints of the AMFR is generated using jets of plasma produced by laser beams. Hence, it is independent of the plasma production mechanism.
iii) The electrical current in the AMFR can be kept below the kink-instability threshold for
a long time (> 50tA ). Therefore, a relatively stable pre-eruption phase of the AMFR can
be maintained for a long time and a source of instability can be applied at a desired time
to trigger the eruption.
iv) The experiment operates with a 0.5 Hz repetition rate and is highly reproducible (normalized shot-to-shot variation in plasma density: n/n ≈ 0.005). Thus, several thousand identical eruptions can be generated and their spatio-temporal evolution can be
recorded by performing a high-resolution three-dimensional (3-D) measurement of
plasma parameters.
The remainder of this paper proceeds with a discussion on the relevance of our laboratory
plasma experiment to actual solar eruptions in Section 2. Details of the experimental set-up
are presented in Section 3, which is followed by results from the experiment in Section 4.
The main points of the paper are summarized at the end in Section 5.
S.K.P. Tripathi, W. Gekelman
2. Relevance of Laboratory Plasma Results to Solar Observations
Relevance of a laboratory plasma experiment for solar observations and its limitations using
a dimensionless form of the two-fluid magnetohydrodynamical (MHD) induction equation
(Hansen, 2001), which can be written as
1
∂B
c
(∇ × B) × B
− ∇ ×∇ ×B−
∇ × (U × B) −
∇×
∂t
S
n
ωpi L
MHD induction
Hall
c
1
c2
d ∇ ×B
+
∇
= 0,
× ∇βe − 2 2 ∇ ×
ωpi L
n
ωpe L
dt
n
hydrostatic
(1)
electron inertia
where B = B/B0 , U = U /vA , n = n/n0 , t = t/tA , and ∇ = L∇ are the normalized magnetic field, plasma velocity, density, time, and length scales with respect to the maximum
√
magnetic field B0 , the Alfvén speed vA = B0 / μ0 nmi , the maximum plasma density n0 ,
the Alfvén transit time tA , and the length L of the AMFR. The Alfvén transit time is the
time taken by the Alfvén wave to travel the length L. n, mi , ωpe , and ωpi are the plasma
density, ion mass, electron plasma frequency, and ion plasma frequency. The dimensionless
parameter S = vA Lμ0 /η is the Lundquist number, which is the ratio between the resistive
diffusion time and the Alfvén transit time. A high value of S indicates a highly conducting plasma where plasma dynamics is primarily dictated by the magnetic J × B force and
the cross-field diffusion is relatively unimportant. Essentially, the plasma is frozen to the
magnetic field when S is high. The primary requirement from the experiment is that the laboratory AMFR retains the geometry of solar AMFRs. This is fulfilled by driving an electrical
current with two electrodes (as shown in Figure 1(a)) along an arched vacuum (potential)
Figure 1 (a) Basic geometry of the laboratory AMFR is depicted in this simplified 2-D view. Electrodes
and laser-generated jets are also indicated. The footpoints of the AMFR are anchored on the electrodes.
An external power supply, connected to the electrodes, drives current I along the arched magnetic field B0
and sustains the AMFR. (b) Arrangement of the AMFR, laser, and ambient plasma sources in a cylindrical
vacuum chamber to achieve the conditions depicted in (a). The z-direction is along the axis of the vacuum
chamber and the ambient magnetic field B0a is in the opposite direction to the z-axis. The AMFR is formed
in the xy-plane at z = 0. The origin of the Cartesian coordinates (x, y, z) is on the vacuum chamber axis
and its approximate location is indicated in panel (a). The footpoints of the AMFR are at x = 11.5 cm and
y = ± 11.5 cm.
Magnetic Flux Rope in a Laboratory Experiment
Table 1 Comparison between
laboratory and solar plasma
parameters. Typical plasma
parameters of an erupting helium
laboratory AMFR are used in the
table.
Lower corona
Coronal loops
Laboratory AMFR
n (m−3 )
≈ 1015
1015 – 1017
3 × 1019
T (K)
≈ 106
106 – 107
2.8 × 105 (≈ 24 eV)
L (m)
≈ 109
107 – 108
0.51
B (G)
≈ 10
≈ 20
r (m)
–
≈ 106
ri (m)
–
≈1
2 × 10−3
–
≈ 106
20
10−3 – 10−1
10−3 – 0.4
≈ 10−14
10−2 – 10−5
–
≈ 10−7
≈ 10−1
–
≈ 10−9
≈ 10−3
–
≈ 10−18
≈ 10−6
r/ri
β
S −1
c
ωpi L
βe c
ωpi L
c2
2 L2
ωpe
≈ 10−3
–
1000
4 × 10−2
magnetic field. The footpoints of the laboratory AMFR are anchored to the electrodes. Following the onset of an instability, the AMFR can expand away from the footpoints without
being affected by the wall of the vacuum chamber. Details of the experimental arrangements
are described in Section 3.
Typical plasma parameters of the laboratory and solar AMFRs are presented in Table 1.
Parameters of a coronal loop (Lang, 2001; Chen and Kunkel, 2010) are presented in the table
to represent solar AMFRs. Even though the absolute plasma parameters (n, T , L, and B) for
laboratory and solar AMFRs significantly differ, the laboratory studies retain the essential
physics of solar AMFRs. This can be explained by examining the terms in Equation (1).
The MHD induction is the dominant term for both laboratory and solar AMFRs because
it is on the order of unity. The resistive term in the MHD induction is on the order of 1/S,
which is ≈ 10−14 on the Sun and 10−2 – 10−5 in the experiment. In both systems the resistive
term is usually ignorable. However, the data presented in Table 1 do not represent localized
regions of the plasma where L is much smaller and the effective resistivity η is orders of
magnitude higher than the Spitzer resistivity. Therefore, S is relatively small and the resistive
term cannot be ignored in these regions. Such a scenario is known to exist in the magneticreconnection regions on the Sun (Lin et al., 2007) and in the laboratory (Stenzel, Gekelman,
and Wild, 1982; Kulsrud et al., 2005), and they strongly affect the dynamics of the AMFRs.
2
L2 ) terms
The Hall (≈ c/ωpi L), hydrostatic (≈ cβe /ωpi L), and electron inertia (≈ c2 /ωpe
−7
−9
−18
for solar coronal loops are on the order of 10 , 10 , and 10 . In the same sequence, these
terms for the laboratory AMFR are on the order of 10−1 , 10−3 , and 10−6 . The magnitude of
these three terms (especially the Hall term) with respect to the MHD-induction term is not
as small in the experiment as on the Sun. Therefore, they can play a more active role in the
experiment than on the Sun. Moreover, the ratio between the AMFR radius and the ion gyroradius (r/ri ) is orders of magnitude higher on the Sun. Hence, laboratory measurements of
waves and structures that are smaller in size than the ion gyro-radius size (≈ 2 mm) may not
be directly relevant to solar AMFRs.
This analysis shows that the MHD induction is the dominant term in both solar and
laboratory plasmas and hence the laboratory AMFR studies are relevant to the Sun. There
are regimes where non-MHD effects can play an important role in the experiment and on the
S.K.P. Tripathi, W. Gekelman
Sun, and they could be relevant to each other. Hence, one should be careful in only stressing
the importance of MHD effects and focusing on plasmas with high S in all situations.
3. Experimental Set-up
The experiment was conducted in a 4.9 m long, 1.0 m diameter cylindrical vacuum chamber that has external electromagnets to produce an axial magnetic field B0a of up to
300 gauss (G) (see Figure 1(b)). The vacuum chamber was filled with neutral helium gas
at 6.0 × 10−4 torr pressure. A rectangular lanthanum hex-boride (LaB6 ) cathode (size:
20 cm × 20 cm) and a molybdenum wire-mesh anode, separated by 27 cm, were placed
at one end of the chamber. The LaB6 cathode was indirectly heated to 1700 ◦ C to obtain
an efficient thermionic emission of primary electrons that are accelerated away from the
cathode when ≈ 100 V discharge voltage is applied between the electrodes. The primary
electrons produce a cylindrical plasma 0.6 m in diameter and 4 m long by ionizing the neutral gas. Details of the plasma production by this mechanism is described by Cooper et al.
(2010). The cylindrical plasma fills the vacuum chamber and serves as an ambient magnetized plasma in which the AMFR evolves. For the results reported in this paper, the ambient
plasma parameters are density na = 2.0 × 1012 cm−3 , electron temperature Tea ≈ 4.0 eV,
discharge current Ia ≈ 600 A, and B0a = 0 – 75 G.
The experimental conditions of the AMFR are shown in a simplified 2-D view in Figure 1(a). The AMFR is produced in the middle of the chamber with two additional electrodes
that are mounted on two radial shafts, as shown in Figure 1(b). The AMFR cathode is circular disk-shaped LaB6 (7.6 cm diameter) that is indirectly heated to 1825 ◦ C. The AMFR
anode is made of copper and is identical to the cathode in shape. The footpoints of the
AMFR are anchored to the electrodes, which are separated by 23 cm. Each AMFR electrode
is surrounded by an electromagnet to produce an arched magnetic field B0 along the axis of
the AMFR (see the direction of B0 and AMFR electrodes in Figure 1). The AMFR is produced when ≈ 200 V discharge voltage is applied between the electrodes from an external
power supply. After producing the AMFR, most of the discharge voltage drops near the electrodes in a thin sheath region (size < 0.1 mm AMFR length) and the bulk of the AMFR
dynamics is not directly affected by the voltage at the footpoints. A stable laboratory AMFR
(as evident from its persistent appearance and stationary n and T profiles) is produced in
≈ 140 µs after applying the voltage on the electrodes. The initial current I , driven by the
power supply, rises to 60 A in 140 µs and remains stationary until a source of instability is
applied to disrupt the persistent appearance of the AMFR. This is accomplished by firing
two identical infrared laser beams (wavelength = 1064 nm, pulse-duration = 8 – 12 ns, beam
energy ≈ 0.8 J/pulse for each beam) toward the footpoints of the AMFR. Each AMFR electrode has a small orifice and a carbon rod placed behind the electrode. The laser beams pass
through the orifices and ablate the carbon targets to produce two plasma jets that emanate
from the footpoints of the AMFR as shown in Figure 1(a). The injection of the plasma jets
into the AMFR causes a sudden increase in the current I and density n of the AMFR, which
results in an impulsive eruption of the AMFR. The injection of the plasma and current I into
the laboratory AMFR is analogous to the plasma and poloidal magnetic flux injection from
the convection zone into the solar AMFRs. The plasma parameters of the stable AMFR are
density n = 6.0 × 1012 cm−3 , electron temperature Te ≈ 14 eV, discharge current I ≈ 60 A,
and B0 = 1000 G at the footpoints. Following the injection of the laser jets, n and Te rise to
a maximum value of 3 × 1013 cm−3 and 24 eV.
At the beginning of the experiment, the vacuum chamber is evacuated to a base pressure of 2 × 10−7 torr and electromagnets for the ambient magnetic field are turned on. Over
Magnetic Flux Rope in a Laboratory Experiment
Figure 2 Timing sequence of the computer-controlled events that form an erupting laboratory AMFR. This
sequence is repeated every 2 s to generate thousands of identical loop eruptions. The displayed timings may
be adjusted to achieve the desired boundary conditions within the experimental limits.
a period of one to two days, the temperature of the two LaB6 cathodes is slowly raised to
the desired value, while a base pressure < 10−5 torr is maintained in the chamber. After
the pressure drops to below 10−6 torr, the chamber is filled with helium gas at the desired
neutral pressure. At this stage, the ambient and AMFR plasmas can be formed in a pulsed
mode with a 0.5 Hz repetition rate with the timing sequence displayed in Figure 2. During
each pulse the following sequence is executed – i) discharge voltage pulse for the ambient
plasma is applied, ii) pulsed magnetic field B0 (not shown in Figure 2) for the AMFR is
turned on, iii) discharge voltage pulse for the AMFR is applied in the afterglow of the ambient plasma (B0 /B0 during AMFR < 0.01), iv) the laser beams are fired in the middle
of the AMFR pulse, and v) data are recorded and the diagnostic probe is moved to a new
location in the plasma. After every AMFR pulse, the laser targets are moved by ≈ 1 mm.
This ensures that a fresh target surface is ablated by the laser beams in the following AMFR
pulse. This is critical for maintaining a good reproducibility of the experiment. The experiment and data acquisition system are automated using computer controls. Hence, several
thousand identical AMFR eruptions are routinely generated and data of high spatio-temporal
resolution are recorded during the course of the experiment, which lasts several hours. Langmuir and magnetic-loop probes are used to measure the plasma parameters (n, Te , and B).
A fast CCD-intensifier camera (5 ns minimum exposure time, 1280 × 1024 pixels, forced
air cooling, 12 bit digital converter) records images of the AMFR evolution.
4. Results and Discussion
The time evolution of a helium laboratory AMFR is shown in Figure 3. The time t was
measured with respect to the laser trigger at t = 0 µs. Two different intensity scales (one
for the top panel and the other for the bottom panel) were used to display the frames in
Figure 3. This way, the relative intensities of the frames in each panel can be compared.
After application of the discharge voltage pulse at t = − 1000 µs on the electrodes, the
AMFR begins to form in 40 µs (see the first frame in Figure 3(a)). During this plasma
breakdown phase, primary electrons ionize the neutral gas and plasma density begins to
build up. After ≈ 140 µs at t = − 860 µs, the AMFR exhibits a persistent appearance and
the discharge current attains a steady-state value of 60 A. This initiates the stable phase of the
AMFR. The Alfvén transit time τA = L/vA in the stable AMFR is 1.2 µs (using L = 0.51 m,
B0 = 1000 G, n = 6 × 1012 cm−3 , vA = 4.5 × 105 m s−1 ). The resistive diffusion time τd is
μ0 L2 /η, where η = η⊥ /2 and η⊥ is transverse Spitzer resistivity. The value of τd for the
S.K.P. Tripathi, W. Gekelman
Figure 3 (a) Formation of a helium laboratory AMFR after application of the voltage on the electrodes at
t = − 1000 µs is displayed in these unfiltered images recorded by a fast CCD camera with 1 µs exposure
time. The laser beams are triggered at t = 0 µs. It takes about 140 µs (t = − 1000 µs to − 860 µs) to form an
AMFR with a persistent appearance. (b) Escape of plasma from the AMFR after the laser-generated flows,
shown on a fast time-scale (2.5 µs ≈ 2 · Alfvén transit time). The exposure time of the fast camera is 50 ns.
laboratory AMFR is ≈ 22 ms (using Te = 14 eV, charge state Z = 1, Coulomb logarithm
ln = 15, η⊥ = 3.0 × 10−5 m). It can be shown that the value of τd in the transverse
direction (μ0 r 2 /η⊥ ) is in the 70 – 200 µs range. The stable phase of the AMFR persists from
t = − 860 µs to t = 0 µs (t = 717tA ), mimicking the behavior of solar AMFRs that remain
stable for several Alfvén transit times before erupting (Abbot, 1911).
The evolution of the laboratory AMFR following the laser trigger is displayed in Figure 3(b). The AMFR current rises to ≈ 500 A within 0.7 µs after the laser trigger. Our
previous study of an argon laboratory AMFR (Tripathi and Gekelman, 2010) has demonstrated that an order of magnitude increase in the current leads to a significant twist in the
magnetic structure of the AMFR (I = 500 A ⇒ average twist (r) 51 3.5π ) and it
becomes kink unstable (Hood and Priest, 1979; Török, Kliem, and Titov, 2004). It was also
confirmed that the laser plasma is primarily composed of C++ ions. The motion of the laser
plasma in the AMFR was observed by attaching a 460 nm narrow-passband filter to the CCD
camera. The filter blocks the light from the helium plasma and transmits the C++ emission
from the laser plasma. The time-of-flight measurements indicate that the laser plasma jets
propagate with an average speed vjet ≈ 5.0 × 104 m s−1 . The ion acoustic speed cs in the
AMFR is 1.8 × 104 m s−1 and vA near the footpoints is 4.5 × 105 m s−1 . Hence, the jets are
supersonic and sub-Alfvénic (cs < vjet < vA ) near the footpoints. A comparison of the bright
regions in the t = − 0.2 µs and 0.7 µs frames implies that the increase in the current leads to
an outward motion of the AMFR and its minor radius decreases significantly. The outward
motion is understood to be caused by the increase in the hoop force, and the decrease in the
plasma radius is generated by the pinch force (Chen, 1996). Our previous study on an argon AMFR (Tripathi and Gekelman, 2010) was performed in the presence of a positive B0a ,
which causes a somewhat stronger outward motion because of the outward J × B 0a force.
For the results reported here, B0a is − 10 G and the J × B 0a force is in the inward direction.
In this configuration, the plasma β value at the leading edge of the AMFR rapidly increases
from 0.1 to 1.3 within 1.4 µs after the laser flows are produced. Magnetic field measurements
(to be discussed below) evince ejection of an axial current from the AMFR at t ≈ 1.4 µs.
Magnetic Flux Rope in a Laboratory Experiment
Figure 4 Plasma density n and
electron temperature Te profiles
of a stable helium AMFR are
measured along the x-axis at
t = − 840 µs. As shown in
Figure 1(a), the positive and
negative x-directions are toward
the inboard and outboard of the
AMFR, respectively.
Following this phase, the AMFR current drops to the steady-state value (40 – 60 A), the outward expansion is inhibited, and plasma from the AMFR escapes to the background. The
escape of plasma from the AMFR leads to the disappearance of the bright regions of the
AMFR, as seen in the last two panels of Figure 3(b). The panels in Figure 3 are unfiltered
and the discharge source is kept on during the entire period. Hence, the light emission from
the newly formed low-density plasma appears in the last two panels in a faint blue color.
The measurement of the plasma density profiles provides more details of the eruption that
are discussed in the following paragraphs. These observations suggest that the laboratory
AMFR erupts on a time scale τexp such that τA < τexp τd .
Density and electron temperature profiles of a stable AMFR were measured with a Langmuir probe and are displayed in Figure 4. The AMFR profile was measured off-axis at
z = 5 cm to ensure that the probes are not destroyed by the laser beams in the z = 0 cm
plane. The maximum in the n-profile exists near the AMFR axis, where n = 6.0 × 1012
cm−3 and Te = 14 eV. A steep decay of n and Te in the inboard region reflects the steep
B0 gradient there. Away from the AMFR axis at x = − 10 cm, the ambient plasma parameters are na = 2.0 × 1012 cm−3 and Tea = 4 eV. In our experiment, the time-resolution of
the evolution of Te during
the Te measurement is ≈ 50 µs, which is too slow to capture √
the eruption. However, the ion-saturation current density ji ∝ n Te can be recorded by the
Langmuir probe with a fast time response (t = 1/25 µs) and the AMFR density can be
extracted from the measurements of ji using an average value Te = 10 eV.
The density fluctuation spectra of the AMFR were sampled in three different time-ranges
that characterize the ambient plasma (t = − 1290 to − 1140 µs), stable AMFR (t = − 400
to − 250 µs), and unstable AMFR (t = 5 to 155 µs). The spectra were averaged over an
xy-plane that intersects the AMFR and is presented in Figure 5. A significant increase in
the fluctuation level in the lower frequency range (f < 1.0 MHz) and appearance of a peak
at 171 kHz (labeled “global kink mode oscillation”) are highlights of the unstable AMFR
spectrum. The currently obtained results are not adequate to characterize the broad lowfrequency spectrum of the unstable AMFR. Although this broad low-frequency spectrum
appears to have characteristics of turbulence modes, the absence of a multitude of coherent
modes cannot be ruled out without performing a more detailed measurement using a fixed
and a moving Langmuir probe pair. The spectra of the stable and unstable phases of the
AMFR above 1.0 MHz frequency are identical, and both indicate the presence of peaks
near the lower hybrid frequency. An analysis of these high-frequency fluctuations reveals
associated modes at the edge of the AMFR, where plasma parameters have steep gradients.
A detailed discussion of these high-frequency modes is beyond the scope of this paper.
The AMFR density profiles with a good spatio-temporal resolution (x = 5 mm, t =
1/25 µs) were produced from the measurements of ji and are presented in Figure 6(a). As
S.K.P. Tripathi, W. Gekelman
Figure 5 Normalized density fluctuation spectra of the AMFR are displayed for the ambient plasma, stable arched magnetic flux rope (AMFR), and unstable AMFR. The spectra have been averaged over 615
equidistant points in an xy-plane (x = − 10 to 10 cm, y = − 3.5 to 3.5 cm, z = 5 cm). Prominent features
of the unstable AMFR spectra are a significant increase in the fluctuation level in the lower frequency range
(f < 1.0 MHz) and the appearance of a low-frequency peak at 171 kHz (labeled “global kink mode oscillation”).
Figure 6 (a) Time evolution of the density profile of an erupting helium laboratory AMFR (see the x-axis
in Figure 1). The striking features are the outward expansion of the AMFR during the eruption and the
damped oscillations with a 165 – 190 kHz frequency in the post-eruption phase. (b) Phase-fronts of a fast
magnetosonic wave are detected in the post-eruption AMFR and presented on an expanded xt -plane. The
data have been processed using a 4.0 – 5.4 MHz digital passband filter before plotting.
Magnetic Flux Rope in a Laboratory Experiment
expected, the density profile before the laser trigger at t = 0 µs appears to be persistent and is
identical to the density profile in Figure 4. Following the laser trigger, the AMFR expands in
the outward direction (see the “Expanding AMFR” label) and a low-density plasma remains
at the location of the stable AMFR. The discharge and magnetic field power supplies for
the AMFR are kept on during the entire period. Hence, a new AMFR quickly forms and
exhibits a damped oscillation in the frequency range 165 – 190 kHz. These oscillations are
also observed on the discharge current and magnetic-field signal, and appear as a 171 kHz
peak in the density spectrum of the unstable AMFR (see Figure 5). With a striking similarity,
the existence of such damped oscillations in post-flare loops on the Sun has been reported by
several groups and has been interpreted as characteristics of global kink modes (Nakariakov
et al., 1999; Chen and Schuck, 2007; Ofman and Wang, 2008). The frequency f of this
mode is given by
1/2
2
vA
,
(2)
f=
2L 1 + na /ni
where na and ni are the ambient plasma and AMFR plasma densities, respectively. The
AMFR magnetic field decays away from the footpoints, and its average value in the AMFR
is 321 G. Hence, the average vA in the AMFR is 1.43 × 105 m s−1 . We used L = 0.51 m and
na /ni = 1/3 to estimate a 171 kHz frequency for the global kink mode in our experiment.
This estimate agrees well with the observed frequency of the oscillations. The oscillation
amplitude decays with an e-folding time of 51 µs, which is much slower than the axial
resistive time and comparable to the transverse resistive time of the AMFR. Detailed measurements are planned in the future to identify the damping mechanism of the global kink
modes in our experiment.
The time evolution of the AMFR density also contains a signal at a 4.7 MHz frequency
with n/n ≈ 0.02. The nature of this high-frequency oscillation was analyzed by applying a 4.0 – 5.4 MHz digital passband filter to the data and plotting them on an expanded
xt scale in Figure 6(b). The density wave at 4.7 MHz frequency and 3.0 cm wavelength
is confirmed from the measurement of the phase fronts displayed in Figure 6(b). The amplitude of the 4.7 MHz peak in Figure 5 is identical for the stable and unstable AMFR.
Hence, the density wave exists even during the stable phase. The wave propagates across
the AMFR magnetic field with a phase speed vφ ≈ 1.41 × 105 m s−1 , which is comparable
to vA ≈ 1.43 × 105 m s−1 and much faster than the ion acoustic speed cs ≈ 1.8 × 104 m s−1 .
Moreover, the wave frequency is lower than the lower hybrid frequency (≈ 10.5 MHz).
These observations indicate that the density wave is a fast magnetosonic wave (Habbal,
Holzer, and Leer, 1979). The global kink mode oscillations and fast waves are also detected
in the magnetic fields of the AMFR. The global kink mode oscillations damp in ≈ 51 µs,
which is followed by a significant increase in the electron temperature (from 14 eV to 24 eV)
of the AMFR. After t ≈ 120 µs, the plasma oscillations disappear, the electron temperature
drops to 14 eV, and the AMFR density profile resembles the density profile of the stable
AMFR at t < 0 µs.
Magnetic-field vectors of the AMFR were measured with a magnetic-loop probe during
the eruption in an xy-plane and are presented in Figure 7. The magnetic loop probe measures the time-varying magnetic field during the eruption and is not sensitive to the vacuum
(potential) magnetic fields. The t = − 0.6 µs panel shows a relatively stronger inboard magnetic field of the AMFR, which links the AMFR footpoints. Following the laser trigger, the
observation of “O”-points in the t = 1.4 and 2.3 µs panels are the main features in Figure 7.
The “O”-points are surrounded by an azimuthal magnetic field, indicating the emergence of
an axial current from the leading edge of the AMFR. Measurement of the Bz profile (not
S.K.P. Tripathi, W. Gekelman
Figure 7 Evolution of the
AMFR magnetic field at four
distinct times is displayed in an
xy-plane at z = 5 cm. The
approximate location of the plane
is indicated by the dotted
rectangle in Figure 1(a). The
vectors are drawn using the Bx
and By components of the
magnetic field. Appearance of the
“O”-points in the 1.4 and 2.3 µs
panels indicates the ejection of an
out-of-plane current (along the
z-direction) from the AMFR. The
inboard point α and outboard
point β are marked here and in
Figure 1(a) to guide us in
approximately locating the
“O”-point in the schematic view
of the AMFR.
presented here) in the xy-plane of Figure 7 shows that the magnitude of Bz is comparable
to Bx and By around the “O”-point. This implies that the emerging axial current-channel
is a large flux rope with a significant twist. The appearance of this axial current channel at
t = 1.4 µs is concurrent with the escape of plasma from the AMFR in Figure 3(b). However, the large flux rope that contains the low-density plasma released from the AMFR is
not visible in the camera images. To verify the presence of this large flux rope, Langmuir
and magnetic loop probes were placed at z = ± 32 cm and unambiguously detected the flux
rope in the xy-planes away from the AMFR. The emergence of a large flux rope from an
erupting laboratory AMFR is relevant to the flux rope models of CMEs (Gibson and Low,
2000), since it demonstrates the transport of magnetic flux into the ambient plasma from the
AMFR. The large flux rope quickly grows to the size of the vacuum chamber. Hence, its
evolution cannot be directly compared with the evolution of CMEs to large spatial scales.
5. Conclusions
A comparison of the solar and laboratory plasma parameters using the dimensionless form
of the two-fluid MHD induction equation in Section 2 has demonstrated that the laboratory
AMFR studies are relevant to the solar AMFRs because the MHD induction is the dominant
term in both plasmas. However, certain non-MHD effects (e.g., the role of the Hall term, the
appearance of structures smaller in size than the ion gyro-radius, and the dynamics of the
plasma on a time scale slower than the resistive diffusion time) can play more active roles in
the laboratory experiments than on the Sun. The experimental results confirmed that a stable
Magnetic Flux Rope in a Laboratory Experiment
arched magnetic flux rope with a persistent appearance lasting several Alfvén transit times
can be generated in the laboratory and that the eruption of the stable AMFR can be triggered
by injecting a dense plasma and magnetic flux into the AMFR. The experiment operates
with a 0.5 Hz repetition rate; hence details of the eruption can be captured by recording
data with a good spatio-temporal resolution. Highlights of the laboratory AMFR eruption
are i) the outward motion of the AMFR due to the hoop force, ii) the decrease of the minor
radius of the AMFR due to the pinch force, iii) the development of a highly twisted magnetic
structure in the kink-unstable AMFR, iv) the appearance of global kink mode oscillations
and a fast magnetosonic wave in the post-eruption phase, v) the ejection of a large magnetic
flux rope from the leading edge of the AMFR, and vi) the observation of strong heating in
the inboard region of the AMFR. We showed that these results resemble the observations
of solar AMFR eruptions. Our future studies will be focused on simulating the storage-andrelease model of the eruption. In these experiments, the AMFR eruption will be triggered
by a programmed increase of the discharge current in the absence of the laser plasma source
and the AMFR force balance will be examined in a variety of scenarios.
Acknowledgements The experiment was conducted at the Basic Plasma Science Facility (BaPSF) at the
University of California, Los Angeles, which is jointly funded by National Science Foundation and Department of Energy. The authors thank James Chen, Alexander Kosovichev, and Steven Spangler for useful
discussions and P. Pribyl, Z. Lucky, and M. Drandell for technical assistance. One of the authors (SKPT)
thanks Paul Bellan for his valuable suggestions in deriving the two-fluid MHD induction equation.
References
Abbot C.: 1911, The Sun, D. Appleton and Company, New York, 128.
Alfvén, H., Carlqvist, P.: 1967, Solar Phys. 1, 220.
Antiochos, S.K., DeVore, C.R., Klimchuk, J.A.: 1999, Astrophys. J. 510, 485.
Arnold, L., Dreher, J., Grauer, R., Soltwisch, H., Stein, H.: 2008, Phys. Plasmas 15, 042106.
Bostick, W.H.: 1956, Phys. Rev. 104, 1191.
Burnette, A.B., Canfield, R.C., Pevtsov, A.A.: 2004, Astrophys. J. 606, 565.
Chen, J.: 1996, J. Geophys. Res. 101, 27499.
Chen, J., Kunkel, V.: 2010, Astrophys. J. 717, 1105.
Chen, J., Schuck, P.: 2007, Solar Phys. 246, 145.
Cooper, C.M., Gekelman, W., Pribyl, P., Lucky, Z.: 2010, Rev. Sci. Instrum. 81, 083503.
Cremades, H., Bothmer, V.: 2004, Astron. Astrophys. 422, 307.
Dennis, B.R., Schwartz, R.A.: 1989, Solar Phys. 121, 75.
Forbes, T.G., Isenberg, P.A.: 1991, Astrophys. J. 373, 294.
Gekelman, W., Maggs, J., Pfister, H.: 1992, IEEE Trans. Plasma Sci. 20, 614.
Gibson, S.E., Low, B.C.: 2000, J. Geophys. Res. 105, 18187.
Habbal, S.R., Holzer, T.E., Leer, E.: 1979, Solar Phys. 64, 287.
Hansen J.F.: 2001, “Laboratory Simulations of Solar Prominences”. Ph.D. thesis, California Institute of Technology.
Hansen, J.F., Tripathi, S.K.P., Bellan, P.M.: 2004, Phys. Plasmas 11, 3177.
Hood, A.W., Priest, E.R.: 1979, Solar Phys. 64, 303.
Hudson, H.: 2011, Space Sci. Rev. 158, 5.
Intrator, T., Sun, X., Lapenta, G., Dorf, L., Furno, I.: 2009, Nat. Phys. 5, 521.
Kosovichev, A., Zharkova, V.: 1999, Solar Phys. 190, 459.
Krieger, A.S., Timothy, A.F., Roelof, E.C.: 1973, Solar Phys. 29, 505.
Kulsrud, R., Ji, H., Fox, W., Yamada, M.: 2005, Phys. Plasmas 12, 082301.
Lang K.: 2001, The Cambridge Encyclopedia of the Sun, 1st edn., Cambridge University Press, Cambridge,
106.
Lawrence, E.E., Gekelman, W.: 2009, Phys. Rev. Lett. 103, 105002.
Leka, K.D., Canfield, R.C., McClymont, A.N., van Driel-Gesztelyi, L.: 1996, Astrophys. J. 462, 547.
Lin, J., Li, J., Forbes, T.G., Ko, Y.K., Raymond, J.C., Vourlidas, A.: 2007, Astrophys. J. Lett. 658, L123.
Low, B.C.: 2001, J. Geophys. Res. 106, 25141.
S.K.P. Tripathi, W. Gekelman
Nakariakov, V.M., Ofman, L., Deluca, E.E., Roberts, B., Davila, J.M.: 1999, Science 285, 862.
Ofman, L., Wang, T.J.: 2008, Astron. Astrophys. 482, L9.
Oz, E., Myers, C.E., Yamada, M., Ji, H., Kulsrud, R.M., Xie, J.: 2011, Phys. Plasmas 18, 102107.
Pevtsov, A.A., Canfield, R.C., Metcalf, T.R.: 1995, Astrophys. J. Lett. 440, L109.
Rosner, R., Tucker, W.H., Vaiana, G.S.: 1978, Astrophys. J. 220, 643.
Stenzel, R.L., Gekelman, W., Wild, N.: 1982, J. Geophys. Res. 87, 111.
Timothy, A.F., Krieger, A.S., Vaiana, G.S.: 1975, Solar Phys. 42, 135.
Török, T., Kliem, B., Titov, V.S.: 2004, Astron. Astrophys. 413, L27.
Tripathi, S.K.P., Bellan, P.M., Yun, G.S.: 2007, Phys. Rev. Lett. 98, 135002.
Tripathi, S.K.P., Gekelman, W.: 2010, Phys. Rev. Lett. 105, 075005.
Tripathi S.K.P., Gekelman W.: 2011, In: Choudhary D.P., Strassmeier K.G. (eds.) The Physics of Sun and
Star Spots, IAU Symp. 6, 483.
Wheatland, M.S.: 2000, Astrophys. J. 532, 616.